09 Evolutionary Theory of Cancer

THE YEAR IN EVOLUTIONARY BIOLOGY 2009

Evolutionary Theory of CancerCamille Stephan-Otto Attolini and Franziska Michor

Computational Biology Program, Memorial Sloan-Kettering Cancer Center, New York,New York, USA

As Theodosius Dobzhansky famously noted in 1973, “Nothing in biology makes sense ex-cept in the light of evolution,” and cancer is no exception to this rule. Our understandingof cancer initiation, progression, treatment, and resistance has advanced considerablyby regarding cancer as the product of evolutionary processes. Here we review the litera-ture of mathematical models of cancer evolution and provide a synthesis and discussionof the field.

Key words: evolution; cancer; mathematical modeling

Introduction

The use of mathematics in the study of med-ical conditions has a long history. The earli-est work dates back to the 8th century, whenAl-Kindi attempted to apply mathematics topharmacological questions such as describingthe strength of drugs (Prioreschi 2002). In the18th century, Bernoulli analyzed the morbid-ity and mortality of smallpox and demon-strated the efficacy of vaccination (Bernoulli &Blower 2004). The interaction between mathe-matics and medicine has proven beneficialfor understanding the underlying biology andfor designing treatments and diagnosing dis-eases. Due to its clinical importance, cancerhas been of particular interest to theoreticalinvestigators. Since the early 1940s (Charles &Luce-Clausen 1942), mathematical approacheshave been developed to explain regularitiesseen in incidence data (Armitage & Doll 1957;Fisher 1958; Knudson 2001), hereditary pre-disposition to disease (Knudson 1986; Frank2007), cancer progression (Tomlinson et al.1996; Desper et al. 1999; Michor et al. 2006c),and response to treatment (Goldie & Coldman

Address for correspondence: Franziska Michor, Computational Biol-ogy Program, Memorial Sloan-Kettering Cancer Center, New York, NY10065. Voice: 646-888-2802; fax: 646-422-0717. [email protected]

1983; Komarova & Wodarz 2005; Michor et al.2006b) (Fig. 1).

Cancer results from evolutionary processesoccurring within the body (Nowell 1976). Sinceevolution describes the temporal changes ofa population of individuals due to variationand selection, the concept is highly relevantto neoplasia. Tumors can be viewed froman evolutionary standpoint as collections ofcells that accumulate genetic and epigeneticchanges, which are then subjected to the selec-tion pressures within a tissue. These normallyheritable variations can lead to adaptationsof the cells such as induction of angio-genesis or evasion of the immune system.Beneficial heritable changes can cause rapidexpansion of the mutant clone since they en-able their carriers to outcompete cells that havenot accumulated similar improvements. Muta-tions advantageous to the cancer cell are nor-mally detrimental to the organism, ultimatelycausing death of both the patient and thetumor. Therefore, neoplastic processes serveas an example for selection acting on dif-ferent hierarchical levels (Buss 1987): clonalevolution generally selects for increased pro-liferation, survival, and evolvability on thecellular level and leads to progression, inva-sion, and resistance; the latter effects are se-lected against on the level of multicellularorganisms.

The Year in Evolutionary Biology 2009: Ann. N.Y. Acad. Sci. 1168: 23–51 (2009).doi: 10.1111/j.1749-6632.2009.04880.x c© 2009 New York Academy of Sciences.

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24 Annals of the New York Academy of Sciences

Figure 1. Contributions to an evolutionary theory of cancer. We show examples of theoretical approachesto cancer initiation (black outline), genetic instabilities (dark gray), progression (gray), and resistance (lightgray). Online version: cancer initiation (purple), genetic instabilities (green), progression (dark blue), andresistance (light blue).

The investigation of cancer evolution re-quires mechanistic, quantitative models that in-corporate realistic properties of biological sys-tems such as stochasticity and nonlinearity. Asthe outcomes of such interactions cannot be de-termined by verbal reasoning alone, they mustbe computed from general integrative mod-els of carcinogenesis (Gatenby & Maini 2003;Michor et al. 2004a; Merlo et al. 2006). Theo-retical approaches to tumorigenesis have led toconsiderable insights into the natural history ofthe disease and have begun to transform cancerresearch into a rational and predictive science.In this review we present a historical and topicalview of mathematical models of cancer evolu-tion. We center our attention on several areasof interest in the analysis of cancer—initiation,progression, genomic instabilities, differentia-tion and heterogeneity, and drug resistance—and provide a synthesis of the field.

Cancer Initiation

An understanding of the mechanisms of can-cer initiation has straightforward implicationsfor prevention, diagnosis, and treatment of thedisease. A genetic cause of cancer was first pro-posed by Boveri in 1914 (Boveri 1914). Charles& Luce-Clausen (1942) presented one of theearliest mathematical models of cancer. Theystudied the incidence of skin carcinomas inmice painted with a carcinogen, finding a lin-ear relationship between the square root of thenumber of tumors and the time since the firstpainting. Their model was based on the as-sumption that each application of the carcino-gen causes a certain number of cells to acquiremutations, but that these cells initiate abnor-mal growth only once both alleles of a partic-ular gene have been mutated. The predictionsof the model were in good agreement with the

Stephan-Otto Attolini & Michor: Evolutionary Theory of Cancer 25

Figure 2. The study of age-specific incidencedata of human cancers. (A) Age at death from var-ious epithelial cancers. The plots show the numberof deaths of male patients per 100,000 versus theage at death for different countries on a doubly log-arithmic plane. According to Nordling (1953), theslopes of the curves indicate that the accumulation offive or six rate-limiting events is required to causea lethal cancer; the slopes differ slightly betweencancer types (data not shown). Figure adapted fromWeinberg (2007). (B) Kinetics of sporadic and hered-itary retinoblastoma. Knudson studied the retinoblas-toma incidence in children and found that in heredi-tary cases, the percentage of cases not yet diagnoseddecreases linearly with age while in sporadiccases, this percentage decreases quadratically with

experimental data, reinforcing the notion thatcancer might have a genetic cause and, in par-ticular, could be explained by the accumulationof two mutations.

In the first half of the last century, the mainsource of cancer-related data was age-specificincidence. For most cancer types in adults, thenumber of cases per age class increases with ahigh power of age. Fisher & Hollomon (1951)presented a multicellular model in which mu-tations were assumed to occur in different cellswithin the same cell population and only thecombination of all mutants led to cancer. De-spite the dismissal of this model as being unableto convincingly explain tumorigenesis, the im-portance of cooperation and co-evolution ofdifferent cell types in tumors, the surroundingstroma, and the immune system has recentlybeen highlighted (Vincent & Gatenby 2008).

As an alternative to the multicellular hypoth-esis, Nordling suggested in 1953 that muta-tions must occur sequentially in the same cellfor transformation into a neoplastic phenotype(Nordling 1953). He noticed that when cancerincidence data are plotted on a doubly logarith-mic plane, the resulting curve is a straight linewith a slope close to six. He hypothesized thatin general, cancer data with a slope of n canbe explained by n + 1 hits (Fig. 2A). In 1957,Armitage and Doll presented a more detailed

←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−time (Knudson 1971). He concluded that a single so-matic mutation is sufficient to cause cancer in hered-itary cases and two somatic mutations are neces-sary for the sporadic cases. Figure adapted fromWeinberg (2007). (C) Colon cancer incidence. Thepanel shows the numerical simulation of equation (1)from Liso et al. (2008) and the adjusted cumulativecolon cancer incidence; data were downloaded fromSEER (www.seer.cancer.gov) and adjustments wereperformed as in Table 1 of Michor et al. (2005a).Parameter values are mutation rate 5 × 10−4, initialpopulation size (i.e., number of stem cells per coloniccrypt) 10, carrying capacity (i.e., final populationsize after clonal expansion) 1013, fitness advantageof mutated cells 1%, average time between stem celldivisions 10 days, and probability of diagnosis permutated cell 10−10.


mathematical model of the successive accumu-lation of mutations (Armitage & Doll 1957).They allowed for different mutation rates atdifferent ages and assumed a fixed order in thesequence of mutations. The model was testedwith age-specific mortality curves of 17 typesof cancer and for patients between ages 25 and74. For a subset of cancer types, the accumula-tion of seven mutations before diagnosis seemedto agree well with the observed data. Never-theless, they noticed that a second subset didnot fit the model assuming n +1 hits; the au-thors attributed this deviation to the influenceof external factors such as exposure to carcino-gens or the influence of endocrine secretions.These considerations led to the formulation oftwo distinct mathematical descriptions of can-cer evolution—one assuming constant muta-tion rates and the other considering variablerates of mutation depending on age, sex, andthe site of the disease (Armitage & Doll 1957).These authors’ hypothesis that cancer resultsfrom the accumulation of multiple mutationsin the same cell eventually became known asthe multistage theory of carcinogenesis.

