A computer model of the lymphatic system.pdf

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(ompsr Bid Med. Pergamon Press 1977 Vol. 7. pp. 1X1-197. Prmted m Great Britain.

A COMPUTER MODEL OF THE LYMPHATIC SYSTEM

NARENDER P. REDDY*, THOMAS A. KROUSKOP and PAUL H. NEWELL, JR.Rehabilitation Engineering Center, Texas Institute for Rehabilitation and Research. P.O. Box 20095.Houston, TX 17025. USA

Received 11 August 1915; in revised form 30 August 1976)

Abstract-A mathematical model of the whole body lymphatic network, which simulates thelymph propulsion along the network from the periphery through the thoracic duct into thevenous system, is developed using the fundamental conservation laws and current notions oflymphology. Only the major lymphatic vessels are considered. The set of mathematical equationsare then translated into a series of FORTRAN statements for the purpose of digital computersimulation of the pressure and flow patterns along the network. The model analysis revealedinteresting characteristics of lymphatic contractility at various points along the network.

Lymph flow and pressure Lymphatic active contractility Lymphangion pump Smoothmuscle Network model Computer simulation

GENERAL NOMENCLATURE

radius of a lymphangion (cm)capillary filtration coefficients (cm5/dyn)filtration coefficients for the terminal lymphatics (cm/dyn)protein concentration (g/ml)modulus of elasticity in the transeverse direction (dyn/cm)acceleration due to gravity (cm/se&

thickness of the lymphangion wall (cm)length of a lymphangion (cm)pressure (dyn/cm)external pressure on the lymphangion (dyn/cm*)flow rate (cm/sec)rate of flow into the terminal lymphatics (cmj/sec)valve resistance (dyn/cm)resting radius of a lymphangion (cm)radi;ll co-ordinate (cm)strainthreshold of strain required to initiate an active contractiontissue pressure (dyn/cm2)time coordinate (set)duration of contraction (set)refractory period (set)interstitial fluid pressurepressure head due to gravity (cm)

SplllOlS

I density gjcm3p viscosity [g/(cm-set)]4 coefficient of active contractility (dyn/cm)4,,1 stress developed due to an active contraction (dyn/cm)D uor hoop stress (dyn/cm)n osmotic pressure (dyn/cxr?) = E,SubscriptsA arterial end of the capillaryV venous end of the capillary.

INTRODUCTION

The development of transplantation procedures and increased clinical problems involv-ing edema and cancer have created a growing need for detailed studies of the lymphaticsystem due to its role as a component in the circulatory system and as a component

* Present address: Postgraduate Research Physiologist, University of California, Cardiovascular ResearchInstitute. San Francisco. CA 94143. U.S.A.


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182 NARENDER P. REDDY THOMAS A. KROUSKOP and PAUL H. NEWELL JR.

in the immunological system Cl]. It is now generally agreed that the lymphocytes whichbecome sensitized at the site of transplantation travel through the afferent vessels tothe regional lymph node where they initiate an immunological response. Thus resto-ration of lymph flow from transplanted tissue is important for the survival of a graft;also restoration of lymph flow between the graft and the host is important in understand-ing the homograft mechanisms [27. Moreover, the lymphatic system is known to playa decisive role in the dissemination of cancer. Malignant cells escape the tissue of originand enter the blood vessels through the lymphatic system [3).

The lymphatic systems role in the circulation of protein molecules and lymphocytesthroughout the extracellular space also makes it important in several common maladies.Filariasis, involving lymphatic obstruction which leads to lymphedema in the associatedregion, is one of the most common diseases found in tropical countries. The impairmentof lymphatic valve function may also lead to lymph stasis and subsequently to edema.With the accumulation of edema fluid and protein molecules, the region involvedbecomes highly susceptible to infection [4]. Another important pathological conditionwhich centers on the lymphatic system is the thoracic duct fistula which leads to lossof protein molecules and fluid [3].

The composition of lymph fluid reflects the composition of interstitial fluid fromwhich it is derived. There is growing evidence that most of the bodys enzymes aretransported from the tissue of origin into the circulatory system via the lymphatics.In addition, most of the metabolic waste products are transported through the lym-phatics. Moreover, as pointed out by Newell et al [IS] the lymphatics play an importantrole in the tissue ulceration and necrosis. Occlusion of the lymphatics leads to theaccumulation of the enzymes and other metabolic waste products in the interstitialspaces, This accumulation then leads to tissue necrosis and ulceration. Thus, in orderto study the pressure effects on soft tissue, it is imperative to understand the mechanismof lymph propulsion.

