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Modeling Renal Physiology Tori Turkington, Jordan Mattheisen, Cinzia Ballantyne, & Caroline Martel 1

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Modeling Renal Physiology Tori Turkington, Jordan Mattheisen, Cinzia Ballantyne, & Caroline Martel 1 Modeling Renal Physiology Tori Turkington, Jordan Mattheisen, Cinzia Ballantyne, & Caroline Martel 1. The Nephron The kidney is an essential organ in vertebrates, involved in removing waste products from the body. In accomplishing this goal, the kidney maintains homeostasis in electrolyte levels, acid/base balance, and salt/water balance. The kidney itself has many parts that serve different purposes, but the large part of the kidney can be understood through an explanation of the nephron. The nephron is the structure that is responsible for filtering water, wastes, and substances such as sodium or potassium from the body and moving them on to redistribute them where appropriate, whether that be where those substances are needed or preparing them for excretion. One kidney itself is made up of about one million nephrons. Figure 1.1: A model of the nephron with labeled parts. Blood enters the afferent arteriole and moves to the glomerulus, where capillaries filter out water and small molecules. From there, fluid moves to the proximal convoluted tubule, where sodium is actively reabsorbed, along with water due to the osmotic gradient that is present. Following that is the Loop of Henle, where the sodium concentrations in the fluid start to change. The Loop of Henle is made up of both an ascending and descending limb. In the permeable descending limb, sodium flows in and water flows out, concentrating the amount of sodium ions in the fluid. Concentration is highest at the end of this process, where the descending limb meets the ascending limb. Sodium is actively pumped out in the impermeable ascending limb, without water following. Because this process moves against a concentration gradient, this is an active process that requires metabolic energy. The end result of this is removal of concentrated or diluted urine, as well as other fluid that gets circulated throughout the body. Many of the functions in the nephron can be explored via mathematical models, and these concepts will be presented in the following sections. 2. Dynamics of Sodium and Water: Transport Along Renal Tubules Let the volume rate of flow along the tubule past � be given by � � . In this model we will assume that the flow is steady and independent of time. The volume flow rate through the walls of the tubule per unit length near � is given by �%&'(�). A positive � will be in the direction of increasing � and positive � will represent the outward flux through the walls of the tubules. We will begin by considering a segment of tubule that lies between the arbitrary points � = � and � = � as shown in the diagram below to derive the equations for sodium ion and water transport along the walls in the renal tubule. Figure 2.1: Diagram of the sodium ion and water transport system (Hoppensteadt) We define the following variables as: � = distance along the tubule � � = concentration of the sodium ion inside the tubule at position x � � = flow of water (volume per unit time) along tubule at x in the direction of increasing x �%&' � = outward transport of water (volume per unit time per length of tubule) across the walls of the tubule �DE � = outward transport of sodium ions (number of ions per unit time per length of tubule) across the walls of the tubule Since we are asumming the flow through the tubule is steady, the volume of the segment cannot change. Therefore the flow into the segment must equal the sum of the flow out of the segment and the flux of water out through the walls of the tubule between � = � and � = �. Using the variables we defined earlier, we can rewrite this as: H (2.1) � � = � � + �%&' � �� E Keeping � fixed and treating � like a variable, we can differentiate with respect to � getting �� 0 = (�) + � � , (2.2) �� %&' which can be written as �� 0 = + � � , (2.3) �� %&' since � is an arbitrary number. Using this equation, we see that if the walls of the tubules are impermeable to water, �%&' = 0 and � is a constant. Considering the same section of tubule, we can let the sodium concentration in the tubule be �(�). Therefore, the amount of sodium ion per unit time transported along the tubule by the flow past the point � is � � �(�). Let the amount of sodium ion per unit time per unit length transported outward through the walls of the tubule be �DE(�). Using the equation derived above, we argue that � 0 = �� + � � . (2.4) �� DE In a tubule whose walls are impermeable to sodium ion, �DE = 0 and �� is a constant. These previous two equations apply to all tubules in the kidneys although the properties of their walls will be