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Why Heterogeneity Matters for Mathematical Modelling

Morgan Craig and Isaac Yule, Sept 18, 2020
CHU Sainte-Justine Research Centre
Université de Montréal

Since the beginning of the SARS-CoV-2 pandemic, the world has been gripped with uncertainty and fear in the face of a virus that has spread faster and farther than any other in recent memory. It is no wonder that discussions of how COVID-19 affects different individuals and how we can treat this novel disease have pervaded the public discourse. With time, our collective and scientific understanding of how the virus impacts those with prior risk factors has quickly expanded. Early on, age was determined to be an important determinant for developing severe COVID-19, and as research has continued, other risk factors including type 2 diabetes, several heart conditions, chronic obstructive pulmonary disease, and being immunocompromised have emerged as complicating conditions. In a previous CAIMS blog, Anita Layton discussed how sex is also frequently an element differentiating disease progression and treatment between individuals. Since it is clear that a variety of biological features contribute to heterogeneity and shape how diseases are ultimately treated, we must ask ourselves how we can approach COVID-19 and other diseases using quantitative approaches. Put another way, given the novelty and variety of current case data, how can we take heterogeneity into account in mathematical models? Include too much information and we may find it difficult to parse out the mechanisms that control disease trajectories and therapeutic responses, but account for too few parameters and we may miss an entire portion of the population.


Image demonstrating population heterogeneity
A closer look at heterogeneity. Image courtesy of Jesse Morris


Lessons from HIV Treatment Research

Predicting how different individuals will respond to treatment in variable conditions is key for bacterial and viral infections, including HIV. Beyond epidemiology, models have been a hallmark of understanding how within-host dynamics and drug treatments dictate heterogeneous responses in HIV. Antibiotics, antivirals, and antiretrovirals must achieve a minimal inhibitory concentration to ensure suppression of bacterial or viral replication and to avoid resistance. In the case of HIV, sufficiently high drug concentrations can suppress the virus to undetectable levels, halting transmission. To maintain viral loads sufficiently low requires adequate adherence to treatment. Here heterogeneity rears its head, as different people comply differently to their prescriptions due to many factors which may limit their ability to have reliable access to and to reliably take their medication. Individuals with HIV who have access to treatment take triple cocktails of different antiretrovirals (known as highly-active antiretroviral therapy, or HAART) to limit the potential for drug resistance. HAART must be taken daily to achieve and maintain concentrations high enough to suppress the virus. Recent focus in HIV drug development has been to design drugs that do not require daily pill taking, to hopefully reduce the impact of poor adherence. One such idea is the oral administration of extended-release drug delivery devices, as medications taken orally are less invasive and easier to administer. Here, modelling again can help predict how likely people are to be successful on such proposed therapeutic modalities. To model this scenario, we integrated pharmacokinetic/pharmacodynamics (PK/PD) models of antiretroviral drugs with an in-host viral dynamics model of HIV, and variable adherence scenarios to predict the likelihood of wild-type and resistance mutation failure in a population of individuals taking conventional HAART or through an extended-release gastric drug delivery device and found a reduction in overall therapeutic failure. Combined with an epidemiological model, we predicted that even in the most pessimistic of cases, 200,000 new cases could be averted if weekly long-acting antiretrovirals were administered as pre-exposure prophylactics, or PrEP. Thus, the incorporation of intracellular, and intra- and interindividual variability into a quantitative framework was able to respond to pressing drug development needs in the global HIV epidemic.


Lessons from Cancer Research

As in HIV, cancer patients’ responses to drugs are often highly variable, especially in diseases with diverse factors and causes. Cancer, for example–which is itself a highly heterogeneous set of diseases–can manifest differently at the cell-to-cell level in addition to between individuals. Chemotherapies, which are largely cytotoxic (deadly to cells), can induce resistance through exerting a selective pressure that leads cancerous cells to adapt and evolve. This can then potentially result in treatment failure and disease relapse, as previously effective drugs are no longer able to successfully kill the cells. It is thus imperative to understand how cells adapt to different drugs and how resistance can emerge to provide improved and durable therapeutic options. Several mathematical approaches can be used to understand the dynamics of resistance (in cancer or other diseases), including game theoretic models like the replicator equation, stochastic models, particularly the Moran model, and agent-based models. Together, these methods have helped us to understand intra- and intercellular responses to changing drug landscapes, leading to improved drug administration and screening protocols. Importantly, we now have a better handle on the mechanisms of resistance which, in tandem with fundamental and clinical research, contributes to being able to better tailor therapies for individual patients.


