Mathematical model of the initial lymphatics

MICROVASCULAR

RESEARCH

12, 121-140 (1976)

Mathematical

Model

of the Initial

Lymphatics

S. ELHAY AND J. R. CASLEY-SMITH Department of Computing Science and Electron Microscope Unit, University of Adelaide, South Australia

Received January 19,1976

A mathematical model of the initial lymphatic cycle has been constructed. This used the observed dimensions, pressures, etc., and the normal physical laws, including the anomaly (for which there is much in vitro and in vivo evidence) that macromolecules exert very substantial effective colloidal osmotic pressures across pores considerably larger than the molecular diameters. The hypothesis, that it is the last factor which actually causes the normal inflow of fluid into the initial Iymphatics, was shown to be quite compatible with the parameters (which can be varied over very wide ranges) and the physical laws. Sets of differential equations describing the model were obtained, and solved, for both the filling and the emptying phases. The model permitted a much better appreciation of the ways in which certain groups of parameters determine the values assumed by the different variables, of which the most important are the concentration of the lymph and its volume. It also pointed to the existence of, and the importance of, an intermediate phase (when the open junctions become close) between the filling and emptying phases. It was found that wherever the mode1 gave results which have also been determined by experiment, the model’s predictions were in substantial agreement with theactual findings. These included a ratio of -3 between the mean concentrations in the lymph and the tissues, a large water leakage back into the tissues during the emptying phase, and concentrated lymph being pumped into the remote colIectors. The model also showed and explained that if the lymph concentration or volume is altered from the “normal” values at which the cycle can continue indefinitely, then there are intrinsic mechanisms in the cycle itself which have the property of causing negative feedback and returning the values of the variables to normal. Thus, there are homeostatic properties in the cycle itself, as well as in the blood vessels, the tissues, and in the segments of the collecting lymphatics.

INTRODUCTION

For a number of years it has been known that material enters the initial lymphatics via their large lengths of open endothelial intercellular junctions, and is retained within them during tissue compression by these open junctions temporarily becoming “close” ones [reviewed Casley-Smith, 1973, 1976a, b). One of the main problems in understanding the functioning of these vessels has become: What force causes the material to enter them? It used to be held that there was a small hydrostatic pressure gradient directed from the tissues into the lymphatics (reviewed Casley-Smith, 1973, 1976a-c; Rusznyak et al., 1967; Yoffey and Courtice, 1970). The discovery that the hydrostatic pressure in the interstitial tissue (THP) is probably usually negative (less Copyright 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain

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than atmospheric; reviewed Guyton, 1969; Guyton et al., 1971) has made this possible mechanism an extremely improbable one, except in a few regions and in oedema, when it is known that the THP is positive. The rationale of this argument is that the micropipettes almost certainly correctly measure the intralymphatic hydrostatic pressures (LHP), as -0 cm of water. In the tissue channels these instruments are likely only to measure the total tissue pressure (TTP), the sum of the THP and solid tissue pressure (STP), since the channels are considerably narrower than the micropipette tips. Since much evidence points to THP being negative, we conclude that the hydrostatic pressure gradient is almost certainly normally directed out of the initial lymphatics (i.e., is negative). Thus, it can not be responsible for the inflow of material into the vessels. Other possible alternatives, suction from the contracting collecting lymphatics (Reddy et al., 1975) and active transport via vesicles, etc., are also very unlikely (reviewed Casley-Smith, 1973, 1976a-c), because the collecting lymphatics are very unlikely to be able to exert a suction force [pressure measurements (Zweifach and Prather, 1971) support this], and the vesicles are essentially passive structures. The only other possible force which has been suggested is the effective colloidal osmotic pressure (ECOP) of the proteins inside the lymphatic (Casley-Smith, 1970, 1972a, b; reviewed Casley-Smith, 1973, 1976a-c). In essence, it is suggested that during compression of the tissues and the lymphatics, the greatly increased TTP is transmitted to the lymph; the resulting increased LHP is greater than the THP and ultrafilters the small molecules out via the close junctions (which are closed to proteins but not to water). When the compression relaxes, the high ECOP of the lymph causes water to enter the vessels, dragging more protein with it, and rediluting the lymph. There is then an intermediate phase (whose significance was first demonstrated in the present work) when the filled vessel rests in equilibrium with “close” junctions, not open ones. There is a growing amount of evidence in favour of this hypothesis. The fundamental and apparently paradoxical point-that macromolecules can exert considerable ECOP’s across pores much larger than the molecules-has been established in vitro (Casley-Smith and Bolton, 1973). It has been suggested (Casley-Smith, 1975) that this is because the macromolecules only slowly diffuse outwards towards the lesser concentration, so that they are dammed back by the fluid flowing swiftly inwards, thus forming a “virtual membrane” at the inner end of the pore. This steep concentration gradient is similar to that occurring at a true semipermeable membrane, when it would produce a COP. Thus, the ratio ECOP/COP is much greater than that which would be predicted by the reflection coefficient as usually calculated from the relative molecular and pore dimensions. It has been objected that theoretically this ECOP still could not cause protein to move up a concentration gradient (Michel, 1974) but this objection has been shown not to be valid (Casley-Smith, 1975; Perl, 1975) and there are good theoretical reasons for believing that solvent drag can indeed cause a net inflow of protein, albeit at a concentration lower than that outside the vessels. This has been confirmed in vitro (Casley-Smith, 1972b, 1976d). Much in vivo supporting evidence has come from the study of a hypothesis, involving a similar mechanism for protein inflow, relating to the functioning of fenestrae on the venous limbs of capillaries (reviewed Casley-Smith, 1973, 1976a-c; Casley-Smith and Sims, 1976).

