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Departments of 1Biostatistics and 4Medical Physiology, University of Copenhagen, Copenhagen, Denmark; 2Department of Physiology and Biophysics, University of South Florida, Tampa, Florida; and 3Department of Molecular Pharmacology, Physiology, and Biotechnology, Brown University, Providence, Rhode Island
Submitted 28 June 2005 ; accepted in final form 29 September 2006
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ABSTRACT |
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 TOP ABSTRACT MATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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stochastic processes; spectral density; nephron; tubuloglomerular feedback mechanism; deterministic chaos
Experimental studies have demonstrated the presence of self-sustained oscillations in proximal tubular pressure (Pt), and experimental studies show that the phenomenon is caused by intrinsic delays in TGF signal transmission (16). In Sprague-Dawley (SprD) and Wistar rats, the oscillations are regular, with only one distinct peak in the frequency spectrum at 
20–40 mHz and an amplitude of 3–4 mmHg. In contrast, in spontaneously hypertensive rats (SHR), the oscillations appear as highly irregular fluctuations, with several frequency peaks within the range 10–50 mHz and a smaller amplitude of only 1.5–2 mmHg (14, 16, 35). Evidently, the dynamic behavior of Pt in SHR is not the same as in normotensive rats.
Why the rhythm in Pt is different in SHR and SprD rats is unknown. We suggested that the irregular fluctuations in the SHR could represent low-dimensional deterministic chaos in the TGF system (16, 34, 35). Low-dimensional deterministic chaos is characterized by its sensitive dependence on initial conditions: a small perturbation (e.g., observation error or a round-off error) at a point leads in a very short time to unpredictability of the trajectory (for review of nonlinear dynamics see Ref. 16). According to this hypothesis, the regular oscillations in the SprD rats and the irregular fluctuations in the SHR arise from the dynamics of the TGF mechanism, the difference in dynamics being a consequence of differences in the system parameters, such as the feedback gain, between the two strains of rats (16). To test this hypothesis, time series from SHR were subjected to a variety of tests (34). The correlation dimension of the phase space attractor was 2.3, a noninteger value consistent with low-dimensional deterministic chaos. The Lyapunov exponent was found to be positive, consistent with sensitive dependence on initial conditions. Both findings are characteristic of systems showing deterministic chaos but may also be found in systems with other types of dynamics (30).
It is well known from nonlinear dynamic systems theory that oscillations and chaos can be induced by varying the value of a single parameter, designated the bifurcation parameter (29). The gain of the TGF mechanism is central to the understanding of Pt dynamics. This parameter plays an important role in the stability of the pressure and flow regulation, and in previous analyses it has been found to be a main bifurcation parameter (2, 5, 13, 16, 19, 20, 23, 24, 26). Computer simulations have consistently shown that if regular oscillations are to occur in the TGF system, the gain has to exceed a critical value, and its magnitude is a major determinant of the amplitude of the oscillations. Furthermore, the gain values necessary for oscillations to occur in the mathematical models have been in excellent agreement with the values found experimentally (9, 16, 26).
The picture has been less clear with the irregular fluctuations. With one exception (5, 24), all the published mathematical models have failed to predict chaotic fluctuations in Pt, even at supraphysiological values of the feedback gain (13, 16, 20, 23, 24, 26). Only in the mathematical model by Mosekilde (24) and Carlsson and Andersen (5) has it been possible to induce chaotic dynamics in the TGF system. It is therefore necessary to consider the possibility that the irregular fluctuations could come from other mechanisms that are either intrinsic to the nephron or external to it, such as blood pressure fluctuations and changes in angiotensin II and nitric oxide (NO) levels, sympathetic nerve activity, and nephron-nephron communication. One approach for representation of the effects of unknown factors is inclusion of a stochastic process in the model. This is well known from regression analysis, where the stochastic component (or the residual error) represents the effects of the unknown factors in determining the value of the dependent variable.
