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MATHEMATICAL MODELING

MATHEMATICAL MODELING. Example 9: Gompertz Model. The ordinary differential equation for this model is, where,

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MATHEMATICAL MODELING

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  1. MATHEMATICAL MODELING

  2. Example 9: Gompertz Model The ordinary differential equation for this model is, where, 𝑉 is the tumor volume, 𝑡 is time, 𝑎 is the initial proliferation rate and 𝛽 is the rate exponential decay of this proliferation rate.

  3. Solution Separating the time and volume variables yields, Integrating from both sides,

  4. Taking exponential from both sides, Rearranging the equation to isolate 𝑉(𝑡) and finding C by putting t=0, yield to,

  5. Gompertz model shows exponential decay of the relative growth rate. • Asymptotically the volume of the tumor converges to the carrying capacity which is, • Defining the Gompertz function in MATLAB and using lsqcurvefit function from optimization toolbox the unknown parameters were found.

  6. Computational Model

  7. Example:

  8. The first part of the code shows the experimental data and this experimental data stored as vectors. • Data in vector format will be used later for fitting the models to the experimental data and plotting the optimized models.

  9. In the next part of the code, the model was defined and lsqcurvefit function was used to optimize the model and fit the model to experimental data. • For using lsqcurvefit function, an initial value should be given to the function for unknown parameters.

  10. where 𝑉(𝑡) is the tumor volume and 𝑉(0)=0.26 𝑚𝑚3.

  11. The unknown parameters in this model are 𝑎, and 𝛽. • In the Gompertz model 𝑎 which is the initial proliferation rate of the tumor cells, was found to be and to be 0.183 day-1 and the rate exponential decay of this proliferation rate 𝛽 was found to be 0.001 day-1.

  12. Results

  13. Example 10: Coulomb Friction Solve the following differential equation for

  14. Solution • To solve the above differential equation by MATLAB, reduce it into two first order differential equations. Let

  15. The above equation reduces to

  16. Computational Model • A M-file must be written to store the derivatives. In this case the file is saved as ‘coulomb.m’. function yp =coulomb(t,y) g=9.81; mu=0; l=0.25; k1=1+mu*sin(y(1));

  17. Computational Model k2=mu*cos(y(1))*(y(2)^2); k3=cos(y(1))*(1-mu^2); k4=mu*sin(y(1)); yp=[y(2);((g/l)*((k3-k4)/k1)-(k2/k1))] Type the given below code in a new M-file.

  18. Computational Model tspan=[0 2] y0=[0;0] [t,y]=ode45('coulomb',tspan,y0) plot(t,y(:,1)) grid xlabel(‘Time’) ylabel(‘Theta’) title(‘Theta Vs Time’) hold on;

  19. Computational Model • This main file calls the function ‘coulomb.m ’ to solve the differential equations. • To plot the results for different values of μ, change the value of μ in the function file, save it and then run the main code.

  20. Computational Model • To plot the angular velocity, change the variable in the plot command line in the main code from ‘y (:,1)’ to ‘y(:,2)’. Also plot the angular velocity with respect to theta.

  21. Plots

  22. Plots

  23. Conclusions • From the plots it can be seen that by increasing the value of the coulomb damping constant the system takes more time to reach θ = π/2 and the maximum value of the angular velocity also decreases.

  24. Thank You!

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