May 20, 2018, Sunday

# MBW:Tumor Modeling

Article review and some more ideas by Jason Hammond, Michael Kochen, and Sekson Sirisubtawee

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

## Background

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 newer 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.a

An extension we make is to use chemotherapy in addition to the immunotherapy in an attempt to find a more efficient technique of erraticating the tumor cells.

## Non-Treatment Model

Variables

• 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.

Variables

The parameters that were used in the paper as well as by us in our analysis can be seen on this table:

ODEs

$\frac {dT}{dt}}=r_{{2}}T(1-bT)-{\frac {aET}{g_{{2}}+2T}$

${\frac {dI_{{L}}}{dt}}={\frac {p_{{2}}ET}{g_{{3}}+T}}-\mu _{{3}}I_{{L}}$

$\sum _{{i=0}}^{\infty }2^{{-i}}$

Parameters

• 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.

Examples with no treatment

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: $c=5*10^{{-5}},/,a=1$

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}} 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, $s_{1}$ and $s_{2}$, are added to the system of ODEs. $s_{1}$ represents treatment with ACI, and $s_{2}$ with IL-2.

${\frac {dE}{dt}}=cT-\mu _{{2}}E+{\frac {p_{{1}}EI_{{L}}}{g1+I_{{L}}}}+s_{1}$

${\frac {dT}{dt}}=r_{{2}}T(1-bT)-{\frac {aET}{g_{{2}}+T}}$

${\frac {dI_{{L}}}{dt}}={\frac {p_{{2}}ET}{g_{{3}}+T}}-\mu _{{3}}I_{{L}}+s_{2}$

Here the authors explore these treatments alone, and in combination. They employ the use of linear stability analysis in which the Jacobian matrix is given by:

$J(E,T,I_{{L}})=\left[{\begin{array}{ccc}{\frac {I_{{L}}p_{{1}}}{g_{1}+I_{{L}}}}-\mu _{{2}}&c&-{\frac {EI_{{L}}p_{{1}}}{(g_{{1}}+I_{{L}})^{{2}}}}+{\frac {Ep_{{1}}}{g_{{1}}+I_{{L}}}}\\-{\frac {aT}{g_{{2}}+T}}&-br_{{2}}T+{\frac {aET}{(g_{{2}}+T)^{{2}}}}-{\frac {aE}{g_{{2}}+T}}+r_{{2}}(1-bT)&0\\{\frac {p_{{2}}T}{g_{{3}}+T}}&-{\frac {Ep_{{2}}T}{(g_{{3}}+T)^{{2}}}}+{\frac {Ep_{{2}}}{g_{{3}}+T}}&-\mu _{{3}}\end{array}}\right]$

ACI only $s_{1}>0$ and $s_{2}=0$

Through linear stability analysis the authors found that at $s_{{1,crit}}=540$ there is a transcritical bifurcation. To do this they first find the equilibrium with a zero value for the tumor, which is:

$(E^{{*}},T^{{*}},I_{{L}}^{{*}})=({\frac {s_{{1}}}{\mu _{{2}}}},0,0)$

The Jacobian for this equilibrium is given by:

$J({\frac {s_{{1}}}{\mu _{{2}}}},0,0)=\left[{\begin{array}{ccc}-\mu _{{2}}&c&{\frac {s_{{1}}p_{{1}}}{g_{{1}}\mu _{{2}}}}\\0&r_{{2}}-{\frac {as_{{1}}}{g_{{2}}\mu _{{2}}}}&0\\0&{\frac {s_{{1}}p_{{2}}}{g_{{3}}\mu _{{2}}}}&-\mu _{{3}}\end{array}}\right]$

and its eigenvalues are $\lambda _{{1}}=r_{{2}}-{\frac {as_{{1}}}{g_{{2}}\mu _{{2}}}},\,\lambda _{{2}}=-\mu _{{2}},\,\lambda _{{3}}=-\mu _{{3}}$

We can see that the bifurcation point is then found by setting $\lambda _{1}=0$, which gives us that

$s_{{1,crit}}={\frac {r_{{2}}g_{{2}}\mu _{{2}}}{a}}$

and by looking at the eigenvalues we can see that the equilibrium is locally stable if $s_{1}>s_{{1,crit}}$ and unstable if $s_{1}.

With more analysis the authors created the following bifurcation diagram from page 244 of the paper:

In this figure the authors depict five regions. They describe how the solutions behave in each of the five regions. I have provided the next plot to illustrate examples in 4 of the 5 regions.

