# Solving the Tumor Growth Problem: Find B_0, a & More

In summary, a tumor may be regarded as a population of multiplying cells. The "birth rate" of cells in a tumor decreases exponentially with time, so that birth rate is given by B(t)=(B_0)*e^(-a*t) where B_0 and a are constants. Consequently, the governing equation for the tumor population P(t) at any time t is dP/dt=(B_o)e^(-at)P. The solution to this differential equation is found by solving for B0 and setting that equal to 2P_0.
Problem: A tumor may be regarded as a population of multiplying cells. It is found empirically that the "birth rate" of cells in a tumor decreases exponentially with time, so that birth rate is given by B(t)=(B_0)*e^(-a*t) where B_0 and a are constants. Consequently, the governing equation for the tumor population P(t) at any time t is
dP/dt=(B_o)e^(-at)P

Solve this differential equation exactly. Suppose the initially at t=0, there are 10^6 tumor cells, and that P(t) is then increase at a rate of 3*10^5 cells per month. After 6 months, the tumor has double in population. Find a, and find the behavior of the tumor population as t goes to infinity

I know from the given information, I can find B_0 by using this equation
3*10^5=(B_0)*e^(-a*0)*10^6
given rate of change is 3*10^5; t=0, and p=10^6
B_0=3/10

I am having trouble finding my constant C, a...

Problem: A tumor may be regarded as a population of multiplying cells. It is found empirically that the "birth rate" of cells in a tumor decreases exponentially with time, so that birth rate is given by B(t)=(B_0)*e^(-a*t) where B_0 and a are constants. Consequently, the governing equation for the tumor population P(t) at any time t is
dP/dt=(B_o)e^(-at)P

Solve this differential equation exactly. Suppose the initially at t=0, there are 10^6 tumor cells, and that P(t) is then increase at a rate of 3*10^5 cells per month. After 6 months, the tumor has double in population. Find a, and find the behavior of the tumor population as t goes to infinity

I know from the given information, I can find B_0 by using this equation
3*10^5=(B_0)*e^(-a*0)*10^6
given rate of change is 3*10^5; t=0, and p=10^6
B_0=3/10

I am having trouble finding my constant C, a...

What do you get if you solve the differential equation to work out P(6 months) in terms of $a$, and set that equal to $2P_0$?

Yup! I figured it out, I had two system of equations with two unknowns which I can solve =) I was trying to use maple program to solve this equation, and finally figured out how to display the answers in approximation instead of weird exact numbers. THANKS!

I'm assuming I'm in your class (lol).

Can you explain how you were able to solve this? I tried solving for B0, got 3/10. Tried solving for P, got e^(-Be^(-at))/a

Plugging B and 2P into the equation for P gives a lambertW function which I'm assuming is far beyond the scope of DE.

## 1. What is the "tumor growth problem" and why is it important to solve?

The tumor growth problem refers to the challenge of predicting the growth and spread of tumors in the human body. It is important to solve because it can aid in early detection and treatment of cancer, potentially saving lives.

## 2. What is B_0 and why is it important in solving the tumor growth problem?

B_0 is a parameter that represents the initial size of the tumor. It is important because it is a key factor in predicting the future growth and spread of the tumor.

## 3. What does "a" represent in the context of solving the tumor growth problem?

"a" represents the rate of tumor growth. It is a crucial factor in determining the speed and extent of tumor growth and spread.

## 4. How can solving the tumor growth problem help in developing new treatments for cancer?

By accurately predicting the growth and spread of tumors, solving the tumor growth problem can help researchers develop new treatments that target specific stages of tumor growth. This can lead to more effective and personalized treatments for cancer patients.

## 5. What are some challenges in solving the tumor growth problem?

Some challenges in solving the tumor growth problem include obtaining accurate and comprehensive data, understanding the complex biological processes involved in tumor growth, and developing reliable mathematical models to predict tumor growth and spread.

Replies
3
Views
789
Replies
3
Views
780
Replies
12
Views
2K
Replies
13
Views
2K
Replies
10
Views
5K
Replies
6
Views
1K
Replies
5
Views
1K
Replies
1
Views
1K
Replies
3
Views
1K
Replies
6
Views
655