Fisher (1958) suggested that the incidencecurves of most cancers can be explained bythe accumulation of only three mutations. Ac-cording to his approach, the slope of the in-cidence curve, s, is proportional to a functionof the number of mutations, m, and the aget at diagnosis, t3(m−1). This formula was de-rived from a modified version of Armitage andDoll’s model that included the possibility ofearly mutations leading to clonal expansion,thereby altering their epidemiological conse-quences. Fisher found that if the radius of thearea covered by the mutant clone grows at aconstant rate, sequential waves of faster thanlinear growth are expected. In the context ofthis model, a slope of six in the incidence curveis caused by three genetic changes, each fol-lowed by a quadratic increase in the populationsize of the tumor.

The next steps in the development of theoriesof cancer incidence were taken by Ashley (1969)and Knudson (1971). Ashley developed the first

mathematical model based on a comparisonof incidence data of sporadic and hereditarycancer (Ashley 1969). He studied the differ-ences between colorectal cancer patients withand without Familial Adenomatous Polyposis(FAP), a hereditary condition now known tobe caused by a germline mutation in the APCtumor-suppressor gene, which results in hun-dreds to thousands of polyps in the colons ofteenage carriers. Ashley proposed that someof the genetic changes leading to cancer couldcause increased ratios of cell division to celldeath. Under this assumption, a slope of n ofthe incidence curve could result from either n

or n − 1 hits since the accelerated net growthrate reduces the number of necessary muta-tions. When comparing the incidence data ofFAP with that of sporadic colorectal cancer,the difference in the slopes was found to beapproximately two for both men and women.Ashley concluded that a mutation in the APCgene accounts for a slope of two or three in theincidence curve of colorectal cancer, depend-ing on whether clonal expansion is considered.Knudson’s investigation in 1971 then led to theidentification of tumor-suppressor genes by the-oretical techniques and became known as the“two hit” hypothesis of tumor-suppressor inac-tivation (Knudson 1971). The model was de-veloped from the observation that retinoblas-toma, a childhood eye cancer, presents in ahereditary unilateral version and a sporadic bi-lateral type. While hereditary cases accumulatelinearly with age, sporadic cases increase withthe second order of age in frequency (Fig. 2B).Knudson developed a statistical model predict-ing that two somatic mutations in an “anti-oncogene,” as he called it, cause the sporadiccases, while a single somatic mutation com-bined with a germline mutation results in thehereditary cases. The gene later identified tocause retinoblastoma if inactivated in both al-leles, RB1, became known as the first tumor-suppressor gene (Friend et al. 1986).

Inspired by Knudson’s work in the 1970s,several investigations into the predisposition forand inheritance of cancer were initiated (Frank


2004a). Nunney (2003) used a probabilisticpopulation genetics model of multistage car-cinogenesis to arrive at three conclusions con-cerning the inheritance of cancer-related muta-tions: lethal or sterilizing cancers are caused byalteration of more than one gene, with the ex-ception of retinoblastoma; cancers that occurin prereproductive ages infrequently have aninherited predisposition; and cancers depend-ing on the mutation of several genes appearwith increased incidence late in life becausethese cancers do not decrease reproductive fit-ness. Frank (2004b) developed a computationalmodel to show that the larger the number ofmutations needed to cause cancer is, the smallerthe mortality rates in the population are; nev-ertheless, the initial mutations in the processof carcinogenesis are easily accumulated be-cause reproductive selection cannot decreasetheir frequency. Frank (2005) presented a math-ematical model that accounts for inherited mu-tations accelerating the rate of cancer initia-tion. He studied the age incidence of hereditaryand sporadic colon cancer as well as retinoblas-toma, arriving at the same conclusions as Ash-ley and Knudson: since cancer progression isdriven by the accumulation of genetic changes,it is expected that cancers in families with in-herited genetic aberrations will progress fasterthrough the different stages of carcinogenesisthan cancers in families with no inherited al-terations. He emphasized that mutations thatare recessive at the cellular level can neverthe-less be inherited in a dominant fashion.

Fisher’s work, together with papers pub-lished in recent years, demonstrated that thenumber of genetic changes necessary to causecancer cannot simply be read off the age-specific incidence curve. In an effort to studythe dynamics of tumor-suppressor gene inac-tivation, Nowak et al. (2004) designed a pop-ulation genetics model and investigated howthe kinetics of mutation accumulation dependson the number of cells in a compartment andthe cellular mutation rates. The authors foundthree different laws: in small populations ofcells, it takes two rate-limiting hits to inactivate

both copies of the gene, while in intermediatelylarge populations, a tumor-suppressor gene isinactivated in a single rate-limiting hit (sincethe rates of these hits limit the evolutionarydynamics, they are called rate-limiting). Thesekinetics are due to a cell with two inactivatedalleles taking over the population before a cellwith one inactivated allele reaches fixation; thisphenomenon also emerges in different scenar-ios and has been called “stochastic tunneling”(Iwasa et al. 2004). In a large population of cells,it takes zero rate-limiting hits to accumulate thetwo mutations because the magnitude of the po-pulation size enables mutations to emergerapidly. Therefore, a tumor-suppressor geneis inactivated with different kinetics depend-ing on the number of cells in the population,and the epidemiological implications are notstraightforward.

Further, evolutionary models investigatingthe incidence of leukemias have shown that asingle mutation is sufficient to explain incidencecurves with slopes of up to four (Michor et al.2006a; Liso et al. 2008). A stochastic model ofcancer initiation and diagnosis leads to threewaiting times, the magnitude of which deter-mines the slope of the incidence curve: (i) thewaiting time until the production of the firstsuccessful mutant cell, (ii) the time for clonalexpansion of its lineage, and (iii) the waitingtime until diagnosis of the disease (Michor et al.2006a). If the fitness advantage of the mutantcell is small and the population size of the tis-sue and/or the carrying capacity of the tumorare large, then a one-mutation model can ex-plain incidence curves with large slopes. Hencea large slope of cancer incidence data may re-sult from a small number of mutations togetherwith clonal expansion of mutant lineages andthe probabilistic process of diagnosis.

Despite the enormous importance of earlywork for establishing the multistage hypothesisof carcinogenesis, interpretations of cancer in-cidence curves must be done with careful con-sideration of the population genetics of muta-tions accumulating in tissues. To demonstratethe inability of cancer incidence data alone to


inform about the number of mutations nec-essary to cause cancer, we fit the age-specificincidence of colorectal cancer with a stochas-tic model assuming that a single mutation issufficient to cause invasive cancer (model fromLiso et al. 2008) (Fig. 2C). For reasonable pa-rameter values, this model can explain the dataeven though it has been demonstrated that sev-eral genetic changes must be accumulated forcolorectal cancer to arise (Fearon & Vogelstein1990). Therefore, the implications of the slopeof incidence data must be re-evaluated, andfinal proof of the number of mutations neces-sary to cause a particular cancer must comeeither from experimental evidence or from adetailed knowledge of the effects of particularmutations together with mathematical model-ing. Additionally, as the investigations into thegenetic causes of cancer advanced (Vogelstein& Kinzler 2002), other questions started toemerge.

The Mutator Phenotype in CancerInitiation and Progression

In 1991, Loeb presented mathematical evi-dence showing that a moderately large numberof mutations cannot be accumulated in the life-time of an individual under the assumption ofnormal mutation rates (Loeb 1991). By multi-plying the baseline mutation rate by the num-ber of cell divisions occurring in a tissue, heshowed that hundreds to thousands of cancercells are generated within a tissue if one or twomutations are sufficient for carcinogenesis. Ifthe accumulation of more than two mutationsis necessary to cause cancer, however, then atumor could not arise within the lifetime of anindividual assuming normal mutation rates. Aspossible solutions to this problem, Loeb pro-posed mutagenic hotspots, mutational eventsaffecting more than one gene (such as chro-mosome amplifications and deletions), the ac-tivity of carcinogens, mutations conferring agrowth advantage to the cell, and, most im-portantly, a mutator phenotype. The latter hy-pothesis has since caused an intense debate not

only in the field of cancer initiation, but alsoin progression and its implications in treatmentand resistance. Several experimental investiga-tions were initiated to study mutator pheno-types, leading to the characterization of twomain types of genetic instabilities (Lengaueret al. 1998) (Fig. 3A): while chromosomal in-stability (CIN) leads to increased rates of los-ing (parts of) chromosomes triggered by geneticalteration of so-called CIN genes (Kolodneret al. 2002), microsatellite instability (MIN) re-sults in elevated point-mutation rates due toa deficiency in the mismatch repair (MMR)pathway (Kinzler & Vogelstein 1996; Perucho1996). CIN can emerge due to one dominantmutation, whereas MIN requires the accumu-lation of two recessive genetic changes (Fig. 3B).

