E. Bruce Pitman

Spectral properties of the tubuloglomerular feedback system

A simple mathematical model was used to investigate the spectral properties of the tubuloglomerular feedback (TGF) system. A perturbation, consisting of small-amplitude broad-band forcing, was applied to simulated thick ascending limb (TAL) flow, and the resulting spectral response of the TGF pathway was assessed by computing a power spectrum from resulting TGF-regulated TAL flow. Power spectra were computed for both open- and closed-feedback-loop cases. Open-feedback-loop power spectra are consistent with a mathematical analysis that predicts a nodal pattern in TAL frequency response, with nodes corresponding to frequencies where oscillatory flow has a TAL transit time that equals the steady-state fluid transit time. Closed-feedback-loop spectra are dominated by the open-loop spectral response, provided that γ, the magnitude of feedback gain, is less than the critical value γc required for emergence of a sustained TGF-mediated oscillation. For γ exceeding γc, closed-loop spectra have peaks corresponding to the fundamental frequency of the TGF-mediated oscillation and its harmonics. The harmonics, expressed in a nonsinusoidal waveform for tubular flow, are introduced by nonlinear elements of the TGF pathway, notably TAL transit time and the TGF response curve. The effect of transit time on the flow waveform leads to crests that are broader than troughs and to an asymmetry in the magnitudes of increasing and decreasing slopes. For feedback gain magnitude that is sufficiently large, the TGF response curve tends to give a square waveshape to the waveform. Published waveforms and power spectra of in vivo TGF oscillations have features consistent with the predictions of this analysis.

A dynamic numerical method for models of renal tubules

We show that an explicit method for solving hyperbolic partial differential equations can be applied to a model of a renal tubule to obtain both dynamic and steady-state solutions. Appropriate implementation of this method eliminates numerical instability arising from reversal of intratubular flow direction. To obtain second-order convergence in space and time, we employ the recently developed ENO (Essentially Non-Oscillatory) methodology. We present examples of computed flows and concentration profiles in representative model contexts. Finally, we indicate briefly how model tubules may be coupled to construct large-scale simulations of the renal counterflow system.

A MODEL FOR GRANULAR FLOWS OVER AN ERODIBLE SURFACE

Geophysical mass flows, such as debris flows, volcanic avalanches, and landslides, can be initiated consequent to volcanic activities. These flows can be tens of meters in depth and hundreds of meters in length and contain

Nonlinear filter properties of the thick ascending limb

O(10^6

A dynamic numerical method for models of the urine concentrating mechanism

–

Mechanisms by which thrombolytic therapy results in nonuniform lysis and residual thrombus after reperfusion

10^8)\,\mathrm{m}^3

A numerical study of a rotationally degenerate hyperbolic system. Part I: The Riemann problem

or more of material. The bulk flow can easily erode the volcanic rocks and soil on the slopes of mountains, thereby increasing their size several fold. The range of scales and the rheology of these flows, especially with erosion, present significant modeling and computational challenges. This paper describes an approach to incorporate the interaction between a mass flow, an erodible bed, and the topography of that bed. The kinetic theory for rapid granular flow is employed to model a “mixing” layer, the interface between the flowing material and the basal surface and the area where erosion occurs. A tractable set of equations is derived, a hyperbolic system that describes the motion of a granular flow and the elevation of erodible bed.

Instability in critical state theories of granular flow

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.

The stability of granular flow in converging hoppers

Dynamic models of the urine concentrating mechanism consist of large systems of hyperbolic partial differential equations, with stiff source terms, coupled with fluid conservation relations. Efforts to solve these equations numerically with explicit methods have been frustrated by numerical instability and by long computation times. As a consequence, most models have been reformulated as steady-state boundary value problems, which have usually been solved by an adaptation of Newton’s method. Nonetheless, difficulties arise in finding conditions that lead to stable convergence, especially when the very large membrane permeabilities measured in experiments are used. In this report, an explicit method, previously introduced to solve the model equations of a single renal tubule, is extended to solve a large-scale model of the urine concentrating mechanism. This explicit method tracks concentration profiles in the upwind direction and thereby avoids instability arising from flow reversal. To attain second-orde...

Forces on bins: The effect of random friction

A transport reaction model describing penetration of plasmin by diffusion and permeation into a dissolving fibrin gel was solved numerically to explore mechanisms that lead to the formation and growth of dissolution fingers through blood clots during thrombolytic therapy. Under conditions of fluid permeation driven by arterial pressures, small random spatial variations in the initial fibrin density within clots (±4 to 25% peak variations) were predicted by the simulation to result in dramatic dissolution fingers that grew in time. Within vitro experiments, video microscopy revealed that the shape of the proximal face of a fibrin gel, when deformed by pressure-driven permeation, led to lytic breakthrough in the center of the clot, consistent with model predictions of increased velocities in this region leading to cannulation. Computer simulation of lysis of fibrin retracted by platelets (where more permeable regions are expected in the middle of the clot due to retraction) predicted cannulation of the clot during thrombolysis. This residual, annular thrombus was predicted to lyse more slowly, because radial pressure gradients to drive inner clot permeation were quite small. In conjunction with kinetic models of systemic pharmacodynamics and plasminogen activation biochemistry, a two-dimensional transport-reaction model can facilitate the prediction of the time and causes of clot cannulation, poor reperfusion, and embolism during thrombolysis.