To test whether the irregular pressure fluctuations could be accounted for by a model with deterministic chaos in the TGF system or whether additional factors were important, we have extended the nephron model of Mosekilde (24) and Carlsson and Andersen (5) to include a stochastic component. The latter is achieved by allowing the feedback gain to vary randomly around a mean value (9). This variation is modeled by an Ornstein-Uhlenbeck process driven by a standard Wiener process, which is a stochastic process with independent and normally distributed increments. It is nondifferentiable, with the generalized derivative being a Gaussian white noise process (25a). Previously, we developed a procedure that allows estimation of the mean value of the gain and the parameters of the associated Ornstein-Uhlenbeck process (9). Depending on the values of the system parameters, this model may show regular oscillations or irregular fluctuations. The latter can again be due to deterministic chaos in TGF or random fluctuations in the system parameters.
The purpose of the present study was to apply the recently developed stochastic model of TGF to determine the set of parameters that gives the best fit to the experimental data. In our previous study (9), the arterial blood pressure (Pa) was fixed to 100 mmHg, since only SprD rats were analyzed. In the present study, differences between SprD rats and SHR, characterized by their differences in Pa, are analyzed; thus the measured value for each rat is used. For each rat, Pa was set to the mean value of the measured Pa for the individual rat, and the normal flow resistance of the afferent arteriole (Ra0) was adjusted accordingly, such that the glomerular pressure (Pg) was the same for SprD rats and SHR, as they are known to be (see APPENDIX). The model is used to estimate the three parameters governing the random behavior of the feedback gain: 
is the mean value, 1/
is the correlation time of the feedback gain process (the magnitude of the "memory" of the process), and 
2 is the variance of the Wiener noise used to drive the process.
Free-flow pressure values were then estimated as the mean value of Pt in a base run of the model incorporating the estimated parameters. Stop-flow pressure (SFP) values and the open-loop TGF gain were estimated from the model with the estimated parameters by setting GFR, the proximal tubular absorption rate (APR), and flow into the loop of Henle to zero in the model and then determining the SFP for different induced flows in the loop of Henle.
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MATERIALS AND METHODS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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Model.
The rat nephron model is described in the APPENDIX, and the model parameters and values are defined in 
Tables 2 and 3. Details of the deduction and physiological justification of the model have been published previously (2, 9, 13–15, 24). The model consists of three coupled parts: a glomerular-tubular model, a TGF-afferent arteriolar model, and a model of the delay in the TGF, representing the delay caused by the time taken by the signal to pass through the loop of Henle and the delay in the signaling process from the macula densa to the afferent arteriole. The model is shown in Fig. 1.
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, , and are unknown parameters governing the process and W(t) is a one-dimensional standard Wiener process. This is an Ornstein-Uhlenbeck process (25a). We assume that Z(0) is normally distributed with mean and variance 2/(2) and is independent of W(t). The process Z is ergodic, with stationary distribution N[,2/(2)] the normal distribution, provided that > 0. It is assumed that, on average, Z is positive, which requires that > 0. In this model, Z can become negative for some time points. This can be justified by experimental evidence that vasodilator stimuli, such as NO, may originate from the macula densa (33). Thus it could be possible that, momentarily, TGF caused positive feedback, even though, on the average, over several minutes the negative feedback is the strongest, as observed in micropuncture experiments.
To enable a comparison between the present results and previous published experimental results, the parameter estimates from the present study were used to simulate open-loop (stop-flow) and closed-loop (free-flow) micropuncture experiments. In a stop-flow experiment, a wax block is inserted into the proximal tubule, and distal flow rate is controlled through a microperfusion pump inserted into the last accessible convolution of the proximal tubule. The responses to variation in microperfusion rate are then measured as changes in SFP. To mimic the experimental setting, GFR, APR, and flow into the loop of Henle were set to zero in the model. Then the SFP was determined in the model for the estimated parameter values and for different induced flows in the loop of Henle. Since a steady state of the model was used to determine the SFP, Z was kept constant at its mean value .
