Spatially discordant alternans (SDA) of action potential duration (APD) has been widely observed in cardiac tissue and is linked to cardiac arrhythmogenesis. Theoretical studies have shown that conduction velocity restitution (CVR) is required for the formation of SDA. However, this theory is not completely supported by experiments, indicating that other mechanisms may exist. In this study, we carried out computer simulations using mathematical models of action potentials to investigate the mechanisms of SDA in cardiac tissue. We show that when CVR is present and engaged, such as fast pacing from one side of the tissue, the spatial pattern of APD in the tissue undergoes either spatially concordant alternans or SDA, independent of initial conditions or tissue heterogeneities. When CVR is not engaged, such as simultaneous pacing of the whole tissue or under normal/slow heart rates, the spatial pattern of APD in the tissue can have multiple solutions, including spatially concordant alternans and different SDA patterns, depending on heterogeneous initial conditions or pre-existing repolarization heterogeneities. In homogeneous tissue, curved nodal lines are not stable, which either evolve into straight lines or disappear. However, in heterogeneous itssue, curved nodal lines can be stable, depending on their initial locations and shapes relative to the structure of the heterogeneity. Therefore, CVR-induced SDA and non-CVR-induced SDA exhibit different dynamical properties, which may be responsible for the different SDA properties observed in experimental studies and arrhythmogenesis in different clinical settings.

Excitable cells, such as cardiac myocytes, exhibit short-term memory, i.e., the state of the cell depends on its history of excitation. Memory can originate from slow recovery of membrane ion channels or from accumulation of intracellular ion concentrations, such as calcium ion or sodium ion concentration accumulation. Here we examine the effects of memory on excitation dynamics in cardiac myocytes under two diseased conditions, early repolarization and reduced repolarization reserve, each with memory from two different sources: slow recovery of a potassium ion channel and slow accumulation of the intracellular calcium ion concentration. We first carry out computer simulations of action potential models described by differential equations to demonstrate complex excitation dynamics, such as chaos. We then develop iterated map models that incorporate memory, which accurately capture the complex excitation dynamics and bifurcations of the action potential models. Finally, we carry out theoretical analyses of the iterated map models to reveal the underlying mechanisms of memory-induced nonlinear dynamics. Our study demonstrates that the memory effect can be unmasked or greatly exacerbated under certain diseased conditions, which promotes complex excitation dynamics, such as chaos. The iterated map models reveal that memory converts a monotonic iterated map function into a nonmonotonic one to promote the bifurcations leading to high periodicity and chaos.

Excitable systems display memory, but how memory affects the excitation dynamics of such systems remains to be elucidated. Here we use computer simulation of cardiac action potential models to demonstrate that memory can cause dynamical instabilities that result in complex excitation dynamics and chaos. We develop an iterated map model that correctly describes these dynamics and show that memory converts a monotonic first return map of action potential duration into a non-monotonic one, resulting in a period-doubling bifurcation route to chaos.

Mathematical models of chronic myeloid leukemia (CML) cell population dynamics are being developed to improve CML understanding and treatment. We review such models in light of relevant findings from radiobiology, emphasizing 3 points. First, the CML models almost all assert that the latency time, from CML initiation to diagnosis, is at most ϳ 10 years. Meanwhile, current radiobiologic estimates, based on Japanese atomic bomb survivor data, indicate a substantially higher maximum, suggesting longer-term relapses and extra resistance mutations. Second, different CML models assume different numbers, between 400 and 10 6 , of normal HSCs. Radiobiologic estimates favor values > 10 6 for the number of normal cells (often assumed to be the HSCs) that are at risk for a CML-initiating BCR-ABL translocation. Moreover, there is some evidence for an HSC dead-band hypothesis, consistent with HSC numbers being very different across different healthy adults. Third, radiobiologists have found that sporadic (background, agedriven) chromosome translocation incidence increases with age during adulthood. BCR-ABL translocation incidence increasing with age would provide a hitherto underanalyzed contribution to observed background adult-onset CML incidence acceleration with age, and would cast some doubt on stage-number inferences from multistage carcinogenesis models in general. (Blood. 2012; 119(19):4363-4371) IntroductionChronic myeloid leukemia (CML) is characterized by Ph ϩ cells, that is, cells having a Philadelphia (BCR-ABL) chromosome translocation. 1,2 Treatment with the tyrosine kinase inhibitor (TKI) imatinib mesylate ("imatinib"), which suppresses bcr-abl oncoprotein action, 3 improves patient prognosis dramatically. 4 However, in some cases this treatment fails, a problem mitigated but not fully solved by the use of more recently developed TKI. 5,6 Moreover, many patients may need to continue TKI treatment indefinitely to avoid relapse. 7 CML is one of the best understood cancers; it has a simpler etiology than most cancers 8 and its time course is comparatively easy to monitor in the clinic. 9,10 Consequently, despite being much less prevalent than major solid tumors, CML has often been regarded as a kind of "model organism" for quantitative modeling of human carcinogenesis. 11,12 CML cell population dynamicsIn this review, we emphasize how radiobiologic studies impact CML models grounded in understanding underlying cell population dynamics. These models track CML time evolution by differential equations and/or stochastic formalisms. Such biologically based quantitative models are more ambitious, more comprehensive, and as yet less definitive than models often used in statistical analyses, which emphasize correlations analyzed by adjusting parameters in functions chosen mainly for mathematical convenience.After work by Rubinow and Lebowitz 13 on hematopoiesis, biologically based, mathematical CML models were pioneered by Clarkson and coworkers. 14 Many additional models have been suggested in the last decade. Recent a...

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