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10.1586/14737140.6.10.1361 © 2006 Future Drugs Ltd ISSN 1473-7140 1361www.future-drugs.com

Mathematical modeling of cancer progression and response to chemotherapySandeep Sanga, John P Sinek, Hermann B Frieboes, Mauro Ferrari, John P Fruehauf and Vittorio Cristini†

†Author for correspondenceUniversity of California, Deptartments of Biomedical Engineering and Mathematics, 3120 Natural Sciences II, Irvine, CA 92697–2715, USATel.: +1 949 824 9132Fax: +1 949 824 [email protected]

KEYWORDS: biobarriers, biocomputation, biosimulation, cancer, chemotherapy, modeling, multiscale, nanotechnology, pharmacokinetics, pharmacodynamics

The complex, constantly evolving and multifaceted nature of cancer has made it difficult to identify unique molecular and pathophysiological signatures for each disease variant, consequently hindering development of effective therapies. Mathematical modeling and computer simulation are tools that can provide a robust framework to better understand cancer progression and response to chemotherapy. Successful therapeutic agents must overcome biological barriers occurring at multiple space and time scales and still reach targets at sufficient concentrations. A multiscale computer simulator founded on the integration of experimental data and mathematical models can provide valuable insights into these processes and establish a technology platform for analyzing the effectiveness of chemotherapeutic drugs, with the potential to cost-effectively and efficiently screen drug candidates during the drug-development process.

Expert Rev. Anticancer Ther. 6(10), 1361–1376 (2006)

Fundamental considerations on cancer biology The physiological processes underlying cancerare highly complex, spanning a wide range ofinterrelated temporal and spatial scales. Thefundamental causes are believed to reside atthe genetic level, where mutations enable cellsto develop a selective advantage, allowingthem to reproduce or prolong life in defianceof normal constraints. In time, these cells formavascular masses limited to approximately afew millimeters in diameter owing to thetransport limitations of oxygen and nutrientsinto tissue [1]. As inner layers of the nascenttumor begin to necrose, tumor angiogenic reg-ulators are released by the tumor mass, whichdiffuse through the surrounding tissue andtrigger a cascade of events upon arrival at localvasculature, culminating in the recruitment ofvessels that supply blood to the burgeoningtumor (i.e., angiogenesis). At this point, thevascularized tumor may remain compact andnoninvasive (i.e., benign), in which case it canusually be successfully removed by surgicalresection or treated with radiation. Con-versely, upon receiving infusion of nutrients

from its newly formed vasculature, a tumormay become malignant and rapidly invadelocal tissue, usually acquiring mutationsenabling navigation through the bloodstreamand lymphatics to metastasize to other loca-tions in the body [2]. The nonlocalized natureof metastatic cancer limits the success ofsurgical and radiation treatment approaches;thus, systemically administered chemother-apy continues to be the standard option inspite of varied outcomes.

Tumor neovasculature plays an integral rolein the administration of such treatment. It isthe first tumor-level barrier that an adminis-tered drug molecule must navigate on itsjourney to its intended intracellular target,and its anatomical and functional irregulari-ties are thought to significantly impair drugdistribution to lesion tissue. For standardtherapy, once drug molecules extravasatethrough vasculature, they must diffusethrough interstitial space, permeate cell mem-branes, survive a gauntlet of intracellularmechanisms designed to detoxify cells andfinally bind to subcellular targets at sufficientcytotoxic concentrations [3]. This series of

CONTENTS

Fundamental considerations on cancer biology

Role of modeling in cancer research & drug development

Fundamentals of cancer modeling

Integrated tumor simulation: foundations for a multiscale simulator

Predictive modeling of chemotherapy & antiangiogenic treatment: implications for drug development

Conclusion

Expert commentary

Five-year view

References

Affiliations

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1362 Expert Rev. Anticancer Ther. 6(10), (2006)

barriers combines to produce an overall reduction in the effi-cacy of many unrelated anticancer drugs, a phenomenoncalled multidrug resistance (MDR) [4], and cannot be over-come simply by administering more drug, as toxicity to hosttissue presents a formidable challenge.

For conventional chemotherapeutic treatment, strategic dos-ing is used to maximize anticancer-drug effects while minimiz-ing host toxicity. However, recent antiangiogenic treatmentstrategies center around targeting tumor neovasculature insteadof the lesion itself. By destroying the vascular bed, the tumor’ssource of oxygen and nutrients is reduced, leading to starvationof the mass. Furthermore, such therapies might reduce metasta-sis by eliminating the escape route into systemic circulation [1].In another innovative strategy known as vascular normaliza-tion, the intent is not to destroy all the vasculature, but ratherto prune it of inefficient branches, thereby normalizing theabnormal structure and function of tumor vasculature andimproving delivery of oxygen, nutrients and drugs [5]. Anotherpromising approach to cancer treatment is the use of nano-vectored therapy, employing nanoscale devices to specificallytarget and deliver drug payloads to cancer cells [6]. These thera-pies, along with conventional treatment, are ideal candidatesfor in silico (computer) modeling, which has the power to offerinsight into their efficacy and potentially develop into a meansfor predicting patient-tailored therapeutic regimens [7,8].

Role of modeling in cancer research & drug developmentDeveloping a detailed understanding of the underlying patho-physiology of cancer, its progression, mechanisms of drugresistance at various scales, as well as the optimization of drugdosing protocols, is the subject of vast amounts of researchdirected towards the development of effective treatment andprevention strategies. Owing to the complexity of this disease,it has proven difficult to assign quantitative weights to eachcomponent. This may be due, in part, to the nature of experi-mental investigation, where mechanisms are often studied in anisolated context. It has been suggested that a conceptual frame-work is necessary to fully understand the data produced inquantity by tumor biologists and clinical oncologists [9–11]. Thechallenges of better understanding the overall cancer phenome-non and its treatment might benefit from an evaluation ofmathematical modeling and biocomputation. This approachcan be integrated with biological experiments and clinical trials.Traditional clinical and biological experiments require costlyinvestments in both time and materials, and are limited byequipment precision, human error and the inability to distin-guish between various underlying mechanisms governingtumor growth [11,12]. In parallel, a critical weakness of theoreti-cal models is their plasticity in uncritically recapitulating train-ing data, without regard to the model’s actual validity and pre-dictive capability. Nevertheless, modeling can provideinvestigators with tools to run computational experiments thatwould otherwise be very difficult or impossible to recreate in anexperimental setting (e.g., varying adhesion forces between cellsor varying membrane permeabilities of a particular cell line);

accordingly, modeling can provide valuable information toplan effective biological experiments for testing theoreticalhypotheses. Data from biological experiments provide neces-sary constraints for choosing appropriate model parameters.Therefore, although pure theoretical or experimental investiga-tions alone have inherent flaws and limitations, an ideal syn-ergy between the two can be approached by using a circular,recursive work-flow methodology.

