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2009;69:1205-1211. Published OnlineFirst January 27, 2009.Cancer ResLeticia R. Paiva, Christopher Binny, Silvio C. Ferreira, Jr., et al.A Multiscale Mathematical Model for Oncolytic Virotherapy

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A Multiscale Mathematical Model for Oncolytic Virotherapy

Leticia R. Paiva,1

Christopher Binny,2

Silvio C. Ferreira Jr.,1

and Marcelo L. Martins1

1Departamento de Fsica, Universidade Federal de Vicosa, Vicosa, MG, Brazil and 2Centre for Molecular Oncology, Institute of Cancer,Barts and the London School of Medicine and Dentistry, Queen Mary University of London, London, United Kingdom

Abstract

One of the most promising strategies to treat cancer is

attacking it with viruses. Oncolytic viruses can kill tumor cells

specifically or induce anticancer immune response. A multi-

scale model for virotherapy of cancer is investigated through

simulations. It was found that, for intratumoral virus

administration, a solid tumor can be completely eradicated

or keep growing after a transient remission. Furthermore,

the model reveals undamped oscillatory dynamics of tumor

cells and virus populations, which demands new in vivo and

in vitro quantitative experiments aiming to detect this

oscillatory response. The conditions for which each one of

the different tumor responses dominates, as well as the

occurrence probabilities for the other nondominant thera-

peutic outcomes, were determined. From a clinical point of

view, our findings indicate that a successful, single agent

virotherapy requires a strong inhibition of the host immune

response and the use of potent virus species with a high

intratumoral mobility. Moreover, due to the discrete and

stochastic nature of cells and their responses, an optimal

range for viral cytotoxicity is predicted because the virother-

apy fails if the oncolytic virus demands either a too short or

a very large time to kill the tumor cell. This result suggests

that the search for viruses able to destroy tumor cells very

fast does not necessarily lead to a more effective control of

tumor growth. [Cancer Res 2009;69(3):120511]

Introduction

One of the most promising strategies to treat cancer is attacking

it with viruses. Oncolytic viruses are biological nanomachines that

can be programmed to specifically target, replicate in, and

ultimately kill cancer cells. The new viruses released from the

lysed cells can subsequently infect adjacent or distant tumor cells

and trigger multiple cycles of infection. All the recombinant DNA

technology can be readily used to enhance the selectivity of

oncolytic viruses for tumor cells, thereby limiting the infection of

normal cells and minimizing therapeutic side effects (1). Several

approaches to selectivity have been explored; for example,

specifically retargeting viruses to cancer cells (2); linking virusreplication to cancer-specific transcription factors such as

telomerase (3) or prostate-specific antigen (4); and deleting viral

genes for advancing cell cycle and/or preventing apoptosis,

typically unnecessary for infecting cancer cells (1). An infected

tumor cell is eventually killed by the oncolytic virus through severalroutes such as lysis, apoptosis, immune-modulated cell killing, or

other forms of programmed cell death (5).

Over the past decade, clinical trials have been conducted with at

least 14 distinct viruses from six different viral strains (6). Although

the arsenal of oncolytic viruses expands continuously, to date only

two viruses (Onyx-015 and H101) have entered randomized phase

III clinical testing (1, 6). The data from these clinical trials have

established the safety, effectiveness, and lack of cross-resistance to

currently available treatments of virotherapy. Hence, in the near

future, this approach has the potential to supplant many of the

current standard anticancer therapies, mainly if combined with

existing treatments. However, several hurdles such as the host

immune response, systemic distribution, and intratumoral spreadmust be overcome. Because our clinical experience is limited,

mathematical models may provide relevant insights about the

nonlinear dynamics ruling the interplay between normal and

cancer cells and their viral parasites. A quantitative understanding

of this complex network of interactions will certainly accelerate

the further development necessary to turn virotherapy into a fully

functional and effective therapeutics.

