A mathematical model of prostate tumor growth and androgen-independent relapse
T.L. Jackson
Discrete & Continuous Dynamical Systems - B 2004, 4(1): 187-201 doi: 10.3934/dcdsb.2004.4.187
A mathematical model is developed that investigates polyclonality and decreased apoptosis as mechanisms for the androgen-independent relapse of human prostate cancer. The tumor is treated as a continuum of two types of cells (androgen-dependent and androgen- independent) whose proliferation and apoptotic death rates differ in response to androgen rich and androgen poor conditions. Insight into the tumor's response to therapies which both partially and completely block androgen production is gained by applying a combination of analytical and numerical techniques to the model equations. The analysis predicts that androgen deprivation therapy can only be successful for a small range of the biological parameters no matter how completely androgen production is blocked. Numerical simulations show that the model captures all three experimentally observed phases of human prostate cancer progression including exponential growth prior to treatment, androgen sensitivity immediately following therapy, and the eventual androgen-independent relapse of the tumor. Simulations also agree with experimental evidence that androgen-independent relapse is associated with a decrease in apoptosis without an increase in proliferation.
keywords: mathematical model. Prostate cancer partial-differential equations
A mathematical model of tumor-immune evasion and siRNA treatment
J.C. Arciero T.L. Jackson D.E. Kirschner
Discrete & Continuous Dynamical Systems - B 2004, 4(1): 39-58 doi: 10.3934/dcdsb.2004.4.39
In this paper a mathematical model is presented that describes growth, immune escape, and siRNA treatment of tumors. The model consists of a system of nonlinear, ordinary differential equations describing tumor cells and immune effectors, as well as the immuno-stimulatory and suppressive cytokines IL-2 and TGF-$\beta$. TGF-$\beta$ suppresses the immune system by inhibiting the activation of effector cells and reducing tumor antigen expression. It also stimulates tumor growth by promoting angiogenesis, explaining the inclusion of an angiogenic switch mechanism for TGF-$\beta$ activity. The model predicts that increasing the rate of TGF-$\beta$ production for reasonable values of tumor antigenicity enhances tumor growth and its ability to escape host detection. The model is then extended to include siRNA treatment which suppresses TGF-$\beta$ production by targeting the mRNA that codes for TGF-$\beta$, thereby reducing the presence and effect of TGF-$\beta$ in tumor cells. Comparison of tumor response to multiple injections of siRNA with behavior of untreated tumors demonstrates the effectiveness of this proposed treatment strategy. A second administration method, continuous infusion, is included to contrast the ideal outcome of siRNA treatment. The model's results predict conditions under which siRNA treatment can be successful in returning an aggressive, TGF-$\beta$ producing tumor to its passive, non-immune evading state.
keywords: siRNA. tumor Mathematical model immune response

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