humoral and cell-mediated responses on cancer immune ... · humoral and cell-mediated responses on...

Bulletin of Mathematical Biology manuscript No.(will be inserted by the editor)

Humoral and Cell-mediated responses on CancerImmune system - a study based on MathematicalModeling

Sumana Ghosh · Sandip Banerjee · HienTran · Alexei Tsygvintsev

Received: date / Accepted: date

Abstract We analyse the mathematical model of interaction between cancercells and immune system, where we have considered both humoral mediatedand cell-mediated Immune response. Our model includes 5 ordinary differentialequations and can be analysed analytically. We have done the sensitivity anal-ysis of our system and estimate the model parameter values using least squaremethod. We study its equilibrium points and find the conditions for globalstability for the cancer-free equilibrium point and also discuss the bifurcationanalysis of the system . We illustrate our results by numerical simulations.

Keywords Cancer cells · Antibodies,B cells · Plasma cells · Cytotoxic Tlymphocytes · Global stability · Transcritical bifurcation.

1 Introduction

Cancer is still a leading cause of death worldwide. The world health orga-nization(WHO) has predicted rise in the number of new cancer cases by 70over the next two decades (WHO (2015)). Cancer is not a single disease; itcomprises more than 200 diseases (Gabriel (2007); Schulz (2007)), which sharecommon characteristics, that is, they are abnormal cells, where the normalprocesses which regulate normal cell proliferation, differentiation and death(cell apoptosis) are interrupted. When certain genes are activated or deacti-

Sumana Ghosh · Sandip BanerjeeDepartment of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, Ut-taranchal, India.E-mail: [email protected]

Hien TranDepartment of Mathematics,Box 8205, NC State University, Raleigh, NC 27695-8205.

Alexei TsygvintsevUMPA, ENS de Lyon, 46, allee d’Italie, 69364 Lyon Cedex 07, France.

2 Sumana Ghosh et al.

vated because of mutation and environmental factors then a normal cell canbe transformed into cancerous cells.

The immune system, the body’s natural defence mechanism, helps the hostin fighting against the cancerous cells. As the cancerous cells grow and reacha detectable threshold number, the body’s own immune system is triggeredinto a search and destroy mode. The defence mechanisms of the human bodyagainst external invaders and other pathogens have two major components,namely, the innate and the adaptive immune-systems. Lymphocytes are theprincipal components of the adaptive immune system, consisting of a classof white blood cells. Adaptive immune system response against external in-vaders and pathogens are mediated by lymphocytes, either through humoral(antibody) mediated or cell mediated immune responses (Adam and Bellomo(1997)). B-cells are the most important elements for humoral response. Millionsof different B-cells are produced by the human body, which secrete antibodiesagainst antigens, thus performing the role of immune surveillance. The anti-gen binds to the immunoglobin receptors, which are present on the surface ofun-stimulated B-cells, resulting in the cell division of B-cells into non-dividingantibody secreting plasma cells. They are the key elements for humoral (an-tibody) mediated immune responses. These B-lymphocytes are stimulated byantigens (pathogenic agents), which in turn release antibodies in the blood.These antibodies bind to the antigens and mark them as foreign objects, sothat they can be eliminated by other components of the immune system, likemacrophages. Antibodies are the key players in humoral immune responses.The antibodies neutralize the cancerous cells by forming a coat over them, thuspreventing them from invading the target cells. The major breakthrough camefrom the research article by Ivanov et al. (2009), who have shown in their lat-est study that antibodies can kill cancerous cells directly. When the antibodiesbind with a cancerous cell, it causes lysosomes (small acid containing sacs) in-side the cell to swell and burst rapidly releasing their toxic contents with fatalresults for cancerous cells, which is non-apoptotic in nature. T-lymphocytesthat is, T Cells are the key element for cell mediated immune response. Thereare two types of T cells, they are CD8+ T cells and CD4+ T cells. CD8+T celltake a major role in cell mediated immune response, it also known as cytotoxicT-lymphocytes (CTL).

One of the most important and challenging questions in immunology andcancer research is to understand how the immune system affects on cancerdevelopment and progression The theoretical study of cancer-immune systeminteraction has a long history in mathematical biology, which has been doneby many authors (Adam and Bellomo (1997); Banerjee et al. (2015); Banerjeeand Tsygvintsev (2015); Forys (2002); Kolev (2003a); Khajanchi and Banerjee(2014); Preziosi (1996); Szymanska (2003); Tsygvintsev and Banerjee (2014);Kuznetsov et al. (1994); Kirschner and Panetta (1998); Nani and Freedman(2000); Kuznetsov and Knott (2001); Bodnar and Forys’ (2000); de Pillisand Radunskaya (2003); Moore and Li (2004); Sarkar and Banerjee (2005);Mallet and de Pillis (2006); Chaplain and Matzavinos (2006); Banerjee andSarkar (2008); Kirschner and Tsygvintsev (2009); de Pillis et al. (2006); Kolev

Title Suppressed Due to Excessive Length 3

(2003b); Dillman and Koziol (1987); Dubey et al. (2008)). In Kuznetsov etal. (1994), the authors presented a mathematical model of the cytotoxic T-lymphocyte and responses to the growth of an immunogenic tumor. Theystudied the immune-stimulation of tumor growth, ”sneaking through” of thetumor and the formation of a tumor dormant state. Kirschner and Panetta(1998) have illustrated the dynamics between tumor cells, immune effectercells and interleukin-2 (IL-2) through mathematical modeling. Their effortsexplain both short -term tumor oscillations in tumor size as well as long-termtumor relapses. Nani and Freedman (2000) proposed a general type mathe-matical model to study cancer immunotherapy. Mathematical analysis of themodel equations with regard to dissipativity, boundedness of solutions, invari-ance of non-negativity, nature of equilibria, persistence, extinction and globalstability were analyzed. Kuznetsov and Knott (2001), proposed a deterministicmodel describing the mechanisms underlying tumor growth, suppression andregrowth and fit to data on B-cell lymphoma. They demonstrated that eithera modest change in the effectiveness of killer suppression or the existence ofa variant non immunogenic clone of the tumor cells can explain the regrowthof a tumor after initial suppression. Bodnar and Forys’ (2000) studied the pe-riodic dynamics in the mathematical model of the immune system. Based onthe experimental results of Diefanbach et al. (2001), de Pillis and Radunskaya(2003) developed a mathematical model of tumor growth to describe the mech-anisms involved in the immune response to a tumor challenge using a systemof differential equations. In their model they mainly focus on the interaction ofNK-cells and CD+

8 8 T-cells with various tumor cells and proposed new formsfor the tumor-immune competition terms, and validate these forms throughcomparison with the experimental data of Diefanbach et al. (2001). In (Mooreand Li (2004)), the authors proposed and analyzed a mathematical model forthe interaction between naive T cells, effector T cells, and CML cancer cells inthe body, using a system of ordinary differential equations. From the analysisthey observed that the parameters, namely, the growth and death rates ofCML, plays great role to the outcome of the system. In Sarkar and Banerjee(2005), the authors expressed the spontaneous regression and progression of amalignant tumor as a prey-predator like system, where the tumor (cancerouscells) is treated as prey and the cytotoxic T-lymphocytes as predator. Thedeterministic model is extended to a stochastic one, allowing random fluctu-ations around the positive interior equilibrium point. The stochastic stabilityproperties of the model are investigated both analytically and numerically.Mallet and de Pillis (2006) have presented a hybrid cellular automaton partialdifferential equation model of moderate complexity, to describe the interactionbetween tumor and the immune systems of the host. Chaplain and Matzavinos(2006) have explained the effect of time and space in tumor immunology us-ing a mathematical model, that is, the spatio-temporal phenomena. The roleof interleukin-2 (IL-2) in tumor dynamics is illustrated through mathemati-cal modeling (a modified version of the Kirshner-Panetta model) in Banerjeeand Sarkar (2008), where the authors have shown that interleukin-2 alonecan cause the tumor cell population to regress. In Kirschner and Tsygvintsev


(2009), the global dynamics of the Kirshner-Panetta model was explored andthe conditions under which tumor clearance can be achieved, were obtained.

Most authors have studied the dynamics of cancer with cell mediated im-mune responses, consisting of the so called T-lymphocytes. There are a few whohave concentrated on humoral immune responses to cancer (Kolev (2003b);Dillman and Koziol (1987); Dubey et al. (2008)). In this paper, the aim is toconsider a mathematical model with both humoral as well as cell mediatedimmune responses to cancer and explore various aspect of the dynamical be-havior. This work will apply techniques from mathematics to gain insight intothe complexities of immune-cancer dynamics.

Schematic diagram and model formulation have been discussed in section2. In section 3 we study the sensitivity analysis of the model parameters alongwith the subset selection and consequently the estimation of the parameters.Quantitative analysis of the mathematical model is done in section 4. In section5 we numerically simulate the model to observe how the dynamics of the systemchanges with the change of parameter values. the paper ends with a discussion.

