The Food and Drug Administration (FDA) is responsible for ensuring the safety and effectiveness of medical devices marketed in the US. For several decades, in a handful of niche applications, medical device industry has used computational modeling to provide evidence for safety or effectiveness, complementing bench, animal, or clinical testing. In recent years, the use of computational modeling in medical device regulatory submissions has grown significantly. FDA’s medical device Center is now tasked with evaluating a wide range of computational models of medical devices, as well as computational models implemented in medical device software (for example, patient-specific model-based software devices, closely related to the concept of a digital twin), and in silico clinical trials. This talk will discuss how computational models are relevant to medical devices, and then delve in model credibility assessment. We will discuss key activities involved in evaluating computational models for medical devices, overview recent FDA-led Standards and Guidances, and summarize recent work expanding these methods to the new frontiers of patient-specific models and in silico clinical trials.
The evolution and maintenance of cooperation is a fundamental problem in evolutionary biology. Because cooperative behaviors impose a cost, Cooperators are vulnerable to exploitation by Defectors that do not pay the cost to cooperate but still benefit from the cooperation of others. The bacteriophage Φ6 exhibits cooperative and defective phenotypes in infection: during replication, phages produce essential proteins in the host cell cytoplasm. Coinfection between multiple phages is possible. A given phage cannot guarantee exclusive access to its own proteins, so Cooperators contribute to the common pool of proteins while Defectors contribute less and instead appropriate proteins from Cooperators. Previous experimental work found that Φ6 was trapped in a prisoner's dilemma, predicting that the cooperative phenotype should disappear. Here we propose that environmental feedback, or interplay between phage and host densities, can maintain cooperation in Φ6 populations by modulating the rate of co-infection and shifting the advantages of cooperation vs. defection. We build and analyze an ODE model and find that for a wide range of parameter values, environmental feedback allows Cooperation to survive.
Several years ago, Dr Kathryn Isaac, a UBC professor and clinical surgeon contacted me with an intriguing problem. In her work on cosmetic reconstructive for post-breast-cancer-surgery patients, she encounters cases of failure that result (weeks or months later) in "capsular contraction" (CC). This painful condition arises when the healing tissue (the "capsule") that forms surrounding a breast implant undergoes pathological contraction and deformation - necessitating a new rounds of surgery. In this talk, I describe work in our team to help understand the root causes of CC, the risk factors, and possible preventative treatments. We use mathematical modeling to depict and investigate hypotheses for cell-level mechanisms involved in initiating CC. I will describe two rounds of modeling, the earliest joint with Cheryl Dyck MacDonald, and the latest joint with Ms Yuqi Xiao (UBC MSc student) and Prof. Alain Goriely (Oxford).
Evolutionary game theory is a discipline devoted to studying populations of individuals that are subject to evolutionary pressures, and whose success generally depends on the composition of the population. In biological contexts, individuals could be molecules, simple organisms or animals, and evolutionary pressures often take the form of natural selection and mutations. In socioeconomic contexts, individuals could be humans, firms or other institutions, and evolutionary pressures often derive from competition for scarce resources and experimentation.
In this talk I will give a very basic introduction to agent-based evolutionary game theory, a bottom-up approach to modelling and analyzing these systems. The defining feature of this modelling approach is that the individual units of the system and their interactions are explicitly and individually represented in the model. The models thus defined can be usefully formalized as stochastic processes, whose dynamics can be explored using computer simulation and approximated using various mathematical theories.
In a linear population model that has a unique “largest” eigenvalue and is suitably irreducible, the corresponding left and right (Perron) eigenvectors determine the long-term relative prevalence and reproductive value of different types of individuals, as described by the Perron-Frobenius theorem and generalizations. It is therefore of interest to study how the Perron vectors depend on the generator of the model. Even when the generator is a finite-dimensional matrix, there are several approaches to the corresponding perturbation theory. We explore an approach that hinges on stochasticization (re-weighting the space of types to make the generator stochastic) and interprets formulas in terms of the corresponding Markov chain. The resulting expressions have a simple form that can also be obtained by differentiating the renewal-theoretic formula for the Perron vectors. The theory appears well-suited to the study of infection spread that persists in a population at a relatively low prevalence over an extended period of time, via a fast-slow decomposition with the fast/slow variables corresponding to infected/non-infected compartments, respectively. This is joint work with MSc student Tareque Hossain.
