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Conclusions

Currently the hardest challenges in modeling tumor growth and treatment are estimating parameters in models that are mathematically simple and are broadly applicable. Some examples of such models in current use were given here. Mathematical generality is always beckoning, but at present the payoff for generalizing is often small and there are too many plausible possibilities, too vaguely tied to observations, for any one generalization to have gained wide acceptance.

Acknowledgements. Research supported by NSF grant NSF #DMS 9532055 (RKS), NIH grant #1RO1CA78496-01 (PH) and a grant to LH from the Gerry and Nancy Pencer Brain Trust.








Figure 1. Anti-angiogenic tumor treatment. The figure shows tumor growth (solid line) and carrying capacity (dotted line) for a possible treatment regimen. Eq. 4 with K regarded as a dynamical variable and Eq. 7 are used. Cell number has been replaced by volume for the dependent variables; for example if the tumor were untreated $ V$ would eventually approach the maximum volume of tumor that the neovascularization, described by $ K(t)$, could support. In the figure, it is assumed that anti-angiogenic treatment is given daily on days 10-19. The concentration $ c$ in Eq. 7 is assumed to obey $ c = c_0 S(t-t_0) \exp [-r(t - t_0)]$ for a bolus administered at time $ t_0$, where S is the step function. Initially the tumor grows and stimulates a rapidly growing neovasculature. On treatment, the latter is decreased and eventually pulls the tumor down as well. After treatment stops growth resumes - slower at first for the tumor than in the comparable situation around day 8 because the carrying capacity is smaller than on day 8.

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Next: About this document ... Up: yakovlev Previous: Some Generalizations
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1999-10-20