In recent years, one of the most important trends of research in the sphere of bioen-gineering has been the modeling of physiological systems. Models are conceptual con-structions which allow formulation of hypotheses and in this way stimulate meaningfulresearch. A mathematical model is a conceptual representation of a real physical system.Mathematical models are widely used in todays scientific world due to the ease withwhich they can be used to analyze real systems. The most prominent value of a modelis its ability to predict as yet unknown properties of the system.

A model can also be used as a powerful education tool since it permits the idealizationof processes. It also allows the study of sub-systems in isolation from the parent system.Model studies are often inexpensive and less time consuming than the correspondingexperimental study. Models of a physiological system often aid in the specification ofdesign criteria for the design of procedures aimed at alleviating pathological conditions.

In a recent investigation [6] we have developed a mathematical model of the lym-phatic vessel using the tools of continuum mechanics together with the currentlyaccepted notion of lymphology, and have identified the equations governing the lymphflow in a lymphatic vessel. The lymphatic vessel model is used in the present investiga-tion to develop a network model of the whole body lymphatic system, consideringthe regional lymphatics. The model is then coupled to a previously developed model [7]used to describe the interstitial fluid dynamics and the resulting composite model issimulated on a digital computer to yield pressure and flow patterns along the lymphaticnetwork.

SYNTHESIS OF THE MODEL

The lymphatic system is a complex network of vessels. A lymph vessel consists ofa number of compartments, called lymphangions, which are separated by unidirec-tional valves (Fig. 2). A growing body of evidence now supports the concept that lymphpropulsion in the lymphatic system is achieved by active and passive contractilities


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A computer model of the lymphatic system

Right heart Left h ea r t

t

Vena cava Aorta

I I

Thoraclcduct

ILymphatic

vessels

ICollectinglymphattcs

I

Lymphcaplllarles

I I

Venous Arteriessystem

I I

Venules Arterioles

I I

Capillaries1

I I

Interstltlal f lulds

Fig. 1. Diagrammatic representation of circulatory and lymphatic systems. The lymph capillariesand other vessels which absorb lymph from the interstitial spaces are collectively referred toas the terminal lymphatics. Not shown in the diagram are the lymph nodes which he along

the lymphatics are the centers of immunological response.

of th.e lymphatic walls and lymph nodes. Passive contractility is due to respiratorymovements, rhythmic changes in the volumes of the intestines, spleen and other organsand also due to movement of the limbs and skeletal muscles. Active contractility isdue to intrinsic contractions of the smooth muscle in the walls of the lymphatics andlymph nodes. A number of investigators [8-l 31 have demonstrated active contractilityof the lymph vessels in uiuo and in vitro The frequency of contractions were foundto be independent of the respiratory, intestinal and cardiac rhythms [8]. Recent findingssuggest that a lymphangion contracts independently and therefore, constitutes anautonomous functional element. Active contractions in the walls of a lymphangion havebeen shown to occur if the diameter of the Iymphangion exceeds certain thresholdvalues [9]. The terminal iymphatics are free of smooth muscle and therefore they donot exhibit active contractility [lo].

In a recent investigation [6] it has been shown that a lymphatic vessel can be charac-terized as a chain of lymphagions and that the adjacent lymphagions can be conceptuallycoupled through the continuity and the momentum equations. This analysis can beextertded to the entire lymphatic system by conceptually coupling the lymphangionsat each branching point to form a network model of the entire lymph system.

Yoffey and Courtice [3] extensively reviewed the literature on regional lymph flowsand noted that under normal conditions, the hepatic and intestinal lymph ducts contrib-ute ;I. major portion of the lymph found in the thoracic duct. However, during conditions

1 III

Fig. 2. A lymph vessel.


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84 NARENDER P. REDDY, THOMASA. KROUSKOP nd PAUL H. NEWELL, R.

Fig. 3. The lymphatic system. [ indicates boundary. The pressure at the arterial and venousends of the capillary (P, = 35 mm Hg and PV = I5 mm Hg in the present case) are the boundary

[ conditions. JV Jugular Venous pressure (6 mm Hg) is the proximal boundary condition.