Many mathematical models have beendeveloped after Loeb’s initial publication thateither support or reject the hypothesis that mu-tator phenotypes are necessary for carcinogen-esis (Tomlinson et al. 1996; Herrero-Jimenezet al. 2000; Luebeck & Moolgavkar 2002;Komarova et al. 2003; Michor et al. 2003a,2004b; Komarova & Wodarz 2004). Tomlinsonet al. (1996) published one of the first responsesto Loeb’s hypothesis, in which they showed thatnormal mutation rates can indeed produce allgenetic changes necessary for carcinogenesisif fitness effects of mutations are considered.By means of computer simulations based ona simple model of cell birth and death, theydemonstrated that in situations in which twomutations must be accumulated for cancer ini-tiation, it is more likely to mutate genes thatconfer a fitness advantage to the cell than toadditionally accumulate a mutation leading toa mutator phenotype. From this observation,the authors concluded that as long as mutationsresult in a growth advantage of tumor cells, ge-netic instability is not the main driving forceof tumorigenesis—the concept of selection issufficient to explain the onset of cancer. Never-theless, as the number of mutations needed forinitiating clonal expansion increases, the im-portance of genetic instability is enhanced. Inparticular, if six neutral mutations are needed to


Figure 3. The mutator phenotype in cancer evolution. (A) Examples of mitotic abnormalities in cancercells. Left, aneuploidy in colorectal cancer cells. Chromosomes have been painted with chromosome-specifichybridization probes, allowing the inspection of chromosome structure and function. Center, abnormal mitoticspindles in oral cancer cells. This cell was stained for kinetochore regions of chromosomes (gray), spindlemicrotubules (white) and chromosomal DNA, revealing an aberrant tetrapolar spindle (arrows). Right, am-plification of centrosomes in prostate cancer. This cell contains six pericentrin-containing centrosomes (whitedots), which nucleate multiple microtubules (gray). Figure from Jallepalli & Lengauer (2001). Online ver-sion: kinetochore regions of chromosomes (red), spindle microtubules (yellow), chromosomal DNA (green).Right, pericentrin-containing centrosomes (yellow), microtubules (green). (B) Pathways to genetic instability.Different types of genetic instability require different numbers of mutational hits to produce the respectiveinstability phenotype. Top, in a heterozygote with one defective MMR (mismatch repair) allele (step 1), allthat is required to begin to develop mutations at a high rate (microsatellite instability, MIN) is the inactivationof the normal allele (step 2). Bottom, chromosomal instability (CIN) has a dominant quality because a singlehit (in a gene such as hBUB1, a component of the mitotic spindle checkpoint (MSC)) is sufficient to producethe CIN phenotype. (C) The role of CIN in tumorigenesis. The early steps of colon cancer initiation occur insmall crypts, each of which is replenished by a small number of stem cells. If the effective population size ofa crypt (i.e., the number of stem cells) is small, then there is a high probability that crypts contain cells ofonly one type at any time. Then a stochastic process can be designed that describes the evolution of the stemcell population toward a more malignant phenotype (by inactivation of the APC tumor suppressor, top row).At any time, a mutation leading to a mutator phenotype (chromosomal instability, CIN, vertical arrows) canemerge. Stochastic tunnels emerge if an intermediate cell type does not reach fixation (gray arrows). Thisstochastic process can be used to evaluate the chance that CIN emerges before inactivation of APC. Figureadapted from Nowak et al. (2002). Online version: Stochastic tunnels emerge if an intermediate cell typedoes not reach fixation (blue arrows).


initiate a neoplasm, then mutation rates mustbe increased by four orders of magnitude inorder for a tumor to emerge within a humanlife span (Tomlinson et al. 1996). The effectsof neutral mutations on cancer progression arefurther discussed below.

Herrero-Jimenez et al. (2000) presenteda statistical analysis of colorectal cancer in-cidence data accounting for demographicstochasticity, that is, heterogeneity in the pa-tient population due to inherited traits andenvironmental exposures. The authors fit atwo-stage initiation–promotion model to thecancer incidence data and calculated adenomagrowth rates, the number of mutations nec-essary for cancer initiation, and the rate ofchromosome loss; the latter was found to besignificantly higher than in normal cells, sup-porting the hypothesis of mutator phenotypes.In contrast to the findings of those authors,a similar initiation–promotion model in thehands of Luebeck and Moolgavkar (2002) ledto the conclusion that genetic instability is notnecessary for fitting the incidence curve of colo-rectal cancer. They found that two rare events(interpreted as the inactivation of both copies ofAPC) followed by a high-frequency event (ac-cumulation of another mutation driving can-cer progression) are sufficient to explain theincidence data. However, a scenario in whichone of the rare early events is accounted forby a mutation causing genetic instability, whichthen accelerates the rate of APC inactivationand progression to cancer, cannot be excluded.

In 2002, a stochastic population geneticsmodel was used to propose that the emergenceof genetic instability is an early event in tu-mors, such as colon cancer, initiated by the in-activation of a tumor-suppressor gene (Nowaket al. 2002) (Fig. 3C). This paper led to otherinvestigations of particular situations in tumori-genesis. Komarova et al. (2003) designed amathematical model of the mutation-selectionnetwork consisting of the tumor suppressorAPC as well as genes causing chromosomaland microsatellite instabilities, CIN and MIN.The authors found that within a broad range

of parameters, a CIN or MIN mutation likelyprecedes the homozygous inactivation of APC.This effect is particularly pronounced if the rateof triggering genetic instability is large and theselective cost of such a phenotype is low. A sub-sequent paper investigated the rate of chromo-some loss optimal for tumor growth that is initi-ated by inactivation of tumor-suppressor genes(Komarova & Wodarz 2004). Evolutionary the-ory predicts that higher mutation rates acceler-ate the rate of evolution but can lead to an errorcatastrophe if the rate exceeds a certain thresh-old (Eigen & Schuster 1977). It is therefore in-teresting to perform a cost–benefit analysis oflarge mutation rates leading to the advantagesof inactivating tumor-suppressor genes as wellas the disadvantages of chromosomal losses.The stochastic approach taken by Komarovaand Wodarz (Komarova & Wodarz 2004) indi-cates that tumor initiation and progression areoptimized if the rate of chromosome loss is ofthe order of 10−2 to 10−3 per cell division—avalue that coincides with experimentally deter-mined rates in CIN cell lines (Lengauer et al.1997). Subsequently, Michor and colleaguespresented a model in which the spatial arrange-ment of cells was explicitly considered (Michoret al. 2004b). Colorectal stem cells were as-sumed to give rise to independent lineages ofdifferentiating cells, which, after undergoing acertain number of cell divisions, are shed intothe gut lumen. By considering mutations in theAPC tumor-suppressor gene and in genes caus-ing chromosomal instability, it was found thatthe presence of a few genes of the latter type issufficient to ensure that the emergence of chro-mosomal instability precedes the inactivationof APC. This finding showed that the hierar-chical structure of colonic crypts reinforces thehypothesis of the mutator phenotype, since theorganization of colorectal stem cells into smallcompartments reduces the protective effect ofnegative selection against genes causing geneticinstabilities (Michor et al. 2003a).

At around the same time, several papers werepublished that investigated the importance andeffect of mutator phenotypes in the progression


of low-grade tumors to more aggressive andinvasive cancers. Sole and Deisboeck (2004)addressed the question of the existence of a mu-tation threshold in cancer by utilizing the qua-sispecies model of Eigen and Schuster (1977).Under the assumption that a mutator pheno-type exists only in a subpopulation of cells, theyconcluded that a limited amount of genetic in-stability is advantageous for a cell clone; if themutation rate surpasses a threshold, the repli-cation rate is reduced and eventually drivesthe clone to extinction. The authors proposedthat cancer cell populations tend to maxi-mize both mutation and replication rates. Lateron, Brumer et al. (2006) used a similar ap-proach to study error thresholds in situationsin which both CIN and MIN are present. Theauthors analyzed the semiconservative quasi-species model of such tumors and, consideringthe role of postmethylation DNA repair in tu-mor cells, found that CIN and MIN tumorsare individually viable, while a cell containingboth types of instability cannot survive. Thisstudy of error thresholds in different situationsprovided an explanation for experimental find-ings that CIN and MIN are mutually exclusive(Lengauer et al. 1998).

