. 2015 Nov 5;5(21):e1644.
doi: 10.21769/bioprotoc.1644.

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# Capturing the Driving Role of Tumor-Host Crosstalk in a Dynamical Model of Tumor Growth

Sebastien Benzekry et al. Bio Protoc. .
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## Abstract

In 1999, Hahnfeldt et al. proposed a mathematical model for tumor growth as dictated by reciprocal communications between tumor and its associated vasculature, introducing the idea that a tumor is supported by a dynamic, rather than a static, carrying capacity. In this original paper, the carrying capacity was equated with the variable tumor vascular support resulting from the net effect of tumor-derived angiogenesis stimulators and inhibitors. This dynamic carrying capacity model was further abstracted and developed in our recent publication to depict the more general situation where there is an interaction between the tumor and its supportive host tissue; in that case, as a function of host aging (Benzekry et al., 2014). This allowed us to predict a range of host changes that may be occurring with age that impact tumor dynamics. More generally, the basic formalism described here can be (and has been), extended to the therapeutic context using additional optimization criteria (Hahnfeldt et al., 1999). The model depends on three parameters: One for the tumor cell proliferation kinetics, one for the stimulation of the stromal support, and one for its inhibition, as well as two initial conditions. We describe here the numerical method to estimate these parameters from longitudinal tumor volume measurements.

### Figures Figure 1. Individual fits of the tumor growth data by the model for each mouse from one of the two groups The blue curve is the volume (V) curve while the red curve depicts the dynamics of the carrying capacity (K). Fits were performed by optimization of the coefficients a, b and d. Initial volume V0 was set to the first volume measured. Initial carrying capacity was set to the double of this quantity. x-axis = time in days, y-axis = tumor size (in mm3). Figure 2. Boxplots of the distribution of three parameters of the model (a, b and d) for the two groups Initial condition V0 and carrying capacity K0 were fixed as in Figure 1. x-axis = Group 1 and Group 2. y-axis = parameter values for a, b and d. * = α < 0.05 (Student’s t-test). Figure 3. Simulated growth curves Left panel: All individual simulations corresponding to each animal-specific fitted parameter set. Right panel: Average growth curves (mean ± standard error). x-axis = time in days, y-axis = tumor volume in mm3. Blue curves = Group 1. Red curves = Group 2.