Modelling the Impact of NETosis During the Initial Stage of Systemic Lupus Erythematosus

by Vladimira Suvandjieva, Ivanka Tsacheva, Marlene Santos, Georgios Kararigas Peter Rashkov

NETosis, or formation of Neutrophil Extracellular Traps (NETs) in response to a pathogenic stimulus, is a suspected contributing factor of autoimmunogenicity in Systemic Lupus Erythematosus. Our ODE-based mathematical model studies the interaction between macrophages, neutrophils and two types of antigen with origin in apoptotic waste and NETs during the initial stage of the disease. Analytical and numerical calculations help classify the bifurcations between equilibria, in particular those with presence of autoantigen. The model predicts that even in parameter regimes with efficient clearance of NETs by immune cells, autoantigen can persist stably in tissue, causing chronic inflammation and loss of immune tolerance in the long run.

Sketch of the interactions between cells and antigens studied by our model.

Morphological Stability for in silico Models of Avascular Tumors

by Erik Blom & Stefan Engblom

We develop a simple, stochastic model of an avascular tumor that displays known behavior at the considered scales. In parallel we develop, analyze, and simulate a surrogate PDE model to explain the growth pattern and shape of the stochastic model. 

We investigate the emergent tumor morphology of the stochastic model (Fig. a) through the PDE model and through comparisons of respective model’s numerical experiments. The stochastic model displays the three characteristic avascular tumor regions -- the proliferative rim, quiescent annulus, and necrotic center -- and sigmoidal volumetric growth pattern in line with the PDE model under radial symmetry (Fig. b). The analysis predicts morphological instability under low oxygen and cell-cell adhesion conditions, as well as an ever-present creeping effect where the tumor as a whole migrates towards the oxygen source -- an effect observed during simulations of both models (Fig. c). The analysis further displays a capacity to predict non-trivial morphological patterns of the model tumor (Fig. d). 

The Unification of Evolutionary Dynamics through the Bayesian Decay Factor in a Game on a Graph

by Arnaud Z. Dragicevic

The study unifies evolutionary dynamics on graphs by employing a decaying Bayesian update in the context of strategic uncertainty. It demonstrates that the replication of strategies leading to shifts between competition and cooperation in well-mixed and Bayesian-structured populations is equivalent under certain conditions. Specifically, this equivalence holds when the rate of transition between competitive and cooperative behaviors matches the relative strength of selection pressures. Our findings help pinpoint scenarios where cooperation is favored, independent of payoff levels, expanding the application of Price's equation beyond its original intent.

Caption: The basins of attraction of the Price-wise unstructured population replicator dynamics

Convex Representation of Metabolic Networks with Michaelis-Menten Kinetics

by Josh Taylor, Alain Rapaport & Denis Dochain

Polyhedral models of metabolic networks are computationally tractable and provide insight into cellular functions. For example, flux balance analysis is a linear program in which reaction fluxes are optimized over polyhedral mass-balance constraints. In this paper, we augment the standard polyhedral model of a metabolic network with a new, second-order cone representation of the Michaelis-Menten kinetics. This enables us to explicitly model metabolite concentrations without losing tractability. We formulate conic flux balance analysis, a second-order cone program in which reaction fluxes are maximized while metabolite concentrations are minimized. While not as tractable as linear programming, second-order cone programs with hundreds of thousands of variables can be solved in seconds to minutes using modern solvers like Gurobi and MOSEK. In addition to predicting both fluxes and concentrations, we can use conic duality to compute sensitivities to kinetic parameters, i.e., maximum reaction rates and Michaelis constants. We also incorporate the second-order cone representation of the Michaelis-Menten kinetics into dynamic flux balance analysis and minimal cut set analysis. These tools provide new, tractable ways to analyze reaction fluxes and metabolite concentrations in metabolic networks. The Python code for each tool is available at https://urldefense.com/v3/__https://github.com/JAT38/conic-metabolic__;!!NVzLfOphnbDXSw!CB8YzXwI0ErdeBFcgljtFA36uhpf2ATRf6MEgYTiLhceaAzDS6gF7M5m047C62AYZH8xjVWlPanu7H7qQcsSzjBGP_RJ4rc$

Discretised Flux Balance Analysis for Reaction-Diffusion Simulation of Single-Cell Metabolism

by Yin Hoon Chew and Fabian Spill

Metabolism comprises thousands of biochemical reactions. It is commonly modelled using Flux Balance Analysis (FBA), a method based on linear programming, because this method requires very few parameters. However, conventional FBA implicitly assumes that all enzymatic reactions are not diffusion-limited though that may not always be the case.

