Problem 10 - Calculus for Biology and Medicine

Question

In Problem 9 we neglected to consider the time delay between a pill being taken and the drug entering the patient's blood. In Chapter 8 we will introduce compartment models as models for drug absorption. We will show that a good model for a drug being absorbed from the gut is that the rate of drug absorption, \(A(t)\), varies with time according to: $$ A(t)=C e^{-k t}, t \geq 0 $$ where \(C>0\) and \(k>0\) are coefficients that will depend on the type of drug, as well as varying between patients. (a) Assume that the drug has first order elimination kinetics, with elimination rate \(k_{1} .\) Show that the amount of drug in the patient's blood will obey a differential equation: $$ \frac{d M}{d t}=C e^{-k t}-k_{1} M $$ (b) Verify that a solution of this differential equation is: $$ M(t)=\frac{C e^{-k t}}{k_{1}-k}+a e^{-k_{1} t} $$ where \(a\) is any coefficient, and we assume \(k_{1} \neq k\). (c) To determine the coefficient \(a\), we need to apply an initial condition. Assume that there was no drug present in the patient's blood when the pill first entered the gut (that is, \(M(0)=0\) ). Find the value of \(a\). (d) Let's assume some specific parameter values. Let \(C=2\), \(k=3\), and \(k_{1}=1 .\) Show that \(M(t)\) is initially increasing, and then starts to decrease. Find the maximum level of drug in the patient's blood. (e) Show that \(M(t) \rightarrow 0\) as \(t \rightarrow \infty\). (f) Using the information from (d) and (e), make a sketch of \(M(t)\) as a function of \(t\).

Step-by-Step Solution

Answer

The drug level follows a modeled increase, peaks, and then declines to zero.

Step 1: Understanding the Problem

We need to solve parts (a) to (f) related to a mathematical model that describes drug absorption and elimination. The model is given by a differential equation with specified conditions and parameters.

Step 2: Formulate the Differential Equation

For part (a), we are given that the rate of drug absorption is governed by \(A(t) = C e^{-kt}\) and the drug elimination follows first order kinetics with rate \(k_1\). Therefore, the rate of change of the amount of drug in the body \(M(t)\) is given by the equation: \[ \frac{dM}{dt} = C e^{-kt} - k_1 M. \] The first term represents the addition of drug, and the second term represents the elimination.

Step 3: Verify the General Solution

For part (b), we need to verify that \[ M(t) = \frac{C e^{-kt}}{k_1 - k} + a e^{-k_1 t} \]is a solution to the differential equation \( \frac{dM}{dt} = C e^{-kt} - k_1 M. \) Plug \(M(t)\) back into the differential equation to check if both sides are equal. First, compute \(\frac{dM}{dt}\) and substitute into the equation; simplify to verify equality.

Step 4: Apply Initial Condition

For part (c), use the initial condition \(M(0) = 0\) to find the coefficient \(a\). Substitute \(t = 0\) into \[ M(t) = \frac{C e^{-kt}}{k_1 - k} + a e^{-k_1 t} \] and solve for \(a\), knowing \(e^0 = 1\). This yields the value of \(a\) based on the initial condition.

Step 5: Analyze the Behavior of M(t)

For part (d), insert the given parameters \(C = 2\), \(k = 3\), and \(k_1 = 1\) into the solution for \(M(t)\). Analyze \(M(t)\) by finding its derivative and checking critical points to determine when the function increases or peaks, and then decreases. Use this to find the maximum level of drug in the blood.

Step 6: Determine Long-term Behavior

For part (e), check the limit of \(M(t)\) as \(t \to \infty\) to show that \(M(t) \to 0\). Since both exponential terms \(e^{-kt}\) and \(e^{-k_1 t}\) decay to zero as \(t\) becomes very large, \(M(t)\) also tends to zero.

Step 7: Sketch M(t)

For part (f), combine insights from previous steps to sketch \(M(t)\). Show the initial increase due to absorption, the peak when the drug level is maximized, and the eventual decrease to zero as \(t\) increases. Use findings of maximum and behavior at \(t \to \infty\) to accurately depict these features in the graph.

Key Concepts

Compartment Models

In pharmacokinetics, a compartment model is a mathematical representation of how drugs move through the body over time. These models divide the body into compartments where each compartment represents a group of tissues with similar drug concentrations.
There are often two main types:

• One-compartment models: Assumes instantaneous mixing of the drug throughout the body.
• Two-compartment models: The body is divided into central and peripheral compartments. The central compartment often includes the blood and organs with rapid uptake.

The idea is similar to simplifying a complex house plumbing system by considering just a few major pipelines. These models are vital for predicting drug concentration over time, which informs dosing and scheduling. They assume first order kinetics, meaning the rate of drug metabolism is proportional to its concentration.

Exponential Decay

Exponential decay describes a process where the quantity decreases at a rate proportional to its current value. In pharmacokinetics, this is often used to model how drugs are eliminated from the bloodstream. The exponential decay formula is given by:

• \(A(t) = Ce^{-kt}\)

Here:

• \(C\) is the initial value or peak concentration.
• \(k\) is the decay rate constant.

This model is intuitive considering a reality where as less of the drug is left in the body, its reduction per unit time tends to decrease. Like watching an ice cube melt, initially, large changes are visible, but as it shrinks, those changes slow down. The exponential decay is a fundamental concept in describing how drugs are cleared from the body.

Initial Value Problems

Initial Value Problems in differential equations involve finding a function that satisfies a given differential equation and meets specified conditions at a starting point. In pharmacokinetics, initial value problems help forecast drug concentration dynamics from a known point.

To solve an initial value problem, you begin with a differential equation:

• \(\frac{dM}{dt} = Ce^{-kt} - k_1 M\)

You also refer to an initial state, such as \(M(0) = 0\), meaning no drug is present in the bloodstream at time zero. Solving involves integrating the differential equation subject to this initial condition.
Practical application allows for tailoring this mathematical approach to individual patients' responses, attacking issues such as overdose, drug interactions, and optimizing drug administration timing.

First Order Kinetics

First order kinetics is a principle in pharmacokinetics where the rate of drug metabolism is directly proportional to its concentration within the body. It implies that:

• The drug concentration decreases exponentially with time.
• The half-life, or the time it takes for the drug concentration to reduce to half, remains constant.

This principle simplifies the modeling of drug elimination by assuming that processes like absorption and elimination happen at rates that are dependent solely on the concentration present. For example, the more a sponge is soaked in water (higher drug concentration), the faster it drips (rate elimination).

Utilizing first order kinetics, predictions can be made about concentration drops and half-lives with varying dosages, enhancing the efficiency of therapeutic regimens.