In this article, a virotherapy model recently proposed (7) is

investigated. It considers normal cells, uninfected and infected

tumor cells, necrotic cells, and free viruses. At the tissue level,

reaction-diffusion equations describe the dynamics of nutrients

and viruses, both small particles in comparison with cells. In these

reaction-diffusion equations, cells act as sinks and sources ofnutrients and viruses depending on their internal states. In turn,

cell dynamics is a stochastic process controlled by the local

concentration of nutrients and free viruses determined at the tissue

level. Thus, the model involves partial differential equations and

probabilistic cellular automata rules to describe the multiscale

dynamics of tumor growth (8, 9). It is worth mentioning that this

model was designed for the avascular stages of cancer growth.

However, the effects of therapies even at these stages are crucial for

the oncologic practice because usually clinically detected tumors

have often already generated micrometastases that are growing

avascularly in other tissues of the organism. Thus, for instance, of

the hundreds of thousand of people who develop colorectal cancer

every year, as many as 50% of them have synchronous or will havelater hepatic metastases (10). Furthermore, the current available

treatments offer essentially no hope for cure to those patients with

unresectable liver metastases.

Here, the aim is to perform an extensive analysis of the system

behavior in a subspace of the model parameter space. Our results

reveal that, for intratumoral virus administration, the solid tumor

can be completely eradicated or keep growing after a transient

remission depending on the parameter values. The regions in

parameter space associated to each one of such tumor responses

were determined. Finally, our findings are compared with the

reported results from clinical trials and the predictions of two other

mathematical models based on systems of ordinary (11) or partial

Note: Supplementary data for this article are available at Cancer Research Online(http://cancerres.aacrjournals.org/).

Requests for reprints: Marcelo L. Martins, Departamento de F sica, UniversidadeFederal de Vicosa, 36570-000 Vicosa, MG, Brazil. Phone: 55-31-3899-2993; Fax: 55-31-3899-2453; E-mail: [email protected] American Association for Cancer Research.doi:10.1158/0008-5472.CAN-08-2173

www.aacrjournals.org 1205 Cancer Res 2009; 69: (3). February 1, 2009

Research Article


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(12) differential equations. Differently from them, the present

model assumes that each cell is an individual agent and uses a

sigmoidal cell infection rate (instead of a linear one).

Materials and Methods

The mathematical model represents the tissue as a square lattice fed by a

single capillary vessel at the top of the lattice. As illustrated in Fig. 1, four

different cell types are considered: normal cells (rn), dead cells (rd), anduninfected (rc) and infected (rv) tumor cells. In contrast to the normal and

dead cells, one or more uninfected or infected cancer cells can pile up in a

given site, reflecting the fact that the division of tumor cells is not

constrained by contact inhibition. Normal cells cannot pile up because they

are not dividing. Piled, infected cancer cells result from infection of

uninfected cancer cells in a pile. The nutrients diffuse from the capillary

vessel through the tissue. Each cancer cell, randomly selected with equal

probability, can carry out one of three actions: replication, death, or

becoming infected if yet uninfected. In contrast, an infected cancer cell does

not divide because its slaved cellular machinery is focused on virus

replication. It is assumed that infected cancer cells sustain their metabolism

until lysis. In addition, infected cancer cells mainly die by lysis, and the

death due to insufficient nutrient supply is neglected. The lysis of each

infected cancer cell releases v0 viruses to the extracellular medium. Because

virions are present throughout the infected cell, a random fraction of these

viruses remains on the site of the lysed cell, and the rest is uniformly

distributed among its four nearest neighbors at the time of lysis. The

infection of a normal cell by the oncolytic virus is neglected. Thus, the

model assumes perfect viral selectivity for cancer cells. At last, the clearance

rate (cv) embodies the complex innate and adaptive immune responses to a

virus. Such response involves the synthesis of antiviral cytokines, activation/

selection of immune cells, and production of antiviral antibodies (13). The

free viruses diffuse through the tissue. The mathematical details of the rules

and partial differential equations controlling cellular actions, nutrient

diffusion, and virus spreading are described in Appendix.

