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The Gompertz differential equation αν a(in b - In VVt is not the only possibility for modeling tumor growth. Suppose a tumor can be modeled as a spherical collection of cells and that it acquires dt 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 M = kм2/3 - μм dt where μ 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→ ∞? 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 = aм1/3 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 dL = 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 = ам1/3 dD =>>> dt dD The von Bertalanffy equation can be written as = A(B-D), then A = and B = dt

The Gompertz differential equation αν a(in b - In VVt is not the only possibility for modeling tumor growth. Suppose a tumor can be modeled as a spherical collection of cells and that it acquires dt 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 M = kм2/3 - μм dt where μ 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→ ∞? 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 = aм1/3 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 dL = 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 = ам1/3 dD =>>> dt dD The von Bertalanffy equation can be written as = A(B-D), then A = and B = dt

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 9T
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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

dL
da
= k(L∞ − L),

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 = aM^1/3 --> dD/dt = ?
 The von Bertalanffy equation can be written as dD/dt = A(B-D), then A = ? and B = ?

The Gompertz differential equation αν a(in b - In VVt is not the only possibility for modeling tumor growth. Suppose a tumor can be modeled as a spherical collection of cells and that it acquires dt 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 M = kм2/3 - μм dt where μ 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→ ∞? 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 = aм1/3 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 dL = 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 = ам1/3 dD =>>> dt dD The von Bertalanffy equation can be written as = A(B-D), then A = and B = dt
Transcribed Image Text:The Gompertz differential equation αν a(in b - In VVt is not the only possibility for modeling tumor growth. Suppose a tumor can be modeled as a spherical collection of cells and that it acquires dt 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 M = kм2/3 - μм dt where μ 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→ ∞? 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 = aм1/3 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 dL = 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 = ам1/3 dD =>>> dt dD The von Bertalanffy equation can be written as = A(B-D), then A = and B = dt
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