To enable the exploration of diffusion effects on cellular metabolism, we present a spatial method that implements FBA on a grid-based system. The method discretises a living cell into a two-dimensional grid; creates variables that represent the rates of reactions within grid elements and diffusions between grid elements; and solves the system as a single linear programming problem.

Simulations using the method suggest that factors such as cell shape, diffusion regime and spatial distribution of enzyme can influence the variability and robustness of metabolism at both single-cell and population levels. We propose the use of this method to explore how spatiotemporal organisation of compartments and molecules in cells affect cellular behaviour.

Yin Hoon Chew is a Research Fellow and Fabian Spill is a Professor of Applied Mathematics at the University of Birmingham, UK. Yin Hoon designed and implemented the method, with feedback from Fabian.

The method can simulate living cells with different shapes and heterogeneous enzyme distribution. Simulations suggest that cell shape (perimeter-to-area ratio) does not affect cellular behaviour such as biomass growth when diffusion is fast, but there is a strong effect at low diffusion.

Mathematical modelling of parasite dynamics: A stochastic simulation-based approach and parameter estimation via modified sequential-type approximate Bayesian computation

by Clement Twumasi, Joanne Cable and Andrey Pepelyshev. 

In an era marked by global health challenges and re-emerging infections, the need for sophisticated and robust mathematical models to better understand infectious diseases has never been more pressing. Our impactful study focused on a biological system known as the gyrodactylid-fish system. While existing modelling studies have fallen short in capturing vital information to reflect the biological realism of this system, our research introduced a novel individual-based stochastic simulation model to realistically simulate the spread of three different strains of Gyrodactylus across three different host populations, enhancing our knowledge of this system given observed experimental data. This study contributed mathematically and biologically to the gyrodactylid-fish system, offering insights that may apply to modelling other biological systems. Expanding on the existing studies, we have added to our understanding of this system and provided answers to open biological questions for the first time through model-based Bayesian analysis. The study also led to robust extensions of likelihood-free Bayesian estimation methods, commonly known as approximate Bayesian computation (ABC), to aid in calibrating complex models with mathematically intractable likelihood. After conducting additional posterior predictive checks, we found our proposed ABC methodologies' efficiency highly compelling and can readily be adapted to fit other highdimensional multi-parameter models.

Second-Order Effects of Chemotherapy Pharmacodynamics and Pharmacokinetics on Tumor Regression and Cachexia

by Daniel R. Bergman, Kerri-Ann Norton, Harsh Vardhan Jain & Trachette Jackson

This paper presents a novel computational framework that constrains high-dimensional ABM parameter space with multidimensional real-world data.  We accomplish this by extending and validating a first-of-its-kind method that leverages explicitly formulated surrogate models to bridge the computational divide between ABMs and experimental data.  We show that Surrogate Modeling for Reconstructing Parameter Spaces (SMoRe ParS) can constrain high-dimensional ABM parameter spaces using unidimensional (single time-course) data.  We then demonstrate that it can constrain parameter spaces of more complex ABMs using multidimensional data (multiple time courses at different biological scales).  To validate our method, we compared the SMoRe ParS-inferred ABM parameter space with ABM parameters inferred by an often computationally expensive direct comparison with experimental data.  A strength of SMoRe ParS is that it allows for exploring ABM parameter space even at points that are not directly sampled and where ABM output was never generated.   Computationally efficient methods to connect ABMs with multidimensional data are timely and important as ABMs are a natural platform for capturing heterogeneity and predicting emergent behavior in multiscale systems.  SMoRe ParS is a robust and scalable computational framework that can explore the uncertainty within multidimensional parameter spaces associated with ABMs representing complex biological phenomena.

Caption: The schematic diagram for using SMoRe ParS to infer ABM input parameters from experimental data via a surrogate model. The solid arrows connecting the Experimental Data and Agent-based Model boxes to the Surrogate Model box represent the direction of information flow in the first few steps of SMoRe ParS. Green (control), yellow (0.75μM oxaliplatin), and red (7.55μM oxaliplatin) colors in the Experimental data box refer to the dosing regimens that generated the experimental data.

