3430
Ind. Eng. Chem. Res. 2008, 47, 3430-3437
Stochastic Analysis of Multistate Systems Eric Sherer and Doraiswami Ramkrishna* School of Chemical Engineering Purdue UniVersity West Lafayette, Indiana 47907
The treatment necessary to remove all cells for a periodic cell cycle specific regimen is analyzed using stochastic equations of a multistate model with state transition governed by residence time distributions. While the objective is to remove all cells, exactly when this occurs is uncertain due to the random timing of state transitions. Previous work has shown that removal of all cells requires great certainty that the number of cells is small; an expected population that is six standard deviations less than one cell is an indicator when cure is nearly likely. In developing the necessary stochastic framework, the expression for the standard deviation for multistate processes is derived and a simple two-state example is presented that illustrates the potential coupling between subpopulations. Finally, the methodology is applied to cell cycle specific chemotherapy and the number of cycles required to nearly be assured that zero cells remain is calculated. This method has computational advantages over Monte Carlo simulations when the initial number of cells is not small and only the variation in the total number of cells is of interest. 1. Introduction A festschrift for Bruce Nauman must feature at least one article on the topic of his early interest: that is, of residence time distributions in continuous flow systems. While Nauman’s concern was solely with chemical reactors on which considerable work has been wrought, the treatment of the topic of residence times must seek a different setting with some twists that are both intellectual and significant for application. This article will then attempt to fulfill this stipulation with the hope that it will suitably adorn a collection meant to felicitate a chemical engineer with outstanding credentials from industry as well as academia. The problem of interest is to model the fate of entities that circulate sequentially through a closed loop of a finite set of environments in each of which the entities spend random amounts of time. A tutorial model will introduce the foundations (see Figure 1) before extending the concepts to a model where one specific environment is considered to randomly eliminate the entity from circulation (see Figure 2). This second scenario includes the facility for the entity to replicate itself in one of the environments and the probability that the entire loop is free of all entities is sought. This abstract scenario is virtually identical to the situation of cancer chemotherapy in which the therapy targets molecules that are present only during a specific phase of the cells life cycle. These cell cycle phases can be distinguished based on a cell’s DNA content. After a cell divides, the resulting two cells each begin in the G0/G1 phase with a single set of DNA. These cells enter the S phase upon initiation of DNA synthesis and have between >1 but
