.. .. A MATHEMATICAL M:lDEL FOR THE ENERGY AND PROTEIN METABOLISM OF HOMEOTHERMS by Kenneth Falter Institute of Statistics Mimeograph Series No. 81; Raleigh - April 1972
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..
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A MATHEMATICAL M:lDEL FOR THE ENERGY
AND PROTEIN METABOLISM OF HOMEOTHERMS
by
Kenneth Falter
Institute of StatisticsMimeograph Series No. 81;Raleigh - April 1972
.,
.v
TABLE OF CONTENTS
LIST OF TABLES .
LIST OF FIGURES
1. INTRODUCTION
2. PHILOSOPHY OF MODELING
3. BACKGROUND AND KEY LITERATURE
30 1 Roles of Feed Constituents3.2 Definition of Physical Compartments3.3 Current Feeding Systems3.4 Mathematical Approaches ....
4. DEVELOPMENT OF THE MATHEMATICAL MODEL
4.1 Preliminaries.. .4.2 Conceptual Framework
iv
Page
vi
viii
1
3
9
99
1416
21
2123
4.2.14.2.2
Nitrogen, Energy and HeatFurther Subdivisions of Nitrogen and
Energy
23
26
4.3 Discussion of the Total Compartment Model4.4 Development of Flow Laws
2734
4.4.14.4.2
NotationThe Flow Laws
3537
4.5 The Mathematical Model
5. TESTING THE MODEL
5.1 A Model for a Growing Steer.5.2 Initial Estimates ..
50
54
5562
5.2.15.2.25.2.3
Evaluation of ConstantsEvaluation of Initial ConditionsInitial Estimates of Parameter Values
626669
5.3 The Goodness of Fit Criterion5.4 The Iterative Estimation Procedure
6. RESULTS AND DISCUSSION. . .
7. CONCLUSIONS AND RECOMMENDATIONS
7984
86
102
v
TABLE OF CONTENTS (continued)
Page ..8. LIST OF REFERENCES . . . • · · · 105
9. APPENDICES . . . · • · · · . · · 108 ."
9.1 Flow Laws for Milk Production · . · · 1089.2 Flow Laws for Heat Loss · · 1129.3 Evaluation of Constants · · 118
9.3.1 Evaluation of PB118
9.3.2 Evaluation of f B, fS · 119
9.3.3 Eva1uatio~ of wBn' Ws 120
9.3.4 Evaluation of r 121s
9.4 Evaluation of Initial Conditions for the Pools 1239.5 Evaluation of c 125
d,o9.6 Evaluation of Kp,B · • · · . · . . . . 128
9. 7 Evaluation of Weights for the Goodness of FitCriterion • . . . · · . . . . . · · · . 130
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vi
LIST OF TABLES
5.4 Constants used to test the model
5.3 Input data for testing the model
5.2 Differential equations for testing the model
5.1 Flow laws for testing the model .•.•..•
Page
51
53
58
61
63
67
10
76
78
79
82
84
. . . . 87
Values of bBW' D, and KBW . .
Initial pool concentrations and values Of. A.B.an.d cd.'oused in testing the model
Initial estimates of parameter values •
Nitrogen and energy balance variables.
Experimental results
4.2 Differential equations for the compartment model
4.1 Flow laws for the compartment model .
5.6
5.5
5.7
5.8
5.9
5.10 Weights for the goodness of fit calculation.
6.1 Final parameter values - averages and coefficients ofvariation . . . . . .. ..•.
6.2 Variance-covariance and correlation matrix of final parametervalues . • . . . . • . . . • • • . . • • • 88
6.3 Parameter values used for sensitivity test 90
6.4 Summary of r values by feeding level and run 92
6.5 Summary of r values, degrees of freedom and mean squarer values . . . • • . . • • . . 94
.. 6.6 Mean square error in the raw data, due to the model andfraction of variation accounted for by the model 95
6.7 Model results 96
6.8 Weighted residuals of final simulation run , . 97
9.1 Simplified composition of energy-containing substances indry skeletal muscle • . . • . • . . • . . . . . . . . • 118
vii
LIST OF TABLES (continued)
9.2 Summary of pool concentration data
Page
125..
9.3 Values of heat production, dry matter fed and fat energybalance . . . . . • • . 126
9.4 Heat production values for zero fat energy balance 127
9.5 Regression constants and values of cd .. 128,0
9.6 Values of sp
9.7 Values of wi' xi' a, b, and s; for each factor
131
132
LIST OF FIGURES
2.1 Iterative model-building
3.1 Structural and energetic roles of feed nutrients
3.2 Physical compartments ...•
4.1 Physico-chemical compartments.
4.2 A compartment model ..
4.3 Flow law from stores to P as a function of fatnessse
6.1 plot of KB P versus ~ B by breed • • . . . . • . . • •, ,6.2 Weighted residuals versus feeding level (FL) for final
simulation run . • . .
9.1 Graph of milk production rate •
9.2 Possible forms of Al
,A2
versus body temperature
9.3 Possible form of K' versus body temperaturePse,Ph
9.4 Possible form of ~ versus body temperaturese,Ph
9.5 Graph of f(x) = xl (x + Of) . . . . . . . . .
9.6 Graph of s versus feeding level and fitted regression1
, pl.nes . . . . . . . . . . . . . . . . . . . . . . . .
viii
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7
10
11
25
28
45
93
98
111
115
116
117
122
133
e.
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1. INTRODUCTION
Since the early 1800's, much effort has gone into .evaluating feeds
for animal functions such as maintenance, growth, fattening and other
production. A number of feeding standards and systems have resulted.
It was early realized that values based solely on feed content were
inadequate, hence the concept of nutrient availability was applied.
These standards have served as useful guides. Starting with these
standards, feeding regimens are empirically adjusted to take into
.'j
account economic factors and factors known to affect feed assessment
and animal production. Many such factors are not handled explicitly
in the standards.
With the advancement of nutritional knowledge, increasing emphasis
has been given to the metabolic processes involved in digestion and
utilization of feed. Efforts have also been made to formulate mathe~
matical equations to describe these processes. Blaxter ~ al. (1956)
derived a two compartment model for the passage of undigested feed
through the digestive tract. Blaxter and Mitchell (1948) used a
statistical procedure for estimating true digestibility of proteins.
Lucas and Smart (1959) successfully applied this approach to calculat-
ing true digestibility of 11 feed components in two types of forage.
Blaxter (1962b) proposed a new system for assessing energy values
of feeds which adjusts the metabolizable energy of a ration for the
plane-of-nutrition effect. Lucas (1964) developed a model to handle
aspects of digestion'and absorption. Waldo (1968) discussed nitrogen
metabolism in a mathematical way. In a recent lecture,
1Lucas has evaluated and reviewed these modeling efforts, and, build-
ing on these, he has outlined the beginnings of a model to handle the
joint metabolism of nitrogen and energy.
The aims of this dissertation are two-fold. First, a conceptual
framework will be developed within which the energy and protein
metabolism of homeotherms may be handled. Second, mathematical formu-
lations will be developed which describe, in a mechanistic sense, the
transport of material from the feed through the metabolic processes
involved in the digestion and utilization of the feed by the animal .••
This model will formally handle some of the considerations used in
the practical application of the feeding standards.
Using experimental data from the literature, facets of the model
will be tested. Further testing needed and kinds of data required for
this will be discussed, as will scientific implications.
1Lucas, H. L. 1968. Mathematical Models in Animal Nutrition.International Summer School on Biomathematics and Data Processing inAnimal Exper iments. Elsinore, Den.mark.
2
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3
2, PHILOSOPHY OF MODELING
A model may be thought of as a representation, or analog, of some
real object or process (Hawthorne, 1964), Models may be categorized
according to the type of material of which they are constructed,
Kleiber (1961) described and illustrated a hydraulic scheme, or model,
for energy utilization, Such a model would be constructed from glass
ware, water, weights, balls J ro1lers J strings, gears and springs,
Energy utilization also can be modeled by ilsing abstract symbols
in mathematical equations, Such a model would be a mathematical model
for energy utilization, Mathematical models may be classified accord
ing to their purpose as empirical models or rational models,
Empirical models are mainly used to describe or to summarize a
body of data, They are attempts to fit curves or relations to data in
order to make predictions. The reason for using an empirical model is
that it fits the data o Predictions are restricted to the domain of
values over which the data were collected, The parameters of the model
and their dimensions, or units, are usually not interpretable in terms
of the system which generated the data o This class of models includes
most statistical regression models,
Rational models are used to explain the behavior of a system in
a mechanistic sense, They are developed by first defining a conceptual
or abstract framework for the system under study, This framework
describes or embodies the major e1em.ents of the structure and behavior
of the system, This framework is then translated into a set of mathe
matical expressions which comprise the model, The parameters and vari
ables in these expressions are identified in terms of elements of the
4
system being modeled} ~'~'} they make sense in terms of the system.
The solution of this model should behave like the system does over a
wide domain of values. Lucas (1964) stated that models based on as
much rationality as possible lead to the best predictability. A
general framework for conceptualizing and mathematizing a problem to be
modeled was presented in the context of grasslands problems (Lucas)
1960). The remarks hold for mathematical modeling in general.
How well a rational mathematical model fits experimental data or
predicts results depends on how well the mathematical expressions
represent the system} and on how close the parameter estimates are to
the true values.
The form and the complexity of the mathematical model depend some
what on the modeler and on his level of competence in mathematics and
in biology. We may consider that rational models lie somewhere on a
continuum with extremes at points A and B.
A ...---------__1 B
Point A represents a purely biological formulation which is
elaborate} very specific and quite extensive. It represents but one
system} with great precision and detail. Such formulations are usually
difficult to handle} and are either incapable of analytical solution or
require an exhorbitant amount of time to solve. An example of such a
formulation might be a model developed to express the biochemical re
actions involved in ruminant digestion and fermentation (Baldwin et a1.)
1970). Forty-five biochemical reactions were defined} leading to 40
simultaneous nonlinear differential equations. These were solved
. ~
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5
numerically by a digital computer and the results presented in tables
and graphs. The point A may be labeled nearly complete reality .
At the other extreme, point B represents a purely abstract
formulation which is quite elegant and represents many systems in
general, but no one system in particular. An example (Kalman et a1.,
1969) is the representation of a dynamical system by an octup1e (T, X,
U, 0, Y, r, _, 11> where T is a time set, X is a state set, U i~ a set
of input values, 0 is a set of input functions, Y is a set of output
values, r is a set of output functions, _ is a state~transition func
tion and ~ is a readout map. Though this is a far cry from biology,
by making certain assumptions about the structure of the model, a body
of theory can be developed. Point B may be labeled complete abstrac
tion.
We want to develop or construct a mathematical model lying between
points A and B, !.~., a tradeoff. The model will have two general
characteristics (Hawthorne, 1964):
(a) Similarity - in some sense, our model is like the real thing.
By ignoring relatively unimportant details, it is practical to
construct and test, or experiment with, our model; and
(b) Nonidentity - the model is not the real ;hing. It is a 1ess
than~comp1ete copy of reality. Results of testing or experi
menting with the model may be false.
The model should be.as abstract as possible, but still related to
the biology. The model may be intractable in one or more of the follow
ing senses:
6
(a) Mathematically - we are unable to formulate the model at the
desired level of abstraction;
(b) Analytically - we are unable to solve the mathematics we have
formula ted;
(c) Experimentally - we either are unable to experiment with the
system modeled or are unable to collect data pertinent to the
model.
These intractabilities are handled by making assumptions or
simplifications to accommodate them. Possible accommodations to the
above would be:
(a) Diminish the degree of abstraction;
(b) Simplify the formulation or obtain an approximate solution;
(c) Simplify the model to one that is tractable or design new
experiments or experimental techniques.
These accommodations must be consciously made. Then} if the model
does not work} ~'~'} it does not behave like the system it purports to
represent} the assumptions or simplifications can be critically re
examined.
Once a model is formulated} it must be tested. It is set up to
represent a particular situation for which experimental results are
available) is solved} and the results from the model are compared to
the experimental results. If the model results do not compare favor
ably with the experimental results) we are then involved in an
iterative process (Figure 2.1). We must apply our technical knowledge}
of the model and of the process being modeled} to the situation and
modify the model (re-formu1ate) and/or modify parameter values
• 4
7
Initialization
..
Initial parameter values(a) from literature(b) derived estimates
Iteration
Testmodel
...."-
\
\\
Modify modeland/or
modify parametervalues
Model valid
AcceptableII
II
/,.-_ ....
Compare withexperimental
results
NotAcceptable
Technicalknowledge
Figure 201 Iterative model-building
8
(re-estimate). This cycle continues until the model results compare
favorably with the experimental results over a wide range of experi
mental situation~. Then it can be judged a valid model. The model may
then be extended to a broader range of experimental situations or to
additional species. It then must be tested again, as shown by the
dotted line in Figure 2.1, and again enters the iterative cycle.
The model is then used to learn about the system which was
modeled. It may be used to evaluate the prediction error in order to
estimate the confidence in the predictions. In those cases in which
there are insufficient data to completely test or completely elucidate
the model, the model will indicate the need for further experimenta
tion and point out which data need to be collected. The model is also
useful in checking out the assumptions made in developing the model.
In cases in which more than one model appears to be valid, examination
of the results of the competing models will indicate for which input
values their results differ and hence define the critical experiment
needed to select one of the models.
The model may be used to predict performance under conditions not
previously explored experimentally, or it may be used to investigate
performance under conditions not previously considered in order to
identify probable fruitful areas of experimentation. In the case of
systems which are expensive or difficult to experiment with, it may be
used to characterize the behavior of such systems.
• A
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A.
.,.
9
3. BACKGROUND AND KEY LITERATURE
3.1 Roles of Feed Constituents
One of the empirical adjustments made when using feeding standards
is for the fact that protein may have an energetic role as well as a
structural one. As part of the abstraction process in the development
of a conceptual framework, only two roles of feed constituents will
be considered, namely structural and energetic (Figure 3.1). It will
be assumed that supplies of regulatory nutrients (~.~., vitamins) are
adequate.
As shown in the figure, the two roles are not mutually exclusive.
Minerals may be considered as almost purely structural. Proteins have
a dual role, but usually function as structural materials. Carbohy-
drates and fats are predominantly energetic materials.
3.2 Definition of Physical Compartments
It is important, in developing a conceptual framework, that
definitions be such as to simplify the ensuing mathematical development
of the model. Often, certain "well-known" concepts or terms are
redefined to provide working definitions which are to be used in the
context of the conceptual framework. Physical compartments, derived
from Lucas,l which are used to describe animal nutrition and metabolism
are presented in Figure 3.2. Descriptions and working definitions as
used throughout this dissertation follow.
1Lucas, H. L. 1968. Mathematical Models in Animal Nutrition.
International Summer School on Biomathematics and Data Processing inAnimal Experiments. Elsinore, Denmark.
10
.~
Structuro-energeticAlmostpurely
structuralUsually mostly
structuralPredominantly
energetic
mineralsproteins
some lipidscarbohydra tesfats
Energy SupplyBody Proper
External Production~.~. milk
•
Figure 3.1 Structural and energetic roles of feed nutrients
e • 4' e .. ~. e
(M)
B
,....--.
......--. ,
> )I
Body Body Ext.Wear Proper Stores Prod
(P) (BW) (B) (S)Pool
(H)(GH) (BH)(U)(F)(FE)(GP)(C) (V) (R) ~
~ ~Ll i t
Fecal Components Heat Prod.Food Gas Feed Gut Excre-Fecal Urine Gut Body HeatCone. Loss Residue Products tion Loss Loss Ferm Process Loss
,II
~ [3< <,
B~,-
P
FE
v
.-....
, , '~~~~ --G
EJ"'
~
"'"BHI ,~~ _'~T r ~
rGJG
Figure 3.2 Physical compartments I-'I-'
12
Food consumed (C) is partitioned as a result of digestive proc
esses into combustible gas loss (V), a residue in the feces (R), heat
of fermentation (GH) and nutrients absorbed into the pool (F).
The pool (F) represents materials in the circulating and inter
changing fluids of the body, or~ in the case of heat, the total heat
content of the body.
External production (M) includes such things as living offspring,
milk, eggs, wool, fur and work.
The body proper (B) is the muscle, skeleton, vital organs, and
other tissues, exclusive of adipose. That is, it comprises the
systems or structure for existence, growth, fat storage and utiliza
tion, external production and reproduction.
The energy stores (8), which are here differentiated from and
exclusive of the body proper (B), are mainly adipose tissue, ~.~.,
the depot fat.
It is very important, for purposes of developing the model, to
differentiate between the body proper and the stores. The body proper
consists of a mixture of structural and energetic materials. The
stores are considered to be energetic material only. Any structural
components usually associated with the depot fat are defined as being
part of the body proper. The functions of the body proper and the
stores are different. In addition, there are constraints on the
maximum size of the body proper, but virtually none on the stores.
Different mathematical formulations are developed for the flow of
material into and out of these compartments. The abstraction process
involves grouping together those items which can be mathematically
...
..
..
..
..
13
handled in a similar fashion. Since the body proper and the stores
cannot be handled similarly, they must be carefully defined and
differentiated. For brevity, the term, body, may be used in place of
the term, body proper.
In addition to the feed residue, the feces (F) contain metabolic
waste products, or an excretion component (FE), and a gut product
component (GP), both of which have the pool as their source. The gut
product component, associated with digestion of food, includes tissue
debris from abrasions of the walls of the digestive tract, mucus and
materials secreted in digestive juices.
The urine waste products (U) and the heat loss (H) are eliminated
from the pool.
The pool is the proximate source of materials for nonfermentative
digestive processes, for body building, for energy stores and for
external production. The body proper and the stores are constantly
interchanging materials with the pool. The net balance is an in
crease, maintenance or decrease of body proper and/or of fat stores.
Associated with the various aspects of processing pool materials
is wear and tear on the body (BW), which is analogous to the gut wear
resulting from digestion of food. Also related to these processes are
heats of reaction, or body heat (BH). The body heat and the gut heat
(GH) supply the heat pool. Materials oxidized to drive the heat
producing reactions are supplied by the pool •
Figure 3.2 generally follows the traditional framework for energy
metabolism. However, certain sub-compartments, which are not usually
identified, have been distinguished. These are the three fecal
14
components (R, GP, and FE), the two heat production components (GH and
BH) and the distinction between the backflow of material from body to
pool and the body wear (BW).
This representation, which is quite general, may be applied to
proteins and to almost any other nutrient by deleting certain compart
ments. For example, by excluding gas loss (V), heat production (GH,
BH), heat loss (H) and stores (S), the remainder is a framework for
protein metabolism.
3.3 Current Feeding Systems
Feed evaluation systems, reviewed and discussed by Kriss (1931)
and Maynard and Loos1i (1962), may be expressed in terms of the physi
cal compartments shown in Figure 3.2.
The digestible nutrients (DN) system states the apparently
digestible amount of each nutrient per unit of feed. If C represents
the amount of a given nutrient consumed, F the amount in the feces,
and W the weight of the feed, then digestible nutrients may be
expressed as
..
'.
(3. 1) DN = (C-F)/W.
If the digestible nutrient calculation were done individually
for crude protein, ether extract, nitrogen-free extract and crude
fiber, and the TDN values for these nutrients labelled CP, EE, NFE and
CF, respectively, then total digestible nutrients (TDN) , in terms of
energy, may be expressed as
(3.2) TDN = CP + 2.25EE + NFE + CF
where EE has 2.25 the energy v~1ue of the others, per unit weight.
