A HYDRODYNAMICAL STUDY OF THE FLOW IN RENAL TUBULES

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1 Bulletin ~?f Mathematical Biology, Vol. 43, No. 2, pp. 15 l-i 63, 1981 Printed in Great Britain /81/ $02.00/0 Pergamon Press Ltd. Society for Mathematical Biology A HYDRODYNAMICAL STUDY OF THE FLOW IN RENAL TUBULES G. RADHAKRISHNAMACHARYA and PEEYUSH CHANDRA Mehta Research Institute of Mathematics and Mathematical Physics, 26 Dilkusha, New Katra, Allahabad , India and 9 M. R. KAIMAL'~ Department of Applied Mathematics, Indian Institute of Science, Bangalore , India The hydrodynamical problem of flow in proximal renal tubule is investigated by considering axisymmetric flow of a viscous, incompressible fluid through a long narrow tube of varying cross-section with reabsorption at the wall. Two cases for reabsorption have been studied (i) when the bulk flow, Q, decays exponentially with the axial distance x, and (ii) when Q is an arbitrary function of x such that Q-Qo can be expressed as a Fourier integral (where Qo is the flux at x = 0). The analytic expressions for flow variables have been obtained by applying perturbation method in terms of wall parameter e. The effects of ~ on pressure drop across the tube, radial velocity and wall shear have been studied in the case of exponentially decaying bulk flow and it has been found that the results are in agreement with the existing ones for the renal tubules. 1. Introduction. The study of the hydrodynamical aspect of flow in the nephrons plays a significant role in the discussion of glomerular tubular balance apart from its importance in the understanding of the mechanism of nephrons. The nephrons are the basic functional units of the kidney which accomplish the entire process that finally results in the formation of urine. Ludwig (1881) put forward a theory of urine formation which consisted of filtration through the walls of glomerular capillaries and reabsorption, which takes place in the renal tubules. This theory was modified by Cushney and has since been confirmed by many experiments tpresent address: Department of Mathematics, Cochin University, Cochin, U.S.A. Gbvernment. 151

152 G. RADHAKRISHNAMACHARYA, P. CHANDRA AND M. R. KAIMAL (Babsky et al., 1970). According to this theory, the quantity of glomerular filtrate is 150-17

2 152 G. RADHAKRISHNAMACHARYA, P. CHANDRA AND M. R. KAIMAL (Babsky et al., 1970). According to this theory, the quantity of glomerular filtrate is litres per day, of which only litres comes out as urine after its passage through the long thin renal tubules and the rest gets reabsorbed into the tubules to pass into the blood. This reabsorption takes place mainly in the proximal renal tubule. The hydrodynamical problem in the renal tubule has been studied by several authors (Wesson, 1954; Kelman, 1962; Macey, 1963, 1965) considering different models for reabsorption in the tubules. Macey (1963) formulated the problem as the flow of an incompressible viscous fluid through a.circular tube with linear rate of reabsorption. In 1962, Kelman noted that the bulk flow in the proximal tubule exponentially decays with the axial distance. This led Macey (1965) to improve his model under general conditions applicable to the flow in renal tubule. Recently, Marshall and Trowbridge (1974) studied this problem by considering the permeable wall model of the tube. However, in the above analyses the cylindrical tube is taken to be of uniform cross-section, while in general such tubes may not have uniform cross-section throughout their length. Moreover, it has been remarked in Gray's Anatomy (p. 1533, 34th ed., 1967) that the neck, where the proximal tubule is connected to the Bowman's capsule, is slightly narrower than the rest of tube. Thus the tube can be considered initially to be of diverging nature. Therefore in this paper we have made an attempt to understand the flow through the renal tubule by studying the hydrodynamical aspect of an incompressible viscous fluid in a circular tube of varying crosssection with reabsorption at the walls. The reabsorption has been accounted for by considering the bulk flow as a function of axial distance i.e. x, which is decreasing with x. The radius a(x), of the circular tube is assumed to vary with x in the following manner: a(x)=a o "R(x) =ao (C x- 1)] (1) where e and fi are wall parameters and a o is the radius of the tube at x = O. Solution for flow variables has been obtained by the perturbation method in terms of the wall parameter e. The analysis has been carried out for two cases, (i) when the bulk flow Q decays exponentially with x and (ii) when Q is an arbitrary function of x such that Q-Qo (where Qo is the quantity of fluid at the cross-section x = O) is Fourier integrable. The effects of ~ and fl on pressure drop, radial velocity and wall shear have been

