ANALYZING TUMOR GROWTH The Gompertz Model

The Gompertz Function Special case of the generalized logistic function Asymmetrical: right-hand asymptote is approached more slowly than left Originally created to describe human mortality Common model to predict growth of cancerous tumors

F(t)=ae-be The General Equation -ct b, c are positive numbers a: asymptote b: displacement along the x- axis (translates the graph to the left or right) c: growth rate

The Recursive Formula In S(t + r)= a+b*InS(t) Recursive formula + linear regression analysis gets the most accurate fit for breast, lung cancer S(t+r): remaining population r : the constant age increment between two consecutive measurements b: initial growth rate, 0<b<1 a: survival fraction after one iteration

Standard Tumor Growth

Significance Able to find carrying capacity and key turning points Optimize treatment Varying constants allows model to work across many types of tumors Cyan: uninhibited growth Magenta: early stage treatment Black: medium stage treatment Yellow: late stage treatment

References Bassukas, Ioannis D. “Use of the Recursion Formula of the Gompertz Survival Function to Evaluate Life-Table Data.” Mechanisms of Ageing and Development, vol. 89, no. 3, 24 July 1996, pp. 155–163., doi:10.1016/0047-6374(96)01747-2. Benzekry, Sébastien, et al. “Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth.” PLoS Computational Biology, vol. 10, no. 8, 2014, doi:10.1371/journal.pcbi.1003800. Tjørve, Kathleen M. C., and Even Tjørve. “The Use of Gompertz Models in Growth Analyses, and New Gompertz-Model Approach: An Addition to the Unified-Richards Family.” Plos One, vol. 12, no. 6, 5 June 2017, doi:10.1371/journal.pone.0178691.