Organs

E

:ef

i?i

:I

mn0

Vessel No123456

89

1011121314151617

18192021

22232425262-I2829

liverstomach and other portions drained by hepatic ductsmall intestinelarge intestineright legright part of the trunkright kidneyleft legleft part of the trunkleft kidneythoracic wallheartlungshead and neckleft handNamehepatic ductvessels from the stomachhepatic ductintestinal ductintestinal ductintestinal ductintestinal duct

left leg lymphaticright part of the trunkright lumbar ductright kidney lymphaticright lumbar ductleft leg lymphaticleft part of the kidneyleft lumbar ductleft kidney lymphaticleft lumbar ductcistemal chylithoracic ductintercoastal lymphaticsthoracic ductheart lymphatic

lung lymphaticbranchio-mediastinal lymphaticthoracic ductcerircal ducthand lymphaticsjugular lymphaticthoracic ductjugular vein

Compartmentsl-89-16

17-2621-3637-5 I52-6162-66

67-106107-I 16117-121122-131132-141142-181182-191192-196197-206207-2 16217217-224225-232233-252253-258

259-266267-270271-274215-279280-257288-293294296297


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A computer model of the lymphatic system 185

such as exercise, the contribution from the other organs increased substantially. In thepresent analysis, lymph flows from the limbs, the trunk, the kidneys, the heart, thelungs, the liver, small and large intestines, the thoracic wall, the cervical vessels andthe thoracic duct are considered. Due to the lack of sufficient quantitative anatomicaldata concerning the lymph capillary network, the present investigation considers onlythe primary lymphatic vessel from each organ. The second and higher generation lymphvessels are assumed to be lumped such that they constitute a series of compartmentsin the primary vessel. Furthermore, it is assumed that all the terminal lymphatics ofeach organ can be lumped into a single compartment of the primary vessel drainingthe organ. Unfortunately, there is no information present in the existing literature regard-ing the microcirculation within a lymph node. This investigation assumes that the flow-characteristics of a lymph node can be treated as a lymphangion within the primaryvessel. Moreover, the ratio of the number of lymph nodes to the number of lymphan-gions is very small and the lymph nodes are localized structures.

The lymph vessels considered in this study are shown in Fig. 3. The present modeldoes not consider the right side of the head, neck, thorax and upper right extremity.These tissues are drained by the right lymph duct which directly empties into the venoussystem. It should be noted that organs, such as the limbs, have superficial and deeplymphatics; however, they are lumped into a single vessel.

In order to facilitate the conceptual handling of the system of vessels and the lymphan-gion:s, each vessel considered in this investigation, and each lymphangion are assignedspecific numbers. The numbers assigned to vessels are shown in Fig. 3. Columns 3and 4 of Table 1 indicate the numbers assigned to the lymphangions which compriseeach lymphatic vessel. The lymphangions listed in column 4 empty into the respectivelymphangions listed in column 6. The last compartment of the thoracic duct is coupledto the venous system. To further facilitate the mathematical handling, each of the organsis identified with a serial number. Table 2 shows the number of the compartment drain-ing #::ach organ. The compartments representing the terminal lymphatics are coupledto the respective tissue spaces through the equations expressing the flow of fluid into

Table 1

j N, B(j-l)+ 1 B(j) mj B(m,-l)+l mj B(mj)

1 82 83 IO4 105 156 10

7 58 409 IO

10 5II IO12 1013 4014 IOI5 516 1017 1018 819 820 2021 6

22 823 424 425 526 827 628 3

917273752

6267

107117122132142182192197207217225233253259267271275280288294

8 316 326 736 651 661 7

66 18106 10116 10121 12131 12141 18181 15191 I5196 17206 17216 18224 20232 20252 24258 23266 23270 24274 28279 27287 27293 28296 29

171762525262

217117117132132217192192207207217233233271267267271294288288294297

4

8 106

10 121

I3

1512

18

21 25820 252

25 21924 274

8

36

26

181

171141

224


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186 NARENDER P. REDDY, HOMAS A KROUSKOP nd PAUL H. NEWELL, R.

Table 2

Compt. draining Vessel drainingOrgan no. the organ organ

k D(k)1 123456789

101112131415

9213167

107122142182197225253259275280

124589

111314161921222526

the terminal lymphatics. Finally, the interstitial spaces are coupled to the blood circula-tory system through the equations describing the capillary exchange and the interstitialfluid dynamics. The complete set of equations are given below.