3 HYDRODYNAMICAL STUDY OF FLOW IN RENAL TUBULES 153 studied for the case (i) and also comparison has been made with the available data. 2. Mathematical Formulation. Consider the steady motion of an incompressible, viscous Newtonian fluid through a long narrow circular tube of varying cross-section as given by equation (1). The motion is assumed to be axisymmetric. The inertial and end effects are neglected in view of the low Reynolds number which is of the order 10-2 for the flow in renal tubules (Macey, 1963). Thus the equations of motion governing such flow are as follows [in the cylindrical polar co-ordinates (r,o,x) such that r=0 is the axis of symmetry]: (~2U (~2U 1 OU 1 Op +?Tr =o (2) ~X 2 r Or # Ox &-0 (3) ~?u 1 c~ (rv)=0 (4) where u,v are the velocities along the x and r axes respectively, p the pressure, and # the constant fluid viscosity. The boundary conditions are- (a) the tangential velocity at the wall is zero i.e. da U+~xxV=0 at r=a(x); (5) (b) the flux across a cross-section is a function of x (this is in view of the reabsorption of the fluid at the wall) i.e. f a(x) Q= 2rcru(x, r)dx=qo "F(~x) (6) do where F(ex)=l for c~=0 and decreases with x; ~> 0 is the reabsorption coefficient and is constant; Qo is the flux across the crosssection at x=0; (c) the regularity condition requires c~u v=0, Ur=0 at r=0. (7)

4 154 G. RADHAKRISHNAMACHARYA, P. CHANDRA AND M.R. KAIMAL Introduce the stream function ~p such that U = r 0r' t~-- r 0x along with the following non-dimensionalized quantities (8) 0' =~-o ~ ' ' 2rCao 3 P =~o p, c~' = C~ao, [3' = [3ao, r' = J]a o and x' = x/a o. Thus the equations of motion (2)-(4) transform to equation, which is written after dropping the primes, the following where V40=0, V2= 1 0 0X 2 "t'-~f2 r Or (9) and the boundary conditions (5)-(7) become Or -(efler at r=r(x), O=-F(ex) at r=r(x), 0=~-=0 at r=0. (10) (11) (12) In the following, we shall first consider the case decaying bulk flow, i.e. F(ax)=e -~x of exponentially (13) and then a general case where F (~x ) = I + G(c~x ) (14) and G(ex) is such that its Fourier transform say, q(7), exists, i.e. G(ex)= f oo --O0 q(7)e-i~xdx. (15)