Estimation method. The adequacy of a mathematical model can be evaluated by simulating a time series from the model and then comparing the result with a time series from a physiological experiment. Model parameters are estimated by minimizing some distance function between the experimental and the simulated data over the parameter range of the model. The most important characteristics of the dynamics of Pt can be characterized by the spectral density, because period and amplitude and their fluctuations characterize the dynamics. In our previous study (9), we proposed to estimate the parameters by the least squares distance between the logarithmically transformed spectral densities estimated from simulated and experimental data.
The spectral density of the experimental data was estimated as follows. Simultaneous time series from two nephrons have been obtained for each rat. It appears reasonable to assume that the dynamics of the nephrons within the same kidney are subject to the same variability in factors affecting renal function. To gain precision in the estimation of the spectral density, the empirical periodograms from the two nephrons within the same kidney were averaged. Moreover, a moving average over three points [a Daniell window (28)] has been applied to further reduce the variance. The spectral density of the model as a function of the parameters was numerically approximated by averaging over 10 empirical periodograms calculated on simulated time series with the given parameter values. Details can be found in our previous study (9).
In our previous work (9), the bias of the estimation procedure was shown to be on the order of 1% for , 15% for , and 10% for . The coefficients of variation of the estimates were shown to be on the order of 1.3% for , 33% for , and 15% for . We would thus expect to be the best determined and the least.
We estimated the delay in the loop of Henle and macula densa (T), together with the parameters , , and . Since T is the major determinant of the oscillatory period, which is clearly recognizable only in SprD rats, it was fixed at 12 s as estimated in previous studies (9, 14, 16). Previous studies have found the same GFR, tubular flow, and tubular length in SHR as in SprD or Wistar-Kyoto rats (1, 31). Since the delay in the loop of Henle is mainly determined by its structure and the flow rate of tubular fluid, it appears likely that there are no major differences in this parameter between the strains.
Statistical methods.
Estimates from SprD rats and SHR were compared by Student's t-test for unpaired data (Welch test). P < 0.05 was considered statistically significant. Since the estimates of , , and and the slopes of the TGF function showed skewed distributions, these estimates were logarithmically transformed before the analysis. Values are means ± SD.
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RESULTS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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Figure 3 is a box plot of the parameter estimates for the two strains of rats.
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, is estimated to be 12.5 ± 4.2 for SprD rats and 21.6 ± 11.7 for SHR. The estimates of from the two strains of rats were significantly different. The drift parameter, , is estimated to be 0.10 ± 0.18 and 0.29 ± 0.87 s–1 for SprD rats and SHR, respectively; the difference is not significant. These estimates correspond to a correlation time of the feedback gain variation, 1/, of 5–10 s. Finally, the standard deviation of the driving Wiener noise, , was estimated to be 2.1 ± 1.8 s–1/2 for SprD rats but was much larger (4.8 ± 4.7 s–1/2) for SHR; the difference was significant. This result indicates that the contribution of random noise had to be substantially greater in the SHR than in the SprD rats to achieve an adequate fit of the model to the power spectra obtained from experimental data.
The value of at which chaos is present depends on the value of T in the TGF (5). In the preceding estimations, we used T = 12 s, which is the value previously estimated in SprD rats (9). However, the lowest value of where chaos is present in the model is T = 16 s (5). We therefore performed a second set of parameter estimations at this value of the delay, which resulted in nearly identical values of the estimates for , , and (results not shown).
The magnification, M, is a measure of the efficiency of a regulatory system, with values between 0 and 1. It is defined as the change in late flow rate per change in the inflow into the proximal tubule, and, correspondingly, the open-loop gain (OLG) is defined as (1 – M)/M (12). Simulating from our model with the mean values of the estimated parameters, we find M = 0.123 and 0.107 in SprD rats and SHR, respectively. This corresponds to an OLG of 7.1 for SprD rats and 8.3 for SHR.
Parameter estimates vary much less for SprD rats than for SHR (Fig. 3), perhaps because SHR represent a more inhomogeneous group than SprD rats and, also, because of the more noisy behavior of Pt in SHR, which makes parameter estimation less precise.
