The countercurrent mechanism works slightly differently in nephrons because the two parallel tubes, the descending branch and the ascending branch of the loop of Henle, are connected at their bottom end. Thus the flow and concentration of solute out of the descending tube must match the flow and concentration of solute into the ascending tube.
Mathematical models of the urine-concentrating mechanism have been around for some time, but all make use of the same basic physical principles, namely, the establishment of chemical gradients via active transport processes, the movement of ions via diffusion, and the transport of water by osmosis. The unique feature of the nephron is its physical organization, which allows it to eliminate waste products while controlling other quantities. In what follows we present a model similar to that of Stephenson (1972, 1992) of urinary concentration that represents the gross organizational features of the loop of Henle. A number of other models are discussed in a special issue of the Bulletin of Mathematical Biology (volume 56, number 3, May 1994), while two useful reviews of mathematical work on the kidney are Knepper and Rector (1991) and Roy et al. (1992).
We view the loop of Henle as consisting of four compartments, including three tubules, the descending limb, the ascending limb, and the collecting duct, and a single compartment for the interstitium and peritubular capillaries (Fig. 20.12). The intersti-tium/capillary bed is treated as a one-dimensional tubule that accepts fluid from the other three tubules and loses it to the venules. It is an easy generalization to separate the peritubular capillaries and interstitium into separate compartments, but little is gained by doing so. In each of these compartments, one must keep track of the flow of water and the concentration of solutes. For the model presented here, we track only one solute, Na+, because it is believed that the concentration of Na+ in the interstitium determines over 90 percent of the osmotic pressure.
We assume that the flow in each of the tubes is a simple plug flow (positive in the positive x direction) with flow rates qd,qa,qc,qs for descending, ascending, collecting, and interstitial tubules, respectively. Similarly, the concentration of solute in each of these is denoted by cd, ca, cc, cs. The tubules are assumed to be one-dimensional, with glomerular filtrate entering the descending limb at x = 0, turning from the descending limb to the ascending limb at x = L, turning from the ascending limb to the collecting x = 0
qd qc
Figure 20.12 Diagram of the simple four-compartment model of the loop of Henle.
duct at x = 0, and finally exiting the collecting duct at x = L. We assume that the interstitium/capillary compartment drains at x = 0 with no flow at x = L.
Descending limb: The flux of water from the descending limb to the interstitium is controlled by the pressure difference and the osmotic pressure difference; hence
kd dx where Pd and Ps are the hydrostatic pressures in the descending tubule and interstitium, ns is the colloidal osmotic pressure of the interstitium, and kd is the filtration rate for the descending tubule. The factor two multiplying the osmotic pressure due to the solute is to take into account the fact that the fluid is electrically neutral, and the flow of Na+ ions is followed closely by a flow of chloride ions, both of which contribute to the osmotic pressure. The transport of Na+ ions from the descending limb is governed by simple diffusion, so that at steady state we have
dx where hd is the permeability of the descending limb to Na+ ions. Ascending limb: The ascending limb is assumed to be impermeable to water, so that
dx and the flow of Na+ out of the ascending limb is by an active process, so that d(qaca)
The pump rate p certainly depends in nontrivial ways on the local concentrations of various ions. However, for this model we take p to be a constant. This simplifying assumption causes problems with the behavior of the model at low Na+ concentrations, because it allows the Na+ concentration to become negative. Although the Na+ ATPase is actually a Na+-K+ pump, the epithelial cells are highly permeable to K+, and so we assume that K+ can be safely ignored. For simplicity, we also ignore the fact that the Na+ transport properties of the thin ascending limb are different from those of the thick ascending limb, and we assume active removal along the entire ascending limb.
Collecting duct: The flow of water from the collecting duct is also controlled by the hydrostatic and osmotic pressure differences, via
1 eke: = Ps — n — pc + iRT (cc — cs), (20.27)
kc dx and the transport of Na+ from the collecting duct is governed by d(qcCc)
Here, kc and hc are the permeability of the collecting duct to water and Na+, and are controlled by ADH and aldosterone, respectively. Conservation equations: Finally, because total fluid is conserved, dqs d „
and because total solute is conserved, d(qscs) d
dx dx
To complete the description, we have the relationship between pressure and flow in a tube, dPj
for j = d, a' c, s. However, for renal modeling it is typical to take each pressure to be constant. Typical values for the pressures are Pd = 14-18 mm Hg, Pa = 10-14 mm Hg, Pc = 0-1° mm Hg, Ps = 6 mm Hg, and ns = 17 mm Hg.
