J Bmmrclm~~.
lY77.
Vol.
IO. pp
807-814.
Pergamon Press.Pnntedm GreatBntam
MATERNAL, PLACENTAL BLOOD FLOW: A MODEL WITH VELOCITY-DEPENDENT PERMEABILITY* F. F. ERlANt Department
of Mechanical and Industrial Engineering, Clarkson College of Technology, Potsdam, NY 13676, U.S.A. and
S. CORRSIN and S. H. DAVIS Department of Mechanics and Materials Science, The Johns Hopkins University. Baltimore, MD 21218, U.S.A. Abstract-As a first step towards mathematical modeling of the maternal blood flow in a human placental circulatory unit, the “villous tree” containing the fetal capillaries is represented as a continuous, deformable porous solid whose local permeability increases with increasing local flow speed. The use of this direct ad hoc dependence allows simple treatment of the large distortions which are observed. Results on quasi-steady flows with “Darcy Law” pressure drops are reported here.
INTRODUCTION The placenta is an organ of chemical exchange between fetal and maternal circulations. Maternal blood is brought close to fetal capillaries within placental sinuses (i.e. cavities) called “cotyledons”. The capillaries are housed in flexible, bush-like structures called “villous trees”, and the maternal blood spasmodically spurts from arterial orifices into the cotyledons. It oozes among the branches and twigs of the “‘villous trees”, diffusing oxygen, nutrients and wastes in appropriate directions through the villous membranes surrounding the fetal capillaries, The maternal blood returns to the venous system through, orifices in the same surface as the arterial jets. Langman (1969). Bloom and Fawcett (1968), Gruenwald (1966), Freese (1966), Ramsey (1959, 1960, 1969 and 1971); Ramsey et al. (1963 and 1966); Metcalf et al. (1955); Whitmore (1968); and Aherne and Dunhill ( 1966) provide anatomical and physiological descrip tions, which are sometimes divergent in minor details. Figure 1 is a qualitative sketch of a typical circulatory unit, adapted from several of the foregoing references. One minor divergence between published anatomical descriptions is on the location of the arterial orifices relative to the villous trees. Some observers place a typical arterial jet so that it spurts into a relatively branch-free space, while others place it so that it spurts into a region densely populated with well ramified and flexible branches. In any case, it appears that each cotyledon has typically several arterial jets, and usually more venous drainage outlets. Research has been done on overall placental mass transfer
(see. for example.
Lardner,
man, 1972), but little is known of the detailed flow and mass transfer processes. The goal is to develop a plausible mathematical model of the maternal blood flow within a cotyledon, since such information is required for detailed study of the mass transfer. The only theoretical hydrodynamic analyses appear to be those of Donaldson (1972) and Yoshinaga (1972).$ An arterial jet Reynolds number can be estimated at roughly 20 based on average speed (see below), so inertia is expected to be important in this part of the flow. However, the Reynolds numbers of the flows past individual villi, based on their diameters and on interstitial velocities of the oozing blood, are likely to be unity or smaller. It seems. therefore, that a small-Reynolds-number analysis can serve as a useful first approximation. This approximation, together with replacement of the villous tree by a continuous porous medium (a representation suggested by Donaldson), and restriction to quasi-steady flow solutions, characterize the present formulation. The maternal blood is treated
MBILICALVEIN MBILICALARTERIES
CHOBlON
1975; Middle
* Rccc4rw/ I Scptcrnhc~r I976 t On leave at The Johns Hopkins University during the principal phase of the work. $ This MS. thesis appears to have a mathematical error, judging from the numerical results.
MATERNAL SIDE Fig. 1. Schematic
807
, sketch of primate/human anatomy.
placental
808
F. F. ERIAN,S. CORRSIN and S. H. DAVIS
as a Newtonian fluid (density p t l.O6gm/cc; viscosity p I 0.04 P). an approximation well-established for channels wider than a few red cell diameters. Typical inter-villous distances have been estimated at 30-50 pm, typical terminal villous diameters at 40 pm (Aherne and Dunhill, 1966). The number of circulatory units. in a placenta is cu. 25. Although there is disagreement, the total number of arterial jets seems to be roughly 200. An estimated 500 ml/min of maternal placental flow gives an average of 2.5 ml/min per arterial jet. With an estimated orifice radius r of 0.03 cm, the jet Reynolds number is R, E pu,(Zr)/p o 20. 0, is the averaged flow speed. The Reynolds number must be considerably larger at the peak moment of a spurt, and it is, of course, zero during part of a “cycle”. There appear to be no published data on the relative durations of spurts and shut-downs for a typical arterial jet, but Ra&ey’s X-ray motion pictures of a monkey placenta* suggests a “duty cycle” (i.e. relative time during one period that each arterial jet flows) of less than 0.2. Since the present analysis is quasi-steady, that number is relevant as a warning that the accelerations may at times be appreciable, and to indicate that the maximum jet Reynolds number may be on the order of 100. Static pressure data of Coldeyro-Barcia (1957) indicate that the mean maternal arterial pressure is ca. 1OOmm Hg, and the average intervillous pressure is ca. I5 mmHg. Ramsey (1959) mentions maternal pelvic venous pressure of cu. 8 mm Hg. Without knowing exactly where Coldeyro-Barcia’s arterial pressure value occurs relative to the arterial jet exits, it is assumed to occur just at the exit; hence, it is a local boundary value for the quantitative hydrodynamic boundary-value problem. MATHEMATICALMODELS 1 A single circulatory unit is represented as a twodimensional rigid square box, 1.8 cm on a side, filled with a deformable porous solid [the villous trees(s)]. For simplicity, the geometry is chosen symmetrical, with a single arterial exit at the center and two venous drains, one on each side, as indicated in Fig. 2. Additional arterial exits and venous drains may be * Shown at a lecture at The Johns Hopkins University several years ago. t The most rational derivation of continuum model (volume-averaged) equations for fluid flow through a porous solid may be that of Saffman (1971). : It should bc noted in passing that “slip” occurs at fixed walls in this kind of model because the Darcv Law is a
first-order differential equation. Physically, the-blood does not slip, but this no-slip effect extends only a few pore widths into the medium. Thus, when a porous solid has substructure fine enough to allow a continuum representation, a solid wall “boundary layer” may be ignored. Brinkman (1949) has shown how a no-slip condition can be included, by adding an ad hoc, second-derivative, viscous term to Darcy’s Law.