different as reflected by their fluxes �%&' and �DE. 3. The Loop of Henle We start with the Loop of Henle to construct a model of the nephron. Using the diagram below as reference, the descending limb of the loop is tube 1 and the ascending limb of the loop is tube 2. The concentration of sodium ion is �K(�) and the water flow in the tubules are �K(�) where � = 1,2. The external concentration of sodium ion is � � . Let the flow be positive in the descending limb and negative in the ascending limb. Figure 3.1: Schematic diagram of the Loop of Henle. The descending limb is on the left side (called tube 1) and the ascending limb is on the right side (called tube 2). (Hoppensteadt) 1) The first simplifying assumption we will make is that the walls of the descending limb are permeable to water but not to sodium ions. Although this is not entirely true in the body, we will ignore the slight permeability to sodium ions since it is not essential in the function of the Loop of Henle. Furthermore, we will assume that the permeability to water is so large that the flux makes the internal and external concentrations of sodium ion equal. This gives us the following equations: �� 0 = O + � O � , (3.1) �� %&' � 0 = � � , (3.2) �� O O �O � = � � . (3.3) 2) Next we assume that sodium ions are pumped from the ascending limb at a ∗ steady rate �DE per unit time and that the ascending is impermeable to water. This gives: 0 = ��&/�� (3.4) ∗ 0 = (�/��)(�&�&) + �DE (3.5) 3) At the turn in the Loop of Henle given when � = �, we assume that all sodium ions and water leaving the descending limb enter the ascending limb. This gives us �O � = �&(�) (3.6) �O � = −�& � . (3.7) 4) Additionally, we account for the peritubular capillaries where sodium ions are actively pumped from the ascending limb and water passively flows from the descending limb of the loop. Making the assumption that the capillaries pick up sodium ions locally and since the model is in steady state, we name the rate at O which the peritubular capillaries pick up water as �%&'(�) and the rate at which ∗ they pick up sodium ions as �DE. We assume that the driving force for the reverse filtration is due to oncotic pressure, a form of osmotic pressure exerted by proteins in the surrounding plasma. The proteins are abundant in the peritubular capillaries because they were held back during the process of filtration by the glomerulus. The reverse filtration process that occurs at the peritubular capillaries allows sodium ion to passively be carried by water. The flux of sodium ions and the flow of water is related by the equation: ∗ O �DE = � � �%&' � . (3.8) Using these assumptions and rewriting equations 3.1, 3.2, 3.3, and 3.8, we can derive a differential equation for the interstitial concentration � � . ∗ 0 = (��O/��) + �DE/� � , (3.9) 0 = (�/��) + (�O�), (3.10) where �O� is a constant. To get �O � in terms of � � , we have the equation: �O � = �O 0 �(0)/� � . (3.11) Here, � 0 is the sodium ion concentration of the fluid entering the Loop of Henle via the proximal tubule which is the same as the sodium ion concentration in the blood plasma so it can be considered a given value. �O 0 is the volume rate of flow entering the Loop of Henle which is less than the filtration rate by the fluid reabsorbed in the proximal tubule per unit time. From above, we also get the equation �� �� ��/�� � O = −� = −� 0 �(0) , (3.12) �� O �� O �(�) which can be written as: �� �∗ � = DE . (3.13) �� (�O 0 � 0 ) This implies that �∗ � � � = � 0 ��� DE , (3.14) �O 0 � 0 So �∗ � � � = � 0 ��� DE , (3.15) �O 0 � 0 ∗ where �DE� is the total rate that the sodium ions are actively pumped out of the walls in the ascending limb in the Loop of Henle and �O 0 � 0 is the rate that the sodium ions enter the loop from the proximal tubules. �∗ � � = DE < 1 (3.16) �O 0 � 0 The ratio of these fluxes can be renamed as α which determines the maximum sodium ion concentrating ability of the nephron given through the equation � � = � 0 exp � . (3.17) Considering the ascending limb of the Loop of Henle which we assumed was impermeable to water, we get �& � = �& � = −�O � = −�O 0 � 0 /�(�) = −�O 0 exp (−�). (3.18) Since �& can be thought of as a constant, equation 3.5 becomes: �� �∗ �∗ exp (�) = DE = DE . (3.19) �� −�& �O(0) Therefore, ∗ �& � = �& � + � − � �DE exp � /�O 0 . (3.20) Because �& � is equal to equation 3.17.. ∗ �& � = � 0 exp � + � − � �DE exp � /�O(0), (3.21) Specifically when � = 0, ∗ �& 0 = � 0 exp � + 0 − � �DE exp � /�O 0 , (3.22) which simplifies to �& 0 = � 0 exp � 1 − � . (3.23) We see that exp � 1 − � < 1 when � ≠ 0. �& 0 < � 0 so the fluid leaving the top of the ascending limb of the Loop of Henle is more dilute than the blood plasma. The nephron, however, actually adjusts the �O(0) to achieve a certain �&(0) at the top of the ascending limb.
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