Wave propagation in a network of interacting nephrons

Highlights

• Kidneys are vital in maintaining a proper environment for cells.

• Nephron is the single function unit of kidney.

• A network of nonlinear coupled nephrons is considered.

• Nephrons are linked by the connection between the radius of their afferent arterioles.

• The results shows spiral waves in the periodic behavior of each single nephron.

Abstract

Nephrons are the fundamental units of the kidneys, functioning for the regulation of the blood pressure and maintaining a normal environment for the body’s cells. The nephrons, as the other biological cells, are not isolated and interact with each other. According to the physiological facts, the nephrons have two types of coupling, known as hemodynamic and vascular propagated coupling, which can effect on the neighboring nephrons activities. In this paper, a 2-dimensional network of interacting nephrons is studied by considering a local linear coupling. Spiral waves are special spatiotemporal patterns, having strong relations with the neural and cardiac cells’ activities. Here, we investigate the spatiotemporal patterns of the coupled nephron models in different cases, for finding the possibility of occurrence of spiral waves. The obtained results confirm the appearance of small spiral seeds in the network, when the nephrons have periodic behavior and strong coupling strength. However, it is observed that the small spiral seeds are not stable in time and disappear and reappear in different places.

Introduction

The most crucial role of a kidney is regulating the blood pressure by excreting salt (NaCl) and water, making a stable and normal environment for body cells. This function is achieved through selective reabsorption and secretion and also a very complicated feedback mechanism. This mechanism protects the nephron and its function against changes in the blood pressure that may occur due to the variation in compactness of blood constituents. Therefore, it is very important to precisely understand its nonlinear and complicated dynamics [1], [2], [3]. Nephron, the functional unit of a kidney is responsible for all the mentioned mechanisms and tasks. Nephron is a complex tube made of a layer of closed tissue that is closed at one end and opened at the other end of the pelvis. There are about one million nephrons in each human kidneys. Each nephron is made up of a Bowman capsule, a close-up tube, a hinged tube, a round tube, and a collector duct. The most important kidney activity in humans is the release of nitrogenous substances such as urea. Nephrons can increase urea up to 100 times in urine, but the urine can only contain 5% urea and become toxic in higher urea concentrations [4].

The ability of nephrons in compensating the changes in the arterial blood pressure is primarily attributable to the well-known feedback mechanism, known as Tubuloglomerular feedback (TGF) [2]. In this mechanism, the nephron controls the flow rate through the ion concentration and composition of the flow that leaves the Henle ring [1], [5], [6]. Studies by Leyssac and Baumbach [7], as well as Leyssac and Rathlou [8], [9] have shown that this regulatory feedback mechanism can be unstable and produce self-sustained oscillations with a specified period of about 30 to 40 s and a variable domain and phase in the proximal tubular pressure [10], [11]. Similar fluctuations are observed in the distal tubular pressure as well as in the concentration of chloride in the inlet section of the loop. In normal rats, this behavioral fluctuation is expressed as a limit cycle, that is, periodic behavior. However, for rats with high blood pressure, these oscillations are chaotic [1], [12], [13]. The mechanism of TGF regulation is in fact the result of the anatomy of the nephrons, which is shown in Fig. 1. The entrance duct of the ascending segment of the Henle loop passes from the same distance to the afferent arteriole of the neighboring nephron. At the contact point, certain cells – known as Macula Densa cells – measure the concentration of sodium chloride in the tubular fluid and produce a signal that activates smooth muscle cells in the arteriolar wall [6]. As the glomerular filtration rate goes higher, the fluid will flow more rapidly in the loop of Henle, and there will be more concentration of sodium chloride in the Macula Densa cells. The high concentration of sodium chloride leads to the activation of smooth vascular cells in the arteriolar wall and consequently the reduction of the vessel radius. Therefore, the blood flow and the resulting glomerular filtration rate decrease. Hence, the feedback mechanism of TGF is negative feedback [6].

In biological systems, due to their main characteristics such as complexity, self-organization, hierarchy, etc., it is important to consider the interactions among the agents, besides identifying a single agent [14]. In other words, it is important to consider and impart complex architectures for biological systems [15], [16], which can lead to complex collective behaviors such as synchronization [17], [18], chimera states [19], [20], spiral waves [21], [22], stochastic resonance [23], etc. Therefore, it is necessary to consider the interactions among the kidney’s functional units (nephrons). These interactions can considerably affect the dynamics of nephrons network.

Pattern formation is an important phenomenon among natural phenomena due to the interaction between components [24], [25], [26]. The appearance of different patterns in a system is directly related to the system’s dynamics. Therefore, studying the formation and selection of patterns has been considered in many physical, chemical and biological systems [27]. Spiral waves are one of the enchanting patterns chosen by nature, in many organisms including plants and animals, formation of weather patterns, clouds, sea waves, and so on. There are some evidence that spiral waves play a significant role in many biological systems [28]. Majority of researches have proved that the most dangerous cardiac arrhythmias are related to reentrant waves, one of which is spiral wave [29]. Some experimental results confirm that the spiral wave can also be seen in the cerebral cortex [30]. Due to its importance, there has been a growing interest in the study of spiral waves in biological systems [31], [32], [33], [34].

Target waves and spiral waves are assumed to have the ability to adjust and control the collective behavior of the network. Besides, spiral waves are mostly identified self-sustained. For instance, in the cortex, its emergence is assumed to be a modulation of electrical activities [26]. The emergence of spiral waves can improve the whole neuronal activities and thus can be helpful in the understanding of initiations of neural illnesses [26]. Spiral waves have also been studied in a mathematical model of human ventricular tissue with myocytes and Purkinje fibers. Investigating a coupled bilayer tissue, composed of one layer of Purkinje cells and a layer of endocardial cells, resulted in fixing the spiral wave turbulences [35].

As a consequence, investigating the occurrence of spiral waves in the network of coupled nephrons can give rise to new insides in this field. In this paper, the emergence and maintenance of the spiral wave are studied in a coupled nephrons network. Basically, appearing of spiral waves relies on several factors including the amplitude and frequency of the external stimulus, the local dynamics of the units and the coupling strength between them. Generally, the proper adjustment of all these factors is needed to support the spiral wave to grow and cover the whole lattice. Here, we examine the effects of the local dynamics of every single nephron and also variation in the coupling strength.

According to physiological studies, nephrons have two types of couplings. The first type of coupling, called vascular propagated coupling, is derived from an electrical or chemical signal, which is still not fully understood. In this type of coupling, which is the effect of the feedback signal of the activation feedback from the Macula Densa cell, there is a mist detector signal that only affects a limited number of its neighbors due to its damping. The second type of coupling, known as hemodynamic, stems from the fact that nephrons receive blood from common branches and the neighboring nephrons use close connections. Hence, an increase or decrease in the amount of blood received by the neighbors affects the dynamics of the other nephrons [6]. According to the physiological evidence, to get the proper response from the computational model, a square network of nephrons is taken into account in this paper. Each nephron is considered to be coupled with its four surrounding neighbors, and the spatiotemporal patterns of the network are investigated.