added to the system at arbitrary boundary locations. The extension to a three-dimensional. axisymmetric configuration is reasonably straightforward. As approximate equations describing the behavior of local volume-averaged velocity u(X) and pressure P(X) fields (see, for example, Bear, 1972). the commonly accepted, negligible-inertia limit called the “Darcy Law” is used. X is spatial coordinate. Also, for simplicity, the porous solid is taken to be isotropic, so the permeability K is a scalar function: IL(X)
=‘- Kf
and the mass conservation v.u
VP,
equation is = 0.
(2)
U is velocity local average over volume “elements” containing both fluid and solid, the velocity of the solid being zero in this steady flow case; P is local volume-average pressure. It is defined only in the fluid phase, but the full volume element, including the solid part, is used in defining the average. This latter operation is equivalent to specifying that the pressure is zero in the solid.? Equation (1) may be viewed as the definition of K. It seems clear that in future, moresophisticated models, the local directedness of the villous trees (both hydrodynamic and elastic) will dictate a non-isotropic permeability. As written, equations (1) and (2) constitute a set of linear partial differential equations for _V and P when K(X) is prescribed. Since the villous trees are considerably deformed by maternal blood flow, K should, however, depend on local velocity. For simplicity, it is supposed that K depends only on local speed, so that K = FclU(xIl.
(3)
Equation (1) is then non-linear. Of course, a more accurate representation could be expected to involve not only the magnitude of the velocity, but also its direction, as well as its gradient. The boundary conditions are as follows: (a) the component of velocity normal to the impermeable parts of the walls is zero;: . (b) the static pressures are prescribed at the arterial jet (source) and at the venous drains (sinks). The geometry is indicated in Fig. 2. ANALYSESAND SOLUTtONS Linear cases
As a preliminary illustration to compare with the non-linear case, the permeability field K(X) is first prescribed. The equations are put into dimensionless form by using the “height” Y of the box as the characteristic length, and the average arterial exit velocity o0 as the characteristic speed. Then the dimensionless variables are urn:. oo,
x=x; y
PY P’p[i,;
ksF.
K
(4)
Maternal. placental blood flow Axis of symmetry )\‘\\\\\~\\‘\“\\\\\\\\\\\\\ .uc\-.\\\\\\\\\\\\\\.~~,,..... $ =O.VP.n=O inis breo unit normal)
/
Two-regions
model
bo/mdories,
:
kh
k, in remoi,ning
0
I
II
Cl
.
Porous
medium
;
for
0
* 3
througFt
the
differejlt Eqs.
i
domain
k =‘k(x)
I
ond
17 :
with
k defined
follows: (2-regions
model)
k -_I constant ’
k=‘kf&f(lyll /
! i
I !
!
(p40)
port
jegions
models’os
/
II p 12
Eqs. 13 6 14 Eq.
in rentrol
(
(p.100)
b=lO)
Fig. 2. The geometry of the mathematical model of a single cotyledon. Linear and nonlinear cases operator) is equal to the time rate of increase of k-’
Thus,
(5) as “seen” by an observer travelling with the local
V.&f = 0.
and 1 VP = - k(y)u.
Equations (5) and (6), together with the boundary conditions outlined above, belong to a well-studied class of boundary value problems. especially in hydrology [see Bear (1971)]. In the absence of an explicit analytical solution, the principal ditticulties are only computational. As a matter of genera1 interest, in two or three dimensions, equations (5) and (6) can be transformed into u -= 0, (7) k(Y)
vx1 I
volume-averaged velocity. Needless to say. k is not a convected quantity. Since this model is two-dimensional, it is convenient to introduce the stream function $, defined by &I s*
V.g = 0 is thus automatically satisfied. When the forms in equation (10) are substituted into equation (7). there follows an equation for $:
The boundary values come from the fact that the walls are streamlines, so constant t+G-valuescan be assigned there, with appropriate jumps across the by forming the curl of equation (6). and into source and sinks. 1 With the solution to equation (11) in hand. equa-V'P = Cu.Vl k(y) (8) [ tion (8) is a Poisson equation for p: a “Poisson equation” for p (after u has been determined), by forming the divergence of equation (6). Alternatively, the latter operation gives The specifications of II/ and p or their derivatives V-[k(.u)Vp] = 0. (9) along the boundaries will complete the formulation of the problem. It may be noted that the right side of equation Without loss of generality, the value $ = 0 may (8) (the “source” term of this Poisson equation) is the be assigned to the streamline which starts at the convective derivative of the flow resistance. In other source, runs upward along the centerline. then along words, the difference between the pressure at any the upper boundary, down the sides and along the point and the average pressure in an isotropic. infinilower boundary to the sink. The value of $ on the tesimal neighborhood (the “meaning” of the Laplace
1
F. F. ERIAN,S. CORRSMand S.
810
H. DAVIS
00 8
(0 I
i 0. (0 39
-2;53
0I1
i
39.721
4i .54 +
0.