Figure 5: I: Shows a Stable node $(E^{{*}},T^{{*}},0)$ where the Tumor survives, II: Converges to a limit cycle, III: A stable spiral node with a surviving tumor, V: The tumor is cleared, and the immune system survives
Figure: shows the Tumor behavior for region V

IL-2 only

The authors did linear stability analysis again with $s_{1}=0$ and $s_{2}>0$ To do this they first find the equilibrium with a zero value for the tumor, which is:

$(E^{{*}},T^{{*}},I_{{L}}^{{*}})=(0,0,{\frac {s_{{2}}}{\mu _{{3}}}})$

The Jacobian for this equilibrium is given by:

$J(0,0,{\frac {s_{{2}}}{\mu _{{3}}}})=\left[{\begin{array}{ccc}-\mu _{{2}}+{\frac {p_{{1}}s_{{2}}}{(g_{{1}}\mu _{{3}}+s_{{2}})}}&c&0\\0&r_{{2}}&0\\0&0&-\mu _{{3}}\end{array}}\right]$

and its eigenvalues are $\lambda _{{1}}=r_{{2}},\,\lambda _{{2}}=-\mu _{{2}}-{\frac {p_{{1}}s_{{2}}}{g_{{1}}\mu _{{3}}+s_{{2}}}},\,\lambda _{{3}}=-\mu _{{3}}$

We can see that the bifurcation point is then found by setting $\lambda _{2}=0$, which gives us that

$s_{{2,crit}}={\frac {\mu _{{2}}\mu _{{3}}g_{{1}}}{p_{{1}}-\mu _{{2}}}}=6.35*10^{7}$

and by looking at the eigenvalues we can see that the equilibrium is always unstable since $\lambda _{1}>0$. Below the critical number, we see that if $s_{2}$ is small, then behavior is similar to the non-treatment model. Above the critical value the IL-2 treatment succeeds in destroying the tumor, but effector cell levels grow uncontrollably. This also has harmful effects on the body and is something to be avoided.

With more analysis the authors created the following bifurcation diagram from page 244 of the paper:

In this figure the authors depict four regions and describe how the solutions behave in each of the regions. I have provided the next plot to illustrate examples in all of the regions.

I: Tumor grows to near carrying capacity. The low amount of s2 as well as low antigenicity have little effect. II: Converges to a stable limit cycle, III: A stable spiral node with a surviving tumor, IV: The tumor is cleared, and the immune system is driven to $\infty$

ACI IL-2 Combination The authors did linear stability analysis again with $s_{1}>0$ and $s_{2}>0$ To do this they first find the equilibrium with a zero value for the tumor, which is:

$(E^{{*}},T^{{*}},I_{{L}}^{{*}})=({\frac {s_{{1}}(\mu _{{3}}g_{{1}}+s_{{2}})}{\mu _{{2}}(\mu _{{3}}g_{{1}}+s_{{2}})-s_{{2}}p_{{1}}}},0,{\frac {s_{{2}}}{\mu _{{3}}}})$

When they required negative eigenvalues of the Jacobian for this equilibrium they obtained the following conditions for stability:

$s_{{2}}<{\frac {\mu _{{2}}\mu _{{3}}g_{{1}}}{p_{{1}}-\mu _{{2}}}}=s_{{2,crit}}=63492063$

and

$s_{{1}}>{\frac {g_{{2}}r_{{2}}}{a}}\left[{\frac {s_{{2}}(\mu _{{2}}-p_{{1}})+\mu _{{2}}\mu _{{3}}g_{{1}}}{\mu _{{3}}g_{{1}}+s_{{2}}}}\right]$

The next figure was created using the two conditions on $s_{1}$ and $s_{2}$

In this figure we replicate the plots in figure 6 in the paper on page 248. The solutions in each region are given in the paper. I have provided the next plot to illustrate examples in all of the regions.

II: The solution tends to a stable spiral node. III: Converges to a cancer free state IV: Converges to a cancer free state, but immune system is growing without bound.

The types of solutions with ACI/IL-2 in combination are described 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.

## Interpretation/Analysis from the paper

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.

