1 Feedback Control in Physiology: The Calcium Homeostatic System Mustafa Khammash Dept. of Electrical & Computer Engineering Iowa State University, Ames, Iowa Joint work with Hana El-Samad, Jesse Goff (NADC)
Jan 15, 2016
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Feedback Control in Physiology:The Calcium Homeostatic System
Mustafa Khammash
Dept. of Electrical & Computer EngineeringIowa State University, Ames, Iowa
Joint work withHana El-Samad, Jesse Goff (NADC)
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Outline
• Blood Plasma Calcium Regulation
• Calcium homeostasis in mammals
• A model for calcium homeostasis
• Hormonal interactions
• Disorders
• Conclusions
Physiological Role of Calcium
• Maintain the integrity of the skeleton.
• Control of biochemical processes:
– Intracellular: • Activity of a large number of enzymes
• Conveying information from the surface to the interior of the cell
– Extracellular: • Muscle and nerve function
• Blood clotting
• The biochemical role of Calcium requires that its blood plasma concentrations be precisely controlled
• Normal concentration of about 9 mg/dl must be maintained within small tolerances despite
– variations in dietary calcium levels– variation in demand for calcium
• Humans and other mammals have an effective feedback mechanism for regulating plasma concentration of calcium [Ca]p
Calcium Regulation in the Cow
• Constant plasma concentrations of calcium are easily maintained during periods on nonlactation (daily need is typically less than 20g/day)
• An especially large loss of plasma calcium to milk takes place during lactation (up to 50 g/day)
• Most animals adapt to the onset of lactation
10 12 14 16 18 20 220.05
0.055
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
10 12 14 16 18 20 220
10
20
30
40
50
60
70
80
90
100
Plasma Ca Concentration (g/l)
Ca Clearance Rate
Parturitiontime (days)
time (days)
Parturient Paresis
• In some cows, the calcium regulatory mechanism fails to meet the increased calcium demands
• These animals become severely hypocalcemic
– Results in disruption of muscle and nerve function– Leads to recumbency
• The clinical syndrome is referred to as Parturient Paresis (Milk Fever)
• It affects 6% of the dairy cows in the US
Plasma Ca (with Parturient Paresis)
Calcium Flow
Calcium poolFiltration
reabsorptionKidney
Milk, fetus
Absorption
Intestine
Secretion
Resorption
Formation
Bone
Mathematical Modeling of [Ca]p
)(1
][ clTp VVVol
Cadt
d
Vol = Plasma Volume (l)
[Ca]p = Plasma Concentration (g/l)
Ca Total Supply Rate
VT (g/day)
Intestinal Absorption
Bone Resorption
Total Ca Clearance Rate Vcl (g/day)
Milk, fetus, urine, etc.Plasma
t
clTp dVVVol
Ca0
)(1
][
k
Vcl
VT pCa][-+
e = error (g/l) = set point (g/l) - pCa][
k
Vcl
VT pCa][-+
-+ ControlSet point e
)(efVT
Standard Model
• A model describing the relation between VT and [Ca]p is given by Ramberg et al.:
Source: Ramberg, Johnson, Fargo, and Kronfeld, “Calcium homeostasis in cows, with special reference to parturient hypocalcemia,” Am. J. Physiol. , 1984.
• This is Proportional Feedback
)/()][..(1770 lgCapsV pT
(g/l)eKV pT
Deficiencies in the Standard Model
• From basic principles of control theory, proportional feedback alone cannot explain:
– The observed zero steady-state error (Perfect Adaptation)
– The shape of the time response of [Ca]p following increased Calcium clearance at calving
t
IpT deKeKV0
Integral Feedback
• In order to account for the zero state-state error integral feedback must be present.
• When combined with Proportional Feedback, Integral Feedback will account for
– The zero steady-state error in response to Ca clearance
– The second order shape of the [Ca]p time response
• We propose the feedback:
k
Vcl
VT pCa][-+
-+
Set point e+
PK
IK
PI Feedback
Implications of PI Feedback
• At any given time, the calcium supply rate VT is not dictated only by the level of calcium deficiency at that time.
• Supply rate depends on both the level and duration of calcium deficiency prior to and until the time of interest.
• Understanding the dynamics of the system is unavoidable.
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Model vs. Experiment
• Data from two groups of normal lactating dairy cowsaround the day of calving (NADC)
• One group was used to determine model parameters
• The model prediction was compared against data from the larger second group (20 animals)
Model Prediction Vs. Actual Data
10 11 12 13 14 15 16 17 18 19 200.05
0.055
0.06
0.065
0.07
0.075
0.08
0.085
0.09simulation output and actual data
How Do Cows Integrate?
• Our model was arrived at through necessity arguments
• Is there a plausible physiological basis?
• Given that calcium is hormonally regulated, what is the mechanism through which integration is realized?
Can a single hormone be at work?
