Mathematical modelling of the renal concentrating mechanism

MUM Compuf. Modehg, Printed in Great Britain

I I,

Vol.

pp. 615-620.

1988

0895.7 177/88 $3.00 + 0.00 Pergamon Press plc

MATHEMATICAL MODELLING OF THE RENAL CONCENTRATING MECHANISM Raymond Mejia Laboratory

of

Mathematical

Kidney

Research

and

Electrolyte

Metabolism,

Branch, NIDDK, National

NHLBI,

and

Institutes of Health,

Bethesda, MD 20892

Abstract.

We describe model equations, a method of solution, and show

examples of steady-state and transient solutions for the urinary concentrating mechanism.

Then we draw some conclusions about the current

successes and failures of the models

and about future directions

in

modelling the mammalian concentrating mechanism. Keywords.

Mathematical model; urinary concentrating mechanism; kidney

model; multipoint boundary value problem; continuation of solutions.

INTRODUCTION The

principal

maintain

function

the chemical

of

partment is separated from the vascular compart-

the

kidney

composition

is

to

of the body

fluids within the narrow limits compatible with life.

Table 1 shows the fluid compartments of

the body and indicates their approximate contributions to body weight. extracellular,

and

These are the cellular,

transcellular

The cellular compartment extracellular branes

of

complicated ties

that

the

by

cells.

These

transport are

the

plasma

mem-

membranes

have

and permeability

responsible

for

the

proper-

Extracellularly,

the

to

impermeable

water

only

and

to

solutes

small

larger molecules

plasma proteins and blood cells). ionic

composition

fluid

is

thus,

of

essentially the

that

plasma the

composition

of

the

and

Hence, the interstitial The

same. of

and

(namely,

the

extracellular

kidney

plasma

and,

and cellular

fluid. Schematically,

we

describe

the

kidney

as

in

Figure 1, which shows two populations of nephro-

differing

vascular

composition inside the cells and in surrounding fluid.

by the capillary walls, which are freely

permeable

regulates

compartments.

is separated from the

compartment

ment

interstitial com-

units, one

one cortical (short).

juxtamedullary

(long) and

This defines a multipoint

boundary value problem with the arterial solute concentration of salt and urea specified as well

Table 1: Total Body Water Weight)

as

I

transcellular

interstitial 16%

arterial

(entering),

(exit) pressures.

venous

and

bladder

The concentration of solutes,

volume flow and hydrostatic

pressure

are com-

puted in each tube and in the interstitium in

cerebrospinal, pleural, peritoneal, intraocular, synovial fluids, digestive secretions, etc. l-3%

order to obtain urine concentrations and flow. Kuhn and Ryffel (1942) first proposed that the loops of Henle multiplier. demonstrated

functioned

as a countercurrent

Subsequently, Wirz that the osmotic

et al.

pressure

(1951) of the

luminal contents rose from the cortico-medullary boundary to the

615

papillary

tip.

Since then the

616

Schematic diagram of two nephrovascular units, one juxtamedullary Open arrows indicate transmembrane water and one cortical (short). See solid arrows indicate salt flux; hatched arrows indicate urea flux. 2 for nomenclature of individual segments.

flux; Table architectural and blood counterflow maintain

and to permit

models

first of

the

by

urine

vice

his

to

the

transport;

fractional

entering

the

is reabsorbed

fractional

urine

flow

flow rate divided at

the

any

fW

namely,

that

hyperosmolality

limb

to

transport

out of the AHL.

fT iS

of Henle fU

is the

the

urine

nephron

flow rate

fractional

divided

solute

loss caused

of ascending

plasma

Here

are the thin the

(tAHL) and thick

ascending

third

limb

(iIMCD)

(tIMCD)

of

and the

concentrating Lemley

and

describe

appear

of

Kriz

the of

study

and

the

terminal

two

thirds

collecting of

Other

papers

concentrating of

Kidney

references

the

for example,

state-of-the-art

31

of

initial

histotopography

are shown,

renal

many

and

medullary

(1987).

volume

in

the

current

the

(1987),

the

process

(TAHL) portions

Henle

inner

Details

ducts.

of

by

that

in

the

mechanism

International are

given

by

(1981).

