A mathematical model for photothermal therapy of spherical tumors

Photothermal ablation is a promising new technique for treatment of some cancers, where metal nanoparticles are introduced into the tumor and the system is locally heated with a laser to destroy the malignant cells. The aim is to have nanoparticles accumulate within the tumor and not in the surrounding healthy tissue, so that the heat source leads to a differential increase in temperature in the cancer and hence cell death. We begin by examining nanoparticle delivery from a single blood vessel into surrounding tissue using asymptotic analysis to construct the extravasation flux of nanoparticles into the tissue. We then incorporate this flux into a spatially one dimensional model which includes interstitial fluid flow to examine the distribution of nanoparticles across the tumor and the surrounding tissue. In this model we also consider nanoparticles conjugated with ligands which selectively bind to tumor cell surface receptors and eventually leads to nanoparticle internalization within the cell. We study how the mass of accumulated nanoparticles within the tumour (and the surrounding tissue) is influenced by the nanoparticle delivery time interval, ligand nanoparticle conjugation and tumor cell capacity for internalized nanoparticles. We then mimic laser irradiation of the tumor and the surrounding tissue (which comprise nanoparticles) estimating the temperature elevation and the induced tissue damage within the tissue. Finally, we validate our predictions against two different sets of experimental data and then discuss how to optimize the outputs of the photothermal therapy. We show that conjugating binding ligands to the nanoparticles can lead to a heterogeneous spatial distribution of nanoparticles and reduce tumor damage induced by laser irradiation. We conclude by discussing strategies to optimize the treatment protocol to induce maximal tumor damage with minimal side effects in the surrounding healthy tissue