## Our New Model (With Chemotherapy)

We now propose a new therapy which combines chemotherapy with the immunotherapy that was described in the paper (for a detailed discussion of optimal chemotherapy treatment see: MBW:Optimal Chemotherapy Strategies). The chemotherapy adds one more equation to the system as well as a few terms. The differential equation for $u$ represents the rate at which the chemotherapy drug is administered as well as the rate at which the body naturally clears the drug. The terms added to the Effector cell and Tumor cell equations represent the annihilation of the tumor and effector cells by the drug. We derived our idea by seeing the model given in [3] The system we suggest to model this reaction is the following:

${\frac {dE}{dt}}=cT-\mu _{{2}}E+{\frac {p_{{1}}EI_{{L}}}{g1+I_{{L}}}}+s_{1}-a_{{1}}(1-e^{{-u}})E$

${\frac {dT}{dt}}=r_{{2}}T(1-bT)-{\frac {aET}{g_{{2}}+T}}-a_{{2}}(1-e^{{-u}})T$

${\frac {dI_{{L}}}{dt}}={\frac {p_{{2}}ET}{g_{{3}}+T}}-\mu _{{3}}I_{{L}}+s_{2}$

${\frac {du}{dt}}=v(t)-d_{{2}}u$

For the purposes of this lab we only analyze the case when $v(t)=d_{1}$, a constant. We realize that this is realistically not feasible since it represents a constant supply of the drug over the whole time. However we hope to obtain results that we could use to determine a good scheme for pulsed dosing( administering chemotherapy for 4-12hrs biweekly or monthly)

Just Chemo: $s_{1}=0,\,s_{2}=0$

In this case we find the effects of administering chemotherapy while $s_{1}=0$ and $s_{2}=0$. In this case the only equilibrium we obtain with no tumor is the trivial equilibrium: $(E^{{*}},T^{{*}},I_{{L}}^{{*}},u^{{*}})=(0,0,0,{\frac {d_{{1}}}{d_{{2}}}})$

Through experimentation we notice an interesting relationship between the behavior of the solutions and the values of $s_{1}$ and $d_{1}$. With more time we would attempt to make a bifucation diagram between the two variables. With $d_{1}=0$ we are just at the non-treatment case in which there were two bifurcation points in $c$. One, $c_{0}$, at which solutions change from tending to a stable nonzero equilibrium to tending to a stable limit cycle, and the other, $c_{1}$, at which solutions go from tending to stable limit cycles to tending to stable spiral nodes. We noticed through experimentation that as we add more $d_{1}$ to the system the $c_{1}$ bifurcation point decreases. And there is a bifurcation at $d_{1}=0.446$ at which solutions go from tending to a nontrivial to steady state to tending to the trivial one. The next figure shows some example solutions for various combinations of parameters.

Figure: I: Shows with no drug a stable limit cycle arises. II: With some drug a stable spiral node is obtained but still have the tumor. III: The trivial node becomes stable and everything goes to zero(i.e. dead patient)

Chemo with Immunotherapy: $s_{1}>0,\,s_{2}>0$

We did linear stability analysis again with s1 > 0 and s2 > 0 To do this we first found the equilibrium with a zero value for the tumor, which is

$(E^{{*}},T^{{*}},I_{{L}}^{{*}},u^{{*}})=(s_{{1}}(g_{{1}}\mu _{{3}}+s_{{2}})/(g_{{1}}\mu _{{2}}\mu _{{3}}+s_{{2}}\mu _{{2}}+a_{{1}}(1-e^{{-d_{{1}}/d_{{2}}}})(g_{{1}}\mu _{{3}}+s_{{2}})-p_{{1}}s_{{2}}),\,0,\,s_{{2}}/\mu _{{3}},\,d1/d2)$

Here is the Jacobian matrix of the system :

$J(E,T,I_{{L}},u)=\left[{\begin{array}{cccc}-a_{{1}}(1-e^{{-u}})+{\frac {I_{{L}}p_{{1}}}{g_{{1}}+I_{{L}}}}-\mu _{{2}}&c&-{\frac {EI_{{L}}p_{{1}}}{(g_{{1}}+I_{{L}})^{{2}}}}+{\frac {Ep_{{1}}}{g_{{1}}+I_{{L}}}}&-a_{{1}}e^{{-u}}E\\-{\frac {aT}{g_{{2}}+T}}&-a_{{2}}(1-e^{{-u}})-br_{{2}}T+{\frac {aET}{(g_{{2}}+T)^{{2}}}}-{\frac {aE}{g_{{2}}+T}}+r_{{2}}(1-bT)&0&-a_{{2}}e^{{-u}}T\\{\frac {p_{{2}}T}{g_{{3}}+T}}&-{\frac {Ep_{{2}}T}{(g_{{3}}+T)^{{2}}}}+{\frac {Ep_{{2}}}{g_{{3}}+T}}&-\mu _{{3}}&0\\0&0&0&-d_{{2}}\end{array}}\right]$

and its eigenvalues are

$\lambda _{{1}}=d_{{2}},\,\lambda _{{2}}=-a_{{1}}\left(1-e^{{-d_{{1}}/d_{{2}}}}\right)-\mu _{{2}}+{\frac {p_{{1}}s_{{2}}}{\mu _{{3}}\left(g_{{1}}+{\frac {s_{{2}}}{\mu _{{3}}}}\right)}},\,\lambda _{{3}}=-\mu _{{3}},\,$