Michor and colleagues (2005b) extendedtheir investigation of colorectal cancer to ex-plore the role of genetic instability during theaccumulation of all mutations considered nec-essary to cause invasive cancer. They foundthat the conditions for early instabilities are meteven more easily when a larger number of mu-tations are considered, since the cost of accu-mulating a mutation causing a mutator pheno-type is balanced by the larger benefit of thatphenotype accelerating the inactivation of ev-ery successive tumor-suppressor gene.

One argument against the hypothesis thatmutator phenotypes are necessary for tumori-genesis was that with high mutation rates, theprobability of accumulating deleterious mu-tations increases. This effect leads to nega-tive clonal selection of mutator phenotypes.Beckman and Loeb (2005) developed a dif-ferential equation model including considera-

tions such as the number of cell divisions thathave occurred in a tumor, the number of dom-inant and recessive genes conferring a fitnessdisadvantage, and the percentage of genomicmutations affecting them, as well as mutationrates and the number of genes present in thegenome. The authors proposed that althoughdeleterious mutations do indeed occur, the ef-fect of negative selection is negligible sincedisadvantageous mutants are lost from the can-cer cell population while the main clone con-tinues to proliferate. However, other authorshave suggested that most of the genes in thehuman genome function to constrain cellu-lar growth and coordinate differentiation path-ways (Rajagopalan et al. 2003). It might be pos-sible, they argued, that most genetic changesarising during tumorigenesis are beneficial to aneoplastic cell. Finally, in 2007, Enderling andco-workers used a system of differential equa-tions to emphasize the necessity of genetic in-stability for driving cancer progression unlessthe presence of a large number of stem cellsand/or tumor-suppressor genes in the genomeis postulated (Enderling et al. 2007).

By now it has become generally acceptedthat genetic instabilities play an important rolein the initiation and progression of cancers.The widespread presence and clinical effectsof instabilities lead to questions about phar-macological strategies that may be used toexploit this cancer trait to the advantage ofpatients. For example, genomically unstable tu-mors could be treated with agents that inflictfurther DNA damage on cells (such as alkylat-ing drugs) that serve to push the tumor cellsacross the error threshold. However, in ge-netically stable tumors the amount of DNAdamage incurred might be tolerable and hencecould potentially accelerate cancer progressionand the evolution of resistance. A more exten-sive theoretical investigation of optimum treat-ment strategies would be useful to assess therisks and benefits of DNA-damaging agents uti-lized to treat potentially unstable tumors. Fur-thermore, the number and identities of genescausing genomic instability when mutated are


still mostly unknown, and a systems biologyapproach to the identification of such genes,their tissue specificity, and membership of path-ways is needed.

Considerations of CancerProgression

Understanding and preventing cancer pro-gression is one of the central goals of cancerresearch. The natural history of a tumor is de-termined not only by the genetic and epigeneticchanges accumulating in the cancer cell popu-lation, but also by the tumor’s interactions withthe microenvironment and the immune system,as well as by the dynamics of different cell cloneswithin the tumor.

Bodmer and Tomlinson (1995) showed thatclonal expansion caused by altered cell deathor differentiation rates can result in the popu-lation growing to higher plateaus in size. Us-ing a mathematical model of discrete-time dif-ferential equations, they investigated possiblescenarios for cancer growth when rates of celldeath and differentiation are altered by mu-tations. This model was able to explain longlags in tumor progression and the existenceof benign lesions growing to an equilibriumcell number. In a later contribution, d’Onofrioand Tomlinson incorporated fluctuations in theparameters and nonlinearity in the equationsinto the model (d’Onofrio & Tomlinson 2007).They found that fluctuations of the populationsize at distinct differentiation stages increase theprobability of exponential growth, which is ir-reversible once initiated. In another extensionof Bodmer and Tomlinson’s work, Johnstonand colleagues (2007) relaxed the need for syn-chronous cell divisions and presented a con-tinuous approximation of the model. Theyconcluded that feedback controls of the cellpopulation add stability to the system, ren-dering exponential expansion impossible un-less these regulatory mechanisms are alteredby mutations.

The importance of apoptosis in carcino-genesis has often been emphasized (Hanahan

& Weinberg 2000) and was investigated byKomarova and Wodarz with a stochastic model(Wodarz & Komarova 2007). They argued thathigh rates of apoptosis increase cell turnover,thus generating more mutants and increasingthe probability of cancer progression. Theyfound that an absence of apoptosis would drivethe population to an evolutionary dead end,resulting in benign lesions unable to surpass fit-ness barriers; an optimum relationship betweenrates of cell death and of mutations exists suchthat enough mutants emerge without disrup-tion of other pathways needed for cell viabil-ity. Also in 2007, Wodarz developed a stochas-tic model with which he studied the effect ofcell turnover and mutational mechanisms onthe processes of cancer progression and aging(Wodarz 2007). He found that if mutations oc-cur independently of cell division, an increasedcell turnover results in a higher probability oftumor progression and a lower degree of ag-ing, therefore pointing to the existence of anequilibrium that maximizes the life span of theorganism. If mutations occur only during celldivision, both aging and cancer risk increasewith cell turnover, thus favoring low turnoverrates.

Neutral mutations have been at the centerof interest of evolutionary biologists ever sinceMotoo Kimura introduced the theory of neu-tral evolution in 1968; in this landmark pa-per, he argued that most genetic variation isselectively neutral and hence neither subjectto nor explicable by natural selection (Kimura1968). Rather, most evolutionary change re-sults from random drift of nonselected alle-les. The emergence of neutral mutations dur-ing cancer progression and clonal evolution isof great importance since such changes couldserve as evolutionary “bottlenecks” (Maley &Forrest 2001): after the initial model by Tom-linson and colleagues in 1996 (Tomlinson et al.1996), Maley and Forrest (2001) introduced acomputational model to investigate the rela-tionship between the numbers of neutral andselected mutations as well as those changesthat cause mutator phenotypes. The model was


based on computer simulations of a cell popula-tion proliferating on a two-dimensional lattice.In a large parameter search, the authors foundthat the number of selected mutations positivelycorrelates with the number of neutral muta-tions needed to progress to cancer. Contrary tothe arguments against the necessity of mutatorphenotypes for cancer evolution, the authorsargued that, with the normal mutation rate inhuman cells, a small number of selected muta-tions cannot be accumulated in reasonable timeframes if neutral mutations are also needed forcarcinogenesis.

The identification of neutral and positivelyselected mutants in tumorigenesis has attractedmuch interest since the advent of powerfulgenome-wide analysis tools. The observationthat some genetic aberrations are present inlarge fractions of tumor samples of the same,and sometimes even different, cancer types sug-gests a defining role of the implicated genes inthe process of tumorigenesis, but it has beendifficult to systematically identify such “driver”mutations. To initiate a systematic analysis ofgenetic alterations in cancer, Vogelstein andcolleagues (Sjoblom et al. 2006), as well as Strat-ton and colleagues (Yuen et al. 2007), deter-mined the sequence of protein-coding genesin a total of 232 diverse human tumor sam-ples. The studies identified 189 genes (Sjoblomet al. 2006) and 120 genes (Yuen et al. 2007)that were mutated at significant frequency. Todistinguish genes likely to contribute to tumori-genesis from those in which passenger muta-tions occurred by chance, Sjoblom and co-workers (2006) developed statistical methods toestimate the probability that the number of mu-tations in a given gene is greater than expectedfrom the background mutation rate. For eachgene, this analysis incorporated the number ofsomatic alterations observed in a genomic mu-tation screen, the number of tumor samplesstudied, and the number of nucleotides suc-cessfully analyzed. Because the mutation fre-quencies vary with nucleotide type and contextand are different in different tumor types, thesefactors were included in the calculations. The

output of this analysis was a cancer mutationprevalence (CaMP) score for each gene ana-lyzed (Sjoblom et al. 2006). The CaMP scorerepresents the probability that the number ofmutations observed in a gene reflects a muta-tion frequency higher than expected by chancegiven the background mutation rate. However,this method of estimating selection across thegenome has been criticized by several authors(Forrest & Cavet 2007; Getz et al. 2007; Rubin& Green 2007; Chittenden et al. 2008). Dis-cussions about the P-values used by Sjoblomand colleagues as well as their estimates of thebackground mutation rate have led to the sug-gestion that the CaMP score in its original formseverely overestimated the number of drivermutations in the analyzed datasets. These criti-cisms were countered by the authors of the orig-inal study, who argued that the experimentalsetup warranted a modification of the statisti-cal approach that, if incorporated into the othergroups’ models, would predict almost identicalnumbers of proposed driver mutations acrossthe investigations (Parmigiani et al. 2007).