15
MetaboLizable energy (ME) J when measured directly in an energy
balance trial" is computed as t.he groe's energy of the feed (C) minus
the energy losses in feces (F)~ combustible gases (V) and urine (U), or
."(3.3) ME C = F ~ V = u.
Net energy (NE), when measured directly in an energy balance
trial, is computed as the metabolizable energy less the heat increment.
In terms of Figure 3.2" we have
(3.4) NE = C = F = V - U = H.
.....
The digestible nutrient system considers only fecal loss, ignoring
the digestive gas, urine and heat losses. The fecal loss (F) includes
gut products (GP), which represent secretions resulting from the
digestive process, and fecal excretion (FE), which represents the
metabolic waste products.
In compu.ting total digestible nutrients" it is assumed that all
nutrients are to be used strictly for energetic purposes, although
the TUN system specifies levels of digestible protein to satisfy
structural needs. Insofar as protein is used for structural purposes,
TDN underestimates the ene.rgy supply.
Metabolizable energy accounts for the losses in metabolism except
gut fermentation heat 108S (GH) ~ \\Jhich is a digestive loss, and body
process heat production (BH) ,9 which is a loss of metabolism.
Net energy brings in the concept of the heat increment, but this
quantity can be affected by the state of: the animal and its environ~
ment.
16
3.4 Mathematical Approaches
Lucas 2 has stressed the importance of taking energetic and
structural materials jointly into account in a proper way. McMeekan
(1940a, 1940b, 1940c) for example, in reporting his classical experi-
ments with pigs demonstrated this point. He showed that by varying
the total energy intake and the protein to energy ratio in the feeds
at different stages of the pig's development, he could affect greatly
the composition of the carcass relative to skeleton, muscle, skin and
fat.
Some of the investigators who have contributed to improvements
in the interpretation of studies on digestion and utilization of feed
are discussed below. They have tried to better explain the mechanisms
involved in the passage of food through the digestive tract.
Schneider (1935) concluded from an investigation of the metabolic
fecal nitrogen in the feces of the rat that there were two components.
One, a constant amount that is probably of excretory origin, was found
to be related to body surface and also to endogenous urinary nitrogen.
This component is represented by the fecal excretion compartment (FE)
in Figure 3.2. The second component, which was found to be propor-
tional to food intake, is represented as gut products (GP). The urine
loss (U) includes the endogenous urinary nitrogen.
Blaxter and Mitchell (1948) incorporated the metabolic fecal
nitrogen into their expression for calculating protein requirements.
.....
17
They also used a statistical procedure for estimating true digestibil
ity of feed proteins.
Lucas and Smart (1959) applied Blaxter and Mitchell's procedure
to calculating true digestibility of 11 feed components including
ash, crude protein, crude fiber and nitrogen-free extract in two
types of forage. The equation used is
(3.5) y=a+~
•
in which y is the apparently digestible amount of the feed component,
as a percent of feed dry matter; x is the amount of the feed component
fed, as a percent of feed dry matter; ~ is the true digestibility
coefficient (as a fraction); and a is the intercept, representing
fecal matter other than undigested feed residues. The term a includes
the secretions of the body due to digestion, represented by gut
products (GP) and excretions due to breakdown of the body, included
in the fecal excretion (FE) in Figure 3.2.
Blaxter et al. (1956), investigating the digestibility of food by
sheep, used mathematical analysis to estimate diurnal variation of
feces production and the length of the preliminary period necessary in
digestion trials. They also concluded that digestibility of feed
could be predicted by its passage through the gut. The digestibility
process was assumed to involve two compartments, rumen and abomasum,
with a time lag accounting for action of the duodenum.
A quantitative theory relating the apparent digestibility of
nutrients in feeds to the composition of feeds and feces has been
developed by Lucas 3
18
The theory is based on several postulates relating
to description of the feed in physi.ca 1 ;lnd chemh:al terms.
Blaxter (1962a, ppo 295~296) stated~
Firstly what is needed is a method wh~reby the productiveperformance. of an individt:1al can be predicted wit.h someprecision from a knowledge of the quantities of differentfoods which make up its ration, and of the conditionsunder which i.t is kept. • .• The second considerationis that any such method mu.st be capable of assigning to aparticular food a nutritional worth in a particular setof circumstances.
In line with this statement, Blaxter (1962b) proposed a sys tern,
called performance prediction, for assessing energy value of feeds.
This system accounts for the plane-af-nutrition effect in calculating
the true metabolizable energy of a ration o It then calculates the
energy availahle for production as the difference between the true
metabolizable energy of the ration and the metabolizable energy re-
quired to maintain the animal. The method involves assuming a produc-
tion requirement to be met, estimating a ration to satisfy this
requirement, and then calculating the production that the ration will
support. If this value does not agree with the assumed production
value, the ration is adjusted and the results recalculated o This
method is a step in the right direction; it takes into account some
factors contributing to non-additivity of individual feed values. It
does not, however, encompass the interaction of protein and energy
intakes or their ultimate use for growth, fat storage or external
production.
3Lucas, H. L. 1960. Relations Between Apparent Digestibility and
the Composition of Feed and Feces. Mimeographed Report, Department ofStatistics, North Carolina State Ur..iversity at Raleigh"
...
•
..
19
Lucas (1964), building on his previous work in digestibility
studies, proposed a model to handle digestion and absorption. This
model distinguishes digestion from absorption, accounts for the gas
loss and considers the synthesis and degradation by micro-organisms,
in the ruminant, during digestion. The model introduces the use of
stochastic elements to account for experimental and other errors.
Lucas showed how the stochastic model may be used to evaluate the
predictivity of various chemical fractions in feeds.
Blaxter (1966), in his discussion of the feeding of cows and
the partition of feed in the maximization of the economic return from
milk, recognized that the milk yield, as a function of feeding level,
follows a law of diminishing returns. The strong diminishing returns
phenomenon is the result of looking at only one output of the animal.
If one considers the partitioning of the feed among the body proper,
the stores and external production, the sum of these values is almost
linearly related to feeding level, but still reflect diminishing
returns to some extent. The diminishing returns for one output
apparently are explained by a change in the partitioning of the feed
by the animal as feeding level increases.
Waldo (1968; p. 270) in a review of nitrogen metabolism in the
ruminant stated~
It should be possible to describe nitrogen metabolism interms of pool sizes and concentrations, the order ofrates of transfer or reaction, and the magnitude of theserates. These concepts can aid us in understanding therelationships between many processes.
Waldo looked only at part of the whole picture. Byoignoring energy
metabolism, he omitted the importance of the dual structural-energetic
role of protein.
20
In a recent lecture, Lucas4 dis c us sed the various feed
evaluation methods and efforts in modeling metabolic processes involved
in digestion and in utilization of feed. Taking these efforts into
account, he has outlined the beginnings of a model to account for the
combined metabolism of nitrogen and of energy.
4Lucas, 1968, 0p. cit.
."
21
4. DEVELOPMENT OF THE MATHEMATICAL MODEL
4.1 Preliminaries
A mathemati.cal model for the partition of nitrogenous and non~
nitrogenolls energetic materials in the bomeothenrr is to be developed.
It will characterize, mathemat:lcal1y, how the animal digests, absorbs,
transports, synthesizes, catabolizes, interchanges and excretes these
rna ted.a1s. The model will extend both the concepts embodied in the
feeding standards and the modeling efforts reviewed in Chapter 3.
For a given feed, or input, the model will trace the flow of this
input through the physical compartments of the animal, through its
chemical transformations to other materials, and to its final disposi
tion, or use, by the animaL The mathematical model which defines the
flows of this material will take into account the physiological state
(~'12" age, stage of growth, stage of gestation, stage of lactation)
of the animal and the relative amounts of protein and other energetic
material in the feed.
At the first level of abstraction, all materials not classified
in Figure 3.1 as structuro-energetic are ignoredo Nitrogenous
materials inclu.de protein, free amino acids, urea? ammonia,
creatine, and creatinineo Ncm~nitrogenou.s energetic materials include
starch, glucose, fiber, fatty acids, lipids, depot fat, and methane.
For simplicity of notation, the term Henergetic materials" will be
used to denote and include all non-nitrogenous energetic materials.
Although heat is a form of energy, it is considered separately from
energetic materials.
22
Each nitrogenous material, in addition to its nitrogen content,
also contains an amount of energy which is characterized by its heat
of combustion. The materials can thus be characterized by a binary
system of notation,
(N,E)
where N = gms of nitrogen per gram of material, and E = kca1 of energy
per gram of material o According to figures from B1axter and Rook
(1953), the figures for body protein and depot fat are
body prote~n (0016, 50322)
depot fat (0·00, 9.367)
This notation is easily generalized to handle additional types of
material. To handle a substance like carbon, a ternary notation is
used, ~.~o,
(N,E,C)
where C gms of carbon per gram of material and Nand E are as defined
above. The above examples become~
body protein (0016, 50322, 0.525)
depot fat (0.00, 90367, 00765)
Note that both nitrogenous and energetic materials have an
associated energy value. Hence, both materials can be characterized
by their heat of combustiono Then, for nitrogenous material, the
kcal of heat of combustion can be converted to grams of nitrogen;
~o~., 5.322 kca1 of body protein is equivalent to 0.16 grams of
nitrogen. Thus, each kca1 of body protein is equivalent to
0.16/5.322 00301
23
' ..
..-
grams of nitrogen.
The first step in developing the model, namely the definition of
those physic.al compartments necessary to describe the flow of nitroge
nous and energetic materials in the homeotherm, has been discussed
previously in Section 302.
4.2 Conceptual Framework
4.2.1 Nitrogen, Energy and Heat
A physical subdivision of the animal is not adequate to handle
the partitioning of several types of feed materials. The classifica
tion of feed materials as either nitrogenous or energetic has been
discussed. These materials start out together in the feed, but are
handled differently in the various physical compartments. In the case
of starch, an energetic material, some of it is completely digested
in the mouth. This digested starch is more readily absorbed than is
the protein in the feed. The stores consist solely of energetic
materials, thus no nitrogenous material flows from the pool to the
stores. The physical compartments must be subdivided into chemical
compartments to ha!ldle nitrogenous and energetic materials.
Heat is produced from reactions involving both types of materials
and is dissipated differently than either y hence a third chemical
compartment ffitlSt be included fer hea t.
A further complication is that nitrogenous materials can be
converted to energetic compounds. Amino acids are deaminated to
keto=acids with the concomitant production of urea and the liberation
24
of heat. Thus, the nitrogenous and energetic compartments must be
interconnected and the energetic must be connected to the heat produc-
tion compartment. Figure 4.1 illustrates a general physico-chemical
compartment model.
The physical compartments are labeled as in Figure 3.2 and the
following subscripts used to denote chemical compartmentation:
n - nitrogenous material
e - energetic material
h - heat.
The nitrogen portion of the food consumed, fecal components,
fecal and urine loss and the pool are shown in Figure 4.1. Compartment
C represents both protein and other nitrogenous substances expressedn
as "protein equivalents". The term protein will be used for both
cases. There are no gaseous emissions containing nitrogen, hence this
compartment appears only in the energy part of the figure. The stores
represent depot fat and have no nitrogen content, hence they appear
.. '
only as energy. The path from C to GH represents the heat of digese
tion and, in the ruminant, fermentation. The path from P to BHe
represents the total heat dissipated from all reactions in the body
other than the heat of digestion. Since heat is a form of energy, it
does not appear in the nitrogenous part of the figure. The path from
P to P represents the conversion of nitrogenous material to energeticn e
material.
Several compartments are not subscripted. The gas loss compart-
ment (V) and the stores (S) involve energy only. Since there is no
e ( 6 -- <. ~. e
_ Fecal Components Heat Prod,Food Gas Food Gut Excre- Fecal Urine Gut Body HeatCons Loss Residue Products tion Loss Loss Ferm Process Loss Pool
(C) (V) (R) (GP) (FE) (F) (U) (GH) (BH) (H) (P)
ICn I) i ~'fGJ IGPn \ ~ (
Body Body Ext.Wear Proper Stores Prod
(BW) (B) (3) (M)
1GPe\ < <I Pe
I-€J
P ,DD
5]< J' II >~ EJ ~.----«
~) i I ,1
N\J1
Legend - subscriptsn - nitrogene - non-nitrogenous
energyh - heatPhysico-chemical compartmentsFigure 4.1
EJ~<--
, I .. >~ ~J <
I ~ ~~ J ~ ~EJ
y X IH~5Jc:J
26
corresponding nitrogenous compartment, the subscript would be
extraneous. The heat production compartments (GH, BR) and heat loss
(R) have an H in their designation" To add a subscript would un-
necessarily complicate the notation, Eody proper (B), body wear (BW)
and external production (M) are unsubscripted for a different reason.
Each of these compartments is considered to represent a well-defined
mixture of nitrogenous and energetic materials. They flow together
in fairly constant proportions to form body proper and may break down
in similar proportions as body wear and back flow from body to pool.
External production also is formed from a flow of nitrogenous and
energetic material in proportion determined by the particular product.
4.2.2 Further Subdivisions of Nitrogen and Energy
The subdivision of the physical compartments into nitrogenous,
energetic and heat components still does not account for all complicat-
ing factors"
The energetic material in the food includes a variety of
substances such as polysaccharides (~.~., starch, cellulose), mono-
saccharides, lipids and others. These differ in rate of digestibility,
in proportion digested and in end products of digestion. The fermenta-
tion of carbohydrates prOVides the gut heat, GR. The gas loss, V, is
related to the digestion of carbohydrates. The gut products, GP ,e
consis~ in large part, of lipid material rather than carbohydrates.
To handle these digestive features properly, energy is further
subdivided into carbohydrate and fat components, denoted by subscripts
c and f, respectively,
-.."
27
Then, compartment C repre.sents a mixture of carbohydrates fed,c
and an average digestibility value is used. Once in the pool, the
end products of carbohydrate and of lipid digestion are not
distinguished from one dnot1:xer, but are designated as P .se
The food residue and gut product components of the feces also
are subdivided to correspond to their originating material, ~.~.,
Cf' Rf
and GPf
for fats and Cc and Rc for carbohydrates. The gut
products represented by GP are not strictly from carbohydrates, butc
rather comprise all gut products which are not from ether extract.
Two categories of pool materials must be distinguished, useful
materials and waste products. The first comprises metabolizable
useful materials such as amino acids, glucose and fatty acids. These
compounds are either digested or hydrolyzed forms from the feed or they
are products from the backflow of materials from the body proper and
stores to the pool" These are available for structure and for energy
for body formation, fat storage and external production. They will be
designated as simple compounds and labeled as P and P for thesn se
nitrogenoills and energetic pool compartments, respecti.vely.
The second category is waste products. These result from body
wear and from chemical degradation of food materials. These are
usually excreted by the animal. They will be designated as degraded
compounds and labeled as Pd
and P:l de"
These refinements are shown in Figure 4.2.
4.3 Discussion of the Total Compartment Model
Figure 4.2 contains ~:he proposed compartment model. It does not
contain a compartment for the dfge.s tive tract. The reason for this is
~
Legend - subscriptsn - nitrogenouse - non-nitrogenous
energyh - heatf - fatsc - carbohydrates
Body Body Ext.Wear Proper Stores Prod(BW) (B) (S) (M)
) ~rsl
< <LJ
Pse
@-<: Ph,
~
~) I ~~ I
( )r:t) IL2.I
Fecal Components Heat Prod PoolFood Gas Food Gut Excre- Fecal Urine Gut Body HeatCons Loss Residue Products tion Loss Loss Ferm Proc Loss Degraded Simple(C) (V) (R) (GP) (FE) (F) (U) (GH) (BR) (H) (P)
EJ) • ~ IPsn
r;I ~~ -<.: r;;l < I') i
~ n ~«~ ""'I
T- «n ~Fnl ru::l.. <~
Figure 4.2 A compartment model N00
e . , e ,0,• e
'.
.'
..
29
that we will concern ourselves with a controlled type of feeding in
which after a few days on a constant amount of a standard diet, a
nonruminant will reach a state of dynamic equilibrium. The rate of
passage of food at a given point in the digestive tract will be rela-
tively constant and the partitioning of the food into an absorbed
portion, fecal residueJ etc. will be characteristic of the food and
the animal (Blaxter et al., 1956; Maynard and 1o0sli, 1962). For a
ruminant, a longer time period i~ necessary for equilibrium to be
reached. We will later consider a 28 day nitrogen and energy balance
trial and so the assumption of equilibrium will be a realistic one.
Thus we can concern ourselves with the partitioning of the food into
absorbed, fecal residue, gas loss and gut fermentation heat portions,
and ignore the details of digestion.
Compartment C represents the protein or nitrogenous materialn
fed. The path from C to P represents the digestion of protein andn sn
the absorption of the resultant amino acids into the pool. Fecal
nitrogen (F ) comes from three sources. Feed residue, or undigestedn
protein, is shown by the path from C to R. A second source is gutn n
products (gut wear and secretory materials) shown as the path from Psn
to GP (Lucas, 1964, p. 376)~n
. these are associated with and/or are necessary forthe digestion of feeds (~.~'J mucus and constituents ofthe digestive juices).
Schneider (1935) refers to this as the digestive fraction of the
metabolic fecal nitrogen. It is proportional to the quantity of dry
matter consumed. The third source of fecal nitrogen, the flow from
Pdn to FEn' may be considered as excretory materials (Lucas, 1964,
p. 376):
. waste products of metabolism or excesses in thebody proper for which the gut is one of the paths ofe1imina tion.
Schneider (1935) refers to this as the constant fraction of the
metabolic fecal nitrogen. According to Schneider's data on rats and
swine, this constant fraction is proportional to the metabolic body
size, !.~., body weight to the three-fourths power. The major part
of the fecal nitrogen is usually accounted for by the undigested feed
residue and the excretory fraction. If the dry matter consumed and
nitrogen consumed are low, then the gut products fraction may be a
significant proportion of the fecal nitrogen, Excretion of urinary
nitrogen is shown as the flow from Pd to U. This includes what isn n
usually termed endogenous urinary nitrogen (Schneider, 1935; Maynard
and Loos1i, 1962) as well as all other waste nitrogen.
The feeding of lipids is denoted as compartment Cf
. The diges
tion of these materials is shown as the path from Cf to P . Un-se
digested "fats" flow from Cf to Rf , and lipids secreted into the
feces are shown as the flow from P to GP f .se
The feeding of polysaccharides, including fibrous material, is
shown as C. Digestion of these materials is shown as the path fromc
30
..
'.
C to Pc se Undigested "carbohydrates" are indicated by the flow from
C to R , and carbohydrates secreted into the feces by the flow fromc c
P to GP •se c
Energetic materials in the urine are shown as a flow from Pde
to
U. Two other losses are associated with the digestion of carboe
hydrates, especially with herbivores. The volatile gas loss, ~.~.,
."
31
methane, is shown as a flow from C to V, and heat production from gutc
fermentation is shown as a flow from C to GR.c
According to Schoenheimer (1942), lipids of fat depots constantly
undergo synthesis, interconversion and degradation. Fat and fatty
acids are steadily and rapidly regenerated. These processes are
depicted by the paths from P to S and from S to P , the formerse se
representing the synthesis and the latter the degradation of the
depot fat. Interconversion of lipids is ignored.
The synthesis of nitrogenous and energetic materials for external
production is shown as the paths from P and P to M. We assume that,sn se
for a given species and product, the product has a fairly constant
ratio of energy to nitrogen. This has been shown for milk of various
breeds of cattle (Overman and Gaines, 1933). Sufficient amounts of
nitrogenous and energetic material will react to form the product, to
provide energy to drive the reaction (and be dissipated as heat) and
to provide for any waste products.