ObjectiveThe objective of this article is to first present a brief overviewof current mathematical and biocomputational modeling ofcancer progression and therapy, followed by a description ofthe integrative approach that we envision towards the develop-ment of higher order biocomputational technology, centeredabout a simulator with the capacity to predict in vivo tumorgrowth and response to therapy. A virtual cancer simulatormight provide a means to efficiently and cost-effectively screendrug candidates with the potential of significant savings inresearch and development. An additional, long-term goal ofthis research is to customize clinical cancer therapy by usingcell-specific tumor information, thereby maximizing benefit toboth patients and providers.

The endeavor to develop software packages capable of sophisti-cated, in vivo-like tumor simulation will be based on a modular,multiscale development process where individual components arebuilt upon mathematical models simulating disease progression,anticancer drug pharmacology and drug resistance. These arethen integrated to simulate the disease and possible therapiesthrough a wide temporal and spatial range. This scalable schemeallows the simulator to be enlarged by adding and appropriatelylinking modules (FIGURE 1). This review is structured to familiarizereaders with key modeling efforts over the past 30 years, inspiringthe development of modules in a cancer simulator, examine thecurrent status of cancer simulation, present applications of thesimulator in the areas of drug development and optimization andindicate challenges for advancing cancer simulation towardshigher levels of sophistication.

Fundamentals of cancer modelingMathematical models can provide biologists and clinicianswith tools that might guide their efforts to elucidate funda-mental mechanisms of cancer progression and either improvecurrent treatment strategies or stimulate the development ofnew ones [13]. Many cancer models have been proposed thatfocus on one or more phases of cancer progression(i.e., avascular, angiogenesis or vascular) and can typically becategorized as either a discrete, continuum or hybridapproach [11,13,14]. Continuum models draw upon principlesfrom fluid and continuum mechanics and describe cancer-related variables, such as cell population, nutrient concentra-tion, oxygen distribution and growth factor concentrations ascontinuous fields by means of differential equations [13]. Bycontrast, cellular automaton (CA) models describe thedynamics of discrete elements (e.g., tumor cells) whose states

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are governed by a set of deterministic and/or probabilisticrules. The state evolution of these elements can be trackedthrough both space and time. Hybrid cancer modelingapproaches combine continuum fields with CA descriptions.In particular, substances such as oxygen, nutrient, drug andgrowth factors can be described as a continuum in the tumormicroenvironment, while individual CA elements dynamicallyevolve in response to local substance concentrations.

Multiscale simulationThe literature devoted to the theoretical investigation of solidtumor growth and angiogenesis using continuum and CAmodeling approaches has been reviewed in depth by Araujo andMcElwain [11], Moreira and Deutsch [14], and Mantzaris andcolleagues [15]. While most of the models described within thesereviews are able to provide useful insight into cancer-relatedprocesses occurring at a particular length and time scale, theydo not address the fundamental problem of how phenomena atdifferent scales are coupled [13]. The multiscale complexity ofcancer progression warrants a multiscale modeling approach to

be taken to produce truly predictive tumor simulators. Proc-esses occurring at various length and time scales must be cou-pled appropriately in order to capture all the dynamicsinvolved. Previous works have developed multiscale systemsmodeling complex biological processes, such as cancer [13,16–18],the heart and lung [19–22], and various phenomena relatedto developmental biology [23,24]. In particular, Jiang and col-leagues [18] and Alarcon and colleagues [13,16] present frame-works for building multiscale cancer progression models capa-ble of integrating a hierarchy of processes at varying length andtime scales. Most cancer models and multiscalesystems [11,13,16–18,25–30] primarily produce 1D and 2D simula-tions that are limited in their ability to capture complex in vivotumor morphologies and microenvironment. The aim of thisperspective is to describe strategies for overcoming these limita-tions by developing a framework focusing on accurately pre-dicting the 2D and 3D distribution of oxygen, nutrients andgrowth factors in the microenvironment, evolution of complextumor morphology, and tumor response to various treatmentstrategies. The underlying models of the multiscale system will

Figure 1. Outlook for a multiscale cancer simulation under development in the mathematical oncology community. A multiscale cancer simulation is founded on the integration of experimental data and mathematical models. It should provide valuable insights into the cancer phenomenon and establish a platform information technology for analyzing the effectiveness of chemotherapeutic drugs with the potential to extend patients’ pain-free survival and cost-effectively and efficiently screen drug candidates during the drug development process. Modules are developed and coupled via sharing of information. DC MRI: Dynamic contrast magnetic resonance imaging; EC: Endothelial cell; ECM: Extracellular matrix; RES: Reticuloendothelial system; TAF: Tumor angiogenic factor.

• DC MRI data• In vivo/in vitroexperimentscharactarizing modelparameters

Chemotherapy module

Experimental support• In vivo/in vitrocytotoxicity data• Drug mechanisms oftransport and action• Combination therapy

Pharmacokinetics• Compartmentmodeling describingdrug transport anddelivery to target

Pharmacodynamics• Stochastic modeldescribing drugeffect on cells• Based on drug mechanism of action

Cellular scale• Cell contact• Cell stress• Cell cycle profiles• Nutrient

Subcellular scale• Cell signalingpathways

Genetic scale• Combination ofgenetic mutations

Tumoral scale• Tumor morphology• Material properties• Nutrient

Surrounding tissue• ECM degradation• Vascularity changes• Immune response

Experimental support• In vivo tumors frommice models• In vitro tumorspheroid models• Human patient MRI• Cell-culture studies

Vasculature module

Experimental support

Blood flow• Nutrient delivery• Drug delivery

Angiogenesis• TAF distribution• EC density• Fibronectin densityin ECM• Vasculature morphology

Nanotechnology module

Parameter definitionss• Tumor vasculature pore-size: leakiness• Nanoparticle size• RES uptake rate• Drug release rate• Active transport mechanisms• Tumor versus host affinity

Experimental support• Protein/gene-chip array of tumor biopsydetermines relevantgene expression levels

Genotype module

Genotype model• Determine set ofparameters governingtumor growth andresponse to therapy

Expert Review of Anticancer Therapy

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be founded on biological principles and controlling parameterswill be adjusted according to experimental and clinical data,thus developing a tumor simulator capable of producing morein vivo-relevant predictions.

Modular development of a multiscale cancer simulation: earlier modelsThis section provides a brief overview of a select group oftumor growth, angiogenesis and pharmacology models serv-ing as the inspiration and foundation for efforts to develop amultiscale cancer simulator.

Tumor growth

Cristini and colleagues were among the first to advance mode-ling of complex tumor morphologies beyond the limited capa-bilities of mathematical linear analyses and into the realm ofnonlinear computer simulation [31]. The multifaceted nature ofcancer requires sophisticated, nonlinear mathematical models tocapture more realistic growth dynamics and morphologies.Boundary-integral simulations [31] of classic continuum-basedtumor models [25–30] determined that a reduced set of two non-dimensional parameters (related to mitosis rate, apoptosis rate,cell mobility and cell adhesion) regulate morphology and growth(invasiveness) of avascular and vascularized tumors. In thismodel, there is no morphological representation of vasculature,rather the effect of vascularity is quantified by a parameter relat-ing the concentration of nutrient in the blood, nutrient transferrate from blood to tissue and nutrient consumption by the cells.Essentially, critical conditions were predicted that separate com-pact, noninvasive mass growth from unstable, fingering, infiltra-tive progression [31]. However, further analysis demonstrated thathighly vascularized tumors tend to grow in compact, nearlyspherical shapes showing little or no sign of invasiveness. Thisunexpected prediction suggests that tumors could maintain sta-ble morphology under normoxic microenvironmental condi-tions. This result is supported by experimental observations indi-cating that hypoxia stimulates invasiveness and tumorfragmentation [32,33]. In later sections of the present article,tumor simulators built upon this framework and their impor-tance in studying cancer therapy will be discussed, together withrecent experimental confirmation.

