Hemodialysis is a process by which a machine is used to...

Hemodialysis is a process by which a machine is used to filter urea and other waste products from a patient's blood if their kidneys fail. The concentration of a patient's urea during dialysis is sometimes modeled by supposing there are two compartments within the patient-the blood, which is directly filtered by the dialysis machine, and another compartment that cannot be directly filtered but that is connected to the blood. A system of two differential equations describing this is $\frac{d c}{d t}=-\frac{K}{V} c+a p-b c \quad \frac{d p}{d t}=-a p+b c$ where $c$ and $p$ are the urea concentrations in the blood (in $\mathrm{mg} / \mathrm{mL}$ ) and in the inaccessible pool, respectively, and all constants are positive. (a) Explain each term of the system of differential equations. (b) Construct the phase plane, including all nullclines, equilibria, and arrows indicating the direction of movement in the plane. What happens to the urea concentration as $t \rightarrow \infty ?$

Hemodialysis is a process by which a machine is used to filter urea and other waste products from a patient's blood if their kidneys fail. The concentration of a patient's urea
during dialysis is sometimes modeled by supposing there are two compartments within the patient-the blood, which is directly filtered by the dialysis machine, and another compartment that cannot be directly filtered but that is connected to the blood. A system of two differential equations describing this is

$\frac{d c}{d t}=-\frac{K}{V} c+a p-b c \quad \frac{d p}{d t}=-a p+b c$

where $c$ and $p$ are the urea concentrations in the blood (in $\mathrm{mg} / \mathrm{mL}$ ) and in the inaccessible pool, respectively, and all constants are positive.

(a) Explain each term of the system of differential equations.
(b) Construct the phase plane, including all nullclines, equilibria, and arrows indicating the direction of movement in the plane. What happens to the urea concentration as $t \rightarrow \infty ?$

Key Concepts

Equilibria (Steady States)

Equilibria, or steady states, occur at points where all the rates of change in the system are zero, meaning the system is at rest. Identifying these points in a compartmental system allows one to analyze the long-term behavior of the system, determining conditions under which the system stabilizes or changes qualitatively.

Asymptotic Behavior and Stability

Asymptotic behavior refers to the tendencies of a system as time goes to infinity, describing the eventual state the system approaches. Stability analysis involves determining whether these steady states are attractive (stable), repulsive (unstable), or neutral, which is critical in predicting the outcome of long term processes such as the clearance of substances in biological systems.

Nullclines

Nullclines are curves in the phase plane where the rate of change of one of the variables is zero. They are crucial for understanding the dynamics of the system because their intersections indicate potential equilibria, and the behavior of trajectories near these curves can provide insights into the system's stability.

Phase Plane Analysis

Phase plane analysis is a graphical method used to study the behavior of systems of two differential equations by plotting trajectories in a coordinate system where each axis represents one of the variables. This analysis helps in visualizing how the system evolves over time, revealing the paths taken by the system from various initial conditions.

Differential Equations

Differential equations are mathematical statements that describe how a quantity, such as a concentration in a biological system, changes over time. In the context of modeling processes like hemodialysis, they are used to represent rates of change in different compartments, capturing both the removal of substances and the exchange between interconnected pools.

Compartmental Modeling

Compartmental modeling is an approach used to simplify and analyze complex biological systems by dividing them into interconnected sections or compartments. Each compartment represents a region where the substance under study is assumed to be uniformly mixed, and the model tracks the transfer rates between these compartments as well as removal or input rates.

Transcript

00:01 All right, so part a asks us to explain each term in the equilibrium or in the system of differential equations here.

00:09 So if we notice here, these two terms are the same in both equations because here, c is in the blood.

00:21 It represents an amount in the blood.

00:22 So if we look at ap first, this is the rate of transfer.

00:30 Rate of urea.

00:33 This is from the blood, sorry, not from the blood, from the pool to blood.

00:44 Okay.

00:45 And then bc is going to be the rate of urea going the other direction, from blood to the pool to the pool.

00:56 Now the first term here, now this is the hemodialysis part.

01:03 Okay, the k over vc, this is the rate of hemodialysis.

01:14 Now part b asks us to construct the phase plot.

01:20 So let's do so by first looking at the no clines.

01:26 So for the c nulcline, we have negative kv plus.

01:32 Ap minus b c is equal to 0.

01:37 So we have that p is going to be equal to b over a times c plus k over v also times c.

01:48 Or we can backer out the c and it becomes k over v, all of that times c.

01:57 Now for the y nulcline, we just have here, p is going to be equal to, or sorry, negative ap first plus b c is equal to 0 or we get that p is going to be b over a times c.

02:18 Here, this slope will always be greater than this slope...

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