Stephenson

is, the fraction

dissipative

relative

of

osmolality,

namely,

the

best

ratio

by the AHL.

is

the

1976)

is

the

(I - fW)].

by the total

papilla.

washout;

This

for

rate;

central

multiplication

ascending

(AHL) that

renal

(1973,

plasma

(I - f")

and

the

The

units.

equation

a

from diurisis

versa.

countercurrent

osmolality

solute

in

by Stephenson

r = I/[1 - fT

of

tubules

generate

pressure

nephro-vascular

described

to

a transition

and

proposed

showed

the

the renal

essential

osmotic

antidiurisis

core

of

has been shown to constitute

system

high

medulla to

organization

vessels

vascular by

the

by

fluid solute

whole

Thus,

kidney

models

are

approximations

that

tially

understanding

to

the

have

centrating

mechanism.

sition

flows

and

interstitium

We

time, and we assume be a well-mixed

of

the

describe

in the tubes

as a function

one-dimensional

contributed

substanrenal the

con-

compo-

and the medullary

of axial distance

the cortical

and

interstitium

to

bath.

MODEL In Figure

1 the

shown

with

ties

for

species

their

not

in

acid-base

been

properties

included

active

balance,

approximately

transport

salt

ions

and

proper-

as

K',

equations

that

are

processes

and To

nephron

segments

as

having

distinct

especially

noted

transport

in this

a

system

paper

(Stephenson of

et

partial

al.,

1976)

differential

as follows:

date

(13)

whose

equations

of

"k Fv Ck - A Ok dx

in

thirteen

Segments

model

consist

such

example.

The

the

Among

buffers

transport

are

identified

properties. are

urea.

are

for

segments

transport

and

NH; and H',

important

tubular

transtubular

water,

Cl-, HCO;,

have

individual

1

= -Jk

(solute conservation) dF $Y + $

= -J,

[fluid conservation)

Proc. 6th Int. Conf. on Mathematical Modelling

g

(equation of motion)

+ Rv Fv = 0

with

initial

fied.

and

boundary

speci-

conditions

A is the cross sectional area of tubule;

Ck is the concentration

of the kth solute or

buffer; x is axial distance. and Dk

Jk and J, are

diffusivity.

iS

Fv is volume flow, tranSmUt'a1

P is hydrostatic

pressure;

and R, is

resistance to flow. The transmural

and osmotic

components

defined as f Ilows:

h,

is

An -r-n* n*is

and

is

1

1 where

oncotic

is

pressure

under consideration;

difference;

pressure outside

the

AP = P - P* ok is

temperature;

coefficient

for

the

difference

structure hydrostatic

is

cludes

species;

for

the

The transmural passive

the

kth

= Ck - c;, where the Czis

"k outside.

reflection the

X/Ax

For a mesh

subintervals, where X is the depth of the

medulla, for At so

that

Choose a time

example. t, = n At

increment Let

for n = 0, l,....

of Ck(jAx,t,)

Value

for j = 0, 1, .... J and write the other variables

Then

similarly.

the

finite

difference

FL(j)-$(j-1) ---xx=-

kth

species

is

the concentration

solute flux, Jk, in-

transport,

solvent

= hk ACk + (1 - ok) gk

(11

1

the

and 1 5 k 5 K,

number

of

The difference equations for the other

solutes.

unknowns, F, and P, are written in an analogous This scheme is

form.

accurate and has

0(8x2)

been shown to be stable and accurate in approximation

of

these

equations

by

Mejia

et

al.

(1977).

concen-

drag

and

active transport and is defined as follows:

Jk

divide the interval [0,X] into J =

for 1 2 j 5 J

pressure difference; R is the gas constant; T is

tration

spacing Ax

C$)-Cc-‘(j)+Cz(j-l)-C:-‘(j-1) ____---_-YE--

permeability;

hydraulic

the

oncotic

absolute

bulk flow

AP - An - R-T f ok ACk

Jv = hv

Let the

aCk F, Ck - Ok dx = Fk.

equations for mass conservation are as follow:

fluid flux, J,, includes hydro-

static, oncotic

where for simplicity we have let A = 1.

C:(j) denote the approximate

solute flux and transmural fluid flux, respectively.

611

With flow toward the papilla defined as positive and

toward matrix

the

for

cortex

equations

negative,

the

(1) is sparse,

block bi-diagonal

for the tubes and dense for

the interstitium.