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The essential part of the lymphatic hypothesis, the considerable concentrating of the proteins in the lymph in the initial lymphatics, has been directly shown by a number of methods in different sites and species (Casley-Smith, 1970, 1972b, 1976~; CasleySmith and Sims, 1976; Jonsson et al., 1970; Witte, 1975). The concentration has also been shown to be reduced during filling, and greatly increased during compression (Casley-Smith, 1976~). The mean lymph concentration, over all the cycle, is -3 times that in the tissues (Casley-Smith, 1976~; Casley-Smith and Sims, 1976). Indirect measurements of protein concentrations from adjacent collecting lymphatics (Rusznyak et al., 1967) and by calculation (Johnson and Richardson, 1974) also confirm this. Another indirect method, estimating the “efficiencies” of the initial lymphatics (i.e., the ratio of the fluid leaving them for the collecting vessels, compared with that entering them), has shown that they are likely to be very leaky indeed (Casley-Smith, 1976e). Only -0.1-3 % of the inflow is passed on (except in the intestine where the proportion is -10-40 %); the rest of the fluid is presumably regurgitated back into the tissues. There is also additional evidence relating to the concentrations in the collecting lymphatics. According to the hypothesis, the lymph which is expelled into the collecting lymphatics must be considerably more concentrated than the fluid in the tissues. It has therefore been suggested that the lymph in these vessels will be rapidly diluted because its elevated ECOP will cause water to enter via the close junctions in their walls (Casley-Smith, 1973, 1976a-c; Casley-Smith and Sims, 1976). Although it has been questioned whether these vessels are permeable, there is abundant evidence that small molecules, but not large ones, readily traverse their walls (reviewed Zoc. cit.). It should, however, be noted that this dilution will occur in the “remote collecting lymphatics” (i.e., those outside the region so that they are not compressed at the same time as the initial lymphatics); redilution cannot occur in the “adjacent collecting lymphatics” [i.e., those within the region), since the LHP’s in them will be similar to those in the initial vessels and much greater, during compression, than the THP’s in the adjacent tissues. Hence, the raised concentration will be maintained in these vessels. Indeed, it has been shown (Nicolaysen et al., 1975; Casley-Smith, 1976~) that the lymph in adjacent collectors is about the same concentration as the mean of the initial lymphatics. The redilution in the remote collectors has been demonstrated in different sites and species and by different techniques (Casley-Smith, 1976~; CasleySmith and Sims, 1976). Since the lymph in the adjacent collectors remains fairly concentrated, this overcomes some objections to the hypothesis based on measurements along such vessels (Nicolaysen et al., 1975; reviewed Casley-Smith, 1976a-c). The present paper reports a further test of the hypothesis. A mathematical model of the whole initial lymphatic cycle has been constructed, using the normal laws of physics relating to flow and diffusion, and what quantitative data we have of the initial lymphatics. This model was made for two reasons: firstly, the evidence for the hypothesis would be severely weakened if it were shown to be probably physically impossible; secondly, it has been shown elsewhere (Casley-Smith, 1976b; Casley-Smith et al., 1975a, b) that this combination of physical laws and quantitative morphology, when combined with the results of physiology, often yields very important information about the functioning of a system which cannot be obtained from any of the single approaches, nor by a simple combination of just two of them. Unlike the previous

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models (lot. cit.), that of the initial lymphatic cycle is quite complex. Two types of junctional slits are involved (one of which has a variable width); the ECOP of the lymph varies with the continually varying concentration of the lymph. A cycle is involved, so the variables must return to their initial values at the end of it, as well as being compatible between its various phases, and the length of the cycle is variable. To this must be added our uncertainties about the values assumed by almost all of the fundamental parameters which must be used in the model. Because of the varying ECOP, sets of differential equations had to be used (different for each part of the cycle). A computer was used to get numerical solutions to these, and to search through the many possibilities of the uncertain parameters. In what follows, we will first discuss the details of the three phases of the cycle, then discuss the values to be assigned to each parameter, and finally we will examine the results from the model. The Filling Phase(Figs. 1 and 4)

We assume incompressible flow, in a continuous fluid without any local sources or sinks (i.e., that there is continuity). Directions into the vessel are called positive. We will consider only proteins and water, since the other small molecules will equilibrate

dcJ

Filling-phase

FIG. 1, A drawing of the model (not to scale) during the filling phase. The further end is supposed to be closed, while the nearer one joins with the remote collecting lymphatic. Water and protein can be seen entering via the length of open junction. The “close” junctional lengths are shown as narrow slits, while the tight junctions are shown as impermeable.

very rapidly across these relatively large slits. Denoting C(z, t) as the concentration of protein at time t and distance z, along the depth of the open junction the rate of increase of protein in the lymph because of water flow in through the open junctions is

The term depth (d) is used to refer to the distance from the ablumal to the lumenal aspect of the junctional slit, width (w) is the distance between the two cells, and length (I) is the length of some portion of junction along or around the circumference of

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MODEL

the vessel; V,(t) is the volume of fluid passing through the open junctions at time t. The rate of protein loss from the vessel by diffusion is

where D is the diffusion coefficient of the proteins. Thus, the rate of change of protein inside the slit at point z is

$ [C,(t) V(t)] = C(z,t) 7

+ Dw, lo%

9

where V(t) is the volume of the lymphatic at time t, and C,(t) the concentration inside it. The boundary conditions are C(d, t) = C,(t) and C(o, t) = C, (the concentration in the connective tissue) and z = 0 lies on the exterior surface of the vessel. Equation (3) has been derived and solved in a number of different ways (reviewed Casley-Smith, 1976f; Curry, 1974). The only real difference between this system and those others is that here C,(t) is a variable. Since the depths of the slits are quite large relative to their widths (vide infra, Poiseuille’s equation (in slit form; Perl, 1971) will apply (Duncan et al., 1960). Hence, the rate of change of the volume of the vessel will be the sum of that occurring via the two types of slits : q

=&

t 0

[THP - LHP + LECOP - TECOP]

w,“lc [THP - LHP + LCOP - TCOP].

