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Am J Physiol Renal Physiol 273: F625-F634, 1997;
0363-6127/97 $5.00
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Vol. 273, Issue 4, F625-F634, October 1997

MODELING IN PHYSIOLOGY
Nonlinear filter properties of the thick ascending limb

H. E. Layton1, E. Bruce Pitman2, and Leon C. Moore3

1 Department of Mathematics, Duke University, Durham, North Carolina 27708-0320; 2 Department of Mathematics, State University of New York, Buffalo 14214-3093; and 3 Department of Physiology and Biophysics, State University of New York, Stony Brook, New York 11794-8661

    ABSTRACT
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References

A mathematical model was used to investigate the filter properties of the thick ascending limb (TAL), that is, the response of TAL luminal NaCl concentration to oscillations in tubular fluid flow. For the special case of no transtubular NaCl backleak and for spatially homogeneous transport parameters, the model predicts that NaCl concentration in intratubular fluid at each location along the TAL depends only on the fluid transit time up the TAL to that location. This exact mathematical result has four important consequences: 1) when a sinusoidal component is added to steady-state TAL flow, the NaCl concentration at the macula densa (MD) undergoes oscillations that are bounded by a range interval envelope with magnitude that decreases as a function of oscillatory frequency; 2) the frequency response within the range envelope exhibits nodes at those frequencies where the oscillatory flow has a transit time to the MD that equals the steady-state fluid transit time (this nodal structure arises from the establishment of standing waves in luminal concentration, relative to the steady-state concentration profile, along the length of the TAL); 3) for any dynamically changing but positive TAL flow rate, the luminal TAL NaCl concentration profile along the TAL decreases monotonically as a function of TAL length; and 4) sinusoidal oscillations in TAL flow, except at nodal frequencies, result in nonsinusoidal oscillations in NaCl concentration at the MD. Numerical calculations that include NaCl backleak exhibit solutions with these same four properties. For parameters in the physiological range, the first few nodes in the frequency response curve are separated by antinodes of significant amplitude, and the nodes arise at frequencies well below the frequency of respiration in rat. Therefore, the nodal structure and nonsinusoidal oscillations should be detectable in experiments, and they may influence the dynamic behavior of the tubuloglomerular feedback system.

kidney; renal hemodynamics; spectral analysis; mathematical model

    INTRODUCTION
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References

EXPERIMENTS ON THE RAT RENAL vasculature have revealed that flow and pressure, considered as functions of time, exhibit substantial spectral structure, both in single nephrons and in whole kidneys (3, 11, 29, 30). This spectral structure, which is manifested as the superposition of several oscillatory components distinguished by differing characteristic frequencies, is being used to investigate renal hemodynamic regulatory mechanisms (10). Experimental measurements and theoretical considerations indicate that the spectral structure arises in part from the dynamics of the tubuloglomerular feedback (TGF) pathway; indeed, an oscillation of 20-50 mHz has been identified as arising directly from an instability in the TGF loop (8, 9, 15, 18, 19). An intrinsic oscillation of the afferent arteriole at ~120 mHz has been proposed as another source of spectral structure (2, 30).

Mathematical models have indicated that the frequency response of the TGF pathway may be influenced by the transport characteristics of the thick ascending limb (TAL) (9, 15, 22). Specifically, model simulations showed that when oscillations in nephron flow have a period significantly longer than the steady-state fluid transit time of the TAL, then fluid arriving at the macula densa (MD), which has been in the TAL for widely varying time intervals, exhibits large oscillations in NaCl concentration. On the other hand, when flow oscillations have a period significantly shorter than the steady-state transit time of the TAL, then all the fluid arriving at the MD has been in the TAL for about the same time interval, and consequently there is little variation in NaCl concentration. These results suggest that slow oscillations should be transmitted through the TAL, and fast oscillations should be attenuated, i.e., the TAL operates as a low-pass filter.

In this study, we used a simple mathematical model to investigate the characteristics and mechanistic origins of the TAL low-pass filter. For this model, an explicit mathematical analysis can be carried out in the idealized case where transtubular NaCl backleak is set to zero and NaCl active transport parameters have no spatial dependence. Under these assumptions, luminal NaCl concentration at each location along the TAL is a decreasing function of TAL transit time to that location.

We now summarize four important results of the analysis of this idealized case, all of which arise directly from the luminal concentration dependence on transit time. Numerical studies demonstrate that these exact mathematical results remain valid when NaCl backleak is included.

First, when a sinusoidal component is added to steady-state TAL flow, the model predicts that the amplitude of resulting oscillations in NaCl concentration at the MD will be bounded by a range interval envelope with magnitude that decreases as a function of oscillatory frequency. For sufficiently large frequencies, the range interval magnitude is inversely proportional to frequency. This result, consistent with previous findings (9, 22), characterizes, in part, the low-pass filter action of the TAL.

Second, the model predicts that the amplitude of the oscillations in NaCl concentration at the MD, when plotted as a function of frequency, will exhibit a series of nodes. The nodes will appear at those frequencies where the oscillatory flow has a transit time to the MD that equals the steady-state fluid transit time: because the fluid transit time to the TAL is constant, a constant NaCl concentration will result at the MD. The model further predicts that the nodes will be associated with the formation of standing waves in luminal concentration along the TAL, relative to the steady-state concentration profile. These standing waves have wavelength that is inversely proportional to the frequency of the flow oscillation, and a node in NaCl concentration will be located at the MD when the TAL length is an integer multiple of one-half the wavelength.