$\lambda _{{4}}=-a_{{2}}\left(1-e^{{-d_{{1}}/d_{{2}}}}\right)+r_{{2}}-{\frac {as_{{1}}\left(s_{{2}}+g_{{1}}\mu _{{3}}\right)}{g_{{2}}\left(-p_{{1}}s_{{2}}+s_{{2}}\mu _{{2}}+g_{{1}}\mu _{{2}}\mu _{{3}}+a_{{1}}\left(1-e^{{-d_{{1}}/d_{{2}}}}\right)\left(s_{{2}}+g_{{1}}\mu _{{3}}\right)\right)}}$

When we required negative eigenvalues of the Jacobian for this equilibrium we obtained the following conditions for stability:

If $d_{{1}}\lessgtr 0.269515,$ then $s_{{2}}\lessgtr -{\frac {g_{{1}}\mu _{{3}}\left(-a_{{1}}+a_{{1}}e^{{d_{{1}}/d_{{2}}}}+e^{{d_{{1}}/d_{{2}}}}\mu _{{2}}\right)}{-a_{{1}}+a_{{1}}e^{{d_{{1}}/d_{{2}}}}-e^{{d_{{1}}/d_{{2}}}}p_{{1}}+e^{{d_{{1}}/d_{{2}}}}\mu _{{2}}}}$ and$s_{{1}}>-{\frac {g_{{2}}\left(a_{{2}}\left(1-e^{{-d_{{1}}/d_{{2}}}}\right)-r_{{2}}\right)\left(-p_{{1}}s_{{2}}+s_{{2}}\mu _{{2}}+g_{{1}}\mu _{{2}}\mu _{{3}}+a_{{1}}\left(1-e^{{-d_{{1}}/d_{{2}}}}\right)\left(s_{{2}}+g_{{1}}\mu _{{3}}\right)\right)}{a(s_{{2}}+g_{{1}}\mu _{{3}})}}$ .

The next two figures show the conditions on $d_{{1}}$,$s_{{2}}$ ,$s_{{1}}$ and for stability.

The next figure shows sample solutions for each of the regions.

Figure: I:the solution tends to $(\infty ,\,0,\,I_{{L}}^{{*}})$ , II up and III up: Stable node $(E^{{*}},\,0,\,I_{{L}}^{{*}})$, and tumor is eliminated , II down: Stable spiral and tumor survives, III down : Stable node, and tumor survives

Special case with Chemo: $s_{1}>0,\,s_{2}=0$

The next picture shows the condition on $d_{1}$ and $s_{1}$ for stability. The curve was found by setting the eigenvalues of the Jacobian less than zero and obtaining the following condition for stability:

$s_{{1}}>-e^{{-2d_{{1}}/d_{{2}}}}g_{{2}}\left(-a_{{2}}+a_{{2}}e^{{d_{{1}}/d_{{2}}}}-r_{{2}}e^{{d_{{1}}/d_{{2}}}}\right)\left(-a_{{1}}+a_{{1}}e^{{d_{{1}}/d_{{2}}}}+\mu _{{2}}e^{{d_{{1}}/d_{{2}}}}\right)$

Notice that for any $d_{1}$>0.446287, then any solution tends to $(0,0,0,d_{1}/d_{2})$.

## Project categorization

### Mathematics

This model uses a system of three ordinary differential equations to model cancer development and treatment and steady state analyses are carried through by studying eigenvalues in different bifurcation regions of the system.

### Model type

The system of ODEs are actually modeling three different magnitudes, that is bulk tumor size, number of immune system cells, and IL-2 (a protein) concentration. By this, the model is a combined system which looks at input on one parameter (cells) and output of two others (tumor size and IL-2 concentration). In general, since tumor size can be measured by the number of the cells, this system can be characterized as a cell population model.

### Biological system

The biological system studied is cancer cells and different treatments (cells and chemotherapeutics) in a mammalian body environment.