+ 12qd,

The subscripts o and c refer to the open and “close” junctions; the superscript f refers to the filling phase; q is the viscosity of water, and qt that of the tissue fluid containing protein; the third type of junctional region, the tight junction, is assumed to be impermeable. It can be seen that water, but not protein, will enter via both types of junctions. Later it will be shown that the close junctions are probably negligible during the filling phase, their contribution being only -l/500 of the total inflow. Since the possible flow of water through the cells themselves has been estimated to be only from 0.15 to some 4 times that through the “close” junctions (reviewed Effros, 1974; Perl, 1973), it can be seen that this flow too, if it occurs will also be negligible in this phase. The net pressure (NP) across the slit can be estimated using Eq. (3.5) of Landis and Pappenheimer (1963) : NP = THP - LHP + K,,[284(C, - C,) + 21OO(C; - C;) + 12,2OO(C: - C:)]

(5)

where the concentrations are in g ml-l, and K,, is a coefficient which is the ratio between the ECOP across a given slit (varying with the slit’s cross section in relation to that of the macromolecules) and the true COP across a semipermeable membrane; for close junctions K0 = 1 and the ECOP = the full COP. It should be noted that K0 is much greater than the reflection coefficient, cr, as usually defined, as mentioned earlier.

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Let Eq. (4) be dV(t)/dt = Kl *NP(t), ignoring the second term, and Dw,l,/d, = K2, then solving Eq. (3) for C(z, t) we have the equations of the filling phase: I [Ci(t)exp (-T/K2)] - Ct. $ (C,(t) v(t)) = y [exp (-F/K2)l]

I

dV,(t)/dt = Kl *NP(t) with the initial conditions: V(0) = V(t) at t = 0

C,(o) = Cl(t) at t = 0.

I‘ ,

(6)

The Intermediate Phase (Figs. 2 and 4) It can be seen from Fig. 5 that during the filling phase, C, tends to a value such that NP = THP - LHP + LECOP - TECOP = 0. At this stage there would be no inflow

Intermediate -phase

FIG. 2. During the intermediate phase the openable part of the junctional length has become “close” and the filled vessel is now in equilibrium, with water and protein neither entering nor leaving it.

of fluid. In fact, however, C, is still > C,; hence, there will still be an outwards diffusion of protein. This will lower C,, thus causing NP to be negative, and lymph to start flowing out. In addition, the removal of the inflow of fluid will prevent the formation of the “virtual membrane” (described earlier) at the inner end of the slits even earlier than this point, thus greatly reducing the LECOP and again making NP negative. We see, then, that as C, tends towards its lowest value there will be a tendency for fluid to flow out of the lymphatics via the open junctions. However, this will not go on for long since the very flexible endothelial cells, which form the inner portions of the slits, will be forced outwards (by the fluid flow) against the outer cells, which are supported by the interstitial tissues. Hence, the flap valves will seal to protein, although not to water. Thus, the outwards passage of protein will be prevented. Across these now much narrower slits the proteins in the lymph will again exert their ECOP’s. Hence, NP will again become = 0 and the outflow of water will be

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prevented. (In fact, there may even be a positive NP since the ECOP will now be rather larger, for the same C,, than across the wider slits; Casley-Smith and Bolton, 1973.) Hence, water will again enter the vessels. This, however, will be short-lived, for as soon as the inlet valves reopen, the ECOP will fall and protein will start leaving the vessels again. Thus, an equilibrium will be attained. While this intermediate phase could have been deduced earlier, in fact it was only during the formulation of the present model that its existence and significance was able to be seen clearly. Its significance is that it shows that the filled lymphatics will remain passively in this state until something causes the TTP to rise. From the point of view of the animal, this means that while Ci must necessarily be somewhat greater than C,, the uncompressed, filled lymphatic will not therefore empty via the outwards diffusion of its protein contents (with a consequent loss of its water) as has been suggested previously (Casley-Smith, 1972b, 1973). Obviously this is of great advantage in the functioning of the vessels. In the model this intermediate phase is held to occur as soon as there is outflow via the open junctions, i.e., as soon as dV,/dt < 0. At this point, C,, V, and the total protein contents of the initial lymphatic (I’= C,) are held constant until the end of the intermediate phase. The Emptying Phase (Figs. 3 and 4) There are two outflows from the initial lymphatic to consider. First, let us consider that through the various intralymphatic valves into the remote collecting lymphatics. Since all the adjacent collecting lymphatics (i.e., those in the region) are exposed to similarly raised TTP’s, the increased LHP will cause flow along them into the “sink” of the remote collectors (external to the region), whose LHP’s are much lower than those of the adjacent collectors. Therefore, when estimating this outflow it is necessary to consider the whole lengths of the initial lymphatics plus the adjacent collecting vessels (Casley-Smith, 1976e) and their mean dimensions. In fact, they all have approximately

Emptying-phase

FIG. 3. The emptying phase commences when the total tissue pressure rises and compresses the lymphatic. This forces water out of the “close” junctions, which are sealed to protein. Both water and protein are pumped into the remote collecting lymphatic (lower left).

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elliptical cross sections, which we will approximate same areas. Then, from Poiseuille’s law, dV,o dt

by circular cross sections with the

= -ew4 (L”P _ RCHp) 3 8vllL/2

(7)

where P’,,(t) is the volume of lymph pumped onwards into the remote collector, whose hydrostatic pressure is RCHP; L/2 is the mean length of the system of initial and

FIG. 4. Cross sections (not to scale) of the openable junctions during the three phases of the initial lymphatic cycle. The various pressures are shown, as are the variations in them. (The LECOP and LCOP are shown dashed because they vary during the first and last phases.) During the filling phase, water and protein are shown entering via the open junction; during the emptying phase, water is shown being expelled via the now-“close” junction; during the intermediate phase, the vessel is in equilibrium with the tissue fluid.

collecting lymphatics in the region, while R is their equivalent mean radius with W(tN4 = ( w)/L~n)2, and vi is the viscosity of lymph. L/2 is used in place of L because the TTP is applied all along the vessel rather than just at its distal end. Hence, dV,o dt

= -(V(t))’

(LHP - RCHP) 47cQL3

= -&(

V(t))” (LHP - RCHP).

