Question 12.1: Consider the reabsorption/metabolism of glucose within the n...
Consider the reabsorption/metabolism of glucose within the nephron as modeled by a two-compartment system. Assume that the input of glucose into the nephron is constant and defined by K1. The output of glucose from the nephron can be described by the metabolism of glucose by the nephron epithelial cells (K2) and the reabsorption of glucose into the peritubular capillaries (K3), where the Ks are defined as the transfer rate, which account for the diffusion coefficient and the membrane parameters. The initial quantity of glucose within the nephron is defined as q0. Solve for the time rate of change of the quantity of glucose within the nephron.
The mass balance of glucose quantity can be defined as the accumulation of glucose within the nephron is equal to the input of glucose minus the output of glucose. Therefore, the mass balance equation is
\frac{dq}{dt}= q_{in} – q_{out}
The input quantity is defined as
K1q
The output quantity is defined as
(K2 + K3)q
Using this definition within the differential equation
\frac{dq}{dt}= K_{1}q – (K_{2} + K_{3})q
\frac{dq}{q}= (K_{1} – K_{2} – K_{3})dt
lnq= (K_{1} – K_{2} – K_{3})t + c
q(t)= Ce^{(K_{1} – K_{2} – K_{3})t}= q_{0}e^{(K_{1} – K_{2} – K_{3})t}
where C is the initial concentration of glucose within the nephron. With increasing complexity, these methods can still be applied, but it is necessary to quantify the time rate of change of each substance, within each compartment. This can lead to multiple differential equations that must be solved simultaneously.