Mathematical and Computational Frameworks for Modeling Mass Transport Phenomena Across Biological Systems

Multiscale mathematical modeling of transport phenomena across different levels of biolog- ical systems, such as cells, capillaries, tissues, and organs, has been increasingly helpful in describing how interactions among these systems lead to their function and dysfunction. The development of models across these scales is based on knowledge from various fields, such as engineering, physiology, and biophysics, and it requires significant collaboration among scientists from the relevant areas. Moreover, mathematical modeling of membrane trans- porters and ion channels helps researchers obtain a thorough understanding of the complex regulation process between membrane transporters, estimate the changes and uncertainties of their mechanisms, and study how disorders in these procedures may cause disease. A unified framework to describe the fundamental principles and unite the established models could support this growing research community. In this regard, the first part of this work provides a mathematical and computational frame- work for modeling molecule transport, such as nutrients, inorganic ions, drugs, and toxins, across the cell through membrane transporters and ion channels along with the associated database. Our purpose is to substantially save time and resources needed to find the math- ematical models available for different classes of membrane transporters and improve com- munication between scientists with different backgrounds interested in this field. To achieve these objectives, this work first deals with the essential terminology required to understand and model biological transport mechanisms, as well as to compile currently available mod- els. Afterward, an inclusive mathematical framework for predicting the mechanisms of mass transport in various tissue compartments, such as cells, capillaries, and gland ducts, is devel- oped with a primary focus on membrane-mediated transporters such as channels, uniporters, symporters, pumps, and antiporters. We present a comprehensive and up-to-date set of cur- rently available mathematical equations for modeling ion channel and membrane transporter mechanisms. With this unified mathematical framework that includes most of the available models for ion channels and transporters, scientists can select the right model more conve- niently. Finally, a comprehensive database of parameters relating to mathematical and computa- tional transport models is developed in order to facilitate efficient research on the mem- brane transporters. TransporterDB is a biophysical database that contains data on the kinetic parameters of membrane transporters and ion channel parameters such as conduc- tance and electric mobility for each ion. TransporterDB can be accessed through a web browser (https://transporterdb.org/) and GitHub repository (https://github.com/ stamp-cell/TransporterDB). Equations for the parameters were taken from our previ- ously published comprehensive mathematical toolkit, which contains over 200 sources. While TransporterDB has not been fully completed, it contains information on more than 80 models of human ion channels and membrane transporters. The source of data is from in vivo and in vitro studies on various mammals. Users and readers are encouraged to submit additional data and information. We have completed the first part of this dissertation by a validation study. The developed mathematical-computational framework is validated for simulating two different cases: 1) human mammary epithelial cell in the breast and 2) cardiac action potentials. The second part of this work explores, an integrated machine learning (ML) and Computa- tional Fluid Dynamics (CFD) simulation approach to predict milk flow behavior in lactating breasts. Using CFD to solve fluid flow problems can be both computationally intensive and time-consuming. Artificial neural networks (ANN) are capable of learning complex depen- dencies between high-dimensional variables. This work uses this capability to develop a novel data-driven approach to CFD. To that end, a fully integrated CFD and ML workflow is developed to predict milk flow behavior in lactating breasts. CFD simulations are used to develop the training and validation data sets, and a machine learning workflow is developed to train the ANN. In addition, different ANN designs are proposed and their prediction re- sults are compared. Finally, we employ the design of the experiment method to determine the minimum number of simulations required to determine an accurate prediction. Our work shows that by integrating CFD and ML approaches (ML-CFD), one can train a neural net- work to produce CFD results, thereby saving both time and resources required for running CFD simulations. This ML-CFD approach provides a capability to build ML models capable of predicting milk flow behavior at a per-lactating mother level faster than traditional CFD solvers.