The synthesis of the body proper for growth and for replacement of
cells, which are constantly breaking down, is shown by the paths from
P and P to B. As for production, we postulate a constant energy tosn se
nitrogen ratio for the body proper. There is a certain amount of fat
associated with the protein in the organs of the body. This fat dif~
fers from depot fat in its function. It will be considered as part
of the total fat in the body, but its formation will be related to the
formation of protein. The combination of this protein and fat
comprisesthe formation of the body, B. Schoenheimer (1942) states that
proteins of the body, like lipids of the fat depots, are also in a
cththe amount of body wear per unit of the L flow
32
dynamic state. Thus proteins are constant.ly being degraded to amino
acids as shown by the path from B to P , and are constantly beingsn
synthesized from amino acids. In order to maintain the constant
energy to nitrogen ratio in the body, the breakdown of each unit of
protein will be accompanied by a concomitant breakdown of fat from the
body and this is shown as the path from B to P 0
se
For every reaction or flow of material in the animal, there is a
certain amount of wear and tear on the systemo This wear and tear is
represented as a breakdown of the body, B, since it is the body which
includes the physiological systems in which the reactions take place.
The wear and tear, BW, resulting from each reaction or flow, is propor-
tional to the flow. The total body wear is the sum of the individual
body wear terms·. The nitrogenous and energetic materials are broken
down into simple energetic (Pse)' degraded nitrogenous (Pdn), and
degraded energetic (Pde) materials, and into heat. The total break=
down, BW, is represented as
BW ~o~ofo1. 1. 1.
where: ~i
f. the ith
flow1.
A fasting animal will oxidize its tissues to produce heat to keep
warm. This will entail body weaL A certain amount of nitrogenous
and energetic material in the feed will spare the oxidation of the
tissues and the body wear (Blaxter, 1962a) and replace the gut wear
resulting from the digestion and metabolism of the feed itself.
This amount is the maintenance level of feeding.
33
Every reaction, or flow of material from one comparbment to
another, produces a heat of reaction. The energy which is converted
to this heat is supplied by the pool, P • The amount of heat isse
proportional to the amount of material transported, synthesized or
catabolized.
For an exothermic reaction, the heat evolved is handled as a
transfer of energy from the energy pool, P ,to the heat production,s~
BH. For an endothermic reaction, the heat absorbed is handled as a
transfer of energy from P to the compartment receiving the productse
of the reaction.
The total heat generated from all reactions will thus be the
sum, over all flows, of the product of the flow and the proportionality
factor reflecting the heat of reaction for that flow, ~.~.,
BH = ~.~.f.~ ~ ~
where: BH = total heat production from body processes
~i the heat production factor for the .th flow= 1
f. the.th flow or reaction.= ~
~
In addition to the heat production and body wear accompanying
every reaction, there also is the production of by-products or waste
products from every reaction. These waste products flow to Pdn or Pde
and are then excreted, or in the case of the ruminant, may be utilized
by the rumen microbes.
For each flow, there will be an amount of ni.trogenous and ener-
getic waste per unit of flow. These amounts are denoted by the
constants V and ~ for nitrogenous and energetic waste, respectively, in
34
the pool. The total amounts of these waste products may be formulated
as:
Pdn
waste
Pde
waste
L. 'V .. f.~ ~ ~
LS.f.~ ~ 1.
where: f.~
= the i th flow
'V.,s. = the amount of nitrogenous and energetic waste which~ ~
flow to Pdn and Pde: respectively, per unit of f i
'V. , S.~ ~
o for those f. which produce no waste.1.
The heat production from all reactions, denoted by compartment BH,
combined with the heat production from gut fermentation, GH, forms the
heat pool, Ph. The dissipation of heat is shown as the flow from Ph to
H.
The model as described herein is quite general and includes the
major paths or flows of nitrogen and energy. Conceptually, extension
of the model to handle ruminant digestion is not difficult. Compart-
ments and flows for the digestive tract and for the ruminal micro-
organisms would have to be added.
4.4 Development: of Flow Laws
For each of the paths developed in Figure 4.2, we will now
proceed to develop a flow law or mathematical equation. The simplest
mathematical formulation consistent with nutritional facts and princi-
pIes will be proposed. These formulations are differentials that
express the flow of material per unit of time or the rate of flow of
material.
35
These flow laws are then combLnedt:0' form dlfferential equations
and Pdeo The rate of change in a compartment eqtlals the sum of the
flows into it minus the sum of the fl'JWS OUit oftt •
-.
.-
for seven compartments in the rr:odeL :J.8JId21y B o S P PJ , J sn J se~
The systEm of differer.tial eq~]at::'onst:hlis derived ccmstitClltes the
mathema tical model.
40401 Notation
In Section 401, we discussed the fact r:hat both nitrogenous and
energetic materials could be characterized by their heat of combustion o
Then from the binary notationJ a corresponding vallLi.e in grams of
ni trogen cO\Jld be associa ted with each emit: of nea t for the ni trogenollls
compounds. Hence? the basic tinit of flow will be the kiloca1orieJ
kca10
The arnmmt of kcal in any compartment at time t will be denoted
by Illpper case letters for the compartment designation followed by
lower case subscripts where appropriate, ~o~oJ the kcal of simple
ni trogenous rna terial in the pool at t:Lme t is P ,and the total kcalsn·
of body proper is Bo
The rate of change, or derivat:ivt y of the amoUlnt of kcal in any
compartment is denoted by lower caae letters for the compartment
followed by lowe.r case subscripts where appropriate. For the abuiJe J
the rate of change of kcal in Fsn
is denoted by Pen and the rate of
change of body is bo The units are kcal/timeo
For the flow from one compart:nent to anether, the ccmpartment
from which the material c'riginates :,s designated as a derivative. The
compartment to which the material is flowing ~s designated as a
36
subscript. This subscript consists of an upper case letter or letters
for the compartment followed by lower case letters where appropriate.
For example, the flow from P to P would be designated as p •sn se sn,Pse
It may be read as the rate of flow of amino acids to Obketo acids, The
flow from body proper (B) to body wear (BW) is designated bBW
'
We have discussed coefficients for waste products in the pool
(\I and s), for heat (T]) and for body breakdown (P). The heat coeffi-
".
cient for Psn Pse ' written,generated per kca1 of flow
as T1 , represents thePsn,Pse
from P to P per unit ofsn se
kca1 of heat
time. The firs t
subscript on T1 denotes the originating compartment and the second
subscript, the destination. The same notation applies for \I, S, and p.
The flow laws are characterized by a "rate" constant denoted by
*a Greek kappa with a superscripted asterisk, K. These are denoted in
the same fashion as the waste and heat coefficients discussed above.
Each flow has related to it a certain amount of breakdown and the
production of waste products and heat, The heat will be supplied from
P and the body breakdown will be deducted from B. The waste producedse
must be subtracted from the gross flow to produce a net flow of
material, This is incorporated into the rate constant.
For example, consider the flow from P to Psn se
loss from P is shown assn
The rate of
-1<
Psn,Pse
The function ; will be defined in the next section, The terms
for the waste produced in the pool are
..
,.
37
and
The net flow to P from. P is thusse sn
psn,Pse*(psn pse) (1, \)
Psn,Pse
where
K:Pne\) - ~ )Psn,Pse ~PsnJPse
In general, to simplify matters, the flow laws will be developed
directly as net flows, ignoring the waste, heat and body breakdown
terms. However, for those reactions where these terms playa major
role, they will be developed explicitly, Ignoring these terms does
not necessarily imply that they are zero. For testing thEi! model and
for predicting, these terms must be estimated.
Additional parameters, and constants and exceptions to the above,
are defined as needed in the subsections to follow.
4,4.2 The Flow Laws
• According to the notaticm. dee,ned above, the amount of protein
fed per unit time (i.e., kcal/day) will be designated c ,n
38
The protein fed is either digested or defecated. There is a
well-known effect of feeding level on digestibility (B1axter et a1.,
1956; Forbes et a1., 1928; 1930). As feeding level increases, digesti-
bi1ity decreases. We define the true digestibility coefficient of
*protein, d , as if there were no feeding level effect. The effectn
of feeding level on digestibility is handled as follows.
For a given diet, define the amount of dry matter fed per day
which results in zero energy balance as cd . This corresponds to a,0
feeding level of unity. Then the feeding level, FL, for a given ration
would be the actual dry matter fed, cd' divided by cd,o' or
Then we may define the feeding level effect as
where 0 < aFL < 1, bFL < 0 and 0 < fFL < 1. The digestibility coeffi
cient of protein, corrected for feeding level, is then
Thus the flow law for the digestion of protein is
...
0.
(4.1a) c = (dn
) (cn
)n,Psn
and the flow law for the passage of undigested residue to the feces
(R) is given byn
(4.1b) c = (1 - d ) (c ) •n,Rn.. n n
".
39
The digestion of fats and carbohydrates is handled similarly to
* *protein. We define true digestibility coefficients, df
and dc' for
fats and carbohydrates. Using the same function of feeding level as
above, we have that
The digestion of fats, or passage from Cf to Pse' is given by
(4.2a)
where cf
is the amount of fat fed per day. The undigested fats flow
to Rf according to
(4.2b)
For carbohydrates, there is a complicating factor. Of the cc
kcal of carbohydrates fed per day, (d )(c ) of this is digested. Thec c
amount of methane produced (V) and the gut heat production (GH)
usually are taken to be fractions of the digested carbohydrate
(Blaxter, 1962a; Bratzler and Forbes, 1940). We thus partition the
digested energy among the energy of the methane loss, the gut heat
production and the energy absorbed into the pool. If the fraction
lost as methane is ~c and the fraction lost as gut heat is ~GH'
then the fraction absorbed is (1 - ~c - 'TbH)·
absorption and food residue in the feces are
Then, the equations for
(4.3a) c = (1 - i-Lcc,Pse
40
(4.3b) c = (1 - dc) (cc) .c,Rc
..
*.
The amount excreted as methane is
(4.3c) cc, V
= (i-L ) (d ) (c ) .c c c
The equation for gut heat production is
(4. 3d) cc,GH
Passage of material to the gut products compartments (GPn , GPf
and GPc) is proportional to the amount of dry matter fed per day, cd'
The proportionality factors differ for protein, fats and carbohydrates
and are denoted by KGpn' KGPf and KGpc ' respectively. The flow laws
are
(4.4)
(4.5)
(4.6)
psn,GPn = (KGPr/(Cd) .
Pse,GPf = (KGPf) (cd)
pse,GPc = (KGpc) (cd)
In developing the remaining flow laws, we assume that the flux is
a function of the product of the body size, B, and the basic driving
force at any point as determined by the concentrations of reactants.
The path from Pd to U and FE represents the passage ofn n n
excretory nitrogen (Lucas, 1964) or endogenous fecal and urinary
-.
41
nitrogen (Schneider, 1935). The concentration of the nitrogenous
material in Pdn may be expressed as
and the product of this term and body size, B, is Pdn . Thus the flux
is a function of Pdn
•
Schneider (1935) found in rats that the ratio of endogenous fecal
to endogenous urinary nitrogen was approximately constant over a wide
range of nitrogen intakes. Thus, we define the fraction of the total
nitrogen excreted from Pdn which goes to Un as fUn and that which goes
to FEn as (1 - fUn). If the rate constant for flow from Pdn is Kpdn'
then the flow laws are given by
(4.7a) Pdn,Un
(4.7b) P = (1 - f ) (K ) (P ) •dn FEn Un Pdn dn,
The total rate of flow of nitrogenous material as fecal loss
is expressed as the sum of the rates of flow of the three fecal
components.
Similarly, for the passage of energetic waste material from Pde'
we define the rate constant KPde
and the fraction passing to Ue as
fUe
. Then the flow laws for energetic waste are given by
(4.8a)
(4.8b)
The total rate of fecal loss of energetic material is expressed
as the sum of the rates of the fiv~ fecal components.
42
Next, consider the flow of material from P to P , the conver-sn se
sion of amino acids to energetic material. We must consider the
relationship between the amounts of material in these pool compart-
ments.
The nutritive ratio for a feed is defined as the ratio of digesti-
b1e protein to the sum of digestible fats and carbohydrates (Maynard
and Loos1i, 1962). We modify this concept and define the nutritive
ratio for the pool (NR) as the ratio of the amino acids (i.e., digested
protein) in the pool, P , to the simple energetic materials (~.~.,sn
.~
digested carbohydrates and fat) in the pool, Pse
NR = P Ipsn se
Thus,
We postulate that for a given animal, in a given state, there is an
optimal nutritive ratio, NR. For example, a mature, non-producingo
animal may have a given optimal ratio for maintenance. If it is a
growing animal, the optimal ratio would differ. Since the production
of milk requires different proportions of nitrogen (P ) and energysn
(P ) than body growth or maintenance, there would be a differentse
optimal ratio for the milking animal. Fattening requires much energy
and little nitrogen and again the optimal ratio differs.
As is well known, when protein intake of growing animals exceeds
a certain level relative to energy intake, the ratio of nitrogen
retained in the body to nitrogen in the urine decreases. Thus, when
the pool amino acids (P ) are high relative to pool energy (P ), wesn se
have
NR > NRo
and the rate of conversion of amino acids to pool energy and urea
increases. This action tends to reduce NR to its optimal value, NR .a
When amino acids are low relative to pool energy,
NR < NRo
and conversion of amino acids to pool energy and urea decreases,
tending to increase NR to its optimal value.
A driving force defined as the ratio of the concentration of
amino acids (P IB) to the concentration of pool energy (P IB) givessn se
this behavior. Then the flow law for the flow of amino acids out of
"1<the pool, p .p , is given bv the product of the driving force, thesn, se .
body size and a rate constant or
43
(4.9a)
This reaction provides urea as a by-product (Dukes, 1955) as well as
heat. The amount of urea formed (kca1) per kca1 of amino acids
deaminated is vpne . Thus the formation of urea is given by
(4.9b) Psn,Pdn
The amount of pool energy formed is the difference between (4.9a) and
(4.9b) or
(4.9c) p sn,Pse
44
where
If the amount of body heat produced per kcal of amino acids
deamina ted is g1ven by Tlpne' then the flow 0 f hea t from Ps e to BH
for this reac tion is given by
··k
(1lpne)(psn,pse) .
The flow of energetic material from P to 8 is defined as these
product of the pool energy concentration (P /B), the body size (B) andse
a rate constant. Thus,
".
(4.10)
According to Schoenheimer (1942);, depot fat is constantly being
degraded to fatty acids and these are constantly being synthesized to
depot fat. If an animal is fed insufficient energy, the net effect
is a decrease in depot fat. We postulate two factors which control
the flow from 8 to P ,body size and fatness. The body has these
machinery for the conversion of depot fat to energetic compounds in
the pooL hence, the flow is proportfonal to Bo We define fatness as
the ratio of fat stores to body, 8/Bo The flow from 8 to P will bese
reduced as fatness is decreased and wi.ll approach a maximum value as
fatness increases. We formulate the effect. of fatness as a hyperbolic
relationship and the flow is then defined by
(4011) sPse
- (K:~ p) (B) [ (S/B) / (S/B + r)] .'"'"
Graphically, this flow is represented in Figure 4.30
-.
• 0
45
- - - - - ~--=--------'--'- - ....
S/B
Figure 4.3
r
Flow law from stores to P as a function of fatnessse
The paths from P and P to B represent the synthesis of bodysn se
proper from the nitrogenous and energetic pools. We will first derive
the flow law for body growth and then partition this growth into
nitrogenous and energetic contributions. The driving force behind
body growth is considered to be the product of the two pool concentra
tions, (P P /B2). The fact that body size is limited must also besn se
taken into account. Brody (1945) characterizes growth by
-ytY = Q' - (~) (e )
where y is body weight, t is age from conception, ~ and yare suitable
con3tants and Q' represents the maximum body weight. The true relation-
ship appears to be a logistic or S-shaped curve) but for ages after
sexual maturity, Brody's formulation is adequate.
We will represent this phenomenon, not in terms of age of the
animal, but in terms of a maximum growth potential or body size,
denoted by AB
. In addition to the driving force, the rate of body
46
growth will be proportional to the remaining growth potential of the
animal, the difference between the maximum body size, AB
, and the
current body size, B, or (AB-B). Multiplying the driving force and
the term for remaining growth potential by the body size .• B, and a
rate constant, Kp B' gives the flow law for body growth,,-.
(4.12) (Kp B) (P p /B2
) (AB
- B) (B), sn se
::: (Kp B) (P ) (P ) (AB
- B) /B 0
, sn se
We have previously discussed the idea of the body being a we1l-
defined mixture of nitrogenous and energetic material. If the amounts
of nitrogenous and energetic material in the body are denoted by Bn
and B , respectively, then the total kca1 of body proper, B, ise
related to these quantities by the relationship
(4.13) B B + Bn e
and the ratio of energetic to nitrogenous material, PB
, is defined by.
(4.14) B /Be n
From (4.14), we have
(4. 15)
and substituting into (4.13) gives
(4.16) B
47
or
-,(4017)
.-Substituting (4.17) into (4.15) gives
(4.18)
Thus. given the amoGCnt of body proper, BJ we use (4.17) and (4.18)
to partition this into its nitrogenolls and energetic components 0 We
assume that: the flow of material into the body, given by (4.12), is
similarly partitioned; hence, the flows of nitrogenous and energetic
material into the body are denoted by Pn Band Pe B' respectively, and, )
are expressed by
(4. 19a)
(4. 19b)
P "- [1/(1 + PB
) ](PB
)llJB
where PB is given by equation (4.12)0
According to Schoenheimer (1942), the body protein and amino
acids are in a d)'namic state just as are the fat stores and fatty
acidso We define a flow law for the degradation of the body which
is proportional to the body size, B, or
(4020)
In order to maintain the ratio PB
within the body, this degradation or
backf10w must be partitioned in the same manner as was the synthesiso
Denoting the backflow to P and P by bp and b , respectively, wesn se n Pe
have
48
(4.21a)
(4,21b) bPe
In addition to the degradaticn of a part of the body tissues to amino
acids and simple energetic forms" some of the body breakdown is wear
and tear on the system,. or body wear, due to carrying out all the
previous reactions, The tctal body wear (BW) may be represented as
a sum of breakdown terms from all the reactions, Then the flow law
for breakdown of the body proper due to body wear is
(4.22a) hBW
-, L. ~ . f .. ,l 1 1
where ~o and f,. are as defined in Section 4,3. This material is then1 1.
partitioned am..::'ng Pdn: Pse and Pde·
The nitrogen of the body protein goes to Pdn
as a waste product
(!:.~., urea and other sub3tances)0 Pdn does not ccmtaLn so much
energy per gram of nitrogen as does body protein, hence the additional
energy goes either to P -' .' or L) P where it can be utilized, Forue' se
each kcal of body broken dc:)",m as body wear,we define the following
partition:
The following constraint alsc holds:
Thus the parti,tion equations are
-..
."
49
(4. 22b) b (\)BW) (bBW)BW,Pdn
(4.22c) b (nBW) (bBW)BW,Pse
(4. 22d) b (~BW) (b BW) .BW,Pde
Flow laws for external production are not discussed here. The model
was tested against data from steers, whose production is encompassed
by body growth and addition to fat stores. However, the development
of equations for the production of milk are found in Appendix 9.1.
As for body wear, the heat production from body processes is
accumulated as a sum of heat production terms from each body process.
As discussed in Section 4.3, we denote this by
(4.23) P BH == I: o T]. Lse, 1 ~ 1
where T]. and f. have been previously defined.1 1
If we denote total heat production as HP, then the rate of heat
production, hp, is the sum of the gut fermentation and body process
heat production, or
(4.24) hp c + Pc, GH se,BH
The heat production flows into the heat pool (Ph)' This heat is then
dissipated, as shown by a flow from Ph to H. The data used to test
the model were collected at a temperature in the range of thermo-
neutrality. A mechanism and flow laws for body temperature regulation
under varying environmental temperatures are presented in Appendix 9.2.
