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Heterogeneity in characteristics from one region (sub-population) to another,commonly observed in complex systems, such as glasses and a collection ofcells, is hard to describe theoretically. In the context of cancer, intra-tumorheterogeneity (ITH), characterized by cells with genetic and phenotypicvariability that co-exist within a single tumor, is often the cause ofineffective therapy and recurrence of cancer. Multi-region sequencing (M-Seq),obtained by sampling multiple regions of a single tumor, has vividlydemonstrated the pervasive nature of ITH, raising the need for a theory thataccounts for evolution of tumor heterogeneity. Here, we develop a statisticalmechanical theory to quantify ITH, using the Hamming distance, between geneticmutations in distinct regions within a single tumor. An analytic expression forITH, expressed in terms of cell division probability ($\alpha$) and mutationprobability ($p$), is validated using cellular-automaton type simulations.Application of the theory successfully captures ITH extracted from M-seq datain patients with exogenous cancers (melanoma and lung). The theory, based onpunctuated evolution at the early stages of the tumor followed by neutralevolution, is accurate provided the spatial variation in the tumor mutationburden is not large. We show that there are substantial variations in ITH indistinct regions of a single solid tumor, which supports the notion thatdistinct subclones could co-exist. The simulations show that there aresubstantial variations in the sub-populations, with the ITH increasing as thedistance between the regions increases. The analytical and simulation frameworkdeveloped here could be used in the quantitative analyses of the experimental(M-Seq) data. More broadly, our theory is likely to be useful in analyzingdynamic heterogeneity in complex systems such as super-cooled liquids.

Statistical Mechanical theory for spatio-temporal evolution of Intra-tumor heterogeneity in cancers: Analysis of Multiregion sequencing data