2 Model Formulation

The host immune system has the ability to produce some significant anti-cancerous immune responses, one of which is the B-cells. The human bodyproduces millions of different B-cells each day that circulates in the blood andlymphatic system performing the role of immune surveillance. The principalfunction of B-cells is to secrete antibodies against antigens. Prior to stimula-tion by either antigen or mitogen, B-cells are morphologically small cells. Thebinding of antigen to antigen receptors (that is, the antibodies) on B-cells canresult in the activation and differentiation of small B-cells into large B-cells.A set of immunoglobulin molecules is present on the surface of unstimulatedB-cells (Warner (1974)). By binding to these immunoglobulin receptors andwith a possible second signal from an accessory cell such as the T-cell, theantigen stimulates the B-cell to differentiate and mature into terminal (non-dividing) antibody secreting cells called plasma cells. The plasma cells are mostactive in secreting antibodies at a much faster rate but large B-lymphocytes,which proliferate rapidly, also secrete antibody, albeit at a lower rate (Eisen(1973)). Some of the large B-cells eventually revert back to small B-cells wherethey probably function as memory cells, which can respond vigorously to sub-sequent antigenic challenges (Perelson et al. (1976)). There is two types onimmune system in our body they are humoral mediated and cell mediatedimmunity. Antibody is the key elements of humoral immunity and CytotoxicT lymphocyte(CTL) pay a major role in cell mediated immunity. The se-creted antibodies then circulate in the blood and lymphatic system, and bindto the original antigen, marking them for elimination by several mechanisms,including activation of the complement system, promotion of phagocytosis via


opsonization and mediation of antibody dependent cell-mediated cytotoxicity(ADDC) with effecter cells such as macrophage, NK cells and neutrophils.Also, the antibodies have the ability to kill the cancerous cells through directcontact while binding to them and causing lysosomes inside the cancerous cells,which then swells and burst resulting in the death of cancerous cells. Antibod-ies, being a protein only decays. CTL and cancer cells both killed each otherdue to direct interaction and this interaction will enhance the secreration ofCTL. Based on the above biological scenario, we present a schematic diagramto describe the interaction between the cancerous cells, large B-cells, plasmacells and antibodies (see figure 1). We now propose a mathematical model todescribe the interaction between the host immune system considering bothhumoral and cell mediated immune responses.

If B, P, A,Tc and Tu be the number of large B-cells, plasma cells, antibodiesand cytotoxic T-lymphocytes and the cancerous cells respectively at any timet, then the governing equations are given by :

dB

dt= auB

(1− B

k1

)− b(1− u)B, (1)

dP

dt= b(1− u)B − µ1P, (2)

dA

dt= r1B + r2P − µ2A, (r1 < r2), (3)

dTcdt

= s+ pTcTu − β1TcTu − µ3Tc, (4)

dTudt

= rTu

(1− Tu

k2

)− β2TcTu − β3ATu, (5)

with the following initial conditions :

B(0) = B0 ≥ 0;P (0) = P0 ≥ 0;A(0) = A0 ≥ 0;Tc(0) = Tc0 ≥ 0 and Tu(0) = Tu0≥ 0.(6)

Here, a is the rate at which large B-cells are proliferating , b is the differenti-ation rate of large B-cells into plasma cells, a fractional part u of large B-cellsremains as proliferating large B-cells at fraction u (0 < u < 1) (Perelson et al.(1976))and (1 − u) is the fraction, that differentiates into plasma cells, k1 isthe carrying capacity of large B-cells, µ1 is the decay rate of the plasma cells.Both B-cells and plasma cells secrete antibodies at a rate r1 and r2 respec-tively, however, plasma cells secrete at a higher rate (r2 > r1) and µ2 is thedecay rate of antibodies. s is the constant source term of cytotoxic T lympho-cytes, p is the rate at which the cytotoxic T lymphocytes are recruited in thevicinity of cancer cells and β1 is the rate at which cytotoxic T lymphocytesare killed due to interaction with cancer cells. µ3 being natural death rate ofcytotoxic T lymphocytes. The intrinsic growth rate and carrying capacity ofcancerous cells are r and k2 respectively. Lastly, cytotoxic T lymphocytes andantibodies kill the cancerous cells by direct interactions at a rate β2 and β3respectively.


3 Sensitivity Analysis, Subset selection and parameter estimation

3.1 Initial Parameter Values

We refer to few published work with proper justifications to get appropriate pa-rameter values for the first two model equations. The mean generation time forlarge B lymphocytes is approximatly to be 6 to 48 hours (Davis et al. (1973)).Hence, the growth rate a1 of large B lymphocytes is estimated to be between0.02 and 0.2 hr−1 (Perelson et al. (1976)). Since the immune response to aT-independent antigen generally lacks any detectable immunological memory(Miranda (1972); Paul et al. (1974)), the conversion of large lymphocytes backinto small lymphocytes and the possible subsequent recycling of memory cellsback into large lymphocytes have not been explicitly considered. The life timeof plasma cells ranges from few days to few weeks. Thus, the natural deathrate µ2 of the plasma cell is approximated to vary between 0.002 to 0.02 hr−1

(Perelson et al. (1976)). The plasma cells and large B-cells secrete antibodiesat different rates. It is known that plasma cells may be even 100 fold moreactive in immunoglobulin synthesis than the large B cells (Melchers and An-dersson (1973)). Thus, the rate (r2) at which plasma cells secrete antibodiescan be estimated to be 2 to 100 times that of large B cells (r1). The secretionrate of antibodies by a single cell also varies considerably. Different authorshave estimated the absolute rate of antibody secretion by a single cell. Forexample, Nossal and Makela (1962) have been found in vitro that the rateat which a single cell secretes antibodies ranges from 100 to 1500 antibodiescell−1sec.−1.

To determine the other parameter values of the system, we look into theexperiment by Tutt et al. (2002),where, an investigation has been made in thetreatment of three syngeneic mouse B cell lymphomas, BCL1, A31, and A20on a panel of rats antimouse B cell mAb, including Ab directed at surfaceIgM Id, CD19, CD22, CD40, CD74 and NHC class II. It is noticed that onlythree mAb, namely, anti Id, anti CD19 and anti CD40 are therapeuticallyactive in vivo. They are interested in the immunotherapy of BCL1, A31 andA20 cells from the expeiments by monoclonal antibody Anti-CD40. Groups ofage-matched mice were injected 105 i.v (intravenously) with different tumorcells (BCL1, A31, and A20) on day 0, followed by mAb treatments daily for 4days at the times indicated for each experiments. In vivo T cell depletion wasdescribed by Cobbold (1990), using i.p. injection of 0.5 mg of anti-CD8 mAband/or 1 mg of anti-CD4 mAb.The injections were repeated every 4-5 days.

The data shown in Tutt et al. (2002) (figure 4A, pg No.2723) gives thegrowth of tumor in presence of control IgG mAb and anti-CD40 mAb. As seenin the figure, control IgG is failed to control the tumor growth whereas anti-CD40 eradicates tumor cells between 8-10 days. Since control IgG mAb does


not have any impact on the growth on tumor cells, we assume the effect ofcontrol IgG mAb as the normal growth of tumor cells in absence of antibodytherapeutic effect. In that case, the fourth equation of the model reduces to

dT

dt= rT

(1− T

K2

)To estimate the parameters r and K2, we obtain 8 data points from Tutt etal. (2002)(figure 4A, pg No.2723). Using least square method, the estimatedvalues of r and K2 are respectively r = 0.9889 day−1 and K2 = 1.336×109cells,that closely approximate the data (see Fig.4)

3.2 Sensitivity Analysis and subset selection

Nonlinear dynamical models derived from basic principles often contain pa-rameters whose values cannot be accurately predicted from theory. Missingparameters can be estimated using information from available data by solv-ing a parameter estimation problem. Parameter estimation problems are oftenformulated as optimization problems in which the unknown parameters aredetermined by minimizing the least squares error between the data and themodel output subject to constraints imposed by the model equations andknown bounds on the parameters. However, due to the model structure andpossible lack of measurements, some parameters may not be identifiable. Pa-rameters may have a very weak effect on the measured outputs (sensitivity),or the effect of certain parameters on the measured outputs may be nearlylinearly dependent (i.e., statistically correlated).

Local Sensitivity Analysis.

A first step before rigorous parameter estimation methods can be appliedto an ODE model is to determine which parameters contribute most to theoutput of the system. In the past, sensitivity analysis was primarily used in theanalysis of the forward, or simulation, problem when one needed to understandthe effects of parameter perturbation on the output. Now, in recent years sen-sitivity analysis has become a de-facto part of inverse problem analysis, partlybecause it directly aids in uncertainty analysis. When physiologically basedmathematical models are developed, the parameters typically have biologicalrelevance, and thus one may wish to estimate these parameters from data.However, if these parameters are insensitive then they could be difficult toestimate despite their relevance. Sensitivity analysis determines which stateoutputs are sensitive to given parameters and whether you can expect to esti-mate the parameter from available data, as well as understanding which modelparameters most influence the model. Sensitivity analysis can also aid in theoptimal design of experiments and data collection.

