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Ordinary Differential Equation (ODE) based tumor models in Immunotherapy 

Kannan Thiagarajan

One of the important ways in which cancer cells survive is, with their ability to turn off our own immune system by evading key immune checkpoints (such as CTLA4 and PD-1); described as one of the Hallmarks of Cancer. In this context, one of the recent approaches in fighting cancer is Immunotherapy which is aimed at either training one’s own immune defense mechanism to fight cancer cells with better efficacy or prevent tumor’s evasion mechanism or a combination of both. For a brief introduction to immunotherapy, look at the previous post.

Building a Quantitative Systems Pharmacology (QSP) of immunotherapy can help us to answer some of the pressing questions in this field such as (i) what are the key biomarkers associated with tumor cells that are present in different organs (ii) which combination therapies would be most efficient in treatment (iii) how can we optimize the treatment to achieve maximum efficacy (iv) can we distinguish between responders and non-responders to treatment and predict some biomarker which would help to separate these groups before treatment.

ODE based models are generally preferred in the field of QSP, because of their robustness and simplicity of implementation, although other techniques such as stochastic differential Equations (SDEs), partial differential equations (PDEs), agent based modelling (ABM) and Boolean modelling are used in specific cases.  Many ODE based mathematical models describing tumor growth, tumor-immune interaction and role of therapies exist in the literature. While it is a nearly impossible task to review all of them in detail, we found three models of tumor-immune interactions to be informative.

Model 1: R J DeBoer et al, 1985.

Model 2: D Kirschner et al, 1998.

Model 3: L. de Pillis et al,2009.

Here, we present a brief review of these models by analyzing the results published. We also try to outline the limitations of these models. These models showcase the development of ODE based tumor- immune interaction models over the past three decades. In the following table, we compare the elements included in each of three models.


Model description Model 1 Model 2 Model 3


Major assumption of these models is that immune response originates at the site of tumor.
Compartments Tumor microenvironment Tumor microenvironment Tumor microenvironment
Species Cytotoxic T cell precursors, Helper T cell precursors, Helper T cells, Cytotoxic T cells, Macrophages, Cytotoxic macrophages, tumor cells, tumor cell debris. Antigen presentation, lymphoid factors and Inflammation are regarded quasi steady state variables. Effector cells (cytotoxic T cells), Tumor cells and InterLeukin-2 Tumor cells, Natural Killer cells, CD8+ T cells and Circulating Lymphocytes
Number of differential equations 8+ ODEs with 3 algebraic equations 3 ODEs 5 ODEs with 1 algebraic equation
Therapy Nil IL-2 therapy and Tumor Infiltrating Lymphocytes (TIL) therapy. IL-2 therapy, infusion of CD8+ T cells and Chemotherapy.

In the table to follow, we present our viewpoint of the results presented in the respective publications and the insights that can be inferred from them.


Aspects Model 1 Model 2 Model 3
Dynamics in the absence of treatment Main theme of the model presented here is to reproduce “sneaking through” mechanism and explore the parameter space to recreate the conditions at which this phenomenon occurs. Sneaking through is the phenomenon where low doses of tumor cells grow progressively in recipient animals, whereas large doses were either rejected or grew at a very low rate. Retardation of immune system has also been presented here. Detailed analysis of model dynamics in the absence of treatment has been presented here. Oscillating dynamics of the tumor and failure of immune system to clear tumor in the absence of the treatment have been presented and correspondence to clinical observations have also been established. Main theme of the model is to find the optimal treatment strategy. Authors have not presented results regarding the model behavior in the absence of therapy.
Therapy Effect of therapy have been accounted here based on the population of helper T cells and activated CTLs in the tumor microenvironment. Specificity of therapeutic agent is not described here. IL-2 therapy alone cannot clear tumor. However, in conjunction with Tumor infiltrating Lymphocytes (TIL) therapy tumor can be cleared. With high dose of IL-2 alone, authors have observed the over-activation of immune system in addition to the clearance of tumor. Effects of therapy and favored outcome (i.e., clearance of tumor) have been presented here. However, authors failed to indicate the tumor regression in their model.
Sensitivity Of the 20 parameters used in the model, four of them influence the system dynamics. In addition to that, authors have studied the system with four parameter space. Entire model dynamics can be understood with two parameters (1) tumor antigenicity and (2) immune response. About 40 parameters have been used in the model, making optimization difficult. However, authors indicate about 5 parameters are most sensitive. Parameters related to CD8+ T cell recruitment are fitted to patient’s data.
Time period of the simulations Varies from 10 days to 600 days Varies from 1000 days to 10,000 days 200 days
Not included Immune suppression mechanisms have not been included here. Dendritic cell (DC) maturation, T-cell activation and migration of Immune cells from other compartments such as lymph node, plasma to site of inflammation are not considered in any of the models considered here.
Major Conclusions Model behavior indicates tumor that are weak antigenic are not rejected by the immune system.
Magnitude of cytotoxic T cell response is dependent on the Helper T cell population
Number of CTL generated depends on the number of lymphocyte precursor populations.
Effector cell switch: Macrophages can reject tumor cells that have low antigenicity whereas CTL can reject cells that display high antigenicity
A relatively simple 3 state variable was able to depict the tumor-immune system dynamics (longer periods of tumor dormancy, damped oscillations) in an intuitive manner.
Combinatorial therapy and their efficacy have been demonstrated here.
Over-activation of immune system even after the clearance of tumor have been presented here.
Potential patient-specific efficacy of immunotherapy may be dependent on experimentally determinable parameters, in particular  degrees of CD8þT-cell efficacy


Drawbacks of these models and extension to a QSP model of tumor immune interaction:

The main drawback in all these models in the context of a drug-development scenario, is the lack of a compartmental structure which considers system dynamics in the lymph node, bone marrow, plasma etc. Since QSP focusses on drug delivery and its PK as well as the systemic interactions of the drug with different organs and physiological networks, a compartmental representation including these would be more applicable.Creation of a ‘Virtual Population’ would be another extension of the model that would be very useful from a QSP perspective. This would be particularly useful to address questions related to responders versus non-responders.

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