Third, for any dynamically changing, but positive, luminal TAL fluid flow rate, the luminal TAL concentration profile decreases monotonically as a function of TAL length, relative to the TAL entrance, since the fluid at each location along the TAL has spent progressively more time in transit through the TAL.

Finally, sinusoidal oscillations in TAL flow, except at nodal frequencies, result in nonsinusoidal oscillations in NaCl concentration at the MD. This effect arises from the nonlinear relationship between transit time and flow rate.

In this study, we first describe a simple mathematical model of the TAL. Then, in RESULTS, we summarize calculations that demonstrate the dependence of luminal NaCl concentration on transit time. Using these calculations and numerical solutions of the TAL model, we obtain bounds on MD concentration excursions as a function of frequency, we elucidate the nodal structure of the frequency response, and we exhibit luminal concentration profiles and associated time records of MD concentration. In the DISCUSSION, we consider the adequacy of the model assumptions and the physiological significance of these results.

Glossary Parameters

Co Chloride concentration at TAL entrance (mM)
f Frequency of flow oscillations (mHz)
Km Michaelis constant (mM)
L Length of TAL (cm)
p Period of flow oscillations (s)
Qop SNGFR (nl/min)
r Luminal radius of TAL (µm)
P TAL chloride permeability (cm/s)
to Steady-state TAL transit time (s)
Vmax Maximum transport rate of chloride from TAL (nmol · cm-2 · s-1)
 alpha Fraction of SNGFR reaching TAL
 epsilon Fractional amplitude of flow oscillations
 phi Phase shift of flow oscillations

Independent Variables

x Axial position along TAL (cm)
t Time (s)

Specified Functions

Ce(x) Interstitial chloride concentration (mM)
F(t) TAL fluid flow rate (nl/min)

Dependent Variables

C(x, t) TAL chloride concentration (mM)
S(x) Steady-state TAL chloride concentration (mM)
T(x, t) Fluid transit time from TAL entrance (s)

    MATHEMATICAL MODEL

Model equation. For simplicity, we model the TAL chloride concentration only, and we assume that sodium is absorbed in parallel with chloride. The chloride ion is thought to be the species sensed by the MD in the TGF response (26), and NaCl backleak is limited by the smaller epithelial permeability of chloride, relative to sodium (24).

The principal model equation, a partial differential equation for the chloride ion concentration C in the luminal fluid of the TAL of a short-looped nephron (14, 15, 23), is given by
<FR><NU>∂</NU><DE>∂<IT>t</IT></DE></FR> C(<IT>x</IT>, <IT>t</IT>) = −<IT>F</IT>(<IT>t</IT>) <FR><NU>∂</NU><DE>∂<IT>x</IT></DE></FR> C(<IT>x</IT>, <IT>t</IT>)
− <FR><NU><IT>V</IT><SUB>max</SUB>C(<IT>x</IT>, <IT>t</IT>)</NU><DE><IT>K</IT><SUB>m</SUB> + C(<IT>x</IT>, <IT>t</IT>)</DE></FR> − <IT>P</IT>(C(<IT>x</IT>, <IT>t</IT>) − C<SUB>e</SUB>(<IT>x</IT>)) (1)
Equation 1 is in nondimensional form, i.e., all variables and parameters have been normalized so that each is a dimensionless quantity (see APPENDIX A of the companion study, Ref. 17). The space variable x is oriented so that it extends from the entrance of the TAL (x = 0) through the outer medulla, and into the cortex to the site of the MD (x = 1). We impose a boundary condition given by C(0, t) = 1, which means that the fluid entering the TAL has constant chloride concentration; at time t = 0, the initial function C(x, 0) must be specified for x is in  (0, 1].

The rate of change of concentration in the TAL depends on processes represented by the three right-hand terms in Eq. 1. The first term is axial convective chloride transport at the intratubular flow speed F, taken in this study to be a specified function of time t. The second is the transtubular efflux of chloride driven by metabolic pumps situated in the tubular walls; that efflux is approximated by Michaelis-Menten kinetics, with maximum transport rate Vmax and Michaelis constant Km. The third term is transtubular chloride backleak, which depends on a specified fixed extratubular chloride concentration profile Ce(x) and on membrane chloride permeability P.

A steady-state solution to Eq. 1 may be obtained by setting F = 1 for 1 unit of normalized time (the transit time of the TAL at flow speed 1). We denote the resulting steady-state TAL concentration profile C(x, 1) by S(x).

Model parameters. As already noted, we will establish several exact mathematical results for the idealized case where the TAL has no transtubular NaCl backleak. Numerical calculations and analytical results have shown that the idealized case provides a good qualitative guide to the results obtained when a value consistent with experimental measurements of backleak permeability is used (15, 16). To make the results from the two cases quantitatively comparable, the transport parameters (Vmax and Km) for the no-backleak case were chosen to give a steady-state concentration at the MD nearly equal to that used for the backleak case, as well as to closely approximate the steady-state backleak case response of MD concentration to small flow variations relative to the steady-state flow rate.

