126 research outputs found
Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks
We consider a standard splitting algorithm for the rare-event simulation of
overflow probabilities in any subset of stations in a Jackson network at level
n, starting at a fixed initial position. It was shown in DeanDup09 that a
subsolution to the Isaacs equation guarantees that a subexponential number of
function evaluations (in n) suffice to estimate such overflow probabilities
within a given relative accuracy. Our analysis here shows that in fact
O(n^{2{\beta}+1}) function evaluations suffice to achieve a given relative
precision, where {\beta} is the number of bottleneck stations in the network.
This is the first rigorous analysis that allows to favorably compare splitting
against directly computing the overflow probability of interest, which can be
evaluated by solving a linear system of equations with O(n^{d}) variables.Comment: 23 page
Minimizing Metastatic Risk in Radiotherapy Fractionation Schedules
Metastasis is the process by which cells from a primary tumor disperse and
form new tumors at distant anatomical locations. The treatment and prevention
of metastatic cancer remains an extremely challenging problem. This work
introduces a novel biologically motivated objective function to the radiation
optimization community that takes into account metastatic risk instead of the
status of the primary tumor. In this work, we consider the problem of
developing fractionated irradiation schedules that minimize production of
metastatic cancer cells while keeping normal tissue damage below an acceptable
level. A dynamic programming framework is utilized to determine the optimal
fractionation scheme. We evaluated our approach on a breast cancer case using
the heart and the lung as organs-at-risk (OAR). For small tumor
values, hypo-fractionated schedules were optimal, which is consistent with
standard models. However, for relatively larger values, we found
the type of schedule depended on various parameters such as the time when
metastatic risk was evaluated, the values of the OARs, and the
normal tissue sparing factors. Interestingly, in contrast to standard models,
hypo-fractionated and semi-hypo-fractionated schedules (large initial doses
with doses tapering off with time) were suggested even with large tumor
/ values. Numerical results indicate potential for significant
reduction in metastatic risk.Comment: 12 pages, 3 figures, 2 table
Multifocality and recurrence risk: a quantitative model of field cancerization
Primary tumors often emerge within genetically altered fields of premalignant
cells that appear histologically normal but have a high chance of progression
to malignancy. Clinical observations have suggested that these premalignant
fields pose high risks for emergence of secondary recurrent tumors if left
behind after surgical removal of the primary tumor. In this work, we develop a
spatio-temporal stochastic model of epithelial carcinogenesis, combining
cellular reproduction and death dynamics with a general framework for
multi-stage genetic progression to cancer. Using this model, we investigate how
macroscopic features (e.g. size and geometry of premalignant fields) depend on
microscopic cellular properties of the tissue (e.g.\ tissue renewal rate,
mutation rate, selection advantages conferred by genetic events leading to
cancer, etc). We develop methods to characterize how clinically relevant
quantities such as waiting time until emergence of second field tumors and
recurrence risk after tumor resection. We also study the clonal relatedness of
recurrent tumors to primary tumors, and analyze how these phenomena depend upon
specific characteristics of the tissue and cancer type. This study contributes
to a growing literature seeking to obtain a quantitative understanding of the
spatial dynamics in cancer initiation.Comment: 36 pages, 11 figure
Mutation timing in a spatial model of evolution
Motivated by models of cancer formation in which cells need to acquire
mutations to become cancerous, we consider a spatial population model in which
the population is represented by the -dimensional torus of side length .
Initially, no sites have mutations, but sites with mutations acquire an
th mutation at rate per unit area. Mutations spread to neighboring
sites at rate , so that time units after a mutation, the region of
individuals that have acquired the mutation will be a ball of radius . We calculate, for some ranges of the parameter values, the asymptotic
distribution of the time required for some individual to acquire mutations.
Our results, which build on previous work of Durrett, Foo, and Leder, are
essentially complete when and when for all
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