Computational methods for birth-death processes

Impact of Resistance on Therapeutic Design: A Moran Model of Cancer Growth

Resistance of cancers to treatments, such as chemotherapy, largely arise due to cell mutations. These mutations allow cells to resist apoptosis and inevitably lead to recurrence and often progression to more aggressive cancer forms. Sustained-low dose therapies are being considered as an alternative over maximum tolerated dose treatments, whereby a smaller drug dosage is given over a longer period of time. However, understanding the impact that the presence of treatment-resistant clones may have on these new treatment modalities is crucial to validating them as a therapeutic avenue. In this study, a Moran process is used to capture stochastic mutations arising in cancer cells, inferring treatment resistance. The model is used to predict the probability of cancer recurrence given varying treatment modalities. The simulations predict that sustained-low dose therapies would be virtually ineffective for a cancer with a non-negligible probability of developing a sub-clone with resistance tendencies. Furthermore, calibrating the model to in vivo measurements for breast cancer treatment with Herceptin, the model suggests that standard treatment regimens are ineffective in this mouse model. Using a simple Moran model, it is possible to explore the likelihood of treatment success given a non-negligible probability of treatment resistant mutations and suggest more robust therapeutic schedules.

Birth-death-suppression Markov process and wildfires

Birth and death Markov processes can model stochastic physical systems from percolation to disease spread and, in particular, wildfires. We introduce and analyze a birth-death-suppression Markov process as a model of controlled culling of an abstract, dynamic population. Using analytic techniques, we characterize the probabilities and timescales of outcomes like absorption at zero (extinguishment) and the probability of the cumulative population (burned area) reaching a given size. The latter requires control over the embedded Markov chain: this discrete process is solved using the Pollazcek orthogonal polynomials, a deformation of the Gegenbauer/ultraspherical polynomials. This allows analysis of processes with bounded cumulative population, corresponding to finite burnable substrate in the wildfire interpretation, with probabilities represented as spectral integrals. This technology is developed to lay the foundations for a dynamic decision support framework. We devise real-time risk metrics and suggest future directions for determining optimal suppression strategies, including multievent resource allocation problems and potential applications for reinforcement learning.

Revisiting recurrence criteria of birth and death processes. Short proofs

The paper contains several new transparent proofs of criteria appearing in classification of birth and death processes (BDPs). They are almost purely probabilistic and differ from the classical techniques of three-term recurrence relations, continued fractions and orthogonal polynomials. Let T ∞ {T^\infty } be the passage time from zero to ∞ \infty . The regularity criterion says that T ∞ > ∞ {T^\infty } > \infty if and only if E T ∞ > ∞ \mathbb {E}{T^\infty } > \infty . It is heavily based on a result of Gong, Y., Mao, Y.-H. and Zhang, C. [J. Theoret. Probab. 25 (2012), no. 4, 950–980]. We obtain the latter expectation by using a two-term recurrence relation. We observe that the recurrence criterion is an immediate consequence of the well-known recurrence criterion for discrete-time BDPs and a result of Chung K. L. [Markov Chains with Stationary Transition Probabilities, Springer-Verlag, New York (1967)]. We obtain the classical criterion of positive recurrence using technique of the common probability space. While doing so, we construct a monotone sequence of BDPs with finite state spaces converging to BDPs with an infinite state space.

Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

We considered the time-inhomogeneous linear birth–death processes with immigration. For these processes closed form expressions for the transition probabilities were obtained in terms of the complete Bell polynomials. The conditional mean and the conditional variance were explicitly evaluated. Several time-inhomogeneous processes were studied in detail in view of their potential applications in population growth models and in queuing systems. A time-inhomogeneous linear birth–death processes with finite state-space was also taken into account. Special attention was devoted to the cases of periodic immigration intensity functions that play an important role in the description of the evolution of dynamic systems influenced by seasonal immigration or other regular environmental cycles. Various numerical computations were performed for periodic immigration intensity functions.

Selection or drift: The population biology underlying transposon insertion sequencing experiments

Transposon insertion sequencing methods such as Tn-seq revolutionized microbiology by allowing the identification of genomic loci that are critical for viability in a specific environment on a genome-wide scale. While powerful, transposon insertion sequencing suffers from limited reproducibility when different analysis methods are compared. From the perspective of population biology, this may be explained by changes in mutant frequency due to chance (drift) rather than differential fitness (selection). Here, we develop a mathematical model of the population biology of transposon insertion sequencing experiments, i.e. the changes in size and composition of the transposon-mutagenized population during the experiment. We use this model to investigate mutagenesis, the growth of the mutant library, and its passage through bottlenecks. Specifically, we study how these processes can lead to extinction of individual mutants depending on their fitness and the distribution of fitness effects (DFE) of the entire mutant population. We find that in typical in vitro experiments few mutants with high fitness go extinct. However, bottlenecks of a size that is common in animal infection models lead to so much random extinction that a large number of viable mutants would be misclassified. While mutants with low fitness are more likely to be lost during the experiment, mutants with intermediate fitness are expected to be much more abundant and can constitute a large proportion of detected hits, i.e. false positives. Thus, incorporating the DFEs of randomly generated mutations in the analysis may improve the reproducibility of transposon insertion experiments, especially when strong bottlenecks are encountered.

A Quasi Birth-and-Death model for tumor recurrence

A major cause of chemoresistance and recurrence in tumors is the presence of dormant tumor foci that survive chemotherapy and can eventually transition to active growth to regenerate the cancer. In this paper, we propose a Quasi Birth-and-Death (QBD) model for the dynamics of tumor growth and recurrence/remission of the cancer. Starting from a discrete-state master equation that describes the time-dependent transition probabilities between states with different numbers of dormant and active tumor foci, we develop a framework based on a continuum-limit approach to determine the time-dependent probability that an undetectable residual tumor will become large enough to be detectable. We derive an exact formula for the probability of recurrence at large times and show that it displays a phase transition as a function of the ratio of the death rate μ_A of an active tumor focus to its doubling rate λ. We also derive forward and backward Kolmogorov equations for the transition probability density in the continuum limit and, using a first-passage time formalism, we obtain a drift-diffusion equation for the mean recurrence time and solve it analytically to leading order for a large detectable tumor size N. We show that simulations of the discrete-state model agree with the analytical results, except for O(1/N) corrections. As an example of the use of our model in a clinical setting, we show that a range of model parameters can fit Kaplan-Meier recurrence-free survival data for ovarian cancer. Finally, we show in simulations that extending the duration of chemotherapy increases both the mean recurrence time and the asymptotic (large-time) probability of no recurrence. Our results should be useful in planning optimized chemotherapy dosing and duration for cancer treatment, especially in cancer types for which no targeted therapy is available.