A dynamic model of renal blood flow autoregulation

To test whether a mathematical model combining dynamic models of the tubuloglomerular feedback (TGF) mechanism and the myogenic mechanism was sufficient to explain dynamic autoregulation of renal blood flow, we compared model simulations with experimental data. To assess the dynamic characteristics of renal autoregulation, a broad band perturbation of the arterial pressure was employed in both the simulations and the experiments. Renal blood flow and tubular pressure were used as response variables in the comparison. To better approximate the situationin vivo where as large number of individual nephrons act in parallel, each simulation was performed with 125 parallel versions of the model. The key parameters of the 125 versions of the model were chosen randomly within the physiological range. None of the constituent models, i.e., the TGF and the myogenic, could alone reproduce the experimental observations. However, in combination they reproduced most of the features of the various transfer functions calculated from the experimental data. The major discrepancy was the presence of a bimodal distribution of the admittance phase in the simulations. This is not consistent with most of the experimental data, which shows a unimodal curve for the admittance phase. The ability of the model to reproduce the experimental data supports the hypothesis that dynamic autoregulation of renal blood flow is due to the combined action of TGF and the myogenic response.     

Slow oscillations in blood pressure via a nonlinear feedback model

Blood pressure is well established to contain a potential oscillation between 0.1 and 0.4 Hz, which is proposed to reflect resonant feedback in the baroreflex loop. A linear feedback model, comprising delay and lag terms for the vasculature, and a linear proportional derivative controller have been proposed to account for the 0.4-Hz oscillation in blood pressure in rats. However, although this model can produce oscillations at the required frequency, some strict relationships between the controller and vasculature parameters must be true for the oscillations to be stable. We developed a nonlinear model, containing an amplitude-limiting nonlinearity that allows for similar oscillations under a very mild set of assumptions. Models constructed from arterial pressure and sympathetic nerve activity recordings obtained from conscious rabbits under resting conditions suggest that the nonlinearity in the feedback loop is not contained within the vasculature, but rather is confined to the central nervous system. The advantage of the model is that it provides for sustained stable oscillations under a wide variety of situations even where gain at various points along the feedback loop may be altered, a situation that is not possible with a linear feedback model. Our model shows how variations in some of the nonlinearity characteristics can account for growth or decay in the oscillations and situations where the oscillations can disappear altogether. Such variations are shown to accord well with observed experimental data. Additionally, using a nonlinear feedback model, it is straightforward to show that the variation in frequency of the oscillations in blood pressure in rats (0.4 Hz), rabbits (0.3 Hz), and humans (0.1 Hz) is primarily due to scaling effects of conduction times between species.     

Feedback-mediated dynamics in a model of coupled nephrons with compliant thick ascending limbs

The tubuloglomerular feedback (TGF) system in the kidney, a key regulator of glomerular filtration rate, has been shown in physiologic experiments in rats to mediate oscillations in thick ascending limb (TAL) tubular fluid pressure, flow, and NaCl concentration. In spontaneously hypertensive rats, TGF-mediated flow oscillations may be highly irregular. We conducted a bifurcation analysis of a mathematical model of nephrons that are coupled through their TGF systems; the TALs of these nephrons are assumed to have compliant tubular walls. A characteristic equation was derived for a model of two coupled nephrons. Analysis of that characteristic equation has revealed a number of parameter regions having the potential for differing stable dynamic states. Numerical solutions of the full equations for two model nephrons exhibit a variety of behaviors in these regions. Also, model results suggest that the stability of the TGF system is reduced by the compliance of TAL walls and by internephron coupling; as a result, the likelihood of the emergence of sustained oscillations in tubular fluid pressure and flow is increased. Based on information provided by the characteristic equation, we identified parameters with which the model predicts irregular tubular flow oscillations that exhibit a degree of complexity that may help explain the emergence of irregular oscillations in spontaneously hypertensive rats.     