Such statistical issues cannot be ade-quately resolved without employing theoreti-cal methodologies unrelated to the above tech-niques, as well as a functional validation of themutations that are ranked highly by the algo-rithms. Such validation is essential for identi-fying true driver mutations and also computa-tional techniques that correctly predict whichgenes are functionally relevant for tumorige-nesis. In the following we review alternativeapproaches to the identification of functionallysignificant mutations in cancer.

Maley and collaborators (2004a) used a sta-tistical approach to identify mutations confer-ring selective advantages in Barrett’s esopha-gus. Biopsies sampled from different regionsof patients’ esophagi were analyzed for loss ofheterozygosity, microsatellite shifts, point muta-tions, and methylation of selected loci. Geneticalterations were sorted according to the pro-portion of proliferating cells that carry the al-teration per sample, as well as the frequency ofthat alteration among patients. Highly ranked


changes according to this approach were de-fined as driver mutations, and any geneticalteration that occurred exclusively togetherwith such a driver was interpreted as a hitch-hiker mutation. The authors found that ho-mozygous p16 inactivation followed by p53mutations has a strong selective advantage inBarrett’s esophagus.

In 2007, Sander and colleagues presentedthe Online Mutation Assessor (OMA), a com-putational approach incorporating informa-tion about the sequence evolution and three-dimensional structures of proteins and of theirinteractions in macromolecular complexes, aswell as their placement in molecular path-ways (Reva et al. 2007). Given a nonsynony-mous sequence variant, OMA provides a func-tional report and shows the mutated residuesin the context of a protein family alignmentview and a 3-D structure view. The result isa prioritized list of mutations ranked by func-tional score as well as background informationthat leads to the score, including evolutionarilyconserved patterns across organisms as well asstructure placement of the mutation. This algo-rithm is promising since it incorporates infor-mation about different aspects of the mutationand might therefore lead to a more robust iden-tification of functionally important genes thanapproaches that incorporate only the frequencyof mutations.

Also in 2007, Beroukhim and colleagues de-veloped a statistical methodology called Ge-nomic Identification of Significant Targets inCancer (GISTIC) (Beroukhim et al. 2007).GISTIC was designed to identify signifi-cant copy-number changes in cancer genomesthrough two key steps: it first calculates a statis-tic that takes into account the frequency andamplitude of the genetic change, and then as-sesses the statistical significance of each geneticchange by comparing the statistic derived inthe first step to the results that would be ex-pected by chance (Fig. 4A). This method iden-tifies regions of aberration that are most likelyto drive cancer pathogenesis. Since its publi-cation, GISTIC has been applied to many ge-

nomic datasets and has led to several impor-tant findings (Weir et al. 2007; TCGA 2008).Taylor and co-workers (2008) presented a sim-ilar statistical approach, RAE, to distinguishfunctionally neutral from causal chromosomalalterations in tumors. The key differences be-tween this method and earlier approaches arethat RAE (i) distinguishes between four classesof genomic gains and losses to reflect the bio-logical significances of such alterations, (ii) ren-ders these four scoring models sample-specific,adapting to individual tumors to account fortheir differences, (iii) uses soft discriminationrather than hard thresholds for improved sig-nal extraction, and (iv) generates a randomaberration model using a background of seg-mental DNA rather than independent arraymarkers. When the performance of RAE andGISTIC were compared (TCGA 2008), the ge-netic alterations that scored as statistically sig-nificant were almost identical, suggesting thatboth methodologies are equally good at identi-fying significant aberrations in cancer.

Although many different statistical and evo-lutionary approaches to identifying driver mu-tations have been proposed, the field is stilllacking a comprehensive comparison of thesetechniques together with experimental effortsto validate purported driver mutations. Onlya functional validation in cell line and mouseexperiments can prove that a specific geneticalteration leads to cancer initiation or an accel-eration of malignant growth. Such endeavorswill be of crucial importance in the comingyears.

In 2007, Beerenwinkel and colleagues de-veloped a mathematical model to investigatethe waiting time to cancer (Beerenwinkel et al.2007). Observations of the mutational patternsin colorectal cancer led to an estimation ofthe number of driver mutations of around 20.Based on this hypothesis, the authors studiedtumor growth and genotype dynamics and re-lated the time for a tumor to reach a givensize to the mutation rate, population size, andfitness advantage of the driver mutations. Sim-ulations and analytical results showed that for


Figure 4. Identification of driver mutations and their time of emergence during tumorigenesis. (A) TheGISTIC algorithm. After identifying the locations and magnitudes of chromosomal aberrations in multipletumors (left), GISTIC scores each genomic marker with a score that is proportional to the total magnitude ofaberrations at each location (upper center). In addition, by permuting the locations in each tumor, GISTICdetermines the frequency with which a given score would be attained if the events were due to chance andtherefore randomly distributed (lower center). A significance threshold (vertical line) is determined indicatingthe value beyond which significant scores are unlikely to occur by chance alone. Alterations are deemedsignificant if they occur in regions that surpass this threshold (right). From Beroukhim et al. (2007). (B)A maximum likelihood tree for the karyotypic evolution of clear cell renal cell carcinoma, based on thechromosomal aberrations seen in more than 10% of the cases. Deletions (amplifications) are denoted by aminus (plus) sign followed by the chromosome number and arm. Figure adapted from von Heydebreck et al.(2004).

reasonable parameter values, a fitness advan-tage conferred by driver mutations (assumingnormal mutation rates) could account for thegeneration of a tumor within a human life span.The model also predicts the high level of het-erogeneity seen in colorectal cancer patients,which, in the context of the model, is accountedfor by the emergence of different mutations af-fecting the same mechanistic pathways. Theauthors showed that the assumption of a largenumber of drivers with small fitness advantagesfits well with the data and the incidence of colo-rectal cancer. In 2008, the same group usedmathematical modeling to arrive at some gen-eral conclusions about colorectal carcinogene-sis (Jones et al. 2008): (i) it takes about 17 years

for a large benign lesion to evolve into cancerbut less than 2 years for cells within that cancerto acquire the ability to metastasize; (ii) it takesfew, if any, selective events to transform a highlyinvasive cancer cell into one that has the abilityto metastasize; and (iii) the rates at which pointmutations develop in advanced cancers are sim-ilar to those of normal cells. Such hypotheses,even though interesting, require experimentalvalidation and remain a theory without wet-labefforts.

It is of interest not only to identify all drivermutations implicated in a particular cancer, butalso to uncover the temporal sequence in whichthese mutations arise. The earlier a driver mu-tation emerges during tumorigenesis, the more


likely it is to induce “oncogenic addiction”(the dependence of a cancer cell on the ac-tivity of a particular oncogene) and thereforecould represent the most promising drug tar-get (Weinstein 2002). In 1990, Fearon and Vo-gelstein published their paradigmatic result re-garding mutational pathways, suggesting thatthe accumulation of a linear series of geneticchanges transforms colonic tissue into invasivecancer (Fearon & Vogelstein 1990); however,this so-called “Vogelgram” has since been criti-cized for describing the tumorigenetic processesof only a subset of colorectal cancers (Smithet al. 2002). The identification of evolutionarytrajectories toward cancer has become a hottopic in theoretical research, particularly be-cause the experimental conditions under whichFearon and Vogelstein’s results were obtainedare hard or even impossible to reproduce forother cancer types.

