In this paper, we present a one-dimensional model for blood flow in arteries, without assuming an a priori shape for the velocity profile across an artery (Azer, Ph.D. thesis, Courant Institute, New York University, 2006). We combine the one-dimensional equations for conservation of mass and momentum with the Womersley model for the velocity profile in an iterative way. The pressure gradient of the one-dimensional model drives the Womersley equations, and the velocity profiles calculated then feed back into both the friction and nonlinear parts of the one-dimensional model. Besides enabling us to evaluate the friction correctly and also to use the velocity profile to correct the nonlinear terms, having the velocity profile available as output should be useful in a variety of applications. We present flow simulations using both structured trees and pure resistance models for the small arteries, and compare the resulting flow and pressure waves under various friction models. Moreover, we show how to couple the one-dimensional equations with the Taylor diffusion limit (Azer, Int J Heat Mass Transfer 2005;48:2735-40; Taylor, Proc R Soc Lond Ser A 1953;219:186-203) of the convection-diffusion equations to drive the concentration of a solute along an artery in time.
Mathematical biology and pharmacology models have a long and rich history in the fields of medicine and physiology, impacting our understanding of disease mechanisms and the development of novel therapeutics. With an increased focus on the pharmacology application of system models and the advances in data science spanning mechanistic and empirical approaches, there is a significant opportunity and promise to leverage these advancements to enhance the development and application of the systems pharmacology field. In this paper, we will review milestones in the evolution of mathematical biology and pharmacology models, highlight some of the gaps and challenges in developing and applying systems pharmacology models, and provide a vision for an integrated strategy that leverages advances in adjacent fields to overcome these challenges.
Acid sphingomyelinase deficiency (ASMD) is a rare lysosomal storage disorder with heterogeneous clinical manifestations, including hepatosplenomegaly and infiltrative pulmonary disease, and is associated with significant morbidity and mortality. Olipudase alfa (recombinant human acid sphingomyelinase) is an enzyme replacement therapy under development for the non‐neurological manifestations of ASMD. We present a quantitative systems pharmacology (QSP) model supporting the clinical development of olipudase alfa. The model is multiscale and mechanistic, linking the enzymatic deficiency driving the disease to molecular‐level, cellular‐level, and organ‐level effects. Model development was informed by natural history, and preclinical and clinical studies. By considering patient‐specific pharmacokinetic (PK) profiles and indicators of disease severity, the model describes pharmacodynamic (PD) and clinical end points for individual patients. The ASMD QSP model provides a platform for quantitatively assessing systemic pharmacological effects in adult and pediatric patients, and explaining variability within and across these patient populations, thereby supporting the extrapolation of treatment response from adults to pediatrics.
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