Nephron blood flow dynamics measured by laser speckle contrast imaging

Tubuloglomerular feedback (TGF) has an important role in autoregulation of renal blood flow and glomerular filtration rate (GFR). Because of the characteristics of signal transmission in the feedback loop, the TGF undergoes self-sustained oscillations in single-nephron blood flow, GFR, and tubular pressure and flow. Nephrons interact by exchanging electrical signals conducted electrotonically through cells of the vascular wall, leading to synchronization of the TGF-mediated oscillations. Experimental studies of these interactions have been limited to observations on two or at most three nephrons simultaneously. The interacting nephron fields are likely to be more extensive. We have turned to laser speckle contrast imaging to measure the blood flow dynamics of 50-100 nephrons simultaneously on the renal surface of anesthetized rats. We report the application of this method and describe analytic techniques for extracting the desired data and for examining them for evidence of nephron synchronization. Synchronized TGF oscillations were detected in pairs or triplets of nephrons. The amplitude and the frequency of the oscillations changed with time, as did the patterns of synchronization. Synchronization may take place among nephrons not immediately adjacent on the surface of the kidney.     

Nonlinear interactions in renal blood flow regulation

We have developed a model of tubuloglomerular feedback (TGF) and the myogenic mechanism in afferent arterioles to understand how the two mechanisms are coupled. This paper presents the model. The tubular model predicts pressure, flow, and NaCl concentration as functions of time and tubular length in a compliant tubule that reabsorbs NaCl and water; boundary conditions are glomerular filtration rate (GFR), a nonlinear outflow resistance, and initial NaCl concentration. The glomerular model calculates GFR from a change in protein concentration using estimates of capillary hydrostatic pressure, tubular hydrostatic pressure, and plasma flow rate. The arteriolar model predicts fraction of open K channels, intracellular Ca concentration (Ca(i)), potential difference, rate of actin-myosin cross bridge formation, force of contraction, and length of elastic elements, and was solved for two arteriolar segments, identical except for the strength of TGF input, with a third, fixed resistance segment representing prearteriolar vessels. The two arteriolar segments are electrically coupled. The arteriolar, glomerular, and tubular models are linked; TGF modulates arteriolar circumference, which determines vascular resistance and glomerular capillary pressure. The model couples TGF input to voltage-gated Ca channels. It predicts autoregulation of GFR and renal blood flow, matches experimental measures of tubular pressure and macula densa NaCl concentration, and predicts TGF-induced oscillations and a faster smaller vasomotor oscillation. There are nonlinear interactions between TGF and the myogenic mechanism, which include the modulation of the frequency and amplitude of the myogenic oscillation by TGF. The prediction of modulation is confirmed in a companion study (28).     

Acute renal function in chronically sympathectomized deoxycorticosterone acetate-treated miniature swine

We recently validated a swine model in which chronic treatment with 6-hydroxydopamine (6-OHDA) produced an effective sympathectomy. These sympathectomized swine demonstrated a significantly attenuated hypertensive response when treated with deoxycorticosterone acetate (DOCA). Because renal nerve activity is elevated and important in controlling renal function and blood pressure in the DOCA swine model, we wanted to study the effect of chronic sympathectomy on acute renal hemodynamics and tubular function. Kidney function was assessed in 14 DOCA-treated miniature swine, 8 of which were sympathectomized by chronic treatment with 6-OHDA, while 6 served as controls. Effective renal sympathectomy in this model has been previously confirmed by a significant reduction (97%) of norepinephrine in renal cortical tissue. When anesthetized, mean arterial pressure and renal blood flow were similar between the two groups. Glomerular filtration rate was lower by 43%, urine flow rate by 71%, sodium excretion by 66%, and potassium excretion by 48% in the 6-OHDA DOCA animals. All of these parameters were significantly different from the intact DOCA controls. These results indicate that anesthetized, chronically sympathectomized swine exhibit decreased renal excretory function. The changes in renal function may have been due to the development of a tubular or glomerular supersensitivity to circulating antinatriuretic factors, since the 6-OHDA group had a 28% greater pressor response to the alpha-agonist phenylephrine and a significantly greater fall in mean arterial pressure in response to alpha-blockade with prazosin when compared with the controls. These changes in renal function may also explain why the 6-OHDA animals demonstrated a slight increase in mean arterial pressure in response to DOCA. Because acute renal denervation in DOCA-treated swine produces a diuresis and natriuresis, this study affirms that there may be important functional differences in acutely versus chronically denervated kidneys for which the implications under normal physiologic conditions are unknown.     

