Tumor progression and metastasis are critically dependent on the tumor stroma consisting of fibrous connective tissue, newly formed blood vessels, extracellular matrix, and recruited inflammatory cells. Tumor stromal fibroblasts (activated fibroblasts or myofibroblasts) acquire a specific phenotype that differentiates them from normal resting fibroblasts. They express smooth muscle actin (SMA), and isolated myofibroblasts have higher proliferation rates than their normal counterparts.1 The importance of stromal mesenchymal interactions with adjacent epithelial cells during development is well characterized. Proliferation of mesenchymal cells is tightly controlled by soluble paracrine factors. Conversely, differentiated fibroblasts do not proliferate and acquire a quiescent phenotype.2 In contrast, the complex paracrine regulatory mechanisms between the mesenchymal tumor stroma compartment and the tumor compartment are far from being understood. Coinjection of activated fibroblasts and tumor cells has been shown to enhance the invasiveness of colon tumors.3 Tumor-associated fibroblasts affect migration and invasion of tumor cells by secreting extracellular matrix proteins, matrix metalloproteinases, and growth factors.4,5 Paracrine-acting factors affecting tumor and stromal cell interactions include fibroblast growth factors, hepatocyte growth factor, and transforming growth factor-␤. Correspondingly, elevated plasma levels of transforming growth factor-␤ have been shown to predict early metastasis.

In this article the numerical approximation of solutions of Itô stochastic differential equations is considered, in particular for equations with a small parameter in the noise coefficient. We construct stochastic linear multi-step methods and develop the fundamental numerical analysis concerning their mean-square consistency, numerical stability in the mean-square sense and mean-square convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency. Further, for the small noise case we obtain expansions of the local error in terms of the step-size and the small parameter. Simulation results using several explicit and implicit stochastic linear k-step schemes, k = 1, 2, illustrate the theoretical findings.

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