Figure 1 also illustrates some of the multiple processes involved in the

model. At the tissue level (macroscopic scale), the nutrient molecules and

viruses are both very small particles in comparison with cells. Indeed, virus

species tested in cancer clinical trials form particles with sizes that vary

from 18 to 30 nm (picornaviruses) to 300 to 400 nm (vaccinia viruses; refs. 1,

6), whereas a cell is typically 10 Am in diameter. These molecules and

viruses disperse in the tissue like Brownian particles. Thereby, reaction-

diffusion equations are used to describe their dynamics. In these equations,

normal and cancer cells are sinks of nutrients; cancer cells become sinks of

viruses on infection; and infected tumor cells act as sources of viruses at

lysis. Thus, the evolution in time of cells determines the macroscopic

distribution of nutrients and free viruses. In turn, cell division, death,

infection, and lysis, occurring at the cellular or mesoscopic scale, arestochastic processes controlled by the local concentration of nutrients and

viruses determined at the macroscopic scale. Such cell actions occur at a

time scale (f10 h) much greater than that associated to nutrient and virus

diffusions (f10 s). Hence, cell dynamics evolves in a stationary state of

these concentration fields.

Summarizing, both mesoscopic and macroscopic scales are inexorably

interwoven, although described in terms of distinct physical models: partial

differential equations (macroscopic level) and probabilistic cellular

automata rules (mesoscopic level) coupled in a single model through

division, death, infection, lysis, uptake, release and adsorption rates, and

diffusion coefficients.

Results

We are interested in the effects of virotherapy on solid tumorscommonly used in experimental assays. Figure 2 shows the total

number of uninfected cancer cells and corresponding virus

concentrations as functions of time for different virus diffusion

and clearance rates. The delay between virus and uninfected tumor

cell dynamics is due to the period for cell lysis (Tl). Depending on

the parameter values (Table 1), the tumor can be completely

eradicated few time steps following a single intratumoral virus

administration, undergo damped oscillations leading to eradication

at a later time, or sustain an overall tumor growth. In this case, after

a transient shrinkage of the neoplastic mass, the growth can be

oscillatory and at a reduced rate or monotonous (nonoscillatory)

and even at an increased rate. Without treatment, the progress in

Figure 1. The model considers normal cells, tumor cells, and oncolytic viruses. Uninfected cancer cells can replicate by mitosis or die due to starvation (lack ofnutrients) or be infected by viruses. Such cell actions are stochastic and governed by probabilities Pdiv, Pdel, and Pinf dependent on the nutrient and virus concentrationsper cancer cell. Analogously, normal and infected cancer cells die due to starvation and lysis with probabilities Pdel and Plysis, respectively. The functional formsof such cell action probabilities are stated in Appendix. After mitosis, the new tumor cell piles up (loss of contact inhibition) at the site of its parent cell if the last is insidethe tumor. Otherwise, if the dividing cell is on the tumor border, its daughter cell will randomly replace one of their normal or necrotic nearest neighbor cells. Nutrients andviruses are modeled as continuous concentrations that evolve in space (tissue) and time according to reaction-diffusion equations described in Appendix. Essentially,nutrients diffuse at a rate D from the capillary vessel through the tissue and are taken up by normal and tumor cells at rates c and kc , respectively. In addition,free viruses diffuse at a rate Dv, are cleared at a rate cv, and have as source infected cancer cells, which release v0 new viruses after their lysis. The virus clearancerate represents the simplest way to model viral loss due mainly to the innate and adaptive immune responses mediated by antibodies, CD8 T cells, IFNs, andother cytokines or inherent viral instability. It can also take into account other mechanisms such as the virus outflow from the tumor driven by the gradient ofinterstitial fluid pressure and eventual virus trapping by the extracellular matrix. cv is always a constant, but several distinct values were simulated.

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time of cancer cell populations for all the simulated patterns is

Gompertzian (9, 14). The curves in Fig. 2 C and D correspond to

failed therapies because the cancer keeps growing with time.

A striking and novel result shown in Fig. 2 is the oscillatorybehavior of both cancer cells and viruses seen at higher viral

diffusivity and clearance rates. It contrasts to the long-term,

spatially uniform profiles for cells and free viruses obtained by Wu

and colleagues (15). Instead of only damped oscillations, the total

numbers of viruses and uninfected and infected cancer cells can

also undergo stable aperiodic oscillations superimposed on a

steady growth.