Non-invasive model-based estimation of aortic pulse pressure using suprasystolic brachial pressure waveforms

Elevated central arterial (aortic) blood pressure is related to increased risk of cardiovascular disease. Methods of non-invasively estimating this pressure would therefore be helpful in clinical practice. To achieve this goal, a physics-based model is derived to correlate the arterial pressure under a suprasystolic upper-arm cuff to the aortic pressure. The model assumptions are particularly applicable to the measurement method and result in a time–domain relation with two parameters, namely, the wave propagation transit time and the reflection coefficient at the cuff. Central pressures estimated by the model were derived from completely automatic, non-invasive measurement of brachial blood pressure and suprasystolic waveform and were compared to simultaneous invasive catheter measurements in 16 subjects. Systolic blood pressure agreement, mean (standard deviation) of difference was ?1 (7) mm Hg. Diastolic blood pressure agreement was 4 (4) mm Hg. Correlation between estimated and actual central waveforms was greater than 90%. Individualization of model parameters did not significantly improve systolic and diastolic pressure agreement, but increased waveform correlation. Further research is necessary to confirm that more accurate brachial pressure measurement improves central pressure estimation.     

Fluid waves in renal tubules.

Autoregulation of renal blood flow is ineffective when arterial pressure perturbations occur at frequencies above 0.05 Hz. To determine whether wave propagation velocity to the macula densa is rate limiting, we estimated compliances of the proximal tubule and the loop of Henle, and used these values in a model of pressure and flow as functions of time and distance in the nephron. Compliances were estimated from measurements of pressures and flows in early proximal, late proximal, and early distal tubules in rats under normal and Ringer-loaded conditions. A model of steady pressure and flow in a compliant, reabsorbing tubule was fitted to these results. The transient model was a set of nonlinear, hyperbolic partial differential equations with split, nonlinear boundary conditions, and was solved with finite difference methods. The loop of Henle compliance was larger than the proximal tubule compliance, and impulses in glomerular filtration rate were attenuated in magnitude and delayed in time in the loop of Henle. Simulated step forcings revealed a similar pattern. Periodic variations of GFR were attenuated at frequencies greater than 0.05 Hz, and there was a delay of 5 s between variations in GFR and macula densa flow rate. The high compliance of the loop slows wave propagation to the macular densa and reduces the amplitude of high frequency waves originating in the glomerulus, but other parts of the signal chain also contribute to the slow response of macula densa feedback.     

Predictive feedback in human simulated pendulum balancing

In studies of human balance, it is common to fit stimulus-response data by tuning the time-delay and gain parameters of a simple delayed feedback model. Many interpret this fitted model, a simple delayed feedback model, as evidence that predictive processes are not required to explain existing data on standing balance. However, two questions lead us to doubt this approach. First, does fitting a delayed feedback model lead to reliable estimates of the time-delay? Second, can a non-predictive controller provide an explanation compatible with the independently estimated time delay? For methodological and experimental clarity, we study human balancing of a simulated inverted pendulum via joystick and screen. A two-step approach to data analysis is used: firstly a non-parametric model—the closed-loop impulse response—is estimated from the experimental data; second, a parametric model is fitted to the non-parametric impulse-response by adjusting time-delay and controller parameters. To support the second step, a new explicit formula relating controller parameters to closed-loop impulse response is derived. Two classes of controller are investigated within a common state-space context: non-predictive and predictive. It is found that the time-delay estimate arising from the second step is strongly dependent on which controller class is assumed; in particular, the non-predictive control assumption leads to time-delay estimates that are smaller than those arising from the predictive assumption. Moreover, the time-delays estimated using the non-predictive control assumption are not consistent with a lower-bound on the time-delay of the non-parametric model whereas the corresponding predictive result is consistent. Thus while the goodness of fit only marginally favoured predictive over non-predictive control, if we add the additional constraint that the model must reproduce the non-parametric time delay, then the non-predictive control model fails. We conclude (1) the time-delay should be estimated independently of fitting a low order parametric model, (2) that balance of the simulated inverted pendulum could not be explained by the non-predictive control model and (3) that predictive control provided a better explanation than non-predictive control.     