To solve equations (1) denote

the

J, + Jkact

flow

Jacobian

vector

flows

for

of

concentrations,

the nth

time

step

pressures by y",

and

and the

system of equations by 4 (Mejia and Stephenson, where

hk

is

solute

permeability;

G

is

the

solute and

concentration at the luminal membrane; act Jk is active or metabolically driven usually

transport, kinetics,

so

that

We then seek a solution of the system of

1984).

equations

$h”s Y”-l)=0

defined by Michaelis-Menten act Jk

akCk = v-5.

ak is the

where y

n-l

maximum

rate

of

tranSpOrt.

and

bk

is

the

Michaelis constant.

is known from a previous time step or values.

initial

(2)

At a steady-state y" = y

n-l

To solve equations

satisfies these equations.

That is,

(2) we employ a quasi-Newton method.

given an estimate yi of y"; if I$(y;l,y '-l)I is METHOD OF SOLUTION

less

than

solution.

some

If

preset not,

we

tolerance, improve

we

the

have

a

estimate

The differential equations are discretized using

of y" by repeatedly solving the system of linear

a difference

equations

and backward

scheme that is centered in

time.

For example, consider the

solute conservation equation: "k K+%

c

in space

aCk Fv Ck - Dk ax

1

= -Jk

$(y”,y”-‘)

- I Ayn = 0

where r is an

approximation

to the Jacobian

618

Proc. 6th Int. Co& on Mathematical Modeling

matrix

(d$~~/dy~)

evaluated

at Y".

Sparse

matrix techniques that exploit the structure of the Jacobian have been described by Tewarson et al. (1976). In practice, we note first that for an estimate of

the

global

variables

variables

and

equations

implicit

(i.e.,

interstitial

boundary

conditions)

written in the form defined by equations (5). As shown previously, a quasi-Newton method is employed to solve each of the tube equations. The Jacobian matrix at each tube position is written analytically, inverted using a symbolic manipulator, and FORTRAN code is generated as In this shown by Mejia and Stephenson (1979). way the tube equations are solved very efficiently as adjunct equations to the dense system defined by the global equations (Mejia et al., 1977), and the solution is continued as a function of the parameters or boundary values.

(1) can be solved in the tubes in the

RESULTS

direction of flow as an initial value problem. Due

to

the

application

connectivity of

the

of

implicit

equations function

(2).

theorem

yields

Oi(y:,

yy-', yi)

- 0,

i - 1, Z,...,I, (3)

n-1) - 0, 4G(YY. v;, ...* Y;. Y;. YG where yl is the vector of unknowns

in the ith

tube segment at the nth time step, Yi vector

of

global

variables,

is the

and $i and 4

G the tube and global equations, respectively.

Figures 2-5 below illustrate the ability of this model to develop a solution for a given set of parameters and to permit study of the effect of parameter variations on the renal concentrating The parameters used are given in mechanism. Note first that there is active salt Table 2. transport in TAHL and only passive transport in Secondly, the only portion of the coltAHL. lecting duct that is urea permeable is the Normalized boundary data consists of the IMCD. concentration of salt (1.0). urea (0.05) and large solutes (0.0038) in addition to arterial pressure (0.013). venous pressure (0.001) and bladder pressure (0.00144). In Figure 2 we show the total urine concentration as a function of the ratio of short to long-looped nephrons in a two nephron population. This shows a multiplicity of solutions at a ratio of 3 to 1 and a maximum concentration at Figure 3 shows a ratio between 5 and 6 to 1. the total urine concentration as a function of hydraulic permeability of the collecting duct of the cortical nephron population for a fixed ratio of 3.5 cortical to 1 juxtamedullary nephron. It shows multiple stable states joined by an (time) unstable branch. The time dependent transition from one stable branch to the other, and vice versa, is shown in Figure 4 for a 1% change in the permeability. An increase in the permeability at position A results in a transition to position 8; a decrease at C Figure 5 shows results in a transition to 0. the solution surface for total urine concentration as a function of hydraulic permeability It illustrates how one may and nephron ratio. study the concentrating mechanism as a function of several physiologic parameters.

