Relative Quality of Tumor Growth Equations Describing Volume and Vascular Evolution Assessed with Ultrasound

Abstract

Three ordinary differential equation models were implemented in Matlab to describe tumor growth: the exponential-linear model (EL), the Gompertz model (G) and the dynamic carrying capacity model (DCC). For 11 control (C) and 10 antiangiogenic (AA)-treated mice with ectopic murine colorectal carcinoma (CT26), ultrasonic data were acquired on days 7, 9, 13, 15 and 21 after implantation (therapy started at day 7). The ellipsoidal volume V was estimated from B-mode measurements and used to estimate Relative Growth Rate. Dynamic Contrast-Enhanced Ultrasound data were acquired and regions with no contrast-enhancement were identified to estimate % perfused area (%P). Average contrast intensity vs. time curves in the P zone were fit to a lognormal model. Initial implementation of the models using a fixed (1 mm3) were compared to models adapted for a variable initial tumor size using the Akaike Information Criterion (AIC). The additional free parameter resulted in improved mean AIC (EL: 37 vs. 40; G: 37 vs. 41; DCC: 43 vs. 45-free and fixed initial volume, respectively). AIC was lowest on average for the G model in C mice. The Time To Peak (TTP), Wish In and Wash Out Rate (WIR, WOR) at $\mathbf{t}_{\mathbf{n}}$ were linked with future RGR between $\mathbf{t}_{\mathbf{n}}$ and $\mathbf{t}_{\mathbf{n}+1}$ (RS = −0.51, 0.44 and −0.46 respectively for C (TTP and WIR) or AA (WOR) group, p<0.01, non-parametric Spearman correlation coefficient). This implies that the microvascular function includes additional information related to tumor growth.