Lessons from Pharmocometrics

PK/PD modelling and pharmacometrics have grappled with the issue of heterogeneity (to drugs or within a population) for a long time. Prior to approval, clinical trials comparing control and treatment arms must be completed to demonstrate the safety and efficacy of a drug. PK/PD modelling began in earnest in the 1960s as a way to understand the basic circulating kinetics of the drug, and to delineate how these concentrations are linked to its effect. Realizing that these concepts (pharmacokinetics, i.e. what the body does to a drug, and pharmacodynamics, or what the drug does to the body) are highly variable even among carefully selected cohorts, Sheiner and Beal developed a method to aggregate data together while adjusting for an individual’s parameter using correlations within the data. This approach, known as NONMEM (non-linear mixed effects modelling), is now an integral component of the drug approval process.


Lessons from Model Building and Exploration

However, NONMEM and mixed effects modelling rely upon having data and then estimating a model, whereas mathematical biologists are generally concerned with constructing a model from known biomedical mechanisms and inferring important processes from their prediction. Nonetheless, a good deal of inspiration for modellers wishing to account for heterogeneity can still be drawn from pharmacometrics. In particular, we may want to use our model to reconstruct a clinical trial and leverage our quantitative toolbox to tease out the mechanisms regulating responses. In this way, in silico or virtual clinical trials can explore a variety of questions (What dose is most effective and less toxic? When should drugs be administered? Who is responding to treatment and who isn’t?) during the drug development process and after to help us better understand drivers of disease and therapeutic outcomes. An in silico clinical trial rests upon a mathematical model built from prior knowledge of disease processes. In mechanistic modelling, parameters are usually fixed from previous studies or experimental results as point estimates, but we know that biological parameters live within ranges. The next stage of a virtual trial is to use the information about the model’s parameter distributions to create virtual patients, who are then digitally twinned/cloned and assigned to a number of cohorts. Then the therapeutic outcomes of each individual in each cohort are simulated using the model based on realistic treatment scenarios or optimization techniques. These outcomes are compared and conclusions are inferred using a variety of statistical techniques. In silico clinical trials have increasingly become integrated in mathematical oncology, particularly for the pre-clinical development of immunotherapies, a class of treatments that aims to leverage the body’s own immune response against cancer. We recently developed an in silico trial platform to establish combination therapy against melanoma using oncolytic viruses (genetically modified viruses that preferentially infect and destroy tumour cells, enhancing the immune response against the cancer) and an immune stimulant, and used our virtual cohorts to optimize this combination treatment and predicted improvements in overall survival for patients with late-stage melanoma.


Going forward

Which brings us back to our current reality and the COVID-19 pandemic. Mathematical modelling, both epidemiological and within-host, have already been critical to responding to and understanding SARS-CoV-2. To provide better and more realistic predictions to policy-makers, researchers, and the public, we must incorporate heterogeneity into our modelling approaches so that they are anchored in the reality that each person’s diseases dynamics and responses to potential therapies and vaccines are different. More broadly, taking heterogeneity into account will allow our models to respond to variable environmental, spatial, and genetic (among others) environments, improving their predictive ability and ultimately bettering patient care.


Papers of interest

  • Kirtane, A.R., Abouzid, O., Minahan, D. et al.Development of an oral once-weekly drug delivery system for HIV antiretroviral therapy. Nature Communications 9, 2 (2018).
  • Craig M, Kaveh K, Woosley A, Brown AS, Goldman D, Eton E, et al. Cooperative adaptation to therapy (CAT) confers resistance in heterogeneous non-small cell lung cancer. PLoS Computational Biology 15(8), e1007278 (2019).
  • Altrock, P., Liu, L. & Michor, F. The mathematics of cancer: integrating quantitative models. Nature Reviews Cancer15, 730–745 (2015).
  • Metzcar, J., Wang, Y., Heiland, R., Macklin, P. A Review of Cell-Based Computational Modeling in Cancer Biology. JCO Clinical Cancer Informatics3, 1-13 (2019).
  • Meibohm B, Derendorf H. Basic concepts of pharmacokinetic/pharmacodynamic (PK/PD) modelling. International Journal of Clinical Pharmacology and Therapeutics 35(10), 401-413 (1997).
  • Bauer, R.J. NONMEM Tutorial Part I: Description of Commands and Options, With Simple Examples of Population Analysis. CPT Pharmacometrics Systems Pharmacology, 8, 525-537 (2019).
  • Sheiner, L.B., Beal, S.L. Evaluation of methods for estimating population pharmacokinetic parameters. I. Michaelis-Menten model: Routine clinical pharmacokinetic data. Journal of Pharmacokinetics and Biopharmaceutics8, 553–571 (1980).
  • Cassidy T, Craig M Determinants of combination GM-CSF immunotherapy and oncolytic virotherapy success identified through in silicotreatment personalization. PLoS Computational Biology 15(11), e1007495 (2019).



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