A summary of parameters and variables, with their dimensional units as commonly reported, is given in the Glossary. Parameter values are given in Table 1 for both the idealized case with no transtubular chloride backleak and the case with measured backleak permeability. Detailed parameter selection criteria (for both cases) and supporting references can be found in Ref. 15. The chloride backleak permeability was taken to be 1.5 × 10-5 cm/s, a value consistent with measured chloride permeability for rabbit (1.06 ± 0.12 × 10-5 cm/s in Ref. 25) and estimated NaCl permeability for rat (1.13 ± 0.52 × 10-5 cm/s in Ref. 20). The remaining transport parameters for the backleak case, Vmax and Km, were chosen to give a steady-state intratubular chloride concentration and concentration slope at the MD consistent with experiments. The extratubular chloride concentration Ce is given in nondimensional form by Ce(x) = Co(A1 exp(-A3 x) + A2), where A1 = (1 - Ce(1)/Co)/(1 - e-A3), A2 = 1 - A1, and A3 = 2. The interstitial concentration at the MD, Ce(1), corresponds to 150 mM. Graphs of Ce, the steady-state backleak-case TAL concentration profile, and the backleak and no-backleak steady-state MD chloride concentrations as a function of flow rate were given in figures 1 and 2 of Ref. 15.

                              
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  		Table 1.   Parameter sets with and without chloride backleak


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  		Fig. 1.   Range of macula densa (MD) chloride concentration as a function of oscillatory flow frequency. The marked frequencies n/to, for n = 1, 2, 3, 4, and 5 correspond to frequencies (in mHz) of ~63.7, 127, 191, 255, and 318, respectively [cf. figure 2 in the companion study (17)]. Dashed curves are approximate bounds for chloride concentration excursions; solid curves provide precise bounds for the concentration range, thus revealing the nodal structure of the frequency response. Curves, computed from analytic expressions, are based on the parameters for no chloride backleak. Vertical bars give the range of chloride excursions obtained by numerically solving Eq. 1; parameters for the cases without and with chloride backleak were used to compute these bars in A and B, respectively. Nodal structure for the backleak parameters, indicated by the vertical bars in B, follows the the same pattern as that predicted by the exact results for the no- backleak parameters.

The steady-state transit time of fluid up the TAL, from the TAL entrance near the loop bend to the MD, is given by to = pi r2L/(alpha Qop), the tubular volume divided by the steady-state flow rate. The steady-state transit time, which corresponds to one unit of normalized time, is a key parameter that plays a prominent role in this study and the companion study (17). The corresponding steady-state chloride concentration at the MD is given by Cop, which is computed numerically from Eq. 1 and which equals S(1).

    RESULTS
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References

Chloride concentration at MD depends on TAL transit time. In this subsection we provide mathematical expressions for TAL transit time as a function of TAL flow speed (Eq. 2) and for MD chloride concentration as a function of TAL transit time (Eq. 7). Except where clearly marked, dimensionless variables are used for mathematical simplicity.

If we assume that intratubular flow in the TAL is plug-flow, and if we follow the advance of a water molecule up the (water-impermeable) TAL, after its entry into the TAL at x = 0, then we may compute the time of transit T(x, t) required for the molecule to reach position x is in  [0, 1] at time t. We assume that flow speed F can differ from the steady-state value 1 by no more than an amount epsilon , with 0 <=  epsilon  < 1. Thus flow is always positive with (1 - epsilon ) <= F <=  (1 + epsilon ). With these specifications, transit time is given implicitly by the integral relation
<IT>x</IT> = <LIM><OP>∫</OP><LL><IT>t</IT> − <IT>T</IT>(<IT>x</IT>, <IT>t</IT>)</LL><UL><IT>t</IT></UL></LIM><IT>F</IT>(<IT>u</IT>) d<IT>u</IT> (2)
which asserts that the distance x traveled up the TAL is the integral of speed F, taken over the interval of transit. Bounds on T follow directly from Eq. 2 and the bounds on F
<FR><NU><IT>x</IT></NU><DE>1 + &egr;</DE></FR> ≤ <IT>T</IT>(<IT>x</IT>, <IT>t</IT>) ≤ <FR><NU><IT>x</IT></NU><DE>1 − &egr;</DE></FR> (3)
If flow speed F is constant, then transit time is inversely proportionally to F, with T(x, t) = x/F.

In this study, we will frequently think of the flow speed F as having an average of unity plus a sinusoidal component of amplitude epsilon , 0 <=  epsilon  < 1; in that case
<IT>F</IT>(<IT>t</IT>) = 1 + &egr; sin (2&pgr;<IT>ft</IT> + &phgr;) (4)
where phi  is a phase shift, and f is frequency (cycles per unit nondimensional time). The resulting relationship between x and T, by explicit evaluation of Eq. 2, is
<IT>x</IT> = <IT>T</IT>(<IT>x</IT>, <IT>t</IT>) + <FR><NU>&egr;</NU><DE>2&pgr;<IT>f</IT></DE></FR> (cos (2&pgr;<IT>f</IT>[<IT>t</IT> − <IT>T</IT> (<IT>x</IT>, <IT>t</IT>)] + &phgr;)
− cos (2&pgr;<IT>ft</IT> + &phgr;)) (5)
Equation 5 defines an implicit function T(x, t), which is periodic in t with period p = 1/f.

For sufficiently large frequencies ( f >> epsilon ), the normalized distance is approximately equal to the normalized transit time, i.e., x approx  T. When x = 1, this corresponds to a result previously known: for sufficiently fast oscillations, tubular fluid reaching the MD has a transit time nearly equal to the steady-state transit time (9, 15, 22). (Because the normalized value for the steady-state flow speed is 1, a speed relating x and T does not appear in Eqs. 3 and 5).