5 HYDRODYNAMICAL STUDY OF FLOW IN RENAL TUBULES Case (i); F(~x)=e -~x. In the present analysis we shall seek a solution for the stream function 4'(r, x) in the form of a series in the parameter e, as 4'(/", X)= 4'O(r, X)-{-,~4' 1 (r, X)'~-... (16) Further the boundary conditions (10) and (11) are given at r=r(x)=l +e(e ex- 1) where [e(e ~x- 1)[~ 1, thus we make a Taylor series expansion of equations (10) and (11) at r= 1 in the powers of e(e px- 1). This gives ~7+e(Jx- 1) ~-72r =erie I~x +e(e ~'~- 1) and 024'. ] ~ r at r= 1 (17) 4'+e(eaX-1)~-~r+... F(c~x) at r=l. (18) Substituting equation (16) into equations (9), (12), (17) and (18) and collecting coefficients of various powers of e, we get the following set of equations for 4'o(r, x) and 4'1(r, x). Zero order: V44'o =0 (19) 04'o = Or 0 at r=l, (20a) 4'o= -F(ex)= -e -~x at r=l, (20b) 4'o = 0 at r = 0 (20c) and first order: V44'1 =0 (21) 4'1 epx 84'0 _ 1) 024'~ Or 0x 0r 2 at r= 1, (22a) 4'1=0 at r=l,0. (22b, c) The solution of equation (19) along with equation (20 a,b,c) which remains finite at r = 0 is - e - ~x[j 1 (~)rj 1 (ro~)- Jo (c~)r2j2 (ro~)] 4'o(r, x)= [jz(~x)_jo(oqj2(oo ] (23)

6 156 G. RADHAKRISHNAMACHARYA, P. CHANDRA AND M. R. KAIMAL The boundary condition (22a) along with equation (23) suggests that ~1 should have the form: 01(r, x) = ~1o (r) e -=x + ~/11(r) e(e =)x (24) where Olo(r) and ~11(r) satisfy the following sets of equations: and where V4(Olo(r) e-~) =0, dl)lo, ~-r =~A1 at r=l, ~//10~---0 at r=0,1 (25) V4(Oll(r)e~)=O d~ll dr =~(fi-a1) at r--1 ~11 ~---0 at r=o, 1 (26) JZ(e)+ J2(c 0 6=fi-c~ and Al=~j~(cO_Jo(cOj2(cO.i (27) Solving equations (25) and (26) with the requirement of ~1o and ~911 along with their derivatives being bounded at r = 0, we get Ol~ ~)-Jo A1 [J2 (~ J1 (rcq- Jl (oor2j2(r~)] j (~)J2 (~) (28) and where 011 (r) = A 2 [J2 (~)rjl (ijr) - J1 (O)r2J2 (~r)] (29) (30)

7 HYDRODYNAMICAL STUDY OF FLOW IN RENAL TUBULES 157 The pressure p(r,x) can now be obtained from equations (2), (3) by using equations (8), (16), (23), (24), (28) and (29) in the following form: 2O~Jo(oOJo(o~r) _~ ~ 2o~A1Jl (oqjo(o~r) p(r,x)=jz(cq_jo(oojz(~z ) e +elj~(~_j~j~) e -'x + 26A 2 J 1 (6)J o (3 r) e~ 1 + 0(/~ 2) -I- a constant of integration. (31) Mean pressure across the cross-section x = x o is defined as f ~(~o) 2nrp(r, Xo ) dr P(Xo)- nr 2 (x o) and the mean pressure drop between x = 0 and x = x o is Ap(xo) =p(o)-p(xo). Thus, substituting equation (31), we obtain the mean pressure drop between x = 0 and x = x o, Jo(~ (cq Ap(xo) =4[jz(cQ_jo(OQj2(o Q (1-e -~x~ -Fe jz(oq_jo(oqjz(oq (1 -e-~x~ A2j2(6)(1 -e ~~ ejo (e)j2 (cq e-'xo(jxo- 1 )]; -~- O(E2). (32) -~ j2(~)_ jo(~)j2(~) A) The wall shear z is defined as at r a(x) BMB B