In Fig. 3, snapshots of the simulated spatial patterns of

uninfected tumor cells are shown. The parameter values used are

those for the curve C in Fig. 2. Successive waves of virus infection

and therefore cell death by lysis are neatly evident. However, these

rounds of infection are not strong enough to eradicate the tumor.

Videos S1 and S2 for unsuccessful and successful treatments,

respectively, are provided as supplementary information.

A fundamental feature uncovered by the simulations of the

model is its great sensitivity to the local, stochastic fluctuations ofuninfected cell population and free virus concentration. Thus, for

instance, although the complete eradication of the solid tumor is

the most frequent therapeutic outcome, the other two types of

tumor response are also observed for cv = 0.01, Dv = 0.7, and Tl = 4.

Hence, it becomes imperative to find the dominant tumor response

in every region of the model parameter space. For a given tumor

cell type (uinf and Tl fixed), the dominant response was determined

for each pair of parameters Dv and cv controlling free virus

spreading. Whatsoever the values of Dv and Tl, the therapy always

fails if the virus clearance rate is large (cvz 0.03). However, a small

decrease in cv with all the other parameters (Dv, Tl, and u inf) fixed

can lead to complete tumor eradication (see Supplementary

Figure 2. Temporal evolution of the number of uninfected cancer cells (Nc) and total free virus concentration with fixed virus infectivity u inf = 0.03 and clearance ratecv = 0.01. A single intratumoral virus load is administered at the time indicated by the arrow. The following virus parameters were used: Dv = 0.7 and Tl = 4, rapideradication (A ); Dv = 0.8 and Tl = 2, slow oscillatory eradication (B); Dv = 0.2 and Tl = 2, oscillatory growth at a reduced rate (C); and Dv = 0.9 and Tl = 32, exponentialgrowth (D). These curves represent the most probable behavior observed for the corresponding sets of model parameters. Each one refers to a single simulation.In all of them, the viruses inside infected cells are not counted for determining free virus concentration in the system.

Multiscale Model for Virotherapy




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Fig. S1). Thereby, virus clearance rate is a key factor determining

therapeutic outcome. In turn, in a tissue characterized by

a moderate clearance rate (0.01 V cv < 0.03), ifu inf z 0.1 (i.e., the

virus has a low efficacy to invade uninfected cancer cells), the

tumor mass steadily grows and, again, the treatment fails. In

Fig. 4, the boundaries corresponding to the various therapeutic

outcomes of an aggressive virotherapy are shown.

Finally, in addition to determining the most probable prognosis

after virotherapy, it is relevant to know the chances for the

occurrence of the other tumor responses in every region in the

parameter space. This is especially true for the probability of

therapeutic success. For the aggressive treatment considered here,those probabilities are shown in Fig. 5.

Discussion

Computer simulations of our multiscale model revealed that,

after intratumoral virus administration, a solid tumor can

completely be eradicated or keep growing after a transient

remission. Furthermore, depending on the parameter values, the

tumor cell and virus populations exhibit undamped oscillatory

dynamics around increasing, nonstationary values (Fig. 2). The

most likely therapeutic responses are determined by the oncolytic

activity (u inf and Tl) and spreading properties (Dv and cv) of the

virus. These are the key parameters that control treatment success.

Figure 3. Snapshots of cancer cell patterns at intervals of 3 time steps after virus injection. The first pattern corresponds to the solid tumor immediately before thetreatment. The parameters are the same used for the oscillatory growth at a reduced rate shown in Fig. 2 C. Such spatiotemporal evolution can hardly be observedexperimentally either in vitro or in vivo. Indeed, histologic samples require killing of the cells, and noninvasive methods (PET/fluorescently labeled virus) oftenprovide fuzzy pictures characterized by a low spatial accuracy.

