ARTICLE IN PRESS

0022-5193/$ - se

doi:10.1016/j.jtb

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Journal of Theoretical Biology 238 (2006) 841–862

www.elsevier.com/locate/yjtbi

Mixed immunotherapy and chemotherapy of tumors: modeling,applications and biological interpretations

L.G. de Pillisa,�, W. Gua, A.E. Radunskayab

aHarvey Mudd College, Claremont, CA 91711, USAbPomona College, Claremont, CA 91711, USA

Received 28 October 2004; received in revised form 31 May 2005; accepted 28 June 2005

Available online 8 September 2005

Abstract

We develop and analyze a mathematical model, in the form of a system of ordinary differential equations (ODEs), governing

cancer growth on a cell population level with combination immune, vaccine and chemotherapy treatments. We characterize the

ODE system dynamics by locating equilibrium points, determining stability properties, performing a bifurcation analysis, and

identifying basins of attraction. These system characteristics are useful not only to gain a broad understanding of the specific system

dynamics, but also to help guide the development of combination therapies. Numerical simulations of mixed chemo-immuno and

vaccine therapy using both mouse and human parameters are presented. We illustrate situations for which neither chemotherapy nor

immunotherapy alone are sufficient to control tumor growth, but in combination the therapies are able to eliminate the entire

tumor.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Mathematical modeling; Chemotherapy; Immunotherapy; Vaccine; Cancer; Tumor

1. Introduction and background

1.1. The importance of the immune system and

immunotherapy

Immunotherapies are quickly becoming an importantcomponent in the multi-pronged approaches beingdeveloped to treat certain forms of cancer. The goal ofimmunotherapy is to strengthen the body’s own naturalability to combat cancer by enhancing the effectiveness ofthe immune system. The importance of the immunesystem in fighting cancer has been verified in thelaboratory as well as with clinical experiments. See, forexample, Farrar et al. (1999), O’Byrne et al. (2000),Morecki et al. (1996), Muller et al. (1998), Stewart (1996).Additionally, it is known that those with weakenedimmune systems, such as those suffering from AIDS, are

e front matter r 2005 Elsevier Ltd. All rights reserved.

i.2005.06.037

ing author. Tel.: +909 621 8975; fax: +909 621 8366.

esses: depillis@hmc.edu (L.G. de Pillis), gu@math.

u), aradunskaya@pomona.edu (A.E. Radunskaya).

more likely to contract certain rare forms of cancer. Thisphenomenon can be interpreted as providing furtherevidence that the role played by the immune response inbattling cancer is critical. See, for example, Dalgleish andO’Byrne (2002), Cancer Research Institute (2000).

Through the mathematical modeling of tumorgrowth, the presence of an immune component hasbeen shown to be essential for producing clinicallyobserved phenomena such as tumor dormancy, oscilla-tions in tumor size, and spontaneous tumor regression.The mathematical modeling of the entire immunesystem can be an enormously intricate task, as demon-strated in Perelson and Weisbuch (1997), so models thatdescribe the immune system response to a tumorchallenge must necessarily focus on those elements ofthe immune system that are known to be significant incontrolling tumor growth. In the work of de Boer andHogeweg (1986), a mathematical model of the cellularimmune response was used to investigate such animmune reaction to tumors. It was found that initiallysmall doses of antigens do lead to tumor dormancy. The

ARTICLE IN PRESSL.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862842

mathematical model of Kirschner and Panetta (1998),which also focuses on the tumor–immune interaction,indicates that the dynamics among tumor cells, immunecells, and the cytokine interleukin-2 (IL-2) can explainboth short-term oscillations in tumor size as well aslong-term tumor relapse. The model developed byKuznetsov (Kuznetsov and Makalkin, 1992; Kuznetsovet al., 1994), in which the nonlinear dynamics ofimmunogenic tumors are examined, also exhibitsoscillatory growth patterns in tumors, as well asdormancy and ‘‘creeping through’’: when the tumorstays very small for a relatively long period of time, andsubsequently grows to be dangerously large. In thesemathematical models, the cyclical behavior of the tumoris directly attributable to the interaction of the tumorwith the immune system. There is a host of literaturethat addresses the development of various mathematicaldescriptions of cancer and the immune response, amongwhich we mention just a few: (Bellomo et al., 2004;Bellomo and Preziosi, 2000; Chandawarkar and Guy-ton, 2002; Chester et al., 2000; Derbel, 2004; De Angelisand Jabin, 2003; Cui, 2002; Kolev, 2003; Dalgleish,1999; De Angelis and Mesin, 2001; Delitala, 2002;Ferreira et al., 2002; Friedrich et al., 2002; Garcia-Penarrubia et al., 2002; Gatenby and Gawlinski, 2003;Gatenby and Maini, 2002; Hardy and Stark, 2002; Keilet al., 2001; Kuznetsov and Knott, 2001; Lucia andMaino, 2002; Marincola et al., 2003; Nani and Freed-man, 2000; Owen and Sherratt, 1999; Sharma et al.,2003; Sotolongo-Costa et al., 2003; Stengel et al., 2002;Takayanagi and Ohuchi, 2001; Wallace et al., 2003;Webb et al., 2002; Wein et al., 2003; Wodarz, 2001;Wodarz and Jansen, 2003). In de Pillis and Radunskaya(2001) we also demonstrate the critical role the immunesystem plays in giving rise to cyclical behavior. Thismodel also highlights the crucial role the immune systemplays in the process of tumor elimination.

The clear importance of the immune system incontrolling cancer growth, both clinically and mathema-tically, indicates that models incorporating tumor growthand treatment would do well to include an immunesystem component. Once this component is in place, it isthen possible to model how various immunotherapiesmay affect the system, either singly or in combinationwith one another. Recent clinical data have shown thereis potential benefit in harnessing the power of the immunesystem in combination with traditional chemotherapy.For example, in Wheeler et al. (2004), it is demonstratedthat vaccine therapy in combination with chemotherapymore effectively extends patient survival times than eitherchemotherapy or vaccine therapy alone.

1.2. Immunotherapy

The clinical evidence for the potential of immunesystem control of certain malignancies has motivated new

research into the development of immunotherapies andvaccine therapies for cancers (see, for example Blattmanand Greenberg, 2004; Couzin, 2002; Donnelly, 2003;Pardoll, 1998; Rosenberg et al., 2004; Wheeler et al.,2004). Immunotherapy falls into three main categories:immune response modifiers, monoclonal antibodies, andvaccines (see, for example, Rosenbaum and Rosenbaum,2005). The first category contains substances that affectimmune response, such as interleukins (including IL-2),interferons, tumor necrosis factors (TNF), colony-stimu-lating factors (CSF), and B-cell growth factors. In thenext category, monoclonal antibodies are currently beingdeveloped to target specific cancer antigens. Thesemonoclonals can distinguish between normal and cancercells, and they can then be used to diagnose cancer, aswell as to treat tumors by ‘‘guiding’’ anticancer drugstoward the malignant cells (see, e.g., Qu et al., 2005;Hadj, 2005; Li et al., 2005). In the third category arevaccines, which are generally used therapeutically, andare created from tumor cells. These work by helping theimmune system to recognize and attack specific cancercells. In this work, we implement treatment from the firstcategory in the form of mathematical terms that representIL-2 and tumor infiltrating lymphocyte (TIL) injections,and additionally incorporate treatment from the thirdcategory: new mathematical forms that distinguishbetween specific and nonspecific immune responses,allowing for the incorporation of a vaccine componentinto the model. Although monoclonal antibody treat-ments are considered promising, they are currently notconsidered in this work.

1.3. Cancer vaccines

There are fundamental differences between the useand effects of antiviral vaccines and anticancer vaccines.While many vaccines for infectious diseases are pre-ventative, cancer vaccines are designed to be usedtherapeutically, treating the disease after it has begun,and preventing the disease from recurring. Cancervaccines are still considered to be highly experimentalas compared with other forms of cancer immunother-apy, but early clinical trials are showing increasingpromise in their ability to improve the immune responseto certain forms of cancer (see, e.g., Rosenbaum andRosenbaum, 2005; Wheeler et al., 2004).

Since cancer vaccines and antiviral vaccines differ intheir application, mathematical models of these vaccinesshould exhibit different dynamics. The goal of this paperis to build on existing models of tumor growth,incorporating an immune system response and expand-ing these models to include the effect of anti-tumorvaccination and immunotherapies in conjunction withchemotherapies. In another work, the authors willextend this model into a larger framework thatincorporates spatial and geometric components.