For over a century, scientists have studied striking spatiotemporal patterns during the continual tooth replacement of reptiles. Aside from the compelling aesthetics of this phenomenon, it is thought that understanding the underlying mechanisms may provide the insight required to trigger adult tooth replacement in humans. Theoretical frameworks have long been proposed to understand the rules behind the observed spatiotemporal order, but have only been analyzed mathematically more recently. Starting from Edmund's observations in crocodiles and proposed theory of replacement waves, we show how a simple model consisting of a row of non-interacting phase oscillators predicts several experimental observations. Next, inspired by the hypothesis put forth by Osborn, we consider a variation of the phase model with ODEs that account for mutual inhibition between tooth sites, and use continuation methods to thoroughly search parameter space for experimentally validated solutions. We then extend the model to a PDE that explicitly accounts for the diffusion of inhibitory signals between teeth, yielding some novel solution types. Using continuation methods once again, we delineate parameter regimes with solutions that closely resemble experimental observations in leopard geckos.
I will report on a simple model for collective self-organization in colonies of myxobacteria. Mechanisms include only running, to the left or to the right at fixed speed, and tumbling, with a rate depending on head-on collisions. We show that variations in the tumbling rate only can lead to the observed qualitatively different behaviors: equidistribution, rippling, and formation of aggregates. In a second part, I will discuss in somewhat more detail questions pertaining to the selection of wavenumbers in the case where ripples are formed, in particular in connection with recent progress on the marginal stability conjecture for front invasion.
The questions of how healthy colonic crypts maintain their size under the rapid cell turnover in intestinal epithelium, and how homeostasis is disrupted by driver mutations, are central to understanding colorectal tumorigenesis. We propose a three-type stochastic branching process, which accounts for stem, transit-amplifying (TA) and fully differentiated (FD) cells, to model the dynamics of cell populations residing in colonic crypts. Our model is simple in its formulation, allowing us to estimate all but one of the model parameters from the literature. Fitting the single remaining parameter, we find that model results agree well with data from healthy human colonic crypts, capturing the considerable variance in population sizes observed experimentally. Importantly, our model predicts a steady-state population in healthy colonic crypts for relevant parameter values. We show that APC and KRAS mutations, the most significant early alterations leading to colorectal cancer, result in increased steady-state populations in mutated crypts, in agreement with experimental results. Finally, our model predicts a simple condition for unbounded growth of cells in a crypt, corresponding to colorectal malignancy. This is predicted to occur when the division rate of TA cells exceeds their differentiation rate, with implications for therapeutic cancer prevention strategies.
Mathematical biomedicine is an area of research where questions that arise in medicine are addressed by mathematical methods. Each such question needs first to be represented by a network with nodes that includes the biological entities that will be used to address the medical question. This network is then converted into a dynamical system for these entities, with parameters that need to be computed, or estimated. Simulations of the model are first used to validate the model, and then to address the specific question. I will give some examples, mostly from my recent work, including cancer drug resistance, side effects and metastasis, autoimmune diseases, and chronic and diabetic wounds, where the dynamical systems are PDEs. In each example, I will write explicitly the biological network, but will not the details of the corresponding PDE system.
While mathematical models have classically been used in the study of physics and engineering, recently, they have become important tools in other fields such as biology, ecology, and sociology. In this talk I will discuss the use of partial differential equations and dynamical systems to shed light onto social and ecological phenomena. In the first part of this talk, we will focus on an Ecological application. For an efficient wildlife management plan, it is important that we understand (1) why animals move as they do and (2) what movement strategies are robust. I will discuss how reaction-advection-diffusion models can help us shed light into these two issues. The second part of the talk will focus on social applications. I will present a few models in the study of gentrification, urban crime, and protesting activity and discuss how theoretical and numerical analysis have provided intuition into these different social phenomena. Moreover, I will also point out the many benefits of utilizing a mathematical framework when data is not available.