T ab le 1. Parameter values used for a compactmorphology (udiv = 0.3, udel = 0.03 without cell migration)

Parameter Description Value

D Glucose diffusion constant 3.6 101 mm2 h1

c Nutrient uptake rate of

normal cells

1065 104 s1

k cc/c 10Dv Virus diffusion coefficient 10

2 mm2 h1

cv Virus clearance rate 2.5 102 h1

Tl Characteristic time for cell lysis 20 h

u inf 0.911 MOI, assuming Pinf = 70%

just after intratumoral injection

MOI e [0.01,0.1]

D Linear cell size (lattice constant) 10 Am

Abbreviation: MOI, multiplicity of infection.

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For example, independently of the viral diffusivity, infectivity, and

time for cell lysis, a high clearence rate (cv z 0.03) leads to a

neoplastic mass and an invaded area that grow monotonously and

even faster than without treatment. Because the immune response

directed at free virus and viral antigens expressed on the surface of

infected cancer cells enhances cv, this result indicates the necessity

of a strong ablation of the patient immune system. Virotherapy also

fails when the tumor growth is oscillatory and progresses at a small

rate. This is the most frequent outcome of varied treatments.

For instance, the use of an oncolytic virus with a low infectivity

(u infz 0.1) in a tissue characterized by a moderate virus clearance

rate (0.01 V cvV 0.03). Here, the failure is caused by the inefficiencyof the viruses at infecting tumor cells, not by the strength of the

host immune response. The same outcome can be obtained in such

a tissue for oncolytic viral agents with greater infectivity (0.01 V

u inf < 0.03) but a small diffusivity (Dv V 0.2; Fig. 3). Now, instead of

the host immune response or virus inefficacy to infect tumor cells,

the therapeutic failure is due to infection waves expanding slower

than the neoplastic mass. Moreover, because a fraction of the

tumor cells are killed by the virus, the nutrient supply available for

the uninfected cells becomes larger. In response, they divide more

frequently, spread, and invade a larger area of the tissue.

With regard to a successful therapy, our results indicate that the

key factor is really the virus clearance rate. Indeed, in a host tissue

with small virus clearance rate (0.005 V cv < 0.01), even mildoncolytic viruses (0.03 V u inf < 0.1) with a rapid virus spreading

(Dv z 0.3) can eradicate the tumor. In agreement, in vivo

experiments reveal that the potency of oncolytic viruses is

markedly reduced when moved to an immunocompetent model

(16). It is also important to note that, in an immunocompentent

host, the evolving immune response to the virus will markedly

change the effective viral clearance rate in a complex way that is

currently difficult to predict (17).

The time for lysis cannot be either too short (Tl < 2 time steps)

o r v er y l on g (Tl > 16 t im e ste ps) . I nd ee d, d ue t o t he

nonhomogeneous virus dynamics, new infection waves cannot

reach the tumor border because either they are infrequent and thus

start far from this border or the large voids generated inside the

tumor by rapid cell lysis cannot provide further fuel for such waves.

Hence, from a clinical point of view, our findings indicate that a

successful, single agent virotherapy requires a strong inhibition of

the host immune response and the use of potent virus species with

Figure 4. Boundaries in parameter space associated to different tumorresponses to an aggressive virotherapy (cv = 0.01, low virus clearance rate;u inf = 0.03, high virus infectivity). In regions labeled A , the cancer is eradicated(absolutely successful treatment); in C, the tumor growth progress at a reducedrate and oscillating, whereas in D the growth is monotonic at a rate that can evenbe greater than that of untreated tumors. In both cases the therapy fails. Thisphase diagram was drawn from simulations involving 200 independentsamples (runs) for 50 points in parameter space arranged as a square lattice.Each simulation lasts for 500 time steps or f3 mo, because a time step is

estimated as f4 h.

Figure 5. Probability for tumor eradication (A ), monotonic growth (B), andoscillatory growth (C) as a function of the parameters Tl and Dv. Both u inf = 0.03and cv = 0.01 remain fixed. The probabilities were evaluated from samplesincluding 200 independent, 500 time-step runs for each pair (Tl, Dv) considered.





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a high intratumoral mobility. A rapid immune-mediated clearance

can be avoided through maintenance or incorporation of virus-

encoded immune response modifier genes or nongenetic

approaches such as coating the viruses with polymers (1, 6).