Testing the fit of a quantal model of neurotransmission

Many studies of synaptic transmission have assumed a parametric model to estimate the mean quantal content and size or the effect upon them of manipulations such as the induction of long-term potentiation. Classical tests of fit usually assume that model parameters have been selected independently of the data. Therefore, their use is problematic after parameters have been estimated. We hypothesized that Monte Carlo (MC) simulations of a quantal model could provide a table of parameter-independent critical values with which to test the fit after parameter estimation, emulating Lilliefors's tests. However, when we tested this hypothesis within a conventional quantal model, the empirical distributions of two conventional goodness-of-fit statistics were affected by the values of the quantal parameters, falsifying the hypothesis. Notably, the tests' critical values increased when the combined variances of the noise and quantal-size distributions were reduced, increasing the distinctness of quantal peaks. Our results support two conclusions. First, tests that use a predetermined critical value to assess the fit of a quantal model after parameter estimation may operate at a differing unknown level of significance for each experiment. Second, a MC test enables a valid assessment of the fit of a quantal model after parameter estimation.     

Oscillatory structure of the parameters of the pulse signal of the radial artery

The typology analysis of the spectral density curves of time series obtained from various parameters of unit pulse oscillations was performed. The formal classification (typology) of spectral curves was performed using correlations of peak values of the spectral density of VLF, LF, and HF slow waves. To include a spectral density curve into a particular type, the criteria were introduced that characterize the intensity of these oscillatory components. The experimental data was collected during examinations of children and adolescents that were performed to detect the early stage of arterial hypertension (AH). The distribution of the subjects among the types of spectral density based on various parameters of pulse oscillation was evaluated. The results of the comparative analysis of spectral type distribution for various parameters are shown. This analysis has revealed the most informative spectral parameters for differential diagnosis of conditions of two types: type 1, essential hypertension, and type 2, various functional disorders with a normal blood pressure. It has been found that the parameters of the oscillatory components of dicrotic pulse wave time characteristics are the most informative for detecting hypertension in children.     

Comparison of various techniques used to estimate spontaneous baroreflex sensitivity (the EuroBaVar study)

This study compared spontaneous baroreflex sensitivity (BRS) estimates obtained from an identical set of data by 11 European centers using different methods and procedures. Noninvasive blood pressure (BP) and ECG recordings were obtained in 21 subjects, including 2 subjects with established baroreflex failure. Twenty-one estimates of BRS were obtained by methods including the two main techniques of BRS estimates, i.e., the spectral analysis (11 procedures) and the sequence method (7 procedures) but also one trigonometric regressive spectral analysis method (TRS), one exogenous model with autoregressive input method (X-AR), and one Z method. With subjects in a supine position, BRS estimates obtained with calculations of alpha-coefficient or gain of the transfer function in both the low-frequency band or high-frequency band, TRS, and sequence methods gave strongly related results. Conversely, weighted gain, X-AR, and Z exhibited lower agreement with all the other techniques. In addition, the use of mean BP instead of systolic BP in the sequence method decreased the relationships with the other estimates. Some procedures were unable to provide results when BRS estimates were expected to be very low in data sets (in patients with established baroreflex failure). The failure to provide BRS values was due to setting of algorithmic parameters too strictly. The discrepancies between procedures show that the choice of parameters and data handling should be considered before BRS estimation. These data are available on the web site (http://www.cbi.polimi.it/glossary/eurobavar.html) to allow the comparison of new techniques with this set of results.     

Signal transduction in a compliant thick ascending limb

In several previous studies, we used a mathematical model of the thick ascending limb (TAL) to investigate nonlinearities in the tubuloglomerular feedback (TGF) loop. That model, which represents the TAL as a rigid tube, predicts that TGF signal transduction by the TAL is a generator of nonlinearities: if a sinusoidal oscillation is added to constant intratubular fluid flow, the time interval required for an element of tubular fluid to traverse the TAL, as a function of time, is oscillatory and periodic but not sinusoidal. As a consequence, NaCl concentration in tubular fluid alongside the macula densa will be nonsinusoidal and thus contain harmonics of the original sinusoidal frequency. We hypothesized that the complexity found in power spectra based on in vivo time series of key TGF variables arises in part from those harmonics and that nonlinearities in TGF-mediated oscillations may result in increased NaCl delivery to the distal nephron. To investigate the possibility that a more realistic model of the TAL would damp the harmonics, we have conducted new studies in a model TAL that has compliant walls and thus a tubular radius that depends on transmural pressure. These studies predict that compliant TAL walls do not damp, but instead intensify, the harmonics. In addition, our results predict that mean TAL flow strongly influences the shape of the NaCl concentration waveform at the macula densa. This is a consequence of the inverse relationship between flow speed and transit time, which produces asymmetry between up- and downslopes of the oscillation, and the nonlinearity of TAL NaCl absorption at low flow rates, which broadens the trough of the oscillation relative to the peak. The dependence of waveform shape on mean TAL flow may be the source of the variable degree of distortion, relative to a sine wave, seen in experimental recordings of TGF-mediated oscillations.     