Tumor-induced angiogenesis

Angiogenesis is the process by which cancers recruit enhancedblood supply to provide the oxygen and nutrients that are com-monly considered necessary to support growth into larger, moreinvasive tumor masses. As is the case with tumor growth,tumor-induced angiogenesis is a topic receiving considerableattention from the biological modeling community and hasbeen extensively reviewed (e.g., Mantzaris and colleagues [15]).

Mathematical model for tumor-induced angiogenesisAngiogenesis is believed to be initiated by proangiogenic pro-teins (PAP), such as vascular endothelial growth factor (VEGF),that have been induced by a lack of oxygen and nutrients to be

released from the necrotic tissue of a tumor lesion into sur-rounding tissue [34]. These proteins create a chemical gradientthat triggers endothelial cells (ECs) from parent vessels in thepre-existing vasculature to migrate towards the tumor (chemo-taxis). Eventually, through a number of complex mechanisms,the accumulation of ECs forms finger-like capillary sproutsextending from a parent vessel. Analogous to plant growth,these sprouts extend and grow towards the tumor along thechemical gradient guided by the migration of the sprout-tip.The interaction between ECs and the extracellular matrix(ECM) itself is also significant in directing the sprout-tips.Fibronectin is generated and adheres to the matrix, serving toguide the direction of endothelial cell progression via a processcalled haptotaxis, similar and complementary to chemotaxis.Once capillary sprouts from the parent vessel extend far enoughtowards the tumor, they tend to lean towards each other andform tip-to-tip and tip-to-sprout fusions called anastomoses(thought to be caused by haptotaxis) [35,36]. Through this proc-ess of anastomosis, an initial network of poorly perfused, inter-connected immature vessels is formed. The previously describedprocess of angiogenesis and subsequent anastomoses occurs in arepetitive fashion using the initial network as parent vessels,thereby producing an extended capillary bed concentrated inthe tumor. However, the neovasculature is irregular and poorlyperfused in comparison with normal tissue vasculature and willbe portrayed as a biological barrier to anticancer treatmentefficacy in upcoming sections.

Anderson and Chaplain developed a tumor-induced angio-genesis model with the ability to follow the motion of ECs atthe capillary tips and control important processes, such as pro-liferation, branching and anastomosis [35]. Their model uses ahybrid approach (i.e., both continuum and discrete modeling)and focuses on three significant variables related to angiogenesis:EC density, and PAP and fibronectin concentrations.

The continuum component of the Anderson and Chaplainmodel describes the evolution of EC, PAP and fibronectindistributions using a reaction-diffusion system of three partialdifferential equations. The equation describing EC distribu-tion accounts for three major components: small amounts ofrandom motion possibly dependent on PAP concentration(i.e., diffusion); general EC migration owing to chemotaxis;and EC adhesion to ECM owing to an interaction withfibronectin (i.e., haptotaxis). The equation describing theevolution of PAP concentration uses an uptake term repre-senting the binding of PAP to EC as they migrate through thetissue. The equation describing the evolution of fibronectinconcentration includes terms representing fibronectin synthe-sis by endothelial cells as they migrate through the ECM, anda presumed degradation. Fibronectin exists in the ECM inbound form and is not freely diffusible, therefore no diffusionterm is included.

The discrete component of the angiogenesis model uses arandom-walk model, which governs the movement of individ-ual ECs at the capillary sprout tips. The model essentiallytracks the migration of sprout tips and presumes the shape of

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the full sprouts based on the sprout tip paths. Using a discre-tized version of the continuum model, the individual motionof the EC sprout tips is governed by probabilities of the ECbeing stationary or moving left, right, up or down; these prob-abilities are functions of local fibronectin and PAPconcentrations [35,36]. Predetermined rules for branching, anas-tomosis and cell proliferation produce the overall morphologyof realistic tumor neovasculature.

Extension of Anderson & Chaplain angiogenesis model: blood flowThe abnormal nature of tumor vasculature compared withhealthy tissue vasculature has been addressed [5]. Irregulartumor vasculature leads to restricted and inhomogeneous drugand nutrient extravasation to tumor tissue, which may exacer-bate the situation by selecting for highly resistant clones.Anderson and Chaplain’s angiogenesis model appears to cap-ture the irregularity of tumor vasculature through appropriateadjustment of the governing mathematical parameters. How-ever, their model only describes the physical structure of thecapillary network. Nutrient, oxygen and drug distributions in atumor can be modeled in a simplified fashion by using theAnderson and Chaplain vasculature as a source boundary con-dition in a diffusion–reaction system. In reality, nutrient, oxy-gen and drug delivery depends on blood flow through the vas-culature; therefore using the entire vasculature as a uniformsource with blood–tissue transfer proportional to local pressuresis a rather elementary description.

McDougall and colleagues developed a direct extension ofthe Anderson and Chaplain angiogenesis model [35] by describ-ing the generated vascular networks as a series of straight, rigidcylindrical capillaries that join adjacent nodes [36]. Blood flow ismodeled through the cylindrical vascular network by modelingthe elemental flow rate in each segment with Poiseuille’s Law,which describes flow rate as a function of capillary lumen, fluidviscosity, capillary length and pressure drop. Using this simpleflow model, McDougall and colleagues identified tumor neo-vasculature as a biobarrier to chemotherapy [36]. Results of theirsimulations indicated that the highly interconnected nature ofirregular vasculature produced by tumor-induced angiogenesiscould cause low rates of drug delivery to the tumor with thepotential for the drug to actually completely bypass the entiremass depending on the tumor shape and consequent proang-iogenic protein distribution. Additionally, the simulationresults suggest that drug delivered by bolus injection suffersfrom severe dilution, thereby reducing drug efficacy.

Stephanou and colleagues [37] extended the work of McDou-gall and colleagues [36] by developing an algorithm that normal-izes vasculature produced by Anderson and Chaplain’s angio-genesis model [35]. They examined how pruning vessels byantiangiogenic drugs might affect blood flow distribution andconsequently drug delivery to tumors. Their work includedblood flow simulations in fully 3D vasculature. Stephanou andcoworkers later included vascular adaptation effects [38], due toshear stress generated by flowing blood [39,40], to the angiogenesis

model to investigate how adaptive remodeling affects oxygenand drug supply to tumors. Alarcon and colleagues also modeledthe vascular adaptation effects in an effort to study inhomogene-ity of oxygen distribution in tumors and the consequential roleof hypoxic cells in tumor invasion [13]. More recently, McDou-gall and colleagues modified their angiogenesis model to simul-taneously couple vessel growth with blood flow to dynamicallyinclude the effects of vascular adaptation [41], rather than adaptthe vasculature a posteriori like Stephanou and colleagues [36] andAlarcon and colleagues [13].