ARTICLE IN PRESSL.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862 843

The outline of this paper is as follows. In Section 2, wepresent the set of assumptions incorporated into ourtumor growth model, and develop the system of ODEsthat form the mathematical model. In Section 3, wediscuss how parameter values for the model are chosen.In Section 4, we carry out an analysis of the non-dimensionalized model, finding equilibrium points,stability criteria, bifurcation points, and basins ofattraction. Section 5 presents numerical experi-ments based on parameters representing data collectedfrom laboratory mice, and includes contrasts betweenimmunotherapies and chemotherapies. In Section 6,we present numerical experiments that reflectparameter values gathered from human clinical trials,again contrasting immunotherapies with chemothera-pies, and simulating combinations of these. In Section 7,we present theoretical simulations highlightingthe potential effect of vaccine therapy on a patient, aswell as vaccine therapy in combination with chemother-apy. In Section 8, we summarize and discuss ourconclusions.

2. Model formulation

One goal for a mathematical model is to allow forsufficient complexity so that the model will qualitativelygenerate clinically observed in vivo tumor growthpatterns, while it simultaneously maintains sufficientsimplicity to admit analysis. The model we present inthis paper can exhibit the following behaviors in theabsence of medical interventions:

(1)

Tumor dormancy and sneaking through. There isclinical evidence that a tumor mass may disappear,or at least become no longer detectable, and then forreasons not yet fully understood, may reappear,growing to lethal size.

(2)

Uncontrolled growth of tumor cells due to acceler-ated growth rates.

(3)

Detrimental effects of immune cells on tumor cellsthrough normal immune function.

(4)

Inactivation of cytotoxic immune cells throughinteraction with tumor cells.

(5)

Global stimulatory effect of tumor cells on theimmune response.

(6)

Both nonspecific and specific immune responses tothe presence of tumor cells.

Additionally, since we wish to determine improvedcombination therapy treatment protocols, we includethe following mathematical model components thatrepresent tumor response to medical interventions.

(1)

System response to chemotherapy (direct cytotoxiceffects on tumor and immune cell populations).

(2)

System response to direct immunotherapy (such asIL-2 and TIL injections).

(3)

System response to vaccine therapy (including directstimulatory effects on the immune system throughinjection of modified autologous tumor cells).

The model is a system of ordinary differential equationswhose state variables are populations of tumor cells,specific and nonspecific immune cells, and concentra-tions of therapeutic interventions. The assumptions thatwere used to determine the model equations are outlinedbelow, followed by a discussion of the model equationsthemselves.

2.1. Model assumptions

The ODE model is based on that originally developedby de Pillis and Radunskaya (2003b), and is nowextended by adding new cell interaction terms as well asterms describing chemotherapy and immunotherapy.For the sake of completeness, we outline the assump-tions of the original model here. We should note thatthere is not universal agreement as to the underlyingdynamics or precise cascades of events that take place inthe immune response process. Our assumptions, there-fore, are based on published statements and conjecturesas well as reasonable suppositions.

�

A tumor grows logistically in the absence of animmune response. This is one accepted growth modelfor tumors (Britton, 2003), and is also based onfittings of the data in Diefenbach et al. (2001). � Both NK and CD8þT cells are capable of killing

tumor cells (see, for example, Diefenbach et al., 2001;Kawarada et al., 2001; Germain, 2004).

� Both NK and CD8þT cells respond to tumor cells by

expanding and increasing cytolytic activity (see, forexample, Osada et al., 2004; Kieper et al., 2001).

� NK cells are normally present in the body, even when

no tumor cells are present, since they are part of theinnate immune response (see, for example, Roittet al., 1993).

� As part of the specific immune response, active tumor-

specific CD8þT cells are only present in large numberswhen tumor cells are present (see, for example, Roittet al., 1993; Kirschner and Panetta, 1998)

� NK and CD8þT cells become inactive after some

number of encounters with tumor cells (see, forexample, Adam and Bellomo, 1997).

The following additional assumptions are used in thedevelopment of therapeutic terms.

�

Circulating lymphocyte levels can be used as ameasure of patient health (see, e.g., Mustafa et al.,1998; Melichar et al., 2001; Glas et al., 2000).

ARTICLE IN PRESSL.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862844

Components of the lymphocyte population areinvolved in the production of antibody and instimulating B-cells. NK cells kill cells to whichantibody has attached (in addition to killing cellslacking MHC-I surface molecules). In this model, thesource of the NK cell population is represented as afraction of the circulating lymphocyte population, asimplification meant to represent the complex cascadeof biological events that leads to NK cell stimulation(see, e.g., Cooper et al., 2001).

� The fraction of the tumor population killed by

chemotherapy depends on the amount of drug inthe system. The fraction killed has a maximum lessthan one, since only tumor cells in certain stages ofdevelopment can be killed by chemotherapy (Pazduret al., 2004).

� A fraction of NK cells, CD8þT cells, and circulating

lymphocytes are also killed by chemotherapy, ac-cording to a similar fractional kill curve (Gardner,2000).

� NK and T cells are components of the process of

stimulation and elimination of activated effector cells, amodel simplification meant to reflect the self-regulatorynature of the immune system (see, e.g., Dudley et al.,2002; Germain, 2004; Jiang and Chess, 2004).

2.2. ODE model: general form of equations

The model describes the kinetics of four populations(tumor cells and three types of immune cells), as well astwo drug concentrations in the bloodstream, using aseries of coupled ordinary differential equations basedon the model developed by de Pillis and Radunskaya(2003b). The populations at time t are denoted by:

�

TðtÞ, tumor cell population, � NðtÞ, total NK cell population, � LðtÞ, total CD8þT cell population, � CðtÞ, number of circulating lymphocytes (or white

blood cells),

� MðtÞ, chemotherapy drug concentration in the blood-

stream,

� IðtÞ, immunotherapy drug concentration in the

bloodstream.

The equations governing the population kinetics musttake into account a net growth term for each population(GT , GN , GL, GC , GM , GI ), the fractional cell kill (FN ,FL, FMT , F MN , FML, FLI , F CM ), per cell recruitment(RN ;RL), cell inactivation (IN ; IL) and external inter-vention with medication (HL;HM ;HI ). We attempt touse the simplest expressions for each term that stillaccurately reflect experimental data and populationinteractions.

Note that the population levels of NK and CD8þTcells actually represent immune cell ‘‘effectiveness’’ in

the sense that an increase in the total population countin the mathematical model may actually be evidencedbiologically either as a greater number of total cells, oras each individual immune cell becoming more efficientat killing the target tumor cells.

The functional forms that we choose for each cell-interaction term are discussed below. We have chosen torepresent parameters by lower case letters alwaysassumed to be positive, and state variables or functionsof state variables by upper case letters.

2.2.1. Growth and death terms

We adapt the growth terms for tumor and CD8þTcells from de Pillis and Radunskaya (2003b). Tumorgrowth is assumed to be logistic, based on data gatheredfrom immunodeficient mice (Diefenbach et al., 2001).Therefore GT ¼ aTð1� bTÞ. Cell growth for CD8þTcells consists only of natural death rates, since noCD8þT cells are assumed to be present in the absence oftumor cells, so GL ¼ �mL.

The model in de Pillis and Radunskaya (2003b)assumed a constant NK cell production rate. In the newmodel, this growth term is tied to the overall immunehealth levels as measured by the population of circulat-ing lymphocytes. This allows for the suppression of stemcells during chemotherapy, which lowers circulatinglymphocyte counts and affects the production rate ofNK cells. Therefore, GN ¼ eC � fN. We assume thatcirculating lymphocytes are generated at a constant rate,and that each cell has a natural lifespan. This gives usthe term GC ¼ a� bC. We assume that the chemother-apy drug, after injection, will be eliminated from thebody over time at a rate proportional to its concentra-tion, giving an exponential decay: GM ¼ �gM. Simi-larly, we assume the immunotherapy drug, IL-2, decaysexponentially: GI ¼ �mI I .

2.2.2. Fractional cell kill terms

The fractional cell kill terms for N and L are takenfrom de Pillis and Radunskaya (2003b). These fractionalcell kill terms represent negative interactions betweentwo populations. They can represent competition forspace and nutrients as well as regulatory action anddirect cell population interaction. The interactionbetween tumor and NK cells takes the formF NðT ;NÞ ¼ �cNT . Tumor lysis by CD8þT cells, onthe other hand, has the form:

F LðT ;LÞ ¼ dðL=TÞl

sþ ðL=TÞlT .

Letting

DðT ;LÞ ¼ dðL=TÞl

sþ ðL=TÞl

we have F LðT ;LÞ ¼ DT .

ARTICLE IN PRESSL.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862 845

The model includes a chemotherapy drug kill term ineach of the cell population equations. Certainchemotherapeutic drugs, such as doxorubicin, areonly effective during certain phases of the cell cycle,and pharmacokinetics also indicate that the effectivenessof chemotherapy is bounded. We therefore use asaturation term 1� e�M to represent the chemotherapyfractional cell kill. Note that at relatively low concen-trations of drug, the kill rate is nearly linear, whileat higher drug concentration, the kill rate plateaus.The mathematical term we use reflects the dose–response curves suggested by the literature (Gardner,2000). We therefore let FMf ¼ Kfð1� e�MÞf, forf ¼ T ;N ;L;C.