Consider the following dynamical system:

dx(t)

dt= f(x(q), t; q), x(t0) = x0, (7)


with observation function

y(t) = g(x(t)). (8)

Here, x ∈ Rn is a vector of the state variables and y ∈ Rp is the outputvector (to be compared with the data). We assume the parameter vector q ∈Rm to be constant over time.

The sensitivity equations are formed by implicitly differentiating both sidesof (8) with respect to q,

∂y

∂q= ∂g

∂x∂x∂q . (9)

These are the output sensitivities, and are in terms of the state sensitivities∂x∂q . These equations are obtained by differentiating (7) also with respect to qand then exchanging the order of differentiation. The mn sensitivity equationsare then given by

d

dt

∂x(t)

∂q= ∂f

∂x∂x∂q + ∂f

∂q . (10)

In addition, if the initial model states are unknown, we must also estimatethe initial conditions for the model. Doing so requires the formulation of initialcondition sensitivity equations, which quantify how the states respond to per-turbations in initial condition. To construct these n2 equations we differentiate(7) with respect to x0. After exchanging the order of integration, we have

d

dt

∂x(t)

∂x0=∂f

∂x

∂x

∂x0. (11)

These equations are coupled with the original sensitivity and state equa-tions yielding a system of n+mn+ n2 differential equations,

dx(t)

dt= f(x(q),q) (12)

d

dt

∂x(t)

∂q=∂f

∂x

∂x

∂q+∂f

∂q(13)

d

dt

∂x(t)

∂x0=∂f

∂x

∂x

∂x0, (14)

which would then be integrated in forward time giving the sensitivity functions.We assume a zero initial condition for the parameter sensitivity equations (10)at time t = t0, as the initial conditions for model states are not parameter de-pendent. In addition, if the initial conditions are to be estimated, a unity initialcondition for the initial condition sensitivities (11) is assumed, in general. Thesensitivity functions can ultimately be calculated using several different meth-ods, including automatic differentiation (AD) or finite differencing.


AD techniques are based on the fact that any function is executed on acomputer as a well-determined sequence of elementary operations like addi-tions, multiplications and calls to elementary functions such exponential, sine,cosine, etc. By repeatedly applying the chain rule to the composition of theseelementary operations one can compute (completely automatically) derivativesof a function correct to machine epsilon. In our work, we used the MATLABAD code, myAD, developed by our collaborator Martin Fink. Another ADcode, which has also been widely available, is the ADIFOR Fortran library de-veloped by Argonne National Laboratory. It has been shown that ADIFOR canproduce exact sensitivity information up to machine epsilon, and can reducecomputer CPU time requirements by up to 57 % compared with brute-forcemethods.

Sensitivity functions can also be computed by using finite difference for-mulas. In our work, we have used a central finite difference approximation,

∂x

∂qi=

x(qi + hei)− x(qi − hei)2h

+ Ø(h2). (15)

The advantage of using finite difference is its relative ease of implementa-tion versus the tedium of direct calculation. Using a central difference approx-imation is computationally more expensive than standard forward differences,however the approximation is accurate to Ø(h2) versus Ø(h). The selection ofstep size h is important; it has been shown that a good choice for h in centraldifferencing schemes is h = (macheps)1/3, where macheps is the relative errorin computing f(x).

To compare the sensitivity functions, we non-dimensionalize the functionsand then compute the norm of the resulting function, referred to as the sensi-tivity coefficients, given by

Cij = ‖∂xi∂qj

qjmaxxi

‖22 =

∫ tf

t0

|∂xi∂qj

qjmaxxi

|2dt (16)

After ranking the functions, the sensitivities can be sorted from least tomost sensitive. Many identifiability techniques have been derived around thesensitivity coefficients. The simplest techniques state that the larger the coef-ficients, then the more dramatic the system output with respect to that pa-rameter. In this case, a parameter is more likely to be identifiable when thereis a greater sensitivity. Finally, because the sensitivity functions are evaluatedat some nominal values for the parameters, the analysis is only local.

Now in our case we calculate the L2 norm of the corresponding parameterof the system by using MATLAB and plotting them we see that the L2 normof the parameter p, β1, r, β3, r2, µ2, µ3, β3 is large compare to the other param-eter of the system(see figure3). So p, β1, r, β3, r2, µ2, µ3, β3 are more sensitiveparameter for the system(1-5).

Local Identifiability Analysis.


The presence of parameters with weak and/or nearly linearly dependenteffects is manifested by the lack of a unique solution to the optimization prob-lem for different initial parameter values. Estimation of such parameters canlead to significant degradation in the predictive capability of the model. Aparameter is said to be practically identifiable if a unique estimate can beobtained from different initial values using the available data.

The correlation between parameters can be analyzed and handled throughthe analysis of the sensitivity function matrix. This matrix is defined for out-puts y1(t), . . . , yk(t) with respect to parameters q = [q1, . . . , qm]T at the timepoints t = (t1, . . . , tn) by the kn×m matrix

S(q, t) =

s1,1(q, t1) . . . s1,m(q, t1)s1,1(q, t2) . . . s1,m(q, t2)

. . . . .

. . . . .

. . . . .s1,1(q, tn) . . . s1,m(q, tn)s2,1(q, t1) . . . s2,m(q, t1)

. . . . .

. . . . .

. . . . .sk,1(q, tn) . . . sk,m(q, tn)

where si,j(q, tk) = ∂yi(tk)

∂qjdenotes the sensitivity of output yi to parameter

qj at time tk. The sensitivity matrix S represents the classical mechanism ofsensitivity analysis. A nominal parameter value is required for the evaluation ofthe sensitivity matrix, thus identifiability analysis based on sensitivity methodsis evaluated with respect to a specific point in the parameter space. Considerthe first order Taylor series expansion of the system response near a nominalparameter value q∗,

y(q) = y(x(tk),q)

≈ y(x(tk),q∗) +∂y(x(tk),q)

∂q|q=q∗ (q− q∗), (17)

where in our notation S = ∂y(x(tk),q)∂q |q=q∗ . Let yk denote a model observation,

e.g. a measurement at time tk and ∆q = q− q∗. The residual sum of squaresbetween the model observations and the linear approximation (17) is

RSS(∆q) =

N∑k=1

[yk − yk(q∗)− ∂y(x(tk,q)

∂q|q=q∗∆q]2

=

N∑k=1

[∂y(x(tk,q)

∂q|q=q∗∆q]2, (18)

where we have assumed that the residuals rk = yk − yk(q∗) are small andnegligible. Rewriting (18) in terms of the sensitivity function matrix S, we


have

RSS(∆q) = (S∆q)TS∆q, (19)

which has a minimum when STS∆q = 0. The matrix F = STS is referred toas the Fisher Information Matrix, and is a first order approximation to theHessian of the cost function in an ordinary least squares sense. It follows thatthe parameters q are locally identifiable if and only if the column rank of thematrix S is equal to m, or equivalently det(STS) 6= 0. If STS is full rank, then(STS)∆q = 0 has a unique solution, and q = q∗ implying that q is locallyidentifiable at q∗. If STS is singular, then there exists a nontrivial solutionq 6= q∗, and the model parameters are not identifiable at q∗.

It is clear that identifying q is contingent on F being full rank. It is typ-ically the case, especially when working with complex compartment modelsthe matrix F is not exactly singular. However F is often nearly singular, inthe sense that its smallest singular value is much smaller than the largest sin-gular value of F . The numerical rank of a matrix offers a measure of linearindependence in the columns of a matrix. To compute the numerical rank ofthe matrix, we use the following measure with ε = ε‖F‖, the floating pointaccuracy at ‖F‖.

rank(A, ε) = maxis.t. |σi|σ1

> ε‖A‖m (20)

where σi are the singular values of the matrix, with σ1 ≥ σi∀i > 1, and mis the number of columns. The calculation of the numerical rank is highlydependent on the tolerance used, especially when the spectrum is relativelysmooth with no obvious breaks or gaps.

When the matrix F is of numerical rank k, we say that only k parametersare identifiable, as k parameters form a mostly linearly independent spanningset. The question then becomes which k parameters are identifiable, and thisis answered through subset selection.

Subset Selection. A number of approaches have been developed and usedto find an identifiable subset of parameters. In this work we utilize the QRfactorization with column pivoting proposed by Golub (1965) and is imple-mented in the MATLAB routine qr, [Q,R,E]=qr(A). It is computationallymore efficient than the singular value decomposition method and is useful inmany applications including the rank-deficient least squares problems. Themethod consists in using a column pivoting strategy to determine a permuta-tion matrix E so that AE = QR is the QR factorization of AE. The indicesin the first k columns of E identify the k parameters that are most estimable.Our case yields the ordering p, β2, β3, β1, µ1, r, µ3, µ2, which means that these8 parameters are most estimable from the given data Tutt et al. (2002)(figure4A, pg No.2723).

As we have already estimated the parameters a, u, b, r1, r2, µ1, s, r and k2(see subsection 3.1), we now estimate rest of the parameter values µ2, µ3, p, β1, β2, β3.


