Mathematical modelling of cancer immunotherapies

Modelling the impact of T-cell exhaustion and growth and treatment of tumours

Tumours evade immune surveillance through a number of different immunosuppressive mechanisms. One such mechanism causes cytotoxic T-cells, a major driving force of the immune system, to differentiate to a state of ‘exhaustion’, rendering them less effective at killing tumour cells.

We have investigated the impact of T-cell exhaustion dynamics on tumour–immune interactions and tumour growth by building a structured mathematical model. Numerical simulations of the model equations reveal how the exhaustion distribution of T-cells changes over time and how it influences the tumour’s growth dynamics. Complementary bifurcation analysis shows how altering key parameters significantly reduces the tumour burden, highlighting exhaustion as a promising target for immunotherapy.

[1] Lai, N., Farman, A. & Byrne, H.M. The Impact of T-cell Exhaustion Dynamics on Tumour–Immune Interactions and Tumour Growth. Bull Math Biol 87, 61 (2025). https://doi.org/10.1007/s11538-025-01433-1

Coarse-grained PDEs of hormone transport in plant leaves

Auxin is a hormone that plays key regulatory roles in plant development. Its diverse functions are enabled by its directional (polar) transport through cells and the feedback between this transport and the intracellular localisation of auxin transporters (PINs). These mechanisms are traditionally modelled using discrete compartmental ODEs, where each compartment represents a single cell.

We develop coarse-grained continuum models of polar auxin transport that are better suited to larger biological domains and enable analytical progress. We investigate the emergence of patterned auxin peaks generated by up-the-gradient PIN polarisation and explore canal-like patterns arising from with-the-flux feedback mechanisms. Finally, we use the continuum framework to identify which mechanisms persist under coarse-graining and which are inherently dependent on the cellular scale, providing a robust comparison between cell-level and tissue-level transport dynamics.

Work in progress

Modelling T-cell migration in microfluidic devices

A pivotal aspect of developing effective immunotherapies for solid tumours is the robust testing of product efficacy inside in vitro platforms. Collaborating with an experimental team that developed a novel microfluidic device at Children’s National Hospital (CNH), we developed a mathematical model to investigate immune cell migration and cytotoxicity within the device.

We study Chimeric Antigen Receptor (CAR) T-cell migration inside the channels, treating the cell as a moving boundary driven by a chemoattractant concentration gradient. We find that certain cell shapes allow for multiple cells to travel inside the channel simultaneously and identify parameter regimes under which cells clog the channel, impairing their movement.

We integrate our model results into a broader model of the device, which also examines the cytotoxicity of CAR T-cells. This provides a tool for distinguishing experimental artifacts from genuine CAR T-cell behaviour.

A. Microfluidic chip

B. Migration channels with CAR T-cells inside

C. CAR T-cells in green, cancer cells in blue and dead cancer cells in purple.

These setups were developped by a team at Children's National Hospital in Washington D.C.

Work in progress

Investigating immune persistence and cancer relapse in CAR T-cell treatments of B-ALL

Amongst the leukaemia patients responsive to CAR T-cell therapy, a significant number relapse due to lack of immune persistence. There are currently two main theories to explain why immune persistence is necessary:

B cells hide in geographical niches that are immune privileged or / and
B cells can evade the immune system when at very low numbers: the likelihood of CAR T-cells encountering the blast cells is very low.

We employ Ordinary Differential Equations (ODEs) and stochastic methods to model the interactions between blast cells and immune cells, and investigate the mechanisms of loss of persistence and subsequent escape of tumour cells.

Work in progress

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