39 -60
3c
0. (0 36
1 .. ..._
I
TiFig. 3. Streamlines and constant pressure lines in linear case with uniform permeability. Upper numbers are lo4 $ and lower numbers are 4 x 10” P. streamline which connects the source to the sink directly is then assigned a value relative to zero, calculated from the arterial flow rate. For the present analysis, the dimensionless stream function value along the boundary from the source to sink is chosen to be $ = I,& = 8.0 x lo-“. The conditions on the pressure are more complicated. Based upon the measured arterial and venous pressures mentioned earlier, the dimensionless pre-ssure at the source is approx. 40 x 104, and at the sink it is 4 x 104. Along all the boundaries no values of pressure can be independently assigned because the velocity at the boundary sets the tangential derivative via Darcy’s Law. Except at the source and sinks, the normal derivative of pressure at the boundary is zero because normal velocity there is zero. Given a function k, the linear cases described above have been solved numerically by iterative relaxation methods. The two-dimensional field was divided into 30 x 30 grid points. The criterion for convergence in the numerical calculations is that the sum of absolute differences between function values at all grid points from two consecutive iterations can be made sufficiently small. The plotted results show streamlines and isobars superimposed. Darcy’s Law, equation (61, indicates that the velocity vector is everywhere parallel to the pressure gradient, so that the streamlines must be perpendicular to the isobars.
Constant permeability case
In the simplest case, k = constant, equations (11) and (12) reduce to Laplace equations: V2$ = 0,
(13) (14) and p becomes the velocity potential function. The resulting boundary value problem can, of course, be solved analytically by conformal mapping but, since this case is a kind of “standard” to contrast with the more relevant ones, it has been solved numerically. The resulting flow net is shown in Fig. 3. The degree of orthogonality of the streamlines and isobars gives one indication of the accuracy of the numerical solution. Most of the flow activity occurs in the lower third of the region. Therefore, if mass transfer were included, it would take place relatively locally. If the source flow penetrated farther, however, the whole volume would be exploited more effectively. This latter fact suggests the utility of a relatively villous-free region situated over the maternal artery (the source) and, in fact, such a tendency is observed. The second linear case (below) describes the consequences of such a region, with prescribed k(s). Parenthetically, it should be remarked that fluid inertia would give greater jet penetration, even with k = constant. vp
= 0,
Cases with prescribed (non-unijbrm) permeability
Solutions of Darcy flows through media of prescribed non-uniform permeability are common in
XII
Maternal. placental blood flow
0.
6;
Fig. 4. Streamlines and constant pressure lines in linear case with two-region permeability. Upper numbers are lo4 JI and lower numbers are 4 x lo3 P.
hydrology. The first non-uniform case here is designed to simulate a penetrating arterial jet by dividing the flow region into two zones of uniform but different permeabilities as shown in Fig. 2. The narrow strip along the symmetry axis has the higher permeability, k,, > k,. Alternatively, this can be viewed as a relatively villous-free region. The numerical solution, shown in Fig. 4. is for the case when the ratio of the permeability in the center region k,, to that in the rest of the circulatory unit, k,, is 10. The flow net clearly shows the additional source penetration. Several ratios k,Jk, were tried. As k,,/k,+ 1.0, the how net approaches that in the uniform case; for k,,/k, > 20, the computational errors were excessive.
Steady flow problems with non-uniform, flowdependent permeability are available in the literature. In more than one dimension, the mathematical modeling in this subject for porous solids seems to have started with the work of Biot (1941). who later generalized the analysis to include anisotropy (1955). viscoelasticity (1956). and small amplitude pressure *The particles in a flowing suspension may also be viewed as a porous medium. but it lacks an “undisturbed” reference state, so the vast literature of that subject is not referenced here.
waves (1962). Bear (1972) mentions the work of Scheidegger (1960) and Verruijt (1969).* The foregoing research appears to be limited to linear theory associated with small displacements of the porous solid, but X-ray motion pictures show that placental villous trees are greatly deformed by maternal blood spurts. The correct formulation of the elastic force balance equations for the porous solid undergoing large deformations would require mathematics considerably more elaborate than can be justified by the modest goals of this paper. Therefore, a proper elastic analysis is replaced by an ad hoc assumption for the dependence of local permeability on local volume-averaged velocity field. If equation (4) is written in dimensionless form as k(x) = fllull~
(15)
a qualitatively reasonable form of / is needed. Then the boundary-value problem to be solved consists of equations (1x3), plus appropriate boundary conditions. f should be. finite at the limits Iu 1- 0 and /MI+ 00. It should be a monotonically increasing function of 1~1because larger local velocities will tend to push the branches aside. It should express the fact that branches pushed aside from any location must go to neighboring locations, which suggests at least an integral conservation condition expressing solid-
F. F. ERLAN, S. Coaasm and S. H. DAVIS
812
mass conservation. In the interests of simplicity, permeability will be conserved instead. One of the simplest forms which meets the first two requirements is
fEtul1 =
ykob)
I
1+
8-M 1
+
a.,y,
9
(16)
where k,(x) is prescribed and a, fl, and y are positive constants. This equation gives a non-linear Darcy Law.
uC1+ alull VP = - ykoOCl + (a + B)lull’
(17)
The additional equations are that for mass conservation, equation (5), and an ad hoc “permeability conservation” condition, ss
A~Clulld.4 =
ko0 dA, IS A
(18)
which in effect determines y after a and /I have been chosen. A is the area of the circulatory unit. The boundary conditions are as before. The computing procedure begins with specification of k,(g), and from this “undisturbed” permeability dis* The slight asymmetry in the top central region is due to the essentially constant value of Jr. The plotter usually chooses the location of the first encounter with the function value being plotted. Furthermore, annotations are written every seven grid points in both directions starting from the upper left corner. This results in some asymmetry in the shown values since the field is divided into 30 x 30 grid points. 0.
3
6.54
1.22
tribution, the solution to the corresponding linear problem is obtained. This “zero-order” flow field is the first step in the stream-function-field iteration. A modified permeability distribution is then obtained from equation (16), to be used in the next iteration for the stream function. An increase in velocity causes an increase in permeability, which, in turn, results in another velocity increase. The condition corresponding to equation (16) prevents this situation from degenerating into a mere binary k-field. The initial permeability field was chosen to be k,(x,y) = Ce’LT+by + Cl {em(x-x~) + err,(~+~~)}+y,
(19)
where C, Ci, a, a,, b, and bl are constants and xi is the distance between the source and either one of the two sinks. This form gives high permeability near the source and sinks, where villi-free regions may exist. Figures 5 and 6 show the initial and final permeability fields for this non-linear model, The broken lines in both figures represent the locus of the average permeability values. It is interesting to note the large distortion which results. The flow, as a result of this distortion, is shortcircuited from the source to the sink as shown in Fig. 7.* The resulting flow pattern would reduce the area of effective “contact” between maternal blood and fetal capillaries. This reduction underscores the importance of having a relatively large Reynolds number for the maternal jet. Given the architecture 6.55
3
Fig. 5. Initial (no-flow) permeability field. Nonlinear case.