50
4.5 The Mathematical Model
The flow laws developed above are the elements of the differential
equations which form the mathematical model. These flow laws, numbered
as in Section 4.4, are listed in Table 4.1. The differential equations
derived therefrom, one for each compartment in Figure 4.2, are listed
in Table 4.2. The differential equation for each compartment is
defined as the sum of the flows into the compartment minus the sum of
the flows out of it. The compartment values, as functions of time,
are evaluated by integrating these differential equations, subject to
appropriate initial conditions.
The development thus far is represented by the light bulb in
Figure 2.1. We have completed our assumption or derivation of a
model. Subsequent sections will be concerned with deriving initial
parameter values and with the iteration part of model-building.
Table 4.1 Flow laws for the compartment model
51
..(4.la)
(4.lb)
cn,Psn
c = (l-dn) (cn)n,Rn
(4.2a)
(4.2b)
cf,Pse
(4.3a)
(4.3b)
(4.3c)
(4.3d)
(4.4)
cc,Pse
cc,Rc
c c,V
cc,GH
psn,GPn
(l-IJ. -11 ) (d ) (c )c GH c c
= (l-d ) (c )c c
(4.5)
(4.6)
(4.7a)
pse,GPc
Pdn,un
(4.7b)
(4.8a)
(4.8b)
(4.9a)
Pde,FEe
*Psn,Pse
(4.9b)
(4.9c)
(4.10)
Psn,Pse
= (Kp s) (P ), se
continued
Table 4.1 (continued)
52
.-(4011) sPse (K:
Sp) (B) [ (SiB) I (S/B+r) ],
(4. 12) PB= (Kp B) (P ) (P ) (AB-B) IB -., sn se
(4. 19a) Pn B [11 ( 1+PB
) ] (PB
),(40 19b) Pe B - [ PBI (1 +P
B) ] (P
B),
(4.20) b = (KB
p) (B)P ,
(4. 21a) bpn = [l/(l+PB)] (bp)
(4.2Ib) bpe = [PBI ( 1+P B) ] (bp)
(4.22a) bBW
= 2:,p.f.~ ~ ~
(4. 22b) b := (\lEW) (b BW)BW,Pdn
(4.22c) b := (~) (bBW
)BW,Pse
(4. 22d) b = (~BW) (bBW)BW,Pde
(4023)Pse,BH := 2:. 'fl. f.
~ ~ ~
(4024) hp := c + Pc,GH se,BH
53
Table 4.2 Differential equations for the compartment model
ok
Pn,M + bpn - Psn,Pse
(4.26)
(4.27)
+b +b -P +s -P +p -cPe BW,Pse se,S Pse se,BH sn,Pse c,GH
Psn,Pdn + bBw,Pdn - Pdn,FEn - Pdn,Un + Li~ifia
(4.28) bBW,Pde
(4.29)
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)
(4.36)
b
s
fn
un
fe
ue
v
hp
sPse
Pdn, Un
c f . Rf + c + P f + P + Pd, c,Rc se,GP se,GPc e,FEe
= Pde,Ue
aSummation is over those fluxes producing nitrogenous waste inaddition to P Pd and b BW Pd which are shown explicitly. Such termssn, n ,nmust appear as losses against the proper compartments in order tobalance the equations.
bSummation is over those fluxes producing energetic waste inaddition to bBw,Pde which is shown explicitly. Such terms must appear
as losses against the proper compartments in order to balance theequations.
54
TESTING THE MODEL
We test: the m<o,dEd <;dth twc' purposes in mind. First? to e~am:ine
the results in a qualitative sense to see if their pattern is reason-
able} and seccl1ld" to derive the best estimates for each parameter in
the model.
FrQIT: our kn':jwledge of animal DrJtrition... the pattern of compartment
values as, a fClnction of time is generally known. If the model results
are qualitatively rea",:onable., we have some assurance that no unreason-
able assumptions were made in deriving the model and that no major
factors have been omitted. Pessimistically;, it might be considered
that a combination of unreasonable assumptions and major factors
omitted might have equal and opposite effects on the results and cancel
out. This view is rejected on the grounds that the model was derived
from nutritioaal concepts,. and it is cmlikely that such a cancelling
Given th<at the resr::dts are qualitati.vely reasonable<J we then must
exami.ne them quantitatIvely. This i.nvolves deriving a best set of
estimates (lif th.e parameters. "Best lY means that over a series of
situations;, for which experimental results are available., the output
for the particular model is closest, in some sense) to the experi=
mental data. Having obtained this best set, the degree of closeness
is evaluated.
The remainder of this section IS cO>:'.ce:r.ned 'Nith a description of
the meldel tel be testEd, the data LO be "-ised; the initial estimates of
the parameters} selection of constar:ts-, and the i.terative method used
to arri.ve at the final parameter estimates.
'.
"
55
5.1 A Model for a Growing Steer
Data available to test the model shown in Figure 4.2 were collect~
ed under a variety of conditions) l.~.) from steers~ from dry cows)
from milking cows J from animals on controlled feeding regimens and from
those on ad 1i.bitum feedi.ng. No one set of data pe.rmits testing the
model in its entirety. A set of data were found? however J that permit
testing of key features of the model except for external production
and body temperature regulation.
Two experiments on growing steers (Forbes et al.) 1928; 1930)
contain data on feed composition and consumption) body weight and
nitrogen and energy balance. Two steers were involved in each
experiment. In the first experiment the steers were Aberdeen=Angus
identified as steers 36 and 47. In the second experiment, they were
Shorthorns identified as steers 57 and 60. They were fed mixtures of
alfalfa hay and corn meal at feeding levels ranging from one=half to
three times maintenance and hay alone at the maintenance level. The
maintenance level as defined in Forbes et al. (1928; 1930) is only
approximate as zero energy balance was not attained. It is used below
as defined in their paper.
Each diet was fed for a 30-day period and observations were made
during the last 20 days. During the period of observationJ the animal
presumably reached a state of dynamic equilibriurnJ where the rate of
passage of food at a given point in the digestive tract was almost
constant, and the partitioning of the food into an absorbed portion)
fecal residue) etc., had effectively stabilized.
56
We thus reduce the model to a simpler version which still retains
the critical features of the joint nitrogen and energy metabolism.
That is, the external production compartment from Figure 4.2 is
eliminated. The energetic material in the gut products component of
the feces is similar to ether extract. Thus, all energetic gut
products are represented by the flow from P to GPf
and compartmentse
GP is eliminated.c
Since the values of the ~., the coefficients for heat production1.
from body processes, are not known, and the data were not sufficient
to estimate them, the handling of heat production had to be simplified.
The heat produced from gut fermentation will still be calculated in
terms of digested carbohydrates as given by equation (4.3d). The
heat loss per day will be taken as the amount of heat dissipation
measured in the calorimeter, say D, or
...
= D .
Then the amount of heat production per day from body processes will be
taken as the difference between the heat dissipated and the gut
fermentation heat, or
= D - c c,GH
Thus, the total heat production, hp, equals the heat dissipation,
since
hp
D
and the heat pool remains at a constant level.
57
The values of the ~iJ the body wear coefficient.s, also were not
known nor estimable. Hence J the degradation of body as body wear, bBw>'
is assumed to be proportional to the heat dissipation.9 DO' and the
proportionality factor will be estimated in the testing process.
The nitrogen flow will be explicitly followed using the binary
notation for nitrogenous compounds. Each compound has an associated
heat of combustion value (kcal/gm nitrogen). Dividing each flow law
for nitrogenous material (kcal) by this heat of combustion value gives
the appropriate flow law in terms of grams of nitrogen. These flow
laws will be designated by prefixing the energy flow law by ~
subscripted lower case n. The heat of combustion value will be denoted
by a lower case h, prefixed by a subscripted n and suffixed by a
subscript to denote the material to which the value applie.s. A heat
of combustion value wi.thout the prefix will have units of. kcal/gm
material. For example, h B denotes kcal equivalent of body proteinn n
per gm body nitrogen, whereas hBn
denotes kcal body p~otein per gm body
protein. With the above explanations, the flow laws in Table 5.1
represent the model to he tested. Values of \) and S, the waste.
production coefficients.9 will be assl'JIIled to be zero except where
explicitly shown.
The differential equations derived from t.hese flow laws are given
in Table 5.2.. Testing the model involves solving this system of
differential equations and comparing the results obtained to the
nitrogen and energy balance tables given in Forbes et ~!: (192.8, 1930).
A computer program (IBMJ 1968) for the 360~Model 75 computer was used
to numerically solve this system of equat:i.ons. The RungeooKutta
58
Table 5.1 F1c", laws for te.sting the model
(5.1a)
(5.lh) cn n,Psn C / hn,Psn n Cn
(5. Ie)
(5. Id)
(5.2a)
(5.2b)
(5.3a)
(5.3b)
(5.3c)
(5.4)
(5.5a)
(5. 5b)
(5.6)
(5. 780)
(5.7b)
(5.7c)
(5.7d)
(5.8a)
(5.8b)
c - (1- d ) (c. )n,Rn n n
c = c / hn nyRn n,Rn' n Rn
c f, Pse (d f ) (c f)-' (d~) [afL+~bfL) (F L) ] (c f)
c f Rf (l-d f ) (c f ).'
c (1=1J, =i1H)(d )(c)::.: (l-IJ. -i1H)(d*)[afL+(bfL)(FL)](C)cyPse c ''C c c. ' c ''G c ' c
c-(l-d)(c)c,Rc c c
P :'" p I hn dn, FEn dn, FEnl n FEn
continued
Table 5.1 (continued)
59
(5.9a)
(5.9b)
(5.9c)
(5.9d)
(5.10)
(5.11)
(5.l2a)
(5.l2b)
(5.l2c)
(5.l2d)
(5.13a)
(5.13b)
(5. Bc)
(5.l3d)
(5.14)
PH
bP
bPo.
b0. Po.
2, P . / h .snJPdnl a-Pdn
( K ) (P I p ) (B).- . Pn.e snl se' .
(Kp~s)(Pse)
.... (KS
.op) (B)[(S/B)/(S/B + r)]
.. (Kp B)(P )(P )(AB=B)/BJ so. se .
- [1/ (1 + PB) ] (P B)
="' Po. B/nhBn:;
- (Ie ) (B)ByP
[1/ (l + PB) ](bp)
"'" bpn/nhB
:= D
(5.15)
(5.16)
Pse,BH
hp
::; D = cc.,GH
(5. Db) bBW.jPdn
continued
Table 5.1 (continued)
60
(5.17c)
(5.17d)
(5. 17e)
bn BW,Pdn
bBW,Pse
bBW,Pde
Table 5.2 Differential equations for testing the model
61
(5. 18a) *Psn,Pse
(5. 18b)
(5.19)
c - P - P + b - Pn n,Psn n sn,GPn n n,B n Pn n sn,Pdn
= c + C - P - P + bf,Pse c,Pse se,GPf e,B Pe
+b -P +s -P +pBW,Pse se,S Pse se,BH sn,Pse
(5.20a) Psn,Pdn + bBW,Pdn Pdn,FEn Pdn,Un
(5.20b)
(5.21)
(5.22)
(5.23a) b
P +b -P -Pn sn,Pdn n BW,Pdn n dn,FEn n dn,Un
bBw,Pde - Pde,FEc - Pde,Ue
hp
(5. 23b)
(5.24)
bn
s
bn Pn
sPse
bn BW,Pdn
(5.25a)
(5. 25b)
(5.20a)
(5.26b)
(5.27)
(5.28)
(5.29)
fn
fn n
un
un n
fe
ue
v
c + P + Pn,Rn sn,GPn dn,FEn
nPdn,Un
- cf,Rf + cc,Rc + Pse,GPf + Pde,FEc
c c,V
62
variable step numerical integration method was used. The program,
entitled Continuous System Modeling Program (CSMP) was run over enough
time periods for the system to approach equilibrium, as we have assumed
the steer does. The final values were then used to compare with the
experimental data.
5.2 Initial Estimates
Use of the CSMP computer program requires that values of all
constants and parameters and initial values for all compartments of
the model be provided. Parameter estimates are refined by an iterative
procedure until an optimal set of parameter values is derived.
5.2.1 Evaluation of Constants
For each experiment (Forbes et al., 1928; 1930), data are provided
on the steers' body weight, dry matter fed (gms), nitrogen fed (gms),
ether extract fed (gms), carbohydrate fed (gms), total energy fed
(kcal) and total heat production (kcal). Constants are needed to
convert the data in grams dry matter into units compatible with the
model, kcal and grams of nitrogen. The given data (Forbes et a1. J
1928; 1930) are summarized in Table 5.3.
The first constant evaluated is the ratio of energy to nitrogen
'0
in the body, PB (Appendix 9.3). Next, a constant, r , is evaluateds
for the flow law sp , equation (5.11) (Appendix 9.3). Then, factorsse
to convert body weight into kcal of body, B, and kcal of stores, S,
are computed. These are denoted f B and fS' respectively (Appendix
9.3) • We also must be able to convert kcal of body protein, B , andn
kcal of fat, S and Be' to body weight. We define constants wBn and wS'
63
Table 5.3 Input data for testi.ng the model
Steer~ Feed Values .fB!!.ls) HeatFeeding Body Size Ether Carbo~ Production
Leve1a (kg) Dry Hatt.er Nitrogen Extract hydrates (kca1)
36-0.5 471.2 1,885 39.5 50.7 1,484.0 8,156.01.0 481.2 3,762 78.8 101.1 2,960.6 9,839.71.0 (hay) 499.9 5,763 145.6 86.8 4,225,9 11,635.01.5 '490.2 5.,353 11.5.9 148.1 4,186.4 11,8.54. 12.0 482.9 7,037 152.2 194.6 5,503.7 13,888.1
47-0.5 474.8 1;>863 39.0 50.0 1,466.1 7,754.51.0 484.8 3,790 79.4 101.9 2,983.3 9,383.01.0 (hay) 499.0 5,771 145.8 87.0 4,231. 8 11,254.61.5 494.6 5,617 121.5 155.3 4,392.8 11,696.92.0 486.2 7,384 159.8 204.2 5,774.9 13,536.3
60-0.5 381.0 1,681 32.9 46.3 1,336.7 7,476.0LO 310.9 2,828 55.6 77.5 2,9 250 . 6 7,252.91.0 (hay) 412.1 4,983 117.1 70.5 3,748.6 9,790. 11.5 332.9 4,237 83.3 117.2 3,370.8 8,82L 42.0 3.58.4 5,704 112.2 158.0 4,538.6 11,156.92.5 39108 7.,520 148.0 207,9 5,982.1 13,976.43.0 426.6 9,489 186.8 267.9 7,544.9 16,133.1
57-0.5 398.3 1.,700 33.3 46.8 1,351. 9 7,939.0LO 359.6 3,085 60.7 84.6 2,455.4 7,908.71.0 (hay) 425.6 5.,125 121.0 72.5 3,854.6 9,953.71.5 384.3 4,612 90.8 127.7 3,669.5 9,493.32.0 403.1 6,233 122.6 172.3 4,958.3 11,851. 22.5 443.7 8,057 158.7 227.3 6,405.6 14,408.2
aFeeding level is expressed as a fraction of the lYmaintenancel!ration.
64
respectively for this (Appendix 9.3). The intake values (gms) are
converted to kcal using appropriate heat of combustion values.
In order to simplify matters, we assume that the heats of combus-
tion of nitrogenous matter (kcal/gm N) in feed, body, Rn~ GPn
and Psn
are equal, 2:.~.,
h ~ h = h = h = h = 34.2n Cn n Bn n Rn n GPn n Psn
where the value is taken from Forbes et ale (1928; 1930).
Assuming that, on the average, body protein contains 16 percent
nitrogen, then the heat of combustion of body protein (kca1/gm protein)
is
hB
= (hB
)(.16) = 5.472 .n n n
The heat of combustion value for ether extract (Maynard and Lcos1i,
1962) is taken as
Since the total energy fed is given (Forbes et aL, 1928; 1930)
and we can calculate the energy fed as nitrogen and as ether extract
from the constants given above, we can solve for the heat of combustion
of Cc from the following equation~
Thus,
he = [(total fed) ~ (c )( h C ) - (cf.)(hCf)J/c •c n n n .. c
Substituting feed values from Table 5.3 and heat of combustion
values derived above into this equation leads to
6.5
h "'. 4.51CC
for mixed diets of alfalfa hay and corn meal, and for a diet of hay
alone, a value of
he "' 4.74c
The heat of combustion for methane (Forbes et: ale, 1928; 1930) is taken
as
For U J we use the value given by Forbes et a1. (1928; 1930) in unitsn --
of kcal/gm nitrogen;
hU
= 7.45 •n n
The nitrogenous waste material flowing to Pdn
as a result of the
deamination of amino acids is urea. Its heat of combustion is 5.414
kcal/gm nitrogen (Blaxter? 1962a) which is lower than the value for U .n
Since the heat of combustion of urinary nitrogenous compounds is 7.45
kca1/gm and the heat of combusticm of the urea nitrogen in the urine is
5.414 kca1/gm, the nitrogenous waste resulting from body breakdown must
have a heat of combustion higher than 7.45. To simplify handling these
flows, we will assume that the heats of combustion ()\f all nitrogenous
material flowing into and out of Pdn equals 7045. Hence
hn Un
~ 7,,45
In the deaminaticm of a.mino acids .• all nitrogen from the amino
acids flows to Pdn' Thus?
66
1 gram P - 1 gram Pdn sn n n
for this reaction. In terms of energy,."
hp kcal in Pn sn sn.,
or
1 kcal in P - ( hpd / hp ) kcal in Pdn •sn n n n sn
Thus
The factor vBW
is derived similarly. In the process of body
breakdown, all nitrogen from the body flows to Pdn'
1 gram B-1 gram Pd .n n n n
One gram of B is equivalent to hB
kcal of B and this is equivalentn n n n n
to (1 + PB) (nhBn) kcal of total body, B.
equivalent to nhPdn kcal of Pdn • Thus
Also, 1 gram of Pd
isn n
or
and thus
.151 •
Constants are summarized in Table 5.4.
5.2.2 Evaluation of Initial Conditions
Initial conditions for all compartments, except B, S, and the four
pools, are taken as zero. For B, the body weight of the animal is
multiplied by the factor fB
• Using a prefixed subscript z to denote
Table 5.4 Constant.s used tCI test the model
67
Constant Valu.e Description
PB .44 ratio of energy to nitrogen in the bodyy B /Be n
r .034 constant in flow law s (Appendix 9.3)s Pse
fB
1.423 kcal B per gram body weight (Appendix 9.3)
fS
.895 kcal S per gram body weight (Appendix 9.3)
wBn .868 grams fat~free body per kcal B (Appendix 9.3)n
Ws .107 grams body per kcal fat (S + B ) (Appendix 9.3)e
h 34.2 kcal feed protein per gram feed nitrogenn Cn
h 34.2 kcal body protein per gram body nitrogenn Bn
h 34.2 kcal fecal food residue (protein) per gram nitrogenn Rn
h 34.2 kcal fecal gut wear per gram nitrogenn GPn
h 34.2 kcal pool amino acids per gram pool nitrogenn Psn
hBn 5.472 kcal body protein per gram body protein
hCf
9.4 kcal feed ether extract per gram feed ether extract
h {4.51 kcal feed carbohydrate per gram feed carbohydrate
Cc 4.74 for mixed diet and hay diet, respectively
hV
13.3 kcal methane per gram methane
h 7.45 kcal uri.ne per gram urine nitrogenn Un
h 7.45 kcal pool material per gram pool nitrogenn Pdn
h 7.45 kcal fecal excretion per gram fecal nitrogenn FEn
\)Pne .218 kcal to Pdn per kcal amino acid deaminated
\)BW .151 kcal to P per kcal body broken downdn
68
i.nitial (CJr zerc time) c,:m:'IitLc'(l and w'B fOl body weight;, we have
For grams of nitrogen at zero time~ we mlist convert zB to kcal of
protein) B whErez n
and then divide by t.he heat of combustion to give
For the initial condition on stores y we multiply body weight by fS
"
Initial conditions for t1:le pools are derived in Appendix 9.4. Initial
values in terms of grams of nitrogen are derived by dividing by the
-,
appropriate heat of combustion.? i. e ••<-= - ,/
P / hz sn nPsn
P / hz dn n Pdn
There are two values which must be derived for each experiment
simulated. The flow law for transport from pool to body requires a
maximum body size in kcal." A.B. For each of the breeds involved
(Shorthorn and Aberdeen~Angus)y a maximJrn bod:v size in grams j wBmax
.?
is assumed.