For sufficiently small frequencies f, the general bound of Eq. 3 provides a more restrictive bound on transit time T than does Eq. 5, if one assumes only that |cos(2pi f [t - T(x, t)] + phi ) - cos(2pi ft + phi )| <=  2. Thus, for oscillatory F as in Eq. 4, Eqs. 3 and 5 taken together, provide bounds (independent of phase shift phi ) on the variation of T as a function of time at each x
max  <FENCE><FR><NU><IT>x</IT></NU><DE>1 + &egr;</DE></FR>  ,  <IT>x</IT>  −  <FR><NU>&egr;</NU><DE>&pgr;<IT>f</IT></DE></FR></FENCE> ≤  <IT>T</IT>(<IT>x</IT>, <IT>t</IT>)  ≤  min  <FENCE><FR><NU><IT>x</IT></NU><DE>1  −  &egr;</DE></FR>  , <IT>x</IT> + <FR><NU>&egr;</NU><DE>&pgr;<IT>f</IT></DE></FR></FENCE> (6)
When x is set to 1, Eqs. 3, 5, and 6 provide information about the transit time to the MD.

A particularly important case arises when x = 1 and Tf = n, n = 1, 2, 3, ... Under these conditions, the cosine terms in Eq. 5 cancel, since cosine is 2pi periodic, and consequently T = 1, which indicates that the transit time equals the steady-state transit time of one nondimensional unit. Since frequency f and oscillatory period p are related by f = 1/p, transit time T will equal the steady-state transit time whenever the steady-state transit time is an integer multiple of the period of an oscillation in flow speed F, i.e., whenever F is given by Eq. 4 with f = n.

Although derived with a sinusoidal perturbation, this result also holds for any general periodic flow oscillation in F that has average speed of 1 and a period that evenly divides the steady-state transit time, because any periodic waveform can be constructed as a constant plus a Fourier series of sine functions with frequencies that are integer multiples of the fundamental frequency.

In APPENDIX A we obtain a formal mathematical expression for the TAL chloride concentration profile, under the assumption that TAL chloride backleak permeability is zero. That expression shows that the concentration at any location in the TAL is a function of transit time T only. Also in APPENDIX A, we show that in the absence of backleak or other spatially inhomogeneous influences on transport rate, the axial TAL concentration profile is monotone decreasing in x at each time t. Numerical calculations, such as those represented in Fig. 2, below, suggest that concentration profiles for the backleak case are also monotone decreasing for parameter values in the physiological range.


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  		Fig. 2.   Chloride concentration profiles in thick ascending limb (TAL) showing nodal points forming and moving to the left as frequency is increased. These profiles were computed from Eq. 1 using a sinusoidal flow perturbation in F with the specified frequencies f and transport parameters for the backleak case. In A-D the wide shaded curve represents the steady-state concentration profile. Other curves correspond to the elapsed times of 0, 1/4, 1/2, and 3/4 of the period of the oscillation. Bar to the right in A-D gives the range of concentration excursion of the oscillation in MD concentration for each period of the oscillation; numbers above and below the bar give the upper and lower bounds of the excursions in mM. B and D correspond to the frequencies of the first two nodes illustrated in Fig. 1. A and C correspond to intermediate frequencies, which produce large magnitude concentration oscillations at the MD. Standing waves, relative to the steady-state concentration profile, can be clearly observed in B-D.

For the particular case of Michaelis-Menten kinetics and no chloride backleak, the analysis in APPENDIX A implies that the concentration C(x , t) and the transit time T(x, t) are related by
(C(<IT>x</IT>, <IT>t</IT>))<SUP><IT>K</IT><SUB>m</SUB></SUP> <IT>e</IT><SUP>C(<IT>x</IT>, <IT>t</IT>)</SUP> = <IT>e</IT><SUP>1 − <IT>V</IT><SUB>max</SUB><IT>T</IT>(<IT>x</IT>, <IT>t</IT>)</SUP> (7)
Because the left side of Eq. 7 is monotone increasing in C, a decrease in transit time T to a fixed location x will cause C to increase, as expected.

TAL frequency response. In this subsection and its sequel we present numerical results obtained from the evaluation of Eq. 1 and the equations in the previous subsection. Numerical calculations were conducted as described in APPENDIX B.

Figure 1 shows the ranges of MD chloride excursions arising from dimensional oscillatory flow of the form
<IT>F</IT> = &agr;Q<SUB>op</SUB>(1 + &egr; sin (2&pgr;<IT>ft</IT>)) (8)
where alpha Qop is the steady-state TAL flow rate (6 nl/min in our model), epsilon  = 3/10 approximates the maximum fractional amplitude permitted by the TGF response (15), and f is frequency, corresponding to the abscissa of the graphs.

The dashed curves in Fig. 1 are theoretical bounds on the chloride concentration range for the parameters corresponding to no chloride backleak; the curves were computed from Eq. 7 and the bounds on transit time given by Eq. 6. The dashed curves in Fig. 1 provide a scaling envelope: the difference between the upper and lower dashed curves (in mM) is about 1200/(f × mHz-1) for f > 64 mHz (64 mHz approx  1/to), indicating that the TAL filter exhibits 1/f scaling. This scaling arises because transit time tends to the steady-state value of 1 with 1/f scaling, i.e., for sufficiently large f and for x = 1, the deviation of transit time from steady-state transit time is inversely related to f, in the sense that |T(x, t) - 1| <=  epsilon /(pi f ) (a consequence of Eq. 5). The deviations in MD concentration resulting from deviations in transit time decrease with essentially the same scaling, relative to the steady-state MD concentration. (This scaling appears to be unrelated to the 1/f scaling that arises in the evolution of certain extended dissipative systems; Ref. 1.)

