In class we examined a growth curve for mouse mammary tumor cells. This question examines the rate of growth of Lewis lung carcinoma from a paper by L. Simpson-Herren et al. (1974)[1] . We will concentrate on their study of the growth of a primary tumor that is implanted in the subaxillary region of BDF mice. They collected data on the tumor weight (in mg) as time passes (in days). Many studies have shown that the best model to fit tumors is the Gompertz growth model, while others use the standard logistic growth model that we studied in an earlier lab. In this lab, we will develop both the logistic and the Gompertz growth curves, then use these models to simulate the data in the paper.

Below we present a table that gives the time (shifted so t = 0 at the first data point), the weight of the tumor (in mg), and the growth rate at each time. (The growth rate uses a 3-point method to approximate the derivative, which is not covered in this course.) We will fit the growth rate data to the logistic and Gompertz growth functions, then simulate each of these models to compare to the data presented in the paper. The table presents the weight of the tumor (p) in mg at times (t) (days)with the growth rate given by g(p) in mg/day.

a. You begin this laboratory exercise by finding the best logistic growth function for the growth of the tumor. In particular, you want to use Excel's trendline polynomial fit of order two with the y-intercept set to zero through the data for p and g(p).

Give the equation for the best fitting growth function, g(p). (Note that you ignore the times listed in the table to find g(p).) Include the sum of square errors between the logistic growth curve and the data. Since g(p) represents the growth rate of the tumor, then the tumor is at equilibrium when g(p) = 0. Find all equilibria according to this model. The largest equilibrium is the maximum size that a particular tumor can reach due to limitations of blood supplies from angiogenesis. The maximum growth rate occurs at the vertex. Find this maximum growth rate and compare to the maximum growth rate in the data. Write these values in your report. Show a graph of the data and the best quadratic g(p) passing through the data, but place this graph below, including the Gompertz grow curve on the graph.

b. The Gompertz growth curve satisfies the equation

G(p) = p(a - b ln(p)).

In this part of the problem, you use the same data as you did in Part a, but you now fit the Gompertz growth curve G(p) through the data. In this case, you want to find the least squares best fit of G(p) to the data by varying a and b and using Excel's Solver routine. Write in your report the best fitting Gompertz growth function, G(p). Include the sum of square errors between the Gompertz growth curve and the data. The Gompertz model has been shown to not hold well for small numbers of tumor cells, and the logarithm function isn't defined at 0. Find the equilibrium corresponding the largest size that this model predicts for the tumor and compare to the value from the logistic growth equation. Use your techniques of differentiation to find the maximum growth rate from the Gompertz growth model and compare this to the maximum growth rate of the logistic growth model. Show a single graph that includes the data, the best fitting logistic growth curve, and the best fitting Gompertz growth curve.

c. In this part of the problem, we want to use the time series data given in the table, then show how our discrete dynamical models can reasonably simulate the data. The discrete logistic growth model is given by the equation:

where g(p_n) is the best quadratic function found in Part a. As an initial guess, start with your initial tumor weight as (p₀ = 324) starting at t = 0 and simulate the growth for 22 days. Use Excel's Solver to find the best possible p₀ value that minimizes the square error between the tumor weight in the data and the tumor weight given by the model. Give this best p₀ value and list the tumor weights at times t = 5, 10, 15, and 20. Give the percent error between the model and the data at these times. Also, include the value of the least sum of square errors between your simulation and the data. Plot both the data from your simulation and the data given in the table above, but again wait to place this graph below after you have included the simulation of Gompertz's model. (Note that this time you need to use only the time data and the population data in the table.)

d. The discrete Gompertz growth model is given by the equation:

where G(p_n) is the best Gompertz function found in Part b. As an initial guess, start with your initial tumor weight as (p₀ = 324) starting at t = 0 and simulate the growth for 22 days. Use Excel's Solver to find the best possible p₀ value that minimizes the square error between the tumor weight in the data and the tumor weight given by the model. Give this best p₀ value and list the tumor weights at times t = 5, 10, 15, and 20. Give the percent error between the model and the data at these times. Also, include the value of the least sum of square errors between your simulation and the data. Add this simulation to your plot for the logistic growth model and present it here.

e. Write a short discussion that compares and contrasts the two models. Which model do you believe is better and why?