Basic stochastic model for tumor virotherapy

A stochastic framework for evaluating CAR T cell therapy efficacy and variability

Based on a deterministic and stochastic process hybrid model, we use white noises to account for patient variabilities in treatment outcomes, use a hyperparameter to represent patient heterogeneity in a cohort, and construct a stochastic model in terms of Ito stochastic differential equations for testing the efficacy of three different treatment protocols in CAR T cell therapy. The stochastic model has three ergodic invariant measures which correspond to three unstable equilibrium solutions of the deterministic system, while the ergodic invariant measures are attractors under some conditions for tumor growth. As the stable dynamics of the stochastic system reflects long-term outcomes of the therapy, the transient dynamics provide chances of cure in short-term. Two stopping times, the time to cure and time to progress, allow us to conduct numerical simulations with three different protocols of CAR T cell treatment through the transient dynamics of the stochastic model. The probability distributions of the time to cure and time to progress present outcome details of different protocols, which are significant for current clinical study of CAR T cell therapy.

Stochastic dynamics of human papillomavirus delineates cervical cancer progression

Starting from a deterministic model, we propose and study a stochastic model for human papillomavirus infection and cervical cancer progression. Our analysis shows that the chronic infection state as random variables which have the ergodic invariant probability measure is necessary for progression from infected cell population to cervical cancer cells. It is shown that small progression rate from infected cells to precancerous cells and small microenvironmental noises associated with the progression rate and viral infection help to establish such chronic infection states. It implicates that large environmental noises associated with viral infection and the progression rate in vivo can reduce chronic infection. We further show that there will be a cervical cancer if the noise associated with precancerous cell growth is large enough. In addition, comparable numerical studies for the deterministic model and stochastic model, together with Hopf bifurcations in both deterministic and stochastic systems, highlight our analytical results.

Hopf bifurcation without parameters in deterministic and stochastic modeling of cancer virotherapy, part II

In part II, we analyze our stochastic model which incorporates microenvironmental noises and uncertainties related to immune responses. Outcomes of the therapy in our model are largely determined by the infectivity constant, the infection value, and stochastic relative immune clearance rates. The infection value is a universal critical value for immune-free ergodic invariant probability measures and persistence in all cases. Asymptotic behaviors of the stochastic model are similar to those of its deterministic counterpart. Our stochastic model displays an interesting dynamical behavior, stochastic Hopf bifurcation without parameters, which is a new phenomenon. We perform numerical study to demonstrate how stochastic Hopf bifurcation without parameters occurs. In addition, we give biological implications about our analytical results in stochastic setting versus deterministic setting.

Deterministic and stochastic modeling for PDGF-driven gliomas reveals a classification of gliomas

Motivated by our study of infiltrating dynamics of immune cells into tumors, we propose a stochastic model in terms of Ito stochastic differential equations to study how two parameters, the chemoattractant production rate and the chemotactic coefficient, influence immune cell migration and how these parameters distinguish two types of gliomas. We conduct a detailed analysis of the stochastic model and its deterministic counterpart. The deterministic model can differentiate two types of gliomas according to the range of the chemoattractant production rate as two equilibrium solutions, while the stochastic model also can differentiate two types of gliomas according to the ranges of the chemoattractant production rate and chemotactic coefficient with thresholds as one non-zero ergodic invariant measure and one weak persistent state when the noise intensities are small. When the noise intensities are large comparing with the chemotactic coefficient, there is only one type of glioma that corresponds to a non-zero ergodic invariant measure. Using our experimental data, numerical simulations are carried out to demonstrate properties of our models, and we give medical interpretations and implications for our analytical results and numerical simulations. This study also confirms some of our results about IDH gliomas.