A few years later, Desper and colleagues(1999) presented one of the first theoreticalapproaches to this question. They introducedbranching mutational trees—so-called oncoge-netic trees—to the investigation of cancer pro-gression. The method was applied to cross-sectional comparative genomic hybridization(CGH) data obtained from tumor samples. Thegoal of the algorithm was to identify the treethat best describes the observed joint proba-bilities of each pair of mutations. Their useof tree reconstruction methods allows for theformulation of temporal relations between ge-netic changes and can lead to the identifica-tion of the order in which mutations occur.One drawback to the approach of Desper andcolleagues is that events in the leaves of thetree can only occur once all previous eventshave happened, resulting in large portions ofthe tree with zero likelihood of occurring. Toovercome this problem, Beerenwinkel and co-workers (2005) extended Desper’s algorithm byintroducing mixture trees. These trees combinesuboptimal trees, including a special topologythat contains all events independently; due tothis modification, all genetic alterations can oc-cur without depending on previous mutations.

It was shown that mixture trees could be used toinvestigate independent mutational pathways,which were enigmatic when using single-treereconstruction. The authors also developed ameasure of progression to estimate the sur-vival times of individual patients and validatedtheir predictions with clinical datasets. In 2002,Newton developed a method to find combina-tions of genomic aberrations from CGH data(Newton 2002). This approach involved a jointprobability distribution of the samples’ profilesfrom which ensembles of genomic abnormali-ties were inferred. One important characteris-tic of the model is that no preselection of rel-evant genetic abnormalities is necessary, sincesuch changes are automatically inferred by thealgorithm.

A variation of these tree reconstructionmethods was presented by von Heydebreckand colleagues in 2004 (von Heydebreck et al.2004). In their model, the leaves of the treerepresent mutational events while intermedi-ate nodes denote “hidden” events, which mightbe interpreted biologically as intermediate (ge-netic) events (Fig. 4B). Starting with the wild-type node, stochastic experiments are realizedthat evolve according to the joint probabili-ties between events. The highest-scoring treeis constructed using a likelihood-maximizationalgorithm, resulting in a tree topology withbranch lengths representing the probability tomove from one node to the other; the treetherefore contains information about the likelytemporal occurrence of events in a mutationalpathway.

Even though these phylogenetic algorithmspromise to reconstruct the temporal sequencein which driver mutations arise during tumori-genesis, they have not been validated using theone dataset for which the answer is known: col-orectal tumorigenesis. Once an algorithm hasbeen validated with available data, the predic-tions of the model should be tested in in vivo

models to investigate the stage at which themutation arises experimentally as well as theeffects it has on cell turnover, physiological de-pendencies, and other issues.


Finally, in 2006, Michor and colleagues de-veloped stochastic mathematical models to in-vestigate the dynamics of metastasis forma-tion in patients (Michor et al. 2006c; Michor& Iwasa 2006). The authors considered situ-ations in which one or two mutations–eithersingle activating mutations in genes such asrat sarcoma virus homolog (RAS) and mye-locytomatosis viral oncogene homolog (MYC)(Michor et al. 2006c) or inactivation of both al-leles of metastasis-suppressor genes like Meta-stasis inhibition factor (NM23) (Michor &Iwasa 2006)—are necessary to enable cells tometastasize. They considered the dynamics ofmetastasis-enabling mutations in a cell popula-tion of constant size and investigated whethermetastatic potential is the property of all cellsin the primary tumor or of only a small subset.They found that if most metastases are causedby mutations that confer a fitness advantage tothe cell in the primary tumor, then most or allcells in the primary tumor will carry metastasis-enabling mutations; this effect arises because, inthat case, a mutant cell is likely to reach largefrequencies in the tumor. In contrast, if mostmetastases derive from disadvantageous muta-tions, then only a small fraction of cells in theprimary tumor will have metastatic potential.To compensate for the selective disadvantage inthe primary tumor, such mutations must be ableto successfully found metastases elsewhere witha probability that is many orders of magnitudehigher than that for advantageous mutations.Since this scenario is unlikely, the authors con-clude, most metastases should arise from mu-tations that have reached a large frequency inthe primary tumor. This finding is supportedby experimental evidence (Ramaswamy et al.2003). These models were later extended tosituations in which the growth of the main tu-mor is described by a branching process andthe tumor may undergo exponential expansion(Dingli et al. 2007a).

Evolutionary approaches to invasion andmetastasis are still underrepresented in the lit-erature, and many open questions remain. Ismetastatic ability conferred exclusively by ge-

netic changes, or do other cellular adaptationsprovide for such capabilities? Can a mathe-matical model incorporating mechanical forceswithin a tumor explain the patterns of metas-tasis seen in patients? How can the evolutionof metastatic sites best be prevented by thera-pies? Future theoretical investigations aimed atreducing the mortality associated with invasionwill be useful for the clinical management oflate-stage cancer.

Stem Cells, Differentiation,and Heterogeneity

The role of tissue-specific stem cells, differ-entiation hierarchies, and the resulting tumorheterogeneity in carcinogenesis has attractedthe interest of both experimentalists and theo-reticians (Fig. 5). Tissues are organized into dif-ferentiation hierarchies of cells: the most prim-itive cells are pluripotent stem cells capableof proliferation, self-renewal, and the produc-tion of many types of differentiated progeny(Reya et al. 2001). Stem cells produce commit-ted progenitors, which in turn produce evenmore committed cells. Differentiated cells typ-ically proliferate to fulfill organ-specific tasks.Once fully differentiated, however, such cellsmay lose the ability to replicate, as illustratedby the loss of nuclei in erythrocytes and ker-atinocytes (Bach et al. 2000). In fact, the orga-nization of tissues into hierarchical structuresmight have evolved to minimize the risk of car-cinogenesis, since stem cells are more proneto accumulating mutations that lead to cancerthan differentiated cells due to their enhancedself-renewal capabilities and should thereforerepresent only a small fraction of cells in a tis-sue (Michor et al. 2003b).

Cairns (1975) was the first to put forward thehypothesis that stem cells retain an “immortalstrand” of genetic material. He proposed that tokeep from accumulating mutations, stem cellscould divide asymmetrically, retaining the orig-inal strand of DNA while passing the newlysynthesized, and possibly mutated, strand to a


daughter cell. This hypothesis has since led tocontroversy, with Conboy and associates pro-viding supporting evidence by showing tem-plate strand cosegregation in muscle stem cells(Conboy et al. 2007), and Kiel and colleagues(2007) insisting that hematopoietic stem cellsdo not asymmetrically segregate chromosomes.This controversy awaits resolution.

In 2002, Cairns presented a mathematicalmodel of cancer initiation based on the as-sumption that mutant stem cells undergo apop-tosis due to deficient DNA repair mechanismsand are replaced by progenitors that may haveaccumulated mutations (Cairns 2002). His setof simple formulas showed good concordancewith observed tumor incidence of mice treatedwith carcinogens. Cairns argued that the bio-logical features represented in his model couldexplain the onset of cancer even with the lowmutation rates of human cells.

Recently, Pepper and colleagues presented amodel of cell differentiation and somatic evo-lution. By means of a system of differentialequations, the authors studied the effects ofmutations as well as changing replication anddifferentiation rates in a hierarchically struc-tured population (Pepper et al. 2007). Severalconclusions were drawn: carcinogenesis canbe initiated both in stem cells and transient-amplifying cells, genetic lesions disrupting dif-ferentiation pathways are critical to tumorigen-esis and should be considered among the mostfundamental hallmarks of cancer, and the struc-ture of serial differentiation is a general strat-egy for the suppression of somatic evolution intissues.

Adding to the importance of differentiationstructures is the fact that long lineages in-crease the probability of accumulating muta-tions, since there are more possibilities for mu-tations to arise if the number of cell divisionsa clone has undergone is large. Frank and col-leagues (2003) developed a model to quantifythe risk of cancer initiation for different tissuearchitectures. The authors found that, if themutation rates of stem and differentiated cellsare of comparable magnitude, a tissue struc-

ture with a shorter lineage has a reduced riskof cancer. If the mutation rates of stem cells aresignificantly smaller than those of progenitorsor differentiated cells, then the risk of cancerdepends on the number of mutations neces-sary to drive tumorigenesis: for large numbersof mutations, more transit and fewer stem celldivisions are favored. As the number of mu-tations decreases, a structure with shorter lin-eage has a decreased risk of cancer. In thesame year, Nowak and co-workers presenteda linear model of mutation accumulation thattakes the spatial relationship between cells ex-plicitly into account (Nowak et al. 2003). Usingstochastic processes, the authors showed that insuch a spatial model, fitness differences betweennon–stem cells are unimportant since they are“washed out” of the system, leaving stem cellsas the only source of lasting variation (Fig. 5B).The authors also concluded that such spatialorganization results in an overall reduction inthe probability of cancer initiation. Later on,Dingli and colleagues investigated the effects ofstochasticity on the dynamics of stem cells giventhe small number of replicating cells in a com-partment (Dingli et al. 2007b). They showedthat remission of tumors and rapid expansionof mutant clones could both be explained bymeans of stochastic dynamics alone.