This description of the nephron consists of eight first-order differential equations in the eight unknowns qj and cj, for j = d, a, c, s. To complete the description, we need boundary conditions. We assume that the inputs qd(0) and cd(0) are known and given. Then, because the flow from the descending limb enters the ascending limb, qd(L) = -qa(L) and cd(L) = ca(L). Furthermore, qs(L) = 0. At x = 0, flow from the ascending limb enters the collecting duct, so that qa (0) = — qc (0) and ca (0) = cc(0). Finally, since total fluid must be conserved, what goes in must go out, so that qd(0) + qs(0) = qc(L).
It is useful to nondimensionalize the equations by normalizing the flows and solute concentrations. Thus, we let q. q.
x = Ly, Qj = —1—, Cj = —'— for j = d, a, c, s, ' qd(0) ' cd(0)
and the dimensionless parameters are qd(0) ^ P Pj + n - Ps H_LhL Pj = 2LRTcd(0)V APj = RT2q-(0) , H = qd(0), tOT 1 = "
Three of these equations are trivially solved. In fact, it follows easily from (20.25), (20.29), and (20.30) that
Two more identities can be found. If we use (20.24) to eliminate cd — cs from (20.23), we obtain dQd 1 d(QdCd) Pd-j— + &Pd = Cd - Cs = -—----, (20.35)
d dy d d s Hd dy from which it follows that
Similarly, we use (20.28) to eliminate cc - cs from (20.27) to obtain dQr 1 d(QcCc)
dy Hc dy which integrates to
Pc(Qc - Qc(0)) + 7^(QcCc - Qc(0)Cc(0)) = -APcy. (20.38)
As discussed above, we assume that the Na+ concentration in the ascending limb is always sufficiently high so that the Na+-K+ pump is saturated and the pump rate is independent of concentration, in which case the solution of (20.26) (in nondimensional variables) is
where P = Q(0pL(0) is the dimensionless Na+ pump rate.
Having solved six of the original eight differential equations, we are left with a system of two first-order equations in two unknowns. The two equations are dQd
dy dQc
dy subject to boundary conditions Qd = 1,Qc = _Qa at y = 0, and Qd = _Qa at y = 1, where Cc,Cs, and Cd are functions of Qd and Qc. Although there are three boundary conditions for two first-order equations, the number Qa is also unknown, so that this problem is well posed. Our goal in what follows is to understand the behavior of the solution of this system.
The primary control of renal dialysis is accomplished in the collecting duct, where the amount of ADH determines the permeability of the collecting duct to water and the amount of aldosterone determines the permeability of the collecting duct to Na+. Impairment of normal kidney function is often related to ADH. For example, the inability of the pituitary to produce adequate amounts of ADH is called "central" diabetes insipidus, and results in the formation of large amounts of dilute urine. On the other hand, with "nephrogenic" diabetes insipidus, the abnormality resides in the kidney, either as a failure of the countercurrent mechanism to produce an adequately hyperosmotic in-terstitium, or as the inability of the collecting ducts to respond to ADH. In either case, large volumes of dilute urine are formed.
Various drugs and hormones can have similar effects. For example, alcohol, cloni-dine (an antihypertensive drug), and haloperidol (a dopamine blocker) are known to inhibit the release of ADH. Other drugs such as nicotine and morphine stimulate the release of ADH. Drugs such as lithium (used to treat manic-depressives) and the antibiotic tetracyclines impair the ability of the collecting duct to respond to ADH.
The second important controller of urine formation is the hormone aldosterone. Al-dosterone, secreted by zona glomerulosa cells in the adrenal cortex, works by diffusing into the epithelial cells, where it interacts with several receptor proteins and diffuses into the cell nucleus. In the cell nucleus it induces the production of the messenger RNA associated with several important proteins that are ingredients of Na+ channels. The net effect is that (after about an hour) the number of Na+ channels in the cell membrane increases, with a consequent increase of Na+ conductance. Aldosterone is also known to increase the Na+-K+ ATPase activity in the collecting duct, as well as in other places in the nephron (a feature not included in this model), thereby increasing Na+ removal and also K+ excretion into the urine. For persons with Addison's disease (severely impaired or total lack of aldosterone), there is tremendous loss of Na+ by the kidneys and accumulation of K+. Conversely, excess aldosterone secretion, as occurs in patients with adrenal tumors (Conn's syndrome), is associated with Na+ retention and K+ depletion.
