ExercisesSolutions

Solution a within the prescribed constraints of the

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Solution: (a) Within the prescribed constraints of the growth function F ( T ) = dT dt , three different graphs ( F 1 ( T ), F 2 ( T ), F 3 ( T )) are shown, each increasing in complexity. Any one of these graphs (and others not shown here) would be a sufficient answer to this question. These solutions will follow through and individually examine these graphs parts (b) and (c) of the question. 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 T Figure 2: F 1 ( T ) - The simplest graph possible given the prescribed constraints. 3
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0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 T Figure 3: F 2 ( T ) - A slightly more complicated graphical representation for Exercise 2a. 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 T Figure 4: F 3 ( T ) - An even more complex possible graphical representation for Exercise 2a. (b) The equations for ( F 1 ( T ), F 2 ( T ), F 3 ( T )) respectively are: F 1 ( T ) = braceleftBigg k 0 T < T a ǫk T a T T MAX F 2 ( T ) = braceleftBigg k 0 T < T a k ( T MAX T ) T MAX T a T a T T MAX F 3 ( T ) = k 2 (1 - tanh(10( T - T a ))) Finding an explicit solution, T ( t ), for each of F 1 ( T ) , F 2 ( T ) and F 3 ( T ) is difficult, so a numerical solver is used. The solution found was used to plot the comparative graphs in part (c). (c) The solution curves of F 1 ( T ) , F 2 ( T ) and F 3 ( T ) (thin line) compared to that of logistic tumor growth F G ( T ) = dT dt = aT ( b - T ) (thick line) can be seen in Figure 5, 6 and 7 respectively. In these plots, T a = 0 . 8, T MAX = 1 and t = 0 .. 70. Varying the tumor size during vascularisation only seems to change the length of time taken to grow to T MAX . No other changes could be detected. 4
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0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 t Figure 5: A comparison between F 1 ( T ) and the logistic equation, F G ( T ). Upon inspection, it can be seen that F 1 ( T ) closely follows the behavior of the logistic equation everywhere except for the region from approximately t = 1 to t = 3. 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 t Figure 6: A comparative plot between F 2 ( T ) and the logistic equation. The behaviour of F 2 ( T ) is quite similar to that of the logistic equation, however there is a slightly more noticeable difference in behavior for the region t = 1 to t = 2. 5
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0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 t Figure 7: Plotting F 3 ( T ) with the logistic equation finds that its behavior varies greatly with that of the logistic equation. 6
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3. Purpose: To come up with a model equation using a different tumor growth function. The solutions are compared with those derived from the logistic growth law. The comparison of tumor growth curves is, continued in the Projects, using published data. See also Exercise 2 Exercise: It has been observed that in certain tumors grown in vitro only a thin layer of cells on the tumor’s surface are actually proliferating. Consider a perfectly spherical tumor, and let T ( t ) denote the tumor population at time t . (a) Write a differential equation for T ( t ), assuming that only the cells on the surface of the sphere proliferate. (Assume that the number of cells, T , is proportional to volume, but that the number of proliferating cells is proportional to the surface area of the sphere. You’ll need to express the number of proliferating cells as a function of T .) This is known as “Von Bertalanffy” growth.
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