The solid curves, also computed for no-backleak parameters, are precise upper and lower bounds for the chloride concentration range; they were computed from Eqs. 5 and 7, and they take into account the variations in transit time that are implicit in Eq. 5. Because they are upper and lower bounds, the two solid curves do not cross at the nodes; rather, they touch at the dimensional nodal frequencies given by f = n/to, n = 1, 2, 3, ... (the frequencies identified as nodal frequencies in the previous section), and then they separate again. For the choice of parameters in Table 1, the first five nodal frequencies are 63.66, 127.3, 191.0, 254.6, and 318.3 mHz. The first four antinodes are at frequencies 90.72, 156.0, 221.2, and 284.9 mHz, while the frequencies intermediate between the nodal frequencies are 95.49, 159.2, 222.8, and 286.5 mHz. Thus the antinodes are nearly halfway between the nodes, and antinodes are found nearer the intermediate frequencies as frequency increases.

The vertical gray bars in Fig. 1A represent MD chloride excursions from the steady-state concentration; we computed these numerically from Eq. 1, using no-backleak parameters, with F given by Eq. 8; these calculations provide an independent check on the analytic results. Excursions computed at predicted nodal frequencies had essentially zero amplitude, as predicted; at other frequencies, the excursions reached exactly to the solid curves in Fig. 1.

The vertical gray bars in Fig. 1B were also computed from Eq. 1, but with transport parameters for the case with a nonzero value for chloride backleak. These excursions from steady-state are somewhat attenuated, relative to those in Fig. 1A, except at nodal frequencies, where the excursions now have non-zero amplitude. Additional calculations showed that the amplitudes of excursions at nodal frequencies are local minima, when amplitude is considered as a function of frequency; thus the nodal frequencies are not displaced by backleak. Additional evidence is provided by figure 2 and table 2 in the companion study (17), where different methods show that the approximate nodes associated with the backleak case are located, as a function of frequency, where predicted by the explicit analysis, with relative error of less than 1.3%, for f <=  1800 mHz.

Table 2 gives the concentration minima, maxima, and range magnitudes, for the no-backleak and backleak cases, corresponding to the frequencies n/(2to), n = 1, 2, 3, ..., 10. Also shown are the analogous values computed in the limit as f approaches zero from above; these limiting values correspond to the maximum range magnitude permitted by the TGF response. Table 2 shows that for the backleak case the range of the excursions at each nodal frequency is approximately 10 times smaller than at the immediately preceding intermediate frequency.

                              
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  		Table 2.   Chloride excursions from the steady state at the macula densa for parameter sets with and without chloride backleak

Chloride concentration profiles. A natural question that emerges from this analysis is: What are the spatial characteristics of TAL chloride concentration profiles that lead to the nodal structure at the MD? Using Eq. 1 with flow given by Eq. 8, we computed axial profiles for the backleak case (Fig. 2). The amplitude of the oscillations was set to epsilon  = 9/10 so that deviations from the steady state would be sufficiently pronounced to be easily observable in graphs. This large amplitude is for illustrative purposes only, inasmuch as the node/antinode structure for amplitude in the physiological range has already been determined. However, these illustrative calculations serve to emphasize that the nodal structure is insensitive to the amplitude of the sinusoidal oscillation, provided that flow remains positive. Moreover, since large amplitude flow oscillations result in large concentration deviations from steady-state, spatial inhomogeneities in transport rate will be magnified; nonetheless, the nodes remain at frequencies predicted by the exact analysis for the idealized case of no chloride backleak.

In Fig. 2, A-D correspond to the frequencies f = n/(2to), n = 1, 2, 3, 4. In each panel the wide, shaded curve is the steady-state profile corresponding to F = alpha Qop. The other curves represent the chloride profiles, after oscillatory flow has been established, at times t = 0, p/4, p/2, and 3p/4, where in each case p is the period of the oscillation. The bar at the right of each panel in Fig. 2 gives the chloride excursion at the MD during each flow oscillation.

In Fig. 2A, for f = 0.5/to, which corresponds to a period twice that of the steady-state transit time to, there are large excursions of the concentration from the steady-state curve, except at the point of entrance (x = 0), which is a fixed boundary value. In Fig. 2B, for f = 1/to, which corresponds to a period that equals the steady-state transit time, there are large excursions from the steady-state curve, except at the TAL entrance and the MD (at normalized length x = 1), where there is an approximate node. The dynamic concentration profiles in Fig. 2B show that a standing wave has been established, relative to the steady-state chloride concentration profile. The standing wave has an internodal distance equal to the length of the TAL.

In Fig. 2C, for f = 1.5/to, we see the effect of increasing frequency. The approximate node moves from the normalized length 1 to 2/3, and the excursions at the MD are again large, as in Fig. 2A. As frequency increases further to f = 2/to in Fig. 2D, the node moves to normalized length 1/2, and a new node appears at length 1. A full standing wave with wavelength equal to the length of the TAL has been established.

This pattern of node formation and displacement to the left, as a function of increasing frequency, is a property of the standing waves that are established as a consequence of the sustained flow oscillations. When Tf = n, with n = 1, 2, 3, ..., Eq. 5 implies that transit time is time independent at locations along the TAL where normalized length x equals n/f. This time independence, combined with concentration dependence on transit time, implies that these TAL locations correspond to nodes (or approximate nodes in the presence of chloride backleak). If we let xn designate these nodal locations and write the relationship between nodal location and frequency in terms of dimensional variables, we obtain xn = n(L/to)/f, where L is TAL length, to is steady-state transit time, and, consequently, L/to is steady-state flow speed. According to this relationship, an oscillatory component with frequency f will produce standing waves with nodes at lengths xn, n = 1, 2, 3, ..., along the TAL. (The wavelength of a standing wave is equal to twice the internodal distance, i.e., 2xn). In addition, the relationship xn = n(L/to)/f implies that when xn = L, the frequencies that will produce nodes at the MD are given by fn = n/to.

