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Copy of Calculus and Tumor Growth
Transcript of Copy of Calculus and Tumor Growth
Tumor Growth measure therapeutic doses
epidemiology in populations How is math related to cancer? Calculus? how fast a tumor is growing/shrinking
size when a tumor will stop growing
when treatment should be given
find volume equations derivaties! Models of Tumor Growth Exponential Growth Model Gompertzian Model Solving the Gompertz Equation Integration Growth Rate? Key Points Gompertz Over 200 types of cancer
no one model for all types
"Direct Curve Fitting" “different growth model equations are fitted to the volume of each individual tumor and the model with the best fitting [is] selected.” volume continuously increases exponentially over time volume at time, t volume when time is zero exponential growth constant time Rate of change = Derivative = change in VOLUME change in TIME Derive volume equation to get
rate of tumor growth Differentiation Constant Multiple Rule u d dt e u = e x u' u substitute! Final Answer: rate at which a tumor grows is directly proportional to volume larger tumor grows faster smaller tumor grows slower growth over a short period of time usually at beginning of formation implies a tumor would grow continuously doesn't take into account factors that could slow tumor growth growth rate declines over time prolonged cell cycle decreased oxygen/nutrient availability increased rate of cell death differential equation volume at a certain time N-number of cells
W-weight growth constant constant for growth retardation horizontal asymptote as time size is limited rate volume number of cells weight (derivative) INTEGRATE! using actual data from
a 1970 study on mouse mammary tumors Gompertz Equation Separate the variables Integrate using u-substitution derivative of ln(N) is 1 N multiply inside and divide outside by -0.4216 replace dN with du Simplify integral du u 1 as a power of e replace u Final Answer: assume Find C to get particular solution substitute into N(t) number of cells Other Calculus Applications equilibrium when growth rate is 0 how tumor will react to treatment most effective when growing fastest critical numbers second derivative
First Derivative Test to find max/min and Shannon Compton Can calculus cure cancer?