From Tutt et al. (2002)(figure 4A, pg No.2723) we obtain 8 data points show-ing the growth of cancer cells in presence of the immune system. Using these8 data points we estimate the parameters µ2, µ3, p, β1, β2, β3. To start the es-timation process, the initial values of the parameters are chosen arbitrarily(within meaningful biological range). We use least square method, which min-imize the sum of the residuals to obtain the estimated values of the system pa-rameters µ2, µ3, p, β1, β2, β3. In practice we use MATLAB function fminsearchto estimate the parameter values. The esimated values of the parameters areas follows:µ2 = 1.3666 sec−1, p = 7.2641 × 10−8 Ab−1hr−1, β1 = 6.8891 ×10−8 Ab−1hr−1, µ3 = 0.1605 sec−1, β2 = 1.9169 × 10−10 Ab−1hr−1, β3 =1.6530× 10−11 Ab−1hr−1. Figure.5) shows the best fit estimate for the modelparameters µ2, µ3, p, β1, β2, β3. All the values of the system parameters aregiven in tabular form in Table 1.

4 Basic properties of the mathematical model

4.1 Positivity

In this section , we study the positivity of the solutions of the system(1-5).We start with a theorem which assures the system is positive in the interval[0,∞).

Theorem 1 Every solution of the system (1-5) with respect to the initial con-ditions (6) exist in the interval [0,∞) and remain positive for all future timet > 0.

Proof. Consider the system of equation (1-5) in vector form

X(t) = col(B(t), P (t), A(t), Tc(t), Tu(t)) ∈ R5+,

and

H=

H1(X(t))H2(X(t))H3(X(t))H4(X(t))H5(X(t))

=

auB

(1− B

k1

)− b(1− u)B

b(1− u)B − µ1Pr1B + r2P − µ2A

s+ pTcTu − β1TcTu − µ3Tc

rT(

1− Tk2

)− β2TcTu − β3AT

which can be written as

X = H(X(t)), (21)

where . ≡ ddt with initial conditionsX(t) = col(B(0), P (0), A(0), Tc(0), Tu(0)) ∈

R5+. Consider the mapping H : R5

+ → R5 and H ∈ C∞(R5+). It is easy to verify


that H(X)|Xi=0 = Hi(0) ≥ 0,∀i = 1, 2, 3, 4, 5. Due to the well known theoremby Nagumo (1942) , the solution of (1-5) with initial conditions X0 ∈ R5

+, issuch that X(t) ∈ R5

+, for all t > 0 , that is, it remains non-negative throughoutthe region R5

+,∀t > 0.

4.2 Boundedness

Theorem 2 The non negative solution of the system (1−5) with respect to theinitial conditions (6) are bounded in the region Ω provided q0 = au−b(1−u) >0 and µ3 > pk2 holds.

From equation(1) we get,

B(t) =q0

auk1

(1− e−q0t) + q0B0e−q0t

, whereq0 = au− b(1− u).

Now if au− b(1− u) > 0, then lim supt→∞B(t) = k1auq0 = B (say).

From equation (2),

dP (t)

dt+ µ1P (t) ≤ b(1− u)B

⇒ P (t) ≤ b(1− u)B

µ1(1− e−µ1t) + C1e

−µ1t

⇒ lim supt→∞

P (t) ≤ b(1− u)B

µ1= P (say).

Now, considering equation (3) we get,

dA(t)

dt+ µ2A(t) ≤ r1B + r2P

⇒ lim supt→∞

A(t) ≤ r1B + r2P

µ2= A(say)

From equation (5) we get,

dTu(t)

dt≤ rTu(t)

(1− Tu(t)

k2

)which implies

Tu(t) ≤ C2k2C2 + e−rt

, C2 being an arbitrary constant.

By using standard kamke’s comparison theory (Kamke (1932)), we get,

lim supt→∞

Tu(t) ≤ k2 = Tu.


From equation (4) we get,

dTc(t)

dt≤ s+ pTck2 − µ3Tc

≤ s− (µ3 − pk2)Tc

⇒ dTc(t)

dt+ (µ3 − pk2)Tc ≤ s

∴ Tc(t) ≤s

µ3 − pk2

(1− e−(µ3−pk2)t

)+ C3e

−(µ3−pk2)t

Now, if µ3 > pk2, then lim supt→∞ Tc(t) ≤ sµ3−pk2 = T c

Hence the region

Ω =

(B,P,Ac, Tc, Tu) ∈ R5

+/ 0 ≤ B ≤ k1q0au

, 0 ≤ P ≤ b(1− u)k1µ1

, 0 ≤ A

≤ k1 (r1q0µ1 + r2aub(1− u))

a u µ1 µ2, 0 ≤ Tc ≤

s

µ3 − pk2, 0 ≤ Tu ≤ k2

is bounded, provided q0 = au− b(1− u) > 0 and µ3 > pk2.

4.3 Existence and Local Stability Analysis of the equilibrium points

The system (1− 5) has four feasible equilibrium points, namely,

(i) the boundary equilibrium E(

0, 0, 0, sµ3, 0)

, which always exist.

(ii) the antibody free equilibrium point E(

0, 0, 0, s

µ3+(β1−p)Tu, Tu

)where Tu is the root of the quadratic equation

r(β1 − p)k2

Tu2

+

(rµ3

k2+ (p− β1)r

)Tu + sβ2 − µ3r = 0 (22)

E exists if equation (22) has positive real root and as well as µ3+(β1−p)Tu > 0.

(iii) the cancer free equilibrium point E(B,P ,A, T c, 0)

where,

B =k1au

(au− b(1− u)) , P =b(1− u)B

µ1

A =r1B + r2P

µ2, T c =

s

µ3


(iv) the interior equilibrium E∗(B,P ,A, sµ3+(β1−p)T∗u

, T ∗u ),

where, T ∗u is obtained from

r(β1 − p)k2

T ∗2

u +

(rµ3

k2+ (β1 − p)(β3A− r)

)T ∗u + sβ2 + µ3(β3A− r) = 0(23)

The interior equilibrium point exist when equation (23) has a positive realroot and (µ3 + (β1 − p)T ∗u ) > 0 .

The local behavior of the system (1 − 5) around each of the equilibriumpoints are obtained by computing the jacobian matrix corresponding to eachequilibrium point.

At the boundary equilibrium point E(

0, 0, 0, sµ3, 0)

, the variational matrix

JE is given by

JE=

au− b(1− u) 0 0 0 0b(1− u) −µ1 0 0 0r1 r2 −µ2 0 00 0 0 −µ3 (p− β1) s

µ3

0 0 0 0 r − sβ2

µ3

The eigen values are λ1 = au − b(1 − u) > 0, λ2 = −µ1, λ3 = −µ2, λ4 =−µ3, λ5 = r − sβ2

µ3, which implies that the point is a saddle and is unstable.

The jacobian matrix calculated at antibody free equilibrium point E is

JE=

au− b(1− u) 0 0 0 0b(1− u) −µ1 0 0 0r1 r2 −µ2 0 0

0 0 0 −(µ3 + (β1 − p)Tu

)(p− β1)Tc

0 0 −β3Tu −β2Tu r − 2rTuk2− β2Tc

Among the five eigen values, one of them is λ1 = au− b(1− u) > 0. hence theantibody free equilibrium point E3 is unstable.

Variational matrix calculated at the disease free equilibrium point E is

JE=

−(au− b(1− u)) 0 0 0 0

b(1− u) −µ1 0 0 0r1 r2 −µ2 0 00 0 0 −µ3 (p− β1) s

µ3

0 0 0 0(r − sβ2

µ3− β3A

)


The eigen values are λ1 = −(au − b(1 − u)) < 0, λ2 = −µ1, λ3 = −µ2, λ4 =−µ3, λ5 = r − sβ2

µ3− β3A. So if r < sβ2

µ3+ β3A , then the system is locally

asymptotically stable about the equilibrium point E and if r > sβ2

µ3+ β3A ,

then the system becomes unstable. For r = sβ2

µ3+ β3A , we get one zero eigen

value of JE . Hence no immediate conclusion about the local stability of Ecan be obtained . The foregoing argument establishes the growth rate of tumorcells, r as a threshold parameter value that characterizes the local stability ofthe ’disease free equilibrium E.

Jacobian matrix calculated at interior equilibrium point E∗ takes the form

JE∗=

−(au− b(1− u)) 0 0 0 0

b(1− u) −µ1 0 0 0r1 r2 −µ2 0 00 0 0 (p− β1)T ∗u − µ3 (p− β1)T ∗c0 0 −β3T ∗u −β2T ∗u − rT

∗u

k2

The characteristic equation is

(λ+ au− b(1− u))(λ+ µ1)(λ+ µ2)(λ2 + σ1λ+ σ2) = 0 (24)

where

σ1 = µ3 + (β1 − p)T ∗u +rT ∗uk2

σ2 =rT ∗uk2

(µ3 + (β1 − p)T ∗u ) + β2(p− β1)T ∗c T∗u

The first three eigen value of the system are λ1 = −(au− b(1− u)) < 0, λ2 =−µ1, λ3 = −µ2, which are negative and the last two eigen value is negativeor negative real part if σ1 > 0 and σ2 > 0. So, the interior equilibrium E∗ islocally asymptotically stable if σ1 > 0 (which is true by existence condition ofendemic equilibrium point) and σ2 > 0.