In Fig. 4, the logarithm of the spectral densities for two SprD rats and two SHR is plotted against frequency (in Hz), together with the logarithm of the spectral densities approximated from the mathematical model incorporating the estimated parameter values. The model captures the slow dynamics of TGF better than the dynamics of the myogenic mechanism, because the fitted parameters primarily affect the dynamics of the TGF mechanism, which is responsible for the slow components in the spectral density, and because the TGF oscillation is the most prominent peak in the experimental data. Because of the smoothing action of the proximal tubular compliance, the faster dynamics due to the myogenic response, which appear in the spectra from the experimental data as peaks at 0.1 Hz, are much less prominent.
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For comparison with the model without system noise, the model has been simulated for four different values of : 10, 15, 25, and 40. In all simulations, the noise was set to zero: = 0, in which case is irrelevant and T = 16 s. In Fig. 5, simulations of the Pt oscillations are plotted against time. Only for unrealistically high values of did the dynamics resemble anything close to experimental results. However, in this case, the amplitude of the Pt oscillations was much larger in the simulations than is found experimentally.
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DISCUSSION |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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Because of the problems proving the presence of chaos in experimental time series, an alternative approach has been to rely on mathematical modeling of TGF. All the various dynamic models of TGF have, without exception, shown that regular oscillations in Pt occur when the OLG in TGF exceeds a certain limit (2, 13, 16, 20, 24). Furthermore, the value at which the bifurcation to an oscillation occurs is within the experimentally determined range for the gain in TGF (2, 16). In contrast, several of the published models have failed to reproduce deterministic chaos, even at high values for TGF gain (16). In the mathematical models of TGF in which chaotic dynamics have been found, parameter values (in the case of the present model, for values of ) significantly greater than experimentally measured values have been required to achieve the bifurcation (5, 16). Because of these apparent discrepancies, it has been suggested that the transition to chaos in TGF could depend on additional factors, such as an increased nephron-nephron coupling in SHR (16, 27).
Consequently, it is necessary to consider the possibility that the irregular fluctuations in Pt in SHR may arise from mechanisms other than deterministic chaos in the TGF system of a single nephron. A method for including unknown factors in a model is to introduce a stochastic component into the system. The stochastic fluctuations represent the influence of the many unmodeled (and possibly unknown) subsystems. The resulting model is known as a stochastic model and will correspond to a system with many degrees of freedom, i.e., a high-dimensional process.
To rigorously test whether the irregular fluctuations in SHR are best described by a low-dimensional deterministic model or a high-dimensional stochastic model, we have applied a recently developed procedure that allows parameter estimation in complex physiological models of TGF (9). By using a model that, in its deterministic version, can have chaotic dynamics (2) and combining it with a stochastic process that allows random fluctuations in one of the key parameters (Z) that determines the gain of TGF, we obtain a model that has a rich variety of dynamics ranging from deterministic to random fluctuations (see APPENDIX). Using experimental data as input, we can then estimate the parameters of the model. In most cases, the value of Z was well below the value (25) where chaotic dynamics are present (cf. Figs. 3 and 5).
It is worth noting that, in case of the SprD rats, the parameters estimated are in good agreement with the values that have been found experimentally with open- and closed-loop approaches (9, 16). For example, Leyssac and Holstein-Rathlou (22) found a slope at IP = 2.3 ± 0.7 and 2.2 ± 1.0 (SE) mmHg·nl–1·min in halothane-anesthetized Wistar-Kyoto and SprD rats, respectively. These values are not significantly different from the value found in the present study (2.6 mmHg·nl–1·min). Also, M = 0.123 in SprD is within the range of values previously reported from our laboratory (12, 18). Maximal compensation values of 0.23 ± 0.02 and 0.07 ± 0.08 have been reported by Holstein-Rathlou (12) and Karlsen et al. (18), respectively.