0.
Maternal, placental blood flow 0.
?
0.48
0.53
0.49
0. 02 +
0.
7
1.53 +
1 .70 +
1 .55 +
0.
c7 +
0.
5
4.15 +
4 .62 +
4.24 +
0.
17 +
0.
30 +
0.
.40 Lb Fig. 6. Final permeability field. Nonlinear case.
Fig. 7. Streamlines and constant pressure lines. Nonlinear case. Upper numbers are lo4 $I and lower numbers are 4 x lo3 P.
F. F. ERUN, S. CORRSINand S. H. DAVIS
814 of a placental
circulatory
essential
to good
for mass
transfer
unit,
utilization can
the jet
penetration
of the full villous
occur
only
with
tree
appreciable
fluid inertia. CONCLUDING REMARKS
The models presented provide a means for assessing one aspect of the blood-villi interaction, namely the role of villi distortion. This prdcedure involves an ad hoc assumption concerning the non-linear permeability, but seems capable of qualitatively describing large deformations
of the villi, a range of behavior
far beyond the capability ing non-linear
elasticity
of a deductive
The predicted villidistortion,
through
ing, develops a fluid flow pattern that fluid inertia
effect, giving jet penetration transport.
would
short-cirtiit-
that would provide
relatively poor diffusive mass exchange. however,
model invok-
theory.
It seems clear,
have the opposite
and so promoting
mass
This effect plus effects of pulsatility are pre-
sently being pursued.
Acknowledgements-This work has been supported primarily by the Engineering Mechanics Program of the National Science Foundation. We should like to thank Drs. E. Ramsey and I. G. Donaldson who introduced us to the problem, M. Yoshinaga, who worked on early phases, and M. J. Karweit, who advised us on computational methods. REFERENCES Aherne, W. and Dunhill, M. S. (1966) Quantitative aspects of placental structure. J. Puth. Bact. 91, 123-140. Bear.J. (1972) Dvnamics of Fluids in Porous Media. Else-
vi&, tiew iort. * Biot, M. A. (1941) General theory of three dimensional consolidation. J. Appl. Phys. 12, 155-164. Biot, M. A. (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182-185. Biot, M. A. (1956) Theory of deformation of a porous viscoelastic anisotrooic solid. J. AD& Phvs. 27. 459-467. Biot, M. A. (1962) Melhanics of defbimation and acoustic propagation in porous media. J. Appl. Phys. 33, 14821498. Bloom. W. and Fawcett, D. W. (1968) A Textbqok of Histology. Saunders, Philadephia. Brinkman, H. C. (1949) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. Al, 27-34.
Coldeyro-Barcia. R. (1957) Physiology of prematurity. Trans. Is? Cor~f..pp. 128-139. Josiah Macy. Jr. Foundation, New York. Donaldson. I. G. (1972) Lecture at The Johns Hopkins University. Baltimore. Freese, U. E. (1966) The fetal-maternal circulation of the placenta-I. Histomorphologic, plastoid injection. and X-ray cinematographic studies on human placentas. Am. J. Obstet. Gynec. 94, 354-360. Gruenwald. P. (1966) The lobular architecture of the human placenta. Bull. Johns Hopkins Hospital 119, 172-190. Langman, J. (1969) Medical Embryology. Williams & Wilkins, Baltimore. Lardner, T. (1975) A model for placental oxygen exchange. J. Biomechanics 8, 131-134. Metcalfe, J., Romney, S. L., Ramsey. L. H., Reid, D. E. and Burwell, C. S. (1955) Estimation of uterine blood Row in normal human pregnancy at term. J. clin. Incest. 34. 1632-1638. Middleman, S. (1972) Transport Phenornenor~ in the Cardiovascular System. Wiley-Interscience. New York. Ramsey, E. M. (1959) Vascular anatomy of the uteroplacental and foetal circulation. In Oxygen Supply to the Human Foetus, CIOMS Symposium, pp. 67-79. Blackwell, Oxford. Ramsey, E. M. (1960) The placental circulation. In The Placental and Fetal Membranes (Edited by Villee, C. A.), pp. 36-62. Williams & Wilkins, Baltimore. Ramsey, E. M. (1969) New appraisal of an old organ: the placenta. Proc. Am. Phil. Sot. 113(4), 296-304. Ramsey, E. M. (1971) Maternal and foetal circulation of the placenta. Jr. J. med. Sci. 140(4), 151-168. Ramsey, E. M., Corner, G. W. and Donner, M. W. (1963) Serial and cineradiographic visualization of maternal circulation in the primate (hemochorial) placenta. Am. J. Obstet. Gynec. 86(S), 213-225.
Ramsey, E. M., Martin, C. B., McGaughey, H. S., Kaiser, I. H. and Donner, M. W. (1966) Venous drainage of the placenta in rhesus monkeys: radiographic studies. Am. J. Obstet. Gynec. 95, 948-955. S&man, P. G. (1971) On the boundary condition at the surface of a porous medium. Stud. appl. Murk Scheidegger, A. E. (1960) The Physics of Flow Through Porous Media, 2nd edn. University of Toronto Press,
Toronto. Verruijt, A. (1969) Elastic storage of acquifers. In FIow Through Porous Media (Edited by de Wiest, R. J. M.), pp. 331-376. Academic Press, New York. Whitmore, R. L. (1968) Rheology of the Circulation. Pergamon Press. Oxford. Yoshinaga. M. A. (1972) A Mathematical Model of Maternal Blood Flow in the Intervillous Space of the Placenta. M.S.E. Thesis, The Johns Hopkins University. Baltimore.
lY77.
