The Gompertz differential equation v = a(In b - In V)Vt is not the only possibility for modeling tumor growth. Suppose a tumor can be modeled as a spherical collection of cells dt and that it acquires resources for growth only through its surface area. All cells in a tumor are also subject to a constant per capita death rate. The dynamics of tumor mass (in grams) might therefore be modeled as dM dt = KM²/3 - μM where is a positive constant. The first term represents tumor growth via nutrients entering through the surface. The second term represents a constant per capita death rate. (a) Assuming that k = 1, find M as a function of t. (Let M(0) = Mo.) M(t) = (b) What happens to the tumor mass as t→∞o? The tumor mass approaches (c) Assuming tumor mass is proportional to its volume, the diameter of the tumor is related to its mass as D = aM¹/3 where a > 0. Derive a differential equation for D and show that it can be written as the von Bertalanffy equation. (The von Bertalanffy equation is d=k(LL), where L(a) is length (in cm) at age a (in years), L. is the asymptotic length, and k is a positive constant whose units are 1/year.) We know that D = aM¹/3 → dD dt grams as t→∞o. = da dD The von Bertalanffy equation can be written as = A(BD), then A= dt and B =
The Gompertz differential equation v = a(In b - In V)Vt is not the only possibility for modeling tumor growth. Suppose a tumor can be modeled as a spherical collection of cells dt and that it acquires resources for growth only through its surface area. All cells in a tumor are also subject to a constant per capita death rate. The dynamics of tumor mass (in grams) might therefore be modeled as dM dt = KM²/3 - μM where is a positive constant. The first term represents tumor growth via nutrients entering through the surface. The second term represents a constant per capita death rate. (a) Assuming that k = 1, find M as a function of t. (Let M(0) = Mo.) M(t) = (b) What happens to the tumor mass as t→∞o? The tumor mass approaches (c) Assuming tumor mass is proportional to its volume, the diameter of the tumor is related to its mass as D = aM¹/3 where a > 0. Derive a differential equation for D and show that it can be written as the von Bertalanffy equation. (The von Bertalanffy equation is d=k(LL), where L(a) is length (in cm) at age a (in years), L. is the asymptotic length, and k is a positive constant whose units are 1/year.) We know that D = aM¹/3 → dD dt grams as t→∞o. = da dD The von Bertalanffy equation can be written as = A(BD), then A= dt and B =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The Gompertz differential equation
a(In b - In V)Vt is not the only possibility for modeling tumor growth. Suppose a tumor can be modeled as a spherical collection of cells
dt
and that it acquires resources for growth only through its surface area. All cells in a tumor are also subject to a constant per capita death rate. The dynamics of tumor mass (in
grams) might therefore be modeled as
dM KM²/3 - μM
dt
=
M(t)
where is a positive constant. The first term represents tumor growth via nutrients entering through the surface. The second term represents a constant per capita death rate.
(a) Assuming that k = 1, find M as a function of t. (Let M(O) = Mo.)
=
=
(b) What happens to the tumor mass as t → ∞o?
The tumor mass approaches
grams as t→∞.
(c) Assuming tumor mass is proportional to its volume, the diameter of the tumor is related to its mass as D = aM¹/3 where a > 0. Derive a differential equation for D
dL
and show that it can be written as the von Bertalanffy equation. (The von Bertalanffy equation is k(LoL), where L(a) is length (in cm) at age a (in years), L is
the asymptotic length, and k is a positive constant whose units are 1/year.)
da
We know that D = aM¹/3
dD
dt
=
dD
The von Bertalanffy equation can be written as
dt
=
A(BD), then A
=
and B =
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