Comparison of SprD rats and SHR revealed a much greater variability in Z in the SHR. Thus of the associated Ornstein-Uhlenbeck process was significantly higher in the SHR than in the SprD rats (cf. Table 1). The larger implies a greater variability in the Ornstein-Uhlenbeck process, which determines Z. Physiologically, this increased variability will correspond to greater fluctuations on a short time scale in the gain of TGF in SHR than in SprD rats. Since TGF is one of the key mechanisms involved in renal autoregulation of blood flow and GFR, an increased variability in feedback gain could cause a less precise and/or a less stable autoregulatory process in the SHR. Zhong and co-workers (36) recently showed that autoregulatory dynamics display greater nonstationarity in SHR than in SprD rats. This observation was based on an entirely different approach, where the spontaneous fluctuations in Pa and renal blood flow were used to assess a time-dependent transfer function between blood pressure and blood flow. Using this approach, they found a greater variability in the time-dependent behavior of the transfer functions in the SHR than in the SprD rats (36). This analysis of experimental results is in excellent agreement with the results of the present study.
The random fluctuations in TGF gain in our model were assigned to the parameter Z in the transfer function of TGF representing the effect of the macula densa on the afferent arteriole. However, it should be noted that the total gain in TGF is dependent on many other factors, including the characteristics of the afferent arteriole. Assigning all the variability to one parameter is therefore only a convenient way to lump several potential sources of variability in TGF gain. The observed increased variability need therefore not necessarily reflect differences in macula densa function between the SHR and SprD rats but could also reflect an increased variability in the other subsystems that determines total feedback gain. In this connection, it is of interest that several studies have found a greater instability in vessels from SHR and stroke-prone SHR (4, 25). When vessels are exposed to a vasoconstrictor, such as norepinephrine, there are greater fluctuations in the developed force in the hypertensive than in the normotensive rats (4, 25). An increased vascular variability would be reflected in the present model by an increased variability in Z. Another source of increased variability could be an increased interaction between nephrons in SHR (7, 32). The interaction strength is increased two- to threefold in hypertension (7, 32). This change in interaction strength will inevitably increase the appearance of signals from adjacent nephrons in measurements from individual tubules and could well contribute to the observed behavior of Pt signals in SHR.
It is apparent from Fig. 2B that, in SHR, Z dips below zero quite frequently. This indicates that, during brief periods, the feedback is actually positive. There are no experimental data that allow us to assess whether this is a reasonable result. As mentioned above, it is known that the macula densa produces vasoconstricting (e.g., adenosine) and vasodilating (e.g., NO) substances. At the level of the individual nephron, it cannot be excluded that random (molecular) fluctuations in the production rates could cause this effect. However, again it has to be stressed that Z is a lumped parameter, and the positive feedback could also be caused by decreased TGF activity in neighboring nephrons, causing vasodilation in the measured nephron, or it could represent variability in local nervous activity. Clearly, this is an issue that warrants further investigation.
From the standpoint of model theory, the main conclusion of the present study is that the type of deterministic chaos generated by our simple model is insufficient to explain the experimentally observed fluctuations in Pt in SHR. Although some of the current models of single-nephron dynamics have shown more complex spectra (21), it seems fair to state that all other reduced single-nephron models in the literature fail to bifurcate beyond regular oscillations when an increase in TGF gain is applied. We suggest that all current models of this type, therefore, are not adequate to describe the irregular fluctuations in Pt in the SHR. In contrast, application of the model to data from SprD rats yields much lower values of than application of the model to data from SHR and excellent agreement between estimated and measured values of the OLG. This result supports the use of the deterministic model to describe nephron dynamics in normotensive animals.
From a physiological standpoint, the results of the present study and the study by Zhong and co-workers (36) suggest a greater variability in the autoregulatory mechanisms in kidneys of hypertensive animals. It is well known that the kidneys play a central role in determining the long-term level of Pa (10) and that this is associated with the kidney's ability to eliminate sodium from the body (11). Clearly, this issue needs further experimental consideration.
In conclusion, the dynamics of the Pt fluctuations in SHR appear to be better described by a model that incorporates random fluctuations in the system parameters than by a model that operates in a chaotic domain. We argue that the results point to the failure of the present models to adequately describe processes that are important for the observed irregular Pt fluctuations and the need to consider the importance of other factors, such as differences in vascular function and/or nephron-nephron interactions in further work on this problem.