Vol.
IO. pp
807-814.
Pergamon Press.Pnntedm GreatBntam
MATERNAL, PLACENTAL BLOOD FLOW: A MODEL WITH VELOCITY-DEPENDENT PERMEABILITY* F. F. ERlANt Department
of Mechanical and Industrial Engineering, Clarkson College of Technology, Potsdam, NY 13676, U.S.A. and
S. CORRSIN and S. H. DAVIS Department of Mechanics and Materials Science, The Johns Hopkins University. Baltimore, MD 21218, U.S.A. Abstract-As a first step towards mathematical modeling of the maternal blood flow in a human placental circulatory unit, the “villous tree” containing the fetal capillaries is represented as a continuous, deformable porous solid whose local permeability increases with increasing local flow speed. The use of this direct ad hoc dependence allows simple treatment of the large distortions which are observed. Results on quasi-steady flows with “Darcy Law” pressure drops are reported here.
INTRODUCTION The placenta is an organ of chemical exchange between fetal and maternal circulations. Maternal blood is brought close to fetal capillaries within placental sinuses (i.e. cavities) called “cotyledons”. The capillaries are housed in flexible, bush-like structures called “villous trees”, and the maternal blood spasmodically spurts from arterial orifices into the cotyledons. It oozes among the branches and twigs of the “‘villous trees”, diffusing oxygen, nutrients and wastes in appropriate directions through the villous membranes surrounding the fetal capillaries, The maternal blood returns to the venous system through, orifices in the same surface as the arterial jets. Langman (1969). Bloom and Fawcett (1968), Gruenwald (1966), Freese (1966), Ramsey (1959, 1960, 1969 and 1971); Ramsey et al. (1963 and 1966); Metcalf et al. (1955); Whitmore (1968); and Aherne and Dunhill ( 1966) provide anatomical and physiological descrip tions, which are sometimes divergent in minor details. Figure 1 is a qualitative sketch of a typical circulatory unit, adapted from several of the foregoing references. One minor divergence between published anatomical descriptions is on the location of the arterial orifices relative to the villous trees. Some observers place a typical arterial jet so that it spurts into a relatively branch-free space, while others place it so that it spurts into a region densely populated with well ramified and flexible branches. In any case, it appears that each cotyledon has typically several arterial jets, and usually more venous drainage outlets. Research has been done on overall placental mass transfer
(see. for example.
Lardner,
man, 1972), but little is known of the detailed flow and mass transfer processes. The goal is to develop a plausible mathematical model of the maternal blood flow within a cotyledon, since such information is required for detailed study of the mass transfer. The only theoretical hydrodynamic analyses appear to be those of Donaldson (1972) and Yoshinaga (1972).$ An arterial jet Reynolds number can be estimated at roughly 20 based on average speed (see below), so inertia is expected to be important in this part of the flow. However, the Reynolds numbers of the flows past individual villi, based on their diameters and on interstitial velocities of the oozing blood, are likely to be unity or smaller. It seems. therefore, that a small-Reynolds-number analysis can serve as a useful first approximation. This approximation, together with replacement of the villous tree by a continuous porous medium (a representation suggested by Donaldson), and restriction to quasi-steady flow solutions, characterize the present formulation. The maternal blood is treated
MBILICALVEIN MBILICALARTERIES
CHOBlON
1975; Middle
* Rccc4rw/ I Scptcrnhc~r I976 t On leave at The Johns Hopkins University during the principal phase of the work. $ This MS. thesis appears to have a mathematical error, judging from the numerical results.
MATERNAL SIDE Fig. 1. Schematic
807
, sketch of primate/human anatomy.
placental
808
F. F. ERIAN,S. CORRSIN and S. H. DAVIS
as a Newtonian fluid (density p t l.O6gm/cc; viscosity p I 0.04 P). an approximation well-established for channels wider than a few red cell diameters. Typical inter-villous distances have been estimated at 30-50 pm, typical terminal villous diameters at 40 pm (Aherne and Dunhill, 1966). The number of circulatory units. in a placenta is cu. 25. Although there is disagreement, the total number of arterial jets seems to be roughly 200. An estimated 500 ml/min of maternal placental flow gives an average of 2.5 ml/min per arterial jet. With an estimated orifice radius r of 0.03 cm, the jet Reynolds number is R, E pu,(Zr)/p o 20. 0, is the averaged flow speed. The Reynolds number must be considerably larger at the peak moment of a spurt, and it is, of course, zero during part of a “cycle”. There appear to be no published data on the relative durations of spurts and shut-downs for a typical arterial jet, but Ra&ey’s X-ray motion pictures of a monkey placenta* suggests a “duty cycle” (i.e. relative time during one period that each arterial jet flows) of less than 0.2. Since the present analysis is quasi-steady, that number is relevant as a warning that the accelerations may at times be appreciable, and to indicate that the maximum jet Reynolds number may be on the order of 100. Static pressure data of Coldeyro-Barcia (1957) indicate that the mean maternal arterial pressure is ca. 1OOmm Hg, and the average intervillous pressure is ca. I5 mmHg. Ramsey (1959) mentions maternal pelvic venous pressure of cu. 8 mm Hg. Without knowing exactly where Coldeyro-Barcia’s arterial pressure value occurs relative to the arterial jet exits, it is assumed to occur just at the exit; hence, it is a local boundary value for the quantitative hydrodynamic boundary-value problem. MATHEMATICALMODELS 1 A single circulatory unit is represented as a twodimensional rigid square box, 1.8 cm on a side, filled with a deformable porous solid [the villous trees(s)]. For simplicity, the geometry is chosen symmetrical, with a single arterial exit at the center and two venous drains, one on each side, as indicated in Fig. 2. Additional arterial exits and venous drains may be * Shown at a lecture at The Johns Hopkins University several years ago. t The most rational derivation of continuum model (volume-averaged) equations for fluid flow through a porous solid may be that of Saffman (1971). : It should bc noted in passing that “slip” occurs at fixed walls in this kind of model because the Darcv Law is a
first-order differential equation. Physically, the-blood does not slip, but this no-slip effect extends only a few pore widths into the medium. Thus, when a porous solid has substructure fine enough to allow a continuum representation, a solid wall “boundary layer” may be ignored. Brinkman (1949) has shown how a no-slip condition can be included, by adding an ad hoc, second-derivative, viscous term to Darcy’s Law.
