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Article review by Michael Kochen

Kirschner, Denise, and John C. Panetta (1998). Modeling immunotherapy of the tumor-immune interaction. Journal of Mathematical Biology 37, 235-252.


There have been many different types of mathematical analyzes of the molecular spread of cancer in the body, methods of improving immutability to it, and methods of curing the disease. This particular article utilizes the stability and bifurcation analysis of systems of ordinary differential equations that represent the rate at which cells become immune to the disease as well as the rate at which the cancer cells can spread. For a general look at the population dynamics of an infectious disease, see Population Biology Of Infectious Diseases.


Advances in the knowledge and treatment of cancer have been numerous in the decade since this paper was published. Even so, conventional chemotherapy and radiation remain widely prescribed by oncologists, and although there are exceptions, they do little for the majority of cancers[1]. Fortunately there are treatment options that are either available today, or will be in the coming years. This paper models one of the more promising ones.

Adoptive cellular immunotherapy (ACI) is a type of cancer immunotherapy. The treatment begins by collecting immune cells from the patient. Cells that have anti-cancer properties are then cultured in the lab. The newly grown cells are subsequently injected back into the patient to fight the cancer.

Interleukin-2 (IL-2) is a cytokine, a signaling molecule produced by immune cells that activates, and stimulates growth and differentiation of T-cells, and other immune system cells.

The immune system model constructed in this paper first explores the dynamics between immune cells, tumor cells and IL-2, and then considers treatment with ACI, and additional IL-2. (For more information on modeling tumors, please see APPM4390:Tumor Modeling).

Non-Treatment Model


  • E(t) are the immune system cells, called effector cells.
  • T(t) is the size of the tumor.
  • IL-2(t) is the concentration of IL-2.



To make the system easier to handle numerically, the authors non-dimensionalized the equations to the following (see page 239 of the paper for the scaling).



  • c is the tumor antigenicity. Its the rate at which effector cells grow in response to a tumor.
  • a represents the rate at which tumor cells are lost due to the immune responses.
  • 1/b is the carrying capacity, or maximum mass of the tumor. b = 1*10^9

These are the parameters of note. The values for these, and the rest of the parameters are given in table 1. on page 238. Notice the logistic growth term in the second equation indicating limited growth of the tumor. Also note that each equation has a term in Michaelis-Menten form indicating limited IL-2, immune, and tumor responses. The antigenicity will be varied in the next section, to see what affect it has on the tumor.


The bifurcation diagram of tumor size vs. antigenicity is given on page 240 of the paper. There are two bifurcation points, one at 8:55*10^5 and the other at 0.032. Tumor behavior for the three different regions of c are demonstrated in the following figures.

Figure 1: c = 8*10^5, a = 1

Figure 1. shows the tumor volume as time progresses (in days) for c < 8:55*10^5 and a = 1. The tumor grows quickly to near carrying capacity and then remains there.

Figure 2: c = 0.02, a = 1

Figure 2. is more interesting. In the region 8:55*10^5 < c < 0:032 there are cycles of tumor growth and abatement. The amplitude and period depend on where c is in the given region. Its thought that this might explain the recurrence of tumors sometimes years after being treated.

Figure 3: 0:032 < c, a = 1

Figure 3. shows tumor progression when c > 0:032. Tumor size goes through damped oscillations leading to a relatively small stable tumor that is not cleared from the body.

Treatment Model

In the treatment model, two parameters, s1 and s2, are added to the system of ODEs. s1 represents treatment with ACI, and s2 with IL-2.


Here the authors explore these treatments alone, and in combination.

ACI only

Through eigenvalue evaluation the authors found that at s1c = 540 there is a transcritical bifurcation. For s1 < s1c the pattern of behavior follows that of the non-treatment model as the antigenicity is increased. When s1c < s1 new behaviors develop.

Figure 4: c = 0:02, a = 1, s1 = 600, s2 = 0

Figure 4. shows tumor size progression when s1, and c are relatively large. Its clear that the tumor has been cleared from the body in this scenario. When s1 is large, and c is small it may be cleared, but it depends on the initial conditions. Starting with small T(0), say T(0) = 1, and small c produces a plot nearly identical to figure 4. However, starting with a large tumor will give a different result.