Several features predicted by the explicit analysis for the idealized case with no chloride backleak can be observed in the concentration profiles shown in Fig. 2. First, despite large excursions in flow, the profiles are monotone decreasing along the TAL (see text near Eq. 7 and near Eq. A7 in APPENDIX A). Second, since increasing frequency results in transit times that more nearly approximate steady-state transit times (as indicated by Eq. 5), maximal excursions from the steady-state concentration profile decrease in amplitude with increasing frequency.

Finally, the asymmetry of concentration profiles around the steady-state profile in Fig. 2, A-D, indicates that the oscillatory time course of chloride concentration at each nonnodal site along the TAL is nonsinusoidal. Figure 3 illustrates the oscillations at the MD corresponding to Fig. 2, A-D. The oscillations at frequencies 0.5/to and 1.5/to (cf. Fig. 2, A and C) are clearly nonsinusoidal and exhibit an asymmetry in which concentration increases more rapidly than it decreases. This asymmetry is a consequence of the implicit nonlinear relationship, between time t and transit time T in Eq. 5, for fixed x and for Tf not equal  n. A second nonlinear feature is a flattening of the trough of the oscillation with largest amplitude ( f = 0.5/to). This is attributable, in large part, to the approach of luminal chloride concentration to the limiting minimal value attainable by transepithelial chloride transport when solute backleak is present (cf. figure 2 in Ref. 15). These distorted waveforms are considered in detail in the companion study (17), where the effects of the TAL low-pass filter on the dynamics of the TGF system are evaluated.


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  		Fig. 3.   Oscillations in MD chloride concentration associated with the cases of Fig. 2, as a function of time. Steady-state MD concentration is ~32 mM. At MD nodal frequencies, concentration excursions have amplitudes contained within the shaded bar (cf. Fig. 2, B and D); at intermediate frequencies, concentration excursions are much larger (cf. Fig. 2, A and C) and are nonsinusoidal as a result of the nonlinear action of the TAL filter. As a result of periodicity, the pattern here is repeated in each time interval of 2to.

Figure 3 also contrasts MD chloride concentration oscillations at nodal frequencies (cf. Fig. 2, B and D). The shaded bar in Fig. 3, which includes the steady-state concentration value of ~32 mM, contains the approximate nodes, which have oscillations in concentration between 30 and 35 mM.

    DISCUSSION
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References

This study provides quantitative predictions of the degree by which the amplitude of oscillations in MD chloride concentration is reduced as a function of frequency of oscillatory fluid flow, and it predicts that the frequency response of NaCl concentration at the MD will have a nodal structure, which arises from the establishment of standing waves in luminal concentration. These results explain and extend earlier model simulations that indicated that the TAL acts as a low-pass filter (9, 15, 22).

In this study we have considered two basic cases, an idealized case where the transepithelial chloride transport rate was assumed to depend only on local luminal chloride concentration, with no spatial inhomogeneity arising from chloride backleak, and a more realistic case with chloride backleak determined by a computed luminal concentration and a time-independent interstitial concentration formulated to approximate the cortical and outer medullary concentration profile (see Ref. 15). The idealized case is important because exact mathematical equations can be obtained in which the effects of all parameters can be readily ascertained. In particular, the equations obtained in the RESULTS section demonstrate that the nodal response pattern emerges from the structural assumptions of the model and are not sensitive to the particular parameter choices given in Table 1: for the assumptions of the idealized case, luminal chloride concentration depends only on transit time, and nodes will appear at spatial locations where transit time is constant.

In an actual TAL, however, the transepithelial chloride transport rate will depend, to some extent, on spatial inhomogeneities such as those introduced by backleak, which will likely be more significant in the cortex than in the outer medulla as a result of a larger transepithelial concentration difference. By finding numerical solutions, we have evaluated the effect of a chloride backleak permeability with magnitude in the upper range of two experimental measurements (see section MATHEMATICAL MODEL, Model parameters). The numerical solutions exhibit approximate nodes rather than the perfect nodes of the exact analysis, but, nonetheless, the nodes arise in the locations predicted by the exact analysis, both as a function of frequency and of TAL length. That is to say, the approximate nodes appear at the sites that would be predicted if luminal chloride concentration were a strict function of transit time only.

This point is most forcefully made by results shown in Fig. 2, C and D, where approximate nodes appear within the model TAL. The exact analysis predicts that the first nodes will appear at normalized lengths 2/3 and 1/2. Despite unrealistically large oscillations in flow, which tend to magnify the spatial inhomogeneity arising from chloride backleak, careful examination of the numerical results showed that the approximate nodes are within 0.2% of the predicted lengths, which is on the order of the error introduced by the numerical methods. In the companion study (figure 2 of Ref. 17), we find for this case with chloride backleak that the nodes differ by less than 1.3% from the predicted locations, as a function of oscillatory frequency, for small-amplitude oscillations below 1800 mHz.

It must be acknowledged that other forms of spatial inhomogeneity in transport may affect these results, e.g., inhomogeneities in luminal diameter and transport capacity. However, in view of incomplete experimental knowledge of these factors and the good agreement between experiments and model predictions described in the companion study (17), simulation studies based on a more detailed mathematical model may not now be warranted.