are

A goal in modelling the renal concentrating mechanism is to understand the transition from one state to another; such as for example, from water diurisis to antidiurisis. In addition, as we shall show below, the solutions of equations (I) are not unique. Thus, for a given set of model parameters, there may exist more than one (time stable) solution. In addition, the domain of convergence may be small for a given set of transport parameters. Hence, we employ a method to continue a steady-state solution of equations (3) as a function of model parameters or boundary conditions and thus compute a connected component of solutions (Mejia and Stephenson, 1984). To do this consider the steady-state analog of equations (2) F(Y; a) - 0

where

F: Rptg x Rq + Rptg

(4)

with unknowns y and parameters a. p is the number of tube equations; g is the number of global equations and q is the number of parameters. By domain decomposition, equations (4) may be written as Fi(yis

yG;

a) - 0

i - 1, 2,

. . . . I,

Pi+9 Fi: R xRq+R

pi

3

,

n

d

1

I

pi 2 3,

FG(Y13Y2, .... YI'YG; a) - D FG: Rp+g x Rq .

i

J

/ (5)

R9.

k Pi -P"g,

i-l

with vectors

pi yi E R

' yG

E Rg and a E Rq.

To investigate the urinary concentrating mechanism we use CONKUB (Mejia, 1986), an interactive path-following algorithm, to evaluate a model

v-v

Total urine concentration is shown as a unc ion of the ratio of cortical to juxtamedullary nephrons. Total filtrate concentration is 1.05 (Mejia and Stephenson, 1984).

Proc. 6th Int. Conf. on Mathematical

I

0.0

0.1

0.2

I

0.3

0.4

0.5

Modelling

619

I

0.6

Yof

Time course for the transition from A Fig. 4. to B and C to 0 shown in Figure 3. The numbers in parentheses are normalized values of the (Mejia permeabilities. See text for discussion and Stephenson, 1984).

Renal models have demonstrated a number of important features of the renal concentrating mechanism. Among them are: 1) the advantage

derived by 'coning'; i.e., by a distribution of nephrons extending to various depths in the medulla, as shown by Lory et al. (1983) and illustrated in Figure 2 for two populations; 2) the feasibility of strictly passive transport in the thin ascending limb of Henle in the concentrating kidney (Stephenson et al., 1974); 3) the prediction, now verified experimentally by Sands

"w.v.z

Total urine concentration as a function t e hydraulic permeability of the collecting duct of the cortical nephrons for a ratio of short to long nephrons fixed at 3.5 to 1 (Mejia and Stephenson, 1984). CONCLUSIONS

0=0.45, b=l 0.2 0 01 6. DT2 od 0, o.o2c 1 6. 0 CD 0.05 0 0 1 E=O. RP 0. OR, = flow reabtance, h, = hydraulic permeability, h, = ralt permeability, h, = urea permeability, u = Staverman reflection coeffkient for Mtered eolutea, u, = reflection coefficient for large, non filtered solute, RA= flow resistanceafferent to glomerul~, Rg = flow redstance &erent to glomeruhq R. = resietanceto flow in the intemtitium, DO, = diiusion constant for salt in the intemtitium, B = fraction of filtrate reabsorbed in the proximal tubule, 4 = maximum rate of transport, b = Michaelii conetant, E = fraction of collecting duct outflow entering renal pelvis. bG = glomerulue, PGC = poetglomerular capillary, DVRl = descending VM rectum for fmt (juxtamedullary) nephrovascular unit, CAVR = cortical wending nephrovwular unit, BC = Bowman’s capsule, PT = proximal tubule, DHL = descending loop of Henle’s lib, AHL = aacendii loop of Hen& lib, DT = diital tubule, CD = collecting duct, RP = renal pelvis. crhc fmt value refers to the outer medulla, where 01~50.6; the eecond refem to the inner medulla, where 0.6~z~X=l. For 0.5<2<0.6 the value vark linearly. r)rhe hydraulic permeability of the cortical nephron population hsll been varied. CThe fort vah~ holL for 0910.4; the second holde for O.tIst
Proc. 6th Inr. Conf. on Mathematical