Additionally, we include CD8þT activation by IL-2immunotherapy. This ‘‘drug’’ is actually a naturallyoccurring cytokine in the human body, and its effect onthe immune system’s efficacy is described mathemati-cally with a Michaelis–Menten interaction term in theequation for L. The presence of IL-2 stimulates theproduction of CD8þT cells, so we let FLI ¼

pI LI=ðgI þ IÞ. This is the activation term developed inKirschner’s tumor–immune model (Kirschner andPanetta, 1998).

2.2.3. Recruitment

The recruitment of NK cells takes on the same formas the fractional cell kill term, D; for CD8þT cells, with l

equal to two, as described by de Pillis and Radunskaya(2003b). Hence the NK cell recruitment term is

RNðT ;NÞ ¼ gT2

hþ T2N.

This is a modified Michaelis–Menten term, commonlyused in tumor models to govern cell–cell interactions(Kuznetsov et al., 1994; de Pillis and Radunskaya,2003b; Kirschner and Panetta, 1998).

CD8þT cells are thought to be activated by a numberof triggers, including fragments of tumor cells thathave been lysed by other CD8þT cells (Huang et al.,1994). The CD8þT cell recruitment term has aform similar to that for the NK cell recruitment,except that the tumor population is replaced with thelysed tumor population from the tumor–CD8þT cellinteraction term, DðT ;LÞ. Thus the new recruitmentterm is

RLðT ;LÞ ¼ jD2T2

k þD2T2.

CD8þT cells may also be recruited by the debris fromtumor cells lysed by NK cells (see, e.g., Huang et al.,1994). This recruitment term is proportional to thenumber of cells killed: RLðN;TÞ ¼ r1NT . The immunesystem is also stimulated by the presence of tumor cellsto produce more CD8þT cells. Recognition of thepresence of the tumor is proportional to the average

number of encounters between circulating lymphocytesand the tumor: RLðC;TÞ ¼ r2CT .

2.2.4. Inactivation terms

Inactivation of cytolytic potential occurs when an NKor CD8þT cell has interacted with tumor cells severaltimes and ceases to be effective. We use the inactivationterms developed by de Pillis and Radunskaya (2003b):IN ¼ �pNT and IL ¼ �qLT . The parameters in theinactivation terms represent mean inactivation rates.

The third inactivation term, ICL ¼ �uNL2, describesregulation and suppression of CD8þT cell activity,which occurs when there are very high levels of activatedCD8þT cells without responsiveness to cytokines pre-sent in the system (see, e.g. Gilbertson et al., 1986; Gettet al., 2003). This term comes into play when the amountof CD8þT cells in the body is high, and experimentaldata document that these cells can become rapidlyinactivated even with a tumor present (Rosenberg andLotze, 1986). The cytokine IL-2 aids in the resistance ofthe CD8þT cell population to this inactivation (Antonyand Restifo, 2005).

2.2.5. Drug intervention terms

The TIL drug intervention term, HL, for the CD8þTcell population represents an immunotherapy in whichthe immune cell levels are boosted by the addition ofantigen-specific cytolytic immune cells. HL is a functionof time denoted by vL ¼ vLðtÞ: Similarly, the drugintervention terms in the equations for dM=dt anddI=dt reflect the amount of chemotherapy and immu-notherapy drug given over time. Therefore, drugintervention terms HM and HI are also functions oftime denoted, respectively, by vM ¼ vM ðtÞ and vI ¼ vI ðtÞ.

2.3. ODE model: specific terms in equations

Bringing together the specific forms for each cellgrowth and interaction term leads to the full system ofequations:

dT

dt¼ aTð1� bTÞ � cNT �DT

� KT ð1� e�MÞT , ð1Þ

dN

dt¼ eC � fN þ g

T2

hþ T2N � pNT

� KN ð1� e�MÞN, ð2Þ

dL

dt¼ �mLþ j

D2T2

k þD2T2L� qLT

þ ðr1N þ r2CÞT ð3Þ

� uNL2 � KLð1� e�M ÞL

þpI LI

gI þ Iþ vLðtÞ, ð4Þ

ARTICLE IN PRESSL.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862846

dC

dt¼ a� bC � KCð1� e�MÞC, (5)

dM

dt¼ �gM þ vMðtÞ, (6)

dI

dt¼ �mI I þ vI ðtÞ, (7)

D ¼ dðL=TÞl

sþ ðL=TÞl. (8)

3. Experimental data and parameter derivation

While a general model with non-specific parameters isuseful in studying the qualitative dynamics of cancergrowth, it is also necessary to study parameter setsmeant to reflect specific cancer types (such as melano-ma). System parameters can vary greatly from oneindividual to another, so multiple data sets can be usedin order to obtain acceptable parameter ranges. In thiswork, we use the data made available from both themurine experimental studies of Diefenbach et al. (2001)and the human clinical trials of Dudley et al. (2002).When necessary, we also use previous model parametersthat have been fitted to experimental curves (de Pillis etal., 2003; de Pillis and Radunskaya, 2003a, b; Kuznetsovet al., 1994). Simulations with several sets of parameterswere performed in order to evaluate the qualitativebehavior of the model.1 Parameters which were esti-mated specifically for this investigation are described inthe sections below.

3.1. Chemotherapy parameters

Chemotherapy strength was assumed to be one log-kill,as described in Perry (2001), and the values of the killparameters KT ;KN ;KL were chosen accordingly. Certainchemotherapy drugs, such as doxorubicin, kill cells bydisrupting the process of division and growth. Rapidlydividing cells, like those of the hair, the stomach lining,and the bone marrow where immune cells are produced,are preferentially damaged by chemotherapy (see, e.g.,National Cancer Institute, 2005; Holland and Emil III,1973). High doses of drug can also damage other tissues inthe body (Holland and Emil III, 1973). Ideally, che-motherapy should more effectively kill tumor cells thanimmune cells. Therefore, KN ;KL, and KC are assumed tobe smaller than KT , but of the same order of magnitude.

The drug decay rate, g, was calculated from the drughalf-life and the relation g ¼ ln 2=t1=2. The drug half-life,t1=2, was estimated to be about 18 h, based on

1See Tables 1 and 2 for a full listing all of the parameters with their

units and descriptions.

experimental data for the chemotherapeutic drugdoxorubicin (Calabresi and Schein, 1993).

3.2. Additional CD8þT cell regulation parameters

In addition to chemotherapy, we introduce immu-notherapy into the system of differential equations. Thecytokine IL-2 is known to stimulate CD8þT cellrecruitment and proliferation (see, e.g., Roitt et al.,1993). IL-2 is produced naturally by the body, and isalso administered therapeutically to boost immunesystem function. IL-2 has been used in clinical trialson its own, in combination with chemotherapy, and inconjunction with TIL injections, in which a largenumber of highly activated CD8þT cells are injectedinto the system (Rosenberg and Lotze, 1986; Curti et al.,1996; Hara et al., 1996; Lumsden et al., 1996).

The three parameters, pI , gI , mI , mirror those ofKirschner’s model (Kirschner and Panetta, 1998) withminor alterations to reflect the fact that we are onlyconcerned with the amount of IL-2 that is not naturallyproduced by the immune system. The half-life, ln 2=mI istaken from Kirschner’s model.

The values of r2 and u are chosen to allow forreasonable simulation outcomes of the model.

4. Non-dimensionalization and analysis

To allow for analysis, we now consider the system ofEqs. (1)–(8) in the absence of treatment. Whenchemotherapy and immunotherapy are eliminated, themodel is reduced to a four-population system ofordinary differential equations. To further clarify thedependence of the system on parameters, and toimprove the performance of numerical methods, wenon-dimensionalize the system as follows. Let the non-dimensionalized state variables be

C� ¼a

aC; T� ¼ bT ; N� ¼

a2

aeN,

L� ¼ bL; D� ¼1

aD; t� ¼ at

and the corresponding parameters be

c� ¼cae

a3; d� ¼

d

a; f � ¼

f

a; g� ¼

g

a; h� ¼ hb2,

j� ¼j

a; k� ¼ ðkb2Þ=a; m� ¼

m

a,

p� ¼p

ab; q� ¼

q

ba; r�1 ¼

r1ae

a3; r�2 ¼

r2aa2

,

s� ¼ s; u� ¼uae

a3b; b� ¼

ba.

Leaving the other parameters unchanged, and droppingthe stars for notational clarity, the non-dimensionalized

ARTICLE IN PRESS

0 0.5 1 1.5 2 2.5 3 3.5 4x 107

-3

-2

-1

0

1

2

3

4

5

6

7

x 104

T in number of cells

L in

nu

mb

er o

f ce

lls

L1E(T) on T−nullsurfaceL2E(T) and L3E(T) on L−nullsurface

PositiveEquilibrium

Fig. 1. The nonzero equilibria are the intersections of the graphs of the

functions LðTÞ obtained by finding the equations of the null surfaces.