Figure 5: T(0) = 100, c = 8:55*10^5, a = 1, s1 = 600, s2 = 0

Here the tumor once again grows to near maximum size. This is a result of bistability. Depending on the initial conditions the tumor will either be cleared from the body or grow to carrying capacity.

IL-2 only

The critical value for s2 (IL-2) is about s2c = 6:35*10^7. Below this number we again get behavior that is similar to the non-treatment model. Above this value we get some interesting results.

Figure 6: T(0) = 1, c = :02, a = 1, s1 = 0, s2 = 7*10^7

IL-2 treatment succeeds in destroying the tumor, but effector cell levels grow uncontrollably. This also has harmful affects on the body and is something to be avoided.

ACI IL-2 Combination

The ACI/IL-2 combination is summed up in figure 6 and table 2, on pages 247 and 248 of the paper. There is little new here, the combination of ACI and IL-2 give the same kinds of results that were shown with the cases above. The only difference is the possibility of low doses of IL-2 boosting the effects of ACI.


The non-treatment model showed that low antigenicity tumors will grow to their carrying capacity. This simply means that the immune system doesn't 'see' the tumor and therefore doesn't respond to it. This suggest an avenue of attack against the tumor, train the immune system to see the cancer for what it is. There are vaccines in development for this, but none currently available.

Moderately antigenic tumors show periodic behavior. On the lower end of this scale the amplitude and period of the oscillations are large, with periods sometimes years long. As the antigenicity gets higher the amplitude and period both shorten. Periodic behavior could account for the recurrence of tumors years after being treated.

Highly antigenic tumors exibit damped oscillations, eventually becoming small dormant tumors. In this model even these tumors are never cleared from the body without treatment. A possible real world example of this would be a benign tumor.

The model for ACI therapy gave impressive results. For moderate to high antigenic tumors This treatment completely cleared the cancer. For low antigenic tumors it was dependent on initial conditions. This would imply that the combination of ACI and early detection would be a powerful way to attack tumors.

The IL-2 therapy model wasn't so impressive. To clear the tumor with IL-2 would take so much of it that it would cause effector cells to grow uncontrolled, causing other problems.

The combined ACI/IL-2 would work but only when using small amounts of IL-2 to amplify the effects of ACI.

For information regarding the use of both immunotherapy and chemotherapy, please see APPM4390:Tumor Modeling.

Project Classification

Mathematics Used

This paper talks about two systems of differential equations, one for a non-treatment model, and one for a treatment model, both of which will be non-dimensionalized. Eigenvalues are analyzed in terms of different parameters, as well as the bifurcations resulting from these changes.

Type of Model

This model is measuring changes in number of immune system cells, the tumor size, and IL-2 concentration. This model is never explicitly solved for, but one can think of it as modeling the three previously mentioned populations in continuous time.

Biological System Studied

This paper tumor growth with either non-treatment or with treatment of just IL-2, just ACI, or both IL-2 and ACI in combination. The paper looks into the macro scale system (size of the tumor, concentration of IL-2, etc.) as opposed to a microbiology approach.

Project Classification 2


This model uses ordinary differential equations to model the populations of immune and tumor cells in continuous time. A logistic limiting-growth equation is used to model the rate of change of the tumor cells. Stability analysis is then used to compare the results of non-treatment paths to initial conditions. Additional analysis can be done on the periodic solutions of the tumor mass when the tumor has very low antigenicity, a measure of how different a tumor is from the ‘self’ and by extension, how easy it is for the immune system to recognize the tumor. Parameters such as the immune response influenced the period and the amplitude of the limit cycle. Sensitivity analysis was used to find proper system scaling that accommodated the large changes of variables over short periods of time and gave insight into the role that tumor antigenicity and immune response on the dynamics of the system. Bifurcation analysis indicated the parameters at which tumors became small and stable.

Type of Model

This model is a tumor- immune model that models the interactions between a growing tumor and the immune system. Broadly, this model uses interacting population dynamics in continuous time. It models the dynamics between populations of tumor cells, immune-effector cells and IL-2 concentration, and the external effects of adoptive cellular immunotherapy on the tumor.

