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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.

External Links

American Cancer Society-Immunotherapy

National Cancer Institute-Biological Therapy

Immunology Models


  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