A Bayesian approach to estimate the diffusion coefficient of Rhodamine 6G in breast cancer spheroids

Multicellular spheroids have emerged as a robust platform to model tumor growth and are widely used for studying drug sensitivity. Diffusion is the main mechanism for transporting nutrients and chemotherapeutic drugs into spheroids, since they are typically avascular. In this study, the Bayesian inference was used to solve the inverse problem of determining the light attenuation coefficient and diffusion coefficient of Rhodamine 6G (R6G) in breast cancer spheroids, as a mock drug for the tyrosine kinase inhibitor, Neratinib. Four types of breast cancer spheroids were formed and the diffusion coefficient was estimated assuming a linear relationship between the intensity and concentration. The mathematical model used for prediction is the solution to the diffusion problem in spherical coordinates, accounting for the light attenuation. The Gaussian likelihood was used to account for the error between the measurements and model predictions. The Markov Chain Monte Carlo algorithm (MCMC) was used to sample from the posterior. The posterior predictions for the diffusion and light attenuation coefficients were provided. The results indicate that the diffusion coefficient values do not significantly vary across a HER2+ breast cancer cell line as a function of transglutaminase 2 levels, even in the presence of fibroblast cells. However, we demonstrate that different diffusion coefficient values can be ascertained from tumorigenic compared to nontumorigenic spheroids and from nonmetastatic compared to post-metastatic breast cancer cells using this approach. We also report agreement between spheroid radius, attenuation coefficient, and subsequent diffusion coefficient to give evidence of cell packing in self-assembled spheroids. The methodology presented here will allow researchers to determine diffusion in spheroids to decouple transport and drug penetration changes from biological resistivity.