We then take
"B= (fB) ("rBmax) •
For steers 36 and 47 J the AberdeenoAngus ~ we assum.e an average breed
value of 650 kg for the maxim;;.m I:>:'d)' sile,? and for steers 57 and 60"
69
the Shorthorns, an average breed value of 750 kg. Thus,
AB = (1.423)(650,000) =
for steers 36 and 47, and
924,950 kca1
AB
= (1.423)(750,000) = 1:067,250 kcal
for steers 57 and 60.
The second required value for each experiment is the feeding level
used in evaluating the parameter d. This value is the ratio of dryn
matter fed, cd' to a base value denoted cd,o. The base value is an
estimate of that amount of dry matter which must be fed to yield zero
energy balance for the steer. For each steer, linear regression was
used to relate heat production to the balance of fat energy and heat
production to dry matter fed (Appendix 9.5). The predicted value of
dry matter for zero fat energy balance was taken as cd • Initial,0
conditions are summarized in Table 5.5.
5.2.3 Initial Estimates of Parameter Values
Initial estimates of parameters are derived in various ways. Data
from many sources are used and- judgments are made to derive these
values. There are 20 parameters to be estimated. From equation (5.la),
(5.30) d = (d*)[f(FL)Jn n
*We will arbitrarily assume that dn = 1 and what we call aFL will
actually represent (d~)(aFL) and similarly for bFL. From equations
(5.2a) and (5.3a),
70
Table 5 • .5 Initial pool concentrations and values of AB
and cd usedin testing the model . ,,0
Term Value Description
P / Bz sn z
P / Bz dn z
P / Bz se z
P / Bz de z
AB-Shorthorn
AB-Aberdeen-Angus
c - Steer 36d,o
c - Steer 47d,o
c - Steer 57d,o
c - Steer 60d,o
.00103 concentration of simple nitrogenousmaterial in the pool
.000665 concentration of degraded nitrogenousmaterial in the pool
.00514 concentration of simple energeticmaterial in the pool
.00331 concentration of degraded energeticmaterial in the pool
924,950 }maximum body size, kcal
1,067,250
3,976
grams of dry matter fed whichcorrespond to zero fat energy balance
2,558
2,313
(5.31)
(5.32) dc
(d;) [ f (FL) J
(d:> [f (FL) ]
71
We will denote d as the digestibility coefficient for carbohydratesc
for a mixed diet of corn meal and alfalfa hay. For the diet of hay
alone) this coeffici.ent is denoted as dc(hay).
The equations for flow of secretory material from the pool to the
feces, (.5.5a) and (5.6) involve parameters KGPn and KGPf" The units
for these paramete.rs are kcal/gm since cd is in gm!day. If f GPn and
fGPf
represent the fraction of the dry matter fed which is secreted
into the feces as grams of crude protein and of ether extract,
respectively, then
(5.33)
and
(5.34)
K:GPn ::::; (fGPn) (hBn)
We define the parameter IJ.c
kcal of digested carbohydrate.
as the kcal of methane produce.d per
In terms of f ,c
We define a new parameter~ fV
' as the grams of methane produced
per gram carbohydrate digested for the mixed diet" and fV(hay) as
the corresponding term for the diet of hay only. Then to determine II. 9,...c·
grams of methane and the gram of digested carbohydrate must be con=
verted to kcaL Thus,
(5.35a)
(5. 35b)
72
Combining those parameters frciffi TablE .solwhie!l are not: functions
of other parameter::> and a'::din,g in tLc' :;e defined in equations (5030)
through (5035) gives the f,,110w'ing 2.0 parameters: aFU
' bFU
KpdnJ
'* *,'; 'f'f Cpn" fUn' KpneJ dc .\' dc (hay).\' de fv;, fV(hay) ,1 f CPe" K:pde) 'Ue"'
Kp ,? S,o KS:' p,? Kp J B~ KB~, p" KBw-" and nBWo
The concept of digestion being affected by feeding level was not
originally incorp"jrated, into the modeL When it was added, the
parameter valu.es "rere ES timated by examining the model output up to
that time. Values for each steer differed slightly, but the average
initial value;5were a FL .'" 09465 and bFL'c ~.01630o
Based on previou.s work of Lucas and Smart (1959)" we estimated
initial values of fCPn
',;0( .04 and fCPe
'" 0016.
The parameter KPdn
was estimated from the ni.trogen and the energy
balance data for Steer 47 ted at maintenance (Forbes et a1. J 192.8) 0
The urinary excretion was 4709 gros ni.trogen/day. Using a heat of
'b . f . ""f' '7 1<; k 1/ . t, " t'h" . 'I t tcom ustl,on 0, urine c., ,O~J .ca" gm nl.,_,r,",gen." ,1.8 IS eqlllVa .en. :0
35608 kcal/day. According to Schneider (l935), endogenous fecal
nitrogen in the rat is abou,t 1/6:)£ the u.rinary nitrogeno Using this
value for steers." fecal nitrogen is about 59047 kcal/day for a total
of 416 03 kcal/dayo Summing equations (507a) and (501c) and solving for
KPdn
gives
We will use the initial condition Pd' LJr Pd,n0 The initi,al, pool value
z n
zPdn equals the product of the initial pool concentration and the
initial body SiZE, or
Pd ". (,Pd ,/ B) ( B) 0z n z n z' Z'
.9
-.
73
From Table 5.5, the first term equals .000665. From Section 5.2.2, we
have
From Table 5.3, wB equals 484.8 kg for Steer 47 at maintenance and from
Table 5.4, f B equals 1.423. Hence,
B = (484,800)(1.423) = 689,870z
and hence,
zPdn = (.000665)(689,870) = 458.76 .
Thus
Kpdn = 416.3/458.76 = .90745 .
Now, total urine energy was 768.3 kca1/day. Subtracting the con-
tribution of nitrogen leaves 411.5 representing Pde,.Ue' Assuming no
flow from Pde to FEc leads to
Applying the same reasoning as above, we have that Pd - 2283 kca1 andz e
thus,
KPde = 411.5/2283 ~ .18024
From the above discussion~ note that we have also partitioned the flow
from Pdn between urine and feces and
fUn 6/7 "" . 8571
f Ue = 1.00
The digestibility factors, like aFL and bFL in the equation for
feeding level effect, were estimated by examination of the model
74
results at the time these parameters were added to the model. Thus,
d* - .8541c
d~ = 1.00 .
It was recognized that digestibility of the carbohydrates in the hay
only diet would be lower than for the mixed diets and so a different
value of d* was used for this case,c
d~(hay) = .7606 .
The factor for methane production, fV
' was estimated from data in
Forbes et al. (1928; 1930). For each steer at each feeding level, the
grams of carbohydrate (crude fiber plus NFE) digested per day and the
grams of methane produced per day were recorded. The ratio of
carbohydrate digested to methane produced gave a value for each steer
at each feeding level and the average of these 23 values was
fV
= .0469 .
Again, for hay, methane production is a higher fraction of digested
carbohydrates due to the higher proportion of fiber in the hay. For
hay, the value is
f =.0527.V(hay)
Flatt et al. (1965) and Blaxter (1966) mention cows losing 16-20
and 20.1 megacalories per day during early lactation. Using this as
an indication of the maximum amount of fat stores which may be
depleted in a day by the steers, we assumed
sp = 21,000 kcal/day •se
.0
..
'.
75
Assuming the fatness to be high enough so that equation (5.11) can be
approximated by
s =:Pse
" then
Using the average body size of the four steers in Forbes et a1.
(1928; 1930) of 611,890 kca1, we have
KS P =: .03432 •,For Steer 47 at maintenance, or more accurat.e1y, at near main-
tenance, the fat energy added is 689 kca1/day and the protein (B )n
added is 208. 7 kca1/day. The fat associated with B isn
thus depot fat added is 689 = 92 ::::: 597 kca1/day. From the differential
equation for stores, equation (5.24), and equation (5.10)
Thus
sc
597 -- (Kp S) (3546) - 21,000 •,
The derivation of ~ is given in Appendix 9.6. The result isJB
Ie B =: .001619 •P,
From equation (5.17a) we see that
'" - b /n"BW - BW •
.'
76
FrQ~ ~he data on steers under fasting conditions (Forbes!! !l., 1928;
193Q; Forbes and Kriss, 1932), the values of bBW and D are taken
(Table 5.6) and a value of KBW i~ c~l~ulated for each steer. The
average v~lue is ~
KBW = .24282 •
Table S.~ Values of bBW' D, aod KBW
,I
Steer bBW D KBW
36 1,857.442 7,651 .2427747 1,645.842 7,396 .2225357 1,725.245 7,482 .2305860 1,901. 270 6,904 .27539
Average .24282
From equations (5.l3a) and (5.23a) we have
and
FrQm above, for Steer 47, protein added per day is 208.7 kcal and the
f~t associated with this is 92 kcal/day, or about 300 kcal/day of B
is ad4ed. Thus, we take b = 300. From Table 5.6, bBW = 1646.
Substituting values already derived for Kp B and initial pool con-,centrattons into (5.l2a) and taking B for Steer 47 as 689,870 kcal
ll:
and As as 1,067,250 kcal gives PB =2231.14. Solving for bp gives
bp = 285.14
'0
77
and
KB P = .0004177 •,The parameter TIBW was introduced into the model after several
simulations had been run. Examination of the data led to the estimate
TIBW = .621 •
From equations (5.9a) and (5.l8a) we have
(5.36)
Based on data from Steer 47 fed at near maintenance, we assume that
p = O. From equat~ons (5.l2b) and (5.l3b), we havesn
Pn B = (1/1.44) (2231.14) = 1549.40,
and
bpn = (1/1.44)(285.14) = 198.01 •
Averaging the true digestibility coefficients of protein for cattle,
sheep and goats (Lucas and Smart, 1959) leads to a value of dn of
.94, and from Table 5.3, c = 79.4 and from Table 5.4, he = 34.2,n n n n
thus
= (.94)(34.2)(79.4) = 2552.55 .
From equations (5.5a) and (5.33) and Tables 5.3, 5.4 and 5.7 J "
= (.04)(5.472)(3790) = 829.56
Substituting into equation (5.36),
Table 5.7 Initial estimates of parameter values
78
"Parameter
aFL
bFL
f GPn
KPdn
fUn
~ned*
c
d*c(hay)
d*f
f v
f V(hay)
f GPe,
~de
Value
.9465,
-.01630
.04
.90745
.8571
.002688
.8541
.7606
1.0
.0469
.0527
.016
.18024
l.0
6.090
.03432
.001619
.0004177
.24282
.621
79
2552.55 - 829.56 - 1549.40 + 198.01
= 371. 6
and substituting for the remaining values leads to
*IL- =.002688.-Pne
The initial estimates of parameter values are listed in Table
5.7.
5.3 The Goodness of Fit Criterion
The goodness of fit of the model to the experimental data will be
based on the nitrogen and energy balances. The nitrogen and energy
balance variables for the steer data and the corresponding terms from
the model are matched in Table 5.8.
Table 5.8 Nitrogen and energy balance variables
Itema Terms in Model
Nitrogen - fed cn n
- in urine un n
- in feces r + ngpn + fen n n n
- in protein (and pools) b + nPsn + nPdnn n
Energy - fed c + c + cfn c
- in urine u + un e
- in feces r + rf
+ r +gp +gp + fen c n f e
- in methane v
- in protein b (= b + b )n e
- in fat (and pools) s + Psn + Pdn + Pse + Pde
aUnits for nitrogen are grams/day and for energy are kca1/day.
80 eThe amount of nitrogen or energy fed per day must equal the
amount excreted per day plus the amount being added to the body,
stores, or pools per day. This balance is expressed as follows:
(5.37)
(5.38)
c = u + r + ngpn + fe + b + nPsn + nPdnn n n n n n n n n n
c + c + cf
= u + u + r + rf
+ r + gpn + gPf + fen c n e n c e
+ v + b + b + s + Psn + Pdn + Pse + Pden e
'\
These two equations are not independent since each term in (5.37)
is related to a corresponding term in (5.38) by its heat of combustion.
Recall that
h = h = h = h = h - 34 2n Cn n Rn n GPn n Bn n Psn - •
7.45 .
If we multiply (5.37) by hC
' the result isn n
(5.39) c = u + ( hn n n Cn 'h ) (u ) + r + gp + (h - h ) (fe )n Un n n n n Cn n FEn n
Subtracting (5.39) from (5.38) gives
(5.40) + U - (h - h ) (fe )e n Cn n FEn n
and rearranging terms and substituting for the heats of combustion
gives
(5.41)
+ fe + v + b + S + P + Pd •e e se e
81
The terms in pare.ntheses represent energy, which originally was in a
nitrogenous compound.. and which has been added to the non~nitrogenous
energy from the deamination of amino acids. Equation (5.41) represents
an energy balance in the system due to non~nitrogenous compounds.
We can assume that the right-hand sides of equations (5.37) and
(5.41) are constant. For equation (5.37) then, if we match the values
of urine and fecal output per day) we will also match the value of
nitrogen remaining in the animal per day (body plus pools). Similarly,
for equation (5.41)) if the model output values match the experimental
values for urine, feces and methane excreted per day, they will also
match the amount retained per day (stores plus body plus pools).
Hence, the goodness of fit criterion will be based on a comparison of
u, f, u , f and v with the corresponding experimental results.n n nne e
The experimental results are given in Table 5.9, with the above five
terms labeled Urine N, Fecal N, Urine E, Fecal E and Methane E,
respectively.
The goodness of fit criterion is in the form of a sum of squares
of deviations between model results and experimental results for each
entry in Table 5.9. A glance at the table indicates, however, that if
each model value were, say, five percent different from the experi
mental value, the difference in fecal energy would dominate the sum of
squares. To overcome this a set of weighting factors are derived to
equalize the variances of the factors. The variances vary over feeding
level, as well as among factors. The values for Steers 36 and 47 are
in good agreement with each other, as are the values for Steers 57 and
60. However, there is a difference between the two grou.ps of steers.
82
Table 5.9 Experimental results
a ,"FL-
Steer Urine Na Fecal N Urine Ea Fecal E Methane E
0.5-36 41. 9 11. 4 253.845 1,936.62 831. 3
47 41. 3 11. 7 229.015 1,,862.16 827.3
57 39.7 10 • .5 216.735 I J 712.90 744.6
60 39.1 11. 6 215.305 1.• 731. 48 737.9
1.0-36 47.1 23.0 386.305 3,679.40 I J 481. 2
47 47.9 23.7 353.345 3J .588.46 l,449.2
57 47.1 17.9 304.535 3,027.02 1,271. 7
60 40.4 17.8 280.720 2,852.64 1,060.8
1.0 (hay)-36 89.0 48.2 597.850 9,632.46 1,656.0
47 89.5 47.6 576.525 9 J 616.88 l,678.7
57 78.7 41. 3 568.78.5 8.. 492.54 lJ593.3
60 72.6 42.2 531. 730 8J 559.26 1,465.2
1.5-36 70.9 39.5 473.495 5.,317.90 1,841. 5
47 74.2 40.6 444.010 5.. 551.78 1,880.2
57 44.9 30.4 423.995 4 J 443.92 1,781. 4
60 39.8 30.0 356.290 40'439.90 lJ482.5
2.0-36 83.4 57.6 512.2'70 7,824.38 2,16804
47 81.1 58.1 497.505 8,155008 2,444.6
57 60.3 43.7 53.5.565 6,534.76 2,335.2
60 49.7 41. 8 492 0035 6,,054.84 2J 085.7
2.5-57 79.3 60.3 627.015 8,8Il.54 2,930.3
60 66.5 58.5 612.975 8,336.30 20'518.0
3.0-60 88.4 76.3 6'17.020 I1j 349.04 3}O74.5
a feeding level fraction of th.e I1ma :i.ntenance ii ration,FL::: as aN = nitrogen (gros/day), E =. energy (kcal/day).
83
This is not unexpected as the two groups of steers correspond to two
breeds and two experiments.
Weighting factors are calculated by the following method. For
each factor~
(1) Calculate the variance between experimental results for
Steers 36 and 47. Call this si;
(2) Calculate the variance between results for Steers 57 and 60,
2and call this s2'
(3)
(4)
2Pool the variances to give s .p"
Calculate the pooled standard deviation, s , as the squarep
2root of s .p"
(5) Calculate a regression line of s versus feeding level;p
(6) Using the regression line from (5), calculate predicted
standard deviations, s*, for each factor at each feeding level.p
To calculate the goodness of fit for each steer J we proceed as
follows:
(1) For each feeding level, and for each of the five factors in
Table 5.9, calculate the difference between the model result and the
experimental result. Call this deviation 8';
(2) Divide 8' by the appropriate weight to give 8 (= 8'/weight);
(3) Calculate the sum of squares, r, as the sum of squares of the
5 values, i.~o,
2r = L:8 0
The computations of the weights are given in Appendix 9.7 and the
weights are summarized in Table 5.10.
84
Table 5.10 Weights for the goodness of fit calculation
.--FeedingLevel Urine N Fecal N Urine E Fecal E Methane E
0.5 .4780 .3347 19.63 8.597 7.864
1.0 2.248 .5174 20.86 79.70 76.61
1. 0 (hay) 4.019 .7001 20.86 79.70 76.61
1.5 4.019 .7001 22.09 168.6 145.4
2.0 5. 789 .8828 23.32 257.5 214.1
2.5 7.559 1.066 24.54 346.4 282.8
3.0 9.330 1. 248 25.77 435.2 351. 6
5.4 The Iterative Estimation Procedure
The iterative estimation procedure involves the method of steepest
descent; i.~., given a vector of parameter values, ~, a corresponding
sum of squares, r, and the vector of derivatives, dr/d~, we determine
that direction in ~-space which results in the greatest reduction in r.
We then move a certain distance in this direction to a new set of
parameter values, re-evaluate r and the derivative and repeat the
process.
Let
~j = an nx1 vector of parameter values following the-n
thj i tera tion,
r j = sum of squares corresponding to ~j
Lj = vector of derivatives, one value for each parameter-n
::: (L~) = or ./o~~~ J ~
The iterative procedure is to take
i = 1,2, ••. ,n •
'.
85
where k is a scalar denoting the distance along the L v'ector which we
move.