Pharmacology & drug efficacy

In the event that a cancer has metastasized, systemic treatmentis generally necessary in the form of chemotherapy delivered tothe primary and secondary tumors through the bloodstream.Drugs must overcome various resistance mechanisms and barri-ers that affect their efficacy en route to their respective targets,thus producing the overall MDR phenomenon. Individualmechanisms and barriers occur at different scales. At the sub-cellular and cellular scale, there exists a range of druginflux/efflux pumps, changes in the expression of topoisomer-ases and alterations in metabolic pathways (e.g., influencingdrug metabolism, DNA repair and/or apoptosis). At the tumorand body scale, resistance can be due to normal clearancemechanisms (e.g., urinary system, reticuloendothelial system orthe blood–brain barrier), abnormal tumor vasculature, tumormicroenvironment, and tumor 3D structure [3]. Consequently,these biobarriers impede the delivery of chemotherapeuticdrugs at effective concentrations to all cancer cells.

Pharmacokinetics

In the study of drug delivery, it is common to conceptualizethe organism as a system of interconnected pools called com-partments. The investigation of the properties of these com-partments and the material fluxes between them is termed‘compartment modeling’ [42]. Conventional pharmacokinetic(PK) models use compartment modeling to investigate cellu-lar drug-uptake and intracellular drug interactions, as well asprovide insight into modeling cellular-scale mechanisms con-tributing to drug resistance. For example, a standard three-compartment model was used by Dordal and colleagues toinvestigate cellular drug uptake [43]. Their objective was toquantify increased efflux, decreased intracellular sequestra-tion and decreased membrane permeability as they relate to areduction in drug effectiveness. Using flow cytometry, theyassessed the cellular uptake of doxorubicin and fluorescentrhodamine-123 in drug-resistant and -sensitive cancer cells.By fitting the experimental data to the compartmentalmodel, kinetic parameters for both inward and outwardtransport were obtained and used to quantify the relativeimportance of the previously mentioned cellular mechanisms.Specifically, their results indicate that of the three cellularmechanisms modeled, decreased intracellular sequestration ina nonexchangeable compartment is quantitatively the mostsignificant contributor towards drug resistance [43]. Similarly,

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compartment modeling can be applied to investigate addi-tional components affecting drug delivery, such as extracellulardrug binding and target repair mechanisms.

Pharmacodynamics

While PK describes drug penetration, pharmacodynamics(PD) describes drug cytotoxicity. Although the mechanismscontributing to drug effects are incompletely understood, sev-eral phenomenological models adequately yield fractional cellsurvival, S, as a function of concentration–time exposure his-tory. The Hill-type model S = (1 + Axm)-1 is often used, wherex is a measure of cellular damage, such as extra- or intracellulararea under the curve (AUC). In turn, AUC is given as the inte-gral of concentration with respect to time . Another pos-sibility is the exponential kill model S = e-kx, where x again is asuitable measure of damage and k is a constant. While theseare perhaps the simplest PD models in use, other equationscan be employed.

A study by El-Kareh and Secomb investigated several meas-ures of cellular damage in conjunction with the Hill-type modelgiven above to determine which provided the best fit to experi-mental data [44,45]. Their investigation was prompted by theobservation that models employing extracellular AUC consist-ently overestimated cytotoxicity in cases of extended exposure tothe drugs cisplatin and doxorubicin. Experimentally, toxicitywould achieve a plateau, above which continued exposure, evento continually refreshed drug, would have no effect. To explainthis, they hypothesized that it was not the time of exposureper se that correlated with cytotoxicity, but rather the peak levelof DNA-bound drug [44]. Accordingly, they used this measureand showed that for short exposure times, the delay in achievingDNA-bound drug equilibrium could explain increasing cyto-toxicity in time. Various experimental cell survival data were fitto determine appropriate values for the constants A and m. Thenew model was compared with previous models describing therelationship between cytotoxicity and exposure time. El-Karehand Secomb’s model consistently proved to be the best fit evenfor long exposure in in vitro datasets [46], establishing that peakDNA-bound cisplatin is a stronger indicator of cytotoxicitythan extra- or intracellular concentrations. Later, they extendedthe model to doxorubicin [45]. Experimental evidence suggeststhat doxorubicin has two cytotoxic mechanisms, one involvingtopoisomerase II inhibition by intracellular drug and the otherinvolving apoptosis induction via extracellular drug. El-Karehand Secomb proposed a model that combines the effects of bothmechanisms into the cellular damage by summing peak intra-and extracellular drug concentrations. Similar to the cisplatinmodel, their doxorubicin PD model provides better fits toin vitro cytotoxicity datasets than previous models [45].

Integrated tumor simulation: foundations for a multiscale simulatorThus far, this perspective has focused on providing a funda-mental overview of key cancer modeling efforts in the areas oftumor growth, tumor-induced angiogenesis, blood flow and

pharmacology. The focus will now be on strategies for thedevelopment of a higher order computer simulator capable ofcustomizing cancer drug therapy based on cancer-specific infor-mation and built upon the fundamental framework describedin the preceding sections. Here, we present current develop-ments in simulation technology, including the determination ofmodel parameter values and validation of their performancebased on experimental in vitro and in vivo data.

Model descriptionsZheng and colleagues produced a first-generation multi-dimensional tumor simulator employing a sharp-interface(level-set) finite-element numerical method for tracking thetumor boundary [47]. This model is capable of simulating 2Dtumor evolution through the major phases of growth, includingavascular dormancy, neovascularization and subsequent rapidexpansion and infiltration of host tissue. Wise and colleaguesrecently produced a second-generation 3D tumor simulatoremploying a more physically accurate diffuse-interface formula-tion on a finite-difference framework, which can more realisti-cally represent tissue interfaces and clonogenicheterogeneity [48]. Both simulators not only serve to modeltumor progression, but also provide test-beds for therapeuticstrategies and hypotheses (FIGURES 2 & 3) [49,50].

These models build on the continuum-based approach usedby Cristini and colleagues, which considers tumor mass as anincompressible and viscous material that locally expands andcontracts in correspondence to variable rates of cell mitosis andapoptosis [31]. Tumor cells themselves are not individually rep-resented. In Zheng and colleagues’s formulation, local lesionenvironment is modeled as three sharply demarcated noninter-secting domains: viable tumor, necrotic tissue and hosttissue [47]. Although Wise and colleagues’ diffuse-interface for-mulation uses a similar partitioning of the lesion environmentinto tissue domains, boundaries are not as strictly defined [48].Instead, at a given location, intermixing of several cancerousclones along with necrotic and host tissue can be represented byspecifying their relative mass fractions. This cannot be done inthe sharp interface model and is a critical improvement towardsrealistically simulating mutation-driven heterogeneity. Pleaseand colleagues were amongst the first to apply multiphase mod-eling to tumor growth by capturing both tumor cells and extra-cellular fluid as separate continuum phases [51,52]. Work byWard and King [53,54] and Breward and colleagues [55] followedsuit by also modeling avascular cancer growth as a two-phasedescription comprised of tumor tissue and dead tissue (extracel-lular space). This multiphase modeling is needed to captureavascular tumor growth as live tumor cells proliferate into thedead tissue space. Breward and colleagues extended their avas-cular model [54] to describe vascular tumor growth, thus incor-porating a third phase to describe the spatial and temporal dis-tribution of blood vessels [56]. Along similar lines, Wise andcolleagues also used a multiphase modeling approach by cap-turing the evolution and interactions between intermixing mul-tiple tumor species, necrotic tissue, host tissue and interstitial

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fluid [48]. Araujo and McElwain recently proposed a multiphasemodel of tumor growth that includes a solid phase representingthe extracellular matrix, in an effort to more accurately captureresidual stresses [57].