EQUATIONS OF THE NETWORK MODEL

In the development that follows, the following notations apply:

1. Each lymphangion in the network and each vessel are identified with unique numbers.The words compartment and lymphangion are used interchangeably. All of the com-partments in a vessel are numbered, consecutively, starting from receiving end tothe emptying end of the vessel.

2. Nj represents the number of compartments in the jth vessel.n=j

3. B(j) = 1 Nn.n=l

B j) represents the compartment located at the emptying end of the J vessel.B(j - 1) + 1 represents the compartment at the receiving end of vessel j. Compartmentsofjlh vessel are represented by B(j - 1) f 1, B(j - I) + 2 - - - - B(j) - 1, and B(j)

4. jth vessel empties into vessel mj such that compartment B(j) empties into compartmentsB(mj - 1) + 1 (Fig. 4).

5. At a converging branch, vessel j is fed by vessels j - 1 and Mj such that compartment

B(j - 1) + 1 receives fluid from B( j - 1) and B(Mj).6. The tissue spaces of kth organ are drained by the terminal lymphatics representedby compartment D k).

The assumptions in derivation and the limitation on the usage of the set of equations(l-3) (6) and (7) are discussed in detail in Ref. 6.

The rate of change of radius of a lymphangion can be obtained using the law ofmass action.

Rate of change of lymphangion radius =flowin-flowout

surface area

If lymphangions i - 1 and i are located in the same vessel, i.e. lymphangion i isnot at the receiving end of a vessel, then flow out of lymphangion i - 1 is the sameas flow into lymphangion i. Therefore, for i = B(j - 1) + 2, B( - 1) + 3, B(j) - 1 andB j ), and j = 1, 2, 3, . ., 28

c i_- 25 CQi- - Qil3dt I,

where Qi is the flowout of ith compartment.


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Fig. 4. Notation at branching points.

AI: a converging branch, two or more vessels feed into a single compartment. Inthe present model (Fig. 3), vessels 3, 6, 7, 10, 12, 15, 17, 20, 23, 24, 27 and 28 receivefluid from two feeding vessels;

For i = B j - 1) + 1, b = B Mj) andj = 3, 6, 7, 10, 12, 15, 17, 20, 23, 24, 27 and 28

3_- .jj CQi- + Qb - Qil~df 1,

(lb)

However, vessel 18 has three feeders;

fori=B(j- l)+ l,b=B(Mj) and j= 18

i_ 1dt 1,

- 2~ {Qi-1 + Qb + QB[TI Qi). (lc)

The above equations describe the rates of change of all the lymphangions in thenetwork except those which represent the terminal lymphatics.

In the case of the lymphangions representing the terminal lymphatics, for i = D(k)and k = 1, 2, 3,. . . , 15

gi_- 2 {Qtl, - Qi 1,dt I, (14

where Qtlr is the flow into terminal lymphatics from the kth organ.The rates of change in flow rates can be-determined using Newtons laws of motion

and the constitutive relation for the fluid [6].

Density x Rate of change of flowrate = hydrodynamic force gradient

+ gradient in gravitation forces

- viscous resistance

- valve resistance;

for

i = B(j - 1) + l,B(j - 1) + 2...Bcj) -2,&j) - 1, and

j= 1,2,...28

dQi- = ;- +pi-pi+l +(q-~~+~)}dt , if1 (2a)

,111 he case of a lymphangion (i) located at the emptying end of a vessel (j) whichempties into lymphangion B(mJ, the pressure difference between lymphangion i andlymphangion B(mj) determines the hydrodynamic forces which drive the fluid out of


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188 NARENDER P. REDDY THOMAS A. KROUSKOP and PAUL H. NEWELL JR.

lymphangion i; therefore,

for

i = B(j), b = B(mj) and j = 1,2,3,. . .28

_ di. (2b)The intraluminal pressure of the lymphangions can be determined by using the

momentum balance for the lymphangion wall [14].Intraluminal pressure = extraluminal pressure + (ratio of wall thickness to

radius) x (hoop stress + stress developed due to active contractility).For i = B j - 1) + 2, B(j - 1) + 2, B(j - 1) + 3,. . . , B(j), j = 1, 2,. . . , 28 and for

i = B(j - 1) + 1, j = 3, 6, 7, 10, 12, 15, 17, 18, 20, 23, 24, 27 and 28

pi = Pext,i + 1 {~hoopi + ~.act,l. 34t

Terminal lymphatics do not have smooth muscle on their walls and therefore donot exhibit active contractility [15].