To see the effect of these controls we examine the behavior of our model in two limiting cases. In the first case, we assume that there is no ADH present, so that pc = to, and that there is no aldosterone present, so that Hc = 0. In this case it follows from (20.37) that Qc = Qc(0) = _Qa and that Cc = Cc(0) = Ca(0). In other words, there is no loss of either water or Na+ from the collecting duct: the collecting duct has effectively been removed from the model.
It remains to determine what happens in the descending and ascending tubules. The flow is governed by the single differential equation
Qd + Qa subject to the boundary conditions Qd(0) = 1, Qd(1) = -Qa. As before, Qa is a constant, as the ascending limb is impermeable to water, and Ca is a linearly decreasing function of y.
We view this problem as a nonlinear eigenvalue problem, since it is a single firstorder differential equation with two boundary conditions. The unknown parameter Qa is the parameter that we adjust to make the solution satisfy the two boundary conditions. It is reasonable to take pd to be small, since the descending tubule is quite permeable to water. In this case, however, the differential equation (20.42) is singular, since a small parameter multiplies the derivative. We overcome this difficulty by seeking a solution in the form y = y(Qd, pd) satisfying the differential equation f (Qd,Qa,y= Pd (20.45)
dQd subject to boundary conditions y = 0 at Qd = 1 and y = 1 at Qd = -Qa.
With Pd small we have a regular perturbation problem in which we seek y as a function of Qd as a power series of pd, which is solved as follows. We assume that y has a power series representation of the form y = y0 + Pdy1 + P2dy2 + o(p\), (20.46)
substitute into (20.45), expand in powers of pd, collect like powers of pd, and then solve these sequentially. We find that
PQd - APdHdQa
Notice that y = 1 at Qd = -Qa. Now we determine Qa by setting y = 0, Qd = 1 in (20.47), and solving for Qa. To leading order in pd we find that
It is now a straightforward matter to plot y as a function of Qd, and then rotate the axes so that we see Qd as a function of y. This is depicted with a dashed curve in Fig. 20.13 (using formulae that include higher-order correction terms for pd). For comparison we also include the curves calculated for the case where ADH is present; the details of that calculation are given below. Note that in either the presence or absence of ADH, Qa is always independent of y, while in the absence of ADH, Qc is also independent of y. Once Qd is determined as a function of y, it is an easy matter to plot the concentrations Cd and Ca as functions of y, as shown in Fig. 20.14.
From these we can draw some conclusions about how the loop of Henle works in this mode. Sodium is extracted from the descending limb by simple diffusion and from the ascending loop by an active process. The Na+ that is extracted from the ascending loop creates a large osmotic pressure in the interstitial region that serves to enhance the extraction of water from the descending loop. This emphasizes the importance of the countercurrent mechanism in the concentrating process. As the fluid proceeds down the descending loop, its Na+ concentration is continually increasing, and during its passage along the ascending loop, its Na+ concentration falls. At the lower end of the loop the relative concentration of the formed urine (i.e., of substances that are impermeable, such as creatinine) is q^Y) . This quantity represents the "concentrating ability" of the nephron in this mode. Since Ca(0) < Cd(0), as can be seen from Fig. 20.14, by the time the fluid reaches the top of the ascending loop, it has been diluted. Furthermore, comparing the value of Qd(1)(= Qc) in the absence of ADH (dashed curve in Fig. 20.13) to the value of Qc(1) in the presence of ADH (solid curve in Fig. 20.13) shows that the flux out of the collecting duct is higher in the absence of ADH. Hence, combining these two observations, we conclude that in the absence of ADH, the nephron produces a large quantity of dilute urine, while in the presence of ADH, it produces a smaller quantity of concentrated urine. This is consistent with the qualitative explanation of nephron function given earlier in the chapter.
In Fig. 20.15 are shown the solute concentration Ca and the flow rate Q at the upper end of the ascending tubule as functions of dimensionless pump rate P. The formed urine is dilute whenever this solute concentration is less than one. The fact that this concentration can become negative at larger pump rates is a failure of the model, since the pump rate in the model is not concentration dependent.