4.4 Global stability of disease free equilibrium point

Theorem 3 The cancer-free equilibrium point E is globally asymptoticallystable, when the system does not contain any other stable equilibrium pointand the following condition holds

r − sβ2µ3− β3

k1[µ1r1 + br2(1− u)][au− b(1− u)]

auµ1µ2< 0


Proof. The inequality r− sβ2

µ3−β3 k1[µ1r1+br2(1−u)][au−b(1−u)]

auµ1µ2< 0 implies that

the cancer-free equilibrium point E2 is locally asymptotically stable. Now wehave to study the global stability of the disease free equilibrium point whenthe system does not contain any other equilibrium point. For that we solvethe first equation of the system and get

B(t) =q0

auk1

(1− e−q0t) + q0B0e−q0t

.

Hence, limt→+∞B(t) = k1(au−b(1−u))au = B, and for all t ≥ 0 we have B(t) <

B, assuming that the initial condition B(0) < B. Since B is an increasingfunction for t ≥ 0, for arbitrary small ε1 > 0, there exists Tε1 > 0 such thatB(t) ≥ B − ε1 for all t ≥ Tε1 . Thus, one has B − ε1 ≤ B(t) < B, ∀ t ≥ Tε1 .From the second equation of the system we get, for t ≥ Tε1

b(1− u)(B − ε1)− µ1P ≤dP

dt≤ b(1− u)B − µ1P, (25)

which can be written as

b(1− u)(B − ε1)eµ1t ≤ (eµ1tP )′(t) ≤ b(1− u)Beµ1t (26)

Integrating the last inequality in the interval [Tε1 , t], one obtains

η(B − ε1)

µ1(eµ1t−eµ1Tε1 ) ≤ eµ1tP (t)−eµ1Tε1P (Tε1) ≤ ηB

µ1(eµ1t−eµ1Tε1 ), (27)

or, after division by eµ1t

η(B − ε1)

µ1(1− eµ1(Tε1−t)) ≤ P (t)− eµ1(Tε1−t)P (Tε1) ≤ ηB

µ1(1− eµ1(Tε1−t)),

(28)where we put η = b(1− u).

If t → +∞, according to the last inequality and since ε1 > 0 is arbitrary,we obtain

limt→+∞

P (t) =ηB

µ1=b(1− u)B

µ1= P .

Now, using the definition of limit superior and limit inferior of a function,we have, for any ε2 > 0, ∃ a large Tε2 such that

r1(B − ε2) + r2(p− ε2) ≤ A(t) ≤ r1(B + ε2) + r2(p+ ε2), ∀ t ≥ Tε2(29)

which can be written as(r1(B − ε2) + r2(p− ε2)

)eµ2t ≤ (eµ2tA)′(t) ≤

(r1(B + ε2) + r2(p+ ε2)

)eµ2t

(30)


Integrating the last inequality in the interval [Tε2 , t], we get

r1(B − ε2) + r2(p− ε2)

µ2(eµ2t − eµ2Tε2 ) ≤ eµ2tA(t)− eµ2Tε2A(Tε2)

≤ r1(B + ε2) + r2(p+ ε2)

µ2(eµ2t − eµ2Tε2 ) (31)

or, after division by eµ2t

r1(B − ε2) + r2(p− ε2)

µ2(1− eµ2(Tε2−t)) ≤ A(t)− eµ2(Tε2−t)A(Tε2)

≤ r1(B + ε2) + r2(p+ ε2)

µ2(1− eµ2(Tε2−t)), (32)

If t → +∞, according to the last inequality and since ε2 > 0 is arbitrary,we obtain

limt→+∞

A(t) =r1B + r2p

µ2= A.

Now we consider the last two equations of our system

dTcdt

= s+ pTcTu − β1TcTu − µ3Tc

dTudt

= rTu

(1− Tu

k2

)− β2TcTu − β3ATu

In order to study the global stability of the above system, we have to studythe asymptotic behaviour of its solutions for t→ +∞ and so one can replacethe variable A by A in the last equation.

So, one obtains the autonomous system of two differential equations

dTcdt

= s+ pTcTu − β1TcTu − µ3Tc

dTudt

= rTu

(1− Tu

k2


which we write in the vector form

Y = F (Y (t)) (33)

where, Y (t)=

(Tc(t)Tu(t)

)and F (Y )=

(s+ pTcTu − β1TcTu − µ3Tc

rTu

(1− Tu

k2


)


Let α = 1TcTu

, then the simple calculation shows

∀ (Tc, Tu) ∈ D : div(αF ) = − s

T 2c Tu

− r

k2Tc

< − s

M21M2

− r

k2M1= −δ

where D =

(Tc, Tu) ∈ R2+/0 < Tc <

sµ3−pk2 = M1, 0 < Tu < k2 = M2

Hence the system (33) satisfies the Dulac-Bendixson condition, which en-

sures that the system (33) has no non-constant periodic solutions and this con-dition obviously is robust under small C1– perturbations of F. Then, it followsfrom Michael and James (1995) that every semi trajectory either tends to anequilibrium point inside D or the boundary of D. Now if we choose the domain

D′ instead of D as in Figure.6

[D′ = D

⋃K,K =

(Tc, Tu)/(Tc−T c)2+T 2

u < ρ

, Tc < 0, ρ > 0

], then in the interior of D′, the only equilibrium is the disease

free one, which is locally asymptotically stable. It is easy to show, that theboundary of D′ can not be a limit of any solutions with initial conditions in D′.Indeed, a solution can not approach asymptotically the boundary ∂K of thehalf disk K since its center is the locally stable equilibrium point. At the sametime, no solution can converge to the boundary part ∂D \ ∂K since ∂D doesnot contain any equilibrium points. Thus, all solutions starting in D′ convergeas t→ +∞ to the disease free equilibrium (see Figure 7).

4.5 Bifurcation Analysis

We now perform bifurcation analysis of the proposed model to obtain thresholdvalues of the parameters where the qualitative behavior of the system changes.To verify this, we use estimated parameter values given in table 1. Though thesystem has four equilibrium points, from clinical point of view, the cancer freeequilibrium point is more important to study because it gives us under whatsituation the patient will be tumor free. Therefore it is necessary to study thebifurcation analysis of the equilibrium point E(B,P ,A, T c, 0).

The cancer free equilibrium point E will be stable if r < β2sµ3

+ β3A = r∗,

otherwise unstable . When r = β2sµ3

+ β3A = r∗, it shows one zero eigen value

of the system (1 − 5), other eigen values are real and negative. Therefore,

the stability of the cancer free equilibrium point E changes as r crosses thethreshold value r = β2s

µ3+β3A = r∗ and the system experiences a transcritical

bifurcation around E, with the tumor growth parameter r as the bifurcationparameter. To verify this analytically, we apply Sotomayor’s theorem Perko


(1991) to obtain the existence of transcritical bifurcation for the system (1−5)

around the tumor free equilibrium E when r = r∗.

The dynamical system (1− 5) can be represented in vector form as

f(B,P,A, Tc, Tu) =

auB

(1− B

k1

)− b(1− u)B

b(1− u)B − µ1Pr1B + r2P − µ2A

s+ pTcTu − β1TcTu − µ3Tc

rTu

(1− Tu

k2


The Jacobian ma-

trix J at E and its transpose have a single eigen value λ = 0 with corresponding

eigen vector vT =[0 0 o (p−β1)s

µ23

1]

and uT = [0 0 0 0 1]

Differentiating the vector expresion with respect to the bifurcation param-eter r about the equilibrium point E, we obtain

fr(E, r) =

(0 0 0 0 Tu

(1− Tu

k2

))T|E = [0 0 0 0 0]T

Now,

(i) uT fr(E, r) = [0 0 0 0 1][0 0 0 0 0]T = 0

(ii) uT [Dfr(B,P,A, Tc, Tu : r)v] =(

0 0 0 0 1)

0000v5

= v5 = 1 6= 0.

where

Dfr(B,P,A, Tc, Tu : r)v =∂Dfr(B,P,A, Tc, Tu : r)v]

∂Bv1

+∂Dfr(B,P,A, Tc, Tu : r)v]

∂Pv2 +

∂Dfr(B,P,A, Tc, Tu : r)v]

∂Av3

+∂Dfr(B,P,A, Tc, Tu : r)v]

∂Tcv4 +

∂Dfr(B,P,A, Tc, Tu : r)v]

∂Tuv5


at the steady state E(B,P ,A, T c, 0)

(iii) uT [D2f(E; r)(v, v)] =(

0 0 0 0 1)

000

(p− β1)v4v5− 2rk2v25 − 2β2v4v5

= −2

[r

k2+

(p− β1)sβ2µ23

]

= −2

[β2s+ β3µ3A

k2+

(p− β1)sβ3µ23

]6= 0

where

D2f(E, r)(v, v) =∂2f(E, r)

∂B2v1v1 +

∂2f(E, r)

∂B∂Pv1v2 +

∂2f(E, r)

∂P∂Bv2v1

+∂2f(E, r)

