Unlocking Cell Secrets: How Mathematics is Helping to Understand Hydrogels
Often hailed as the "heroes" of biomedicine, hydrogels are proving to be game-changers across a number of medical applications. These fascinating materials, essentially three-dimensional networks of polymers, possess an uncanny resemblance to living tissue, making them incredibly versatile.
Exceptional at retaining large amounts of fluids—often up to thousands of times their own weight in water—hydrogels are non-toxic, readily available, and widely used to mimic the body’s natural cell environment in lab-grown cells, disease studies, and tissue repair research.
Although widely used across a number of industries, research into the mechanics of hydrogels is still ongoing. In his work in developmental biology, UBC Mathematics professor James Feng was exploring the mathematical modeling of cell behaviours when he began taking a closer look at hydrogels. Alongside a team including Pengtao Yue [1], Jiaqi Zhang [2], and Lei Li [3], Professor Feng delved into some fundamental questions surrounding hydrogels and the intricate interactions they have with the living cells they encapsulate. The team was particularly focused on three key interactions within hydrogels: how the gel deforms and adapts to fluid flow through it, how it transmits forces, nutrients, and chemicals to those cells, and how cells interact with each other through the gel. These questions led to the team developing a computer program that accurately shows how fluid moves through a gel that has a biological cell inside of it.
The Mathematical Challenge of Jiggling Gel
Understanding the mechanics of hydrogels is no simple feat; it’s a fascinating, often complex, mathematical puzzle. Professor Feng points to “mixture theory” as the intriguing framework guiding their efforts. Mixture theory is a model that aims to understand how systems behave when their components intermingle and influence each other.
"The solution of these problems is challenging," Professor Feng explains. "The gel is soft and porous and deforms if we try to pass liquid through it.” This inherent characteristic creates a dynamic interplay: the movement of liquid directly influences the gel's shape, and in turn, the gel's changing form dictates how the fluid flows. For Professor Feng, untangling this interwoven relationship—how to separate the liquid's influence from the gel's deformation—proved to be the most compelling challenge.
A New Model for Understanding Cell-Hydrogel Interactions
With this complex interplay in mind, Professor Feng and his colleagues then turned their attention to the fluid and solid mechanics at play in organ-on-chip devices. These microfluidic devices, which contain miniaturized tissues that mimic the functions of human organs, became the perfect testbed. The team created a computer program that simulated an organ-on-chip with the goal of studying how hydrogels behave when fluid flows through them as well as the physical forces and interactions between the hydrogel and a biological cell under realistic conditions.
This new model improves on previous ones by incorporating several important factors that were not accounted for before: the liquid flowing into the hydrogel, the liquid seeping out, the elastic shape changes of both the hydrogel and the cell, and the mechanical interaction between the hydrogel and the cell.
Although this new computational model is a valuable tool for studying the mechanics of hydrogels, it is still a work in progress. Professor Feng’s team has recently published a model that focuses on single cells. Currently they are extending the model to study multi-cell arrays, because most cell cultures contain cell colonies. Additionally, the study was conducted in two dimensions due to the computational costs involved; three-dimensional simulations remain a challenge at present.
Nevertheless, Professor Feng and his team's work offers new insights into the cell dynamics in hydrogels. These insights will undoubtedly be critical for future progress in creating artificial tissues and for better understanding natural bodily processes like embryo development and wound healing.
Professor Feng and his team currently work with experimenters in the UBC School of Biomedical Engineering and at St. Paul’s Hospital who developed organ-on-chip devices. His theoretical know-how helps them in designing and optimizing their in vitro devices. Ultimately, thanks to Professor Feng and his team, the future of hydrogel research is looking bright, and the computational tools that they continue to develop will eventually help design and improve future hydrogel-based organ-on-chip devices.
Professor Feng’s article appears in Biomicrofluidics (March 2025).
James Feng is a professor in the department of mathematics and the department of chemical and biological engineering at UBC. He is the co-director of the Laboratory for Complex and Non-Newtonian Fluid Flow, and his two main areas of research include the mechanics of biological cells and tissues and the interfacial dynamics of complex fluids.
[1] Pengtao Yue, Department of Mathematics, Virginia Tech
[2] Jiaqi Zhang, Research Center for Mathematics, Advanced Institute of Natural Sciences & Guangdong Provincial/Zhuhai Key Laboratory of IRADS
[3] Lei Li, Department of Chemical and Biological Engineering, University of British Columbia