There is only one nonzero positive equilibrium for the estimated

parameter set.

0 0.5 1 1.5 2 2.5 3 3.5x 10-6

4

100

101

102

103

104

105

106

107

108

Value of the parameter, c

Nu

mb

er o

f T

um

or

Cel

ls a

t E

qu

ilib

riu

m

Bifurcation Diagram for NK−kill rate:bifurcation occurs at c ≈ 1.4 × 10-6

StableUnstable

Tumor−Free Equilibrium isUnstable

Tumor−Free Equilibrium BecomesStable

Fig. 2. Bifurcation diagram showing the effect of varying the NK-kill

rate, c: Other parameters for this diagram are from Table 1.

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862 847

system is given by

dT

dt¼ Tð1� TÞ � cNT �DT , (9)

dN

dt¼ C � fN þ g

T2

hþ T2N � pNT , (10)

dL

dt¼ �mLþ j

D2T2

k þD2T2L

þ ðr1N þ r2CÞT � qLT � uNL2, ð11Þ

dC

dt¼ 1� bC, (12)

D ¼dðL=TÞl

sþ ðL=TÞl. (13)

4.1. Determination of equilibria and their stability

In order to study the equilibria of the system and theirstability, we note first that Eq. (12) decouples from Eqs.(9)–(11), so that at equilibrium, we have CE ¼ 1=b. Thisleaves a system of three equations, (9)–(11), which we setsimultaneously equal to zero in order to find theequilibria.

Eq. (9) has one zero at the ‘‘tumor-free’’ equilibriumat TE ¼ 0, and possibly several non-zero tumorequilibria. The tumor-free equilibrium for all four statevariables is given by ðTE ;NE ;LE ;CEÞ ¼ ð0; 1=bf ; 0; 1=bÞ.Let E0 represent the point ð0; 1=bf ; 0; 1=bÞ.

In the case where TEa0; the equilibria are stilldetermined by finding the simultaneous solutions ofEqs. (9)–(11), but in this case, values of equilibriumpoints for a non-zero tumor must be found numerically.

In particular, setting Eq. (10) to zero and solving forN yields

NE ¼CEðhþ T2Þ

fhþ ðf � gÞT2 þ phT þ pT3. (14)

Similarly, requiring that Eq. (9) equal zero gives

DE ¼ 1� T �NE . (15)

Using this expression in Eq. (13) gives an expressionfor L:

LE1 ¼DEsTl

d �DE

� �1=l

. (16)

Finally, setting Eq. (11) to zero gives

L2ðuNEÞ þ m�jD2

ET2

k þD2ET2þ qT

� �L

� ðr1NE þ r2CEÞT ¼ 0, ð17Þ

which is quadratic in L. Equilibrium points of system(9)–(13) are found by intersecting Eqs. (16) and (17).

Observe from Eqs. (14)–(17) that there can bemultiple non-zero tumor equilibrium points. For theestimated set of mouse parameters, although there aretwo solutions, there is only one positive (biologicallyrelevant) solution to Eqs. (14)–(17). These solutions aregraphed in Fig. 1. As a system parameter is changed,other non-zero equilibria can appear (see Fig. 2), ornegative equilibria can become positive, and thereforebiologically feasible.

Stability of the equilibria are determined by lineariz-ing the system about the calculated values, and bydetermining the stability of the linearized system. Thestability of equilibria is important from a physiologicalviewpoint. If the system is in equilibrium, but the

ARTICLE IN PRESSL.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862848

equilibrium point is unstable, a small perturbation fromequilibrium will cause the system to move away fromthat point and evolve toward the stable equilibrium.This effect is illustrated in Fig. 3. At the estimatedparameter values, the tumor-free equilibrium is un-stable, while the high-tumor equilibrium is stable. Thestability of the high-tumor equilibrium implies that, iftreatment is stopped, the system will inevitably return tothe high-tumor state, i.e. the tumor will escape immunesurveillance unless every single tumor cell is killed. Thus,in a case such as ours for which there are only twoequilibria, if the tumor-free equilibrium is unstable, thenin order to realistically effect a cure, any treatment mustnot only reduce the tumor burden, but it must alsochange the parameters of the system itself. The role ofimmunotherapy, therefore, might be interpreted in thiscontext as a treatment which changes system parametersby, for example, permanently raising the cytolyticpotential of the natural killer cells. We note that if the

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After bifurcation, zero-tumor equilibrium becomes stable

System goes to high tumor equilibriumSystem goes to zero tumor equilibrium

c = 2 × 10-6

c = 1 × 10-6 One Tumor Cell

Same parameter values, slightly different initial tumor values

Fig. 3. Simulations illustrating system behavior for three values of

parameter c. The bifurcation point is c � 1:4� 10�6: Top: After the

bifurcation point, at c ¼ 4� 10�6; the system shows sensitivity to

initial conditions. In this simulation, a one cell difference in the initial

number of tumor cells, T0, results in very different outcomes. Initial

values for the state variables are T0 � 6:89� 106; N0 � 3:97� 109;L0 � 7:15� 103; C0 � 1:65� 109: All parameters besides c are from

Table 1. Bottom: Before the bifurcation, at c ¼ 1� 10�6; one tumor

cell grows to the high-tumor equilibrium of approximately 2:1� 1015

in 60 days. After the bifurcation, at c ¼ 2� 10�6; an initial value of

one tumor cell is drawn to the now stable zero tumor equilibrium.

Immune recruitment parameters, g; j; r1 and r2 are set to zero. With the

exception of the values for c and the immune recruitment parameters,

all other parameters are from Table 1. Initial values for the state

variables are T0 ¼ 1; N0 ¼ 1:73� 1010; L0 ¼ 0; C0 � 1:65� 109.

system were one that admitted a very small but stabletumor, then another ‘‘healthy’’ state might be one forwhich it is possible to maintain the system at this lowtumor level.

4.2. Bifurcation analysis

At the tumor-free equilibrium, E0 ¼ ð0; 1=bf ; 0; 1=bÞ,the Jacobian matrix becomes

1�c

bf0 0

�p

bf�f 0

r1

bfþ

r2

b0 �m

26666664

37777775.

The eigenvalues of the system linearized about thetumor-free equilibrium are therefore:

l1 ¼ 1�c

bf; l2 ¼ �f ; l3 ¼ �m.

Since f ;m40, l2 and l3 are always negative, the tumor-free equilibrium E0 is stable if and only if l1 ¼1� c=bfo03c4bf .

The bifurcation diagram in Fig. 2 shows thedisappearance of two unstable non-zero tumor equili-bria and the stabilization of the tumor-free equilibriumas the parameter c is increased. Recall that theparameter c represents the effectiveness of the NK cellsin lysing the tumor cells. Fig. 2 shows a bifurcationpoint near c ¼ 1:4� 10�6, after which the progression ofthe disease is very sensitive to the initial tumor size. Thissensitivity is shown in Fig. 3, Top, which displays twodifferent scenarios for initial tumor sizes that differ by 1cell. In one case, the tumor grows to the highequilibrium, and in the other case the tumor populationgoes to the zero equilibrium. Fig. 3, Bottom, shows twosimulations. The dotted line illustrates the case in whichc is smaller than the bifurcation value, so that the zeroequilibrium is unstable. In this case, one tumor cell cangrow to a large tumor mass greater than 2� 1015 cells in60 days. However, as illustrated by the solid line, if c islarger than the bifurcation value, the zero tumorequilibrium becomes stable and a single tumor cellwill die.

Similar analyses can be performed using any of thesystem parameters in order to determine conditions forthe appearance or disappearance of equilibria and todetermine equilibrium stability. For example, Fig. 4shows a bifurcation diagram for the CD8þ recruitmentparameter, j. In this case, the mouse parametersare again used, except that all the tumor-specificimmune-cell recruitment terms are initially set tozero (that is, g ¼ r1 ¼ r2 ¼ 0), in order to isolate theeffect of the antigen-specific CD8þ response influenced

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tumor disappearsBASIN BOUNDARY

Fig. 5. Basins of attraction of the zero-tumor and high-tumor

equilibria when the CD8þ T-cell recruitment parameter, j ¼ 4:5.

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862 849

by j. The parameter j is then gradually increased from0 to 7.