In the presence of ADH, the collecting tube is highly permeable to water, so that, since the concentration of Na+ in the interstitium at the lower end of the tube is high, additional water can be extracted from the collecting duct, thereby concentrating the dilute urine formed by the loop of Henle.
To solve the governing equations in this case is much harder than in the case with no ADH. This is because the equations governing the flux (20.40) and (20.41) are both singular in the limit of zero pd and pc. Furthermore, one can show that the quasi-steady solution (found by setting pd = pc = 0 in (20.40) and (20.41)) cannot be made to satisfy the boundary conditions at y = 1, suggesting that the solution has a boundary layer. To avoid the difficulties associated with boundary layers, it is preferable to formulate
JO CD
\ |
^v,^ s |
-Qa | |||||||||||||||
Ql | |||||||||||||||||
Figure 20.13 The flux of fluid in the loop of Henle, with ADH present (solid curve, pc = 2.0) and without ADH present (dashed curve, pc = to). Parameter values are P = 0.9, APd = 0.15, Hd = 0.1, pd = 0.15, Hc = 0. JO CD Figure 20.13 The flux of fluid in the loop of Henle, with ADH present (solid curve, pc = 2.0) and without ADH present (dashed curve, pc = to). Parameter values are P = 0.9, APd = 0.15, Hd = 0.1, pd = 0.15, Hc = 0. Figure 20.14 The solute concentration in the descending (Cd) and ascending (Ca) tubules with no ADH present (pc = to), plotted as a function of distance y for the parameter set as in Fig. 20.13. the problem in terms of the solute flux Sd = QdCd, because according to (20.35) this function is nearly linear and does not change rapidly when pd is small. In the case that ADH is present but there is no aldosterone (Hc = 0), the governing equations are dSd = Hd(S - Sf) = HdFd (Sd, Qc), (20.49) Figure 20.15 The solute concentration and the flow rate at the upper end of the ascending tubule plotted as functions of pump rate P when there is no ADH or aldosterone present (pc = <x,Hc = 0). Dilution occurs if the solute concentration is less than one. Parameter values are APd = 0.15,Hd = 0.1,pd = 0.15. subject to boundary conditions Sd(0) = 1, Qc(0) = 1 + 1-Sd-dHpdHd, and Qd(1) = -Qa. These equations are difficult to solve because there are two unknown functions, Sd and Qc, and an unknown constant Qa, subject to three boundary conditions. One way to solve them is to introduce the constants Qa and Qc (1) as unknown variables satisfying the obvious differential equations Q = dQdy1^ = 0, and to solve the expanded fourth-order system of equations in the four unknowns Sd, Qc, Qa, Qc(1) with four corresponding boundary conditions (adding the requirement that Qc = Qc(1) at y = 1). These equations were solved numerically using a centered difference scheme for the discretization and Newton's method to find a solution of the nonlinear equations (see Exercise 8). Typical results are shown in Fig. 20.16. Here we see what we expected (or hoped), namely that the collecting duct concentrates the dilute urine by extracting water. In fact, we see that the concentration increases on its path through the descending loop, decreases in the ascending loop, and then increases again in the collecting duct. This behavior is similar to the data for Na+ concentration shown in Fig. 20.9. The effect of the parameter pc is shown in Figs. 20.17 and 20.18. In these figures are shown the solute concentrations and the flow rates at the bottom and top of the loop of Henle and at the end of the collecting duct. Here we see that the effect of ADH is, as expected, to reconcentrate the solute and to further reduce the loss of water. Figure 20.16 Solute concentrations in the loop of Henle and the collecting duct, plotted as functions of y for P = 0.9,APd = 0.15,APc = 0.22,Hd = 0.1,pd = 0.15 and with pc = 2.0,Hc = 0. Figure 20.16 Solute concentrations in the loop of Henle and the collecting duct, plotted as functions of y for P = 0.9,APd = 0.15,APc = 0.22,Hd = 0.1,pd = 0.15 and with pc = 2.0,Hc = 0. 1/Pc Figure 20.17 Solute concentrations at the bottom and top of the loop of Henle and at the end of the collecting duct plotted as functions of inverse permeability —, with P = 0.9, APd = 0.15, APc = 0.22, Hd = 0.1, pd = 0.15, and Hc = 0. The asymptotic value of Cc(1) as pc ^ 0 is the maximal solute concentration possible and determines, for example, whether or not the individual can safely drink seawater without dehydration. The asymptotic value of 1/Qc(1) represents the highest possible relative concentration of impermeable substances such as creatinine. Further generalizationsThis model shows the basic principles behind nephron function, but the model is qualitative at best, and there are many questions that remain