Possibly, viral genes associated with immune evasion could be used

to reduce the viral clearance rate (18). In contrast, deliberate up-

regulation of the immune response to target cancer cells is also

being considered (19). Approaches to increasing oncolytic efficacy

include arming the viruses with directly lethal (20) or prodrug-converting (21) enzymes. Enhanced intratumoral diffusivity is

possible if, for instance, the virus is engineered to incorporate a

gene encoding for a matrix metalloprotease. This enzyme is

secreted by the infected cancer cell before its lysis and breaks down

the surrounding extracellular matrix, facilitating the spreading of

the viruses released after cell death (22).

The above scenario about the most frequent therapeutic

outcomes is a simplified one because their corresponding regions

in the model parameter space are irregular and intertwined (Fig. 4).

Furthermore, the model is sensitive to the local, stochastic

fluctuations of uninfected cell population and free virus concen-

tration. As a result of such sensitivity, the history of cancer

progression and therapeutic outcome can drastically vary even fora given set of fixed parameters.

A relevant issue is how our results compare with those obtained

from clinical trials (1, 6). Even with intratumoral administration,

wild or engineered adenoviruses, herpes viruses, and reoviruses

have characteristically induced only transient tumor remissions (6).

Thus, for instance, single intratumoral injection of hrR3 virus into

HT29 flank tumors in nude mice has only reduced the rate of

monotonic growth of this human colon carcinoma (10), a response

predicted in Fig. 2D. In addition, transient antitumoral responses

were observed after single intratumoral administration of Reolysin

(reovirus) and dl1520 (adenovirus ONYX-015) into superficial (23)

and head and neck (2426) cancers. Because intratumoral spread

seems to be inefficient with adenovirus (ref. 6; small diffusivity),

Figs. 4 and 5 can explain why virotherapy using dl1520 fails.Unfortunately, in these phase I and phase II clinical trials with 11%

and 14% of positive responses, respectively, it was not reported

if the tumor growth progress either in a monotonic or oscillatory

way. A neat indication of such oscillatory tumor growth is observed

in an ovarian cancer xenograft model attacked with MV-CEA

(an engineered measles virus) administered intratumorally (27). By

contrast, vaccinia viruses have induced durable complete responses

in patients after intratumoral or even i.v. injection. Specifically, the

BP (28) and the engineered JX-594 (29) vaccinia viruses eradicated

melanomas in 4 (13%) and 2 (29%) patients, respectively. These

successful therapeutic outcomes, corresponding to Fig. 2A or B,

have frequencies consistent with eradication probabilities smaller

than 35% obtained from the model simulations for Dv V 0.6 andTl > 5 h (Fig. 5A). Again, a long-term oscillatory eradication was not

evidenced in reported clinical trails. Accurate monitoring of tumor

burden is often difficult due to irregularly shaped and inaccessible

tumors. This, combined with the contribution of dead cells to the

tumor mass, would make detection of oscillating responses very

difficult. New noninvasive imaging techniques such as high-

sensitivity fluorescence imaging and positron emission tomography

(PET) scanning (30) could facilitate quantitative monitoring of

both living tumor mass and viral load in vivo . This approach could

be used to test the predictions of our model.

Very recent results obtained by Nagano and colleagues (31)

reveal the spatial distribution of herpes simplex virus inside

tumors developed in mice. These authors show that the induction

of cancer cell apoptosis before the virus administration enhances

intratumoral virus spreading and antitumor efficacy by generating

voids and channels inside the tumor. With the main barriers

against virus spreading disrupted, intratumoral diffusion is

likely to be closer to the regimen assumed in our model (i.e.,

viruses can always diffuse through the neoplastic mass). Indeed,

the spatial patterns of virus distribution shown in Fig. 2A and B

of Naganos experiment are very similar to the free virussimulated patterns exhibited in our Supplementary videos S1

and S2.

Although the present model does not consider the natural

barriers to virus spreading (fibrillar collagen in the extracellular

matrix, narrow space between tumor cells, and contrary

interstitial pressure gradient) or intratumoral heterogeneity in

the virus diffusivity (as in necrotic areas, for instance), it can

predict that virotherapy has a great chance to fail if virus

diffusivity is very low, in complete agreement with the findings

of Nagano and colleagues (31). Furthermore, our results indicate

that virus diffusivity is only part of the puzzle. Larger diffusivity is

desirable, but a low virus clearance rate and an adequate lysis

time are also necessary to sustain successive waves of infectionrequired for tumor eradication.