Two bifurcations are evident in the diagram: the firstis a transcritical bifurcation, where the negative equili-brium becomes positive, and the zero-tumor equilibriumchanges its stability. Before the bifurcation, the zero-tumor equilibrium is strictly unstable: even one tumorcell will result in the system moving toward the high-tumor equilibrium. After the bifurcation, the immunesystem is able to control small initial tumor populations,as long as there is some antigen-specific immuneresponse. Only when the tumor-specific CD8þ popula-tion is zero does a very small initial tumor populationescape immune surveillance. Initial tumor populationsthat are controlled are said to be in the basin of

attraction of the zero-tumor equilibrium. On the otherhand, those which escape immune surveillance, leadingthe system toward the high-tumor equilibrium, are saidto be in the basin of attraction of the high-tumorequilibrium. These basins are shown in Fig. 5. Note thatsince the state-space of the system is actually four-dimensional, what is depicted in Fig. 5 is the projectionof the basins onto the Tumor–CD8þ plane, where thevalues of N and C are kept at their zero-equilibriumvalues. Fig. 5 highlights the fact that even if the tumor isvery small, extremely low CD8þT levels will allow thetumor to escape immune surveillance. The location ofthe basin boundary is therefore crucial in determiningthe outcome of the disease. In the case of a patient whohas undergone chemotherapy which reduces both tumorand CD8þT cell levels, if these levels place the systembelow the basin boundary then even an undetectabletumor will regrow. However, if the patient is givenimmunotherapy subsequent to chemotherapy, thereby

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Stableequilibria

UnstableEquilibria

Transcritical bifurcation:non−zerounstable equilibriumbecomes positive

Saddle−nodebifurcation:equilibriadisappear

Zero−tumor equilibrium becomes semi−stable

Fig. 4. As the parameter representing the strength of the antigen-

specific immune response is varied, system equilibria appear and

change their stability.

boosting CD8þT levels above the basin boundary, thesystem will evolve toward the stable zero-tumorequilibrium, and the tumor will not regrow. Thishypothetical scenario emphasizes the potential impor-tance of combination therapy.

5. Numerical experiments: mouse data with

immunotherapy, chemotherapy and combination

treatments

In this section, we test the behavior of our modelusing the data and results from a set of murineexperiments presented in Diefenbach’s work (Diefen-bach et al., 2001). These data have also been employedin the simulations of de Pillis and Radunskaya’s model(2003b). We examine cases for which the immune systemcannot fight a growing tumor on its own as well as casesfor which neither chemotherapy nor immunotherapyalone can kill the tumor. We also present a case forwhich the administration of a combination of bothchemotherapy and immunotherapy is necessary in orderto cause a large tumor to die. For the following in silico

experiments, we use the mouse parameters provided inTable 1.

5.1. Immune system response to tumor: mouse data

In the first set of experiments for the mouse model,we present a situation in which the tumor escapesimmuno-surveillance. The tumor reaches carrying capa-city, and we assume the mouse dies under this extremetumor burden. The initial conditions for this situationare chosen to be a tumor of size 106 cells, a circulatinglymphocyte population of 1:1� 107, a natural killer cell

ARTICLE IN PRESSL.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862850

population of 5� 104, and a population of 100 CD8þTcells. With the set of parameters in Table 1, the outcomeof the simulation is sensitive to the initial conditionschosen. This set of initial conditions is meant to reflect alaboratory mouse experiment, in which an initial tumorchallenge of 106 cells is directly implanted into themouse, and then the progression of the tumor isobserved. Simulation results are presented in Fig. 6,Top Left.

5.2. Chemotherapy or immunotherapy: mouse data

As seen, with parameters as currently set, a tumorchallenge of size 106 cells is too large for the innateimmune system to control. The following set ofexperiments is useful for testing simulated treatmentoptions.

The first treatment approach employs seven pulseddoses of chemotherapy, each dose represented by settingvMðtÞ ¼ 1 in Eq. (6) for 1 day, and given in a 14 daycycle. The second treatment approach employs immu-notherapy in the form of an injection of 8� 108 highlyactivated CD8þT cells from day 7 to day 8. This CD8þ

injection is meant to represent the TIL treatments usedfor certain patients (see, for example, Dudley et al.,2002).

For an initial tumor challenge of 106 cells, the tumorsurvives despite either method of intervention. Theseexperiments are pictured in Fig. 6, Top Right andBottom Left.

5.3. Combination therapy: mouse data

Some laboratory experiments have indicated thepotential benefits of combining chemotherapy withimmunotherapy (see, for example, Machiels et al.,2001). In light of this, we next present the results of anin silico experiment simulating combination therapy,displayed in the Bottom Right image in Fig. 6. In thisexperiment, the chemotherapy and immunotherapytreatments outlined above are given simultaneously.Initial population sizes are set to the same values as inthe previous experiments.

According to our model, combination therapy is farmore effective at killing a tumor than either individualtreatment alone. We note that the drop in tumorpopulation for this case is extremely steep. It is currentlynot clear whether the time-scale for this tumor decline inresponse to combination therapy reflects realisticdynamics. However, the qualitative result highlightsthe synergistic effect of combination treatment, whichdoes reflect the outcomes of certain studies (see, e.g.Machiels et al., 2001; Wheeler et al., 2004).

In some cases, if chemotherapy alone or immunother-apy alone are to be used effectively against tumorgrowth, it is thought that very high or more frequent

doses must be employed. Higher medication levels cansometimes cause undesirable side-effects. Heavy che-motherapy will harm other tissues in the body, and toomany TILs administered over a short period of timemay be attacked and killed by suppressor immunecells before the TILs have a chance to significantlyimpact tumor cell growth. It seems reasonable thatcombination therapy that can employ lower doses ofeach therapy element has the potential to allow forfaster tumor elimination with milder side-effects. Thecirculating lymphocyte level in the combination therapyexperiment can be used to indicate that the immunehealth of the mouse has not suffered too greatly duringtreatment.

6. Numerical experiments: human data with

immunotherapy, chemotherapy and combination

treatments

In this section, we test the behavior of our modelusing parameters taken from experimental results of twopatients from Rosenberg’s study on metastatic melano-ma (Dudley et al., 2002). In Dudley et al. (2002), bothpatients responded to treatment, so the in silico

simulations here are experiments meant to explorevarious scenarios using human parameters.

First we examine the model with the set of parametersrepresenting patient 9 in Table 2. We present a case forwhich a certain tumor burden can be controlled by ahealthy immune system, but not by a slightly weakenedimmune system. We also present a case for which eitherchemotherapy alone or immunotherapy alone can kill atumor, and a case for which combination therapy isessential to the survival of the patient. We then comparethe results using parameters from patient 9 of Dudleyet al. (2002) to the simulated behavior of the modelusing patient 10 parameters of Dudley et al. (2002). Thispatient comparison provides insight into patient-specificparameter sensitivity.

6.1. Immune system response to tumor: human data,

patient 9

In the first set of human experiments, we examine aninitial tumor burden of 106 cells. This experimentrepresents a situation in which the immune system hasnot become activated against the tumor cell populationuntil the population has reached 106 cells, a size which inmany cases is still considered to be below the thresholdof clinical detectability in a human. For this tumor,immune system strength is very important in determin-ing whether or not the body alone can kill a tumor.The first experiment specifies what we will denote asan initially ‘‘healthy’’ immune system with 1� 105

natural killer cells, 1� 102 CD8þT cells, and 6� 1010

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Immuno Treatment: Single T-cell Pulse Combination Chemo and Immuno Treatments

TumorNK CellsCD8+T CellsCirculating Lymphocytes

TumorNK CellsCD8+T CellsCirculating Lymphocytes

Fig. 6. Mouse data. Top Left: No treatment. Immune system without intervention where the tumor reaches carrying capacity and the mouse ‘‘dies’’.

Top Right: Chemotherapy. The immune system response to high tumor with chemotherapy administered for 1 day in a 14 day cycle. Bottom Left:

Immunotherapy. Immune system response to high tumor with the administration of immunotherapy from days 7 to 8. Bottom Right: Combination

therapy. Chemotherapy and immunotherapy as previously described given simultaneously effectively. control of the tumor. Parameters for all

simulations are provided in Table 1.

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862 851

circulating lymphocytes. As seen in Fig. 7, Left, theinnate immune response is sufficiently strong to controlthe tumor. However, when the immune system isweakened, a tumor of the same size grows to adangerous level in the absence of treatment interven-tions. Simulated results for this weakened immune case,with initial conditions set to 1� 106 tumor cells, 1� 103

NK cells, 10 CD8þT cells, and 6� 108 circulatinglymphocytes, are pictured in Fig. 7, Right.

6.2. Chemotherapy: human data

For cases in which the tumor would grow to adangerous level if left untreated (such as in the depleted

immune system example shown in Fig. 7, Right), wemodel pulsed chemotherapy administration into thebody, but only after the tumor is large enough to beconsidered potentially detectable. Examining the tu-mor’s response to pulsed chemotherapy given overdifferent time courses, we determined that for atumor burden of 2� 107 cells, chemotherapy pulsesadministered once every 5 days are effective at killingthe tumor, which dies by day 50 (see Fig. 8, Top Left).The system is clearly sensitive to the chemotherapydosing regimen. A treatment with the same totalchemotherapy, but administered less frequently, onceevery 10 days (see Fig. 8, Bottom Left) will allow thetumor to regrow.