Flow into terminal lymphatics is due to the instantaneous pressure difference betweenthe tissue pressure and i&alumina1 pressure of the terminal lymphatics [15].

for i = D k) and k = 1, 2, 3,. . . , 15

Qt~k Cl yrnphk Tk Pi ) >(4)

where QIlr is the flow into lymphangion i which represents the terminal lymphaticsof organ k; Tk is the tissue pressure of organ k; Clymphkis the filtration coefficientof two terminal lymphatics of kth organ.

The tissue pressure is dynamic and is a function of the interstitial fluid volume. Therate of change of interstitial fluid volume may be expressed as:

Rate of change of interstitial fluid volume = Net rate of filtration into the tissueacross the blood capillary-rate of flow into the terminal lymphatics

for k = 1, 2, 3, .,., 15

d&, _- - Cfap*PAl, + pvrdt - Tk - n,, + fl,,> - Qt~p 54

where Ccapk is the capillary filtration coefficient of kth organ; PA* is the pressure atarterial end of kch organ; Pv, is the pressure at venous end of kfh organ; II,, is theosmotic pressure in the capillary of kth organ; BT, is the tissue osmotic pressure in kthorgan.Osmotic pressure is a function of the protein concentration [16].

n = 133.24 210 C,, + 1600 C& + 9000 Cz, dyn/cm, (5b)

where C,, is the protein concentration in g/ml.The dynamic changes in the tissue protein concentration may be represented byRate of change of protein = net rate of leakage into the tissue across the bloodcapillary wall - the rate of removal via the lymphatics

d&l, 1~ = - WM&~, - C,,,) - Qt ~, Cpr t Jdt VW,

(5c)

The stress developed in the walls of a vessel due to active contractility is a functionof the strain history. The mathematical representation of the stress due active contracti-lity, presented in [6], is given below.


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A computer model of the lymphatic system Ih9

If,

then

si 2 Sthr,i and t > (7i + t,)

Ti = t + t ,

i = 4i l + Si - Sthr,i)Ka.

Here Ka is a constant.

If t -:s TV, hen

(r = yiSitl Ti - t)

act,,tP

where

tp = ;.

If

or il:

Si - S,hr,i and (Ti + tp) > t > Ti.

Si < Sthr,i and t > (Ti + t,).

(64

(6b)

(64

the],.

0.& = 0. (6dl

Where: Si be the engineering strain in the wall of the iIh lymphangion ; Sthr, bethe threshold of strain required to initiate an active contraction in the lth lymphangion;

I$+be the amplitude of the stress developed due to an active contraction when initialfilling is such that Si = Sthr,i; pi be a parameter which denotes the amplitude of stressdeveloped due to an active contraction and is dependent on the initial filing; 7; bea parameter which denotes the time at which the contraction ceases, t, be the durationof contraction; and t, be the refractory period. The parameters cbi and Sthr,i dependon the amount of smooth muscle present in the wall of the ith lymphangion.

In addition, the presence of valves along the lymphatics imposes the constraint that

Qi 2 0 for all i (74

and

Qtlt 2 0 for all k. 7b)

In all, there are 296 lymphangions and hence 296 equations each of type 1, 2, 3,6, 2nd 7. There are 15 organs and 15 equations each of type 4 and 5.

THE BOUNDARY CONDITIONS

The pressure in the jugular vein forms one of the boundary conditions of the presentmodel. The pressures at the arterial and venous ends of the capillaries of each organand the capillary protein concentrations form the second set of boundary conditions.The third set of boundary conditions include the external pressures on the individuallymphangions. These boundary conditions are the input parameters to the presentmodel.

RESULTS

The differential system described above is translated into a FORTRAN program forthe purpose of digital computer simulation. In developing the program reported here,an effort was made to make the program sufficiently general so that it can be easilymodified when further quantitative anatomical data are available, especially with regard


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NARENDER P. REDDY. THOMAS A. KROUSKOP and PAUL H. NEWELL JR.