Finally, our results are compared with those provided by two

other mathematical models. Wein and colleagues (12) modeled the

dynamics of oncolytic viruses through a deterministic set of partial

differential equations. They found that, in the case of a uniform

virus injection, the tumor mass either grows exponentially or is

eradicated if the waves of infection can or cannot outpace the

tumor proliferation. Oscillations in time of the virus population

were explicitly shown. Furthermore, because the wave speed of the

infection is inversely proportional to Tl (the time for cell lysis),

these authors suggest that the most efficacious viruses will be

those that kill an infected tumor cell faster. In contrast, our results

indicate that there is an optimal range for Tl. Indeed, if the time

for cell lysis is very short, the waves of infection cannot control theneoplastic growth. A few isolated cancer cells, distant from the

large voids produced by rapid cell lysis and plenty of free viruses

looking for targets, provide the opportunity to tumor escape.

This fundamental and novel prediction seems to emerge from the

stochastic nature of our model in which the entry of a virus in a

cell, the lysis of an infected cell, the replication or death of a

uninfected cell, etc., are all random, individual events. Cell

discreteness and stochasticity of cellular responses deeply affect

the strength of the fluctuations, particularly at tumor regions in

which the number of cancer cells is very low. Surprisingly, the

deterministic, ordinary differential equation model proposed by

Wodarz (11) also predicts an optimal viral cytotoxicity. In this

model, the infection of a cancer cell occurs only through contactwith an infected cell, which, in turn, is killed by the virus at a rate

a , the viral cytotoxicity. According this model, if there are

uninfected cancer cells, an increase in the viral cytotoxicity

increases the tumor mass because infected cells are eliminated

before the virus had a chance to significantly spread. The

reciprocal of a should be interpreted as the time for cell lysis Tl.

However, this model can only exhibit damped oscillations in tumor

cell populations.

In summary, we investigate a multiscale model for virotherapy of

cancer in which the cells are discrete agents. Our predictions are in

qualitative agreement with results from clinical reports. Through

extensive simulations, it was found that, for a single intratumoral

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virus administration, a solid tumor can completely be eradicated or

keep growing after a transient remission. Furthermore, the model

reveals undamped oscillatory dynamics of tumor cells and virus

populations, which demands for new i n vivo and in vitro

quantitative experiments aiming to detect this oscillatory response.

The conditions (regions in the model parameter space) for

domination of each one of the different tumor responses, as well

as the occurrence probabilities for the other nondominant

therapeutic outcomes, were determined. From a clinical point ofview, our findi ngs indicate that a successful, single agent

virotherapy requires a strong inhibition of the host immune

response and the use of potent virus species with intratumoral high

mobility. Moreover, due to the discrete and stochastic nature of

cells and their responses, an optimal range for viral cytotoxicity is

predicted because the virotherapy fails if the oncolytic virus

demands either a too short or a very large time for killing the

tumor cell. This finding suggests that the virus that kills cancer

cells most rapidly is not necessarily the more effective agent to

eradicate the tumor. The implications of such a result for the

design of new replication-competent viruses are clear.

Disclosure of Potential Conflicts of Interest

No potential conflicts of interest were disclosed.

Acknowledgments

Received 6/10/2008; revised 8/28/2008; accepted 10/3/2008.Grant support: Brazilian Agencies Coordenacao de Aperfeicoamento de Pessoa l

de N vel Superior, Conselho Nacional de Desenvolvimento Cient fico e Tecnologico ,and Fundacao de Amparo a` Pesquisa do Estado de Minas Gerais.

The costs of publication of this article were defrayed in part by the payment of pagecharges. This article must therefore be hereby marked advertisement in accordancewith 18 U.S.C. Section 1734 solely to indicate this fact.

We thank M.L. Munford (Department of Physics, UFV) for preparing Fig. 1.

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