(8)

The second outflow from the lymphatics is via the “close” junctions, both those which are continually “close,” and those which are open duringfillingand “close” during emptying. There is no doubt also that some outflow through the “close” junctions of the adjacent collecting lymphatics, but since they are relatively much less frequent than the initial lymphatics, we shall essentially ignore them, just considering the regional system as a whole complex of initial lymphatics, of length L and volume V(t). If water flow also occurs through the cells themselves (Effros, 1974; Perl, 1973), this would increase dV,(t)/dt, but ions would have to traverse the junctions to equilibrate with the water. If this flow is present, it would reduce the length of the emptying phase. It would be equivalent to increasing w: or w,. As mentioned later, the values used for these widths would easily be “enlarged” by the amount needed to allow for this flow, which might about halve the calculated length of emptying, from the figures given (Zoc. cit.).

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MODEL

As in the filling phase, the rate of change in the volume of the vessel, due to ultrafiltration, VU, will be :

dvdt) -=-dt

[

(%)310+ w; 2, (LHP - THP + TCOP - LCOP) = -K4 NP(t), do

d,

I

121

(9)

where w: is the width of the openable junctions during the emptying phase, and the full COP’s are used (not the ECOP’s) since the slits are now assumed to be impermeable to protein, i.e., K. = 1 in Eq. (5). From Eqs. (8) and (9) we can obtain a set of equations describing the emptying dV (t> i.e., the rate of change of the concentration of the lymph phase (using2 = -v ‘2 a*; is determined by the rate of water loss via the junctions; the flow to the collecting vessels does not affect the lymph concentration). Thus,

W,(t) v(t)1 = -K3( dt

V(t))” C,(t) (LHP - RCHP)

dV(t) = dV,W + dVu(t) = -K3( V(t))’ (LHP - RCHP) - K4 NP(t) dt

dt

C,(h)= Cl,

dt

t,

where VI and Cl1 are the values taken during the intermediate phase, and is the start of the emptying phase, and dV/dt = dVJdt + dV,,/dt.) To obtain numerical values for the systems, Eq. (6) was rewritten in terms of new variables: x = Ci V/C, V(O), y = V/V(O) and z = tKJV(O), which are dimensionless. Equation (10) was rewritten in terms of the variables : x = C, V/C, VI, y = V/V,, r = - t,)K3 VI, which are also dimensionless. Both systems were solved on a CDC computer using a fourth-order Runge-Kutta method. It is important to briefly discuss what happens to the water which is ultrafiltered back into the tissues. As Taylor et al. (1973) and Taylor and Gibson (1975) have pointed out, if the water merely remains outside the junctions it will dilute the protein which enters at the next filling phase so that there will be effectively no lymphatic pumping of protein. This, however, only applies to the fluid which passes out of the openable junctions; as will be seen later, quite a considerable amount passes out via the permanently “close” junctions (and perhaps the cells themselves), so that it enters tissue well removed from the openable junctions and will not have this diluting effect. Apart from this, it appears very unlikely that the fluid would simply stay near the openable junctions. It is probable (reviewed Casley-Smith, 1976a-c; Casley-Smith and Sims, 1976) that the protein is contained in channels (~0.1 ,um; Casley-Smith et al., 1975a) through the interstitial tissue. These have been observed ending at open junctions (Collan and Kalima, 1975). The (THP - TECOP)‘s of the solution in them must be in equilibrium with those ofthe surrounding gel phase which forms their walls. This, while impermeable to protein, is readily permeable to water. Hence, as the water is expelled from the lymphatics it will dilute the proteins, lowering their ECOP’s, and also raising the THP. Thus, it will quickly be moved into the gel phase, into other more distant channels, the tissues in general, and into the blood capillaries.

(t

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It can be seen from Fig. 5 that during this phase, C, tends towards an asymptote, when NP = (LHP - THP + TCOP - LCOP) = 0, i.e., when the concentration inside the vessels rises so high that no further ultrafiltration occurs since the forces across the junctions are in equilibrium. Unlike in the intermediate phase, however, the whole

FIG. 5. An individual initial lymphatic cycle, using the chosen parameters (see text). The time scales are different for the filling and emptying phases, and one is not shown for the indeterminate intermediate-phase. C, and C, are shown x 100, i.e., in g (100 ml)-‘. The volume is shown relative to V(O), while the product (the protein concentration of the lymphatic) is shown x 50.

vessel is not in equilibrium, but continues slowly to lose fluid (and protein) as the lymph is expelled to the remote collectors. Valuesfor the Parameters

One of the main problems in formulating this model has been our uncertainty about many of the values which should be used to correspond to what actually occurs in the many tissues of the many species. While some are known with some accuracy for certain special tissues in a few species, many are uncertain. Therefore, while we had to finally decide on certain specific values, we also tried various extreme ones. The extremes for which the model performed well (when most other values were the specific ones) are indicated in brackets after each figure. In fact, by choosing mainly extreme values for the other parameters we could usually make the model satisfactory for considerably more extreme conditions than those we cite here, but there seemed little point in seeing just how widely we could vary a parameter once we had reached what we could conceive to be its likely upper and lower limits in life. The length (L) of the initial-lymphatic/adjacent-collector plexus was taken to be 10 cm (O.l-IOO), its semimajor axis 40 pm (10-IOO), and its semiminor axis, at its