Fig. 5. Pressure patterns in the thoracic duct lymphangion 250. Active contractions are super-posed over the low frequency respiratory rhythm.

to the number of compartments and their arrangement. The model has been simulated.The parameters, and boundary conditions used in the model together with the listingof the computer program are documented in [16]. Figures 5-14 depict pressure pat-terns and flow patterns in intestinal, hepatic, and leg lymphatics, thoracic duct. andcistera chyli.

Fig. 6. Pattern of flow-out of thoracic duct lymphangion 250.


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~~+--_ .

klo 1.m 2. 00 5. w I . 00 5. 00 6. 00I

7. 00TI HE IN SECUNDS

D. 00 n 00 10. 03

Fig. 7. Pressure pattern in cisterna Chyli.

DISCUSSION

Il is extremely difficult to measure pressures and low rates of flow in small collapsibletubes such as the lymphatics. Consequently, most measurements of flow have beenmade with cannulated lymphatics. The cannulation procedure distorts the normal mic-roanantomy and pressure flow relationships that exist in the intact lymphatics. Hallet al [S] recorded pulsatile pressures in the intestinal, hepatic and lumbar lymphaticducts and the efferenct lymphatic duct of popliteal lymph nodes of sheep. It shouldbe noted that the pressure pattern differs from one lymphangion to another in the

Fig. 8. Pattern of flow-out of Cisterna Chyli.


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NARENDER P. REDDY THOMAS A. KROUSKOP nd PAUL H. NEWELL JR.

-_-._--t__-_-t--_-+-__-+ Ia 1 00 2.00 3.uo LOO 5.00 6.00 7.00 e.00 o.orl 111.00

TIWE IN SECBNDS

Fig. 9. Pressure pattern in intestinal duct lymphangion 66.

same lymphatic vessel. Hall and his associates recorded the pressure pattern by cannulat-ing the lymphatic against the direction of flow. In their experiments they used cannulaswith i.d. ranging from 0.8 to 1.0 mm and outer diameter ranging from 1.20 to 2.00 mm.

The cannulas were tied to the lymphatics. These procedures cause trauma to the vessels.Furthermore, it should be pointed out, from the experiments of Smith [12] and Mis-lin [9] that when a lymphangion is sufficiently dilated there is an increase in the pressuredue to increased intrinsic contractility. Therefore, the pressure recorded by them maynot be assumed as absolute. Instead, the concepts derived from their recordings andthe pressure patterns are important. Considering the various factors discussed above,

Fig. 10. Pattern of flow-out of intestional duct lymphangion 66.


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Fig. 11. Pressure pattern in leg lymphatic (lymphangion 106).

the pressure patterns in various lymphatic simulated in present investigation are inaccord with the experimental observations by Hall et al. The frequencies observed inthe present investigation are in the neighborhood of those reported. The frequency

of contraction depends on the amount of filling and also on the rate of fluid flowout of the lymphangion; the insertion of a cannula of the size which is comparableto the size of the vessel may alter the frequency of the intrinsic contraction.

The flow rates from the intestinal duct are shown in Fig. 9. The average lymphflops from the vessel (1.48 ml/min), computed in the present analysis, are consistentwith the average flow rates for sheep reported by Hall et al., when their results are

Fig. 12. Flow pattern in leg lymphatic (lymphangion 106).


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NARENDER P. REDDY, THOMASA. KROUSKOP nd PAUL H. NEWELL, R

0 1.00 2.00 4s.00TI%?IN &J&S

6.00 7.00 a.w WJO 1o.m

Fig. 13. Pressure pattern in hepatic duct lymphangion 26.

extrapolated to the human. It should be pointed out that Hall and his associates havereported the how rates after averaging over a period of hours. In their experimentsthe lymph was collected in graduated clyinders. The flow rates recorded by Hall et

al. should by no means be assumed to be the normal physiological flows with thevessel intact. On the other hand, in the absence of better instrumentation techniques,their observations should be commended. The average flow rate from the intestinalduct decreased by 50% when the intestinal motility is removed. This data is in agreementwith the observations of Lee l-171 that intestinal lymph flow increases when the intestinalmotility is established.

Fig. 14. Flow pattern in hepatic duct lymphangion 66


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Quantitative anatomical data of the lymphatic system are scarce. In particular, thereis very little information in the present literature regarding the dimensions of variouslymphangions. Moreover, the diameter of a lymphangion varies with intraluminal pres-sure, and it is difficult to obtain the exact unstressed or the resting diameter of avessel. Although, the lymphangion at the proximal end of a vessel may be smallerin diameter than the lymphangion at the distal end of the vessel, a uniform unstresseddiameter and a uniform length are assumed over the entire length of the vessel inthe present analysis.