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APPENDIX |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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The variables are categorized as follows: constants, the values of which are unchanged during model study; system variables, with evolution in time that is defined through differential equations; auxiliary variables, which change in time as functions of the system variables; a random variable modeled as an Ornstein-Uhlenbeck process (25a); and parameters that are supposed constant and with values that are estimated individually for each rat from the data.
All model variables are listed and values for the constants are given in Tables 2 and 3.
Glomerular-tubular model. The transit time for a volume element in the glomerular capillaries is several orders of magnitude smaller than the period length of the oscillations in Pt, so we assume that GFR equilibrates instantaneously and is defined by the instantaneous plasma flow (GPF), the hydrostatic glomerular capillary pressure (Pg), and Pt.
GPF is given by the conventional flow equation, where GBF is blood flow, Hct is the arterial hematocrit, and Ra is the flow resistance in the afferent arteriole
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is the fraction of the afferent arteriole having a fixed resistance. An expression for Ce is obtained by assuming that filtration equilibrium is established before the blood leaves the glomerular capillaries and idealizing the glomerular capillary bed as a single tube of equivalent surface area. This leads to a third-order polynomial in Ce with coefficients depending on Pt and Ra. The value for Ce is the positive root of the third-order polynomial, which can be shown to have a unique solution.
Pg is determined by mass conservation, such that GBF equals the sum of GFR and the flow through the efferent arteriole
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TGF-afferent arteriolar model.
The TGF function, 
, is assumed to be mediated only through the afferent arteriole, where NaCl concentration at the macula densa causes a signal to be transmitted to the afferent arteriole, which, in response, will constrict or dilate (3). In the model, the delayed flow into the loop of Henle, instead of the NaCl concentration at the macula densa, has been used as the signal to TGF. This simplification is motivated by the fact that the two are directly proportional and by the fact that it is the flow into the loop of Henle that has been the independent variable in the vast majority of published experiments on TGF (3). It is well established empirically that the activation level of the vascular smooth muscle cells in the wall of the afferent arteriole, , can be described by a logistic equation (16)
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min and max are the lower and upper activation limits, respectively; Z is a randomly varying parameter that determines the slope of the S-shaped TGF function and, thereby, the gain of the TGF mechanism; F3/Fh0 is flow at the macula densa normalized with respect to the equilibrium flow in the loop of Henle; and IP, the inflection point of the curve, is the flow at which the feedback response is half-maximal and is given by
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eq is the equilibrium activation level.
The dynamics of the afferent arteriole are modeled by a second-order differential equation to mimic the tendency of the arterioles to perform damped, oscillatory contractions in response to stimuli
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is the mass-to-elasticity ratio for the arteriolar wall and controls the natural frequency of the oscillations, and 
is the damping of the arteriolar dynamics.
Pav is determined by assuming a linear pressure drop across the variable part of the arteriolar resistance
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Delay.
Flow at the macula densa does not change immediately after a change in Pt, but only after a delay. Furthermore, it seems likely that an additional delay occurs in the transmission of the signal from the macula densa to the afferent arteriole. We have conflated these two delays into one and modeled it as a third-order lag, where the input is the flow into the loop of Henle and the output is the delayed flow, which serves as the input to the TGF function, F3. The delay is defined by the equations
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Pa. In previous applications of the present model, Pa has been fixed to 13.33 kPa (100 mmHg) (2, 5, 9, 24). Inasmuch as Pa was measured in the present data set and we are looking at differences between SprD rats and SHR that are characterized by their increased Pa, it is natural to use the actual measured value for each rat when estimating the parameters. It has been shown previously (6) that, despite the increased Pa in SHR, afferent arteriolar blood flow and Pg are the same in SHR and SprD.
Therefore, Ra0 (=2.44 kPa·s·nl–1 in the previous models) has been adjusted accordingly, such that
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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