∂P 2v2v3 +

∂2f(E, r)

∂B∂Av1v3 + +

∂2f(E, r)

∂A∂Bv3v1

+∂2f(E, r)

∂A2v3v3 +

∂2f(E, r)

∂B∂Tcv1v4 +

∂2f(E, r)

∂Tc∂Bv4v1

+∂2f(E, r)

∂T 2c

v4v4 +∂2f(E, r)

∂B∂Tuv1v5 +

∂2f(E, r)

∂Tu∂Bv5v1

+∂2f(E, r)

∂T 2u

v5v5 +∂2f(E, r)

∂P∂Av2v3 +

∂2f(E, r)

∂A∂Pv3v2

+∂2f(E, r)

∂P∂Tcv2v4 +

∂2f(E, r)

∂Tc∂Pv4v2 +

∂2f(E, r)

∂P∂Tuv2v5

+∂2f(E, r)

∂Tu∂Pv5v2 +

∂2f(E, r)

∂A∂Tcv3v4 +

∂2f(E, r)

∂Tc∂Av4v3

+∂2f(E, r)

∂A∂Tuv3v5 +

∂2f(E, r)

∂Tu∂Av5v3 +

∂2f(E, r)

∂Tc∂Tuv4v5

+∂2f(E, r)

∂Tu∂Tcv5v4

Thus all three properties (see appendix) for the existence of transcritical bi-furcation holds and we conclude by Sotomayar’s theorem, that the system un-dergoes transcritical bifurcation as the tumor intrinsic growth rate r, crossesthe threshold value r∗.

5 Numerical Results

We perform the numerical simulation of system (1-5) using the parameter val-ues for Table 1. We like to see how the dynamics of the system changes with


the change of some parameter value, keeping the other parameters fixed.

Equilibrium and Stability:

• The boundary equilibrium pointE(

0, 0, 0, sµ3, 0)

= E(0, 0, 0, 80996.9, 0)

exists for the parameter values in Table 1. The eigen value of the vari-ational matrix implies that it is a saddle point and hence unstable.

• The antibody free equilibrium point is E(0, 0, 0, 4.9936×109, 4.2799×107)for the parameter values in Table 1, it is unstable.

• Since au−b(1−u) = .011 > 0, the cancer free equilibrium point exists and is

given by E = (6.875×106, 3.0938×106, 2.7669×109, 80996.9, 0).. It is locally

asymptotically stable if r < sβ2

µ3+β3

k1[µ1r1+br2(1−u)][au−b(1−u)]auµ1µ2

= 0.0457 (=

r∗, say). The locally asymptotically stable cancer free equilibrium point isalso globally stable, which has been shown numerically in figure 7.

• For the set of parameter value in Table 1, the interior equilibrium point isE∗(6.875×106, 3.0938×106, 2.7669×109, 4.7549×109, 4.2799×107) and theeigen values of the variational matrix are (−0.011,−0.02,−1.3666,−0.0158+0.3821 i,−0.0158− 0.3821 i). Hence, it is locally asymptotically stable.

We now use the model to predict responses to the treatment at the popula-tion model. The medel is simulated for response to treatment for patients witha range of immune strength. For a given set of immune systems parametersthe effectiveness of immune system against cancer depends on four control pa-rameters namely, p (rate of recruitment of cytotoxic T-lymphocytes),β1 (deathrate of cytotoxic T-lymphocytes due to interaction with tumor cells), β2 (killrate of cancer cells by cytotoxic T-lymphocytes) and β3 (rate of cancer celldeath by antibodies). These values are varied in a biologically reasonable rangeto predict the changes in the dynamics of the system and to ascertain how acancer free state can be reached.

Varying the rate of recruitment of cytotoxic T-lymphocytes (p):

The long term behavior of all the state variables, that is B-Cells , plasma Cells,antibodies, cytotoxic T-lymphocytes and cancer cells are shown in Figure 8for p = 6.8 × 10−8, with other parameter values remaining same (Table 1).B cells grow and reaches its steady state (fig 8A) but the plasma cells showa sharp decline and then increase to reach the steady state (fig 8B). Thismay be due to the fact that the initial plasma cells present in the body isused to secrete antibodies, till replaced by the new plasma cells, obtainedfrom B-cell differentiation. The antibodies also reach the steady state aftersome transient dynamics (fig 8C). With this recruitment value of cytotoxicT-lymphocytes, the cancer free equilibrium is unstable, hence the CTL’s aswell as antibodies fail to control the cancer cells, which attain large cancercell steady state (fig 8E). As p is increased, both CTLs and cancer cells show


oscillations(Figure9(A,B,C,D,E,F)), which increase with the increment of pand ultimately becomes a atable limit cycle (see Figure9(G,H)). Antibodiesshow similar dynamics as seen in (fig 8C).

Varying β1, the death rate of CTL’s by cancer cells:

Figure 10 shows the dynamic of B-Cells , plasma cells, antibodies, CTL’s andcancer cells for β1 = 9.5 × 10−9, other parameter values as in Table 1. Withthis set of parameter values, the cancer free state is unstable. There is nochange of dynamics for B cells, plasma cells and antibody but the dynamicsof the CTL’s and cancer cells are oscillatory as the large cancer cells steadystate (interior equilibrium point) is a stable focus. As we increase the valueof the parameter β1, the changes in the dynamics of antibodies, CTL, andcancer cells is shown in Figure 11 . Figure 11(A, D, G) shows that there isno change in the dynamics of antibodies due to change in the parameter β1.This is because of the fact that there is no interaction between antibodiesand CTLs. But as we increase the value of β1, the oscillation in the cancercells vanishes and reaches a new internal equilibrium(large cancer cells steadystate). This change of dynamics is due to the fact that the interior equilibriumchanges from stable focus(fig.11(C)) to a stable node(Fig.11(I)) as the deathrate of CTLs by the cancer cells increases 11(I).

Varying β2, the kill rate of cancer cells by CTL’s:

By local stability analysis we observed that if β2 >µ3

s (r − β3k1[au − b(1 −u)] [µ1r1+br2(1−u)]

auµ1µ2)(= 1.16 × 10−5 Ab−1hr−1), then the cancer free equilib-

rium is stable otherwise unstable. Therefore, for any value of β2 < 1.16 ×10−5 Ab−1hr−1, the CTL’s and cancer cells reach the interior steady statesthrough damped oscillation (see Figure 12), as it is a stable focus. As the valueof β2 crosses 1.16 × 10−5 and increases, the cancer cells are eradicated veryfast(see Figure 13)(I) since the cancer free equilibrium is stable now and thelarge cancer cells steady state does not exist. The cell count of CTL’s alsodecreases at a faster rate with the increment of β2 (see Figure 13)(H).

Varying β3, the kill rate of cancer cells by antibodies:

Figure 14 shows the dynamics of antibodies, CTL’s and cancer cells for varying

β3. If β3 >auµ1µ2(rµ3−sβ2)

k1[µ1r1+br2(1−u)][au−b(1−u)] (= 3.57× 10−10 Ab−1hr−1), cancer free

equilibrium point is stable(see 14(I)) otherwise unstable. For β3 < 3.57×10−10,the interior equilibrium is a stable focus (see14(B,C,E,F)) , implying high can-cer cells interior point. However for β3 > 3.57 × 10−10, cancer free state isobtained, no matter what the initial size is. Thus, in order to realistically ef-fect a cure when the cancer free equilibrium point is unstable, a therapy or


treatment must ensure that not only the cancer burden must be reduced butthe therapy itself is capable of changing the parameter values of the system.In this context, monoclonal antibody therapy of cancer is suggested as treat-ment, which may be capable of changing the system parameters. Monoclonalantibodies can be used to target a number of cancer associated targets, includ-ing tumor associated blood vessels, vascular growth factors, diffuse malignantcells like leukemia, cancerous cells within a solid tumor and tumor associatedstroma like fibroblasts.

6 Conclusion

We have formulated a mathematical model representing interaction betweencancerous cells and immune system considering both humoral and cell-mediatedimmune system. Using sensitivity analysis we have identified the parametersthat are sensitive and by subset selection, we have identified the one thatare estimable from given experimental data Tutt et al. (2002)(figure 4A, pgNo.2723). We then estimate those parameters using least square method.Though there are four equilibria, we concentrate only on two equilibriumpoints, namely, cancer free equilibrium and high cancer equilibrium. Cancercan be driven to either of these states in simulations, depending on the relativestrength of the patient’s immune systems, considering the role of antibodiesand CTL’s. By simulation, it is revealed that Cytotoxic T lymphocytes andantibodies play an important role in bringing down the cell count of cancercells. This suggest that proper activation of the system parameters in the con-dition obtained, namely β2 and β3(see Theorem 3), the effectiveness of thecytotoxic T lymphocyte and antibodies to kill the cancerous cells respectively.We observe that for certain values of β2 and β3, one can control the unlimitedgrowth of the cancerous cell population. As the values of β2 and β3 crossecertain threshold value, the cancerous cells population always decays to zero,no matter what the initial size is.