Brief description of the roles of the authors (e.g. student, group-leader etc):

Daniel R. Bergman, postdoc

Kerri-Ann Norton,  computational modeling collaborator, and developer of SMoRe Pars

Harsh Vardhan Jain, co-senior author and co-developer of SMoRe Pars

Trachette Jackson, co-senior author and co-developer of SMoRe Pars

Second-Order Effects of Chemotherapy Pharmacodynamics and Pharmacokinetics on Tumor Regression and Cachexia

by Luke Pierik, Patricia McDonald, Alexander R.A. Anderson & Jeffrey West. 

Read the paper

Second order effects describe changes in a system which result from introducing variability or fluctuations in a system’s input. Quantifying second-order effects relies on an understanding of the convexity of an underlying function determining system output, and this has been effectively used in several fields, notably financial risk management. Previously, the vocabulary of fragile or antifragile has been used: fragile systems are harmed by variability while antifragile systems benefit from variability. The key insight here is that oncologists can control the input variability of treatment schedules, and therefore it is critical to define the fragility (or antifragility) of tumors. In cancer, second-order effects have been studied through dose response curves, which are ubiquitous theoretical and clinical tools in the field. However, these curves do not incorporate knowledge about how long dosages remain near the tumor (i.e. pharmacokinetics), which influences treatment outcomes. In this paper, we explore this relation between second-order effects and pharmacokinetics through standard mathematical models as well as a previously parameterized tumor model with 5-fluorouracil. By studying second-order effects with pharmacokinetics, more efficient treatment schedules may be devised which utilize the underlying convexity of dose response to produce greater patient outcomes.

Nonlinear Regression Modelling: A Primer with Applications and Caveats

by  Timothy O'Brien & Jack Silcox

In their applied studies, researchers often find that nonlinear regression models are more applicable for modelling various biological, physical, and chemical processes than are linear ones since they tend to fit the data well and since these models – and especially the associated model parameters – are usually more scientifically meaningful.  For example in relative potency, drug synergy, and similar compound interaction modelling, key model parameters aid researchers in making important decisions regarding comparisons of drugs or compounds and/or whether combinations of these substances would enhance effects.

These researchers may be at a loss for how best to perform this nonlinear modelling, including choosing between various growth models or binary logistic models, how these work and which analysis methods are best and why.  Working through several key examples, this paper provides a gentle yet informative hands-on introduction to nonlinear modelling, provides key R code which can be easily adapted to fit ones own nonlinear models, and underscores key caveats regarding often-problematic Wald confidence intervals and p-values as well as the lack of penalizing for overfitting in a certain large-sample likelihood-based approach.

About the Authors: Tim O’Brien is a professor of Mathematics and Statistics (with a joint appointment in Environmental Sustainability) at Loyola University Chicago.  Jack Silcox is a postdoctoral researcher in the Department of Psychology at the University of Utah.

Predicting Radiotherapy Patient Outcomes with Real-Time Clinical Data Using Mathematical Modelling

by  Alexander Browning, Thomas Lewin, Ruth Baker, Philip Maini, Eduardo Moros, Jimmy Caudell, Helen Byrne and Heiko Enderling

Mathematical models have the potential to revolutionise clinical practise by providing real-time insights that guide decision-making and predict patient responses. Challenges associated with the application of mathematical models are perhaps, however, most acute for single-patient clinical data of cancer tumour progression. Data are often noisy, sparse, and simplistic; patient responses are often highly variable; and mathematical models may be necessarily complex.

In this work, we develop and present a novel, simple, mathematical model of tumour volume progression in response to radiotherapy that can capture a full gamut of patient responses observed in the clinic. To maximise the utility of data collected from a large clinical cohort whilst accounting for significant patient-to-patient variation, we present alongside the model a Bayesian statistical method that allows for real-time clinical predictions to be drawn throughout a patient's course of treatment.

All model parameters vary between patients, with prior parameter knowledge for new patients informed by a weighted mixture of posterior parameter knowledge from previously observed patients. We demonstrate the ability of our model and statistical framework by considering a subset of patients for which predictions are continuously updated throughout their course of treatment.

The research was led by Alexander Browning (from 2023), a research fellow, and Thomas Lewin (until 2022), a DPhil student.

Caption: Data from a cohort of training data are used to calibrate population-level posterior distributions that account for patient-to-patient variability. Individual-level predictions are then drawn and then updated throughout a patients’ course of treatment