Such theoretical investigations of differenti-ation structures provide important insights intothe evolutionary effects of tissue design. How-ever, in the absence of a clear connection toexperimental results, these endeavors remainabstract. It would be interesting to see mathe-matical approaches to situations in which dif-ferentiation is disrupted by specific mutations.Such investigations can lead to suggestions oftherapeutic intervention based on predicted ef-fects of perturbations of the system.

Genetic and phenotypic heterogeneity oftumors is not only caused by differentiationhierarchies, but also by cancer progression(Maley et al. 2006), mutator phenotypes and theevolution of resistance (Komarova & Wodarz2003), and interactions of cancer cells withtheir microenvironment (Gatenby & Vincent


2003a) (Fig. 5A). Historically, special empha-sis has been devoted to studying the mecha-nisms of generation and maintenance of suchheterogeneity, and a number of computationalapproaches have been developed. In 2002,Gonzalez-Garcıa and colleagues investigated astochastic spatial model of tumor growth andshowed that coexistence of diverse clones withdifferent fitness values in the same tumor is pos-sible due to the complexity of the microenviron-ment (Gonzalez-Garcıa et al. 2002). The evolu-tion of a tumor in three-dimensional space wassimulated utilizing simple mutation–selectionrules. The authors showed that maintenance ofdiversity can be explained by spatial dynam-ics since the exclusion of competitors takes along time as compared to the time scale oftumor growth. A similar approach was usedby Zhang and colleagues (2009). They imple-mented a three-dimensional model of tumorgrowth as a computer simulation and consid-ered several levels of environmental interac-tions. They investigated the accumulation ofa linear sequence of mutations, each capable ofchanging the phenotypic response of the cell tothe environment. The model predicted consid-erable heterogeneity at the molecular level anddifferent rates of evolution in distinct regions ofthe tumor. For instance, in those clones locatednear blood vessels, diversity was reduced dueto the expansion of fast-growing phenotypeswhich led to an increased rate of progression tomore aggressive variants; regions close to thecore of the tumor show a sustained increase indiversity since in that environment coexistenceof different clones is possible. In 2004, Nagydesigned a differential equation model to de-scribe three aspects of a solid tumor: changein cell mass over time, change in tumor vascu-larization over time, and competition betweentwo different cancer cell types (Nagy 2004). Themodel also describes the number of immaturevascular endothelial cells, which build bloodvessels. The model predicts that natural selec-tion always favors more aggressive cancer cellphenotypes, even if the growth of such pheno-types eventually destroys all or part of the tu-

mor. The author suggested that this “hypertu-mor” mechanism may be the cause of necrosisobserved in many vascularized tumors. A simi-lar mechanism was later applied to study tumorincidence across species (Nagy et al. 2007).

In 2006, Maley and colleagues demonstratedfor the first time a direct relationship betweenclonal diversity in premalignant lesions andprogression to cancer (Maley et al. 2006). Theyinvestigated the heterogeneity of cellular clonesin patients with Barrett’s esophagus using eco-logical measures of diversity, finding that di-versity was highly correlated with risk of pro-gression to esophageal carcinoma (Fig. 5C).Diversity was measured by three different in-dices, which incorporated information aboutloss of heterozygosity (LOH), microsatelliteshifts, and sequence mutations across thegenome. A high level of clonal diversity wasfound to increase the risk of progression to ma-lignancy, but to stem from genetic instabilityonly if such variation leads to viable pheno-types. Normal apoptotic pathways may thusprevent progression of the tumor by limitingpopulation diversity and eliminating cells witha mutator phenotype.

Theoretical investigations of this natureare important both for an evolutionaryunderstanding of tumorigenesis as well as fora guideline for clinical interventions: it is im-portant to predict which patients with prema-lignant lesions are most likely to develop malig-nant tumors such that the risks of cancer andof unnecessary surgery can be minimized. Un-fortunately, few systems are amenable to ap-proaches relying on measurements of prema-lignant tissue. However, it is an urgent goal ofthe field to identify methodologies and histolo-gies in which such important studies can beperformed.

In 2008, Vincent, Gatenby, and Gillies pro-posed that cancer cells must surpass severalenvironmental barriers in order to acquirean aggressive–invasive phenotype (Vincent &Gatenby 2008; Gatenby & Gillies 2008). Theauthors presented an evolutionary game the-ory model to investigate the characteristics of


Figure 5. Differentiation hierarchies, spatial arrangements, and heterogeneity in cancer.(A) Cancer stem cells and clonal succession. Cancer stem cells (middle) can differentiate intotransit-amplifying cells (horizontal arrows), but can also accumulate further mutations that drivecancer progression (vertical arrows). Both differentiation and progression processes contributeto tumor heterogeneity. From Weinberg (2007). (B) “Linear process” of somatic evolution.


tumor initiation, promotion, and progression(Vincent & Gatenby 2008). Their analysis wasbased on the idea of adaptive landscapes, inwhich selective traits are altered as the microen-vironment changes; the microenvironment isin turn shaped by the evolving populationsof cells. Selective barriers arise from cell–cellinteractions, tissue organization, blood supply,the stage of carcinogenesis, and other intra-and extra-cellular factors. Numerical simula-tions coincided with the observed biologicalfeatures of a multistep process: initially, muta-tions occur mostly in tumor suppressor genes—these mutations are selectively (almost) neutralbut set the fitness landscape for the secondphase of carcinogenesis, in which limited pro-liferation is initiated. This slow expansion issurpassed only in the third stage of carcinogen-esis when glycolytic and acid-resistant pheno-types evolve. The authors concluded that nor-mal tissue is situated in an adaptive landscape,which allows for the coexistence of multiple celltypes—a condition needed for the evolutionof multicellularity. While normal conditions inthis landscape impose strict limitations on pro-liferation, populations of benign cells are vul-nerable to invasion of fitter genotypes, whichinitiate tumor growth. Further progression islimited by available blood supply, but accel-erated by promoters such as inflammation orwounding which can increase the flow of nu-trients, as well as mutations leading to adapta-tions to acidic–hypoxic environments. The au-thors proposed that the stages of tumorigenesis

strongly depend on changes in tumor microen-vironment, rather than exclusively on variationof intracellular processes such as cell cycle reg-ulation or apoptotic mechanisms. In a similarsetting, Gatenby and Gillies made several im-portant observations: the environment selectsfor phenotypes instead of genotypes, the fullcancer phenotype is attained only once all thebarriers have been surpassed, and adaptationto one barrier may change the adaptive fitnessresponse to subsequent ones (Gatenby & Gillies2008).

Basanta (2008) devised a model based onevolutionary game theory to analyze the inter-actions between three different tumor cell phe-notypes defined by autonomous growth, anaer-obic glycolysis, and tissue invasion. He foundthat the invasive phenotype is more likely toevolve after appearance of the glycolytic pheno-type, which would explain the presence of inva-sive growth in many malignant tumors. Theseresults suggest that therapies which increase thefitness cost of switching to aerobic glycolysismight decrease the probability of emergenceof more invasive phenotypes; when applied toglioma progression, the model explains aspectssuch as the emergence of diffuse tumor cell in-vasion in low-grade tumors.

The application of concepts from evolu-tionary game theory to cancer enables re-searchers to investigate the effects of frequency-dependent fitness on tumorigenesis. Suchmodels may be superior to approaches us-ing frequency-independent fitness in situations

←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−In this process, cells are arranged in a row and labeled i = 1, . . ., N. For reproduction, cells are chosenproportional to their reproductive rate. The reproducing cell is replaced by two daughter cells, and all cellsto the right are shifted by one position. The right-most cell undergoes apoptosis (falls off the edge of theone-dimensional table). This architecture can delay the onset of cancer since only mutations arising in theleft-most cell, the stem cell, remains in the population. From Nowak et al. (2003). (C) Clonal diversity inindividuals with Barrett’s esophagus. Barrett’s esophagus is an abnormal change in the cells of the lower endof the esophagus thought to be caused by damage from chronic acid exposure and is considered to be apremalignant condition associated with an increased risk of esophageal cancer. It is important to identifythose lesions likely to progress to cancer since unnecessary surgery should be avoided. In the panel, weshow genetically distinct fractions of a biopsy with their frequencies. Clones and total frequencies within asegment were used to calculate the Shannon diversity index and mean pairwise genetic divergence scores.Such measures of diversity are able to predict the risk of progression to esophageal cancer. From Maley et al.(2006).


in which the cellular growth rate or someother cell-specific parameter is a function ofthe behavior of other cells. For example, sce-narios in which some cells secrete substancesthat are beneficial to the population as awhole but costly to produce for the cells them-selves (such as production of vascular endothe-lial growth factor [VEGF] to initiate vascu-larization) require game-theoretic approaches(Axelrod et al. 2006). However, for some sit-uations arising in cancer it has not beenfound that a trait depends on the compo-sition of the population. In those cases, theconsideration of frequency-dependent fitnessdoes not lead to appreciable additional insightsand should be neglected due to unnecessarycomplexity.