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minimum volume, 1 pm (0.1-10). Thus, the model vessel was conceived to have a elliptical cross section whose minor axis varied during filling and whose length (from the depths of an organ to its periphery) included all the network of initial lymphatics and adjacent collectors. In fact, there would be no difficulty in using a network as a model, except that it is easier to think of a single vessel, and knowledge of the dimensions of the initial lymphatics per 100 grams of tissue is scanty (Casley-Smith, 1976e). In these circumstances it seemed that modelling a single vessel was sufficient. From the work of Dobbins and Rollins (1970), one obtains that 2.5 % of the junctional length is openable (0.5-25) 10 % is “close” (5-25), and 88 y0 is tight (50-95), assuming their uncertainties were only between “close” and tight junctions, and not about open ones. The total junctional length for muscle blood capillaries in the dog leg is -2000 cm/cm’ of capillary surface (Casley-Smith et al., 1975b) and the initial lymphatic endothelial cells appear to have considerably greater interdigitations in the plane of their surface (Altshul, 1954; Jdanov, 1965; French et al., 1960; Collins, 1969). Using the extra cell periphery and the figure for capillaries yields -5500 cm of junctional length per cm2 of lymphatic surface. The surface area of the model is 0.16 cmz; thus, I,, = 21 cm, 1, = 86 cm, and the tight junctional length is 880 cm, for the model. We assumed the lumenal to ablumenal depth of the open junctions (d,) was 500 nm (loo5000) and their mean width (w@was 100 nm (ZO-lOOO), which appear to be reasonable values from electron microscopy. During the emptying phase (and the intermediate one), we assumed that the widths (w;) became 8.2 nm (420), i.e., similar to the mean along “close” junctions in blood capillaries (see below). For the “close” junctions, we used the (NJ:/&), which had been found by measurement in blood capillaries and which allows close approximation to the actual results (Casley-Smith et al., 1975b), viz. 1.1 nmz (0.5-10). (This is of course a mean for the whole depth of the slit and implies a mean w, = 8.2 nm along the 500-nm slit, but it must be emphasised that this is only a mean and, in fact, for part of the depth the width has a mean of only -6 nm). The viscosity of the inflowing tissue fluid (ylt) was assumed to be 0.01 poise (i.e., slightly higher than water, because of its protein content). From these parameters :

K = 1

t

= [4.2 x 10-r’ + 9.5 x lo-r3]/0.12 = 3.5 x 10e9inc.g.s. units. From this it can be seen that the filling of the initial lymphatics, in This is contrary to the conclusions Azzali (1965), but occurs because conductivities. Using a diffusion coefficient (D)

close junctions will play no significant part in the spite of the relative paucity of the open junctions. of Dobbins and Rollins (1970) and Ottaviani and of the importance of the w3 term in the hydraulic for proteins of 6 x lo-’ cm set-r yields :

K2 = Dw, lo/do = 2.5 x 10e6, in c.g.s. units.

For the filling phase, we used a THP of -5 cm of water (-10 to $5) (Guyton, 1969; Guyton et al., 1971) and a LHP of 0 cm (-5 to +5). The concentration of protein in the tissue fluid (C,) was taken as 0.02 g ml-l (0.005-0.07), while the maximum concentration in the lymphatics (C,(O)) was taken as 0.1 g ml-’ (0.02-0.15). The TECOP and LECOP

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were found in Eq. (5) using K0 = 0.8 (OS-l), which would apply over most of the range of w,’ (Casley-Smith and Bolton, 1973). For the emptying phase we used (LHP - THP) = 55 cm of water (10-200). This value is perhaps the least certain in any of the parameters; it was adjusted so that C, came to an asymptotic value at the end of the emptying phase equal to that used at the start of filling. It was essential to adjust one of the parameters and this was arbitrarily chosen to be the one. In life, as pointed out later, there are negative-feedback mechanisms which ensure that the system is stable, but to incorporate them directly would have meant a much more complicated model. Thus, the present model may be looked upon as the mean of several runs. The LHP has not been measured during compression of a tissue. [The measurements of Wiederhelm (1969) unfortunately do not apply because the bat’s wing is abnormal (Casley-Smith, 1975, 1976a).] Values of -15 cm of water have been measured in contracting remote collectors (Zweifach, 1972; Zweifach and Prather, 1971), but these were only in the isolated, uncompressed mesentry which is a tissue that suffers little compression compared with a muscle. No other measurements appear to have been made during, e.g., muscular compression and respiration. Indeed, the movement of the tissues would make it rather hard to do so. This stricture applies with even more force to the THP. No such measurements have been made of THP during muscular contraction at all, nor would they be easy to do without artefactual compression of the capsules or wicks, Guyton (1975, personal communication) has measured TTP and THP variations in subcutaneous tissue under externally applied compression and has found that the increase in THP was ~80% of TTP. There are estimations of TTP in muscle obtained by the needle and balloon methods, which show TTP can rise to -1400 cm of water, although it is often rather less (reviewed Hesse, 1971; Ludbrook, 1966; Guyton el al., 1971; Sylvest and Hvid, 1959). Since this would represent initially an increase in STP which was only transmitted to the THP, it would seem reasonable that the TTP should increase a certain amount above THP, as fluid was forced through the interstitial tissue to regions of least resistance. The TTP is probably transmitted nearly completely to the lymph, greatly raising the LHP. We have arbitrarily chosen a value of 55 cm for (LHP - THP) but, as the range shows, many values are compatible with a reasonable modelling of the hypothesis. The remarks in the last paragraph apply almost as much to the value of (LHP RCHP), except that we know (Zweifach, 1972; Zweifach and Prather, 1971) that RCHP is only a few centimeters, although in the larger vessels it can be higher (Hall et al., 1955). Again, we have arbitrarily chosen 50 cm of water (l&200), although if LHP-TTP, it would be higher at times, Using wi = 8.2 nm, as mentioned earlier, and q, = 2 x 10m2 poise (because of its high projection content) and qw = 7 x 10v3 poise (because the flow out of the junctions is protein-free), we obtained : K3 = 1/4n~ L3 = 4.0 x 10e3, and

K = M)3 - 4 10+ wf d,1,II 12rl 4

= [2.3 x lo-l3 + 9.5 x 10-13]/0.084 = 1.4 x 10-rl, in c.g.s. units.