620

Modelling

and Knepper (1987), that for passive concentration in the inner medulla, urea must be delivered to be reabsorbed by the terminal segment of the inner medullary collecting duct; 4) the description of a path from one stable state to another that resides on a highly convoluted solution surface, and which might be traversed due to changes in a number of membrane parameters. At the same time, the models fail to reproduce maximal urine concentrations measured in vivo with the transport parameters measured experimentally. This seems to suggest the need for better models, better data, or both. Steps toward better models include: 1) the incorporation of the cell compartment into the models and the consideration of ionic species and electrical potential gradients, especially in the proximal tubule, in the thick portion of the ascending limb of Henle, and in the cortical collecting duct (for a model of electrolyte transport see Stephenson, in press); 2) the ability to study the role of the whole kidney in regulating acid-base balance, for example, including secretion of ammonium and absorption of bicarbonate by proximal straight tubules (Garvin et al., 1987), and proton and amnonia secretion by collecting duct (Star et al., 1987); 3) models that take into account more details of the histotopography of the kidney, including the cortical labrynth and the medullary rays. REFERENCES Garvin, J.L., M. Burg and M.A. Knepper

(1987).

NH3 and NH; transport by rabbit renal proximal straight tubules. 252, F232. Kuhn, W. and K. Ryffel (1942). zentrierter Losungen aus blosse Membranwirkung. zur Funktion der Niere.

Am. J. Physiol., Herstellung Konverdunnten durch Ein odellversuch

Lemle$??%%~.%~z14~i987) 7:: separations: The histotopoqraphv of the urinary concentrating process.Kidney Internat., 2, 538. Lor‘Y ,TxGilg and M. Horster (1983). Renal countercurrent system: Role of collecting duct convergence and pelvic urea predicted from a mathematical model. J. Math. Biol., 16, 281. Mej a, R. B. Kellogg and J. L. Stephenson (1%). Comparison of numerical methods for renal network flows. J. Computational m., 23, 53. Mejia, R. anTJ. L. Stephenson (1979). Symbolics and numerics of a multineohron kidnev model. In V. E. Lewis (Ed.),'1979 MACSYMX User's Conference, pp.596-603. Mejia, R. and J. L. Stephenson (1984). Solution of a multinephron, multisolute model of the mammalian kidney by Newton and continuation methods. Math. Biosci., 68, 279. Mejia, R. (1986). CONKUB: A conversational path-follower for systems of nonlinear equations. J. Computational Phys., 63, 67. Sands, J. M. and M. A. Knepper (1987). Urea permeability of mammalian inner medullary

[URINE]

V 0

1 hCD,v

Three dimensional representation of the so u ion surface showing total urine concentra%

tion as a function of the hydraulic permeability of the collecting duct of the cortical nephrons (Mejia and the ratio of short to long nephrons and Stephenson, 1984). and papillary duct system collecting J. Clin. Invest., 79, surface epithelium. 138. and M. A. Knepper M. B. Burg Star, R. A., Luminal disequilibrium pH and (1987). medullary outer in aimnonia transport collecting duct. Am. J. Physiol., 252, F1148. Stephenson, J. L. (1973). Concentrating engines I. Central core model of and the kidnev. Biophyb J., 15, 512. the renal medclla. engines and (1973). Concentrating II. Multisolute central core th k' Steph$$?.e~~~w&~~ 54zid R. Mejia 11974i. Ouantitative analysis of mass and energy balance in non-ideal models of the renal counterflow system. Proc. Nat. Acad. Sci. USA, 11, 1618. Stephem.L. (1976). ;;;centrating engines . Canonical mass and the kidney. balance equation for multinephron models of the renal medulla. ._Biophy;.i: 3T;;iz;on Stephenson J. L., R. Mejia an (1976). Model of solute and water movement in the kidney. Proc. Nat. Acad. Sci. USA, 73, 252. Steohenson. J. L. (19811. Case studies in renal In F.C. and epitheli‘al physiology. Hoppensteadt (Ed.), Lectures in Applied sets 0 Mathema ica Mathematics, ~s;;!o;+~_/ Mtdthamatita? kciety! Steohenson J. L.. Y Zhanq, A. Eftekhari and R. ' Tewarson (in press);. Electrolyte transport in a central core model of the renal TewarzEdnulha'P %$&%$?J: L. Stephenson and Use of sparse matrix R. 'Mejia"(1676). techniques in numerical solution of differ eauations for renal counterflow ential systems. Come. Bio;e$: Re&,('9G5;r):. Loka_ Wirz, H.. B. Hargl ay an -1isation des Konzentrierungsprozesses in der Niere durch Kryoskopie. Helv. Physiol. Acta 9, 196.