added to the system at arbitrary boundary locations. The extension to a three-dimensional. axisymmetric configuration is reasonably straightforward. As approximate equations describing the behavior of local volume-averaged velocity u(X) and pressure P(X) fields (see, for example, Bear, 1972). the commonly accepted, negligible-inertia limit called the “Darcy Law” is used. X is spatial coordinate. Also, for simplicity, the porous solid is taken to be isotropic, so the permeability K is a scalar function: IL(X)
=‘- Kf
and the mass conservation v.u
VP,
equation is = 0.
(2)
U is velocity local average over volume “elements” containing both fluid and solid, the velocity of the solid being zero in this steady flow case; P is local volume-average pressure. It is defined only in the fluid phase, but the full volume element, including the solid part, is used in defining the average. This latter operation is equivalent to specifying that the pressure is zero in the solid.? Equation (1) may be viewed as the definition of K. It seems clear that in future, moresophisticated models, the local directedness of the villous trees (both hydrodynamic and elastic) will dictate a non-isotropic permeability. As written, equations (1) and (2) constitute a set of linear partial differential equations for _V and P when K(X) is prescribed. Since the villous trees are considerably deformed by maternal blood flow, K should, however, depend on local velocity. For simplicity, it is supposed that K depends only on local speed, so that K = FclU(xIl.
(3)
Equation (1) is then non-linear. Of course, a more accurate representation could be expected to involve not only the magnitude of the velocity, but also its direction, as well as its gradient. The boundary conditions are as follows: (a) the component of velocity normal to the impermeable parts of the walls is zero;: . (b) the static pressures are prescribed at the arterial jet (source) and at the venous drains (sinks). The geometry is indicated in Fig. 2. ANALYSESAND SOLUTtONS Linear cases
As a preliminary illustration to compare with the non-linear case, the permeability field K(X) is first prescribed. The equations are put into dimensionless form by using the “height” Y of the box as the characteristic length, and the average arterial exit velocity o0 as the characteristic speed. Then the dimensionless variables are urn:. oo,
x=x; y
PY P’p[i,;
ksF.
K
(4)
Maternal. placental blood flow Axis of symmetry )\‘\\\\\~\\‘\“\\\\\\\\\\\\\ .uc\-.\\\\\\\\\\\\\\.~~,,..... $ =O.VP.n=O inis breo unit normal)
/
Two-regions
model
bo/mdories,
:
kh
k, in remoi,ning
0
I
II
Cl
.
Porous
medium
;
for
0
* 3
througFt
the
differejlt Eqs.
i
domain
k =‘k(x)
I
ond
17 :
with
k defined
follows: (2-regions
model)
k -_I constant ’
k=‘kf&f(lyll /
! i
I !
!
(p40)
port
jegions
models’os
/
II p 12
Eqs. 13 6 14 Eq.
in rentrol
(
(p.100)
b=lO)
Fig. 2. The geometry of the mathematical model of a single cotyledon. Linear and nonlinear cases operator) is equal to the time rate of increase of k-’
Thus,
(5) as “seen” by an observer travelling with the local
V.&f = 0.
and 1 VP = - k(y)u.
Equations (5) and (6), together with the boundary conditions outlined above, belong to a well-studied class of boundary value problems. especially in hydrology [see Bear (1971)]. In the absence of an explicit analytical solution, the principal ditticulties are only computational. As a matter of genera1 interest, in two or three dimensions, equations (5) and (6) can be transformed into u -= 0, (7) k(Y)
vx1 I
volume-averaged velocity. Needless to say. k is not a convected quantity. Since this model is two-dimensional, it is convenient to introduce the stream function $, defined by &I s*
V.g = 0 is thus automatically satisfied. When the forms in equation (10) are substituted into equation (7). there follows an equation for $:
The boundary values come from the fact that the walls are streamlines, so constant t+G-valuescan be assigned there, with appropriate jumps across the by forming the curl of equation (6). and into source and sinks. 1 With the solution to equation (11) in hand. equa-V'P = Cu.Vl k(y) (8) [ tion (8) is a Poisson equation for p: a “Poisson equation” for p (after u has been determined), by forming the divergence of equation (6). Alternatively, the latter operation gives The specifications of II/ and p or their derivatives V-[k(.u)Vp] = 0. (9) along the boundaries will complete the formulation of the problem. It may be noted that the right side of equation Without loss of generality, the value $ = 0 may (8) (the “source” term of this Poisson equation) is the be assigned to the streamline which starts at the convective derivative of the flow resistance. In other source, runs upward along the centerline. then along words, the difference between the pressure at any the upper boundary, down the sides and along the point and the average pressure in an isotropic. infinilower boundary to the sink. The value of $ on the tesimal neighborhood (the “meaning” of the Laplace
1
F. F. ERIAN,S. CORRSMand S.
810
H. DAVIS
00 8
(0 I
i 0. (0 39
-2;53
0I1
i
39.721
4i .54 +
0.
39 -60
3c
0. (0 36
1 .. ..._
I
TiFig. 3. Streamlines and constant pressure lines in linear case with uniform permeability. Upper numbers are lo4 $ and lower numbers are 4 x 10” P. streamline which connects the source to the sink directly is then assigned a value relative to zero, calculated from the arterial flow rate. For the present analysis, the dimensionless stream function value along the boundary from the source to sink is chosen to be $ = I,& = 8.0 x lo-“. The conditions on the pressure are more complicated. Based upon the measured arterial and venous pressures mentioned earlier, the dimensionless pre-ssure at the source is approx. 40 x 104, and at the sink it is 4 x 104. Along all the boundaries no values of pressure can be independently assigned because the velocity at the boundary sets the tangential derivative via Darcy’s Law. Except at the source and sinks, the normal derivative of pressure at the boundary is zero because normal velocity there is zero. Given a function k, the linear cases described above have been solved numerically by iterative relaxation methods. The two-dimensional field was divided into 30 x 30 grid points. The criterion for convergence in the numerical calculations is that the sum of absolute differences between function values at all grid points from two consecutive iterations can be made sufficiently small. The plotted results show streamlines and isobars superimposed. Darcy’s Law, equation (61, indicates that the velocity vector is everywhere parallel to the pressure gradient, so that the streamlines must be perpendicular to the isobars.
