The work of this thesis can be divided into two parts. In the first part, a numerical method is presented capable of solving nonlinear elliptic partial differential equations(PDEs) where the diffusion coefficient, the source term, the solution and its flux are discontinuous across an irregular interface. The interface is represented by the zerolevel set of a signed distance function, empowering a natural and systematic approach to generate adaptive cartesian grids which drastically reduces computational cost by focusing on regions near the interface. The methodology uses the p4est library of Burstedde et al., SIAM J. Sci. Comput., 33(3) (2011) and is an extension of the work of Bochkov et al., JCP 407(2020). An application of this solver is illustrated by solving the nonlinear poisson Boltzmann equation to calculate the electric field around large biomolecules. Another computational method is presented for the sharp interface simulation of multicomponent solidification coupled with incompressible fluid flow. A Newton-type approach to solve the nonlinear system of coupled PDEs arising from the time discretization of the governing equations of diffusion-driven multi-component solidification was introduced by Bochkov et al. arXiv:2112.08650(2021). Experiments have shown complex natural convection phenomena in solidifying ternary systems due to the presence of two porous zones (cotectic and primary mush) and the rejection of two differently dense solutes, which thus far has not been captured in any simulation framework. To study this rich physics, the existing method is coupled with a pressure-free projection method to solve the incompressible Navier-Stokes equation while taking special care to enforce the sharp interfacial conditions on temperature, multiple scalar concentrations composing the alloy, flow velocity, interface velocity and pressure. In the second part, we address the long standing challenge in cancer biology for tracking engineered cells in deep tissues non-invasively by integrating monte carlo diffusion simulations(MCDS) with genetic reporters for diffusion weighted magnetic resonance imaging(DW-MRI). The genetic reporter used in this approach is a membrane protein called aquaporin-1(Aqp1) which upon expression in cells increases water transport across cell membranes and produces a negative contrast in DW-MRI. To use this reporter for cell tracking in tissues, we first try to understand the relationship between diffusion changes in mixed cell populations of Aqp1-expressing cells and wild type cells with the volume of engineered cells present using DW-MRI and MCDS.