The stress developed due to an active muscular contraction is dependent on theamount of smooth muscle present in the wall of a lymphangion. Nisimaru [13] studiedthe relative amounts of smooth muscle, and leg lymphatics have higher muscle quantity.However, the absolute muscle mass per unit of lumen diameter decreases with the sizeof the vessel. High muscle quantity has been reported by Mislin for the intestinal lym-phatics. Nisimaru further observed that the thoracic duct has higher muscle quantityin the upper part than in the lower part. These findings suggest that a lymphangionhas lower amount of smooth muscle than the lymphangion into which it empties. Simi-larly, a lymphangion develops lower stress during active contractility than the lymphan-gion into which it empties. Mislins experiments reveal that a pressure of 2 mmHgis necessary in the mesenteric lymphatics to initiate an active contraction. His findingsalso suggest that the threshold varies with the vessel. The distribution of smooth muscleobserved [13] suggests that a lymphangion has a lower threshold than the lymphangioninto which it empties. This increase in the threshold from the upstream end to thedown stream end of a vessel further facilitates the contraction co-ordination betweenthe iymphangions. A parametric study of the present model revealed that the gradientsin thresholds are important for the efficient propulsion of the lymph along the lym-phat its.

This analysis assumes that the duration of a single contraction and the durationof the refractory period are the same for all the lymphangions regardless of their locationwithin the body. In actual case this may not be true for all the lymphangions. Moreover,thesl.: durations may be significantly altered by several pharmacological agents [ 1 S].Experiments are in progress in several laboratories which are aimed at studying theeffects of various neurotransmitters. It is further assumed that all the lymphangionshave: the same elastic modulus in the transverse direction. The relative amount of elastictissue present in the lymphangion may vary from one lymphangion to another.

Furthermore, a uniform intervalvular distance is assumed for all the lymphangionsin a vessel. We hope that the present analysis would stimulate lymphologists to conductexperiments aimed at obtaining quantitative anatomical data which would make it poss-ible for better estimation of the parameters involved in lymph propulsion.

SUMMARY

A computer model of the whole body lymphatic system is developed consideringonl , the major lymph vessels and using a previously developed model of a lymphaticvess,el. The terminal lymphatics of each organ are lumped into a single compartmentof the primary vessel draining the organ. In order to facilitate conceptual handlingof the system of lymph vessels, each lymph vessel and each lymphangion (compartmentwithin the vessel) are identified with specific numbers. A set of differential equationsare presented which describe the flow and pressure patterns along the lymphaticnetwork. The jugular venous pressure, the pressures at the arterial and venous ends

of the capillaries of each organ and the external pressures on the individual lymphan-gions form the set of boundary conditions of the present. model. The mathematicalmodel is then translated into a series of Fortran Statements for the purpose of digitalcomputer simulation. Simulation results are consistent with the available experimentaldata. A parametric study of the model revealed that gradients in thresholds of distention,along the vessel, are important for efficient lymph propulsion along the network.


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I i, NARENDER P. REDDY, THOMASA. KROUSKOP nd PAUL H. NEWELL, R

Acknowlrdgemenrs-This investigation was partially supported by the Social and Rehabilitation Servicethrough a Grant (SRS-23-P-55-823-6). This work is a part of a Ph.D. dissertation submitted by the seniorauthor to Texas A & M University.

REFERENCES

1. P. Malek and J. Vrubel. Lymphatic system and organtransplantation, Lymphology 1. l-22 (1968).2. P. R. Koehler. Injuries and complications of the lymphatic system following renal transplantation, Lympho-lo


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A & M University. Dr. Newell has served Texas A & M University as associate dean of engineer-ing and as head of Biological and Industrial engineering. was also a professor of bioengineeringin the departments of Physiology, Physical Medicine and Rehabilitation at Baylor College ofMedicine. He is currently the president of New Jersey Institute of Technology.

Dr. Newell is a member of many professional organizations and heads a number of technicalcommittees. He has received a number of awards for excellence in engineering education and

has been credited with numerous publications.

A computer model of the lymphatic system.pdf

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