unanswered and many generalizations that might be pursued. For example, the model could be improved by incorporating a better representation of the interstitial/capillary bed flow, taking into account that the peritubular capillaries issue directly from the efferent arteriole of the 1/Pc Figure 20.18 Fluidflowratesatthe bottom of the loop of Henle and at the end of the collecting duct plotted as functions of inverse permeability -1, with P = 0.9,APd = 0.15,APC = 0.22,Hd = 0.1,pd = 0.15, and Hc = 0. glomerulus, thus determining the hydrostatic and osmotic pressures in the capillary bed. The model is also incorrect in that the active pumping of Na+ out of the ascending limb is not concentration dependent, and as a result negative concentrations can occur for certain parameter values. It is a fairly easy matter to add equations governing the flux of solutes other than Na+, as the principles governing their flux are the same. One can also consider a time-dependent model in which the flow of water is not steady, by allowing the cross-sectional area of the tubules to vary. Nonsteady models are difficult to solve because they are stiff, and there is a substantial literature on the numerical analysis and simulation of time-dependent models (Layton et al., 1991). Nephrons occur in a variety of lengths, and models describing kidney function have been devised that recognize that nephrons are distributed both in space and in length. These models are partial differential equations, and again, because of inherent stiffness, their simulation requires careful choice of numerical algorithms (Layton et al., 1995). 20.3 Exercises_ 1. The flow of glomerular filtrate and the total renal blood flow increase by 20 to 30 percent within 1 to 2 hours following a high-protein meal. How can you incorporate this feature into a model of renal function and regulation of glomerular function? Hint: Amino acids, which are released into the blood after a high protein meal, are cotrans-ported with Na+ ions from the filtrate in the proximal tubule. Thus, high levels of amino acids leads to high reabsorption of Na+ in the proximal tubule, and therefore, lower than normal levels of Na+ at the macula densa. 2. How much water must one drink to prevent any dehydration after eating a 1.5 oz bag of potato chips? (See Exercise 10 in Chapter 2.) Remark: A mole of NaCl is 58.5 grams and it dissociates in water into 2 osmoles.
Figure 20.19 Diagram of a countercur-rent flow mediated by an interstitium, for Exercise 4. 3. Why is alcohol a diuretic? What is the combined effect on urine formation of drinking beer (instead of water) while eating potato chips? What is the combined effect on urine formation of drinking beer while smoking cigarettes? Hint: Alcohol inhibits the release of ADH, while nicotine stimulates ADH release. 4. Construct a simple model of the countercurrent mechanism that includes an interstitial compartment (Fig. 20.19). Show that inclusion of the interstitium has no effect on the overall rates of transport. Allow the solute to diffuse in the interstitium, but not escape the boundaries. Hint: View the interstitium as a tube with zero flow rate. 5. Generalize the four-compartment model for the loop of Henle by separating the interstitium and peritubular capillaries into separate compartments, allowing no flow across x = 0 or x = L for the interstitium. 6. What changes in the exchange rates of the four compartment model for the loop of Henle might better represent the geometry of the loop of Henle, as depicted in Fig. 20.2? Remark: Some features you might want to consider include the location of the thickening of the ascending and descending limbs and the location of the junction of the peritubular capillaries with the arcuate vein. 7. Formulate a time-dependent four-compartment model of urine concentration that tracks the concentration of both Na+ ions and urea. 8. Develop a numerical computer program to solve the equations of renal flow in the case that both ADH and aldosterone are present. It is preferable to formulate the problem in terms of the unknowns Sd and Sc and to expand the system of equations to a fourth-order system by allowing Sd(1) and Sc(1) to be unknowns that satisfy the simple differential equations ^dy1 = 0 and dS|(1) = 0. With the 4 unknowns, Sd(y),Sc(y),Sd(1), and Sc(1), the Jacobian matrix is a banded matrix, and numerical algorithms to solve banded problems are faster and more efficient than full matrix solvers. 9. Generalize the renal model to include a concentration-dependent Na+ pump in the ascending tubule. Does this change in the model guarantee that the flux and concentrations are nowhere negative? CHAPTER 21 |
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