Other nonideal factors that may affect the application of model results to the physiological setting include the elastic compliance of the TAL and axial diffusion of NaCl within TAL luminal flow. However, experimental evidence suggests that tubular compliance becomes a significant effect at frequencies of 1 Hz (8); thus spectral characteristics below 500 mHz are not likely to be significantly affected [see APPENDIX C in the companion study (17)]. Standard estimates indicate that diffusion of NaCl along the axis of the lumen will not be a significant factor (14).

A final nonideal factor is the use of a simple, single-barrier, pump-leak representation of TAL NaCl transport that ignores the response dynamics of the TAL cells. This approach implicitly assumes an instantaneous response to changes in luminal NaCl concentration, whereas the response of epithelial cells should exhibit at least two time constants. The first represents the initial response of NaCl uptake through the apical membrane. This response should be very rapid, as it is only limited by the molecular dynamics of the cotransport systems and ion channels in the apical membrane. Most epithelial cell models assume that this step has an instantaneous response (see, e.g., Refs. 5 and 13). The second component of the cellular response represents the transition of cytosolic ion concentrations and cell volume to new steady states in response to the altered apical membrane ionic fluxes. This response is slower (ca. 2-3 s; Ref. 7) and is determined by the relationship between the magnitude of the ionic fluxes and cell volume. However, these slower changes in cytosolic ion concentrations alter the driving force for apical membrane NaCl uptake.

These effects should have only minor, if any, effects on our analysis for two reasons. First, the disappearance of NaCl from the tubular lumen relies on apical uptake of Na+ and Cl-, and this response is very rapid. In addition, cytosolic Na+ levels are well regulated by Na-K-ATPase in the basolateral membrane, so that the slower secondary adjustments in apical ionic flux will be small relative to the magnitude of the initial change in apical uptake. The second reason is that processes with a 2- to 3-s time constant are likely to have little effect on spectral structure below 300-500 mHz, which is above the frequencies corresponding to the first few nodes (see APPENDIX C in Ref. 17).

If, as we expect, the nonideal factors considered above have only a minor impact on the results of our model analysis, then the predicted nodal structure should be observable in open-feedback-loop experiments employing oscillatory perturbations in TAL flow that are ~30% of steady-state flow. The first two approximate nodes shown in Fig. 1B have oscillatory amplitudes of 1.6 and 0.8 mM (see Table 2), whereas the near-antinodal amplitudes at the surrounding intermediate frequencies are 30.1, 9.6, and 5.7 mM. Differences of this magnitude should be measurable with existing methods (8). The detection of these nodes would validate the simple equation for mass conservation in the TAL used in this study (Eq. 1), a formulation that has been widely used to obtain both steady-state and dynamic simulation results (see, e.g., Refs. 6, 8, 12, 15, 21, 27).

In summary, the nodal structure of the TAL frequency response curve, which arises from the establishment of standing waves in luminal concentration, illustrates the nonlinear character of the TAL filter. This unusual filter not only attenuates high-frequency perturbations, but it also distorts the waveform. Simulations in the companion study (17) indicate that TGF-mediated oscillations in nephron flow will be affected by the TAL filter, and, in some cases, the spectral structure of the TAL filter will be imposed on that of the entire TGF system. Considerations in the DISCUSSION of the companion study (17) suggest that the nodal structure predicted in this study may have already been observed in some experimental records.

    APPENDIX A
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References

TAL Concentration is a Function of Transit Time

We use the method of characteristics (4) to show that, under appropriate hypotheses, TAL concentration is a function of transit time. The method of characteristics has been used previously to compute profiles of key variables in proximal tubule flow (28).

For simplicity and generality, let K(C) denote the transtubular active transport of chloride, and assume that K is Lipschitz continuous in C, that K > 0 for C > 0, and that K decreases monotonically to 0 as C decreases to 0. In the absence of backleak, the rate of transepithelial chloride transport at position x and time t depends only on C(x, t), the local intratubular concentration at time t, and Eq. 1 becomes
<FR><NU>∂</NU><DE>∂<IT>t</IT></DE></FR> C(<IT>x</IT>, <IT>t</IT>) = −<IT>F</IT>(<IT>t</IT>) <FR><NU>∂</NU><DE>∂<IT>x</IT></DE></FR> C(<IT>x</IT>, <IT>t</IT>) − <IT>K</IT>(C(<IT>x</IT>, <IT>t</IT>)) (A1)
The fluid at location x and time t entered the TAL at time so = t - T(x, t). At time s is in  [so, t], the fluid passed through the location
<IT>y</IT>(<IT>s</IT>) = <LIM><OP>∫</OP><LL><IT>s</IT><SUB>o</SUB></LL><UL><IT>s</IT></UL></LIM><IT> F</IT>(<IT>u</IT>) d<IT>u</IT> (A2)
where 0 <=  y(s) <=  x, and y(t) = x. If g(s) is defined to be the chloride concentration along the path determined by Eq. A2, then g(s) = C( y(s), s), and g(so) = C(0, so) = 1. If we differentiate the expression for g(s) and employ Eq. A2 with so considered as a fixed (constant) time, then we obtain
<FR><NU>d</NU><DE>d<IT>s</IT></DE></FR> g(<IT>s</IT>) = <FR><NU>∂C</NU><DE>∂<IT>y</IT></DE></FR> <IT>F</IT>(<IT>s</IT>) + <FR><NU>∂C</NU><DE>∂<IT>s</IT></DE></FR> (A3)
Comparing Eqs. A1 and A3, we obtain an ordinary differential equation (indeed, an initial value problem) for the concentration along the TAL transit path
<FR><NU>d</NU><DE>d<IT>s</IT></DE></FR> g(<IT>s</IT>) = −<IT>K</IT>(g(<IT>s</IT>)),  g(<IT>s</IT><SUB>o</SUB>) = 1 (A4)
Because of the properties assumed for K, g is positive and monotone decreasing in s.

