A finite element formulation to approximate the behavior of lipid membranes is proposed. The mathematical model incorporates tangential viscous stresses and bending elastic forces, together with the inextensibility constraint and the enclosed volume constraint. The membrane is discretized by a surface mesh made up of planar triangles, over which a mixed formulation (velocity-curvature) is built based on the viscous bilinear form (Boussinesq-Scriven operator) and the Laplace-Beltrami identity relating position and curvature. A semi-implicit approach is then used to discretize in time, with piecewise linear interpolants for all variables. Two stabilization terms are needed: The first one stabilizes the inextensibility constraint by a pressure-gradient-projection scheme (R. Codina and J. Blasco, Computer Methods in Applied Mechanics and Engineering 143:373-391, 1997), the second couples curvature and velocity to improve temporal stability, as proposed by Bänsch (Numerische Mathematik 88:203-235, 2001). The volume constraint is handled by a Lagrange multiplier (which turns out to be the internal pressure), and an analogous strategy is used to filter out rigid-body motions. The nodal positions are updated in a Lagrangian manner according to the velocity solution at each time step. An automatic remeshing strategy maintains suitable refinement and mesh quality throughout the simulation.Numerical experiments show the convergent and robust behavior of the proposed method. Stability limits are obtained from numerous relaxation tests, and convergence with mesh refinement is confirmed both in the relaxation transient and in the final equilibrium shape. Virtual tweezing experiments are also reported, computing the dependence of the deformed membrane shape with the tweezing velocity (a purely dynamical effect). For sufficiently high velocities, a tether develops which shows good agreement, both in its final radius and in its transient behavior, with available analytical solutions. Finally, simulation results of a membrane subject to the simultaneous action of six tweezers illustrate the robustness of the method.

Immunotherapy is currently regarded as the most promising treatment to fight against cancer. This is particularly true in the treatment of chronic lymphocytic leukemia, an indolent neoplastic disease of B-lymphocytes which eventually causes the immune system's failure. In this and other areas of cancer research, mathematical modeling is pointed out as a prominent tool to analyze theoretical and practical issues. Its lack in studies of chemoimmunotherapy of chronic lymphocytic leukemia is what motivates us to come up with a simple ordinary differential equation model. It is based on ideas of de Pillis & Radunskaya and on standard pharmacokinetics-pharmacodynamics assumptions. In order to check the positivity of the state variables, we first establish an invariant region where these time-dependent variables remain positive. Afterwards, the action of the immune system, as well as the chemoimmunotherapeutic role in promoting cancer cure are investigated by means of numerical simulations and the classical linear stability analysis. The role of adoptive cellular immunotherapy is also addressed. Our overall conclusion is that chemoimmunotherapeutic protocols can be effective in treating chronic lymphocytic leukemia provided that chemotherapy is not a limiting factor to the immunotherapy efficacy.

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h i g h l i g h t s • A mean field model for antiangiogenic chemotherapy is proposed. • Metronomic and maximum-tolerated schedules are compared by numerical simulations. • Metronomic schedules are found to be more effective in eliminating tumour cells.

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