While it is the growth and regression of lesion tissue that is ofprimary interest, other processes support and interact with thisgrowth, necessitating a modular design in which simulator com-ponents are dedicated to process management. The major compo-nents essential to basic lesion simulation are specific to growthand regression, nutrient delivery and angiogenesis. Beyond these,modules pertaining to genetic mutation, cell-cycle and phenotypespecifics, and therapy can be added to provide pertinent informa-tion from smaller spatial and temporal scales. The growth andregression component postulates that cell velocity is proportionalto pressure gradient (Darcy’s law), which is commonly used to

model motion through porous substrates (i.e., a continuum ofcells flowing through the extracellular matrix). Morphology, espe-cially as it pertains to invasiveness, is affected by parameters thatmodel cell adhesion, for instance through the definition of anequivalent surface tension at the tumor boundary [27]. Based oninputs from other components, the growth module produces acell displacement (velocity) field and advects the tumor boundaryin the case of Zheng and colleagues’ method [47], and the speciesmass fractions in the case of Wise and colleagues’ method [48].

Vasculature is incorporated into these simulators as an angio-genesis module inspired by Anderson and Chaplain’s angiogen-esis model [33,47], linked to tumor growth through the release oftumor angiogenic regulators by necrosing tumor tissue. Thetransition between avascular and vascular tumor growth ismarked by the recruitment of microvasculature from local

Figure 2. Examples of simulated tumor evolution. (A) Byrne and Chaplain’s necrotic tumor model with a specific apoptosis rate. This model describes the shrinkage of tumor and necrotic rim. Reprinted from [29], © 1996, with permission from Elsevier. (B) Cristini and colleagues’ nonlinear continuum-based boundary-integral model, which was among the first to capture complex tumor morphologies. Adapted from [31], © 2003 Kluwer Academic Publishers. With kind permission from Springer Science and Business Media. (C) Zheng and colleagues’ [47] sharp-interface level-set model of tumoral lesion, which captures complex tumor progression including tumor-induced angiogenesis; A: Tumor boundary; B: Neovasculature; C: Viable tumor tissue; D: Necrotic tumor tissue; E: Host tissue; and F: ‘Free’ endothelial cells migrating from parent vessel (not shown). Reprinted from [45], © 2005, with permission from Elsevier. (D) Wise and colleagues’ [48] diffuse-interface model, which captures complex tumor morphology and clonogenic heterogeneity in 3D (angiogenesis not shown).

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blood vessels (i.e., angiogenesis). The governing processes ofangiogenesis are still very much in debate, but one proposedmechanism followed by Anderson and Chaplain’s model growsnew vessels from parent vessels due to chemotaxis of endothe-lial cells along an angiogenic regulator gradient towards thetumor [33]. ECs also interact with the extracellular matrix in aprocess known as haptotaxis.

Underlying these components are sophisticated numericalalgorithms, including adaptive computational meshes [58,59],that enable high-resolution rendering of complex tumor morph-ologies, including fingering, invasion and reconnection, acrossmultiple length scales at the minimum computational expense.

Simulation as an investigative tool: integration of experimental data with theoretical modelingA key aspect of the successful development of an advanced simu-lator of in vivo tumor growth is to both derive fundamental para-meter values from, and validate simulator performance through,in vitro and in vivo experimental (and clinical) results. Often, theprocess begins with elementary properties of cells growing inmonolayer, such as doubling time, oxygen and glucose consump-tion, response to confluence, and uptake and response to drugs.

Maintaining awareness that these properties probably changein vivo, it is believed that when enough information is knownregarding the mechanisms that give rise to a cell’s behavior,whether that be in a dish or in a living organism, the appropriateinterplay of these mechanisms via a computer model can predictnot only individual cell behavior, but also that of cellular aggregateswithin a given set of environmental conditions. This line of reason-ing is justified by the success of cell automaton models in predict-ing cell and tissue behavior, suggesting that gene expression infor-mation and cell-signaling mechanisms involved in determining cellbehavior can be reduced to a subset of governing rules [60–62].

Antiangiogenic therapies target tumor neovasculature with theintention of disabling a tumor’s source of oxygen and nutrients.However, a recently proposed concept termed ‘diffusional insta-bility’ suggests that antiangiogenic therapies may actually triggertumor instability and tissue invasion [49]. Based on simulations oftumor growth, Cristini and colleagues hypothesized that complextumor morphologies leading to local tissue invasion are caused byoxygen and nutrient gradients [49]. These gradients result inregions of hypoxia and acidosis, which directly increase tumorinvasiveness by increasing cell motility through the production ofautocrine motility factor, expression of tumor urokinase

Figure 3. Correspondence between in silico tumor progression and in vitro tumor spheroid growth. (A) Example of a fairly compact in vitro glioma spheroid with high levels of cell adhesion. (B) An in silico representation of a compact tumor based on appropriate input parameters. (C) Example of an in vitro glioma spheroid with low levels of cell adhesion. The formation of subspheroids indicates an overall morphological instability. (D) An in silico representation of an unstable tumor with low cell adhesion. Bar: 130 mm. Adapted from [49] with permission from the American Association for Cancer Research.

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plasminogen activator receptor, production of cathespsin B andupregulation of hepatocyte growth factor [63–65]. Frieboes and col-leagues tested the hypothesis using experimentally derived para-meters as inputs to the simulator [66]. The experiments varied glu-cose and serum conditions for in vitro human and ratglioblastoma tumor spheroids, yielding parameters relating tumorshape and relative nutrient concentration. Subsequently, simula-tions were used to predict and interpret in vitro tumor spheroidmorphology and invasiveness. Accordingly, simulations using theexperimentally determined parameters suggested that tumorsmight form shapes that maximize cellular exposure to oxygen,nutrients and growth factors by correspondingly adjusting cellproliferation and adhesion. Macklin and Lowengrub furthersupported these predictions in their simulations [67].

Results of these three studies exemplify the usefulness ofintegrating experimental and theoretical research to developvaluable insight into explaining biological phenomena. Exper-imentally derived parameters were used to drive the realism ofthe tumor simulators, while also interpreting nutrient gradi-ents, resulting from the presence of multiple cell species,inherent diffusion limits and therapy as a mechanism for localtumor infiltration.

Predictive modeling of chemotherapy & antiangiogenic treatment: implications for drug developmentChemotherapy simulations using a 2D tumor modelTherapy simulations performed recently by Sinek and colleaguesusing Zheng and colleagues’ tumor simulator [47] along withAnderson and Chaplain’s vasculature model [33] were among thefirst to be performed in a true multidimensional (rather thancylindrically or radially symmetric) and vascularized setting.While these simulations used an elementary one-compartmentmodel (lesion mass with no cellular resolution) with the vascula-ture serving as the nutrient and drug source, they were neverthe-less useful in highlighting potential adverse characteristics oftumor response to intravenously supplied cytotoxic drug [50].Among these were some of the previously mentioned biobarriers,including the susceptibility of lesion vasculature to collapse underpressure generated by cellular proliferation and its consequentcontribution to nutrient and drug heterogeneity. More impor-tantly, it was demonstrated that these heterogeneities may impairtherapeutic efficacy by precipitating invasive morphology, even ina best-case scenario assuming uniform drug delivery from vascula-ture (e.g., by using constant-release ‘smart’ nanovectors uniformlyextravasated along vessels), a single, drug-sensitive cell-type andnegligible host toxicity. Other simulations in this study demon-strated how angiogenic normalization might be of benefit [5,50].Although this work provided qualitatively compelling results,more detailed PK and PD are needed for quantitative analysis.

Incorporation of a multicompartmental pharmacokinetics & pharmacodynamics modelIn more recent work by Cristini and coworkers [UNPUBLISHED

DATA], a multicompartmental tissue- and cell-level PKPDmodel for cisplatin and doxorubicin was incorporated based on

experimentally derived parameter estimations, thereby estab-lishing a more rigorous platform for analyzing the effective-ness of chemotherapeutic drugs (FIGURE 4). PKPD modeling iswell established, with representative examples given byPanetta [68], Gardner [69], Lankelma [70] and Jackson [71]. Thefirst two researchers closely examined cellular behavior withregard to drug, such as the fraction and sensitivity of cyclingversus quiescent cells, response to cycle phase-specific andnonspecific drugs, and response to cytotoxic and cytostaticdrug combinations. The latter two are more concerned withlesion scale PK effects, such as drug exposure of inner regionsof spheroids and cancer islets, and the dynamics of tumorresponse to various dosing regimes. While the model per sedeveloped by Cristini and coworkers is not necessarily anextension or refinement of these previous efforts, its couplingwith Zheng and colleagues’ tumor growth and angiogenesissimulator represents advanced modeling technology, ena-bling the simulation and analysis of chemotherapy applied tovascularized in vivo tumors [47].

While this PKPD model was designed explicitly for cisplatinand doxorubicin, since their primary mechanisms of action havebeen well characterized and both drugs are widely used, theintention is that it may be adapted to simulate therapy usingother drugs. During a simulated intravenous drug bolus admin-istration, concentration is assumed constant along the vascula-ture, which acts as a source throughout the tumor. Concentra-tion subsequently decreases as drug diffuses throughinterstitium and is taken up by cells. The model accountsexplicitly for tissue- and cell-level biobarriers, tracking drugfrom the point of extravasation through lesion interstitium andcell membranes, and into intracellular organelles and target.The amount and time of exposure of DNA-bound drug thendetermines its effect throughout the lesion. Each element of thedrug pathway contributes a measure of transport resistance, theend result being diminished and nonuniform drug-to-targetpresentation and consequent compromised efficacy. By provid-ing precise control over parameters corresponding to PKPD ele-ments, the chemotherapy model can deliver powerful and flexi-ble hypothesis-testing capability. More strategically, it isenvisioned as part of a larger scheme intended to optimize estab-lished clinical therapies and to serve as a screening phase towardsimproving overall efficiency and cost effectiveness during thedrug development process.

Special attention is paid to intracellular compartments andassociated fluxes so that resistance mechanisms can be accu-rately modeled. In the case of cisplatin, glutathione conjuga-tion and removal from cytosol, as well as DNA repair, are keymechanisms. The model for doxorubicin includes an addi-tional lysosomal compartment owing to the affinity anthracy-clines express for these organelles and their hypothesized rolein expelling drug via exocytosis. Intercompartmental fluxes aregoverned by rates (the k’s in FIGURE 4), which are themselvesderived from underlying tissue and cell parameters governingthe transport of drug molecules. Some of the more importantparameters are drug diffusivity, membrane permeability,

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transmembrane carrier activity (e.g., PGP and MRP),organelle sequestration and DNA repair. The PDE system ofequations modeling PK (e.g., cisplatin model in FIGURE 4) iscapable of tracking the amount of drug both spatially andtemporally through multiple compartments based on govern-ing rate parameters. Parameters can be tuned not only to drugand cancer type, but also to cancer grade as well as cellularsubtleties of individual patients. Initially, parameter valueswere obtained through experimental data reported in pub-lished literature; however, it is intended that more targetedhistological, cellular and genetic analysis of tissue biopsies willprovide information necessary to dynamically determinemodel parameter values, yielding a truly predictive tool.

The PD component is a phenomenological model along thelines of El-Kareh and Secomb’s work [44,45], which also takesinto account the extended exposure plateau of cytotoxicity.However, Cristini and coworkers hypothesized that this effectwas due to a resistant fraction of cells (F), and that, for a givenDNA-bound level of drug, the remaining sensitive fraction,S - F, would follow classical exponential kill kinetics. Theresulting PD model for both cisplatin and doxorubicindescribing the surviving fraction of cells S is as follows:

where λs is the effect rate of DNA-bound drug s3, M is the maximum possi-ble effect rate of the drug and A, B, nand m are parameters fit specifically toeach drug/tissue. The model has beencompared with various experimentalcytotoxicity data available in theliterature and proves to be an acceptablemethod for modeling cell death. ThisPD model, in combination with the pre-viously mentioned cisplatin anddoxorubicin PK models, integrates as achemotherapy module in a tumorsimulator to describe drug delivery fromthe vasculature to its target, andcorresponding tumor regression. Specifi-cally, the drug effect rate (λs) links thePD model with the tumor growth (andregression) module, which tracks localcell densities and resulting cell velocities,and consequently determines whether asimulated tumor grows or regresses.

Chemotherapy simulationsFIGURES 5 & 6 are typical chemotherapysimulations using the model in FIGURE 4,demonstrating some recent findings ofCristini and coworkers (UNPUBLISHED

DATA) regarding heterogeneous nutrient and drug distribution,corresponding cell survival, nonuniform tumor regression andstabilization at a mass far above detectable levels [50]. In particu-lar, PKPD input parameters for doxorubicin and cisplatindetermined through literature and experiments have been usedto compare the performance of these two drugs. Doxorubicin isknown to quickly penetrate cell membranes and avidly bind tointracellular organelles and DNA, whereas cisplatin enters cellsmuch more slowly and exhibits far less binding [72,73]. Thesedifferences reveal themselves dramatically in FIGURE 5, wherestrong gradients of doxorubicin stand in contrast to minor gra-dients of cisplatin, a behavior governed by the ratio of cellularuptake to diffusive flux. Furthermore, the level of DNA-bounddoxorubicin declines almost immediately after the 2-h bolusinfusion, whereas cisplatin within the cytosol continues toaccrue on DNA for several hours longer, only showing amarked decline 8 h after the cessation of infusion. The PDmodel furthermore predicts a survival gradient of approxi-mately 10% from vessel source to distal tissue face in bothcases. Taking advantage of the precise control of experimentalparameters available in silico, the model suggests that this is dueto the decrease in nutrients and oxygen, which penetrates

dS( ) dt( )⁄ λ– smax S F 0,–{ }=

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Figure 4. Multicompartmental pharmacokinetics modeling. The top depicts the three-compartment model used to develop the pharmacokinetics (PK) equations for cisplatin. Compartment 1: Extracellular fluid/matrix; 2: Cytoplasm; 3: Nuclear/DNA-bound drug. Doxorubicin PK is based on a four-compartment model similar to cisplatin PK with the addition of a lysosomal compartment. The bottom depicts the PK equations for cisplatin and doxorubicin. Si represents drug concentration in compartment i, where i = 1, 2, 3 and 4 represent extracellular volume, intracellular cytosolic volume, target (e.g., DNA) and organelles (e.g., lysosomes), respectively, while SM is a DNA saturation parameter relevant to doxorubicin. The parameters kij and kij represent the rate of transfer between compartments, and ki represents a rate of permanent removal from compartment i; these parameters account for important phenomena, such as efflux pumps, cell permeability and DNA repair. VC is the volume of a cell and appears in the first equation of each system to reconcile the dimensions of S1 with the dimensions of the other compartment-specific equations. DS is the diffusivity of the drug through interstitial space.

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approximately 150 µm into tissue before being fully con-sumed. Hypoglycemia and hypoxia affect cells in ways thatmay create drug resistance [74] and these substrate gradientsproportionally decrease cell cycling in the model [47,50]. Sim-ulation can be used to investigate a tumor’s morphologicalresponse to chemotherapy, as well as quantify drug, nutrientand pressure distributions within a tumor (FIGURE 6).

Antiangiogenic therapy simulationsAs mentioned earlier, antiangiogenic therapies target tumorneovasculature with the intention of disabling a tumor’s sourceof oxygen and nutrients. However, these therapies may triggertumor instability and tissue invasion through a recentlyproposed concept termed diffusional instability [49]. Using 2D

simulations of tumor growth linked to angiogenesis, Cristiniand colleagues hypothesized that complex tumor morph-ologies leading to local tissue invasion are caused by oxygenand nutrient gradients [49]. Frieboes and colleagues tested thishypothesis through in vitro experiments using simulations topredict and interpret in vitro tumor spheroid morphologyand invasiveness [66]. Results from this (simplified) modelindicated that diffusion gradients of nutrients and growthfactors could indeed influence tumor morphology and inva-siveness (FIGURE 3). Simulations predicted tumor shape as afunction of cell proliferation and adhesion based on cellularaccess to oxygen, nutrients and growth factors: low prolifera-tion rates and high cell adhesion are sufficient to maintaincompact tumor shapes, whereas high proliferation rates and

Figure 5. Drug delivery to target and tumor response. (A) Simulation of 2-h bolus infusion of doxorubicin penetration, where t corresponds to time after initiation of the bolus. The inset illustrates a slab of tissue next to a vessel releasing drug. (B) 2-h bolus infusion of cisplatin, where t corresponds to time after initiation of the bolus. (C) cell survival after doxorubicin dosing of 0.30 µM whole blood concentration. (D) cell survival after cisplatin dosing of 6.20 µM whole blood concentration.

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low cell adhesion promote tumor fingering and eventual for-mation of separate tumor clusters. Additionally, there is com-putational evidence that the presence of multiple tumor cell

species with varying nutrient uptake and death rates can alsolead to aggressive fingering into surrounding tissue in vivo asa result of therapy [67].

Figure 6. Computer simulation of response to chemotherapy. Top three frames show three stages of regression of a simulated tumor undergoing chemotherapy. The next three frames, from left to right, show nutrient, drug and tissue pressure contours corresponding to the second stage of regression. Distribution heterogeneity is significant, underpinning key phenomena that diminish drug efficacy. One such result is the stabilization of regression, as depicted by curve B in the graph. Mass of tumor without chemotherapy is shown by curve A for comparison. Mass and time are in dimensionless units.

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ConclusionCancer is a multifaceted disease with complex governingprocesses occurring at a wide range of temporal and spatialscales. Biological barriers and MDR mechanisms, such asbody clearance systems, tumor neovasculature, druginflux/efflux pumps, DNA-repair mechanisms, tumor micro-environment and 3D structure, exist at each of these scalesand hinder the efficacy of anticancer therapies. Biosimulationbased on mathematical modeling of cancer-related phenom-ena might prove to be a valuable tool for integrating the vastamount of data produced by experimental and clinical cancerspecialists, thus providing a method to analytically investigatecancer progression and drug resistance. Multiscale tumor sim-ulation work by Arakelyan and colleagues [75] and Alarconand colleagues [13,16] has examined effects of oxygen/nutrientheterogeneity and antiangiogenic therapies on tumor growthand invasion. These works have proposed that experimentsproviding in vivo-evaluated parameters and validating modelpredictions are necessary. In this regard, highly sophisticatedtumor simulators are currently underway relying on variousmathematical models and using experimentally and clinicallyderived parameters. Simulation studies have already providedinsight into tumor morphology and metastasis [49,66] and areexpected to help optimize treatment strategies and new drugdevelopment [50].

Expert commentaryChallenges to the successful design, in silico implementation & utilization of a cancer simulatorTechnically, the design and in silico implementation of a cancersimulator is a monumental task and one that has been tacitlyongoing for the past few decades. Beyond this, the question of itsuse in clinical practice and pharmaceutical development is open.

To say that cancer is a complex disease is an understatement.With an estimated 30,000 genes, each comprised of 3000 baseson average, the human genome is a fantastically complicatedenigma. Its physical description has only recently been eluci-dated and, while some of the functions of genes or the proteinsthey encode are known, the magnitude of their innumerablepotential interactions is certainly overwhelming. A model thatattempts too boldly to include everything biologists knowregarding the cell and tissue would be far too intractable toeither analyze or simulate, the lightening pace of computationalimprovement notwithstanding. Of course, this has neverstopped scientific inquiry from succeeding, to a very highdegree, in situations that may be somewhat analogous. After all,one need not fret over the exact shape of an apple, nor thepositions and quantum states of each of its electrons, in orderto predict its trajectory to earth.

Thus, the challenge in designing a cancer simulator is inchoosing appropriate scales of resolution and subsumingunnecessary detail within the chosen level. Since cancer isinherently multiscalar, this must be done at each scale ofinterest. At the level of the genome and its protein products,for example, one could find duplication of pathways, or could

find that a set of pathways converges on a particular phenotypicresult. Thus, it would not be necessary to model each gene andeach protein; rather, a surrogate functional token might repre-sent the net function of a related set of genes. For instance, withrespect to the apoptotic program, recognizing that a cell’s finalcommitment to apoptosis is ultimately determined by mito-chondrial pore disposition of Cytochrome C, the openers,including Bax and Bid, could be represented by a single token,while the closers, including Bcl-2 and Bcl-xL, could be repre-sented by another [76]. In this way, numerous and similarly func-tioning pathways could be braided into far fewer and more eas-ily managed strands. Another device to reduce complexity whilemaintaining accuracy in cell repertoire is to employ stochasticmodeling. For example, while it may be asking for too much totrack all the chemical reactions giving rise to a glial cell’scytoskeletal flexions, its migratory trajectory may be adequatelyrepresented by a biased random walk.

The use of a cancer simulator in clinical practice or pharma-ceutical development is a more difficult question. In the clinic, itmust first be decided what role such a simulator would play.Cancer is not a deterministic disease; its driving force, after all, isgenetic mutation. One role currently envisioned – that of indi-vidually tailored therapy – might be a difficult one, even if veryspecific information on a given patient’s cancer is extractedthrough genetic or proteomic analysis. Perhaps the statisticalpredictions would not be substantially better than today’s empir-ically based prognoses. The history of using in vitro chemosensi-tivity assays to assist in patient therapy selection reminds us tobe cautious [77].

More promising is the use of simulation to enhance pharma-ceutical and novel therapy development. If drug/cell character-istics could be determined adequately through traditional wet-lab techniques, such as cytotoxicity assays, uptake/efflux assaysand blot analyses, the resulting functional parameters could beused as simulation input. The resulting simulation outputwould therefore carry weight, at least in terms of hypothesisgeneration, and guide the next phase of inquiry. It is importantto note that the power of this method lies in the relative easewith which simulation could be carried out, employing param-eter perturbation to thoroughly explore the response spacebefore any cells, animals or humans are ever needed. The sav-ings in resources and lives is certainly powerful motivation inpursuing this line of development.

Five-year view The pharmaceutical industry spends an average ofUS$500–800 million over a span of 10–15 years per drug devel-oped and released into the market [78]. The overall drug-develop-ment process is highly inefficient due to the high failure rate(>75%) of the majority of drug candidates in costly clinicaltrials [79]. These considerations are apparently having an impacton the industry, as important players, such as Novartis and Glaxo-SmithKline, are devoting increasingly greater resources to compu-tational modeling. While the informational aspects of modeling,such as gene sequencing, expression and the statistical inference of

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their effects, have dominated this discipline for the last decade, theimportance of more mechanistic modeling, linking subcellularsignaling with cellular phenotype and gross tissue response, iscoming to light [80]. The results described in this review exemplifythe power of this approach in isolating and explaining determi-nants underlying cancerous progression and therapeutic success.Simulation can provide a powerful hypothesis test-bed, examiningand clarifying recurring patterns critical to the growth of cancer,shedding light on the most promising avenues towards successfulanticancer therapies. This translates into an effective tool toimprove drug development by preclinically screening drugs

in silico, predicting response before a single subject has beenrecruited and assisting in the efficient designs of clinical tests.Therefore, it is probable that the rational development of anti-cancer drugs will depend heavily on the type of computationalmodeling described in this paper, and that no serious pharmaceuticalcontender will be without some level of in silico research.

AcknowledgementsThe authors are very grateful to the referees for their thoroughand helpful reviews. This work was funded by the NationalScience Foundation and the National Cancer Institute.

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•• Computer simulations illustrate a connection between the morphology of the tumor boundary and tumor invasion. This connection could be exploited to predict invasive capability from clinical observation of tumors. The models also predict invasive response to antiangiogenic therapy.

50 Sinek JP, Frieboes H, Zheng X, Cristini V. Two-dimensional chemotherapy simulations demonstrate fundamental transport and tumor response limitations involving nanoparticles. Biomed. Microdevices. 6(4), 297–309 (2004).

• Biobarriers to drug delivery are investigated using nonlinear simulation. Among these biobarriers, invasion is predicted as a result of antiangiogenic therapy.

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56 Breward CJW, Byrne HM, Lewis CE. A multiphase model describing vascular tumour growth. Bull. Math. Biol. 65, 609–640 (2003).

57 Araujo RP, McElwain DL. A mixture theory for the genesis of residual stresses in growing tissues I: a general formulation. SIAM J. Appl. Math. 65(4), 1261–1284 (2005).

58 Cristini V, Blawzdziewicz J, Lowenberg M. An adaptive mesh algorithm for evolving surfaces: simulations of drop breakup and coalescence. J. Comp. Phys. 168, 445–463 (2001).

59 Anderson A, Zheng X, Cristini V. Adaptive unstructured volume remeshing – I: the method. J. Comp. Phys. 208, 616–625 (2005).

60 Markus M, Bohm D, Schmick M. Simulation of vessel morphogenesis using cellular automata. Math. Biosci. 156(1–2), 191–206 (1999).

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63 Cairns RA, Kalliomaki T, Hill RP. Acute (cyclic) hypoxia enhances spontaneous metastasis of KHT murine tumors. Cancer Res. 61, 8903–8908 (2001).

64 Postovit LM, Adams MA, Lash GE, Heaton JP, Graham CH. Oxygen-mediated regulation of tumor cell invasiveness: involvement of a nitric oxide signaling pathway. J. Biol. Chem. 277, 35730–35737 (2002).

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66 Frieboes HB, Zheng X, Sun CH, Tromberg B, Gatenby R, Cristini V. An integrated computational/experimental model of tumor invasion. Cancer Res. 66(3), 1597–1604 (2006).

•• Connection between morphology and invasion [49] is demonstrated in vitro.

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78 Workman P. How much gets there and what does it do?: The need for better pharmacokinetic and pharmacodynamic endpoints in contemporary drug discovery and development. Curr. Pharm. Des. 9(11), 891–902 (2003).

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Affiliations

• Sandeep SangaUniversity of California, Departments of Biomedical Engineering and Natural Sciences II, Irvine, 3120, CA 92697–2715, USATel.: +1 949 824 6439Fax: +1 949 824 [email protected]

• John P SinekUniversity of California, Department of Mathematics, Irvine, CA 92697–2715, USATel.: +1 949 824 6439Fax: +1 949 824 [email protected]

• Hermann B FrieboesUniversity of California, Department of Biomedical Engineering, Irvine, CA 92697–2715, USATel.: +1 949 824 6439Fax: +1 949 824 [email protected]

• Mauro FerrariThe University of Texas Health Science Center, Brown Institute of Molecular Medicine, Department of Biomedical Engineering, Experimental Therapeutics, The University of Texas MD Anderson Cancer Center, Bioengineering, Rice University, Houston, TX. 7000 Fannin, Ste. 1565, USATel.: +1 713 500 3573Fax: +1 713 500 [email protected]

• John P FruehaufUniversity of California, Clinical Medicine, Clinical Pharmacology and Developmental Therapeutics, Dept. of Medicine (Hematology/Oncology), Irvine, Building 23, Room 244, Rt. 81, UCI Medical Center, 101 The City Drive, South, Orange, CA 92868, USATel.: +1 714 456 7669Fax: +1 714 456 [email protected]

• Vittorio CristiniUniversity of California, Deptartments of Biomedical Engineering and Mathematics, Irvine, 3120 Natural Sciences II, Irvine, CA 92697–2715 , USAand School of Health Information Sciences, University of Texas, Health Science Center, Houston, Texas, USATel.: +1 949 824 9132Fax: +1 949 824 [email protected]

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