We cannot estimate the L vector analytically as the system of
differential equations cannot be solved analytically. Therefore, the
derivati.ves are approximated by finite differences. We define the
following:
r j = sum of squares corresponding to ~jo
llj =n
a diagonal matrix of increments on ~
ll~ the .th column of j- ~ II
-~
r~ = sum of squares based on ~j + ll~~ -~
i = 1,2, ... ,n
Then
1 = an nxl vector of l's •-n
As ~j+1 ~j becomes smaller, the size of the elements (o~) of II~
and of k are reduced in order to converge closer to the minimum value
of r and the optimal set of parameter values, ~.
86
6. RESULTS AND DISCUSSION
The model was fitted for each steer. For each steer a series of
iterations were carried out using the method of steepest descent. A
point was reached where further reduction in the swn of squares, f,
appeared virtually impossible.
The parameter values corresponding to the final runs are given in
Table 6.1. The parameters are also averaged over the four steers and
the coefficient of variation given. Examination of the coefficients of
variation reveals that all but, five of the parameters are remarkably
coq,sistent. Except for K;ne~ f Gpe ' KS p: Kp Band KB p all but one of'J ,
the remaining coefficients are less than 3 per cent and that for f GPn
is 8.06 per cent. For the five listed above, the breed averages are
also given (Table 6.1). Steers 36 and 47 are Aberdee.n~Angus and
Steers 57 and 60 are Shorthorns.
In addition to estimating the values of the parameters, we are
also interested in estimating their variances and covariances. These
are necessary in estimating the variance of the compartment values or
of the derivatives of the compartment values. A correlation and co-
variance matrix is given (Table 6.2). The underlined elements are the
diagonals of the matrix and contain the variances of the parameters.
The upper right part of the matrix contains the correlation coefficients
and the lower left part contains the covariances.
The variances and covariances are coded values. Alongside the
row headings and below the column headings are numbers in parenthese,s.
These stand for negative powers of 10. To decode the ijth covariance,
say s*(1j) to the actual covariance;, s(ij), where Pi is the number in
.0<
e ., .. e .. 1 e
Table 6.1 Final parameter values = averages and coefficients of variation
By BreedAverage over Steers Steers
Parameter Steer 36 Steer 47 Steer 57 Steer 60 Four Steers eVa 36 J 47 57, 60a
FL .987943 1.00200 .974097 ,978085 ,985531 1,26
bFL =,0412381 =.0395800 =.0408788 =,0405472 =,0405610 1, 76
f .0243660 .0232201 .0215682 ,0202752 .0223574 8.06GPnb
I1>dn .438196 ,438200 .438200 .438199 .438199
fUn .870704 .861498 ,887197 ,887290 .876672 1.46
~ne .00722799 .0106301 .0100345 .00975440 .00941175 15.95 ,00892904 .00989445
d* .855239 .855398 .861397 .864024 .859014 .51c
d* .656320 .661565 .665774 .635601 .654815 2.04c(hay)d* 1.0 1.0 .993897 .993993 .996972 .35ff V .0488357 .0494198 ,0482809 .0516488 .0495463 2.98
f C ) ,0548425 .0554984 .0542194 .0580016 ,0556405 2.98V hayf GPe ,0505173 .0513194 .0394395 .0390366 .0450782 14.98 .0509184 .0392380
KPde .0863266 .0863291 .0864183 .0864159 .0863725 .06
fUe .965594 .965198 .961197 .961283 .963318 ,25
Kp S 1.5 1.5 1.5 1.5 1,5 0.0J
KS P .0378494 ,0431698 .0615897 .0625355 .0512861 24.64 .0405096 .0620626J
Kp B .000129204 .000101692 .0000435954 .0000282780 .0000756924 62.98 .000115448 .0000359367J
K .000430977 .000285667 .000985627 .000774249 .000619130 51.50 .000358322 .000879938BJP
KBW 0197585 .191300 .197197 .196884 .195742 1. 52
nBW .620603 0624098 .613297 .613401 0617850 .8700
a eV = coefficient of variation (per cent).......
bev < 001.
Table 602 Variance=covariance and correlation matrix of final parameter valuesa
Parameter aFL
b f Ie f K* d* d* d*
(Code) FL GPn Pdn Un Pne c c(hay) f(2) (3) (3) (6) (2) (3) (2) (2) (2)
aFt (2) 1.,')44 068 .65 -.03 .97 .09 =.81 .26 088
bpL (3) .6018 .5073 =008 .67 051 077 =.18 006 .25
f Gpo (3) 1.453 =00972 30246 =.58 =.80 =056 =097 .56 ,92
K . (6) =7.336 08982 =1.984 3.583 .18 .98 .40 017 =0 46Pdn ~-
f U(2) =L546 =04681 =1.850 .4418 L 631 .08 .92 =.39 =.96fn
K;'( (3) .1709 08269 =L513 2.797 01579 2.253 ,36 .86 =037Pne -~
d* (2) =04440 =.0.560 =20894 .3364 .5189 .2346 01937 ~.57 =097c.".
d~ (2) L 326 01855 4.020 1. 319 =20001 .51'78 =1. 002 50371 .34c (hay)d* (2) .3817 .0619 2.357 03044 =04267 =01953 =.1488 .4854 01222ff V (3) =.1661 00263 =3.407 .2466 .5055 04434 -.3553 =5.619 ~.1636
f (3) = • .559? .8851 =5.520 .8308 L 703 1.494 L 197 ~6. 310 =.5510V(hay)f GPe (2'> o 7512 .1383 40561 =.5238 =.8355 =.3285 =02894 1.046 .2359
~de (4) =.5574 =.0848 =08538 04669 .6251 .3025 .2189 =.6940 =01802
fU
(3) 2.528 .3164 16.12 =2.356 =2.868 =1. 566 =1.021 30197 08390e
KS P (2) 1.226 =00661 =8.327 10416 1.437 .9908 .5366 =1. 782 =04349.'
~;B(4) .4258 =.0039 08466 =.5554 =05184 =04047 =.2034 08153 .1602
IeB.'p (3) =03764 =01047 =04259 .1992 .3896 . 00953 .1189 =01498 =01056
KBW(3) =3.128 =2.017 =1. 219 =20839 20795 =2. 713 00098 =30321 =.5249
TTBW (2) 06491 01884 07926 =02188 =.6880 =00920 =.2204 08190 .1817
continued 0000
e . \ .. - , .• e
e • • .. e ~ T tit
Table 6.2 (continued)
Parameter f V f f GPe K:Pde fUe KS P Kp B KB P KBW 'TTBW(Code) V(hay) , :; :;
(3) (3) (2) (4) (3) (2) (4) (3) (3) (2)a
FL(2) ~.09 ~.09 .90 -.87 .85 =. 78 .72 =.95 =.85 .97
bFL
(3) .25 .25 .29 =.23 .18 -.07 =.01 =.46 =.95 .49f' (3) =062 =062 091 =.92 093 =.96 .98 =0 74 =.23 .82~GPn
KPdn (6) 009 . 009 =.41 .48 =.52 .59 =.62 033 "".50 =.21
fUn (2) 027 .27 =097 .95 =.93 .89 =085 .96 .74 -LO
K* (3) .20 020 =.32 .39 =.43 ,52 =.56 .12 =.61 =.llPned* (2) 055 055 =.97 096 =.96 ,96 =097 .85 .46 =.93cd* (2) =.95 =.95 .38 =.34 .33 =.35 .43 =.18 =.28 .38c(hay)d* (2) =.32 =.32 1.0 ~LO LO =.98 .96 =.95 ~.50 .96ff (3) 2.181 1.0 =.34 .31 ~.32 .38 =.48 .03 -.01 -.26Vf (3) 7.348 8.251 =.34 .31 ~.32 .38 =.48 .03 -.01 =.26V(hay)f GPe (2) =.3407 =1. 148 .4561 =1.0 .99 ~.98 .95 ~.94 =.54 .98
KPde (4) .2380 .8018 =.3470 .2657 -LO .99 ~.96 .94 .49 =.96
fne (3) =1. 150 =30874 1. 612 =1. 238 50785 ~.99 .97 ~.93 =044 094
KS P (2) 07063 2.379 =.8327 .6430 =3.020 10597 =.99 .89 .35 =.90:;
Kp B (4) =03346 =1. 127 .3069 =.2371 1.117 =.5980 .2273 =.83 ~028 .86:;
KB P (3) 00128 .0£.29 =.2036 .1554 =.7160 03585 =.1262 .1017 065 =.96:;
KBW (3) -.0626 =02108 =L093 07480 =3.189 1.304 =03924 .6150 8.850 =.71
'TTBW (2) =.2113 =07119 .3551 -.2664 L224 ~. 6155 .2223 -.1653 =1. 146 .2904
aparameter Kp S is omitted as its variance and covariances equal zero.:; 00
\0
90
h h . th h d' d . h b .parent eses next to t e 1 row ea 1ng an p. 1S t e num er 1nJ
th below the J.th 1 h d' thparen eses co umn ea 1ng, en
-(p.+p.)( .. ) s*(~J') x 10 1 Js 1J = ....
For example, the covariance between KP B (power = 4) and fUn,(power = 2) is given by
= -.5184 x 10-(4+2) = -.5184 x 10- 6 •
These values are tabulated to show their estimates based on the
limited number of trials we investigated. Each correlation coefficient
and covariance is based on four observations. The mean for each factor
has been estimated, hence these values have only two degrees of free-
dome Similarly, the variances have three degrees of freedom. No
inferences or hypothesis tests are called for or warranted.
A small study was made to evaluate the sensitivity of the model
to averaging parameters over all four steers and for the five param-
eters previously shown to have large coefficients of variation, over
breed. Seven trials were run, with the parameters varied as shown in
Table 6.3.
Table 6.3 Parameter values used for sensitivity test
Run
1
2
3
4
5
6
7
KP B' KB,p' KS P K f GPe OthersPne', ,Individual Individual Individual
Individual Individual Averaged
Individual Breed average Averaged
Individual Averaged Averaged
Breed average Breed average Averaged
Breed average Averaged Averaged
Averaged Averaged Averaged
..
91
For each steer and for each ru.n;> values of the sum of squares of
residuals) f J are tabu.lat.ed for each feeding level and the total given
for each steer (Table 6.4).
Looking at the total r values;> run one is superior for all but
Steer 60. For Steer .36, runs two and five are almost as good as run
one. For Steer 47;> runs four;> six and seven were fairly close. to run
one and for Steer 60, runs four and six were far superior to run one.
We obviously did not reach the minimum r value for Steer 60 when
fi.tting that steer individually, for substituting the values of run
four for those of run one gave great improvement. It is probable that
the minimum value has not been attained for any of the steers. We were
forced to halt the search for the minimum value of r due to precision
problems in the computer program.
There also appear to be compensating factors in the model. For
instance, Kp Band K can be varied quite a bit) without changing, B;>P
the sum of squares very much. If one is lowered, the other can also
be lowered and the pool and body sizes maintained relatively constant.
The covariance between the two is negative, but this is due to breed
differences. A plot of the values (Figure 6.1) indicates positive
correlation within breed.
We believe that ~ Sand KS p act similarly. However) over the:I ;>
iterations done, Kp S remained constant. It is possible that the rates;>
of breakdown of body and stores p KB p and KS
p;> respectively y are;> )
measures of the rate at which all reactions occllr.o ..!..~. y they act H.ke
clocks in the system. Fix.ing values of these two parameters for all.
steers and then evaluating animal to animal and breed to breed varia-
tion in the other parameters may be very meaningfuL
92
Table 6.4 Summary of r values by feeding level and run
Steer- RunFLa 1 2 3 4 5 6 7
360.5 146.03 90.19 104.41 247.52 96.10 234.32 243.88
1.0 115.79 80.27 99.15 77.67 94.92 73.26 76.95
1.5 35.77 44.97 42.74 27.97 43.15 28.27 27.94
2.0 60.97 101. 64 92.32 85.56 93.41 86.54 84.90
1.0Hb
104.05 145.56 138.00 83.32 140.46 85.33 85.79
Total 462.61 462.63 476.62 522.04 468.04 507.72 519.46
470.5 93.62 151.42 140.52 101.15 138.31 102.40 106.38
1.0 85.84 128.69 113.36 78.91 114.96 80.60 84.04
1.5 36.40 61.50 59.99 39.10 59.93 39.07 39.02
2.0 37.20 52.10 55.10 43.34 55.04 43.32 42.24
1.0H 81.77 151. 86 153.30 83.86 152.16 82.96 83.22
Total 334.83 545.57 522.27 346.36 520.40 348.35 354.90
570.5 124.72 217.18 225.33 105.22 228.34 102.11 137.48
1.0 54.47 48.20 47.79 63.34 48.52 63.62 65.24
1.5 38.44 28.66 27.92 43.41 31.06 46.24 40.48
2.0 30.22 22.77 22.22 34.64 24.44 36.64 32.88
2.5 43.62 33.39 32.76 47.34 34.92 49.30 45.62
1.0H 126.88 134.84 133. 70 207.32 124.77 197.37 224.44
Total 418.35 485.04 489.72 501.27 493.05 495.28 546.14
600 •5 224.72 297.76 286.49 75.04 290.07 87.23 163.36
1.0 12.96 15.10 15.40 23.93 14.35 23.38 26.58
1.5 27.10 15.47 15.83 23.10 14.14 21. 72 22.11
2.0 24.97 18.88 19.38 28.98 17.10 26.94 25.77
2.5 33.07 29.38 29.99 42.38 27.52 40.12 38.52
3.0 41.46 48.01 48.79 62.82 44.84 59.07 54.59
1.0H 160.99 68.52 69.08 116.47 78.34 127.04 153.16
Total 525.27 493.12 484.96 372.72 486.36 385.50 484.09
aFL = feeding level as a fraction of the "maintenance" ration.
bH = hay.
93
Aberdeen-Angus
.00100
.00080
.00060
.00040
/Shorthorn
.000204
o0.'----L:----I.:----l-:----':---:-7--7-=---:;-&-:-~ Kp , B(x10 ).2 .4 .6 .8 1.0 1.2 1.4
Figure 6.1 Plot of KB p versus Kp B by breed, ,
At any rate, discussion of the goodness of fit will be based on
the final runs with individual parameter values (run one). Each total
r value shown in Table 6.4 is based on a simulation run in which 20
parameters were estimated. Thus the degrees of freedom associated with
r is the difference between the total number of observations and 20.
There are five observations per feeding level, hence for Steers 36 and
47, r is based on five degrees of freedom; for Steer 57, 10 degrees of
freedom; and for Steer 60, 15 degrees of freedom. To put the r values
on an equal footing, divide by the degrees of freedom, giving a mean
square r value (Table 6.5).
94
Table 6.5 Sunnnary of r va1ues J degrees of freedom and mean squarer valu.es
Steer r dfa
Mean Square
36 462.61 .5 92 • .522
47 334.83 5 66.966
57 418~ 35 10 41. 835
60 525.27 15 35.018
adf degrees of freedom.
To evaluate the goodness of fit J we must compare r to the sum of
squares of the raw data. The experimental results (Table 5.9) and the
weights associated with them (Table 5.10) are used. Dividing the
results by the weights gives a new tab1eJ like Table 5.9 J but with
values of approximately equal variance. Considering feeding levels as
replications and Urine NJ Fecal NJ etc. J as factors, an analysis of
variance is performed on the weighted variables. We are interested in
the error term which represents the sum of squares within each factor,
corrected for the mean of that factor. If ~L denotes the number of
feeding levels for a steerJ then in the analysis of varianceJ the error
term has (5)(~L~1) degrees of freedom. Carrying out the analysis of
variance and dividing the sum of squares for the error term by the
degrees of freedom gives the mean square for error for the raw data.
It is this number which is compared to the mean square error for the
model. The ratio of the mean square error from the model to that for
the data indicates how much of the variation in the data has been
accounted for by the model results (Table 6.6).
95
Table 6.6 Mean square error in the raw data, due to the model andfraction of variation accounted for by the model
Steer MSE Data MSE Model % Accounted For
36 1974.015 92.522 95.31
47 1830.065 66.966 96.34
57 1257.152 41. 835 96.67
60 1292.113 35.018 97.29
Thus even though the best fits may not have been obtained, the
model sti.ll accounted for over 95 per cent of the variation in each
steer's data.
The model results for each steer have been tabulated (Table 6.7)
against feeding level, for each factor. From these and the experi~
mental results (Table 5.9), weighted residuals for each steer have
been tabulated (Table 6.8) and plotted (Figure 6.2) against feeding
level, for each factor. For completeness of presentation, weighted
residuals are also given for protein (B ) and stores (S), even thoughn.
these do not enter into the calculation of f.
The urine N residuals indi.cate a change in urine excretion at the
0.5 and 1.0 feeding levels. The model has averaged out the excretion
patterns and predicts low at the lower feeding levels and high at the
upper levels.
The fecal N residuals are not consi~tent among steers. All
residuals for the hay diet are low. Steer 36 seems to show a linear
trend.
96
Table 6.7 Model resuits
FL-a •Steer Urine N
aFecal N Urine Ea Fe,.:al E Methane E
,~- ~----_...~ ..-.-"--="------~0.5-
36 41. 2. 14. 7 356. 040 1.,945.55 798.3
47 40.4 14.1 322..051 1)836.61 808.9
5'7 39.2 12,.6 3.'55 . .548 l.~ 683.07 708.1
60 38.8 12..1 ;334.307 1,615.46 7.52.6
1.0-36 55.5 26.9 428.362 4~171.42 1,560.6
47 .56.8 26.4 389.442 4.,066.02 1,610.3
57 41.9 20.5 352.927 3) 349.70 1,256.0
60 40.2 18.3 323.038 2,980.17 1,240.3
1. 0 (hay)-36 87.0 45.8 505.118 10,271. 98 1,,877.7
47 90.6 45.0 466.027 9.,190.48 1,418.8
57 66.6 39.2 442.897 9 y 168.25 1,651. 1
60 71,3 38.1 43.5.190 9~355.95 1)637.2
1.5-36 71.0 39.2 515.425 6,193. '74 2,168.3
47 76.3 40.4 484.612 6.369.19 1,321. 1
57 52.4 31.6 423.024 5,.394.36 1,827.5
60 51.0 28.2 392.404 4;828.06 1,808.2
2.0-36 83.8 52.8 602. 786 8.'))2.61 2 1 797.2
47 89.9 54.6 561. 406 8 ;1865.37 2 J 987.9
5'7 66.0 45.3 527.586 7.842.89 2,398.4
60 65.2 40.5 495.995 7,.011003 2,365.4
2.5-57 82.5 62.8 641. 30 '7 10,935.72 2? 995.2
60 82.7 57.1 621. 220 10) 0,'59.80 3,9°04 • 6
3.0-60 98.2 78.4 716.743 13.,831.32 3,635.0 4
aFL feeding level, N nitrogen (gm.s/day)y E energy (kcal/day).
