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.
Keywords: Mathematical oncology, Tumor growth, Parameters estimation
Software
Procedure
This protocol describes how to use a dedicated mathematical model for analysis of tumor growth kinetics. Specifically, it deals with a method for determination of the coefficients of the model from longitudinal measurements of tumor growth. In the data example provided, the measures were obtained by using calipers to determine, every other day, largest (L) and smallest (w) diameters of subcutaneously implanted tumors (Benzekry et al., 2014) for additional details. The formula V = w^{2}Lπ/6 was then used to compute the tumor volume. Researchers can use this protocol in order to compare two groups (or more) of tumor kinetics data and identify which of the coefficient(s) of the mathematical model significantly differ(s) (or not) among the groups. Longitudinal measurements of growth data are required. A two-dimensional ordinary differential equation model for tumor growth as a function of the tumor volume, V, and its carrying capacity, K (formally, the maximum tumor volume that level of host environmental support could sustain), was used (t denotes time): with V_{0} = the volume of the tumor at t = t_{0}, K_{0}= the carrying capacity at t = t_{0}, a = proliferation of the tumor cells, b = carrying capacity stimulation, and d = carrying capacity inhibition. Numerical solutions of the model are fitted to experimental measurements of tumor volume at several time points, and the set of parameters (a, b, d) generating the best fit of the model are thereby determined. In other words, the values of (a, b, d) are determined as the ones minimizing the distance between the model simulation and the data (thus maximizing the likelihood of the data under the hypothesis that it has been generated by the model). The quantity V_{0} is fixed to the first observed data point and K_{0} is fixed to 2V_{0}. For instance, for the first animal of group 1 of the data example, the resulting values of the parameters were: V_{0} = 836 mm^{3}, K_{0} = 1672 mm^{3}, a = 0.224 day^{-1}, b = 0.710 day^{-1} and d = 0.0018 mm^{-2} day^{-1}. This protocol is composed of several Matlab scripts and functions that are freely downloadable at the following address: https://github.com/benzekry/fit_tumor_growth. In the following, we detail each step that should be sequentially launched (by typing the script name in a Matlab command window) in order to perform the full task. The associated script is indicated in bold font. We perform these on an illustrative data set example composed of two tumor growth curve groups, each n = 20 mice.
Acknowledgments
This project was supported, in part, by the National Aeronautics and Space Administration under NSCOR grants NNJ06HA28G and NNX11AK26G and by Award Number U54CA149233 from the National Cancer Institute, both to L. Hlatky. This study also received support within the frame of the LABEX TRAIL, ANR-10-LABX-0057 with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the “Investments for the future” Programme IdEx (ANR 10-IDEX-03-02).
References
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