Biological System Studied

This model studies the cells in the immune system that can be used to orchestrate an immune response that fights tumors. It specifically focuses on T-cell activation, growth, and differentiation in the presence of a tumor. The model is then used to explore how adoptive cellular immunotherapy can aid the immune system in effectively eradicating tumor cells.

External Links

American Cancer Society-Immunotherapy

National Cancer Institute-Biological Therapy

Immunology Models

Importance of Math Modeling in Immunology

A Biological Look at Immunology

| Hematology/Oncology (Cancer) Approvals & Safety Notifications

| Journal for ImmunoTherapy of Cancer

See Also

APPM4390:Optimal Chemotherapy Strategies

APPM4390:HIV Dynamics


  1. Morgan G., R. Ward, and M. Barton, The contribution of cytotoxic chemotherapy to 5-year survival in adult malignancies. Clin Oncol (R Coll Radiol). 2004 Dec;16(8):549-60.

Additional Citations

This article, "Modeling immunotherapy of the tumor-immune interaction" by Kirschner, Denise, and John C. Panetta (1998) has been cited in many recent papers. One of which is "Viruses as Antitumor Weapons Defining Conditions for Tumor Remission" by Dominik Wodarz (Institute for Advanced Study, Princeton, New Jersey 08540).

  • Wodarz."Viruses as Antitumor Weapons: Defining Conditions for Tumor Remission" Cancer Research 61, 3501-3507, April 15, 2001.

This paper is a mathematical analysis of a more recent type of treatment having to do with utilizing the certain viruses that attack a specified location in the body (in particular, a cancerous tumor). There are so many methods of treating cancer that are very invasive to the host, like chemotherapy, which target the cancer, but also destroy the quality of life of the person who is undergoing the treatment. The hope of the mathematical analysis of this method is to show that it will target cancerous cells and will be a less invasive procedure for the individual. Wodarz cites the article by Kirschner, Denise, and Panetta in his "Results" section, where he briefly mentions that there are now many more mathematical models of tumor growth and inhibition.

  • Bellomo N., and Preziosi L. "Modelling and Mathematical Problems Related to Tumor Evolution and its Interaction with the Immune System." Mathematical and Computer Modelling, 2000
  • LG De Pillis, A Radunskaya - A Mathematical Tumor Model with Immune Resistance and Drug Therapy: an Optimal Control Approach." 18 Apr 2000

Another paper that cites “Modeling immunotherapy of the tumor-immune interaction” by Kirschner and Panetta is “Immune Checkpoint inhibitors: An introduction to the next-generation cancer immunotherapy” by Lee, Gupta, and Sahasranaman. They discuss the limitations in the model, citing that there have been many different math models looking at tumor growth, but only a minority are used (n.p.). The thought is that Kirschner and Panetta’s model does not keep track of a wide enough range of parameters (Lee, Gupta, and Sahasranaman, np). Due to the complexity of the problem it seems that Kirschner and Panetta’s model may be oversimplified. This may be because the authors were writing it in 1998 when information about patients’ specific tumor was possibly less accurate. This would give a higher parameter error, meaning it would be advantageous to use as few as possible.

  • L. Lee, M. Gupta, S. Sahasranaman (2015, September 29). “Immune Checkpoint inhibitors: An introduction to the next‐generation cancer immunotherapy”. Journal of Clinical Pharmacology.

This paper, “Modeling immunotherapy of the tumor-immune interaction” by Kirschner, Denise, and Panetta, John (1998) has been recently cited in the paper “Addressing current challenges in cancer immunotherapy with mathematical and computational modelling” by Konstorum, Vella, Adler, and Laubenbacher(2017). The paper surveys various modelling approaches, assessing them under the major challenges currently facing immunotherapy such as tumor classification, optimal treatment scheduling, and combination therapy designs. The Panetta-Kirschner model is the first model explored in the paper. It describes the applications of optimal control theory to the model and solutions that minimized tumor mass and therapy administration, and maximized effector cell and IL-2 concentration.

  • Konstorum, A., Vella, A., Adler, A., & Laubenbacher, R. C. (2017). Addressing current challenges in cancer immunotherapy with mathematical and computational modelling. J R Soc Interface.