INITIAL

LYMPHATIC

MODEL

133

From this it can be seen that the outflow via the “close” junctions (w,) is greater than that via the ones which were previously open (~3, as was mentioned earlier. It shows that much of the ultrafiltered water passes out junctions other than those through which protein will next pass in. (To this may well be added water passing out via the cells; see above.) The parameters mentioned above, and the values of the K’s, allowed us to use the sets of Eqs. 6 and 10. We started at the beginning of the filling phase, stopping this when the volume started to decrease rather than increase. This marked the start of the intermediate phase, when the now-“close” junctions prevented escape of protein (and hence, of water). The emptying phase was started an arbitrary time after the start of the intermediate phase. It was stopped when the values of C,, V, and the protein content (C,. V) had all returned to within 10% of their original values at the start of filling. This was more than close enough since there is a negative feedback (see below) which corrects a discrepancy in any of these values over repeated cycles. (C, had often achieved an asymptote, made to equal C,(O) by adjusting (LHP - THP) as mentioned above.) Figure 5 shows one such cycle, using the assumed values mentioned earlier. The other cycles are represented by the ranges given for the parameters. An Individual Initial Lymphatic Cycle

As has been mentioned, Fig. 5 and situation 1 in Table 1 represents the results obtained with the particular parameters we chose as representative. The filling-phase is very short relative to the emptying phase, 6 set vs 78 set, for all the variables in the emptying phase to return to within 10% of the initial conditions. (They are within 1.6 % at 100 sec.) The intermediate phase is of an indeterminate length. In fact, altering the various parameters (especially the two values of w, and of wc, and the relevant pressures) alter the times very considerably, and in so doing alter the slopes of the graphs, although not their basic shapes. In the filling phase, it can be seen that C, falls rapidly at first then continues to fall (dC,/dt < 0) until the NP across the junctions = 0; here this occurs at C1 = 0.0345 g ml-‘. Meanwhile, Y has been increasing, rapidly at first when C, is high, then more slowly, so that dV/dt > 0, but -+ 0. Their product, the protein content of the lymph, usually reaches a maximal value and then falls a little as the effect of diffusion outweighs that of the very diminished inflow, until this latter stops altogether. At this point, the intermediate-phase commences. In the emptying phase, C, + an asymptote as the outflow via the junctions falls to zero, when the NP across them is = 0. Here the value of (LHP - THP) has been adjusted so that this occurs at C, = C,(O) = 0.1 g ml-l. The S-shape of the graph of C, is not always evident since it depends on the relative values of K3, K4, and the HP’s and COPS. C1 usually reaches its asymptote before V and V. C, return to their initial values, again depending on the precise parameters used. V always falls (dV/dt < 0), quickly at first when C, is low. Eventually the outflow through the junctions stops, but Vcontinues to fall because of the flow to the remote collecting lymphatics. The protein content of the lymph (V. C,) behaves similarly and continues to fall, although d( I/. C,)/ dt may be >O, early, with some parameters. The total volume lost during emptying was 4.85 x lo-’ ml, of which 1.67 x lows was pumped into the collecting lymphatics and 3.18 x 10d5 was ejected via the junctions.

Condition

FOR

a “Normal” values of V(0) * Abnormalities

1.51

1.51

1.01

12.0

8.0

12.0

3.46

3.46

3.46

3.46

3.46

3.46 3.46

3.46 3.46

6.13

5.52

9.20

3.68

7.66

7.36 4.60

6.13 4.90

was defined as the cycle, using the set of parameters and C,(O). were produced by varying V(O) nt Cl(O) by 20%

1.26

1.01

8.0

1.51 1.26

10.0 8.0

12.0

1.26 1.01

of filling

in the text, their“norma1”

given from

6.1

5.4

9.0

3.6

7.6

7.3 4.5

6.0 4.9

-

INITIAL

Time to end of filling (ECl

IN THE

Volume of bwh (x 105 ml)

End

OF LYMPH

Concn of hwh (x 102 g ml-j)

VOLUME

Initial volume of bmph (x 105ml) r V(O)1

of filling

AND

10.0 10.0

[Cl(O)1

Initial concn of lymph (xl02 g ml-‘)

Start

CONCENTRATIONS

Normal Reduce I’(o) Increase V(0) Reduce C\(O) Increase Cl(O) Reduce V(0) and Cl(O) 7. Increase V(0) and Cl(O) 8. Increase l’(0) and reduce C,(O) 9. Reduce I’(0) and increase Cl(O)

1. 2. 3. 4. 5. 6.

VALIJES

values.

for which

9.98

9.99

9.84

10.0

9.93

9.95 10.0

9.98 10.0

Concn of lymph (x 102 g ml-j)

1 AFTER

After

AND

the initial

1.28

1.21

1.54

0.941

1.42

1.39 1.09

1.28 1.13

lymphatic

1.67

1.35

3.65

0.569

2.60

2.41 0.917

1.67 1.05

ABNORMAL~

cycle

repeated

3.18

2.96

4.01

2.17

3.64

3.56 2.59

3.18 2.72

STARTING

continuously,

0.34

0.31

0.48

0.21

0.42

0.40 0.26

0.34 0.28

always

63

58

78

43

71

70 52

63 54

Volume passing to collecting lymphatics as proportion of total Time when leaving Cj > 0.09 vessel g ml-’

100 set of emptying

NORMALS

Volume pumped to Volume Volume of collecting nut via lymphatic junctions lymph (x 10s ml) (s 105 ml) (x 105 ml)

