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The Gompertz growth model is commonly used to model tumor
Chapter 2, Problem 62(choose chapter or problem)
The Gompertz growth model is commonly used to model tumor cell growth. Let v(t) be the tumor's volume, then
\(\frac{d v(t)}{d t}=\lambda e^{-\alpha t} v(t)\)
where \(\lambda\) and \(\alpha\) are two appropriate constants (EdelsteinKeshet, 2005).
(a) Verify that the solution to this equation is given by \(v(t)=v_0 e^{\lambda / \alpha\left(1-e^{-\alpha t}\right)}\), where \(v_0\) is the initial tumor volume.
(b) This model takes into account the fact that when nutrients and oxygen are scarce at the tumor's core, its growth is impaired. Find the final predicted tumor volume (let \(t \rightarrow \infty\) ).
(c) For a specific mouse tumor, it was experimentally found that \(\lambda=2.5\) days, \(\alpha=0.1\) days with \(v_0= 50 \times 10^{-3} \mathrm{~mm}^3\) (Chignola, 2005). Use any method available to make a plot of \(v(t)\) vs. \(t\).
(d) Check the result obtained in Part b with the results from the graph found in Part c. A muscle hanging from a beam is shown in Figure P2.37(a) (Lessard, 2009). The \(\alpha\)-motor neuron can be used to electrically stimulate the muscle to contract and pull the mass, m, which under static conditions causes the muscle to stretch. An equivalent mechanical system to this setup is shown in Figure P2.37(b). The force \(F_{i s o}\) will be exerted when the muscle contracts. Find an expression for the displacement \(X_1(s)\) in terms of \(F_1(s)\) and \(F_{\text {iso }}(s)\).
Questions & Answers
QUESTION:
The Gompertz growth model is commonly used to model tumor cell growth. Let v(t) be the tumor's volume, then
\(\frac{d v(t)}{d t}=\lambda e^{-\alpha t} v(t)\)
where \(\lambda\) and \(\alpha\) are two appropriate constants (EdelsteinKeshet, 2005).
(a) Verify that the solution to this equation is given by \(v(t)=v_0 e^{\lambda / \alpha\left(1-e^{-\alpha t}\right)}\), where \(v_0\) is the initial tumor volume.
(b) This model takes into account the fact that when nutrients and oxygen are scarce at the tumor's core, its growth is impaired. Find the final predicted tumor volume (let \(t \rightarrow \infty\) ).
(c) For a specific mouse tumor, it was experimentally found that \(\lambda=2.5\) days, \(\alpha=0.1\) days with \(v_0= 50 \times 10^{-3} \mathrm{~mm}^3\) (Chignola, 2005). Use any method available to make a plot of \(v(t)\) vs. \(t\).
(d) Check the result obtained in Part b with the results from the graph found in Part c. A muscle hanging from a beam is shown in Figure P2.37(a) (Lessard, 2009). The \(\alpha\)-motor neuron can be used to electrically stimulate the muscle to contract and pull the mass, m, which under static conditions causes the muscle to stretch. An equivalent mechanical system to this setup is shown in Figure P2.37(b). The force \(F_{i s o}\) will be exerted when the muscle contracts. Find an expression for the displacement \(X_1(s)\) in terms of \(F_1(s)\) and \(F_{\text {iso }}(s)\).
ANSWER:Step 1 of 6
Consider the given equations:
\(v(t)=v_{0} e^{\frac{\lambda}{\alpha\left(1-e^{-\alpha t}\right)}}\) …… (1)
\(\frac{d v(t)}{d t}=v(t) \lambda e^{-\alpha t}\) …… (2)
\(\begin{array}{l} \alpha=0.1 \text { days } \\ \lambda=2.5 \text { days } \\ v_{0}=50 \times 10^{-3} \mathrm{~mm}^{3} \end{array}\)