Question

In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations. The growth of cancer tumors may be modeled by the Gompertz growth equation. Let \(M(t)\) be the mass of a tumor, for \(t \geq 0 .\) The relevant initial value problem is $$\frac{d M}{d t}=-r M(t) \ln \left(\frac{M(t)}{K}\right), M(0)=M_{0}$$ where \(r\) and \(K\) are positive constants and \(0

Short answer

#Answer#: The limiting size of the tumor as time approaches infinity will be K.

Step-by-step solution

1. Verify the solution by substitution

The given solution to the initial value problem is: $$M(t)=K\left(\frac{M_{0}}{K}\right)^{\exp (-r t)}$$ To verify this is the correct solution, we should take the derivative of the suggested solution, \(M(t)\), with respect to \(t\) and substitute it into the given initial value problem. The derivative of the suggested solution \(M(t)\) is: $$\frac{dM}{dt} = K \frac{-r}{M_0} \left(\frac{M_{0}}{K}\right)^{\exp(-rt)}M_{0}\exp(-rt)$$ $$\frac{dM}{dt} = -r M(t) \exp(-rt)$$ Now, we will substitute \(M(t)\) and \(\frac{dM}{dt}\) into the differential equation: $$\frac{d M}{d t}=-r M(t) \ln \left(\frac{M(t)}{K}\right)$$ $$-r M(t) \exp(-rt)=-r M(t) \ln \left(\frac{K\left(\frac{M_{0}}{K}\right)^{\exp (-r t)}}{K}\right)$$ Since both sides are negative, we can cancel \(-rM(t)\): $$\exp(-rt) = \ln \left(\frac{M_{0}}{K}\right)^{\exp (-r t)}$$ Therefore, the given solution is a correct solution of the initial value problem.

2. Substitute the given values and graph the solution

Now, we will substitute the given values \(M_{0} = 100\) and \(r = 0.05\) into the given solution: $$M(t) = K\left(\frac{100}{K}\right)^{\exp(-0.05t)}$$ We cannot plot the function directly since the value of \(K\) is not given. However, it is given that \(M_{0} < K, \) so the function will have an asymptote at \(M(t) = K. \)

3. Estimate the limiting size of the tumor

As we see in step 2, the function \(M(t) = K\left(\frac{100}{K}\right)^{\exp(-0.05t)}\) has an asymptote at \(M(t) = K\). Since \(M_{0} < K\), the tumor's size never reaches \(K\) and will always remain less than \(K\). Therefore, as \(t\) approaches infinity, the limiting size of the tumor will be \(K\).

Key concepts

Differential Equations

Differential equations are mathematical equations that involve derivatives, which represent rates of change. These types of equations are essential in modeling situations where change occurs continuously over time, such as population growth or chemical reactions.
Differential equations can be classified as either ordinary or partial. Ordinary differential equations (ODEs) involve functions of a single variable and their derivatives, which is what we deal with in the Gompertz growth equation for tumor modeling. The equation given in the Gompertz model is a first-order ODE because it includes the first derivative of the tumor mass with respect to time, represented as \(\frac{dM}{dt}\).
To solve a differential equation means to find a function that satisfies it. In our context, solving the given ODE provides insight into how the tumor mass changes over time. The process typically involves simplifying the equation or checking given solutions by substitution, as seen in the original exercise, to verify if they satisfy the differential equation.

Initial Value Problem

An initial value problem (IVP) is a type of differential equation that, besides the equation itself, includes additional information about the value of the unknown function at a particular point. This extra piece of information is known as the initial condition.
In the context of the Gompertz Growth Equation for modeling tumor growth, we have an IVP where the initial mass of the tumor \(M_0\) at time \(t=0\) is provided. This helps us determine the specific solution to the differential equation that aligns with the initial circumstances of the problem. The value \(M(0) = M_0\) essentially sets the starting point of our solution curve and enables us to map the progression of tumor growth accurately over time.
Understanding and solving an IVP is crucial because it guarantees a unique solution that reflects the real-world scenario modeled by the differential equation. In our exercise, confirming the solution through substitution ensures it adheres to both the differential equation and the given initial condition.

Tumor Growth Modeling

Tumor growth modeling is pivotal in understanding how cancer progresses within the body. Accurate models, like the Gompertz growth equation, help predict the future behavior of a tumor’s growth.
The Gompertz equation is particularly useful in this regard because of its realism in capturing initial rapid growth that transitions into slower growth as the tumor reaches its carrying capacity \(K\). The equation is given by:

• \(\frac{dM}{dt} = -r M(t) \ln\left(\frac{M(t)}{K}\right)\),

where \(r\) is the growth rate and \(K\) is the theoretical maximum tumor size.
In this model, as the size of the tumor \(M(t)\) approaches \(K\), its growth rate decreases, making it an ideal model for situations where growth is self-limiting. This is characteristic of many biological processes, where constraints like space and nutrients limit the size a population or tumor can reach.
Estimating the long-term behavior of tumor growth is important for developing treatment strategies. By understanding that \(M(t)\) approaches \(K\) as \(t\) goes to infinity, researchers and doctors can approximate the maximum potential size of the tumor and use this information to plan interventions accordingly.