LYMPHATICS

TABLE

returning

8.40

7.04

16.7

3.32

12.4

11.6 5.04

8.48 5.70

Protein pumped to collecting lymphatics (x 10’ 9)

volume

increased

CYCLE

cycle

cycle

cycle

1

to the initial

with

close to 1 Identical

Very

See 3 for next

See 2 for next

See 3 for next

Final volume reduced See 2 for next cycle

Final

THE

Remarks

FOR

to very close

CONDITIONS

3:

g

E Y

s

i

2

F

INITIAL

LYMPHATIC

MODEL

135

(Again these relative volumes are very dependent on the parameters chosen.) The amount of protein pumped into the collectors was 8.48 x lo-’ grams, i.e., the mean concentration of the lymph entering them was 5.15 x 10d2 g ml-‘, which is -2.5 times C,. This protein also represents the net amount entering the initial lymphatic during the filling phase. Rather more entered at first (8.77 x lo-’ g), but a part (0.29 x lo-’ g) diffused out during the end of this phase. When this is compared with the amount of water entering during filling (equal to that lost during emptying), it can be seen that the mean concentration in the inflowing fluid (1.8 1 x 1Oe2g ml-‘) is less than C,. So indeed are the concentrations entering the vessel at any individual time, as is demanded by theoretical considerations (Casley-Smith, 1976f). The mean C, over the whole cycle is obviously very dependent not only on the various parameters, but also on the length of the intermediate phase. The ratio of the mean C/C, has been found experimentally to be -3 (Casley-Smith, 1976~). In the model the ratio is -3.6 if the intermediate phase lasts 0 set, 3.0 if it last 46 set, and --f 1.7 as this phase + m. The question naturally arises, what if the filling or emptying phases are not allowed to proceed to their ends, as shown here, but are interrupted earlier, or in the case of the emptying phase is permitted to last longer? In the last case, as shown in Fig. 5, C, remains constant, but V and V-C, tend slowly to zero. Both this and the early interruptions are equivalent to either V or C, (or both) not returning to their initial values at the end of the emptying phase. (Early interruption to the filling phase would cause this also.) It will be shown in the next section that there is a negative-feedback mechanism inherent in the cycle, which will tend to restore these variables to their “optimal” values over a few cycles. Homeostatic

Mechanisms

(a) Contained within the initial lymphatic cycle. The initial lymphatic cycle itself contains a very powerful mechanism which will readjust the two crucial variables, V(O) and C,(O), so that they tend to return to their “optimal,” normal values. Here “normal” is defined as the values which, given the parameters we used, allow the cycle to continue indefinitely, each time with I’ and C, at the end of the emptying phase returning to the values they had at the start of the filling phase. (Obviously, if one of many of the parameters were altered, V(O) and C,(O) would have different optimal values.) In Table 1 are shown the results of altering either or both V(0) and C,(O) upwards or downwaRds by 20% of their normal values. In every case, the cycle produces values of V and C, at the end of the (arbitrarily chosen) emptying phase of 100 set which are considerably closer to the normal values than those with which the cycle started. The effect of increasing C,(O) or V(0) on the filling phase is to increase the volume at the end of filling and to increase the time taken to achieve this; the reverse occurs when these initial conditions are reduced. In neither case is the final value of C, altered. In fact, C, is always almost normal at the end of the whole cycle; it is V which alters to accommodate the altered initial conditions. If C,(O) or V(0) is increased: the final volume is increased; the volume pumped to the collecting lymphatics (both absolutely and as a proportion of all the fluid leaving the vessel) is increased; the volume escaping via the junctions is increased (absolutely, although its proportion of the whole fluid outflow is reduced); the amount of protein pumped to the collecting vessels

136

ELHAY

AND

CASLEY-SMITH

is increased; and C, tends more slowly towards its asymptote. If C,(O) or V(U) is reduced, all these variations are reversed. It should be noted that in both filling and emptying (assuming they go on for long enough), C, will always tend to the same asymptotic values (3.46 gram/100 ml for filling and 10.0 grams/100 ml for emptying, with our chosen parameters), because these are determined only by C,, LHP, and THP. If V(0) is the initial condition which is altered, one cycle is enough to approximately halve the amount of disturbance. If C,(O) is what is altered, the first cycle corrects this completely, but at the expense of a grossly altered V. This is -15 % away from normal when C,(O) alone is altered by 20 %, and -25 % away when both C,(O) and V(O) are altered by 20% in the same direction. However, these alterations in V(O) are largely corrected in the next cycle. Thus, alterations in V(O) tend to be corrected in the cycle in which they occur. Alterations in C,(O) are, in the first cycle, transformed to rather smaller alterations in the V(O) of the next cycle, in which these are then corrected. In either case, one or two cycles produce very substantial corrections. If both C,(O) and V(O) are altered in opposite directions, the final result of the first cycle is almost identical with a normal one. Why do these corrections occur? Alterations in C,(O) are transformed into alterations in V(O) because C, tends to its asymptotic values at the end of both filling and emptying. If C,(O) is increased (say), more fluid enters the vessel and, hence, V is raised at the end of filling, and this additional volume is less easily removed during emptying. Therefore, an alteration in C,(O) is converted to one only in V(O), since C, tends towards its constant asymptote. An increase in V(O) also increases the volume at the end of filling, because [if C,(U) is constant] this means that there is more protein contained in the vessel at the start of filling and, hence, more fluid will be caused to enter the vessel. During emptying, however, the extra volume means that C, will not increase so greatly when a certain volume of fluid is expelled via the junctions. Hence, more fluid will be expelled than normal, thus reducing this excessive volume. (This process will continue as a progression, tending closer to the normal V(O) with each succeeding cycle.) It can thus be seen that the initial lymphatic cycle, by itself, will tend to correct any deviations from the optimal values for the existing parameters. Thus, if the cycle is interrupted, e.g., before normal filling or emptying have occurred, the situation will be rapidly restored to normal in a few cycles. In fact, if the emptying phase is interrupted, it will be similar to situation 8 in Table 1 which produces very similar conditions to those with a normal cycle (situation 1). Similarly, allowing the emptying phase to continue for longer than “normal” will produce situation 2, which tends to be corrected in the next cycle. (b) Contained in the tissues. While the intrinsic negative feedback in the initial lymphatic cycle is of great interest, we must not forget the other homeostatic mechanisms which reside in the physical conditions of the tissues, and which have been known for many years. These include the alterations of THP with the amount of fluid removed from the tissues (Guyton, 1969; Guyton et al., 1971; Taylor et al., 1973), the lowering of LHP in filling during oedema by the pull of fibrils attached to the lymphatic endothelium (Casley-Smith, 1973, 1976a), and the variation in the amounts of protein entering the vessels according to its concentration in the tissues. This last is not quite