Structural time series models with feedback mechanisms

Structural time series models have applications in many different fields such as biology, economics, and meteorology. A structural times series model can be represented as a state-space model where the states of the system represent the unobserved components and the structural parameters have clear interpretations. This paper introduces a class of structural time series models that incorporate feedback from the latent components of the history. An iterative procedure is proposed for estimation. These models allow flexible and robust feedback mechanisms, have clear interpretations, and have a computationally efficient estimation procedure. They are applied to hormone data to characterize hormone secretion and to explore a potential feedback mechanism.     

Biomass estimation using satellite remote sensing data—An investigation on possible approaches for natural forest

Vegetation type and its biomass are considered important components affecting biosphere-atmosphere interactions. The measurements of biomass per unit area and productivity have been set as one of the goals for International Geosphere-Biosphere Programme (IGBP). Ground assessment of biomass, however, has been found insufficient to present spatial extent of the biomass. The present study suggests approaches for using satellite remote sensing data for regional biomass mapping in Madhav National Park (MP). The stratified random sampling in the homogeneous vegetation strata mapped using satellite remote sensing has been effectively utilized to extrapolate the sample point biomass observations in the first approach. In the second approach attempt has been to develop empirical models with satellite measured spectral response and biomass. The results indicate that there is significant relationships with spectral responses. These relationships have seasonal dependency in varying phonological conditions. The relationships are strongest in visible bands and middle infrared bands. However, spectral biomass models developed using middle infrared bands would be more reliable as compared to the visible bands as the later spectral regions are less sensitive to atmospheric changes It was observed that brightness and wetness parameters show very strong relationship with the biomass values. Multiple regression equations using brightness and wetness isolates have been used to predict biomass values. The model used has correlation coefficient of 0.77. Per cent error between observed and predicted biomass was 10.5%. The biomass estimated for the entire national park using stratified and spectral response modelling approaches were compared and showed similarity with the difference of only 4.69%. The results indicate that satellite remote sensing data provide capability of biomass estimation     

The estimation of growth yield and maintenance parameters for photoautotrophic growth

Methods are presented for examining the consistency of experimental data for microbial growth where light energy is converted to chemical energy through photosynthesis. True growth yield and maintenance parameters are estimated for several sets of available experimental data. Methods of parameter estimation are presented which allow all of the measured variables to be used simultaneously for parameter estimation. The results show that a wide range of values have been found for the true growth yield and maintenance parameters. Values of the true growth yield range from 0.04 to values above those predicted by the Z-scheme model for photosynthesis.     

Application of 1D blood flow models of the human arterial network to differential pressure predictions

A new application of 1D models of the human arterial network is proposed. We take advantage of the sensitivity of the models predictions for the pressure profiles within the main aorta to key model parameter values. We propose to use the patterns in the predicted differences from a base case as a way to infer to the most probable changes in the parameter values. We demonstrate this application using an impedance model that we have recently developed (Johnson, 2010). The input model parameters are all physiologically related, such as the geometric dimensions of large arteries, various blood properties, vessel elasticity, etc. and can therefore be patient specific. As a base case, nominal values from the literature are used. The necessary information to characterize the smaller arteries, arterioles, and capillaries is taken from a physical scaling model (West, 1999). Model predictions for the effective impedance of the human arterial system closely agree with experimental data available in the literature. The predictions for the pressure wave development along the main arteries are also found in qualitative agreement with previous published results. The model has been further validated against our own measured pressure data in the carotid and radial arteries, obtained from healthy individuals. Upon changes in the value of key model parameters, we show that the differences seen in the pressure profiles correspond to qualitatively different patterns for different parameters. This suggests the possibility of using the model in interpreting multiple pressure data of healthy/diseased individuals.     