Constant permeability case
In the simplest case, k = constant, equations (11) and (12) reduce to Laplace equations: V2$ = 0,
(13) (14) and p becomes the velocity potential function. The resulting boundary value problem can, of course, be solved analytically by conformal mapping but, since this case is a kind of “standard” to contrast with the more relevant ones, it has been solved numerically. The resulting flow net is shown in Fig. 3. The degree of orthogonality of the streamlines and isobars gives one indication of the accuracy of the numerical solution. Most of the flow activity occurs in the lower third of the region. Therefore, if mass transfer were included, it would take place relatively locally. If the source flow penetrated farther, however, the whole volume would be exploited more effectively. This latter fact suggests the utility of a relatively villous-free region situated over the maternal artery (the source) and, in fact, such a tendency is observed. The second linear case (below) describes the consequences of such a region, with prescribed k(s). Parenthetically, it should be remarked that fluid inertia would give greater jet penetration, even with k = constant. vp
= 0,
Cases with prescribed (non-unijbrm) permeability
Solutions of Darcy flows through media of prescribed non-uniform permeability are common in
XII
Maternal. placental blood flow
0.
6;
Fig. 4. Streamlines and constant pressure lines in linear case with two-region permeability. Upper numbers are lo4 JI and lower numbers are 4 x lo3 P.
hydrology. The first non-uniform case here is designed to simulate a penetrating arterial jet by dividing the flow region into two zones of uniform but different permeabilities as shown in Fig. 2. The narrow strip along the symmetry axis has the higher permeability, k,, > k,. Alternatively, this can be viewed as a relatively villous-free region. The numerical solution, shown in Fig. 4. is for the case when the ratio of the permeability in the center region k,, to that in the rest of the circulatory unit, k,, is 10. The flow net clearly shows the additional source penetration. Several ratios k,Jk, were tried. As k,,/k,+ 1.0, the how net approaches that in the uniform case; for k,,/k, > 20, the computational errors were excessive.
Steady flow problems with non-uniform, flowdependent permeability are available in the literature. In more than one dimension, the mathematical modeling in this subject for porous solids seems to have started with the work of Biot (1941). who later generalized the analysis to include anisotropy (1955). viscoelasticity (1956). and small amplitude pressure *The particles in a flowing suspension may also be viewed as a porous medium. but it lacks an “undisturbed” reference state, so the vast literature of that subject is not referenced here.
waves (1962). Bear (1972) mentions the work of Scheidegger (1960) and Verruijt (1969).* The foregoing research appears to be limited to linear theory associated with small displacements of the porous solid, but X-ray motion pictures show that placental villous trees are greatly deformed by maternal blood spurts. The correct formulation of the elastic force balance equations for the porous solid undergoing large deformations would require mathematics considerably more elaborate than can be justified by the modest goals of this paper. Therefore, a proper elastic analysis is replaced by an ad hoc assumption for the dependence of local permeability on local volume-averaged velocity field. If equation (4) is written in dimensionless form as k(x) = fllull~
(15)
a qualitatively reasonable form of / is needed. Then the boundary-value problem to be solved consists of equations (1x3), plus appropriate boundary conditions. f should be. finite at the limits Iu 1- 0 and /MI+ 00. It should be a monotonically increasing function of 1~1because larger local velocities will tend to push the branches aside. It should express the fact that branches pushed aside from any location must go to neighboring locations, which suggests at least an integral conservation condition expressing solid-
F. F. ERLAN, S. Coaasm and S. H. DAVIS
812
mass conservation. In the interests of simplicity, permeability will be conserved instead. One of the simplest forms which meets the first two requirements is
fEtul1 =
ykob)
I
1+
8-M 1
+
a.,y,
9
(16)
where k,(x) is prescribed and a, fl, and y are positive constants. This equation gives a non-linear Darcy Law.
uC1+ alull VP = - ykoOCl + (a + B)lull’
(17)
The additional equations are that for mass conservation, equation (5), and an ad hoc “permeability conservation” condition, ss
A~Clulld.4 =
ko0 dA, IS A
(18)
which in effect determines y after a and /I have been chosen. A is the area of the circulatory unit. The boundary conditions are as before. The computing procedure begins with specification of k,(g), and from this “undisturbed” permeability dis* The slight asymmetry in the top central region is due to the essentially constant value of Jr. The plotter usually chooses the location of the first encounter with the function value being plotted. Furthermore, annotations are written every seven grid points in both directions starting from the upper left corner. This results in some asymmetry in the shown values since the field is divided into 30 x 30 grid points. 0.
3
6.54
1.22
tribution, the solution to the corresponding linear problem is obtained. This “zero-order” flow field is the first step in the stream-function-field iteration. A modified permeability distribution is then obtained from equation (16), to be used in the next iteration for the stream function. An increase in velocity causes an increase in permeability, which, in turn, results in another velocity increase. The condition corresponding to equation (16) prevents this situation from degenerating into a mere binary k-field. The initial permeability field was chosen to be k,(x,y) = Ce’LT+by + Cl {em(x-x~) + err,(~+~~)}+y,
(19)
where C, Ci, a, a,, b, and bl are constants and xi is the distance between the source and either one of the two sinks. This form gives high permeability near the source and sinks, where villi-free regions may exist. Figures 5 and 6 show the initial and final permeability fields for this non-linear model, The broken lines in both figures represent the locus of the average permeability values. It is interesting to note the large distortion which results. The flow, as a result of this distortion, is shortcircuited from the source to the sink as shown in Fig. 7.* The resulting flow pattern would reduce the area of effective “contact” between maternal blood and fetal capillaries. This reduction underscores the importance of having a relatively large Reynolds number for the maternal jet. Given the architecture 6.55
3
Fig. 5. Initial (no-flow) permeability field. Nonlinear case.
