Discrete network models of interacting nephrons
Introduction
A complex system is comprised of many interconnected parts, whose behaviours are dependent on the behaviour of the other parts [1]. Such systems exhibit emergent behaviour that, although it may be deterministic, has properties that can only be examined at a higher level than the individual parts. The difficulty in understanding such systems is that the behaviour of the whole is highly dependent on the individual parts, and each part is highly dependent on other parts [2].
The kidney is a complex system [3], [4], [5], resilient to the effects of renal diseases and the loss of renal tissue. The extent to which the basic units–nephrons–operate independently or as interacting populations is not well understood and has proved difficult to study. Indeed, although some of the regulatory mechanisms in the kidney are well understood, little is understood about how the stability of the entire kidney arises from the individual nephrons and their interactions.
This paper presents a multi-nephron model for studying the interactions among nephrons and the emergent dynamics that arise from them, which renders the simulation of an entire kidney (106 nephrons) tractable. This model lays the foundations for a virtual kidney, capable of predicting whole-kidney function and providing insight into how whole-kidney stability arises from the interactions between nephrons. Such a model would facilitate novel research possibilities–such as studying the emergent effects of progressive renal disease–and our ultimate aim is to develop a kidney model for use in clinical applications.
Existing models have focused on the chemical processes and fluid mechanics of the nephron at the cellular and tubular levels, and the hemodynamics of the renal arterial tree. Mathematical models of renal tubular function have been developed for most nephron segments [6]. Models of coupled nephrons have been used to study the oscillations in nephron filtration rates [7], [8], [9], although only a few [10], [11] examine more than two nephrons. More general studies of coupled oscillators also exist [12], but these circuit models do not allow for interactions such as vascular signalling (described in Section 2). Multi-nephron models have also been used to study the concentrating effects of the loop of Henle and the collecting ducts [13], [14], [15].
An observation common to the multi-nephron and multi-oscillator studies is that the dynamics of a single nephron/oscillator in isolation differ significantly from those of the same unit placed in a multi-unit system [10], [11], [12]. Observed changes include the regions of stable and unstable dynamics, and the nature of the steady-state solutions. Such a behaviour is to be expected of a complex system.
These models are all formulated as systems of coupled differential equations, an approach that is well suited to modelling solute transport and fluid mechanics at the cellular and tubular levels. However, this approach is not amenable to modelling and analysing the kidney as a complex system [2], [16], as the large number of functional units (106 in the human kidney) and the associated coupling mechanisms would lead to 107 or more highly-coupled differential equations. Since the computational cost of solving each individual equation can increase dramatically due to coupling, and unique solutions are not guaranteed to exist, a whole-kidney model based on this approach would almost certainly be computationally intractable.
The model introduced here is a discrete-time network automata model, combining cellular automata and complex networks into a single model. The network structure explicitly represents the connectivity and interactions in a multi-nephron system, from which the associated difference equations are derived. The nephron tubule function is modelled at a higher level of abstraction than most existing models, as the focus is on dynamics at much higher scales than the cellular level, and the difference equations can be solved explicitly without resorting to complicated approximation techniques. By refining the model structure and deriving difference equations from equations in existing models, this model is capable of approximating existing continuous models.
This paper begins by introducing the physiology and function of the kidney, then presenting the multi-nephron model, whose structure mimics the renal physiology. The dynamics of a single-nephron system are validated, and a two-nephron system is used to show that, as mentioned in [17], the strength of the anti-phase coupling mechanism and the arterial blood pressure have equivalent effects. The same two-nephron system is then used to demonstrate the interactions that arise between the two coupling mechanisms. Finally, the computational cost of the model is demonstrated to render the simulation of a whole-kidney model (106 nephrons) computationally tractable.
Section snippets
Nephrons and the kidney
The kidney is one of the major organs involved in whole-body homeostasis. It is responsible for regulating solute concentrations in the blood, the volume of extra-cellular fluid, the acid–base balance and blood pressure in the body. It is also a part of the body’s endocrine system, as it secretes a number of hormones.
The functional unit of the kidney is the nephron, a long tubule into which blood plasma is filtered (see Fig. 1). As the filtrate flows along the nephron, solutes and water are
Network models
Two common classes of network model are: cellular automata[24], [25], [16], which have fixed, regular topologies (typically a square or hexagonal grid) and homogeneous update rules; and complex networks [26], [27], [28] with complex, varying topologies, which allow a state to propagate over the network based on the network topology (known as percolation[29]).
Cellular automata have been used to study biological phenomena such as computational operations in reaction–diffusion systems [24] and
Discrete time
The kidney exhibits multi-rate dynamics, from a matter of seconds (e.g., the TGF mechanism) to responses on the order of hours (e.g., the secretion of hormones). The filtration rates of individual nephrons tend to oscillate with a period of around 30–40 s [7], [21], [32] and the TGF signal has a delay of 3–5 s [21], [9], so we chose to use a time-step of one second. This sampling rate is sufficient to reconstruct signals with frequencies less than 0.5 Hz [34] and therefore meets our needs.
Fluid
Model analysis
The behaviour of the single-nephron model is validated by examining the filtration rate and the stability of the sodium gradient produced by the model. We investigate the behaviour of a two-nephron model, showing that the arterial pressure and steepness of the TGF response have similar effects on the model. We show that the two-nephron model produces a larger sodium gradient than the single-nephron model, and that the gradient produced by an eight-nephron model is larger again. Finally, we
Conclusion
We have presented a novel multi-nephron model, combining graph automata and network approaches into a single model, allowing the connectivity and interactions in the model to be easily altered. The underlying equations explicitly calculate the next state of the model, unlike systems of continuous equations, which provide constraints on the state for which a solution must be found. The model is interactive, as parameters can be altered at any point of a simulation.
The single-nephron model is
Acknowledgements
The first author’s sincere thanks to Donald J. Marsh and S. Randall Thomas for providing commentary on the manuscript.
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2014, American Journal of Physiology - Renal Physiology