Etiology and treatment of hematological neoplasms: stochastic mathematical models

Leukemias are driven by stemlike cancer cells (SLCC), whose initiation, growth, response to treatment, and posttreatment behavior are often "stochastic", i.e., differ substantially even among very similar patients for reasons not observable with present techniques. We review the probabilistic mathematical methods used to analyze stochastics and give two specific examples. The first example concerns a treatment protocol, e.g., for acute myeloid leukemia (AML), where intermittent cytotoxic drug dosing (e.g., once each weekday) is used with intent to cure. We argue mathematically that, if independent SLCC are growing stochastically during prolonged treatment, then, other things being equal, front-loading doses are more effective for tumor eradication than back loading. We also argue that the interacting SLCC dynamics during treatment is often best modeled by considering SLCC in microenvironmental niches, with SLCC-SLCC interactions occurring only among SLCC within the same niche, and we present a stochastic dynamics formalism, involving "Poissonization," applicable in such situations. Interactions at a distance due to partial control of total cell numbers are also considered. The second half of this chapter concerns chromosomal aberrations, lesions known to cause some leukemias. A specific example is the induction of a Philadelphia chromosome by ionizing radiation, subsequent development of chronic myeloid leukemia (CML), CML treatment, and treatment outcome. This time evolution involves a coordinated sequence of > 10 steps, each stochastic in its own way, at the subatomic, molecular, macromolecular, cellular, tissue, and population scales, with corresponding time scales ranging from picoseconds to decades. We discuss models of these steps and progress in integrating models across scales.