97
Table 608 Weighted residuals cf final simulation run
• Steer FLa Urine N Fecal N Prine E Fecal E CH4 E P!'iJL E Fat E----, ---- --~-- ---, ----
36 0.5 ,~L49 9.90 5021 1.04 ,~4. 20 =5.49 .68
1.0 30 73 7.60 2002 6.25 L04 ~12.5l =049
LOH ~o 49 -3.36 -4044 8002 2.89 2078 -1.29
1.5 .03 =.36 1090 5.19 2025 .06 -L83
200 .07 -So 41 3.88 2,83 2094 2010 -2.09
47 00) -1.98 7,28 4.74 -2.97 "2.. 33 ",3 011 ,47
LO 3095 .5. 18 1.73 ]099 2. 010 -110 7S =052
LOR .27 ~3067 ,-50 30 5035 3.39 091 -L83
1.5 053 ~022 1.83 4.84 3.03 -1. 34 -,1. 65
200 .49 -3092 2.74 2076 2..54 024 -L 79
57 005 -L 11 6032 7007 -3 047 ·,4064 =3035 .09
LO -2059 5.09 2.32 4005 -.20 3.26 ·,1.11
LOH - 3. 00 -3000 ~"6. 03 8048 075 9.41 -2. n1.5 L87 1 '''0: -.04 ]064 032 -5.85 -L45l'J &' ,J
200 .98 1.80 -.34 .5.08 030 -3064 .,1. 27
205 .42 2.38 .58 60 n 023 -2.28 -2.17
60 0.5 '~. 56 L42 6006 -13.50 1.87 ~.41 025
LO -.07 089 2.03 1.60 2.34 '~. 29 -088
LOH =.31 -5087 =.46 1000 2024 3056 -2.84
L5 2.78 -2.'52 1.63 2030 2024 ~6. 30 c> • 85
200 2067 ~L52 017 3071 1.31 ,~7 .02 ~L06
205 2.15 '~o 79 034 4098 I. 72 =6010 -]. 79
300 L05 10 70 ].54 So 70 1.59 -3094 -2032
aFL :=:; feeding level J LOR -= hay diet at ma intenance leveL
4+ 4 Urine N
LEGEND3+
X X Steer ~ymbol
2+ X36 •C
C X1+47 ~A A DA , • I ~ FLo I Ii )(
~I I
0l~ 2 2~ 3 57-1 ! ~
c60 X•
-2 ~
C-3 + 0
lOI Fecal N 8 Urine E•C
6 X8+ • •AA6-1- c 4 •e 2 Ax A4-+.
4 XD 02+
0 0 X ~ 1 II l~Q ~ 2~ 3X 't-201 I I ! JI i h I .. FL
~ 1 lH l~ 2 3X -4-2+
X-6t
A=4+ ~ A c
X • -8-6 t
\000
Figure.602 Weighted residuals versus feeding level (FL) for final simulation run
e ~ . .. e l' ~ e
e ~ . I' e .. ., e
10+ X. Fecal E 4 Methane E0 A t!!.8+ • 3 ••
0 ! X ~A.
6+ • 2 XA 6- iii X- XA
0 X )( X4..1. CI 1 •
~ 0X 0 [] Q
2 4- 0 FL• X ~ 1H 1~ 2 2~ 3
o I I I , I , , ' .. FL =1~ 1 IH 1~ 2 2~ 3
=2 + -2~
I A
=41[] -3
)( -4I .c
': ! DProtein E 2 t Fat E
1
0 ~ I •• A
0 I p *.,
f i" e ,~ ! ~ FL 0 L1H 1~ 2 ~ 3 ~ 1 IH 1~ 2 2~ 3
AD 1:.0 X X X-5 40- •~ X -1 C X• lJX 0
b.A A X..
-10 +-2 I 0 0
A )(•~-15 + -3
I -0
Figure 6.2 (continued) -0
100
Urine E shows higr" predicU::ms at the lower fe.eding leve.ls and a
low prediction for the hay. Since urine E resu.lts from body break·~
down" we possibly must improve the formulatiom of that flc\N' lawo
The fecal E residuals are all high except for the 0.5 fe~ding
level. The hay values are higher than. the rest and t1:'.ere :s:eems to be
a linear trend with feeding leveL
The methane E residuals are all high, but are small in magnitude
compared to the others. The 0.5 feeding level is low for three
steers. A slight adjustment
that is needed,. assuming that
tion, is correct.
in the parameter fV
would seem. to be all
d*J which controls carbohydrate diges=c
The protein E residuals seem. to have less patte.rn than the~~t~Rel·s,
but are larger in magnitude. Improvement in t~ie body~relat:ed param=
eters will probably correct this.
The fat E residuals are negative for all but the 0.5 feeding
level. They trend downward as feeding level increases. Improvement
in the stores-related parameters shou.ld solve this.
Trends in the res idua Is seem to indicate tha t there is an effE'c t
of feeding level on the compartment whose value is being predicted.
The fonuulation for di.gestibility, which decreasE's digestibility as
feeding levE,l increases.? is an attempt to handle this problem. The
parameters f'Jr this formula;, aFL
and bFV
prcbably need SOlme imprcn7e=
ment. When they are at their optimum values, most of the linear trends
in the residuals should disappear. If not J then some additional flew
laws may need to be modified as was the digestibility fC?ImlJla.
101
Even thcugh the optimum parame.ter valoes have not been cttained y
and there might be imprcv'ement in some cf the f10,,1 laws" it is
apparent from th,e high percentage of variation in the experimental
results accounted for that the model wo~ld be lisehd in predicting
animal performance over wide ranges of protein and energy intakes and
their ratio.
If desired9 an economic framework can be easily imposed.
W2
7. CONCLUSIONS AND RECOMMENDATIONS
A conceptual framework defining the physical and chemical com
partments necessary for handling protein and energy metabolism of
homeotherms has been developed. A mathematical mode~or representa
tion, has been developed in this framework. This framework and model
treat the energy and protein metabolism as an input-output system.
They trace the flow of feed components (input) through the physical
compartments, through chemical transformations and ultimately to the
final uses by the homeotherm (output) such as body gain, fat storage
Or external production. The model takes account of the physiological
state of the homeotherm, of its maximum capacities for growth and
production, and also of feed composition, the amount of feed and the
energy to protein ratio in the feed. The model deals jointly with
the structural and energetic needs of the homeotherm and with the
structural and energetic roles of the feed constituents.
The model was tested against experimental data from the litera
ture. These data are from direct calorimetry studies on four steers
which were fed mixtures of hay and corn meal at feeding levels ranging
from one-half to three times maintenance and hay alone at the main
tenance level. The model explained more than 95 percent of the
variation in the data.
Although the patterns in the plots of residuals from the fit
indicate that the optimum set of parameter values was not attained,
they were not such as to lead to questioning of the model; hence the
conceptual framework proposed and the model developed are valid.
W3
With some improvements, and extension to milking cows, this
model can be used to study the partitioning of nitrogen and energy
among the various body processes for animals on a controlled feeding
regimen, whose intake is measured accurately. Further work on
factors affecting consumption of feed is necessary in order to extend
the model to the case of ad libitum feeding.
The model should be tested over a broader range of experimental
situations, ~.~., milking cows, mature cows, calves, and on other
species such as rats, chickens, sheep and goats.
Data on pool concentrations and fatness would simplify future
testing of the model and make the model output more accurate.
A mathematical analysis of the system would be useful.
Identification of stationary points and prediction of the long range
behavior of the system under constant input would be useful.
The handling of digestibility coefficients should be generalized.
Two coefficients were identified for carbohydrates; one for the mixed
diet and one for the hay. The methane production coefficients were
similarly identified. A more general approach would be to define
coefficients for each type of material, 1.~., protein, carbohydrate
and ether extract, which would apply to the hay as well as to the
grain, and treat the inputs separately. They would then be combined
in the same proportions as they appear in the diets.
The search for the optimum set of parameter values should be
pursued. More precision must be introduced into the computer program
and more judgment is called for also.
should be fixed and held constant.
The coefficients KB,P
This will allow ~ Band,
and KS P,~S~,
W4
vary and indicate animal to animal and breed to breed variation.
This should also improve the behavior of the steepest descent method
as the compensating effect of these pairs of parameters will be re
moved. A combination of more precision in the steepest descent method
and more intuition would probably lead to this optimum set.of values.
After testing the model over a broader range of situations, an
attempt should be made to estimate the heat (~ and the body breakdown
(~) coefficients as described earlier.
105
8. LIST OF REFERENCES
Albritton, E. C. (ed.). 1952. Standard Values in Blood. W. B.Saunders and Company, Philadelphia, Pa.
Baldwin, R. L., H. L. Lucas and R. Cabrera. 1970. Energeticrelationships in the formation and utilization of fermentationend-products, pp. 319-334. In A. T. Phillipson (ed.), Physiologyof Digestion and Metabolism in the Ruminant. Proceedings of theThird International Symposium, Cambridge, England. Oriel Press,Newcast1e.Upon-Tyne, England.
B1axter, K. L. 1962a. The Energy Metabolism of Ruminants. Charles C.Thomas, Springfield, Illinois.
B1axter, K. L. 1962b. Progress in assessing the energy value offeeding-stuffs for ruminants. Journal of the Royal AgriculturalSociety of England 123:7-21.
B1axter, K. L. 1966. The feeding of dairy cows for optimal production. The George Scott Robertson Memorial Lecture. Queen'sUniversity Press, Belfast, Ireland.
B1axter, K. L., N. McC. Graham and F. W. Wainman. 1956. Someobservations on the digestibility of food by sheep, and on related problems. British Journal of Nutrition 10:69-91.
B1axter, K. L. and H. H. Mitchell. 1948. The factorization of theprotein requirements of ruminants and of the protein values offeeds, with particular reference to the significance of themetabolic fecal nitrogen. Journal of Animal Science 7:351-372.
B1axter, K. L. and J. A. F. Rook. 1953. The heat of combustion ofthe tissues of cattle in relation to their chemical composition.British Journal of Nutrition 7:83-91.
Bratz1er, J. W. and E. W. Forbes. 1940. The estimation of methaneproduction by cattle. The Journal of Nutrition 19:611-613.
Brody, S. 1945. Bioenergetics and Growth. Reinhold PublishingCorporation, New York, N. Y.
Crampton, E. W. and L. E. Lloyd. 1959. Fundamentals of Nutrition.W. H. Freeman and Company, San Francisco, California.
Dukes, H. H. 1943.tion, revised.N. Y.
The Physiology of Domestic Animals. Fifth EdiComstock Publishing Company, Inc., Ithaca,
Dukes, H. H.Edition.
1955. The Physiology of Domestic Animals. SeventhComstock Publishing Company, Inc., Ithaca, N. Y.
106
Flatt: W. P., L. A. Moore., N. W. Hooven and R. D. Plowman. 1965.Energy metabolism studies with a high producing lactating dairycow. Journal of Dairy Science 48:797-798.
Forbes, E. B., W. W. Braman and M. Kriss. 1928.of cattle in relation to plane of nutrition.cultural Research 37~253-300.
The energy metabolismJournal of Agri-
Forbes, E. B., W. W. Braman and M. Kriss. 1930. Further studies ofthe energy metabolism of cattle in relation to the plane ofnutrition. Journal of Agricultural Research 40~37-78.
Forbes, E. B. and M. Kriss. 1932. The analysis of the curve of heatproduction in relation to the plane of nutrition. Journal ofNutrition 5~183=187.
Hawthorne, G. B., Jr. 1964. Digital simulation and modeling.Datamation, October~25-29.
IBM Corporation. 1968. System/360 Continuous System Modeling Program(360A-CX-16X). User's Manual No. H20-0367-2, New York, N. Y.
Kalman, R. E., P. L. Fa1b and M. A. Arbib. 1969. Topics in Mathematical System Theory. McGraw-Hill Book Co.: New York, N. Y.
Kleiber, M. 1961. The Fire of Life. John Wiley and Sons, Inc.,New York, N. Y.
Kriss, M. 1931. A comparison of feeding standards for dairy cows,with especial reference to energy requirements~ editorial review.Journal of Nutrition 4~14l~161.
Lucas, H. L., Jr. 1960. Theory and mathematics in grassland problems,pp. 732-736. Proceedings of the Eighth International GrasslandCongress, Reading, Berkshire.1 England.
Lucas, H. L., Jr. 1964. Stochastic elements in biological models;their sources and significance: pp. 355-385. In John Gur1and(ed.), Stochastic Models in Biology and Medicine. The Universityof Wisconsin Press, Madison.
Lucas, H. L., Jr. and W. W. G. Smart, Jr. 1959. Chemical compositionand digestibility of forages. Proceedings of the SixteenthSouthern Pasture and Forage Crop Improvement Conference, StateCollege, Mississippi.
Maynard, L. A. and J. K. LoasH. 1962. Animal Nutrition. FifthEdition. McGraw-Hill Book Company,. Inc., New York, N. Y.
McMeekan, C. P. 1940a. GroW"th and development in the pig, withspecial reference to carcass quality characters. I. Journal ofAgricultural Science 30;276-343.
McMeekan» C. P. 1940b. Growth and developmer.t in thespecial reference bJ carcass quality characters.influence of the plane. of nutrition on growth andJournal of Agricultural Science 30~387=436.
pig., withII. Thedevelopment.
107
McMeekan, C. P. 1940c. Growth and development in the pig, withspecial reference to carcass quality characters. III. Theeffect of the plane of nutrition on the form and composition ofthe bacon pig. Journal of Agricultu.ral Science 30~511=569.
Overman, O. R. and W. 1. Gaines. 1933. Mi1k~energy formulas forvarious breeds of cattle. Journal of Agricultural Research46~ 1109-11200
Reid, J. T., G. Ho Wellington and Ho 00 Dunno 1955.ships among the major chemical components of thetheir application to nutritional investigationsoDairy Science 38:1344-1359.
Some relationbovine body andJournal of
Schneider, B. Ho 19350 The subdivision of the metabolic nitrogen inthe feces of the rat, swine and man. Journal of BiologicalChemistry 109: 249~278.
Schoenheimer~ R. 1.9420 The Dynamic State of Body Constituents.Harvard University Press J Cambridge. Massachusetts.
Waldo J D. R. 1968. Nitrogen metabolism in the ruminant. Journal ofDairy Science 51:265-275.
108
9. APPENDICES
9.1 Flow Laws for Milk Production
Milk production is represented in Figure 4.2 by the paths from
compartments P and P to M. The flow laws for milk production aresn se
derived somewhat differently than the other flow laws derived in
Section 4.4. The quantity of interest is not milk production, M, but
the milk production rate~ denoted Pw and having units of kca1/time.
The notation PM refers to the total milk production arising as a
result of the flow of materials from both the nitrogen and energy
pools, P and Psn se
We postulate that the rate of change of the milk
production rate is proportional to two factors:
(a) the rate of change of the pool concentrations, and
(b) the difference between the "genetic maximum" milk
production rate for the cow and the current rate.
Since nitrogen and energy are needed jointly for milk production,
we consider the product of the two pool concentrations as the driving
force, and the rate of change of the pool concentrations is thus given
by
where d denotes derivative.
Brody (1945, p. 703) says the following abou.t the life curve of
lactation:
The rise in milk production up to seven or eight yearsin dairy cattle parallels in shapeJ although it lagsin time, the rise in body weight..
109
In addition to the life curve of lactation, we must consider the
lactation-period curve of lactation. This curve rapidly rises to a
maximum and then declines. According to Brody (1945, p. 703):
• • • it appears that the decline in milk productionfollowing the attainment of the maximum yield at theprime of life or the prime of the lactation period isexponentia1 .
We must postulate a genetic maximum production curve which takes these
features into account.
For each lactation period, we can define a maximum milk production
curve, per unit of body size, asa difference of two exponentials
(Brody, 1945),
Then, for the first seven or eight lactations, multiplying this by
body size will give the maximum milk production curve, ~B. Then the
difference between the genetic maximum milk production rate and the
current rate is given by (\iB - PM). Letting the proportionality factor
be Kp M and representing the rate of change of milk production rate by,d(PM)' our differential equation is:
Exact solution of this equation requires an appropriate initial condi-
tion. We postulate that a certain minimum concentration of P andsn
P is necessary to "drive" the milk production. However, the valuese
of ~ also affects this minimum value. A large value of ~, ~.~., a
large potential to produce, requires a lower concentration to "drive"
the production. If ~ is an appropriate constant, then we define the
initial condition as follows: the rate of milk production, PM equals 0
110
when (P P /B2) ::: ~/A, or:sn se 'M
(9.2)
Solving the differential equation (9.1), subject to the initial
condition (9.2), yields the differential equation for milk production
rate,
We consider milk to be a fairly we11~defined mixture of nitrog-
enous and energetic material and denote the nitrogenous material, !.~.,
casein, by M and the energetic material, e.~., lactose, by M. Thenn ~~ e
the total amount of milk produced, M, is related to these quantities
by:
(9.4) M ::: ~ + Me
and the ratio of energetic to nitrogenous material, ~. is defined by:
(9.5) 0.. ::: M /M •'M. e n
From (9.5) we have
(9.6) M = (PM)(M)e n
and substituting into (9.4) gives
(9. 7)
or •
(9.8)
111
Substituting (9.8) into (9.6) gives
(9.9)
."Thus, given the amount of milk, M, we use (9.8) and (9.9) to
partition this into its nitrogenous and energetic components. We
assume that the flow of material to form milk, given by (9.3), is
partitioned similarly. Denoting the flows of nitrogenous and energetic
material for milk production as p M and p respectively, the flown, e,M'
laws are:
(9. lOa)
(9. lOb)
For a given value of ~, the graph of the milk production rate,
PM' as defined by equation (9.3) is shown in Figure 9.1.
~B ---------------
p psn se
~ B2
~
Figure 9.1 Graph of milk production rate
112
9.2 Flow Laws for Heat Loss
Heat in the heat pool, Ph' which cannot be used by the animal to
keep warm, is dissipated as shown by the flow from Ph to H, the
compartment for heat loss.
Four factors in heat loss are radiation, conduction, convection
and water evaporation. Consider the heat loss per unit of surface
area. According to Kleiber (1961), radiation is proportional to
(Ti - T~) where Tl ,T2 are the temperatures of the sender and receiver
of the radiation in degrees Kelvin. For small differences in
4 4temperature, (Tl
- T2) may be approximated by (T
l- T
2) and
radia tion ex 6T
where 6T = Tskin - Tair, the difference between skin temperature of
the animal and air temperature, in of or °C. Heat loss due to con-
duction is proportional to 6T, or
conduction ex 6T
Heat loss by convection is proportional to air velocity (v) and
temperature difference, or
convection ex (v) (6T)
For a given experimental situation, we assume that the air velocity
is constant and write
convection ex 6T .
Heat loss by water evaporation is proportional to air velocity and
the difference in vapor pressure at the surface and in the air, t:NP,
or,
113
evaporation a (a + bv) (6VP)
where a and b are suitable constants. Again assuming that air velocity
is constant, we have
evaporation a 6VP •
It is reasonable to assume that at the evaporating surface (lungs,
skin), the vapor pressure is equivalent to 100 per cent relative
humidity. Denoting this as VP lOO and the vapor pressure of air as
VP • we writea'
evaporation a (VP lOO - VPa) •
Now, surface area is proportional to body size to the two-thirds
power, and kcal of body (B) is proportional to body size; hence,
surface area is proportional to BZ/ 3•
Combining the four factors above and multiplying by surface area
leads to the equation for heat lossJ
(9.11)
where Al is an emission factor for the animal and AZ
is a vaporization
factor and each are functions of the body temperature. The product
Al
BZ/ 3 may be considered an "effective" surface area for heat transfer
and AZ
BZ/ 3 as an "effective" surface area for evaporation. In order
to prevent heat from flowing into the animal, we will not allow (6T)
or (VP lOO = VPa) to be negative. Thus, the notation in equation (9.11)
might be more correctly written as
(6T) =: max (0 , 6T)
114
However, for simplicity, it will be left as shown. Al and AZ must be
defined to satisfy the following conditions:
(a) when the animal is cold .. Al and AZ
are minimized,
(b) when the body temperature starts to rise above some optimal
valueJ a small increase in water evaporation occurs, but conduction,
radiation and convection will handle most of the dissipation of the
heat, and
(c) further above the optimal temperature, conduction, radiation
and convection reach their maximum levels and water evaporation
increases to dissipate the extra heat.