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TumorNK CellsCD8+T CellsCirculating Lymphocytes

TumorNK CellsCD8+T CellsCirculating Lymphocytes

Fig. 7. Human data, Patient 9. Left: A healthy immune system effectively kills a small tumor. Initial Conditions: 1� 106 tumor cells, 1� 105 NK

cells, 100 CD8þT cells, 6� 1010 circulating lymphocytes. Right: A depleted immune system fails to kill a small tumor when left untreated. Initial

Conditions: 1� 106 tumor cells, 1� 103 NK cells, 10 CD8þT cells, 6� 108 circulating lymphocytes. Parameters for these simulations are provided in

Table 2.

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862852

6.3. Immunotherapy: human data

In addition to pure chemotherapy treatments, weexamine pure immunotherapy treatments. One of themajor advantages of mathematical modeling is thatwhile we can model therapies for which clinical trialshave been performed (e.g. chemotherapy alone) therebyvalidating our model, we can also use our model tosimulate theoretical treatment strategies, such as im-munotherapy alone, a treatment approach not yetemployed in most clinical trials. Our model parametershave been calibrated by comparing simulated outcomesto results from chemotherapy and combination therapytrials. Clinical trials for melanoma, against which wemight at a future point directly compare simulations, arecurrently underway (see, for example, National CancerInstitute, 2005, for a current list of ongoing trials).

In this section we present in silico experiments with aTIL injection followed by short doses of IL-2. Thismirrors the treatment that was given to patients 9 and 10in Rosenberg’s experiments (Dudley et al., 2002), thedifference being that the patients in the clinical trial werefirst administered immuno-depleting chemotherapy be-fore the administration of TIL therapy.

In Fig. 9, we investigate a 106 cell tumor, a tumor levelthat the in silico innate immune system as specifiedcannot control on its own. However, as we have seen inour previous in silico experiments, a chemotherapyregimen can effectively handle this tumor challenge inpatient 9. Fig. 9, Left, shows the effect of immunother-apy alone against this tumor. The dosing of TILs andIL-2 administered here are the same as those shown in

Fig. 10. One advantage of this treatment approach isthat the immune system is directly strengthened, and notdepleted as it is with chemotherapy.

We note that immunotherapy effectiveness may belimited to smaller tumor sizes. Fig. 9, right, shows thatimmunotherapy alone is not effective at treating thetumor of size 107: This size tumor could, instead,be eliminated by combination therapy, as is shown inFig. 10.

6.4. Combination therapy

While we have presented cases for which chemother-apy alone or immunotherapy alone can kill a tumor,there are situations in which these treatments inisolation are not sufficient to eliminate the tumor. Wemeasure the patient’s immunological health by thenumber of circulating lymphocytes in the body and donot allow the circulating lymphocytes to drop below athreshold where risk of infection may be too high. In ourexperiments, we chose that threshold to be on the orderof 108 cells. This amount reflects a fraction ofapproximate normal white blood cell levels in an adulthuman (normal levels are about 5� 109 per liter ofblood, see, e.g. Roitt et al., 1993).

For the case in Fig. 10, Left, combination treatment isnow able to eliminate a tumor of 107 cells. Thecombination treatment given is simply a superpositionof the separate chemotherapy and immunotherapy regi-mens (unsuccessfully) applied in the previous experiments(see Fig. 8, Bottom Left and Fig. 9, Right). All initialconditions and parameter values remain the same as in

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Fig. 8. Human data, Patient 9. Top Left: A case for which nine doses of chemotherapy over 45 days are sufficient to eliminate a tumor. Initial

Conditions: 2� 107 tumor cells, 1� 103 NK cells, 10 CD8þT cells, 6� 108 circulating lymphocytes. Top Right: Drug administration pattern. Nine

doses, strength vM ðtÞ ¼ 5, 1 day per dose on a 5 day cycle. Bottom Left: A case where nine doses over 90 days is given and the tumor shows regrowth.

Initial Conditions: 2� 107 Tumor cells, 1� 103 NK cells, 10 CD8þT cells, 6� 108 circulating lymphocytes. Bottom Right: Drug administration

pattern: Nine doses, strength vM ðtÞ ¼ 5, 1 day per dose on a 10 day cycle. Parameters for these simulations are presented in Table 2.

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862 853

these previous experiments. IL-2 and chemotherapyconcentrations are provided in Fig. 10, Right.

The simulation shown in Fig. 10, Left, is consistentwith the Rosenberg data (Dudley et al., 2002) for patient9, in the sense that patient 9 responded to treatment, andour simulation also reflects a positive response tocombination therapy.

We performed multiple additional experiments, in-cluding those for which we examined the samecombination therapy for tumors of sizes 108 and larger.In these cases, the tumors were not controlled, except incertain cases for which additional doses of chemother-apy and IL-2 were given in varying combinations(figures not included).

6.5. Comparison with patient 10

In order to examine whether these treatment simulationsvary from patient to patient, we change patient-specificparameters extracted from Rosenberg’s study, and runsimilar simulations with the parameters for patient 10(Dudley et al., 2002). These parameters are given in Table 2.

First we repeat the experiment simulated in Fig. 7 (nointervention) for a tumor of size 106. In this case, theimmune system conditions that killed the tumor inpatient 9 are ineffective at managing the same size tumorin patient 10 (see Fig. 11, Left). However, patient 10’simmune system is able to handle a smaller tumor of size105 given the same initial immune cell count, as we show

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Fig. 9. Human data, Patient 9. Left: Immunotherapy is able to kill a tumor of size 106 cells. 109 TILs are administered from day 7 through 8. IL-2 is

administered in 6 pulses at strength 5� 106 from day 8 to day 11. Right: Immunotherapy is unable to kill a tumor of size 107 cells. 109 TILs are

administered from day 7 through 8. IL-2 is administered in 6 pulses from day 8 to day 11. Parameters for these simulation are provided in Table 2.

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Fig. 10. Human data, Patient 9. Combining the separately unsuccessful therapies for a 107 size tumor (see Fig. 8, Bottom Left and Fig. 9, Right)

succeeds in eliminating the tumor. Left: A 107 cell tumor is successfully eliminated by combining nine 1-day chemotherapy doses of strength

vM ðtÞ ¼ 5 every 10 days, with a boost of TILs and IL-2. 109 TILs are administered from day 7 through 8. IL-2 is administered in 6 pulses from day 8

to day 11 at concentration vI ðtÞ ¼ 5� 105 per pulse. Initial conditions are as in previous simulations: 2� 107 Tumor cells, 1� 103NK cells, 10

CD8þT cells, 6� 108 circulating lymphocytes. Patient 9 parameters for these simulations are in Table 2. Right: Drug concentrations for IL-2 and

chemotherapy.

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862854

in Fig. 11, Right. Clearly, the immune system’s tumorhandling capacity is patient specific. This is notsurprising, since the combination therapy administeredto 13 patients in Rosenberg’s study (Dudley et al., 2002)gave rise to objective clinical responses in only six of thethirteen patients.

In addition to the differing results of the immunesystem alone, we examine the conditions under which

combination therapy can control a 107 size tumor inpatient 10 with a compromised immune system (asspecified by initial conditions). As opposed to thepositive outcome for patient 9 (see Fig. 10), theparameter set for patient 10 leads to the growth ofthe tumor, as seen in Fig. 12, Top Left.

In this case, one treatment approach is to administeradditional immunotherapy in the form of IL-2 doses.

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Fig. 11. Human data, Patient 10. Left: Patient 10 cannot kill a 106 cell tumor with a healthy immune system of 105 NK cells, 100 CD8þT cells, and

6� 1010 circulating lymphocytes. Right: Patient 10 kills a 105 cell tumor with a healthy immune system of 105 NK cells, 100 CD8þT cells, and

6� 1010 circulating lymphocytes. Patient parameters for these simulations are provided in Table 2.

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862 855

This expansion in treatment does lead to tumor deathin silico, as shown in Fig. 12, Top Right. We note in thiscase tumor behavior seems to reflect tumor dormancyfollowed by relapse. The tumor appears to havecompletely died out by day 22. However, around day79, the tumor begins to re-emerge. Without theadditional IL-2 treatment given at day 80, the tumorwould regrow. We also ran the simulation for 2000 days,and the tumor did re-emerge, but at levels generallyconsidered below detectability thresholds. The tumorsubsequently died out again and did not reappear. SeeFig. 12, Bottom Left. This leads us to believe that thetumor population has been drawn into the stable zero-tumor equilibrium at this point. Such a case in the clinicwould likely be viewed as a successful case of completeremission.