8 158 G. RADHAKRISHNAMACHARYA, PI CHANDRA AND M. R. KAIMAL where eu =2ff eu' \at This gives z w (non-dimensionalized wall shear) as 2c~e -~x % = J~ (~) - Jo (a)j2 (a) {Jo (~)Jx (c~)- ~[Jo 2 (c~) + J~ (cq]) _~xa 1 e -~,x +2e j~(cq_-~o~-)72(cq J~(cQ+ A2(SJ~(5)e ~x e -~x J~(a)- Jo(cOJ2(e) = {(e a~ - 1)J1 (c~)(jo (c~) - c~j 1 (~))}, (33) where 2rta 3 27 w -- ~. #Qo Discussion. It may be observed that the zero order solutions correspond to the solution of Macey (1965) i.e. to the e=0 case when the tube is of constant radius. Also, 13=0 implies the uniform cross-section of the tube and the solution reduces to Macey's solution for the flow in a tube of constant radius and reabsorbing walls. Further, to study the effects of and fl more clearly we have carried out numerical calculations using the available data for rat kidney (as given by Macey, 1965). Thus, Qo=4 x 10-Tcm3/sec, ao=10-3cm, #=7x 10-3dyne sec/cm 2 and the length of the tube= 1 cm, which give the reabsorption coefficient c~ as 1.6cm-1. The results are studied for divergent, straight and convergent tubes which are described by equation (1) with e<0,=0 and >0 respectively. The mean pressure drop over the whole length of the tube is calculated for various values of e and fl and is given in Table I. It can be noticed that for divergent tubes the pressure drop is less than its corresponding values (for given fl) for a straight tube while it is more for converging tubes. It can also be observed that for a convergent tube the mean pressure drop increases as /3 increases for a given value of e, but it decreases for the divergent tubes.

9 HYDRODYNAMICAL STUDY OF FLOW IN RENAL TUBULES 159 As a passing remark, it may be mentioned that Marshall and Trowbridge (1974) have calculated the value of mean pressure drop lap (/5o -p(l) in their notation] in their analysis for a normal nephron, as mm Hg, which is more realistic than the value 3.2 mm Hg given by Macey (1965). It can be observed from Table I that the values of mean pressure drop Ap for divergent tubes are closer to the values given by Marshall and Trowbridge (1974). This might mean that proximal tubules diverge initially. TABLE I Values of the Mean Pressure Drop (mmhg) Across the Tube (Length = 1.0 cm., = 1.6cm -1 ) The non-dimensional radial velocity is plotted at different cross-sections in Figure 1 for /~=-0.04 and various values of 5. It can be observed that for a straight tube (e=0) radial velocity attains its maximum value at r ~-0.8 all along the axial distance, while for the divergent tubes maximum is obtained at r =0.6 near the entrance and it shifts towards the boundary downstream. The corresponding results for convergent tubes are also given in Figure 1. Figure 2 depicts the non-dimensional shear stress (-%) at various values of the axial distance. The absolute value of wall shear % is more for a convergent tube and less for a diverging tube as compared to the values for a straight tube at a given cross-section. 4. Case (ii); F(ex)=l+G(ex). Now, we shall consider a general case where the bulk flow can be considered as an arbitrary function of x. It can be seen from the equations (19) and (20a,b,c) that the zero order set of equations gives a solution for flow in a tube of constant radius with reabsorbing walls even in the general case. Thus, following Macey (1965) the zero order flow variables will have the form f =f'+ f" (34)

10 160 G. RADHAKRISHNAMACHARYA, P. CHANDRA AND M. R. KAIMAL ol= ~ r p=-o.o4 r " I X - - E =-0.1 / I i rl~o, "~'\.J /,Y / / / I / /o i 2 x I x=o.02 (v x 103) Figure 1. Non-dimensionalized radial velocity (v) vs r/a o at different crosssections 10 or,= r~ lb=-o.o4 - - E= E=O.O ~ ~ = +0.1 \ \\\\\ I x la o Figure 2. Non-dimensional Jf~'w~J vs axial distance x/a o