Estimated mechanical properties of synergistic muscles involved in movements of a variety of human joints

One of the most challenging aspects of biomechanical modelling is parameter estimation. Parameter values that define the nonlinear relations within the classic Hill-based muscle model structure have been estimated for a large number of muscles involved in movements of a number of joints. The technique used to estimate these parameters is based on combining information on muscle as a material with geometrical data on muscle-joint anatomy. The resulting relations are compatible with available human experimental data and with past modelling estimates. An estimation of the relative importance of the various synergistic muscle properties during dynamic movement tasks is also provided, aided by examples of muscle load-sharing as a function of optimization criteria including measures of position error, muscle stress and neural effort.     

Nonlinearities and asymmetries of the human cardiovagal baroreflex

To determine whether an approach such as the modified Oxford technique can consistently produce data that reveal the nonlinear nature of the cardiovagal baroreflex and to ascertain whether the model parameters provide unique insight into baroreflex function, we retrospectively examined 91 baroreflex trials (38 subjects, 27 men and 11 women, ages 22-72 yr). The modified Oxford technique (bolus sodium nitroprusside followed by bolus phenylephrine) was used to perturb blood pressure, and the resulting systolic blood pressure-R-R interval responses were plotted and modeled using a linear, a four-parameter symmetric, and a five-parameter asymmetric model. Several issues, such as the effect of data averaging, various approaches to gain estimation, and the predictive value of model parameters, were examined during reflex modeling. Sigmoid models accounted for a greater amount of the variance than did the linear model: linear r2=0.81+/-0.01, four-parameter r2=0.90+/-0.08, and five-parameter r2=0.90+/-0.08 (P<0.05, linear vs. sigmoid models). Data averaging did not affect model fits. Although the four gain estimates (linear remodel, 1st derivative, peak, and set point) were statistically related, the set point gain was significantly lower than other estimates (P<0.05). Subgroup comparisons between young and older healthy subjects revealed differences in all indexes of cardiovagal baroreflex gain, as well as R-R interval operating range and curvature parameters. In conclusion, the modified Oxford technique consistently reveals the nonlinear nature of the human cardiovagal baroreflex. Moreover, of the parameters produced by the symmetric sigmoid model, only the response range provides unique information beyond that of reflex gain.     

Adaptation of reflexive feedback during arm posture to different environments

 In this study we have examined the ability of the central nervous system (CNS) to use spinal reflexes to minimize displacements during postural control while continuous force perturbations were applied at the hand. The subjects were instructed to minimize the displacements of the hand from a reference position that resulted from the force perturbations. The perturbations were imposed in one direction by means of a hydraulic manipulator of which the virtual mass and damping were varied. Resistance to the perturbations came from intrinsic and reflexive stiffness, and from the virtual environment. It is hypothesized that reflexive feedback during posture maintenance is optimally adjusted such that position deviations are minimal for a given virtual environment. Frequency response functions were estimated, capturing all mechanical properties of the arm at the end point (hand) level. Intrinsic and reflexive parameters were quantified by fitting a linear neuromuscular model to the frequency responses. The reflexive length feedback gain increased strongly with damping and little with the eigenfrequency of the total combined system (i.e. arm plus environment). The reflexive velocity feedback gain decreased slightly with relative damping at the largest eigenfrequency and more markedly at smaller eigenfrequencies. In the case of highest reflex gains, the total system remained stable and sufficiently damped while the responses of only the arm were severely underdamped and sometimes even unstable. To further analyse these results, a model optimization was performed. Intrinsic and reflexive parameters were optimized such that two criterion functions were minimized. The first concerns performance and penalized hand displacements from a reference point. The second one weights afferent control effort to avoid inefficient feedback. The simulations showed good similarities with the estimated values. Length feedback was adequately predicted by the model for all conditions. The predicted velocity feedback gains were larger in all cases, probably indicating a mutual gain limiting relation between length and velocity afferent signals. The results suggest that both reflex gains seem to be adjusted by the CNS, where in particular the length feedback gain was optimal so as to maximize performance at minimum control effort. Received: 23 July 2001 / Accepted in revised form: 15 January 2002