7. Numerical experiments: human data with vaccine and

combination treatments

In addition to T-cell boosts and IL-2 injections, wenow simulate cancer vaccines. Cancer vaccines are aspecial case of immunotherapy for treating cancer. Witha vaccine, the body is challenged with some modifiedform of the cancer, consequently sensitizing the immunesystem to the presence of the cancer, allowingthe immune system more effectively to find and lysecancer cells.

7.1. Vaccine therapy and a change of parameters

In order to simulate vaccine therapy for humanpatients, we change the values of five parameters at the

time of vaccination. These parameter changes aredocumented by the experimental data found in Diefen-bach’s results on mouse vaccine trials (Diefenbach et al.,2001) and the experimental curves produced by thesedata are fitted to de Pillis’s model (de Pillis andRadunskaya, 2001). According to these laboratoryexperiments in the context of our model, the parametersthat would change to reflect the administration of atherapeutic vaccine are c, the fractional tumor cell killby natural killer cells, g, the maximum NK-cellrecruitment rate by tumor cells, j, the maximumCD8þT cell recruitment rate, s, the steepness coefficientof the tumor–CD8þT competition term, d, the satura-tion level of fractional tumor cell kill by CD8þT cells,and l, the exponent of fractional tumor cell kill byCD8þT cells. Of these six parameters, l is the only onefor which there is no clear relationship between itsmagnitude and the response of the system to vaccine,since this relationship is highly nonlinear and non-monotonic. We therefore leave l unchanged, and focuson modifications to the remaining five parameters.

For this set of experiments, we use the patient 9parameter set to simulate theoretical in silico vaccinetherapy by altering parameters in the same direction asthey change in Diefebach’s murine model (Diefenbachet al., 2001; de Pillis and Radunskaya, 2003b). Weincrease the values of c, g, j, and d and decrease the valueof s.

7.2. Vaccine and chemotherapy combination experiments

We first present a theoretical case in which a patienthas a detectable tumor of size 2� 107 and a ‘‘healthy’’immune system of 3� 105 NK cells, 102 CD8þT cells,

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Fig. 12. Human data, Patient 10. Top Left: Combination therapy fails to eliminate the 107 cell tumor in patient 10 with a compromised immune

system of 103 NK cells, 10 CD8þT cells, and 6� 108 circulating lymphocytes. 109 TILs are administered from days 7 through 8. IL-2 is administered

in 6 pulses from day 8 to day 11 at concentration vI ðtÞ ¼ 5� 106 per pulse. Top Right: Combination therapy kills the 107 cell tumor in patient 10.

Treatment is identical to that in the Top Left panel, with the exception that pulses of IL-2 are administered from days 8 through 13, 20 through 25

and 80 through 90. Bottom Left: The same effective combination therapy as given in the Top Right panel and as shown in the Bottom Right panel,

but viewed over 2000 days. Bottom Right: Concentrations for IL-2 and chemotherapy implemented in the simulations shown in the Top Right and

Bottom Left panels. Patient 10 parameters are provided in Table 2.

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862856

and 1010 circulating lymphocytes. This is a case inwhich the patient’s immune system is not strongenough to handle a 107 size tumor on its own (notpictured). As shown in Fig. 13, Left, the body cannothandle this size tumor even when treated with aggressivepulsed chemotherapy for 50 days. Additionally,vaccine therapy, represented by changes to theoriginal parameter set after 10 days, fails to controlthe tumor. This simulation is pictured in Fig. 13,Middle. Only the combination of both treatmentscan kill a tumor of this magnitude, as shown inFig. 13, Right.

Such dramatic tumor regression is still uncommonamong most patients, and sensitive to the choice oftumor and patient parameters, as well as to the timing ofthe treatments (see the next section). It has been noted,however, in the clinical studies of Wheeler et al.(2004), that over a period of 50 days, combinationtherapy provoked a drastic (50%) tumor regression inthree of 13 glioma patients. Wheeler points out thatthese tumor regression outcomes are rare in theliterature. However, our model is able to describe thesepositive clinical responses to emerging combinationtherapies.

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0 5 10 15 20 25 30 35 40 45 50100101102103104105106107108109

1010

100101102103104105106107108109

1010

Time

Cel

l Co

un

t

Circulating Lymphocytes

0 5 10 15 20 25 30 35 40 45 50100

102

104

106

108

1010

1012

Time

Cel

l Co

un

t

0 5 10 15 20 25 30 35 40 45 50

Time

Cel

l Co

un

t

Circulating Lymphocytes

Circulating Lymphocytes

Fig. 13. Human data, Patient 10. Left: Chemotherapy alone cannot kill a 2� 107 tumor with an immune system of 3� 105 NK cells, 100 CD8þT

cells, and 1010 circulating lymphocytes. Chemotherapy at strength vM ðtÞ ¼ 2 is administered for 3 consecutive days in a 10 day cycle. The parameters

for this simulation are provided in Table 2. Middle: Vaccine therapy alone cannot control the tumor with the same initial conditions as given in Left

figure. The parameters for this simulation are provided in Table 2, with parameters, modified after 10 days to reflect vaccine therapy, given by

c ¼ 7:131� 10�9; g ¼ 0:5; j ¼ 1; s ¼ :0019; d ¼ 15. Right: Combination therapy effectively controls the tumor.

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862 857

7.3. Vaccine therapy time dependence

There are cases for which vaccine therapy alone isable to control a growing tumor, but our simulationsindicate that timing is an important factor in determin-ing the effectiveness of the vaccine treatment. Since wemodel vaccine therapy as a change in parameters, andnot as a drug population, the amount of vaccine cannotbe altered, but the timing of administration can becontrolled. In the case for which an initial tumor is set tobe half the size as that in the previous experiments (107

cells), and the immune system initial conditions are setto be the same, vaccine therapy effectively eliminates atumor. However, this successful tumor eliminationhappens only if the vaccine is administered to thepatient no more than 13 days after the tumor ishypothetically detected at 107 tumor cells. If the vaccineis administered any later than that, it is ineffective. Weshow how the timing of vaccine therapy affects the finaloutcome in Fig. 14.

8. Discussion and conclusion

We have extended previous mathematical models thatgovern cancer growth with chemotherapy treatments toinclude immunotherapy and vaccine therapy. Thisstructure can be used to test combination treatmentapproaches. The model is formulated as a system ofordinary differential equations that describe the globalstimulatory effect of tumor cells on the immuneresponse in conjunction with chemotherapies.

Through an analysis of the system of equations in theabsence of chemotherapy or immunotherapy, we deter-mined the equilibrium points of the system along withthe criteria for stability. For a specific parameter set, wefound two equilibria. One was a tumor-free equilibrium,which was shown to be unstable, and the other was ahigh-tumor equilibrium, shown to be stable. Theinstability of the tumor-free equilibrium implies that

any successful treatment must be able to change thesystem parameters in order to force this desirableequilibrium point to become stable. The stability ofthe high-tumor equilibrium, on the other hand, impliesthat reducing the tumor burden through chemotherapyalone is not sufficient to drive the tumor level to zero.Once chemotherapy is stopped, a system with even anundetectable tumor remaining will eventually return tothe high-tumor state. However, system parameters maybe altered through treatments such as vaccination,thereby changing the stability nature of the zero-tumor equilibrium, and allowing a combination treat-ment protocol to eliminate the patient’s tumor. Wenote that in Gatenby and Vincent (2003), a similarconclusion is reached through a different modelingapproach: in this case, too, it is shown that unless systemparameters are altered in some way, cytotoxic drugsalone are often not sufficient to control even a very smalltumor.

A bifurcation analysis was performed for two ofthe system parameters. These analyses show that forcertain parameter values, the long-term behaviorof the system can be very sensitive to the initialconditions. One important implication of this resultfor tumor treatment is that if both the zero- and high-tumor equilibrium points are stable then for cellpopulations that are close to the boundary separa-ting the basins of attraction of these equilibria, verysmall changes in either the initial tumor size or theantigen-specific CD8þT levels can have drastic con-sequences on the outcome of the disease. This wouldindicate that combination immuno-chemo therapieswhich push the system into the tumor-free basin ofattraction should be employed to achieve a stabledisease-free state.

Results of our model simulations have been validatedby comparing the outcomes both to mouse (Diefenbachet al., 2001) and to human (Dudley et al., 2002) data.Simulations using parameter sets from two differentpatients show that treatment efficacy depends strongly

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0 5 10 15 20 25 30 35 40 45 50100

106

104

106

108

1010

1012

100

106

104

106

108

1010

1012

Time

Cel

l Co

un

t

0 5 10 15 20 25 30 35 40 45 50Time

Cel

l Co

un

t

Fig. 14. Human data, Patient 10. Left: Vaccine administered after 13 days. The parameters for this simulation are provided in Table 2 with

parameters modified on day 13 to reflect vaccine therapy given by c ¼ 7:131� 10�9; g ¼ 0:5; j ¼ 1; s ¼ :0019; d ¼ 15. Right: Vaccine administered

on day 14, parameters modified as indicated.