For any so, define a function h by
h(<IT>u</IT>) = g(<IT>u</IT>+<IT>s</IT><SUB>o</SUB>) (A5)
Then h(0) = g(so) = 1, and h is a uniquely determined function that is independent of the choice of so. Since for each so, t - so = T, it follows that
C(<IT>x</IT>, <IT>t</IT>) = C( <IT>y</IT>(<IT>t</IT>), <IT>t</IT>) = g(<IT>t</IT>) = h(<IT>t</IT> − <IT>s</IT><SUB>o</SUB>) = h(<IT>T</IT>(<IT>x</IT>, <IT>t</IT>)) (A6)
We conclude from Eq. A6 that C(x, t) depends only on the transit time T and not on the distribution of time that solute-laden fluid has spent at the locations between 0 and x. This result does not depend on a specific form for K but, rather, only on the general properties of K set forth before Eq. A1.

Now we show that, under our hypotheses, the profile of C at each time t is monotone decreasing. If we differentiate Eq. A6 with respect to x, and evaluate partial T/partial x by differentiating Eq. 2 with respect to x, we obtain
<FR><NU>∂C</NU><DE>∂<IT>x</IT></DE></FR> = − <FR><NU><IT>K</IT>(h(<IT>T</IT>(<IT>x</IT>, <IT>t</IT>)))</NU><DE><IT>F</IT>(<IT>t</IT> − <IT>T</IT>)</DE></FR> (A7)
Since h > 0, we have K > 0, and since we assume that F > 0, it follows that partial C/partial x < 0.

In the case were K is Michaelis-Menten kinetics, Eq. A4 has an implicit solution, analogous to Eq. A6, given by Eq. 7.

    APPENDIX B
Top
Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References

Numerical Methods

Numerical methods are identified by corresponding figure numbers.

Figure 1. Dashed lines providing the outer envelope for the MD chloride concentration range were computed for discrete values of f, incremented by 1/(40to). Dimensionless lower and upper bounds of T(1, t) (Tm and TM, respectively) corresponding to each value of f were computed from Eq. 6; the resulting range of C(1, t) was computed from Eq. 7, using the dimensionless iteration map Cn+1 = exp((1 - VmaxTCn)/K m), where C1 = 0.1 and T = Tm or TM. The iteration was terminated when successive iterates of Cn differed by <= 10-8. The plotted values were obtained by multiplying dimensionless values of Cn by the reference concentration Co.

In Fig. 1 the solid curves for the precise bounds were computed from the same discrete values of f used above. Bounds on T(1, t) were computed from Eq. 5, for each f, by finding extremal values of T as t varied in increments of to/100 from 0 to 40to, the period of one oscillation at the lowest frequency evaluated. Bounds on C(1, t) were then computed by the iteration map described above.

The vertical bars in Fig. 1 were obtained from the numerical solution of Eq. 1, which was computed in double-precision arithmetic using a second-order essentially nonoscillatory (ENO) scheme, coupled with Heun's method for the time advance; this algorithm yields solutions that exhibit second-order convergence in both space and time (14). The numerical space and time steps, in nondimensional units, were Delta x = 1/640 and Delta t = 1/214, respectively, corresponding to dimensional values of 7.8125 × 10-4 cm and 9.58738 × 10-4 s. The initial data for each numerical run was the steady-state profile S(x). Sampling of values of C(1, t) began after a computational time interval that was an integer multiple of both the transit time to of the TAL and the period of the oscillation in F. This served both to expel the initial profile S(x) and to initiate the sampling interval at onset of an oscillation. The range of the oscillation in C(1, t) was then determined over a time interval corresponding to the period of the oscillation.

Figures 2 and 3. The profiles in Fig. 2 and the MD concentrations in Fig. 3 were generated from the numerical solution of Eq. 1 using the ENO scheme. The space and time steps, in nondimensional units, were Delta x = 1/5120 and Delta t = 1/214. High resolution in space and time was required because axial profiles contain segments where partial C/partial x is very large. In each case, preceding the time marked t = 0, the numerical solution was computed for an interval that was an integer multiple of both the transit time to of the TAL and the period of the oscillation in F. For Fig. 2, sampling was conducted for one period of an oscillation in F; for Fig. 3, sampling was conducted for an interval 2to, which is an integer multiple of the periods of the four oscillations: 2to, to, to/3, and to/2.

    ACKNOWLEDGEMENTS

We thank Niels-Henrik Holstein-Rathlou for the suggestion that we examine the simulated TAL chloride concentration profiles. We thank John M. Davies for assistance in preparation of Figs. 1-3.

    FOOTNOTES

This work was supported in part by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-42091.

Address for reprint requests: H. E. Layton, Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320 (E-mail: layton{at}math.duke.edu).

Received 1 April 1996; accepted in final form 19 June 1997.

    REFERENCES
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Abstract
Introduction
Results
Discussion
Appendix A
Appendix B
References

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AJP Renal Physiol 273(4):F625-F634
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