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MBW:A Validated Mathematical Model of Cell-Mediated Immune Response to Tumor Growth

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This wiki page is a review of a paper by L. G. de Pillis, A. E. Radunskaya, and C. L. Wiseman, titled: A validated mathematical model of cell-mediated immune response to tumor growth. [1]


Mathematics Used: Sensitivity analysis, data fitting to find parameter values.

Type of Model: Population dynamics. Similar to a predator prey model except for the given assumptions.

Biological System Studied: The immune system's response to tumor cells.

Executive Summary

In this paper the authors modify and build on existing tumor growth models with the specific goal of gaining more insight into the immune systems role. Three ordinary differential equations are used to model the levels of the tumor cells, and two different types of immune cells, natural killer and CD8+ T cells. After justifying the thinking behind the overall structure of the differential equations as well as some of the individual terms, the authors propose a change to one of the terms with motivation stemming from experimental data. The authors make use of data from both mouse and human studies as well as numerical fitting techniques to then estimate values for parameters in the equations. The model is then compared to the experimental data. Lastly, sensitivity analysis is used to determine the most important parameters and also draw clinical hypothesis about how to better fight cancer in humans.

Context/Biological Phenomenon

The immune response is thought to play a very important role in cancer prevention that is not yet fully understood. In this paper two specific immune cell types are singled out for the important role they are thought to play. The first type of immune cell considered in this model, the natural killer cells, are part of the body’s innate defense system and are always prepared to respond to general invasions. These natural killer cells have no specificity but are tightly regulated because of the high cytotoxicity. The mechanism by which they kill malignant cells, which can be either virulent or cancerous, occurs through the release of toxic proteins into the space around the target cell causing their death. The other type of immune cell considered in this model, the CD8+ T cells are part of the body’s adaptive immune system and therefore are more specific to invasions yet they have a slower response. These cells also carry with them a memory and are therefore prepared to respond to invasions on much quicker time scale given a prior acquaintance with the malicious cell in question.

A large part of the motivation for better understanding the role that the immune system can play in cancer prevention and elimination is the hope that cancer immunotherapy might one day become an effective treatment of cancer. Cancer immunotherapy works by stimulating the immune system in some way as to enhance and better the immune systems ability to recognise and fight the cancer cells. By stimulating the body's immune system, you could potentially avoid having to use chemotherapeutics and the process of finding an optimal chemotherapy strategy.

In this paper, the authors use data from two studies, one of which uses a mouse and another of which uses a human as its base platform. In the mouse study, the immune system is simulated to be stimulated by injecting the mouse with tumor cells that express higher levels of a chemical marker on their surface that the immune system recognises as corresponding to malignant cells that need to be removed. In the human study, a stimulated immune system is achieved by treating the human with immune cells that are already configured to recognise particular tumor cells for removal.


There have been many papers published and many mathematical models created around cancer. A large part of these include a immune system interaction. The authors in this paper chose to base their model off of the models presented in two specific papers published in 2001 and 2003 that they themselves published.

The first paper is titled A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, and was published by the authors of the current paper being reviewed [2]

The second paper was also published by the authors of the current paper being reviewed and is titled The dynamics of an optimally controlled tumor model: A case study [3]

Experimental Data

The author used data from two published studies to acquire values for their parameters and also to compare the predictions of their model to real world data. The studies used two different animals as a model for the tumor-immune interaction.

The first study published by Diefenbach A, Jensen E, Jamieson A, and Raulet D [4] recorded data after injecting syngenic mice with modified tumor cells expressing murine NKG2D ligands that give their immune system the ability to recognize the cells as foreign.

The second study from which data was acquired was published by Mark E. Dudley et al. [5], in which humans with metastatic melanoma are treated with immune cells that are highly specific towards tumor cells. The patients in this study receive these specific immune cells only after being subjected to immune depleting chemotherapy which is thought to disrupt the regulation of the immune system and increase the efficacy of the treatment. It should be noted that the authors choose to take data from only two of the patients in this study that responded well to this treatment out of a total of six that had a response and thirteen that participated in the study.


Before the authors state the model that they use in their paper, they state the assumptions they made. They are the following:

- The tumor cell growth is of a logistic from in the absence of a response by the immune cells.

-Both types of immune cells considered in this paper, the natural killer cells and the CD8+ T cells, have the ability to kill tumor cells

-Tumor cells can cause a response in immune cells such that cells that were previously non-hostile towards tumor cells can become hostile and kill the tumor cells through a cytotoxic response

-Natural killer cells are always present and active in the system, even if their are no tumor cells

-Tumor specific CD8+ T cells present themselves once the tumor cells are present

-Natural killer cells and CD8+ T cells will become inactivated after a certain number of encounters with the tumor cells

Mathematical Model

The authors in this paper use a system of three ordinary equations to model the tumor-immune cell interaction. The first differential equation models the tumor cell population as a function of time, T(t).

The second differential equation models the total natural killer cell effectiveness as a function of time, N(t). Natural killer cell effectiveness is used instead of just a simple population count since the effect of the immune system on the tumor cell count is a function of both the population of the immune cells as well as their ability to release cytotoxic chemicals into the environment.

The third and last differential equation used models the amount of tumor-specific CD8+ T cell effectiveness as a function of time, L(t).