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862858

on patient-specific characteristics. This observation mayactually be used to our advantage: it is already possibleto measure some of the significant patient-specificparameters. In particular, through chromium releaseassays we can measure patient-specific immune-tumorlysis rates, while tumor growth rates can also bemeasured in the laboratory. We have seen that changesin these measurable parameters clearly affect systemoutcomes. As a result, although not all system para-meters are yet measurable, those that are help to providea good start in designing customized treatment proto-cols for each patient.

The development of combination immunotherapy–chemotherapy protocols for treating certain forms ofcancer is an exciting new strategy in cancer treatmentresearch. In some preliminary clinical studies, immu-notherapy has been found to be most effective whenadministered in conjunction with chemotherapy(Machiels et al., 2001), and this qualitative result hasbeen borne out in our mathematical simulations. In thiswork, we have contributed to the emerging body ofcancer modeling research by developing and analysing anew mathematical model of cancer growth, immuneresponse, and treatment that includes vaccine therapy,

Table 1

Estimated mouse parameter values

Parameter Units Description

a day�1 Tumor growth rate

b cells�1 1=b is tumor carrying capacity.

c cell�1 day�1 Fractional (non)-ligand-transduced tumor ce

kill by NK cells

d day�1 Saturation level of fractional tumor cell kill b

CD8þT cells. Primed with ligand-transduced

cells, challenged with ligand-transduced cells

activated anticancer-cell transfer (TIL injections), andactivation-protein injections (IL-2 injections) in combi-nation with chemotherapy. The mathematical modelpresented in this paper should serve to enrich thestudy of cancer treatments and aid in guiding thedevelopment of new patient-specific combination treat-ment protocols.

Acknowledgements

We would like to thank the following Harvey MuddCollege students for their contributions to this work:Lindsay Crowl, Lorraine Thomas and Michael Vrable.This work was supported in part by the W.M. KeckFoundation and the National Science Foundation NSF-0414011.

Appendix A. Nomenclature and parameter values

Here we list all of the parameters used in the model,their meaning and their estimated values. Table 1 reflectsthe experiments run to simulate the mouse experimentsfrom Diefenbach et al. (2001). Table 2 reflects humandata from Dudley et al. (2002).

Estimated value Source

4:31� 10�1 Diefenbach et al. (2001)

2:17� 10�8 Diefenbach et al. (2001)

ll 7:13� 10�10 Diefenbach et al. (2001)

y 8:17 Diefenbach et al. (2001)

.

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Table 1 (continued )

Parameter Units Description Estimated value Source

l None Exponent of fractional tumor cell kill by CD8þT 6:57� 10�1 Diefenbach et al. (2001)

cells. Primed with ligand-transduced cells,

challenged with ligand-transduced cells.

s None Steepness coefficient of the tumor-(CD8þT cell) 6:18� 10�1 Diefenbach et al. (2001)

lysis term D. Primed with ligand-transduced

cells, challenged with ligand-transduced cells.

(Smaller s ) steeper curve)

e day�1 Fraction of circulating lymphocytes that become NK

cells.1:29� 10�3 Kuznetsov et al. (1994)

f day�1 Death rate of NK cells. 4:12� 10�2 Kuznetsov et al. (1994)

g day�1 Maximum NK cell recruitment rate by ligand-

transduced tumor cells.4:98� 10�1 Kuznetsov et al. (1994);

Diefenbach et al. (2001)

h cell2 Steepness coefficient of the NK cell recruitment

curve.2:02� 107 Kuznetsov et al. (1994)

p cell�1 day�1 NK cell inactivation rate by tumor cells. 1:0� 10�7 Diefenbach et al. (2001)

m day�1 Death rate of CD8þT cells. 2:0� 10�2 Yates and Callard (2002)

j day�1 Maximum CD8þT cell recruitment rate. Primed with

ligand-transduced cells, challenged9:96� 10�1 Kuznetsov et al. (1994);

Diefenbach et al. (2001)

with ligand-transduced cells.

k cell2 Steepness coefficient of the CD8þT cell recruitment

curve.3:03� 105 Kuznetsov et al. (1994);

Diefenbach et al. (2001)

q cell�1 day�1 CD8þT cell inactivation rate by tumor cells. 3:42� 10�10 Kuznetsov et al. (1994)

r1 cell�1 day�1 Rate at which CD8þT cells are stimulated to be

produced as a result of tumor cells killed by NK

cells.

1:1� 10�7 Yates and Callard (2002);

Lanzavecchia and Sallusto

(2000)

r2 cell�1 day�1 Rate at which CD8þT cells are stimulated to be 3:0� 10�11 No data found.

produced as a result of tumor cells interacting

with circulating lymphocytes.

u cell�2 day�1 Regulatory function by NK-cells of CD8þT-cells. 1:80� 10�8 No data found.

KT day�1 Fractional tumor cell kill by chemotherapy. 9:00� 10�1 Perry (2001)

KN , KL,

KC

day�1 Fractional immune cell kill by chemotherapy. 6:00� 10�1 Perry (2001)

a cell day�1 Constant source of circulating lymphocytes. 1:21� 105 Bannock (2002); Hauser (2001)

b day�1 Natural death and differentiation of circulating

lymphocytes.1:20� 10�2 Bannock (2002); Hauser (2001)

g day�1 Rate of chemotherapy drug decay. 9:00� 10�1 Calabresi and Schein (1993)

Table 2

Estimated human parameter values

Patient 9 Patient 10 Source

a ¼ 4:31� 10�1 a ¼ 4:31� 10�1 Diefenbach et al. (2001)

b ¼ 1:02� 10�9 b ¼ 1:02� 10�9 Diefenbach et al. (2001)

c ¼ 6:41� 10�11 c ¼ 6:41� 10�11 Dudley et al. (2002); Diefenbach et al. (2001)

d ¼ 2:34 d ¼ 1:88 Dudley et al. (2002)

e ¼ 2:08� 10�7 e ¼ 2:08� 10�7 Kuznetsov et al. (1994)

l ¼ 2:09 l ¼ 1:81 Dudley et al., 2002

f ¼ 4:12� 10�2 f ¼ 4:12� 10�2 Kuznetsov et al. (1994)

g ¼ 1:25� 10�2 g ¼ 1:25� 10�2 Dudley et al. (2002); Diefenbach et al. (2001)

h ¼ 2:02� 107 h ¼ 2:02� 107 Kuznetsov et al. (1994)

j ¼ 2:49� 10�2 j ¼ 2:49� 10�2 Dudley et al. (2002); Diefenbach et al. (2001)

k ¼ 3:66� 107 k ¼ 5:66� 107 Dudley et al. (2002); Diefenbach et al. (2001)

m ¼ 2:04� 10�1 m ¼ 9:12 Yates and Callard (2002)

q ¼ 1:42� 10�6 q ¼ 1:59� 10�6 Kuznetsov et al. (1994)

p ¼ 3:42� 10�6 p ¼ 3:59� 10�6 Diefenbach et al. (2001)

s ¼ 8:39� 10�2 s ¼ 5:12� 10�1 Dudley et al. (2002)

r1 ¼ 1:10� 10�7 r1 ¼ 1:10� 10�7 Yates and Callard (2002); Lanzavecchia and Sallusto (2000)

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862 859

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Table 2 (continued )

Patient 9 Patient 10 Source

r2 ¼ 6:50� 10�11 r2 ¼ 6:50� 10�11 No data found.

u ¼ 3:00� 10�10 u ¼ 3:00� 10�10 No data found.

KT ¼ 9:00� 10�1 KT ¼ 9:00� 10�1 Perry (2001)

KN ¼ KL ¼ KC ¼ 6� 10�1 KN ¼ KL ¼ KC ¼ 6� 10�1 Perry (2001)

a ¼ 7:50� 108 a ¼ 5:00� 108 Bannock (2002); Hauser (2001)

b ¼ 1:20� 10�2 b ¼ 8:00� 10�3 Bannock (2002); Hauser (2001)

g ¼ 9:00� 10�1 g ¼ 9:00� 10�1 Calabresi and Schein (1993)

pI : Maximum CD8þT-cell recruitment

rate by IL-2. Units: day�1

pI ¼ 1:25� 10�1 pI ¼ 1:25� 10�1 Kirschner and Panetta (1998)

gI : Steepness of CD8þT-cell recruitment

curve by IL-2. Units: cell2

gI ¼ 2:00� 107 gI ¼ 2:00� 107 Kirschner and Panetta (1998)

mI : Rate of IL-2 drug decay. Units: day�1.

mI ¼ 1:00� 101 mI ¼ 1:00� 101 Kirschner and Panetta (1998)

L.G. de Pillis et al. / Journal of Theoretical Biology 238 (2006) 841–862860

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