{Use of Tech} Tumor growth The Gompertz growth equation is often used to model the growth of tumors. Let M(t) be the mass of a tumor at time t≥0. The relevant initial value problem is dM/dt=−rM ln (M/K), M(0)=M0, where r and K are positive constants and 0\u003cM0\u003cK.b. Solve the initial value problem and graph the solution for r=1, K=4, and M0=1. Describe the growth pattern of the tumor. Is the growth unbounded? If not, what is the limiting size of the tumor?

Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

All textbooks

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.3.51b

Chapter 9, Problem 9.3.51b

{Use of Tech} Tumor growth The Gompertz growth equation is often used to model the growth of tumors. Let M(t) be the mass of a tumor at time t≥0. The relevant initial value problem is
dM/dt=−rM ln(M/K),M(0)=M0,
where r and K are positive constants and 0<M0<K.
b. Solve the initial value problem and graph the solution for r=1,K=4, and M0=1. Describe the growth pattern of the tumor. Is the growth unbounded? If not, what is the limiting size of the tumor?

Verified step by step guidance

Recognize that the given differential equation is a Gompertz growth model: \[\frac{dM}{dt} = -r M \ln\left(\frac{M}{K}\right), \quad M(0) = M_0,\] where \(r\), \(K\) are positive constants and \(0 < M_0 < K\).

Rewrite the equation by separating variables. Divide both sides by \(M \ln\left(\frac{M}{K}\right)\) and multiply both sides by \(dt\) to get: \[\frac{dM}{M \ln\left(\frac{M}{K}\right)} = -r \, dt.\]

Make the substitution \(u = \ln\left(\frac{M}{K}\right)\), which implies \(M = K e^u\) and \(dM = K e^u du\). Substitute these into the integral to simplify the left side integral in terms of \(u\).

Integrate both sides: the left side with respect to \(u\) and the right side with respect to \(t\). After integration, solve for \(u\) as a function of \(t\), then back-substitute to find \(M(t)\) in terms of \(t\).

Apply the initial condition \(M(0) = M_0\) to determine the constant of integration. Finally, analyze the solution to describe the tumor growth pattern, noting that the growth is bounded and approaches the limiting size \(K\) as \(t \to \infty\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gompertz Growth Model

The Gompertz growth model describes growth processes that slow down as they approach a limiting size. It is characterized by a differential equation where the growth rate decreases exponentially with the size of the population or mass. This model is often used in biology to represent tumor growth, capturing the initial rapid growth that slows as the tumor nears a maximum carrying capacity.

Solving Initial Value Problems (IVPs)

An initial value problem involves a differential equation along with a specified value of the unknown function at a starting point. Solving an IVP means finding a function that satisfies both the differential equation and the initial condition. Techniques include separation of variables, integrating factors, or substitution, depending on the equation's form.

Long-term Behavior and Stability of Solutions

Analyzing the long-term behavior of solutions to differential equations helps determine if growth is bounded or unbounded. Stability analysis identifies equilibrium points and whether solutions approach these points over time. For the Gompertz model, the tumor mass approaches a limiting size (carrying capacity), indicating bounded growth and a stable equilibrium.

Recommended video:

05:21

Finding Limits by Direct Substitution

Related Practice

Textbook Question

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).

y′(t) = -2y, y(0) = 1; Δt = 0.2, T = 2; y(t) = e⁻²ᵗ

views

Textbook Question

Properties of stirred tank solutions

b. Verify that M(0) = M₀

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. Euler’s method is used to compute exact values of the solution of an initial value problem.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. The solution of a stirred tank initial value problem always approaches a constant as t→∞

views

Textbook Question

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.

y′(t) = 2t + 1, y(0) = 0; y(t) = t² + t

views

Textbook Question

brOrthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection. A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. Use the following steps to find the orthogonal trajectories of the family of ellipses 2x² + y² = a²

b. The family of trajectories orthogonal to 2x² + y² = a² satisfies the differential equation dy/dx = y/(2x). Why?

views