The equation to model the tumor cell population is set up in the following way:

rate of change of tumor cell population = (growth and death rate) -(cell-cell kill rate)

The authors represent the growth and death rate logistically by aT(1-bT), where a and b are just proportionality constants for the growth and death rate respectively. They then represent the cell-cell kill rate with two different terms. One term accounts for the natural killer cells and another accounts for the CD8+ T cells. The cell-cell kill rate for the natural killer cells is represented by a simple proportionality relation between the natural killer cell effectiveness and the tumor cell population. The cell-cell kill rate for the CD8+ T cells is represented by a more complex term (represented by D) that is a novel piece of this paper.

{\frac  {dT}{dt}}=aT(1-bT)-cNT-D

Where D is given by:

D=d{\frac  {(L/T)^{\lambda }}{s+(L/T)^{\lambda }}}T

The authors propose the above form for the cell-cell kill rate caused by the CD8+ T cells instead of just a proportionality term after analysing the data from the two studies described above. Specifically, data generated from chromium release assays was the motivating factor. The proportionality term for the natural killer cell-cell kill rate fit well with this data but the cell-cell kill rate for the CD8+ T cells in the authors opinion did not, so the new form seen as equal to D above was created.

The authors first used an iterative numerical process to try and find parameters to fit the data to a simple product term such as dL^{{\lambda }}T and cN^{{\lambda }}T to describe the cell-cell kill rates. They found that the product term for the cell-cell kill rate fits well with the data of the natural killer cells, with lambda equal to one, but not with that of the CD8+ T cells. The authors also note that experimental evidence suggest that the percent cell lysis never exceeds a maximum value, suggesting there is a saturation effect, something that this new form takes into account.

The equations to model the immune cell effectiveness are set up in the following way:

rate of change of active effector cell populations = (growth and death rate) + (recruitment rate) - (inactivation rate)

The differential equations then become:

{\frac  {dN}{dt}}=\sigma -fN+{\frac  {gT^{2}}{h+T^{2}}}N-pNT

{\frac  {dL}{dt}}=-mL+{\frac  {jD^{2}}{k+D^{2}}}L-qLt+rNT

Parameter values have been derived from experimental results for two systems. A mouse model system and a human model system.

A summary of all the parameters used in the mouse and human model and there estimated values derived from experimental results are defined below:


The main benefit of creating this model that the authors state immediately is its ability to model the effect of increased ligand expression on tumor cells which has the effect of simulating an stimulated immune system.

The following image below shows how the data from the mouse study compares to the model in the case of the natural killer cells. The image also shows how the model correctly predicts how a change in expression of the ligands from the control will effect the data.


The next image below shows how the new function form created in this paper to model the cell-cell kill rate of the CD8+ T cells fits the data better in the case when tumor cells with the specific ligands expressed are introduced into the system.


The two figures on the left show data resulting from the system being primed with normal tumor cells and the challanged with the same tumor cells. The two figures on the right represent data from ligand expressing tumor cells being primed and the challenged. The model predictions agreed with experimental data which shows that the least effective combination is to prime the system with normal tumor cells and the introduce those same cells into the system, and the most effective combination is to both prime and challenge the system with ligand-transduced tumor cells.

The next image below shows how the new functional form also fit the data from the human study.


The results from the sensitivity analysis are shown below. They were calculated by running a simulation with a time of 25 days and observing the change in the tumor volume. The results below show that the model is most sensitive to the lambda parameter that is an exponent of the CD8+ T cell-cell kill rate and the tumor growth rate parameter a. It is alos interesting to note that the results below show the the cell-cell kill rate parameter for the natural killer cells is not nearly as important in determining the final size of the tumor.


The last image shown below shows model simulations run under different scenarios.


Recent Extension

This article was cited in a 2006 article titled Cancer Immunotherapy by Interleukin-21: Potential Treatment Stregies Evaluated in a Mathematical Model by Antonia Cappuccio, Moran Elishmereni, and Zvia Agur. The article discusses a newly characterized member of the interleukin-2 family of cytokines, Interleukin-21, and its role in tumor regression. Cappuccio, et al. studied Interleukin-21’s effects on tumor erradication through the use of mathematical models. Similar to the article by Pillis, et al., their mathematical model focused on NK and CD8+ T mediated lysis of tumor cells. They used the citation to briefly mention how Pillis, et al. had “emphasize[d] the role of certain effectors in anticancer responses.”

External Links

  1. de Pillis, Lisette G., Ami E. Radunskaya, and Charles L. Wiseman. “A Validated Mathematical Model of Cell-Mediated Immune Response to Tumor Growth.” Cancer Research 65.17 (2005): 7950-7958. Web.
  2. De Pillis, L. G., and A. Radunskaya. “A mathematical tumor model with immune resistance and drug therapy: an optimal control approach.” Journal of Theoretical Medicine 3.2 (2001): 79-100. Print.
  3. De Pillis, L. G., and A. Radunskaya. “The dynamics of an optimally controlled tumor model: A case study.” Mathematical and Computer Modelling 37.11 (2003): 1221-1244. Web.
  4. Diefenbach, Andreas et al. “Rae1 and H60 ligands of the NKG2D receptor stimulate tumour immunity.” Nature 413.6852 (2001): 165-171. Web.
  5. Dudley, Mark E. et al. “Cancer Regression and Autoimmunity in Patients After Clonal Repopulation with Antitumor Lymphocytes.” Science 298.5594 (2002): 850-854. Web.