Problem 52 - Calculus: Early Transcendentals

Question

A model for tumor growth is given by the Gompertz equation \( \frac {dV}{dt} = a (\ln b - \ln V) V \) where \( a \) and \( b \) are positive constant and \( V \) is the volume of the tumor measured in \( mm^3. \) (a) Find a family of solution for tumor volume as a function of time. (b) Find the solution that has an initial tumor volume of \( V(0) = 1 mm^3. \)

Step-by-Step Solution

Answer

The family of solutions: \( V = b e^{-e^{-(at + C)}} \). For \( V(0) = 1 \): \( V(t) = b e^{-e^{at - \ln(\ln b)}} \).

Step 1: Understand the Gompertz Equation

The given Gompertz equation is \( \frac{dV}{dt} = a(\ln b - \ln V)V \). It describes the rate of change of tumor volume, \( V \), as proportional to the volume itself and the difference between the logarithm of a constant \( b \) and the logarithm of \( V \).

Step 2: Separation of Variables

To solve the differential equation, we first separate variables: \( \frac{1}{V(\ln b - \ln V)} dV = a \, dt \). This allows us to integrate both sides individually.

Step 3: Integrate Both Sides

Integrate the left-hand side with respect to \( V \) and the right-hand side with respect to \( t \). For the left side, let \( u = \ln V \), then \( du = \frac{1}{V}dV \). We have \( \int \frac{1}{\ln b - u} \, du \), which simplifies as \( \int -\frac{1}{u - \ln b} \, du \). This gives a natural logarithm: \( -\ln|u - \ln b| \). The right side simply integrates to \( at + C \), where \( C \) is the integration constant.

Step 4: Solve for \( V \)

After integrating, we have \(-\ln|\ln V - \ln b| = at + C \). Integrate both sides, leading to \( |\ln V - \ln b| = e^{-(at + C)} \). To eliminate the absolute value, note that \( \ln V - \ln b \) must remain consistent in sign over time.

Step 5: Express \( V \) in Terms of \( t \)

We express \( \ln V = \ln b - e^{-(at + C)} \). This gives \( V = e^{\ln b - e^{-(at + C)}} \), or equivalently, \( V = b e^{-e^{-(at + C)}} \), which is the general form of the solution for tumor volume as a function of time.

Step 6: Apply Initial Condition \( V(0) = 1 \)

Using the initial condition \( V(0) = 1 \), substitute into the solution: \( 1 = b e^{-e^{-C}} \). Solving for \( C \) gives \( e^{-C} = \ln b \), so \( C = -\ln(\ln b) \).

Step 7: Write the Specific Solution

Substitute \( C = -\ln(\ln b) \) back into the general solution: \( V(t) = b e^{-e^{at - \ln(\ln b)}} \). This is the specific solution with the initial condition \( V(0) = 1 \).

Key Concepts

Differential Equations

Differential equations are mathematical expressions that involve functions and their derivatives. They are crucial in modeling systems that change over time, such as in physics, engineering, and biology. A differential equation expresses a relationship between a dependent variable and its rates of change (derivatives) with respect to one or more independent variables. In the context of the Gompertz equation for tumor growth, the differential equation relates the rate of change of the tumor's volume, \( \frac{dV}{dt} \), to the current volume \( V \) and two constants \( a \) and \( b \).
This equation type allows us to predict how the volume will change over time. Understanding this fundamental concept helps us comprehend complex natural and engineered systems.
Differential equations can be solved using various techniques, depending on their form. One common method is "separation of variables," which we'll discuss next.

Tumor Growth Modeling

Tumor growth modeling is an essential step in understanding how tumors evolve and expand. The Gompertz equation is frequently used for this purpose due to its realistic representation of biological growth patterns. This model captures the slowed growth rate of tumors after an initial rapid expansion, mimicking actual tumor behavior in living organisms.
In the Gompertz model, the growth rate slows as the volume \( V \) increases, due to the \( \ln V \) term influencing \( \frac{dV}{dt} \).
This representation is crucial for predicting how a tumor might behave in response to different conditions or treatments. By modeling tumor growth effectively, researchers and clinicians can develop better strategies to understand the progression of cancer and possibly improve treatment outcomes.

Separation of Variables

Separation of variables is a technique used to solve certain types of differential equations. The method involves rearranging the equation so each variable and its differential appear on opposite sides of the equation. This allows you to integrate each side separately.
In the Gompertz equation, separation of variables begins by rearranging terms: \( \frac{1}{V(\ln b - \ln V)} dV = a \ dt \). This setup enables the integration of \( V \)-related expressions on one side and \( t \)-related expressions on the other.

• Integration Process: Once variables are separated, integrate both sides. The left side is handled by substituting \( u = \ln V \), which simplifies the integral.
• Solving Integrals: The integration of \( \int \frac{1}{\ln b - u} \ du \) yields \(-\ln|u - \ln b|\), while \( \int a \, dt \) integrates to \( at + C \).

This method makes solving the differential equation more straightforward, leading directly to finding the function detailing the tumor volume as a function of time.