Abstract
Mathematics is often used to model biological systems. In mammary gland development, mathematical modeling has been limited to acinar and branching morphogenesis and breast cancer, without reference to normal duct formation. We present a model of ductal elongation that exploits the geometrically-constrained shape of the terminal end bud (TEB), the growing tip of the duct, and incorporates morphometrics, region-specific proliferation and apoptosis rates. Iterative model refinement and behavior analysis, compared with biological data, indicated that the traditional metric of nipple to the ductal front distance, or percent fat pad filled to evaluate ductal elongation rate can be misleading, as it disregards branching events that can reduce its magnitude. Further, model driven investigations of the fates of specific TEB cell types confirmed migration of cap cells into the body cell layer, but showed their subsequent preferential elimination by apoptosis, thus minimizing their contribution to the luminal lineage and the mature duct.
Highlights
Mathematical models have been used to inform basic biological research for well over 100 years [1]
We present a model of ductal elongation that exploits the geometrically-constrained shape of the terminal end bud (TEB), the growing tip of the duct, and incorporates morphometrics, region-specific proliferation and apoptosis rates
We defined the stereotypical morphology of a single TEB in a moving frame and divided the TEB into eight regions (Fig 1A and S1 Fig)
Summary
Mathematical models have been used to inform basic biological research for well over 100 years [1]. Mathematical and computational models are useful as a complementary approach to lab-based studies in developmental biology. Such models have been developed to investigate morphogenetic phenomena [2, 3] in a broad sense, and in particular, for developmental pattern formation Grant et al developed a model of mammary ductal morphogenesis adapted from a cellular-automaton model of vascular morphogenesis [27]. While this model was capable of approximating mammary branching gross morphology, it did not take into account the arrangement of the cells at small scales, or use any experimentally derived measurements. At smaller scales of interest, Tang et al created an agent based model of breast acinus formation in vitro, in which they were able to determine the proliferation and apoptotic dynamics required for proper lumen formation and DCIS development [11], whereas Rejniak et al have proposed a sophisticated single deformable cell based model to derive conditions for acinus structure and lumen stability [30]
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