In order to satisfy the aboveJ we first define body temperature
as TB
• Body temperature will be proportional to body heat divided by
body mass. We take Ph' the kcal of heat in the heat pool, as our
measure of body heat. B is proportional to body mass, hence,
We define the optimal body temperature for the animal as T. Ino
order to simplify the initial formulations, we also assume that the
body, skin and rectal temperature of the animal are equal and use TB
to denote them. No attempt will be made to derive formulae for Al and
AZJ but a graph (Figure 9.Z) will illustrate possible forms which
satisfy points (a) - (c).
The dissipation of heat requires energy to drive the reaction.
This energy is supplied by P and results in a transfer of heat fromse
Pse to BH and then to Ph. ObviouslYJ the amount of heat thus trans
ferred must be less than the amount of heat dissipated, else the
animal will not be able to keep cool. This is what happens at very
115
J-------+-------------"~TB
o
Figure 902 Possible forms of Al
,A2
versus body temperature
high environmental temperatures when the body is unable to dissipate
its heat without working very hard (producing much heat) and thus it
produces more than it loseso Kleiber (1961, p. 162) states~
Under these latter circumstances) the body temperaturerises, as also does the metabolic rate) because thecellular processes are now uncontrolled and operateaccording to Van't Hoff's law. If this positive feed~
back continues J it becomes a fatal vicious cycle.
We will not include a fatal point i.n the model1 but wi.11 note that if
the heat produced in dissipating heat exceeds the heat dissipated, the
animal is in trouble. This energy transfer may be formulated as
(9. 12) p Y ::0; K'se,Ph Pse,Ph
where K;seyPb is a 'function of body temperature, TB• This function,
in shape, will resemble the sum of Al and AZ
of equation (9.11),
except that it will not level out at higher temperatures as Al and A2
116
do. In magni.tude: it will be quite a bit less than Ph H except atJ
high body temperatures. A graphical representation of a possible form
of K:~se,Ph is shown in Figure 9.3.
K'Ps€;) Ph
To
Figure 9.3 Possible form of Ki versus body temperaturePse,Ph
When body temperature falls below T J the animal becomes cold.. 0
To warm u.p it increases its metabolic rate? or generates more heat.
The animal oxidizes energy from P to provide heat to BH and thencese
to Ph to raise the body temperature. We formulate this as
(9.13) _ l("Pse, Ph
where K~se:,Ph is a function of body temperature which is high at low
temperatures and minimal above T. A possible form is shown ino
Figure 9.4.
117
Ie"Pse,Ph
I------~T---------------"~TBo
Figure 9.4 Possible form of Ie" versus body temperaturePse,Ph
In Table 4.2, the differential equations for the comparbment model
were given. The differential equation for the heat pool (4.29) was
p = hp - Ph h,H
where hp represented the rate of heat production from both gut fermenta-
tion and body processes, and Ph H represents dissipation of heat or.,heat loss. We must now augment hp by the heat produced in dissipating
heat and in warming up. Thus, we may write~
(9.14)
where pI Ph and p" Ph have been defined above and are functions ofse, se,
body temperature. The flow law for Ph H has been given as equation,(9.11).
118
9.3 Evaluation of Constants
9.3.1 Evaluation of PB
Combining data from several sources (Crampton and Lloyd, 1959;
Maynard and Loosli, 1962; Blaxter and Rook, 1953) yields the following
table:
Table 9.1 Simplified composition of energy-containing substances indry skeletal muscle
Component
P. a
rote~n
bFat
Fraction
.8
.2
kcal/gmof Component
5.322
9.367
kcal/gmof Dry Tissue
c
4.2576
1. 8734
aMixed, deposited material containing, on the average, 16 per centnitrogen.
b . .Conta~ns no n~trogen.
CEqual to fraction times kcal/gm of component.
Carbohydrate is omitted since it comprises less than 1 per cent of
the total body. Now, one gram of body, B, consists of 4.2576 kcal Bn
and 1.8734 kcal B. Therefore,e
(9.15) P = B /B = 1.8734/402576Ben.44 •
Some useful relationships derived from (9.15) are
and
119
or
(9.16)
9.3.2 Evaluation of f B, f S
Given the body weight of a steer, we want to determine those
fractions of the total which can be taken as depot fat and as body.
We then will convert these fractions to kca1.
Let wB designate body weight in grams. We assume an average
fatness of 14.2 per cent (Reid et a1., 1955). Thus,
fat (grams) = .142 wB
and by subtraction, the remainder is fat-free body, therefore
fat-free body (grams) = .858 wB .
The fat is partitioned between stores, S, and body, B , with a heat ofe
combustion of 9.367 kca1/gm. Thus,
(9.17) S + Be = (9.367) (.142) (wB) = 1.33 wB .
The average protein content of the fat~free body is 21.64 per cent
(Reid et a1., 1955), hence
protein (grams) = (.2164) (fat-free body)
= (.2164) (.858) (wB)
and using a heat of combustion value of 5.322 kca1/gm for protein, the
value of B isn
Bn = (5.322)(.2164)(.858) (wB)
From (9.15),
.988 wB •
120
B = P oB = (.44)(.988) (wB) = .435 owBe B n
and since
then,
B = .988owB + .435owB = 1.423owB •
Therefore,
(9. 18) fB
= 1.423 •
Also, from (9.17),
and therefore,
(9.19) f S = .895 •
9.3.3 Evaluation of wBn' Ws
Given the energy content of the body,~ and of the stores, S, we
want to calculate the body weight. We use B to determine the weightn
of the fat-free body and Band S to determine it for the fat. Onee
kca1 of S or B corresponds toe
1/9.367 = .107
grams of fat. Hence
(9.20) wS =·107.
One kca1 of B corresponds ton
1/5.322 = .1879
..
121
grams of protein. Assuming that protein comprises 21.64 per cent of the
fat-free body, the .1879 grams of protein imply
.1879/.2164 = .8683
grams of body weight. Hence
(9.21)
and
wBn
= .868
= .868·Bn+ .107· (S + B) •e
9.3.4 Evaluation of r s
The flow law for s contains a term of the formPse
(9.22) f(x) = x/ (x + Ol)
where x SiB and Ol = r s • The graph of this function is a hyperbola
(Figure 9.5) with asymptotes at x = -Ol and x = 00. For a given initial
condition, we can evaluate Ol. Suppose that the condition is
Then
f = x / (x + Ol)o 0 0
and solving for Ol gives
Thus to evaluate Ol in (9.22), we must specify x = (S/B) and f. It000
was decided that when the amount of fat in the stores, S, equalled the
amount of fat in the body, B , of an average animal (i.e., one withe
14.2 per cent fat in its body), then f(x) = 0.9.
122
f(x)
1 ------------f
o
----...----..,..----.......---------------3~x
Figure 9.5 Graph of f(x) = x/(x + 01)
From Section 9.3.2 above, one gram of body weight contains 1.423
kca1 (B) of which .988 kca1 is in Band .435 kca1 is in B. Thus,n e
B /B = .435/1.423 = .3057 •e
Thus, the initial condition is that when
SiB = .3057 ,
f =.9.o
Then,
(9.23) r = 01 = (.3057) (1 - .9)/.9 = .3057/9 = .034 .s
•
123
9.4 Evaluation of Initial Conditions for the Pools
Dukes (1955) gives normal ranges of several chemical constituents
of the blood of mature domestic animals. For the cow, pertinent values
are (in mg per 100 ml whole blood) ~
Amino acid nitrogenTotal non-protein nitrogenUrea nitrogenSugarLactic Acid
4 - 820 - 40
6 - 2740 - 70
5 - 20
Dukes (1943) gives a value of .0567 gms fat per 100 gms blood.
Taking the specific gravity of whole blood as 1.052 for cattle (Dukes,
1955), this is equivalent to
(.0567) (1.052) = .0596
grams of fat per 100 ml whole blood.
For P , we consider amino acid nitrogen. Taking the midpoint ofsn
the range as an average figure, we have 6 mg amino acid nitrogen per
100 ml whole blood. Assuming amino acids to contain 16 per cent
nitrogen, we have
6/.16 = 37.5
mg amino acids or .0375 gm amino acids per 100 ml whole blood.
For Pdn, consider the average of the total NPN values to be 30 mg
NPN per 100 mI. Subtracting the amino acid nitrogen value of 6 mg per
100 ml leaves 24 mg of nitrogen per 100 ml in Pdn
• Of this, approxi
mately 16.5 mg per 100 ml is urea nitrogen. Thus to convert the mg of
nitrogen to mg of Pdn material, we assume that all the nitrogen is
present in urea. Urea consists of approximately 45.75 per cent
nitrogen. Thus we have
124
24/.4575- 52.46
mg of Pdn or .0525 gms Pdn per 100 ml whole blood.
The energy pooL P • will contain substances like sugar, volatile, se-
fatty acids, lactic acid and fat. Taking average figures for sugar and
lactic acid as was done above.? and using the sugar value to estimate
the fatty acid content gives
55 + 55 + 12.5 - 122.5
mg or .122 gms non-fat P per 100 ml whole blood. From above, wese
have .0596 gms fat pe.r 100 ml whole blood.
Given the above va1ues J we must now convert to kcal of pool
materials per kcal of B to give initial pool concentrations. Several
conversion factors will be used. Albritton (1952) gives a value of
85 gms water per 100 ml whole blood for cattle. Reid et al. (1955)
provide average concentrations in the fat-free body of water as .7291
and protein as .2164. Hence, we consider the ratio. 7291 gms water
per .2164 gms protein. The heat of combustion of protein is taken as
5.322 kcal/gm (Blaxter and Rook, 1953) and the factor 1. 44 kcal B
per kcal B was derived in Section 9.3.n
If k denotes the concentration of materials in blood (gms/lOO ml
whole blood) and h denotes the heat of combustion of these materials
(kcal/gm), then the conversion formula is~
kca1 materialkcal B
_ ( k gros material) (100 ml whole blood) ( 07291 gIn water)100 ml whole blood' 85 gm water .2164 gm protein
1 kcal B(1 gm protein n )(h kcal material5.322 kca1 B ) (1.44 kca1 B 1 gm material) •
n
•
125
Multi.plying the terms out and cancelling units give the concentrations
as
kcal materi.al (0051'72) (k') (h)- kcal B -- 0 . . . '. - 0
If we consider Pdn as mostly urea J then we may take h "" 2.450 For fat
in the blood, we. will take h ,= 9000 Summarizing the above (Table 9.2),
we have~
Table 9.2 Summ.ary of pool concentration data
Pool k (gros/IOO ml blood) h (kcal/gm) Concentration
P 00375 5.322 .00103sn
Pdn.0525 2.45 .000665
fat 00596 9.0 .00277 } .00514Pse other 0122 3.75 .0023'7
Pde .00331
No infonnation was available for Pde
., so it was evaluated to
satisfy the following ratio~
Pde concentrat.ion
P concentrationse
Pdn concentration-co
P concentrationsn
905 Evaluation of Cd,0
For each steerJ data were given on heat producti.on, dry matter
fed and fat energy balance per dayo We wish to derive a value of dry
matter fed which corresponds to zero fat ene.rgy balance. The data are
given in Table 9.30
126
Table 9.3 Values of heat production, dry matter fed and fat energybalance
Feeding Steer 36 Steer 47
Level Da DMa EBa D DM EB
0.5 8,155.8 1,885 -3,082.8 7,754.5 1,863 -2,625.8
1.0 9,839.7 3,762 -15.2 9,382.8 3,790 688.8
1.0 (hay) 11,635.0 5,763 -685.0 11,254.6 5,771 210.9
1.5 11,854.1 5,353 2,377.9 11,692.9 5,617 3,288.0
2.0 13,888.1 7,037 4,431.8 13,536.3 7,839 5,598.4
Feeding Steer 57 Steer 60Level D DM EB D DM EB
0.5 7,939.1 1,700 -3,098.0 7,476.2 1,681 -2,720.8
1.0 7,908.7 3,085 530.1 7,252.9 2,828 472.2
1.0 (hay) 9,953.7 5,155 474.1 9,790.1 5,013 89.3
1.5 9,493.3 4,612 2,761. 7 8,821.4 4,237 2,240.1
2.0 11,851. 2 6,233 3,987.2 11,156.9 5,704 3,371.6
2.5 14,408.2 8,057 6,071.5 13,976.4 7,520 4,967.6
3.0b
16,133.1 9,489 7,171.6
aD = heat production (kca1/day), DM = dry matter fed (gms/day),EB = fat energy balance (kca1/day).
bNo observations for Steer 57 at this feeding level.
•
•
127
The method used was to estimate that value of heat production
correspondi.ng to zero fat energy balance.. Linear regression was used
with heat production as the dependent variable and energy balance as
the i.ndependent variable. The values for the hay feedings were omitted
from this regression for all steers, and for Steers 57 and 60, the 0.5
feeding level data were also omitted. The heat production values thus
derived are given in Table 9.4.
Table 9.4 Heat production values for zero fat energy balance
Steer
36475760
Heat Production (kcal/day)
10,228.39,351. 86,892.56,348.4
Then, a regression line was fit, for each steer, relating heat
production (dependent variable) to dry matter fed (independent
variable). Using the regression line, the value of dry matter fed
which corresponded to the heat production value of Table 9.4 was
calculated. If we represent the regression line by
y = a + bx
where y = heat production and x ._. dry matter fed, and if Yo is the
value in Table 9.4, then
c = x = (y ~ a)/bd,o 0 0
The values of a, band c thus derived are given in Table 9.5.dJo
128
Table 9.5 Regression constants and values of cd;0
Steer a b cd,o
36 5,937.9 1.07900 3,976 •47 5,845.6 .98064 3,57557 3,478.2 1. 33462 2,55860 3,136.6 1. 38861 2,313
9.6 Evaluation of K BP2
The flow law for body growth, equation (5.12a) is
Brody (1945) considers body growth to follow an exponential law. This
holds fairly well after birth, but starting at conception; the logistic
law seems applicable. The term in the second bracket of the above
equation is a logistic type term. If we denote the term in the first
bracket by K' and use wB (grams) instead of B (kca1), we may represent
body growth by
Note that we are temporarily ignoring body breakdown, bBW
and bp•
Dividing both side.s of this equation by wB leads to the following
equivalent form,
d (In wB
) / d t = (K') (A - w )w
BB
where K1 =: (K I) (A ) and K2 =: - K' •wB
Constants K1 and K2 are estimated from the data in Brody (1945)
on Holstein growth, pages 571-572. The derivative is estimated by
•
129
differences, hence,
or
where t1
, t2
are values of time at which observations are made. From
data ranging over values of wB
from 660 to 1760 kg, a rough estimate
of K' is
-1K' = ,0001446 (kg=months) ,
Converting units leads to
K: '- ,48182 x 10=8 (gram= days) - 1
and converting grams B to kca1 (1 gram = 1. 423 kca1) gives
K' = ,68563 x 10=8 (kcal=days)=l ,
Now
P P(K ) ( sn) (~)
P,B B B
Substituting initial values of pool concentrations gives
K - .001295 (kca1=days)=1 ,P,B
In order to compensate for body breakdown, hBW
and bp
which were
ignored, we mus t increase '1> B to sus taiD the growtho If we arbi="
trari1y increase the above value by one=fourth, the initial estimate
will be taken as
Kp B .~. ,001619 .J
l~
9.7 Evaluation of Weights for the Goodness of Fit Criterion
The evaluation of weights follows the procedure outlined in
Section 5.3 and uses the data in Table 5.9.
If for any factor, we let x 36' x47 ' x57 and x 60 denote the value
given in Table 5.9 for the steer indicated in the subscript, then the
variance between Steers 36 and 47 is given by
and between Steers 57 and 60 by
The pooled variance is then
or
2s
p
and s is the square root ofp
observations on Steers 36 and
2s .P
47,
For feeding level 2.5, there are no
hence
The values of s are given in Table 9.6.p
When regressing s on feeding level for each of the factors, wep
must recognize two points. First, for feeding levels 0.5 through 2.0,
the pooled variances are based on two sets of two values each, and
hence, have two degrees of freedom. For feeding level 2.5, we have
one set of two values and hence, one degree of freedom. Thus, we must
fit a weighted regression line with weights equal to the degrees of
131
Table 9.6 Values of sp
• FeedingLevel Urine Na Fecal N Urine Ea Fecal E Methane E
I 0.5 .4242 .5700 12.4355 38.3715 3.90161.0 3.6718 .3535 20.3317 98.3342 106.65691.0 (hay) 3.0602 .5408 21. 3765 34.2563 65.04781.5 3.0372 .5852 36.9233 116.9572 150.69742.0 5.6135 .9823 22.9829 291. 4128 186.10252.5 9.0509 1.2727 9.9277 340.2880 291. 5401
aN = nitrogen, E :;:: energy.
freedom. The second point is that for the hay diet, the amount of
nitrogen fed is close to that of feeding level 1.5 for the mixed diets.
Hence, for Urine N and Fecal N, the hay data will be treated as feeding
level 1.5 in determining the regression line and in predicting the
*value of s to be used for the weight.p
f d h · th f d' 1 1 b . . h (d fI we enote t e ~ ee ~ng eve y x., ~ts we~g t egrees 0~
freedom) by w. and the value of s by y~j then the regression line is~ p ~
y:;::a+bx
and by least squares theory,
- / - 2b :;:: [Dl. (x. - x) (y.)] [Dl. (x. - x) ].~~ ~ .~~~ ~
a = y - bi = (Dl.y.. ~ ~~
bDl.x.) /Dl.. ~ ~ . ~~ ~
Using the values of x., a and b, we derive the predicted values,~
* which are used as weights for the goodness of fit criterion. Thesp'
values of w., band s* summarized in Table 9. 7. TheXi' a, are~ p
observed (Table 9.6) and predicted (Table 9.7) values of s and thep
132
regression line for each factor are given in Figure 9.6. There was
no observed value of s for feeding level 3.0 as there was only onep
observation per factor, that for Steer 60. Using the regression line,•
however, a predicted value is calculated. \
Table 9.7 Values of wi' xi' *a, b, and s for each factorp
Feedings*
w. x. pLevel ~ ~ Urine N Fecal N
0.5 2 0 . .5 .4780 .3347
1.0 2 1.0 2.248 .5174
1.0 (hay) 2 1.5 4.019 .7001
1.5 2 1.5 4.019 .7001
2.0 2 2,0 5.789 .8828
2.5 1 2.5 7.5.59 1.066
3.0 9.330 1. 248
a' -1. 2924 .1520
b: 3.5407 .36541
Feedings*
w. X. pLevel ~ ~ Urine E Fecal E Methane E
0.5 2 0.5 19.63 8.597 7.864
1.0 2 1.0 20.86 79.70 76.61
1. 0 (hay) 2 1.0 20.86 79.70 76.61
1.5 2 1.5 22.09 168.6 145.4
2.0 2 2.0 23.32 257.5 214.1
2.5 1 2. .5 24.54 346.4 282.8 ,.
3.0 25. 77 435.2 351. 6
a: 18.399 - 98.065 - 61. 5684 ~
b· 2.458 177.77 137.49
e • • I: -1 e .~ . .. e
] Urine N
~ 1. 6~ Fecal N
sp I ~ - I •
4
0 , , • sp
40t- Urine E •
• 020 II
SP
o t • •I ! I I , I 4001- Fecal E
300f Methane E 300
200f s 200P
sp
100L ./ I /' •100
01 v' I I , , I~
00 0.5 1.0 1.5 2.0 2.5 3.0 0 0.5 1.0 1.5 2.0 2.5 3.0
Feeding Level Feeding Levelt-'I.J-l
Figure 9.6 Graph of s versus feeding level and fitted regression lines I.J-lp.
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