The parameter sensitivity analysis yield results that are intuitively rea-sonable, which also highlight which parameter(s) to target so that the cancerburden is reduced. For example, if we get a better idea of how to influencethe parameters β2 and β3 biologically (parameters, which effect the kill ratesof cancer by CTL’s and antibody respectively), a large increase in both ofthem will result in our immune system more efficient to eradicate cancer cellsthan the immune system resulting from a change in the other immune systemparameters.

There are several notable clinical observations that are important for thenext stage of model development. Though the antibodies have the ability to killcancerous cells through direct, the process is not instantaneous and followedby some time lag. This can be captured by introducing interaction delay intothe model, which may mimic some notable clinical observations missing in thismodel.


Acknowledgements This study was supported by the Ministry of Human Resource De-velopment (MHRD) (Grant No. ).

Appendix: Sotomayor Theorem (Perko (1991))

Consider the system of ordinary differential equations

dx

dt= f(x, η). x ∈ Rn,

where η ∈ R R is a system parameter. It is assumed that the function f issufficiently differentiable so that all the derivatives appearing in that theoremare continuous on Rn × R. We denote the matrix of partial derivatives of thecomponents of the vector field f with respect to the components of x by Dfand the vector of partial derivatives of the components of f with respect toparameter η denoted by fη.

Theorem 4 (Perko (1991))(Sotomayar):Suppose that f(x0, η0) and that then× n matrix B ≡ Df(x0, η0)has a simple eigen value λ = 0 with eigen vectorv and that BT has an eigen vector u corresponding to the eigen value λ = 0.Furthermore, suppose that B has k eigenvalues with negative real parts and(n− k − 1) eigenvalues with positive real parts.

(i)If the following conditions are satisfied

uT fη(x0, η0) 6= 0, uT [D2f(x0, η0)(v, v)] 6= 0 (34)

Then there is a smooth curve of equilibrium point of (34) in Rn × R passsingthrough (x0, η0) and tangent to the hyperplane Rn×η0. Depending on the signsof the expressions in (35), there are no equilibrium points of (34) near x0 whenη < η0 (or η > η0) and there are two equilibrium points of (34) near x0 whenη > η0 (or η < η0). The two equilibrium pointss of (34) near x0 are hyperbolicand have stable manifolds of dimension k and k + 1 respectively, that is, thesystem (34)experiences a saddle-node bifurcation at the equilibrium point x0as the parameter η passes through the bifurcation value η = η0. The set of C∞

-vector fields satisfying the above conditions is an open, dense subset in theBanach space of all C∞, one parameter vector fields with an equilibrium pointat x0 having a simple zero eigenvalue.

(ii)If the following conditions are satisfied

uT fη(x0, η0) = 0,

uT [Df(x0, η0)v 6= 0

uT [D2f(x0, η0)(v, v)] 6= 0 (35)

the system(34) experiences a transcritical bifurcation at the equilibrium point(x0, η0) as the parameter η varies through the bifurcation value η = η0.


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Table 1 Parameter values

Parameter Parameter values and Units

a (growth rate of large B-cells) 0.1 hr.−1 (Perelson et al. (1976))b (conversion rate of large B-cellsinto plasma cells) 0.01 hr.−1 (Perelson et al. (1976))µ1 (natural death rate of plasma cells) 0.01 hr.−1 (Perelson et al. (1976))k1 (carrying capacity of large B-cells) 106cells (Perelson et al. (1976))u (the fraction of daughter cellswhich remain as large B-cells) 0.1 (Perelson et al. (1976))r1 (rate at which large B-cellssecrete antibodies) 100 Ab cell−1 sec.−1(Nossal and Makela (1962))r2 (rate at which plasma cellssecrete antibodies) 1000 Ab cell−1 sec.−1(Nossal and Makela (1962))µ2 (natural death rateof antibodies) 1.3666 sec.−1 (estimated)s (natural source term ofCytotoxic T-lymphocyte) 1.3× 104cells/day−1(Kuznetsov et al. (1994))p (increasing rate of Cytotoxic T-lymphocytedue to interaction with cancerous cells) 7.2641× 10−8 Ab−1 sec.−1(estimated)β1 (death rate of Cytotoxic T-lymphocytedue to interaction with cancerous cells) 6.8891× 10−8 Ab−1 sec.−1(estimated)µ3(natural death rate of Cytotoxic T-lymphocyte) 0.1605 sec.−1 (estimated)r (the intrinsic growth rateof cancerous cells) 0.9889 day−1(estimated)k2 (carrying capacityof cancerous cells) 1.336× 109 cells(estimated)β2 (the death rate of cancerous cellsdue to interaction with Cytotoxic T-lymphocyte) 1.9169× 10−10 Ab−1 sec.−1(estimated)β3 (death rate of cancerous cellsdue to interaction with antibodies) 1.6530× 10−11 Ab−1 sec.−1 (estimated)


Fig. 1 The schematic diagram illustrate the interaction between B cells, plasma cells, an-tibody, Cytotoxic T lymphocyte (CTL) and cancer cells.


0 10 20 30 40 50−10

−5

0x 10

8 a

0 10 20 30 40 50−2

−1

0x 10

8 u

0 10 20 30 40 50−20

−10

0

k1

0 10 20 30 40 50−4

−2

0x 10

9 b

0 10 20 30 40 500

2

4x 10

9 µ1

0 10 20 30 40 50−10

−5

0x 10

5 r1

0 10 20 30 40 50−2

−1

0x 10

5 r2

0 10 20 30 40 500

1

2x 10

8 µ2

0 10 20 30 40 50−40

−20

0s

0 10 20 30 40 50−4

−2

0x 10

15 p

0 10 20 30 40 500

2

4x 10

15 β1

0 10 20 30 40 500

5

10x 10

8 µ3

0 10 20 30 40 500

2

4x 10

8 r

0 10 20 30 40 500

2

4x 10

−3 k2

0 10 20 30 40 50

−4

−2

0x 10

15 β2

0 10 20 30 40 50−2

−1

0x 10

16 β3

Fig. 2 The figure shows the sensitivity graph for different parameter of the system.

0 2 4 6 8 10 12 14 160

10

20

30

40

50

60

70

µ3 β

2

β1

r

β3 µ

2r2

p

Fig. 3 The figure indicate the L2 norm of different parameter.L2 norm of the param-eter p, β1, r, β3, r2, µ2, µ3, β2 is large compare to the other parameter of the system. Soµ3, r, p, β1, k2, β2, a, β3, µ2 are more sensitive.


0 1 2 3 4 5 6 70

2

4

6

8

10

12

14x 10

8

Time

Can

cer

Cel

ls

Fig. 4 The figure shows that best fit line for estimation of the parameter r&k2 using realdata Tutt et al. (2002) in absence of immune system. Applying least square method theestimated value of r and k2 are r = 0.9889 day−1 and k2 = 1.336× 109 cells.

4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

8

Time

Can

cer

Cel

ls

Fig. 5 The figure shows the curve of best fit for estimation of the parameterµ2, p, β1, µ3, β2, β3 using real dataTutt et al. (2002). Applying least square method, the es-timated value of the parameters are µ2 = 1.3666 sec−1, p = 7.2641×10−8 Ab−1hr−1, β1 =6.8891 × 10−8 Ab−1hr−1, µ3 = 0.1605 sec−1, β2 = 1.9169 × 10−10 Ab−1hr−1, β3 =1.6530× 10−11 Ab−1hr−1.


Fig. 6 The domain D′ where the cancer free equilibrium point is globally asymptoticallystable.

0 5 10 15 20 25 30 350

2

4

6

8

10x 10

7

Time

Num

ber

of C

ance

r C

ells

Tu0

=9× 107

Tu0

=7× 107

Tu0

=5× 107

Tu0

=3× 107

Tu0

=1× 107

Tu0

=3× 106

0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

8x 10

4

Time

Num

ber

of C

ance

r C

ells

Tu0

=8× 104

Tu0

=5× 104

Tu0

=2× 104

Tu0

=9× 103

Tu0

=3× 103

Tu0

=1× 103

A B

Fig. 7 The figures show the global stability of the cancer free equilibrium point. The cancerfree equilibrium is globally stable for r < 0.0457. The values of r = 0.0099

.


0 10 20 300.9

0.95

1

1.05

1.1x 10

4

Time

CT

L

0 100 200 300 400 5002

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y

0 10 20 301

1.1

1.2

1.3x 10

9

Time

Can

cer

Cel

ls

0 200 400 600 800 10006

6.2

6.4

6.6

6.8

7x 10

6

Time

B C

ells

0 200 400 600 800 10002.85

2.9

2.95

3

3.05

3.1x 10

6

TimeP

lasm

a C

ells

ED

A B C

Fig. 8 The figure shows the dynamics of all the state variables, namely B cells, Plasmacells, Antibodies, CTL and Cancer cells for p = 6.8×10−8 Ab−1hr−1 with initial conditionB(0) = 6× 106, P (0) = 3× 106, A(0) = 2× 109, Tc(0) = 9× 104, Tu(0) = 1× 109, otherparameter values are same as given Table 1.