0.
Maternal, placental blood flow 0.
?
0.48
0.53
0.49
0. 02 +
0.
7
1.53 +
1 .70 +
1 .55 +
0.
c7 +
0.
5
4.15 +
4 .62 +
4.24 +
0.
17 +
0.
30 +
0.
.40 Lb Fig. 6. Final permeability field. Nonlinear case.
Fig. 7. Streamlines and constant pressure lines. Nonlinear case. Upper numbers are lo4 $I and lower numbers are 4 x lo3 P.
F. F. ERUN, S. CORRSINand S. H. DAVIS
814 of a placental
circulatory
essential
to good
for mass
transfer
unit,
utilization can
the jet
penetration
of the full villous
occur
only
with
tree
appreciable
fluid inertia. CONCLUDING REMARKS
The models presented provide a means for assessing one aspect of the blood-villi interaction, namely the role of villi distortion. This prdcedure involves an ad hoc assumption concerning the non-linear permeability, but seems capable of qualitatively describing large deformations
of the villi, a range of behavior
far beyond the capability ing non-linear
elasticity
of a deductive
The predicted villidistortion,
through
ing, develops a fluid flow pattern that fluid inertia
effect, giving jet penetration transport.
would
short-cirtiit-
that would provide
relatively poor diffusive mass exchange. however,
model invok-
theory.
It seems clear,
have the opposite
and so promoting
mass
This effect plus effects of pulsatility are pre-
sently being pursued.
Acknowledgements-This work has been supported primarily by the Engineering Mechanics Program of the National Science Foundation. We should like to thank Drs. E. Ramsey and I. G. Donaldson who introduced us to the problem, M. Yoshinaga, who worked on early phases, and M. J. Karweit, who advised us on computational methods. REFERENCES Aherne, W. and Dunhill, M. S. (1966) Quantitative aspects of placental structure. J. Puth. Bact. 91, 123-140. Bear.J. (1972) Dvnamics of Fluids in Porous Media. Else-
vi&, tiew iort. * Biot, M. A. (1941) General theory of three dimensional consolidation. J. Appl. Phys. 12, 155-164. Biot, M. A. (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182-185. Biot, M. A. (1956) Theory of deformation of a porous viscoelastic anisotrooic solid. J. AD& Phvs. 27. 459-467. Biot, M. A. (1962) Melhanics of defbimation and acoustic propagation in porous media. J. Appl. Phys. 33, 14821498. Bloom. W. and Fawcett, D. W. (1968) A Textbqok of Histology. Saunders, Philadephia. Brinkman, H. C. (1949) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. Al, 27-34.
Coldeyro-Barcia. R. (1957) Physiology of prematurity. Trans. Is? Cor~f..pp. 128-139. Josiah Macy. Jr. Foundation, New York. Donaldson. I. G. (1972) Lecture at The Johns Hopkins University. Baltimore. Freese, U. E. (1966) The fetal-maternal circulation of the placenta-I. Histomorphologic, plastoid injection. and X-ray cinematographic studies on human placentas. Am. J. Obstet. Gynec. 94, 354-360. Gruenwald. P. (1966) The lobular architecture of the human placenta. Bull. Johns Hopkins Hospital 119, 172-190. Langman, J. (1969) Medical Embryology. Williams & Wilkins, Baltimore. Lardner, T. (1975) A model for placental oxygen exchange. J. Biomechanics 8, 131-134. Metcalfe, J., Romney, S. L., Ramsey. L. H., Reid, D. E. and Burwell, C. S. (1955) Estimation of uterine blood Row in normal human pregnancy at term. J. clin. Incest. 34. 1632-1638. Middleman, S. (1972) Transport Phenornenor~ in the Cardiovascular System. Wiley-Interscience. New York. Ramsey, E. M. (1959) Vascular anatomy of the uteroplacental and foetal circulation. In Oxygen Supply to the Human Foetus, CIOMS Symposium, pp. 67-79. Blackwell, Oxford. Ramsey, E. M. (1960) The placental circulation. In The Placental and Fetal Membranes (Edited by Villee, C. A.), pp. 36-62. Williams & Wilkins, Baltimore. Ramsey, E. M. (1969) New appraisal of an old organ: the placenta. Proc. Am. Phil. Sot. 113(4), 296-304. Ramsey, E. M. (1971) Maternal and foetal circulation of the placenta. Jr. J. med. Sci. 140(4), 151-168. Ramsey, E. M., Corner, G. W. and Donner, M. W. (1963) Serial and cineradiographic visualization of maternal circulation in the primate (hemochorial) placenta. Am. J. Obstet. Gynec. 86(S), 213-225.
Ramsey, E. M., Martin, C. B., McGaughey, H. S., Kaiser, I. H. and Donner, M. W. (1966) Venous drainage of the placenta in rhesus monkeys: radiographic studies. Am. J. Obstet. Gynec. 95, 948-955. S&man, P. G. (1971) On the boundary condition at the surface of a porous medium. Stud. appl. Murk Scheidegger, A. E. (1960) The Physics of Flow Through Porous Media, 2nd edn. University of Toronto Press,
Toronto. Verruijt, A. (1969) Elastic storage of acquifers. In FIow Through Porous Media (Edited by de Wiest, R. J. M.), pp. 331-376. Academic Press, New York. Whitmore, R. L. (1968) Rheology of the Circulation. Pergamon Press. Oxford. Yoshinaga. M. A. (1972) A Mathematical Model of Maternal Blood Flow in the Intervillous Space of the Placenta. M.S.E. Thesis, The Johns Hopkins University. Baltimore.
