Problem 70 - Calculus for Biology and Medicine

Question

The Gompertz equation is used to model the growth of a tumor. We will study it in Chapter \(8 .\) In this model the number of cells \(N(t)\) in a tumorgrows over time at a rate that depends on \(N\), that is, tumors of different sizes grow at different rates, producing a differential equation: $$ \frac{d N}{d t}=a N \ln (b / N) $$ where a and b are positive constants that depend on the type of tumor, whether the tumor is being treated, and on the kind of treatment. In Chapter 8 we will see that the solution to this equation is given by evaluating the integral $$ t=\int \frac{d N}{a N \ln (b / N)} $$ Evaluate the integral (7.13), keeping \(a\) and \(b\) as unknown constants

Step-by-Step Solution

Answer

The integral evaluates to \( t = -\frac{1}{a} \ln|\ln(b/N)| + C \).

Step 1: Simplify the Integrand

The integral we need to evaluate is given by: \[ t = \int \frac{dN}{aN \ln(b/N)}. \] First, notice that \( \ln(b/N) = \ln b - \ln N \). Thus, the integrand can be rewritten as: \[ \frac{1}{aN (\ln b - \ln N)}. \]

Step 2: Substitution Setup

We will use substitution to solve this integral. Let \( u = \ln N \). Then, the derivative \( du = \frac{1}{N} dN \), or equivalently \( dN = N du \). Substitute these into the integral: \[ t = \int \frac{N du}{aN (\ln b - u)} = \int \frac{du}{a(\ln b - u)}. \]

Step 3: Integrate with Simple Substitution

The integral has the form \( \int \frac{du}{a(\ln b - u)} \). This is a standard form, where the integral of \( \frac{1}{c-x} \) is \( -\ln|c-x| \). Apply this result: \[ t = -\frac{1}{a} \ln|\ln b - u| + C, \] where \( C \) is the integration constant.

Step 4: Re-substitute for Original Variable

Recall our substitution \( u = \ln N \). Plug this substitution back into our expression: \[ t = -\frac{1}{a} \ln|\ln b - \ln N| + C. \] Simplify the expression inside the logarithm: \( \ln b - \ln N = \ln(b/N) \). Thus, the expression for \( t \) becomes: \[ t = -\frac{1}{a} \ln|\ln(b/N)| + C. \]

Key Concepts

Tumor Growth Modeling

When it comes to understanding how tumors develop, mathematical models provide significant insights. One widely used model is the Gompertz equation. This model captures how the growth rate of a tumor changes based on its size. Tumors don't grow at a constant rate; they often grow slowly at first and then accelerate before slowing down again as resources become limited. This can be visualized similarly to how a sigmoid curve behaves.
This equation models the number of cells in a tumor over time, represented as \(N(t)\). The growth rate is a function of the current size \(N\), which is typical for biological processes due to resource limitations and other constraints. This dependence on the size of the tumor is what makes Gompertz's equation a handy tool for realistically predicting tumor growth dynamics over time. It provides researchers and healthcare professionals with valuable information on how different stages of treatment might affect tumor progression.

Differential Equations

Differential equations are mathematical equations that involve an unknown function and its derivatives. They are used to model systems that change over time or space and are fundamental in expressing real-world phenomena. In the case of the Gompertz equation for tumor growth, we use a differential equation to describe how the number of tumor cells, \(N(t)\), changes over time.
Specifically, the differential equation used is \(\frac{dN}{dt} = aN \ln(b/N)\). Here, \(a\) and \(b\) are constants based on the type of tumor and the conditions under which it is growing or being treated. This equation illustrates that the rate of change of \(N\) over time \(t\) is affected by both the number of cells \(N\) and the logarithmic relationship between a constant \(b\) and \(N\).
Understanding and solving such differential equations enables us to predict how a system evolves over time, which is crucial for fields dealing with dynamics, like biology and physics.

Integration Techniques

Integration is a core mathematical technique used to find the area under a curve or to solve differential equations. In our Gompertz equation, integrating allows us to find how the number of tumor cells changes over time given initial conditions. The expression \(t = \int \frac{dN}{aN \ln(b/N)}\) represents the process of integrating the function with respect to \(N\).
To evaluate the integral, simplifications are often needed. By rewriting \(\ln(b/N) = \ln b - \ln N\) for the integrand, the expression becomes easier to handle. Successful integration allows us to solve the differential equation and get meaningful predictions regarding tumor progression over time. This process highlights how robust mathematical tools are essential for applications ranging from medical predictions to engineering problems.

Substitution Method

Substitution is a powerful technique used in integration to simplify complicated integrals. In this context, the substitution method lets us transform the integrand of a complex expression into a simpler standard form that is easier to integrate.
For the Gompertz equation, we let \(u = \ln N\). This choice transforms our original differential form: since \(du = \frac{1}{N}dN\), it allows us to substitute \(dN = N du\) back into the integral, simplifying the process: \(t = \int \frac{du}{a(\ln b - u)}\).
This new integral resembles a known form which can be solved more straightforwardly. After solving, we substitute back to the original variables, reversing our initial substitution. This method showcases the power of appropriately chosen substitutions in transforming difficult problems into manageable ones, a classic trick in tackling mathematical challenges.