0 100 200 3003.5

4

4.5

5x 10

9

Time

CT

L

0 50 100 150 200 250 3001

1.5

2x 10

8

Time

Can

cer

Cel

ls

0 500 1000 1500 20003

4

5

6x 10

9

Time

CT

L

0 500 1000 1500 20000

1

2

3x 10

7

Time

Can

cer

Cel

ls

0 500 1000 1500 2000 2500 30002

4

6

8x 10

9

Time

CT

L

0 500 1000 1500 2000 2500 30000

5

10x 10

4

Time

Can

cer

Cel

ls

3.5 4 4.5 5 5.5 6

x 109

0.5

1

1.5

2

2.5x 10

7

CTL

Can

cer

Cel

ls

3.5 4 4.5 5 5.5 6 6.5

x 109

0.5

1

1.5

2x 10

6

CTL

Can

cer

Cel

ls

C

HG

A B

D

FE

Fig. 9 The figures(A,B,C) show the dynamics of Antibody, CTL and Cancer cells fordifferent values of p keeping other parameter value fixed as in Table 1.(i) figure A is forp = 7 × 10−8 Ab−1hr−1 with initial condition B(0) = 6 × 106, P (0) = 3 × 106, A(0) =2×109, Tc(0) = 4×109, Tu(0) = 1×108 (ii) figure B is for p = 8.1×10−8 Ab−1hr−1 withinitial condition B(0) = 6×106, P (0) = 3×106, A(0) = 2×109, Tc(0) = 4×109, Tu(0) =1 × 107 (iii) figure C is for different p = 4.5 × 10−6 Ab−1hr−1 with initial conditionB(0) = 6 × 106, P (0) = 3 × 106, A(0) = 2 × 109, Tc(0) = 4 × 109, Tu(0) = 3 × 104.Figure(D,E,F) shows the formulation of limit cycle as we increase the value of p. Thevalues of p are in figure (D) p = 7.2 × 10−8, figure (E) p = 8.1 × 10−8 and figure (F)p = 2× 10−7.


0 200 400 600 800 10006

6.2

6.4

6.6

6.8

7x 10

6

Time

B C

Ells

0 200 400 600 800 10002.85

2.9

2.95

3

3.05

3.1x 10

6

TimeP

lasm

a C

ells

0 100 200 300 400 5001.8

2

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y

0 1000 2000 30003

4

5

6

7x 10

9

Time

CT

L

0 1000 2000 30001

2

3

4

5x 10

6

Time

Can

cer

Cel

l

ED

B CA

Fig. 10 The figure shows the dynamics of all the state variables, namely B cells, Plasmacells, Antibodies, CTL and Cancer cells for β1 = 9.5×10−9 Ab−1hr−1 with initial conditionB(0) = 6× 106, P (0) = 3× 106, A(0) = 2× 109, Tc(0) = 4× 109, Tu(0) = 2× 106 , otherparameter values as in Table 1.

0 100 200 300 400 5002

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y

0 200 400 600 800 10003.5

4

4.5

5

5.5

6x 10

9

Time

CT

L

0 500 1000 15001

1.5

2

2.5

3

3.5x 10

7

Time

Can

cer

Cel

ls

0 100 200 300 400 5002

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y

0 20 40 60 80 1002.5

3

3.5

4

4.5x 10

9

Time

CT

L

0 20 40 60 80 1000

5

10

15x 10

8

Can

cer

Cel

ls

0 100 200 300 400 5002

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y

0 20 40 60 80 1001

1.1

1.2

1.3x 10

5

Time

CT

L

0 5 10 15 201

1.1

1.2

1.3x 10

9

Time

Can

cer

Cel

ls

CB

D E F

G H

A

I

Fig. 11 The figures show the dynamics of Antibody, CTL and Cancer cells for differentvalues of β2 keeping other parameter value fixed as on Table 1.(i) figure A is for β1 =6.5 × 10−8 Ab−1hr−1 with initial condition B(0) = 6 × 106, P (0) = 3 × 106, A(0) =2× 109, Tc(0) = 4× 109, Tu(0) = 2× 107 (ii) figure B is for β1 = 7.2× 10−8 Ab−1hr−1

with initial condition B(0) = 6 × 106, P (0) = 3 × 106, A(0) = 2 × 109, Tc(0) = 3 ×109, Tu(0) = 2×108 (iii) figure B is for β1 = 7.26×10−8 Ab−1hr−1 with initial conditionB(0) = 6× 106, P (0) = 3× 106, A(0) = 2× 109, Tc(0) = 1× 105, Tu(0) = 1× 109.


0 100 200 300 400 5002.5

3

3.5

4

4.5x 10

9

Time

CT

L

0 100 200 300 400 5002

3

4

5

6

7x 10

7

TimeC

ance

r C

ells

0 200 400 600 800 10006

6.2

6.4

6.6

6.8

7x 10

6

Time

B C

ells

0 200 400 600 800 10002.85

2.9

2.95

3

3.05

3.1x 10

6

TimeP

lasm

a C

ells

0 100 200 300 400 5002

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y

A B C

D E

Fig. 12 The figure shows the dynamics of all the state variables, namely B cells, Plasmacells, Antibodies, CTL and Cancer cells for β2 = 2.5× 10−10 Ab−1hr−1 with initial condi-tion B(0) = 6 × 106, P (0) = 3 × 106, A(0) = 2 × 109, Tc(0) = 3 × 109, Tu(0) = 4 × 107,other parameter values as in Table 1.

0 10 20 30 40 508

8.02

8.04

8.06

8.08

8.1

8.12x 10

4

Time

CT

L

0 5 10 150

2

4

6

8

10x 10

5

Time

Can

cer

Cel

ls

β2=1.7×10−5

β2=2×10−5

β2=2.9×10−5

β2=6.5×10−5

β2=1.7×10−5

β2=2×10−5

β2=2.9×10−5

β2=6.5×10−5

0 200 400 6008

8.1

8.2

8.3

8.4x 10

4

Time

CT

L

0 200 400 6001

1.1

1.2

1.3

1.4x 10

6

Time

Can

cer

Cel

ls

0 50 100 150 2001

1.1

1.2

1.3

1.4x 10

5

Time

CT

L

0 50 100 150 2001

1.5

2

2.5x 10

7

Time

Can

cer

Cel

ls

0 100 200 300 400 5002

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y

0 100 200 300 400 5002

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y

0 100 200 300 400 5002

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y

B

E

F

H

A

D

G

C

I

Fig. 13 The figures show the dynamics of Antibody, CTL and Cancer cells for differentvalues of β2 keeping other parameter value fixed as on Table 1.(i) figure A is for β2 =7.5 × 10−6 Ab−1hr−1 with initial condition B(0) = 6 × 106, P (0) = 3 × 106, A(0) =2×109, Tc(0) = 1×105, Tu(0) = 1×107 (ii) figure B is for β2 = 1.13×10−5 Ab−1hr−1 withinitial condition B(0) = 6×106, P (0) = 3×106, A(0) = 2×109, Tc(0) = 8×104, Tu(0) =1× 106 (iii) figure C is for increasing β3 where( β2 > 1.16× 10−5 Ab−1hr−1) with initialcondition B(0) = 6×106, P (0) = 3×106, A(0) = 2×109, Tc(0) = 8×104, Tu(0) = 1×106.


0 10 20 30 40 508

8.02

8.04

8.06

8.08

8.1

8.12x 10

4

Time

CT

L

0 5 10 150

2

4

6

8

10x 10

5

Time

Can

cer

Cel

l

0 100 200 300 400 5002

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y β3=5.6×10−10

β3=1.7×10−9

β3=6.9×10−10

β3=9.5×10−10

β3=5.6×10−10

β3=9.5×10−10

β3=1.7×10−9

β3=6.9×10−10

0 200 400 600 800 10000

2

4

6

8x 10

8

Time

CT

L

0 200 400 600 800 10002

4

6

8x 10

7

Time

Can

cer

Cel

ls

0 100 200 300 400 5002

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y

0 100 200 300 400 5001

2

3

4x 10

9

Time

CT

L

0 100 200 300 400 5002

2.2

2.4

2.6

2.8x 10

9

Time

Ant

ibod

y

0 100 200 300 400 5002

4

6

8x 10

7

Time

Can

cer

Cel

ls

A B C

D EF

GH

I

Fig. 14 The figures show the dynamics of Antibody, CTL and Cancer cells for differentvalues of β3 keeping other parameter value fixed as on Table 1.(i) figure A is for β3 =1.5 × 10−10 Ab−1hr−1 with initial condition B(0) = 6 × 106, P (0) = 3 × 106, A(0) =2×109, Tc(0) = 2×109, Tu(0) = 4×107 (ii) figure B is for β3 = 3.4×10−10 Ab−1hr−1 withinitial condition B(0) = 6×106, P (0) = 3×106, A(0) = 2×109, Tc(0) = 7×107, Tu(0) =4× 107 (iii) figure C is for different β3 where( β3 > 3.574× 10−10 Ab−1hr−1) with initialcondition B(0) = 6×106, P (0) = 3×106, A(0) = 2×109, Tc(0) = 8×104, Tu(0) = 1×106.

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