INITIAL

LYMPHATIC

MODEL

137

as simple as it appears, because as the protein concentration is lowered, more fluid is caused to enter the vessels since the TECOP is lowered. In fact (Casley-Smith, 1976f), there is an optimal value of C, (approximately 3 of C,, depending on LHP - THP), above and below which less protein enters the vessel. For most normal circumstances, C, < Cl/2 and the amount of protein removed from the tissues varies with C,. Variations in C,, or in the amount of fluid in the tissues (and hence of THP and, in oedema, of LHP), will correspondingly alter the amounts of proteins or fluid entering the vessels during filling. (Variations in C, will also cause variations in fluid accumulation and THP.) As mentioned, eventually the cycle will correspondingly adjust the values of C,(O) and V(0) to the optimum for these altered conditions, thus correspondingly altering the amounts of protein or fluid pumped to the remote collecting lymphatics. This adjustment in the cycle will itself be occasioned by the variations in the amounts of these two substances entering the lymphatics, as determined by the new parameters. In addition, alterations in the amounts of protein or fluid pumped during any one cycle will cause inverse changes in the amounts of these in the tissues awaiting removal in the next cycle, with corresponding alterations in C, or THP. The alterations will assist the intrinsic mechanisms of the cycle to return C,(O) and V(0) to their optimal values. We thus have an interacting series of homeostatic mechanisms. First, we have alterations to the interstitial tissue itself; these cause alterations to the amounts of material entering the initial lymphatics; finally, we have alterations in the initial lymphatic cycle. Actually these three negative-feedback systems are only part of an even greater series: homeostatic mechanisms in the blood capillaries are well known, and Mislin (1972) has shown that the individual segments in the collecting lymphatics adjust their contractile force to vary with the amount of their filling. At each stage, the mechanisms work to restore the tissues to normal, and the alterations in each successive stage are caused by those both in the preceding and succeeding ones.

CONCLUSION The mere fact that a mathematical model accords with the known facts and with a hypothetical mechanism does not of itself imply that this mechanism is correct. It does, however, lend weight to the evidence in favour of the hypothesis, just as the model’s failure to accord would seriously detract from it. We believe, therefore, that the success of this model does in fact support the hypothesis. If the hypothesis is assumed to be correct, the model is of great value in allowing us to understand the details of the mechanisms which allow it to function. The occurrence of the intermediate phase, the relative importances of the open and “close” junctions in the filling and emptying phases, the ways in which various groups of parameters determined the values of the different variables, and the presence and explanation of the intrinsic negative feedback, were all things which were discovered from the model. We have emphasized that the actual results from the model (e.g., the lengths of the different phases) are greatly dependant upon the parameters chosen. Some of these are fairly well established, others are more like guesses; all of them vary considerably from site to site and species to species. The construction of the model has emphasized the need for much more

138

ELHAY AND CASLEY-SMITH

quantitative data, but the wide ranges over which it has been found to operate that it will do so under all the in vivo situations which are likely to arise.

APPENDIX

show

NOMENCLATURE

C = concentration; C, = concentration in lymphatics; C, = concentration in tissue channels; C(z,t) = concentration at a distance, z, along the depth of a junctional slit, at time t; COP = colloidal osmotic pressure; d= depth of a junctional slit, from lumenal to ablumenal surfaces; D = diffusion coefficient of plasma proteins; ECOP = effective colloidal osmotic pressure, across an open junction; HP = hydrostatic pressure; I = length of junctional region, in the plane of the lumenal surface; L = length of model initial lymphatic, to remote collecting lymphatics outside organ; LCOP = colloidal osmotic pressure of lymph; LECOP = effective colloidal osmotic pressure of lymph; LHP = hydrostatic pressure inside initial lymphatic; NP = net pressure, of both HP’s and ECOP’s (or COP’s) across a junction; R = equivalent radius of initial lymphatic; RCHP = hydrostatic pressure in the remote collecting lymphatics outside organ; STP = solid tissue pressure; TCOP = colloidal osmotic pressure in tissue channels outside initial lymphatic; TECOP = effective colloidal osmotic pressure in tissue channels outside initial lymphatic; THP = hydrostatic pressure in tissue channels outside initial lymphatic; TTP = total tissue pressure (= THP + STP); V= volume of initial lymphatic; V, = volume passing in through close junctions during filling-phase; V, = volume passing in through open junctions during filling-phase; V, = volume being pumped to remote collecting lymphatics during emptying-phase; Vu = volume being ultrafiltered back into tissues during emptying-phase; w = width of junctional slit; w,’ = width of openable junctional slit during emptying-phase; IV: = width of openable junctional slit during filling-phase; g, ,v = variables defined in the text; z = distance along dfrom outside to inside of initial lymphatic junction; q= viscosity; T = related to time, as defined in the text. Subscripts c= I= o= p= t= u= 1=

Relating to close junctional slits; relating to the lymph; relating to openable junctional slits; relating to lymph pumped to remote collecting lymphatics during the emptying-phase; relating to the tissues outside the initial lymphatics; relating to the water ultrafiltered out of the initial Iymphatics during the emptying-phase; at the start of the emptying-phase.

ACKNOWLEDGMENTS We are very grateful to Ms. R. Altman for the drawings, and to the Australian Commission for support.

Research Grants

INITIAL

LYMPHATIC

MODEL

139

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