11 HYDRODYNAMICAL STUDY OF FLOW IN RENAL TUBULES 161 where f represents the zero order flow variables; single prime implies the contribution due to Poiseuille's motion and double prime indicates the contribution purely due to reabsorption. Thus, the bulk flow Q(x) will be given as Q(x) = Qo + Q, (~x) (35) where Qo is the flow flux at x = 0 and Q~(0)=0. Thus, in view of equation (34) we will have the following form of ~ and p: O=O;+O;+e t P= Po + P~ + epl +... where ~) and p~ give the contribution of Poiseuille flow and ~ equation (19) along with the boundary conditions (20 a,c), satisfies ~"o = - G(c~x) at r = 1 (36) [G(ex) is as restricted by equations (14) and (15)], 01 is the solution of equation (21) with the boundary conditions (22b,c) and c~r=(/~'e px) -(ep~-l)xk~-rz ~?rz ) at r=l. (37) Again, ~'1 will have contributions from Poiseuille's flow as well as due to pure reabsorption. Thus separating ~1 into 0] and ~, we have 01 = 0', + ~ where ~k] and 0~ satisfy equation (21) and boundary conditions (22b,c) along with ~?~'i 0r =- (e px - 1) ~ ~72~,; at r=l (38)

12 162 G. RADHAKRISHNAMACHARYA, P. CHANDRA AND M. R. KAIMAL and &P~ =flet3x 00; (e px- 1) 020; ~r ~x - x ~ at r=l. (39) Solving for 0; and 0~ by the Fourier transform method, we get 0;(r, x)= oo f --OO H a (7){7Io(7)r2I, (yr) - 7 [11 (7) + 21o (V)] ri1 (7r)} e- i,~ 47 (40) 0'~(r, x) = f~- 0o Hl(7)Hz(7'fl)q(7) [11 (7)r2Io (7 r) -- Io (?)ri1 (Yr)] + e- i~x dt, (41) where q(7) H1 (7) = 210 (7)11 (7) + 7 [ 12 (7) - I2 (7)3 (42) and H2( 7, fl) is the Fourier transform of the r.h.s, of equation (39). Thus, the stream function for this case is given by 0 = (0; ) + (0; ) where 0~ and 0~ are given by equations (40) and (41) and 0~) and 0] correspond to the flow in tube of varying cross-section [as given by equation (1)] with no reabsorption. 0~) and 0] may, therefore, be obtained from case (i) by taking the limit ~--*0. Thus, + [J2 (fi)rj1 (flr)-j1 (~)r2j2(fir)]e~x'~. (43) ~J~)~ Jo(fi)J2(fi)] J [ Further, the expression for pressure is obtained by substituting the expression for 0 in equations (2), (3) and (7), separating the purely reabsorption part. Thus, p is given by the following expression p=p'-2i f oo H(~)[7Io(7)+eH2(7, fi)ii(7)]io(tr)e-irxd7 (44)

13 HYDRODYNAMICAL STUDY OF FLOW IN RENAL TUBULES 163 where p' is obtained from equation (31) under the limit e~0. For a given suitable form of G(ex) equations (40)-(44) may be solved to get the expressions for velocity components and pressure drop. The first two authors acknow!edge the financial support from the Department of Science and Technology (Govt. of India) under the Project No. 12(35)/76-SERC. Also, the authors are grateful to the referees for their valuable comments. LITERATURE Babsky, E. B., B.L Khoolorov, G. I. Kositsky and A. A. Zubkov Human Physiology, Vol. I, pp Moscow: Mir Publications. Kelman, R. B "A Theoretical Note on Experiment Flow in the Proximal Part of the Mammalian Nephron." Bull. Math. Biophys., 24, Macey, R. I "Pressure Flow Patterns in a Cylinder with Reabsorbing Walls." Bull. Math. Biophys., 25, "Hydrodynamics of Renal Tubule." Bull. Math. Biophys., 27, Marshall, E. A. and E. A. Trowbridge "Flow of a Newtonian Fluid Through a Permeable Tube: the Application to the Proximal Renal Tubule." Bull. Math. Biol., 36, ( Wesson, L. C., Jr "A Theoretical Analysis of Urea Excretion by the Mammalian Kidney." Am. J. Physiol., 179, RECEIVED REVISED