• P feedback:
• PI feedback: Error) Error A] [Hormone (dt
dk
dt
d
]AHormone[ T V
Error A] [Hormone
A Two Hormone Solution…
]AHormone[ ]BHormone[ dt
d
Error]AHormone[
;BAT VVV ]BHormone[ ];AHormone[ BA VV
ErrorkErrorVT
Hormonal Regulation
PTH stimulates renal calcium reabsorption and bone resorption
(1,25 OH2 D3) Hormone stimulates calcium absorption from the intestine
Bone resporption and intestinal absorption account for the entire calcium supply
The Parathyroid Gland monitors blood calcium and secretes Parathyroid Hormone (PTH) in proportion to [Ca] deficiency
Error]PTH[
]PTH[ AV
]OH 1,25[ 32DVB
BAT VVV
Setpoint Origin: The Parathyroid Glands
The Integral Term
PTH25 (OH)D 1,25 (OH)2D
]PTH[ D](OH) [1,25 2 dt
d
For a given [25 (OH)D]:
• Two forms of Vitamin D: 25 (OH)D and 1,25 (OH)2 D
• PTH activates 25 (OH)D in the kidney to form 1,25 OH2 D
Understanding Parturient Paresis
• In normal animals, a linear model was adequate for describing observed regulatory response
• However, the linear model alone cannot account for
– Breakdown in [Ca] seen in cows with Parturient Paresis
– Recovery after Calcium IV infusion
Nonlinear Effects
• The supply of calcium from the bone cannot be increased indefinitely in response to an increases in [PTH]
Set point
k
Vcl
VT pCa][-+
-+e
+PK
IK
Absorption Nonlinear Effects: Rumen Motility
• When [Ca]p is significantly reduced, the processes responsible for intestinal absorption will be impacted
• The net result is a “slowing” of intestinal absorption when it is most needed
• A clear example is the impact of reduced plasma calcium levels on rumen motility
Hypocalcemia Affects Motility
Source: R.C. Daniel, “Motility of the Rumen and Abomasum During Hypocacemia,” Can. J Comp Med 1983.
Rumen Contractions
Abumasal Contractions
Normal
DuringHypocalcemia
Normal
DuringHypocalcemia
Set point
k
Vcl
VT pCa][-+
-+e
+PK
IK (.)fx
Absorption Reduction Factor
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
calcium concentration in g/l
mot
ility
Exploring the Model Properties
• With both nonlinear effects included, calcium break-down does take place
• Breakdown depends on the saturation level, absorption reduction function, and the linear model parameters
• Fixing the nonlinear elements, breakdown depends entirely on the values of
• Larger values of lead to smaller undershoot in the linear model
Ip KandK
Ip KandK
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
0
10
20
30
40
50
60
70
80
90
100
x1
x2Phase portrait for Kp=5000 and Ki=3000
Phase Portrait for Kp=5000, Ki=3000
Equilibrium(high clearance rate)
Initial condition (low clearance EP)
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
0
10
20
30
40
50
60
70
80
90
100Phase portrait for Kp=3000 and Ki=1200
x1
x2
Phase Portrait for Kp=3000, Ki=1200
Equilibrium(high clearance rate)
Initial condition (low clearance EP)
A Sufficient Condition for Breakdown
then, will be monotonically decreasing, and
for some ,
)(
]))(()([1
1
.
2
121
.
1
xrKx
VxrKsatxxfvol
x
I
clpS
.0)( , some
for and decreasinglly monotonica be will)0()(Then
,:0 where.
)(
If.)0( and ,)0( Suppose
110
1
210
2101
Txx
T
ttx
vol
SV
rKx
volxf
VS
xxrxx
cl
I
cl
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Summary & Future Work• Calcium homeostasis is achieved through integral feedback.
Integral action is realized by the dynamic interaction among 1,25 (OH)2D and PTH
• Sequence of discovery: Perfect adaptation necessity of integral action specific action at molecular level
• The dynamic interactions give a new perspective on calcium homeostasis disorders and disease trajectories
• Future work:
– Osteoporosis
– Other homeostatic mechanisms, e.g. blood sugar, diabetes
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Control Theory in Biological Systems• Feedback regulation mechanisms are ubiquitous
• Bring out the dynamic nature of biochemical interactions
– Explain interactions in the context of regulation
• Pathologic behavior when systems operate at their extremes. Capturing the dynamics will
– lead to better understanding of the trajectory of disease
– suggest more effective courses of treatment
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• Identify functional biological modules
– Reveal structural constraints on the dynamics
– Structural constraints impose functional requirements on biological modules
– Easier to understand/predict the function of sub-modules
• New understanding of the behavior of biological subsystems
– Notions such as robustness, adaptation, amplification, isolation, and nonlinearity are required for a deeper understanding of biological function
– Many similarities with engineering systems
– Ask the right questions