One of the most challenging tasks in constructing a mathematical model of cancer treatment is the calculation of biological parameters from empirical data. This task becomes increasingly difficult if a model involves several cell populations and treatment modalities. A sophisticated model constructed by de Pillis et al., Mixed immunotherapy and chemotherapy of tumours: Modelling, applications and biological interpretations, J. Theor. Biol. 238 (2006), pp. 841-862; involves tumour cells, specific and non-specific immune cells (natural killer (NK) cells, CD8 þ T cells and other lymphocytes) and employs chemotherapy and two types of immunotherapy (IL-2 supplementation and CD8 þ T-cell infusion) as treatment modalities. Despite the overall success of the aforementioned model, the problem of illustrating the effects of IL-2 on a growing tumour remains open. In this paper, we update the model of de Pillis et al. and then carefully identify appropriate values for the parameters of the new model according to recent empirical data. We determine new NK and tumour antigen-activated CD8 þ T-cell count equilibrium values; we complete IL-2 dynamics; and we modify the model in de Pillis et al. to allow for endogenous IL-2 production, IL-2-stimulated NK cell proliferation and IL-2-dependent CD8 þ T-cell self-regulations. Finally, we show that the potential patient-specific efficacy of immunotherapy may be dependent on experimentally determinable parameters.

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Variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value. Rather, the gradients of minimizing sequences do not converge pointwise and can have oscillations. However, the gradients do converge in the weak topology, i.e., their local spatial averages converge.Such functionals arise in the description of equilibria of crystals or other ordered materials.Stable configurations for solid crystals which have symmetry-related (martensitic) energy wells have a fine-scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit. An analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals.