Multiscale models for the growth of avascular tumors
M.L. Martins a,∗, S.C. Ferreira Jr. a, M.J. Vilela b
a Departamento de Física, Universidade Federal de Viçosa, 36570-000, Viçosa, MG, Brazilb Departamento de Biologia Animal, Universidade Federal de Viçosa, 36570-000, Viçosa, MG, Brazil
Received 21 March 2007; received in revised form 16 April 2007; accepted 18 April 2007
Cancer is a disease derived, with few exceptions, from mutations on single somatic cells that disregard the normalcontrols of proliferation, invade adjacent normal tissues, and give rise to secondary tumors (metastasis) on sites dif-ferent from its primary origin [1]. This excessive, autonomous and nonhomeostatic cell growth occurs in multicellularorganisms, i.e., animals and plants. In the animal kingdom it was observed even among the most primitive inverte-brates of the Arthropoda and Mollusca phyla [2]. Paleopathologists reported neoplasias in dinosaurs whose fossilswere dated from around 100 million years. In human beings, authentic cancer—an osteosarcoma—was found in amandibular fragment of a Pleistocene hominid alive approximately 100 thousand years ago [2].
In the human population, cancer refers to more than 100 forms of a disease that can develop in almost every tissuein the body [1,2]. Today, cancer is the second leading cause of death in the United States and will probably become theleading one there and in some other parts of the world within a few years [3]. The incidence of every type of cancer ismodulated by genetically inherited traits and external factors (lifestyle, diet, or environment) affecting the body [4].
Although each cancer type has unique features, all these diverse tumors evolve according to a universal scheme ofprogression [5] which involves genetic and epigenetic events as well as an intricate network of interactions among cellsand the extracellular matrix in the host tissue. Complexity and universality are main concerns in modern statisticalphysics and dynamical system theory [6]. In turn, the successful treatments for cancer emerge from an iterative processthat depends on both experimental research advances and feedback from clinical trials in which it is possible tolearn whether fundamental ideas and associated theoretical and biological models are having a therapeutic benefit[7]. Hence, an increased use of mathematical modeling to understand tumor progression, guide the design of newexperiments by indicating promising candidates for further clinical investigation, and suggest novel approaches tocancer therapy can be projected.
The aim of this Review is to discuss some of the multiscale approaches and simulation methods used for modelingthe dynamics of in situ tumor growth before angiogenesis. This Review is divided in two parts. Initially, a shortintroduction to the basic biology of cancer is presented. In the second part of the Review, the phenomenologicalapproach to cancer modeling based on multiscale frameworks is discussed in general terms and illustrated throughspecific models. Finally, some applications of multiscale models to study the effects of therapies on cancer growth arepresented. Even on the avascular stages of cancer progression, these effects are crucial for the oncological practicesince usually clinically detected tumors already generated micro-metastasis which are growing avascularly in othertissues of the organism.
2. Cancer biology: Basic features
2.1. How cancer arises
Cancer arises from an interaction of the environment upon the genome. In truth, the double helix structure of theDNA has a limited chemical stability [8] and, over time, DNA accumulates damage caused by external mutagens(mutation-causing agents), spontaneous reactions which disintegrate some chemical bonds in DNA under physiolog-ical conditions, and errors made during DNA copying in proliferating cells. In Fig. 1, the sites on the DNA strandsusceptible to spontaneous intracellular decay are shown. Also, in this figure the main DNA lesions generated byhydrolytic, oxidative, and nonenzymatic methylation processes, as well as the mutations induced by these changesafter DNA replication are illustrated. The slow transcription and replication in mammalian cells, which exposes DNAlocally in a single-stranded form for longer periods, increases the rates of both hydrolytic depurination and cytosinedeamination. So, since the DNA is submitted to millions of replications during the lifespan of an individual, evenDNA damage triggered by such natural decay processes has a large chance to occur.
External mutagens include environmental agents (ultraviolet light, ionizing radiation and numerous chemical com-pounds such as benzo(a)pyrene present in cigarette smoke) and infectious pathogens (the bacterium Helicobacter
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Fig. 1. Reactive oxygen species (superoxide anions, hydroxyl radicals and hydrogen peroxide), (by)products of normal cell aerobic metabolism andof lipid peroxidation, as well as other small reactive molecules promote hydrolytic attack to, oxidative damage in, and nonenzymatic methylationof DNA. In (a) the target sites for these spontaneous reactions in one strand of the DNA double helix are shown. The relative frequencies of eachdamage event are indicated by the widths of the arrows. (Adapted from Ref. [8].) (b) Hydrolytic depurination generates apurinic sites in DNA and,in consequence, a loss of genetic information.
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Fig. 1. (continued) (c) Hydrolytic deamination of cytosine to uracil and (d) oxidation of guanine residues to 8-hydroxyguanine in DNA are the twomajor spontaneous premutagenic events in living cells. 8-hydroxyguanine base-pairs preferentially with adenine rather than cytosine. (e) Mutationsproduced after replication by such chemical alterations of the nucleotides.
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Fig. 2. Many and varied external carcinogens (cancer-causing factors) trigger cancer by creating mutations on DNA. Therefore, they are reallymutagens. For instance, (a) a single point mutation in normal human ras gene (five thousand DNA bases long) transforms it into the bladdercarcinoma oncogene. This minimal change can be the result of an unrepaired oxidation of guanine to 8-hydroxyguanine, which pairs preferentiallywith adenine rather than cytosine. (b) Exposure to ultraviolet light causes formation of dimers between two successive pyrimidinic bases (cytosineor thymine) which induce copying errors. Here, a thymidine dimer is shown. (c) Uncorrected rejoining of chromosomal breaks caused by, forinstance X-rays or radiation, can juxtapose DNA segments that were previously in different chromosomes (translocations). In Burkitt’s lymphoma,a normal myc proto-oncogene becomes improperly fused to sequences regulating the expression of antibodies in B cells. The translocated mycgene, now regulated by the antibody enhancer, is persistently expressed and drives the excessive proliferation of the mutated B cells. This kind ofmutation explains why cancers of the immune system are the most frequent in people exposed to radiation.
pylori, more than 100 human papillomaviruses—HPVs, hepatitis-C virus—HCV, Aids virus—HIV, etc.). The damagethey create in the genome frequently induces either point mutations affecting single genes or large-scale chromoso-mal mutations which may involve multiple genes [9,10]. Specifically, chemical carcinogens typically cause pointmutations in the DNA sequence. Ionizing radiation causes double-strand DNA breaks which presumably lead to chro-mosomal mutations when incorrectly repaired. Oncogenic viruses introduce exogenous (alien) DNA in normal cells.In Fig. 2 are shown examples of point mutations and translocations which play a major role in cancer formation.
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The rate of mutations induced by external mutagens depends on the exposure level to these agents and, in the caseof radiation, the distribution of energy density [11]. In turn, during DNA replication as many as one in every thousandbases is incorrectly inserted in the newly copied DNA strand by the DNA polymerase. The overwhelming majorityof these copying errors are corrected by the complex machinery of DNA repair [12], so that fewer than one basein a million appears to have been mutated at the end of DNA replication [13]. Similarly, hydrolysis, oxidation andnonenzymatic methylation of DNA as well as damage inflicted on DNA by mutagens are counteracted by specificDNA repair pathways.
So, as claimed by Weinberg [13], the rock-solid stability of the genome is a mirage. Its constance results of “a per-manent struggle in which an ever-vigilant repair apparatus continuously fights off genetic chaos”. Moreover, defects(themselves mutations) in the DNA repair machinery responsible for erasing DNA lesions created by mutagens, in-tracellular decay or miscopying, increase the rate at which mutations accumulate and drive this competitive dynamictowards genetic instability.
The outcome of DNA damage is generally adverse on its metabolism, blocking transcription, triggering cell-cyclearrest or cell death and posing problems for proper chromosome segregation during mitosis. Hence, natural evolutionhas elicited a tapestry of sophisticated, interwoven DNA repair systems that cover almost all the damage inflicted onthe genome of living organisms [12]. Blocked transcription or replication or specialized sensors may lead to lesiondetection, and thus to the arrest of the cell-cycle machinery at specific checkpoints in G1, S, G2 and M to allowrepair of lesions. Persistent blockage of RNA synthesis or extensive damage in DNA trigger p53-dependent apoptosis,the ultimate model of rescue resulting in cell sacrifice. So, we are faced with a basic process involved in genomicinstability. Inherited or acquired deficiencies in genome maintenance systems contribute widely to the onset of cancerby perpetuating mutations. Indeed, various familial cancers are caused by inherited defects in DNA repair [12].
Rarely, unrepaired damage is converted into a permanent mutation and will cause problems in subsequent rounds ofreplication of the host cell and its progeny. Central events for carcinogenesis are mutations that occur in genes crucialfor the control of cell growth. These mutations activate proto-oncogenes and inactivate tumor-suppressor genes, thetwo main kinds of growth-controlling genes. In Ref. [14] an extensive list of cancer genes and the pathways theycontrol is presented. Single oncogenes cannot transform normal cells into cancer cells, while the concerted actionof various pairs of oncogenes induce further cell transformation and proliferation. But in order to produce a fullymalignant phenotype, the antagonist, coupled circuitry that responds to external growth-inhibitory signals and maytrigger regulatory mechanisms such as apoptosis, differentiation or senescence, must be disrupted by a sequence ofmutations in tumor suppressor genes. A paradigm for tumor formation is colon cancer in which an oncogene (suchas ras) is activated and various suppressor genes (APC, DCC, and p53) are inactivated. There seems to be a universalpattern of mutations in human cancers: all derive from the activation of at least one oncogene and the deactivationof tumor-suppressor genes. Deregulated cell proliferation and suppressed apoptosis (programmed cell death) togetherconstitute the minimal platform upon which all neoplastic progression occurs [15].
At the tissue level, the initial lesion caused by a carcinogen is the clonal proliferation of a single mutated cell(orange, in Fig. 3). This clonal growth, called hyperplasia, is focal, ordered in form, temporally restricted and tends todisappear by a programmed pathway of differentiation. Rarely, one of these cells (pink, in Fig. 3) accumulates anothermutation, giving rise to a new clonal expansion. The corresponding cell progeny constitutes a subpopulation thatexhibits aberrant differentiation, growth disorganization, cytologic abnormalities, and further proliferative advantages.Once again, atypical cells (purple, in Fig. 3) appear within the areas of aberrant differentiation. Such cells, altered byan additional mutation, are large with prominent and usually hyperchromatic nuclei, and tend to be separate fromone another. The tissue is now said to be dysplastic. Further progression comes with the clonal expansion arisingfrom such a focal event. The resulting lesion, called an in situ cancer, is confined to the tissue compartment of originand may remain contained indefinitely. Its growth is normally quite slow but rarely ceases. Thereby, some cells mayeventually suffer additional mutations (blue, in Fig. 3). If the genetic changes allow these cells to traverse the basementmembrane and grow in the subjacent mesenchyme, the lesion becomes a primary cancer. The ultimate step towards afull malignant phenotype involves further mutations that convey metastatic competence to a cell (red, in Fig. 3) withinthe primary tumor. Now, it and its progeny can escape from the primary site by entering into blood and lymph vessels,circulate freely in the bloodstream, adhere to and penetrate the capillary wall again, and grow in the mesenchyme ofa secondary tissue. Invasion and metastasis are the cardinal features of a malignant neoplasm, a cancer.
So, at the tissue level, carcinogenesis and tumor progression are evolutionary processes in which natural selectionacts upon the diversity of somatic clones, promoting the expansion of those with the best inherited or acquired form of
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Fig. 3. (a) Schematic illustration of the main stages observed in cancer progression. Carcinomas, tumors in epithelial tissues that represent morethan 90% of all human neoplasms [17], are the paradigm. In (b) a real pattern of a canine skin carcinoma is shown. At the right of the arrow, theepithelial layer is normal, but at left its normal architecture was lost due to the invasive tumor growth. The stages in (a), adapted from Refs. [18,19],are explained in the text. (For interpretation of the references to color, the reader is referred to the web version of this article.)
proliferative advantage. The evolutionary trajectories of cancers are shaped by the selective pressures imposed uponthem by differing somatic environments in which they arise, grow and spread [16]. Hence, the heterogeneity anddiversity observed in cancers are the signatures of dynamic and stochastic evolutionary forces that shift accordinglytheir previous living history.
2.2. Hallmarks of cancer
Although cancers are extremely diverse and heterogeneous, a small number of pivotal steps associated with bothderegulated cell proliferation and suppressed cell death is required for the development of any and all tumors. In a
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seminal paper, Hanahan and Weinberg [20] claimed that all types of tumors are ultimately a collective manifestation ofsix essential transformations in cell physiology: self-sufficiency in growth signals, insensitivity to growth-inhibitorysignals, evasion of apoptosis, limitless replicative potential, sustained angiogenesis, and tissue invasion and metas-tases.
The alterations associated with each of these physiological changes are successively acquired during tumor devel-opment through either oncogene activation or loss of function of tumor suppressor genes. So, for instance, constitutiveactivation of the SOS-Ras-Raf-MAP kinase pathway can lead tumor cells to synthesize many of their own mitogenicgrowth signals. In turn, disruption of the retinoblastoma protein (pRb) pathway allows cell proliferation, renderingcells insensitive to antigrowth factors that operate to maintain cellular quiescence and tissue homeostasis. Since trans-formed cells exhibit various signals (DNA damage, signaling imbalance caused by oncogene activity, hypoxia, etc.)that trigger apoptosis, acquired resistance toward apoptosis is a hallmark of cancer cells. At the molecular level, themost common strategy to evade apoptosis is the functional inactivation of the p53 gene through mutation. However,those three acquired traits—growth factor autonomy, insensitivity to antigrowth signals, and resistance to apoptosis—do not ensure expansive tumor growth. Independently of the disruption of cell-to-cell signaling pathways, neoplasticcells need to become immortalized, i.e., acquire the ability to replicate without limit. Upregulation of the telomeraseenzyme expression in 85% to 90% of malignant cells maintains the length of the telomeres above a critical threshold,leading to unlimited proliferation of descendant cells.
Later on along their progression, incipient neoplasias must acquire the abilities to promote angiogenesis in orderto nourish increasing populations of cancer cells, promote migration and invasion into nearby stroma, penetrate bloodvessel walls, and travel to distant sites where they may settle new colonies—metastases. Each one of these newcapabilities demands additional cellular changes such as the functional inactivation of E-cadherin, shift of the integrinexpression patterns in favor of integrins that preferentially bind to the products of the extracellular matrix degradationby proteases, upregulation of protease genes, downregulation of protease inhibitor genes, and active proteases dockedon their surface.
2.3. The driving force: genetic instability or selection?
As mentioned earlier, cancer is a multistep process involving numerous mutations that convey selective growthadvantages. Since the probability of a cell accumulating sufficient mutations in order to acquire the full, malignantphenotype is too low, it has been proposed that some form of inherent genomic instability is the driving force ofcarcinogenesis [21]. According to this view, mutations in genes that are involved in genomic maintenance (such asDNA repair and chromosomal segregation genes) occur as the initiating events in cancer. Such mutations, althoughhaving no direct selective advantage or disadvantage, increase the mutation rates of other genes. The very existenceof genomic instability is neatly demonstrated through the inherited cancers caused by mutations in genes involved inmaintaining genomic integrity [12]. However, the controversial issue is the relative importance of genomic instabilityand Darwinian selection for tumorigenesis in sporadic cancers [22].
Nowadays, no convincing experimental evidence indicates that mutations causing genomic instability precedesthe two tumor-suppressor mutations required for initiate clonal expansion in the most simple models of sporadictumorigenesis [22]. Also, from a theoretical point of view, hypermutation significantly increases the risk of eitherdisruption of essential cellular pathways or induction of cell-cycle arrest or apoptosis in response to mutations sensedas damages by checkpoints. Hence, the cost of genomic instability could overcome the benefits, mainly in the initiallesions in which transformed cells maintain largely intact the cell-cycle checkpoints of the normal cells from whichthey derive. Accordingly, it is only at the latter stages of tumor progression that cancer cells can acquire genomicinstability as an effect of other mutations, defective checkpoints and severe genetic constraints imposed by radicallynovel environments.
Recent analysis of genetic alterations in human cancer indicates that there is a continuous generation of variantsin proliferative potential among growing cells. Under persistently applied selective growth conditions, these cellsprogress towards a frankly neoplastic state, but much of the selection occurs before they become capable of produc-ing neoplastic foci [23–25]. This observation in culture and the development of a cancer in multicellular organismsdemonstrates the selection of genotypes exhibiting growth advantage over the normal cells. Such advantage manifestsitself as increased proliferation or decreased apoptosis. Further evidence for selection is the tissue-specific characterof tumor associated mutations. As a consequence, only a constrained set of changes convey selective advantages to a
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transformed cell in a given tissue and leads to a nonrandom mutation pattern. Both facts cannot readily be explainedby hypermutation and it seems more likely that Darwinian selection shapes initiation and drives cancer progression.
In outline, the Darwinian model of cancer progression assumes that an initial mutation occurs in one cell amongsta large cell population. This mutant cell acquires a proliferative and/or survival advantage, and its clonal descendantsdominate in a localized area of the neoplasm. Eventually, a second mutation strikes a cell in this first clone transform-ing it into a doubly mutated cell endowed with an even greater advantage. Then, starting from such a newly mutatedcell the process of clonal expansion repeats itself, resulting in a large descendant population in which a third mutationoccurs, and so forth. Actually, this Darwinian model of clonal succession can be improved in order to include theexistence of tumor stem cells and their progeny, the so called transit-amplifying cells [26,27]. Now, cancer stem cellscan differentiate into transit-amplifying cells, but the converse, i.e., dedifferentiation of transit-amplifying cells intocancer stem cells, generally does not occur. Furthermore, the tumor masses are composed mainly of transit-amplifyingcells whereas cancer stem cells are largely in the minority. Mutations in tumor stem cells can be transmitted to de-scendant tumor stem cells which can launch new clonal expansions. On the contrary, since transit-amplifying cellshave a limited replicative potential, mutations in these cells almost always cannot be transmitted further. As cancerprogresses, the genomic instability of tumor cells increases and their mutation rates may exceed the rate at which Dar-winian selection eliminates less-fit clones. Hence, the neoplasm exhibits an increasing number of genetically distinctsubclones, each one dominating a different sector of the tumor mass.
Although conceptually attractive and undoubtedly correct in its global framework, the experimental validationof the Darwinian model of tumor development is very difficult. The main reason for this is the sparse knowledgeabout the nature and kinetics of the key genetic and epigenetic changes in cell genomes that trigger each clonalexpansion occurring in each stage of multi-step carcinogenesis. Epigenetic transformations in cell phenotype withoutchanges in its DNA sequence raises a great challenge to uncover those key alterations directly from the sequenceanalysis of the genome of cells in different stages of cancer development. Virtually, every step in tumor progression isaffected by such epigenetic events, mainly aberrant promoter hypermethylation [28]. Indeed, through this mechanisman increasing number of possible tumor-suppressor genes is inappropriately silenced in certain cancers and this mightbe the possible source of genetic instability. In turn, epigenetically mediated gene silencing is an appealing explanationfor the diversity of gene expression patterns observed in cell populations comprising a tumor. Indeed, the degreeof transcriptional silencing of CpG-containing promoters depends on the number of methylated sites in the CpG-island. The extent of methylation might vary from one DNA strand to another and among different cells, leading toheterogeneous levels of protein synthesis throughout the tumor cell population. By contrast, all subsequent clones of acell carrying a permanent genetic mutation will harbor that mutation and, consequently, the protein expression patternwill be uniform across that pool of tumor cells. However, there is also no direct evidence that CpG-island methylationoccurs as the initial step in carcinogenesis (see Fig. 4).
3. Modeling cancer through multiscale approaches
The aim of this section is to briefly review some of the multiscale models that have been used to study cancergrowth. It is focused on the avascular stage of tumor progression, and other aspects of the disease modeling can bereviewed, for example, in references [31–33]. Also, an extensive historical survey of the contribution of mathematicalmodeling of solid tumor growth is provided in Ref. [34].
3.1. Multiple scales in cancer growth: Phenomenological description
All those basic features of cancer biology reviewed in the previous section demonstrate that tumor growth is in-trinsically multiscale in nature. It involves phenomena occurring over a variety of spatial scales ranging from tissue(for instance, tissue invasion and angiogenesis) to molecular length scales (for example, mutations and gene silenc-ing), while the timescales vary from seconds for signaling to months for tumor doubling times. Moreover, all thoseprocesses, many of which may still be unknown, are strongly coupled. Indeed, an oncogene activation may confer aproliferative advantage to a given cell, promoting its clonal expansion and the depletion of the nutrient and oxygensupply which, in turn, affect the growth of cell clones. To survive in a hypoxic (low level of oxygen) environment, thetransformed cells may acquire new traits such as resistance to apoptosis by a tumor suppressor gene inactivation or
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Fig. 4. The progression of colon cancer involves an ordered succession of mutations and epigenetic changes that strike the genome of colonicepithelial cells. (a) The four alterations registered by the Johns Hopkins researchers [29,30]. The identity of the inactivated tumor suppressor gene(TSG) on chromosome 18q remains unclear. The precise contribution of hypomethylation (loss of methylated CpG-islands) to tumor progression isunknown. Some evidence suggests that it creates chromosomal instability. (b) There are alternative genetic paths followed by all colon carcinomas.The common initiation event is the loss of APC function, but the identities of the altered genes and the precise order of these changes vary insubsequent steps, making feasible the multiple paths suggested here. Taken from Ref. [30].
activated synthesis of growth factors that stimulate angiogenesis [35,36]. Thus, information flows not only from thefiner to coarser scales, but between any pair of scales.
The complexity of cancer progression manifests itself at least in three scales that might be distinguished anddescribed in mathematical models: microscopic, mesoscopic and macroscopic [31]. Specifically:
• The microscopic scale refers to molecular and sub-cellular phenomena occurring within the cell or at its plasmamembrane. Examples are gene mutations or changes in gene expression patterns, alterations of signaling cascadesand/or metabolic pathways, cytoskeleton rearrangement and altered membrane activity, progression through andcontrol of the cell cycle, etc.
• The mesoscopic scale refers to cellular interactions between tumor and host cells such as endothelial cells,macrophages, lymphocytes, and even the local components of the extracellular matrix (ECM). This patholog-ical communication between tumor and stroma is mediated through the synthesis and secretion of stimulatorygrowth factors and cytokines by neoplastic and host cells [16]. In addition, this level includes cell–cell and cell–matrix adhesion mechanisms that determine cell aggregation properties, remodeling of the adjacent ECM thatseems to be a necessary step in the microinvasion of cancer cells, etc.
• The macroscopic scale is concerned with processes occurring at the tissue level such as cell migration, convectionand diffusion of nutrients and chemical factors, mechanical stress, rupture of capsules or basement membranesand invasion of nearby tissues, etc.
So, cancer growth is neatly a multiscale, nonlinear dynamical problem whose fundamental evolution cannot bequantitatively described without the help of mathematical models. It sets up one of the major challenges in mod-ern biology: formulate models in which the rapidly increasing amount of information obtained at the various scales(molecular, sub-cellular, cellular, tissue, organ, etc.) is integrated in accurate models for use in prediction, control
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and simulations of the system response, the macroscopic description level. A direct numerical solution of the multi-ple scale problem aiming to predict macroscopic properties requires an unfeasible computational effort, even usingmodern supercomputers. Hence, it is imperative to develop models able to capture the small scale effects on the largescales and provide the macroscopic responses of the systems accurately and efficiently without resolving the smallscale details. In a multiscale approach, each scale of interest is described in terms of distinct physical models and allof them are coupled in a single model [37–39]. It is partially due to this coarse-grained description of the small scalesthat the resulting multiscale models become computationally more tractable.
In mathematical terms, the tumor and its host environment are described by a macroscopic state vector �u(�x, t)
whose components can include cell population densities, concentrations of the nutrients and chemical factors presentor synthesized in the tissue by interacting cells, and mechanical quantities, such as the pressure, describing the re-sponse of the tissue to external forces. In addition, cell populations are characterized by the internal, functional stateof each of their cells specified by a discrete or continuous variable σ , in turn, either a scalar or a vector quantity.
At the microscopic scale, interacting networks of molecular species operating within sub-cellular compartments(e.g., nucleus, cytosol and plasma membrane) are generally modeled through stoichiometric and reaction kineticsapproaches. In a continuous description, coupled ordinary differential equations (ODEs) are established on the basisof mass-action kinetic principles for both reactions comprising the interaction network and molecular flows betweenneighboring sub-compartments and from some of them to the extracellular microenvironment. Such subcellular mod-els can supply parameters to and interface with the cellular (mesoscopic) and tissue (macroscopic) models. Indeed,molecules synthesized by gene expression, and their derivatives, determine the cell phenotype (state σ ). Furthermore,these gene products may reside on the cell surface or, when secreted, diffuse or advect throughout the tissue. Both,cell surface and secreted molecules generate cell–cell and cell–matrix interactions.
This modeling approach is well illustrated by the study of Athale et al. [40] concerning the cellular decision-processleading up to either a proliferative or a migratory phenotype in human glioma cells. They considered a molecularregulatory network consisting of the epidermal growth factor receptor (EGFR), its ligand transforming growth factor-α(TGFα), the downstream enzyme phospholipase C-γ (PLCγ ) and the associated mitosis pathway. The proposedreaction network model involves 13 state variables and 30 constants. The evolution equation for the first variable is,for instance,
(1)dX1
dt= k−1X3 − k1X1X2 + k9X7,
where X1, X2, X3, and X7 are the concentrations of extracellular TGFα protein, EGFR cell–surface receptor, dimericTGFα–EGFR cell surface complex, and cytosolic TGFα ligand, respectively. In turn, k−1 and k1 are the rates ofdissociation and formation, respectively, of TGFα–EGFR cell–surface complex, and k9 is the rate of insertion into themembrane and secretion of TGFα. Similar ODEs describe the other molecular concentrations. Furthermore, the in-and out-flows of a molecular species m in a given sub-cellular compartment j cause its spatial redistribution accordingto the expression
(2)dX
(j)m
dt= kin
[X
(j−1)m + X
(j+1)m
] − 2koutX(j)m ,
where (j − 1) and (j + 1) refers to the neighboring components before and after j , and kin and kout are the flux rateconstants into and out of the compartment j .
Finally, the functional state σ(t) of a cell is defined by the cellular concentration X4 and the X11 (PLCγ active,phosphorylated, Ca-bound) temporal profile, i.e., its change over time dX11/dt . If dX11/dt is greater than a thresholdθPLC for a given cell, then its functional state corresponds to a migratory phenotype. In turn, a cell for which θn �dX11/dt � θPLC and X4 � θEGFR has a proliferative phenotype. Finally, a quiescent state is associated with a cell inwhich dX11/dt � θPLC and X4 < θEGFR. So, the spatio-temporal dynamics of the molecular processes specify thefunctional state of a cell and determine the stochastic actions (proliferation, migration and quiescence) to be carriedout by each cell, thereby impacting cancer progression and growth at the tissue level.
Also, from this example, other spatially explicit, discrete microscopic descriptions can be easily designed in whichthe molecular network dynamics relevant for cancer progression are, for example, represented in terms of latticereaction-annihilation processes [41].
The mesoscopic scale is usually described in terms of ordinary differential equations [42,43] or cellular automataevolution rules [33,44] for the size Ni (i = 1,2, . . . , n) of each cell population and the functional state σ of every cell
M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156 139
comprising such populations. Both dynamics are determined by intra- and intercellular interactions that can modifythe state of the interacting cells and generate cell replication and death. From general arguments based on balanceprinciples, Ni(σ, t)—the number of cells of the type i and in the state σ at time t—evolves as( Rate of
change ofNi(σ, t)
)=
( Internalevolution
of i-cells: Ii
)+
( Conservativeintercellular
interactions: Ji
)+
(Proliferative
interactions: Pi
)
(3)−(
Death due tocell interactions: Di
)+
(Source/sinkof i-cells: Si
).
Here, the internal evolution Ii takes into account all those intracellular phenomena, such as phenotypic transitionstriggered by mutations, that change the functional state σ of a cell. In turn, the conservative interactions Ji representcell–cell interactions which change the functional state of one or both interacting cells, but do not lead to cell repli-cation or death, as is the case for proliferative interactions Pi or destructive encounters Di , respectively. Finally, thecontribution Si , associated with external sources or sinks, refers to processes such as the production of immune cellsby the bone marrow, the destruction of tumor cells by medical therapies or the injection of cells, for example.
Assuming that Ni is a continuous population density and suitably regular, the balance equation (3) can be writtenas
where Ii is the internal evolution operator, Si is the source/sink operator, and the collision terms Ji , Pi and Di areassociated with conservative, proliferative and death intercellular interactions, respectively. In the realm of classicalpopulation dynamics [45], the collision terms are the “functional responses” dependent on the encounter rates perunit volume and unit time between cells with states η of the ith-population and other cells with states ξ in the j th-population. Thus, for instance, in its simplest form the conservative interactions Ji are given by a quadratic form
(5)Ji =n∑
j=1
q−1∑η=0
p−1∑ξ=0
Tij (η, ξ ;σ)Ni(η, t)Nj (ξ, t) − Ni(σ, t)
n∑j=1
p−1∑ξ=0
Tij (σ, ξ ;η)Nj (ξ, t),
where Tij (η, ξ ;σ) is the transition rate of the ith-cell type to the state σ , given its initial state η and the state ξ ofthe interacting cells belonging to the j th-population. Also, it was assumed that the ith- and j th-cell types have q
and p discrete functional states, respectively. The first term for Ji takes into account those cells that, as a result ofintercellular interactions, end up with a state σ , whereas the second one is a loss term, i.e., related to those cells thattransit from the state σ into another state after their pair interactions with other cells. Clearly, if instead discrete,the functional state σ is a real variable in the interval [a, b], then the sums over such states become integrals, Ji isrewritten as
(6)Ji =n∑
j=1
b∫a
b∫a
Tij (η, ξ ;σ)Ni(η, t)Nj (ξ, t)dη dξ − Ni(σ, t)
n∑j=1
b∫a
Tij (σ, ξ ;η)Nj (ξ, t)dξ
and the evolution equation (4) transforms into an integro-differential equation for cell populations. In Ref. [42], equa-tion (3.35) is a concrete example of this general framework applied to a specific model of the tumor-immune systeminteraction.
An alternative and more realistic approach at the cellular scale is based on cellular automata models [46,47]. Suchan approach can be better illustrated through an example. Recently, Alarcon et al. [48] studied the influence of bloodflow and red blood cell (RBC) heterogeneity on the growth of a colony of normal and cancer cells. The colony evolvesin a two-dimensional square lattice endowed with a vascular network. The vasculature has highly inhomogeneousdistributions of radii, blood flow and haematocrit (i.e., volume fraction of RBC in the blood). Once generated, thevascular network does not evolve further and determines the dynamic of the cell colony in response to the associateddistribution of oxygen in the tissue. The rules iterated sincronously to update the state of the automata are:
(1) A site of the square lattice occupied by a vessel does not evolve.
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(2) A normal cell dies if the oxygen concentration in its site is below a threshold value. Otherwise, the cell divides ifthere is at least one empty nearest neighbor, and the daughter cell will occupy the free neighbor with the largestoxygen concentration. If there is no empty space, then the normal cell fails to divide and dies.
(3) A cancer cell divides as a normal cell but constrained by a different threshold. If the local oxygen level falls belowthis threshold, a cancer cell enters a quiescent or latent state. While the oxygen level remains below threshold, acancer cell can stay in the quiescent state for at most a given number of time steps, otherwise it dies. However, ifduring this time interval the oxygen level overcomes the threshold, then the cancer cell returns to the proliferatingstate and its “internal clock” resets to zero.
(4) The oxygen level threshold for a normal cell is NT 1 if it is surrounded by more normal than cancer cells. If, on thecontrary, there are more cancer cells than normal ones, the threshold is NT 2 > NT 1. So, normal cells are assumedto divide more readily if they are mainly surrounded by normal cells. Analogously, a cancer cell has the oxygenthresholds CT 1 and CT 2, again with CT 2 > CT 1.
Thus, in the cellular automata framework one proposes a set of evolution rules for the discrete state variablesinspired by generic features of the system dynamics. These rules (deterministic or stochastic) are local in space and,in general, Markovian in time, i.e., dependent only on the previous state of the system. They replace the set of coupledODEs (4) used in continuous models to describe the evolution in time of the system at the cellular (mesoscopic) level.Also, all the components of the system have their states updated in parallel.
Each modeling approach, discrete or continuous, assumes typical time and length scales and a specific nature of thephysical or biological interactions involved. Indeed, the continuous evolution equation (4) for cell population densi-ties essentially describe mean-field-like cellular interactions. In turn, cellular automata (or agent-based) models relieson local or short-range interactions. Possibly both types of interactions may occur. In addition, the balance betweendeterminism and stochasticity depends on time and length scale on which questions are posed. At the molecular level,protein–protein and protein–DNA interactions are stochastic, but at the cellular level and long time span, average con-centrations can be statistically defined and their evolutions described by deterministic equations. Moreover, unknownor incompletely understood biological mechanisms that play a role in the considered phenomena can lead to stochasticmodel descriptions.
At the macroscopic scale, a continuous approach is generally applied based on the assumption that there is a suffi-ciently large number of cells and relevant molecules in order to define average continuous values for the macroscopicvariables of interest across the whole tissue. Accordingly, continuum models are mathematically formulated in termsof PDEs in the realm of fluid and continuum mechanics. The majority of them establish an evolution equation for thefield �u(�x, t). For those components of �u representing cells and chemical substances, the derivation of the correspond-ing evolution equations follows from the mass balance principle applied to each element volume:⎛
⎜⎝Rate of changeof the densityof cells within
an element
⎞⎟⎠ =
( Generationof cells in
this element
)−
(Advective outflowthrough its
boundary surface
)
(7)−(
diffusive outflowthrough its surface
)−
(Death ofcells in it
).
Under suitable regularity assumptions, Eq. (7) can be written in local form as
(8)∂uj
∂t= −∇ · (�vuj ) − ∇ · (D∇uj ) + Γ (uj ) − δuj
where �v is the convective velocity, D is the diffusion coefficient, Γ (u) is the proliferation term per unit volume, and δ
is the death coefficient of the j th-cell type population density or concentration. The evolution equation (8) depends onsinks, sources and appropriate boundary conditions determined by the distribution of cells and blood vessels, leadinggenerally to the so-called moving boundary problems. As an concrete example, let us return to the Alarcon et al.model [48] in which the extracellular concentration of oxygen P(�x, t), consumed by both normal and cancer cells,evolves accordingly to the PDE
(9)∂P = DP ∇2P − γ (�x)P,
∂t
M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156 141
Fig. 5. Hierarchy of scales and the related mechanisms and modeling approaches. The arrows indicate the mutual interdependence between thelevels in multiscale modeling of cancer growth, implying that models/subsystems at a given scale use information from another scales. See the textfor details.
where DP is the oxygen diffusion coefficient, and γ (�x) is its uptake rate at site �x. The oxygen uptake rate is γN , ifthere is a normal cell at �x, γC , if there is a cancer cell at �x, and 0 if the site �x is empty or occupied by a vessel. Theoxygen concentration field has a crucial importance because it couples the dynamics at the tissue or macroscopic level(blood flow, haematocrit distribution and oxygen diffusion) to the processes of cell division and death occurring at thecellular or mesoscopic scale. In feedback, normal and cancer cells are sinks of oxygen and, therefore, their evolutionin time determine the macroscopic distribution of extracellular oxygen. Hence, both mesoscopic and macroscopicscales are also inexorably interwoven.
Such interwoven levels of description is the main feature typifying multiscale models, neatly evidenced in thegeneral framework expressed through Eqs. (4) and (8). Indeed, as shown in Fig. 5, phenomena at the subcellularlevel (microscopic scale) affect the cell dynamics (mesoscopic scale) since the internal evolution Ii operates on thecellular states σ which determine the conservative Ji , proliferative Pi and destructive Di intercellular interactions.Vice-versa, conservative interactions Ji can also alter cell states σ and consequently the nature and timing of thesubcellular processes at the microscopic scale. In turn, cell division and death and the external introduction of newcells, events associated with the cellular level, lead to new distributions of source/sink of chemical factors, nutrientsand mechanical stress as well as generate moving boundary conditions that specify the boundary-value problem forthese continuous fields at the macroscopic level. Again, in counterpart, the distribution of nutrients, chemical factorsand mechanical stress in the tissue, components of the macroscopic physical state of the system �u, clearly affect boththe sub- and intercellular phenomena, i.e., the microscopic and mesoscopic scales. Mathematically, the link betweenthe macroscopic, mesoscopic and microscopic scales has to be referred to the parameters (growth, death, uptake orabsorption, and degradation rates, threshold densities, diffusion coefficients, drift velocities, etc.) characterizing themodel. Each parameter refers to a given phenomenon and has a particular effect on a specific cell population, or achemical substance, or a subcellular process within a cell type. Some of the parameters can be evaluated from biolog-ical essays, obtained from generic databases or derived from mathematical models, which justify the increasing effortto develop a quantitative and structural information infrastructure suitable to support physiological modeling [52].
Finally, for the complete specification of the macroscopic model, it is necessary to introduce constitutive lawsto characterize, for instance, the mechanical properties of the tumor [49–51]. Also, as emphasized by Bellomo andPreziosi [42], the link between the microscopic and the macroscopic description remains an open problem. The reasonis that sub- and cellular models involve a more detailed dynamics dependent on the state of the cells, information which
142 M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156
is lost in macroscopic models as one can see comparing the evolution equations (4) and (8), even for nonmoving cellsin a spatially homogeneous environment.
3.2. A multiscale model for avascular tumor growth
In order to illustrate the general framework for multiscale modeling of cancer growth presented above, a specificmodel for avascular tumor dynamics will be discussed. The model, proposed by the authors [53], integrates the cellular(mesoscopic) and tissue (macroscopic) scales. Furthermore, it introduces an effective stochastic cell kinetics controlledby local probabilities for cell division, migration and death as a strategy to connect the macroscopic diffusion equationsfor nutrients and/or growth factors to cell response and interactions at the microscopic scale, a central challenge indeveloping multiscale models.
The model consists of a tissue fed by a single capillary vessel. The tissue is represented by a square lattice ofsize (L + 1) × (L + 1) and lattice constant �. The capillary vessel, localized at the top of the lattice at x = 0, isthe unique source from which nutrients diffuse through the tissue towards the individuals cells. Three cell types areconsidered: normal, cancer and tumor necrotic cells by depletion of oxygen and nutrients. Any site, with coordinates�x = (i�, j�), i, j = 1,2, . . . ,L, is occupied by only one of these cell types. In contrast to normal cells, one or morecancer cells can pile up in a given site. In turn, tumor necrotic cells are inert and, for simplicity, will be consideredalways as a single dead cell. Thus, each lattice site can be thought of as a group of actual cells in which the normal,necrotic and cancer cell populations assume one of the possible values σn(�x, t) = σd(�x, t) = 0,1 and σc(�x, t) =0,1,2, . . . , respectively. As initial “seed” a single cancer cell in the half of the lattice (x = L�/2) and at a distanceY from the capillary vessel is introduced in the normal tissue, in agreement with the theory of the clonal origin ofcancer [54]. Periodic boundary conditions along the horizontal axis are used. The row i = 0 represents a capillaryvessel and the sites with i = L constitute the external border of the tissue.
As considered by Scalerandi et al. [55], it is assumed that dividing cancer cells are especially vulnerable to somecritical nutrients such as iron, essential for DNA synthesis and, therefore, for cell division. The many other nutrientsnecessary for eucaryotic cells are supposed to affect mainly the motility and death of the cancer cells. So, the nutrientsare divided into two groups: essential and nonessential for cell proliferation, described by the concentration fieldsN(�x, t) and M(�x, t), respectively. However, it is assumed that both nutrient types have the same diffusion coefficientsand consumption rates by the normal cells. These concentration fields obey the diffusion equations:
in which the nutrient absorption terms are proportional to the cell populations present in each site, and differentnutrient consumption rates for normal and cancer cells are assumed, with factors λN and λM . It is important to noticethat the model assumes the simplest form for the nutrient diffusion phenomena, i.e., linear equations with constantcoefficients. Also, λN > λM is used, reflecting the larger cancer cell affinity for essential nutrients.
The boundary conditions satisfied by the nutrient concentration fields are N(x = 0) = M(x = 0) = K0, repre-senting the continuous and fixed supply of nutrients provided by the capillary vessel; N(y = 0) = N(y = L�) andM(y = 0) = M(y = L�), corresponding to the periodic boundary conditions along the x-axis; finally, Neumannboundary conditions, ∂N(x = L�)/∂y = ∂M(x = L�)/∂y = 0, are imposed at the border of the tissue. The hypoth-esis that a blood vessel provides a fixed nutrient supply to the cells in a tissue is a simplification, which neglects thecomplex response of the vascular system to metabolic changes of cell behavior [56,57].
Each tumor cell can be selected at random, with equal probability, and carry out one of three actions:
• Division. Cancer cells divide by mitosis with probability Pdiv. If the chosen cell is inside the tumor, its daughterwill pile up at that site. Otherwise, if the selected cell is on the tumor border, its daughter cell will occupy atrandom one of their nearest neighbor sites �x′ containing a normal or a necrotic cell. The mitotic probability Pdivis determined by the concentration per cancer cell of the essential nutrients N present on the microenvironment
M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156 143
of the selected cell:
(12)Pdiv(�x) = 1 − exp
[−
(N
σc θdiv
)2].
The Gaussian term is included in order to produce a sigmoidal curve saturated to the unity, and the model para-meter θdiv controls the shape of this sigmoid.
• Migration. Cancer cells migrate with probability Pmov. A selected cell inside the tumor, at a site �xi , will move toa nearest neighbor site �x′ chosen at random. Otherwise, if the selected cell is on the tumor border, the invasionof a normal or necrotic nearest neighbor site will be dependent on the number of cancer cells present in theselected site. If in this site there is a single cancer cell, it migrates by interchanging its position with that of theinvaded cell. If there are other cancer cells in the same site of the one selected, the migrating cell will, as assumedabove for cell division, replace the normal or necrotic nearest neighbor cell. There is no room for the normalcell accommodation. Clearly, this is a model simplification since deformable cells and mechanical stresses in thetissue, as well as an explicit dynamics for normal cells, are not considered. The probability of cell migration Pmovhas the same functional form of Pdiv, but depends on the concentration of the nonessential nutrients M present onthe microenvironment of the selected cell and increases with the local population of cancer cells. So,
(13)Pmov(�x) = 1 − exp
[−σc
(M
θmov
)2]
with the model parameter θmov controlling the shape of this sigmoid.• Cell death. Cancer cells die transforming in a necrotic cell with probability Pdel. The cell death probability Pdel is
determined by the concentration per cancer cells of the nonessential nutrients M present on the microenvironmentof the selected cell:
(14)Pdel(�x) = exp
[−
(M
σc θdel
)2],
a Gaussian distribution whose variance depends on the model parameter θdel.
The cell dynamics rules used in our model take into account that, as cancer growth progress, cell migration increasesnear the border of the tumor due to the high availability of nutrients and the increase in the number of cancer cells,which release a series of enzymes (collagenases, metalloproteinases, etc.) responsible for the progressive destructionof the extracellular matrix. Also, in the regions where there is a high population density and an ineffective supply ofnutrients via diffusion processes, cell division is inhibited and, at the same time, the probability of cell death increases.But, under these rules cell growth and migration is possible even inside the tumor. Finally, the model parameters θdiv,θmov and θdel, which characterize the cancer cell response to nutrient concentrations and embody complex geneticand metabolic processes, should be interpreted in terms of the underlying biochemistry and molecular biology, stillan open problem. The other three model parameters α, λN and λM , associated with the consumption of essential andnonessential nutrients for cell proliferation by the normal and cancer cells, should be more easily determined frombiological experiments.
Typical patterns generated by the model vary from compact and circular to papillary-like shapes, as shown in Fig. 6.The tumor morphology is determined primarily by nutrient consumption rates, controlled by the model parameters α,λN and λM , associated with normal and cancer cells. These patterns are compared with histological samples associatedwith the most commonly observed morphologies in tumor growth, such as papillary, compact and disconnected, inFig. 7. Disconnected patterns are typical of round cell neoplasias such as lymphoma, mastocytoma, and plasmacytoma;papillary morphologies are found in epithelial tumors, such as basal and skin cell carcinomas, and hepatomas. Finally,compact patterns are frequently observed in solid tumors.
In order to simulate disconnected and ramified tumor patterns, typical of round cell tumors and trichoblastoma, achemotactic interaction among cancer cells mediated by growth factors was added to the competition for nutrients.The growth factor (GF) concentrations obeys the diffusion equation:
(15)∂G
∂t= DG∇2G − k2G + Γ σcN(GM − G),
144 M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156
Fig. 6. Simulated patterns generated by this nutrient-limited cancer growth model. They are organized as functions of the dimensionless nutrientconsumption rate α = �(γ/D)1/2 for normal cells and the multiplicative factor λN to the consumption rate of mitotic essential nutrients by thecancer cells. The remaining four parameters of the model were fixed at λM = 10, θdiv = 0.3, θmov = ∞ (absence of cell migration) and θdel = 0.01.The patterns are drawn in a gray scale where the darker regions represent higher cancer cell populations. The tissue size is 500 × 500, with theinitial “cancer seed” a distance 300 sites from the capillary. The total number of cancer cells depends on the tumor morphology and attains up to2 × 105 for compact patterns. The simulated patterns are compact for low λN values and become papillary or finger-like for high λN . For the sameλN the patterns are more papillary for higher α. Since the capillary vessel provides a fixed nutrient supply, the consumption rate of the normaltissue α set up the levels of available resources for which cancer cells compete. So, high α and/or λN values correspond to the limit of strongnutrient competition.
which includes the natural degradation of GF, also imposing a characteristic length ∼ 1/k for GF diffusion, and aproduction term increasing linearly with the local nutrient concentration up to a saturation value GM . The boundaryconditions satisfied by the GF concentration field is G(�x, t) = 0 at a large distance (d > 2/k) from the tumor border.
On the other hand, the cell dynamics has essentially the same rules used previously, but with probabilities for celldivision and migration altered as follows:
• Cell division.
(16)Pdiv(�x) = 1 − exp
[−
(N
σc
− N∗)
G2
θ2div
].
The parameter N∗ determines the nutrient-poor level below which the replication of cancer cells is inhibited.
M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156 145
Fig. 7. Common morphologies observed in cancer growth. (a) Papillary pattern of a squamous papyloma, (c) a compact solid basocellular carcinoma,(e) a disconnected pattern of a plasmacytoma, and (g) characteristic cell filaments of a trichoblastoma. All these histological patterns were obtainedfrom dogs. The corresponding simulated patterns are shown in (b), (d), (f) and (h), respectively.
• Cell migration.
(17)Pmov(�x, �x′) = 1 − exp
[N(�x)G(�x)[G(�x) − G(�x′)]
θmov
],
implying that a cell migrates in a gradient-sensitive way towards sites where the GF concentration is lower thanthat at its starting point. Thus, cancer cells disseminate from the growing tumor mass outwards. The functionalform of Pmov takes into account some key biological features. Cell migration involves large cytoskeleton reor-ganizations which consume energy and is promoted by GFs, such as hepatocyte growth factor, fibroblast growthfactor, insulin-like growth factor and transforming growth factor-β , that trigger the epithelial-mesenchymal tran-sition (EMT) required to convert polarized epithelial cells into motile ones [58]. In addition, extracellular matrix(ECM) degrading enzymes, such as matrix metalloproteinases and cathepsins, usually upregulated in tumor cells,facilitate migration by disrupting the ECM and destroying the adhesivity structures between normal cells. Also,
146 M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156
Fig. 8. Simulated patterns of the nutrient-limited cancer growth model including the influence among cancer cells mediated by growth factors.(a) Compact (k = 0.1 and θmov = 1), (b) disconnected (k = 0.025 and θmov = 0.1), and (c) ramified (k = 0.025 and θmov = 1) morphologies. Theremaining parameters of the model were fixed at α = 3/L, θdiv = 0.5, N∗ = θdel = 0.01, λ = 5 and Γ = 10. The tissue size is 500 × 500, with theinitial “cancer seed” distant 300 sites from the capillary and the total number of cancer cells is 5 × 104. For comparison, a real ramified patternobserved in trichoblastoma is shown in (d).
the loss of cell–cell junctions between cancer cells, a prerequisite for the EMT, can occur through the upregulationof proteases that cleave cadherins [59].
In Fig. 8 are shown typical compact, ramified and disconnected patterns simulated using the extended model whichincludes GFs. The ramified structure shown in Fig. 8(b) should be compared with the pattern of a trichoblastomaexhibited in Fig. 8(d). It should be emphasized that without chemotactic signaling among cancer cells the nutrient-limited model cannot generate stationary disconnected patterns. The reason is that on average, GFs drive cell migrationoutwards, promoting tumor expansion and, as a consequence, generating disconnected patterns for high cell motility.
All the simulated growth patterns, with and without GFs, reproduce a main feature of avascular tumors, namely, theformation of a necrotic core of dead cancer cells due to nutrient starvation, of an outer rim of nutrient-rich, proliferatingtumor cells and, in between these two layers, an intermediate region of quiescent cells. Such multilayered structure wasobserved in multicellular spheroids of cancer cells formed in culture essays [60,61]. In addition, all the growth patternswere characterized by its gyration radius, total number of cancer cells, and number of cells on tumor periphery. Thesequantities could be related to clinically important criteria such as progress curves, rate of growth (volumetric doublingtime) at given radii, proliferative and necrotic fractions of the tumor. In medicine, these data are used to determinetumor malignancy and its prognosis. Essentially the following results were obtained:
• the progress in time of the total number of cancer cells, tumor gyration radius and number of cells on the tumorborder is described by Gompertz curves. So, tumoral patterns exhibit growth saturation and an early exponentialgrowth phase, as frequently observed in clinical practice.
• The generated compact and papillary or finger-like morphologies obey different scaling laws for the number ofperipheral cancer cells. For compact patterns S ∼ N
1/2 as in the Eden model [62], whereas for papillary patterns
C
M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156 147
the exponent in the power-law increases towards unity as the nutrient consumption increases, indicating a fractalmorphology for the tumor. Since in this model version cell migration is not driven by chemotactic signals secretedby the cancer cells, cell motility contributes to round and homogenize the growth patterns.
• The simulated tumors incorporate a spatial structure composed of a central necrotic core, an inner rim of quiescentcells and a narrow outer shell of proliferating cells in agreement with biological data.
3.3. Other multiscale models
In this subsection, we briefly comment on a number of multiscale models that have been developed to describethe avascular phase of tumor growth. The majority of these models uses reaction-diffusion equations for nutrients andgrowth factors [34] and deterministic or stochastic dynamics based on continuous (differential equations) or lattice(cellular automata) approaches for cell populations.
In a series of papers [63–65], Ward and King have made an important contribution to modeling of avascular tumorgrowth. Their initial model [63] considered only live or dead cancer cells whose dynamics were described by a systemof nonlinear PDEs. As the tumor grows, a velocity field develops in response to local volume variations due to cellproliferation and death, both processes are dependent on the concentration of a generic nutrient. In particular, cell deathwas a gradual process in which dying cells contract at a rate dependent on nutrient availability. This model predicted anearly exponential growth phase followed by linear growth, as observed in the intermediate phase of spheroid growth.However, it was unable to generate growth saturation since the products of cell death remained permanently withinthe spheroid. This feature was immediately altered in a new extended version of the model which incorporates theremoval of necrotic mass [64] through two distinct mechanisms: leakage and consumption by neighboring cells. Thelatter mechanism is supported by experimental observations [66]. As a result, the tumor can exhibit either travelingwaves or growth saturation as long-time solutions for different values of the model parameters. Finally, in a secondextension, the effects of mitotic inhibitors released by necrotic cells in dissociation were investigated. Such mitoticinhibition did not alter qualitatively the nature of the growing patterns, but it did significantly increase the chance ofthe tumor reaching a steady state with a reduction in its saturation size, rather than exhibiting a traveling wave solution.
Byrne [67], also motivated by the interplay between apoptosis and cell proliferation, introduced two types of timedelay in cancer cell proliferation. The first one is the time taken for the cells to undergo mitosis, which is regulatedby the cell itself (autocrine control). The second delay mechanism is associated with the time for cells to upregulatethe synthesis of growth factors and for these GFs to modify the rate of apoptotic cell loss. This corresponds to aparacrine control exercised by the neighboring cells. The numerical solutions and asymptotic analysis of the modelingequations revealed that the first type of delay did not affect the behavior of the tumor, but the second one can, beyonda certain critical delay time, destabilize radially-symmetric stationary solutions generating tumor volume oscillations.A few years latter, Sherratt and Chaplain [68] proposed a model for avascular tumors involving continuum densities ofproliferating, quiescent and necrotic cancer cells, as well as a nutrient or growth factor concentration field. In addition,the model incorporated cell migration inhibited by contact in which the presence of one cell type limits the motility ofanother cell type. In turn, the model proposed by Jiang et al. [69] introduces cell–cell adhesion and, at the subcellularlevel, cell divisions are regulated by their position along the cell cycle. Both frameworks were able to generate themultilayered structure typical of solid spheroids. Furthermore, the tumor patterns can be altered significantly by supplyof nutrients coming from the underlying tissue.
Cellular automata models for cancer growth constitute an approach more appealing and familiar to biologicalphysicists. Here, the pioneering work of Smolle and Stettner [70] have inspired several other models. These authorsconsidered a stochastic growth dynamics in which cancer cells can divide, die or move on a square lattice withprobabilities pdiv, pdel and pmov, respectively. (Obviously, pdiv + pdel + pmov = 1.) Only those cells at the tumorborder can move for x sites (1 � x � d) along a random lattice direction. In addition, the concentration of autocrineand paracrine growth factors synthesized by the cancer cells and the stroma, respectively, determine locally the threeprobabilities for cell actions. Thus, a chosen action (division, death or movement) will be carried out by a cancer cellwith a probability Pact = cPI , in which c is the local GF concentration evaluated in a neighborhood of radius ID (theinfluence distance) and the exponent PI is the GF influence power. So, the model used a crude local approximation,instead of a macroscopic diffusion equation, to describe the evolution of the GF field in the tissue. A variety of growthpatterns, ranging from compact cancerous mass to disseminated cells, have been generated by the model dependingon the parameters used. However, these authors provided only a qualitative description of such patterns.
148 M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156
A series of discrete models for cancer growth in which CA rules at the cellular level were coupled with reaction-diffusion equations for nutrients and GFs at the tissue level has been proposed. These approaches became known ashybrid cellular automata models. In particular, Ferreira Jr. et al. [71,72] extended the model of Smolle and Stettnerconsidering a macroscopic diffusion equation for the GFs, distinct functional forms for the local probabilities forcell actions (including Michaelis–Menten-like responses) and the motility of all cells, even those inside the growingtumor. Furthermore, the generated tumor patterns were characterized by its gyration radius, surface roughness [73,74], total number of cancer cells, and number of cells on the tumor periphery. Simulations of this model show thatvaried morphological patterns follow Gompertz growth curves and their gyration radii increase linearly in time andscale, in the asymptotic limit, as the square root of the total number of cancer cells, as found for the Eden model [75].However, for connected and disconnected morphologies the number of cells on the tumor periphery increases in timeaccording to a Gompertz law and scales linearly with the number of cancer cells. Thus, those results revealed thatsome biological features of malignant behavior seem to influence particularly the structure of the tumor border, whileits gyration radius and progress curve are described by more robust functions. Finally, for the growth rules used,morphology transitions as well as transient behaviors up to the onset of the phase of rapid growth in the Gompertzcurves were observed.
Two other important hybrid CA models are those proposed by Dormann and Deutsh [76] and Patel et al. [44].Dormann and Deutsh formulated a two-dimensional CA model in which the chemotactic signaling due to a substancereleased by necrotic cells is included. This key feature is responsible for the effects of cell flow towards the center of thegrowing tumor where necrotic cells accumulate and the nutrient is very scarce. It is necessary to generate the growthsaturation observed in multicellular spheroids. Also, this model was able to reproduce the layer structure characteristicof avascular tumor colonies cultured as spheroids [60]. In turn, the model of Patel et al. examines the effects of thenative vasculature and tumor metabolism on the progression of a small number of monoclonal transformed cells upto an invasive cancer. The diffusion of glucose and H+ ions to and from the microvessels are described through PDEswith sinks and sources associated with their consumption and synthesis by cells. Simulations of this model revealedthat a low local pH facilitates tumor growth and invasion. However, fixing the H+ ion production rate by cancer cells,the optimal conditions for tumor growth and invasion are determined by the microvessel density. Moreover, the mainresult is the presence of a sharp transition between states of tumor confinement and invasiveness when H+ productionpasses through a critical value, as experimentally observed [77].
A general discussion of several mathematical models, both discrete and continuous, developed for the study of thevarious stages of tumor progression (avascular growth, angiogenesis, invasion and tumor-host interactions) is providedin Ref. [78].
Finally, we mention some existing models focused on specialized rather than on generic tumors, as is usually thecase in the literature devoted to modeling solid cancer growth. Here, two cancer types will be considered, namely,ductal carcinoma in situ (DCIS) and Glioblastoma multiforme (GBM). In 2004, Xu [79] adapted a nutrient diffusion-limited model with rigid duct wall in a radially symmetric, cylindrical geometry to study DCIS, the initial growthstage of breast cancer. It occurs when a cell within the thin epithelial layer lining internally the duct wall (a basementmembrane) undergo a malignant transformation and proliferates. Xu showed that the spatial patterns associated tothe stationary solutions of his model (spots, stripes and uniform distributions) were consistent with morphologies (forinstance, cribiform, comedo and papillary) commonly observed in DCIS. Later, Franks et al. developed new DCISmodels in which the local cellular density and pressure were taken into account [80–82]. Their results suggest that theelevated pressure that accompanies tumor growth can, in addition to deforming the basement membrane, stimulateprotease production and localization near the duct wall more effectively than hypoxia. Finally, GBM accounts formore than 50% of all primary brain tumors and is the most malignant glioma (cancer derived from glial cells ortheir precursors). Its growth pattern consists basically of a proliferating core surrounded by an invasive zone. Mobiletumor cells continually shedded by the surface layers of the core form branches within the invasive zone. Since thebranch tips are richer in nutrients, enhanced cell proliferation occurs in these branches. So, it seems clear that GBMexhibits a growth instability of the Mullins–Sekerka type [83] which explains in part why GBM is, perhaps, the tumortype mathematically most studied. In reference [84] can be found a survey of the growing number of works dealingwith the mathematical modeling of glioblastoma. Several features of GBM such as its early growth [85,86], tumorinvasion [87–89] and the macroscopic effects of genetic mutations [40] were modeled. The main results are: (i) bothstrong heterotype chemotaxis and strong homotype chemoattraction are required for branch formation within theinvasive zone. (ii) The heterogeneity of brain parenchyma (e.g., grey and white matter differentiation) leads to space-
M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156 149
dependent glioma growth. Invasion into white matter is faster and facilitated by white matter fibers. (iii) Geneticmutations ultimately determine GBM growth patterns and malignancy tumor grades.
4. Applying the models: From theory to therapies
In the last few years promising avenues for tumor treatment such as gene therapy, virus and antiangiogenic drugshave been opened [15,16]. However, most of the clinically-used cancer therapies have been developed empirically[15], and therefore mathematical models might be complementary (maybe necessary) tools for the understanding ofthe dynamics of drug response in the organism. Thereby, in this section the use of multiscale models to analyse theeffects of therapies on cancer growth will be illustrated through an example worked out in our research group. Clearly,several other studies concerning chemotherapy [52,90–92], anti-angiogenic treatments [93–96], macrophage-basedanti-cancer therapy [97] and other therapeutic protocols for cancer can be found in the literature.
4.1. A virotherapy model
Current cancer therapy is based on damaging DNA by irradiation or chemicals. However, cells lacking p53 tumorsuppressor gene are unable to respond to DNA damage by inducing cell cycle arrest or apoptosis, frequently becomingrefractory to chemotherapy or radiation. These cells occur in more than 50% of all human cancers, and hence it isimperative to develop new treatments. Viruses can be used to trigger a DNA damage response without damagingcellular DNA and to selectively replicate in and lyse p53-deficient human tumor cells [98,99]. Furthermore, otherpotential oncolytic agents were recently tested. Among them are the reovirus, which can be used in more than 50%of all human tumors having an activated Ras signaling pathway [100], the avian Newcastle disease virus [101], anda genetically altered herpes simplex virus, which kill cancer cells exhibiting an altered p16/pRB tumor suppressorpathway [102]. Also, since genetically engineered virus can be selected to kill only cancer cells, virus therapy might bemore specific than standard DNA-damaging drugs associated to many toxic effects on the normal cells [14]. A reviewon the recent progress in the use of oncolytic viruses against tumors is given by Parato et al. [103].
In order to address the effects of oncolytic viruses on cancer growth, we extend our model [53] by introducing amacroscopic reaction-diffusion equation for the virus spreading. This extended model [104] seems to be appropriateto study the response of cancers accessible to direct intratumoral virus injection (primary brain tumors and cancers ofthe head and neck, for example), but mainly their distant metastatic tumors.
The virotherapy begins when the tumor attains N0 cells and it consists of a single virus injection. In the directintratumoral (intravenous) administration a uniform virus concentration V = v0 = 1 over the entire tumor (tissue) issupplied. This approach corresponds to the experimental protocols used in severe combined immune deficient (SCID)mice [100] and in vitro essays [98,99]. Here, four different cell populations were considered: normal σn, uninfectedσc, infected σinf, and dead σd tumor cells. It is assumed that once infected by viruses the cancer cells sustain theirmetabolism until lysis (rupture of cell plasma membrane, leading to the release of cytoplasm and the death of the cell).However, the rules for the dynamics of these infected cells (division, migration and death) are modified as follows.
• Division. An infected cancer cell does not divide since its slaved cellular machinery is focused on virus replication.• Migration. For the same reason, only uninfected cancer cells migrate.• Death. Infected cancer cells die by lysis with probability
(18)Plysis(�x) = 1 − exp
(−Tinf
Tl
),
where Tinf is the time since the infection and Tl is the characteristic period for cell lysis. After cell death, σinf(�x) →σinf(�x) − 1 and σd(�x) = 1 when (σc + σinf) vanishes.
• Virus spreading. The lysis of each infected cancer cell releases v0 viruses to the extra-cellular medium. A randomfraction of them is equally distributed among the nearest neighbor sites of the lysed cell. So, the time evolution ofvirus concentration is given by the quasi-stationary solutions of the discrete diffusion equation
(19)v(�x, t + 1) = v(�x, t) + Dv
4
∑�′
[v(�x′, t) − v(�x, t)
] − γvv(�x, t).
〈�x,x 〉
150 M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156
Here, Dv is the virus diffusion constant, γv is the viral clearance rate and the sum, representing the discreteLaplacian, extends over all the nearest neighbors of the site �x. Eq. (19) was iterated 100 times at each time step,and Dv and γv were varied on the interval [0,1]. Periodic boundary conditions along the y-axis and Neumannboundary conditions (∂v/∂x = 0) at the border of the tissue (x = L) and at the capillary vessel (x = 0) were used.
• Virus infection. Viruses can infect a tumor cell with probability
(20)Pinf(�x) = 1 − exp
[−
(v(�x)
σc(�x)θinf
)2],
a sigmoid function of the local virus concentration per cancer cell controlled by the parameter θinf, whose recip-rocal is a measure of the efficacy of virus infection.
It is worthy to note that the model does not consider the infection of normal cells by viruses. Indeed, as shown inRef. [99], an adenovirus mutant is unable to replicate in normal cells containing functional p53 tumor suppressor gene.Furthermore, such adenovirus mutant produces about 100 times less infectious virus than does wild-type adenoviruseven in cancer cells retaining functional p53. So, genetically engineered viruses can be designed to kill only cancercells exhibiting a given altered set of genes.
Fig. 9 shows the total number of cancer cells as a function of time for different virus diffusion and clearance ratesobtained from simulations of this model in lattices with linear size L = 500 and parameter values as estimated inRef. [104]. Without treatment, the progress in time of cancer cell populations for all the simulated patterns growsconsistently with Gompertz curves [53]. In the case of a successful therapy the tumor mass is completely eradicated afew time steps after virus injection. The key features determining the success of the treatment are the virus oncolyticactivity (θinf and Tl) and spreading properties (Dv and γv). Indeed, a smaller virus diffusion constant and a greaterclearance rate lead to an unsuccessful therapy in which, after a significant initial decrease in uninfected cancer cellsfollowing virus injection, the tumor grows faster than without treatment. A tumor submitted to therapeutic approacheswhich do not lead to its complete eradication might become progressively more resistant and malignant (invasive), asit is well known in clinical practice. These results address the central issue concerning the effect of the host immuneresponse on virotherapy, which for human adenoviruses cannot be investigated through any appropriate animal model[99]. An immune response directed at free virus and late viral antigens expressed on the surface of infected cancercells can be modeled by an increase in γv and the introduction of an additional probability to the death of infectedcells, respectively. For very small γv values the tumor is eradicated with a unique dose, in agreement with the essaysperformed for immune deficient (SCDI) [100] and athymic [99] mice. In turn, for the same dose, but higher valuesof γv , more frequent virus injections are necessary to eradicate the tumor, again as it was observed for syngeneicimmune-competent C3H mice [100].
A striking and novel result in Fig. 9 is an oscillatory behavior for both viruses and cancer cells. It contrasts to thelong-term, spatially uniform population profiles reached through damped oscillations obtained by Wu et al. [105].Instead, the total number of virus, uninfected and infected cancer cells undergo growing (unstable) oscillations, asshown in Fig. 10. Similar patterns exhibiting necrotic regions behind expanding fronts of infected cancer cells havebeen found by Heise et al. [106] (Fig. 10(e)). An oscillatory dynamics may become inappropriate to monitor thetherapy at predefined time intervals, hindering clinical prognosis. A central feature of such oscillations is that theyare not generated by whatever time delay between virus infection and cell lysis, as required in Ref. [107]. In fact, oursimulations demonstrate that for a vanishing Tl this oscillatory behavior is sustained and, mainly, the effectiveness ofvirus therapy is largely increased. Furthermore, in this region of parameter space, a single intratumoral virus injectioncan result in a complete tumor regression with a continuously decreasing probability as the clearance rate increases.Thus, the experimentally observed outcomes for reovirus [100] and adenovirus [99] therapies, in which the completeregression of tumors occurs in 60 to 80% of the mice, seems to correspond to that region of the model parameterspace. An extensive study of the complete parameter subspace θinf, γv is currently being performed.
Additionally, our results suggest that virus treatment may be more effective against highly invasive papillary tumors(e.g., glioblastoma multiform or trichoblastoma), since the cells in such tumors grow slowly due to nutrient limitation.This finding contrasts with the conjecture that low cell densities at the periphery of the ramified tumors may preventfurther spreading of the wave of virus infection [107]. The differences between the predictions of our model and thosein Refs. [105,107], namely, oscillations without time delay and sustained wave infection at low cell densities, prob-ably arise from the combined use of macroscopic equations and stochastic microscopic dynamics for discrete tumor
M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156 151
Fig. 9. Temporal evolution of the number of cancer cells Nc with and without virotherapy. The curves correspond to the compact morphologies(λN = 25, λM = 10, α = 2/L, θdiv = 0.3, θdel = 0.03, and θmov = ∞—no migration). The following virus parameters were used: Dv = 0.7,γv = 0.01, θinf = 0.03, and Tl = 2 (successful therapy); Dv = 0.7, γv = 0.03, θinf = 0.03, and Tl = 2 (oscillatory response); Dv = 0.1, γv = 0.03,θinf = 0.03, and Tl = 2 (unsuccessful therapy); Dv = 0.7, γv = 0.01, θinf = 0.1, and Tl = 2 (complex behavior).
cells, instead of partial differential equations (PDEs). The interplay between interactions and fluctuations inherent tothe discrete microscopic components of the system may lead to collective behaviors not revealed by a PDE-basedmodel [108].
The spatio-temporal patterns of the waves of virus infection, leading to oscillatory tumor cell and virus populations,emerge from the local movement behavior of the individual agents (cells and viruses). In an originally homogeneoustissue, the stochastic rules for cancer cell actions and virus spreading generate the required underlying spatial hetero-geneity for those persistent patterns. Otherwise, as in prey-predator or host-parasitoid systems (the general frameworkembracing our model), populations in a spatially homogeneous environment will fluctuate to extinction via Nicholson–Bailey-like oscillations [109].
The oscillatory (periodic or aperiodic) dynamics of cancer cell population and viral loads predicted by the presentmodel has counterparts in antiviral therapies. Plasma viral loads in HIV infected patients under antiretroviral therapydecrease, after an initial delay phase, rapidly for ∼ 1 week and then less rapidly for the remaining treatment. Dixitand Perelson [110] developed a mathematical model of viral dynamics under antiretroviral therapy and found that,depending on the relative magnitudes of the intracellular, pharmacokinetic, and intrinsic viral dynamic time-scales,the viral load decays in an oscillatory, periodic manner, or with step decays superimposed on these oscillations.Interestingly, such large step decays have been observed for the decline in hepatitis B and C virus levels in patientstreated with potent antiviral therapy [111,112]. However, the sampling rate employed in the present experiments isinsufficient to distinguish nonexponential patterns from the exponential decay commonly assumed in the analysis ofclinical viral load data under therapy.
Fig. 11 provides further comparison among model predictions and data for virotherapy using agents with distinctoncolytic activities. It shows the evolution in time of the number of cancer cells for treatments using virus of distinctoncolytic activities. Notice that a simple relationship between the number of cancer cells and virus oncolytic powercannot be derived due to the oscillatory response to virotherapy. The insets show the tumor population and volume sizeratios defined, respectively, as the number of cancer cells and tumor volumes normalized by their values at the timeof virus injection. In the inset (b), the experimental data correspond to the effects of three different virus (a wild-type
152 M.L. Martins et al. / Physics of Life Reviews 4 (2007) 128–156
Fig. 10. Snapshots of cancer cell (left) and virus (right) patterns at the time steps indicated by the arrows in Fig. 9. Virus and cancer cell con-centrations are depicted in gray scale with darker levels corresponding to higher values. Dead cancer cells are depicted in black. In (e) groups ofdl1520-infected HCT116 tumor cells (dark blue) bordering uninfected and necrotic tissue are shown. In the boxed area the necrotic cells correspondto the lighter region. (Taken from Ref. [106].) (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.)
and UV-inactivated adenovirus, and a mutant adenovirus dl1520) on C33A human cervical tumors xenografts grownsubcutaneously in SCID mice [99]. These experiments may be simulated in our approach by associating distinctvalues for θinf to different viruses, whereas the virus diffusion constants and clearance rates are kept fixed sinceonly one unique experimental model (mouse and implanted tumor) is considered. Remarkably, the predictions of thepresent model are in good quantitative agreement with the experimental data.
5. Conclusions and perspectives
This short review has presented an introduction to the multiscale modeling of solid, avascular tumor growth. Thecomplexity and diversity of phenomena underlying cancer growth and invasion, the range of spatial and temporalscales over which they act, extending from the molecular to the tissue levels, and the intricate way in which theyare interwoven, make practically unfeasible the understanding of carcinogenesis through intuition alone. The devel-opment of quantitative theoretical models for tumor growth, such as those reviewed here, might be a very usefulapproach to deduce how distinct mechanisms interact in cancer, to integrate the rapidly increasing amount of bio-logical information obtained at the various scales in accurate models, and to predict the macroscopic response of thesystem to therapeutic interventions. Such mechanistic models can provide real insights into critical parameters that
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Fig. 11. Temporal evolution of the number of cancer cells for treatments using viruses of distinct oncolytic activities. The arrow indicates the timeof virus injection. Inset (A) shows the simulated cancer cell population ratios, and the inset (B), taken from Ref. [99], exhibits the C33A cells(p53−) volume ratios after virus injection. In both insets, the ratios 5 weeks after virus injection were evaluated averaging over 10 different runsand animals for simulations and experiments, respectively.
control cancer progression, guide the design of new essays by indicating relevant physiological processes for furtherclinical investigation, and prevent excessive experimentation needed to develop effective treatments. Hence, the hardcalculations and extended simulations necessary for the solution of multiscale mathematical models can be closerto the patients bed as clinical oncologists never dreamed before. Indeed, as Hanahan and Weinberg [20] asserted,“with holistic clarity of mechanism, cancer prognosis and treatment will become a rational science, unrecognizableby current practitioners. It will be possible to understand with precision how and why treatment regimens and specificantitumor drugs succeed or fail. (. . .) One day, we imagine that cancer biology and treatment—at present a patchworkquilt of cell biology, genetics, histopathology, biochemistry, immunology, and pharmacology—will become a sciencewith a conceptual structure and logical coherence that rivals that of chemistry or physics.”
At this point, it is important to comment on the new requirements that predictability and quantified uncertaintyimpose on multiscale modeling and observation [113]. Realistic simulations for multiphysics, multiscale phenomenasuch as tumor progression will necessarily involve compromises in model specification and accuracy of the analyticor numerical solutions. Simulation errors comprise errors of numerical origin and modeling errors. Numerical errorsarise from round off errors due to the use of finite precision arithmetic in solving the discretized equations, but mainlyfrom the difference between their solutions (exact or approximate) and the exact solution of the continuum equationsthat these discrete equations approximate. Modeling errors are due to either approximations in the equations and thephysics they represent or errors in the physical parameters, constants and constitutive laws used in model specification.The method of a posteriori error analysis, for instance, can provide an absolute or heuristic bound on the error interms of the PDE which it satisfies, at least for linear equations [113]. In turn, modeling errors can be assessed onlythrough comparison to solutions of more accurate models. Simulations of more complex multiscale models are aquantitatively definite approach. Comparison of the simulated full system behavior with experimental measurementsand observations, when possible, is the most definitive test of accuracy of multiscale models. This is specially truefor biological systems, generally nonlinear and not feed-forward, i.e., plenty of feedback loops connecting differentscales or modeling levels. For such kind of systems it is not known how errors in approximations made at one level
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propagate and accumulate at the next level. At most, error based estimates will likely be gross overestimates of actualerrors.
Today, the quantitative successes of multiscale modeling are the exception and the unresolved scientific difficultiesare the rule [37]. However, the increasing computer power, the development of inherently multiscale modeling andtheoretical ideas, and a growing interest from physicists, mathematicians and tumor biologists, will certainly acceleratethe progress and broad applicability of the multiscale program in biological sciences, particularly in the fight againstcancer.
Acknowledgements
We thank Letícia R. Paiva for the critical reading of the manuscript and for very constructive comments. Specialgratitude should be extended to S.G. Alves and A.V.N.C. Teixeira for their help in preparing some of the figures inthe present work. Also, we would like to thank Dr. Lissandro Conceição from the UFV Veterinary Department forkindly providing us with the histological sections of the tumors. The authors apologize for omissions in citations andcoverage. This research was partially supported by the Brazilian agencies CNPq and FAPEMIG.
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