The investigation of cellular heterogeneityin tumors has led to many insightful resultsregarding its emergence, maintenance, andpathological importance. The interest of in-vestigators in tumor heterogeneity was furtherpiqued by the clinical phenomenon of drug re-sistance, and many models were developed tostudy the dynamics of tumors during treatmentand the kinetics of resistance mutations.

Anti-Cancer Therapy and theEvolution of Resistance

The first wave of contributions to mathemat-ical modeling of cancer treatment was startedin 1964 by Skipper and colleagues who inves-tigated the response of murine leukemias tochemotherapy, establishing the so-called “logkill” law: they showed that in an exponentiallygrowing population, the fraction of cells killedby a drug is independent of the total numberof tumor cells (Skipper et al. 1964). Thereafter,other models of anti-cancer therapy were de-veloped to estimate tumor growth rates andto optimize chemotherapeutic dosing sched-ules (Norton & Simon 1977). The investigationof cancer therapy from an evolutionary view-point, however, was initiated only after Goldieand Coldman started thinking about the emer-

gence of resistance in the 1980s (Goldie & Cold-man 1983, 1984; Coldman & Goldie 1986). In1983, they presented a mathematical model ofcancer treatment to investigate the risk of re-sistance, and found that the probability P ofhaving a resistant cell at any given time is asigmoid function of the number of cells in thetumor, N , and the mutation rate, α, given byP = 1 − exp[−αN ] (Goldie & Coldman 1983).The shape of this function plotted against thelogarithm of the population size is constantacross mutation rates—once the probability ofcure starts decreasing from the initial plateau,small delays in the start of treatment lead to sig-nificant reductions in the chance of successfultreatment (Fig. 6A). The authors proposed twostrategies to maximize the probability of suc-cessful therapy: first, treatment must be startedas soon as possible. This conclusion was drawnfrom the fact that the probability of cure de-creases as the size of the tumor increases, andalso from the prediction that larger tumorspresent a higher level of heterogeneity, which isinversely related to the probability of success-ful therapy (Goldie & Coldman 1984); second,multiple drugs should be used in combinationwhenever possible, or otherwise alternated. As-suming that the growth rate of sensitive cells isdecreased by the effect of the drug, short treat-ment breaks to limit the toxicity ensure thatclones do not reach large abundances whenthe drug is not administered. The authors alsoutilized stochastic models of tumor growth tocalculate the risk that clones with multiple mu-tations emerge; their approach lead to the sug-gestion of strategies maximizing the probabilityof successful therapy (Coldman & Goldie 1986).The prediction that early initiation of therapyand the use of multiple drugs could reduce therisk of resistance became known as the Goldie-Coldman hypothesis.

Coldman and Murray presented in 2000 astochastic model to investigate the optimumadministration strategies of chemotherapeuticdrugs for different tumor growth dynamics(Coldman & Murray 2000). They consideredsituations in which a patient is treated with two


Figure 6. The evolution of resistance to anti-cancer therapy. (A) The mathematical inves-tigation of resistance mutations was initiated by Goldie and Coldman, who found that theprobability of cure depends on the mutation rate and the population size of the tumor and isgiven by the exponential of the negative mutation rate times the population size. The panelshows the probability of cure for two different mutation rates, A and B, versus the logarithm ofthe population size. Equation from Goldie & Coldman (1983). (B) A stochastic process modelcan be used to investigate the risk of resistance emerging in a mutation-selection network. If,for example, three positions must be mutated for resistance to anti-cancer therapy and thethree mutations can be accumulated in any order, then the evolutionary trajectories from awild-type cell (three unmutated positions, 000) to an escape mutant (three mutated positions,111) can be studied and the risk of resistance quantified. From Iwasa et al. (2003). (C) Theprobability of producing resistant mutants before treatment depends on the death rate of tumorcells, D. In this example, the mutation rate is set to zero once treatment is started, and mutantsare assumed not to die. The tumor size, N, is shown at which the probability of treatmentfailure due to preexisting resistance equals 0.01. While the dependence is linear for a singledrug, it becomes stronger with an increasing number of drugs. Here the mutation rate is 10−6

and the growth rate of cells is one. From Komarova & Wodarz (2005).


drugs and assumed that mutations conferringresistance to one drug are sensitive to the other.The authors optimized drug dosage strategiesto reduce toxicity and increase the probabil-ity of cure, showing that for equally effectivedrugs, the optimum treatment regimen consistsof combined symmetric doses administered asoften as possible. The model predicts that anearly control of resistant clones—attained byhigh initial doses followed by a decreasing dosefor the remainder of the treatment time—is im-portant. Iwasa and colleagues (2003) designeda stochastic mathematical model based onmulti-type branching processes to investigatethe evolutionary dynamics of resistance againstanti-cancer therapy. They considered a sce-nario in which a heterogeneous population ofcancer cells is subjected to chemotherapy, andcalculated the probability of success or failureof biomedical interventions consisting of oneor more drugs (Fig. 6B). They found that thechance of cure is largest when many drugs areadministered simultaneously since only if tu-mor cells evolve many mutations will they beable to escape from eradication by therapy.

The scenarios investigated in these contri-butions represent just a subset of the situa-tions that can emerge when tumor cells evolvedrug resistance. Other scenarios include thetreatment of a cancer patient with a cocktailof drugs that contains both specific (molecu-larly targeted) and unspecific (general cytotoxic)agents, the necessity of altering several molecu-lar pathways for a cell to evade chemotherapyand immune attacks, dose-limiting toxicity andside effects, and the emergence of radiation- orchemotherapy-induced secondary tumors. Theliterature on models optimizing therapeuticstrategies for such more complicated scenariosis still small, and it is essential to obtain a quan-titative understanding of anti-cancer treatmentstrategies and their long-term risks and bene-fits such that drug resistance can be prevented;the field is in dire need of optimization studiesguiding therapy decisions.

Komarova and Wodarz (2003) investigatedthe effects of mutator phenotypes on the risk

of resistance by considering parameters likeapoptotic efficiency, the cost of DNA repair,and the magnitude of the mutation rate. Theirstochastic model predicted that genetic insta-bility is positively selected for in clones withlow mutation rates since that phenotype confersenhanced evolvability to cells; in situations inwhich mutation rates are high, however, genet-ically stable cells experience a selective advan-tage since cells with a mutator phenotype ac-cumulate deleterious mutations at a high rate.The authors found that the outcome of the sys-tem depends on two parameters: the differencein the ability to repair DNA damage betweenstable and unstable cells and the cost of gen-erating lethal mutations due to genetic insta-bilities. In the same year, Gatenby and Vincentcombined population biology and evolutionarygame theory to study the dynamics of tumorand healthy cells when treated with cytotoxicdrugs (Gatenby & Vincent 2003b). A modelreminiscent of Lotka–Volterra equations wasused to describe the interactions of normal andcancer cells interpreted as a predator–prey sys-tem, and the authors found that, in general,treatment with cytotoxic drugs alone is insuf-ficient to eradicate the tumor. This model wasused to identify two main barriers to achieving acomplete remission of the tumor: evolving pop-ulations tend to produce resistant clones, whichare able to drive the system back to equilibriumabundance, and drug-induced alterations of thetumor microenvironment change the selectionpressure and select for tumor cells with largerevolvability.

The existence of resistance mutations at thetime of diagnosis has been of considerable in-terest to the cancer research community sincesuch pre-existing resistance influences treat-ment choices. The investigation of resistancemutations emerging while a population under-goes exponential expansion was initiated byLuria and Delbruck in 1943 (Luria & Delbruck1943). They were interested in the mutationsof bacteria that confer resistance to phagesand performed experiments as well as cal-culations to quantify the rate at which such


mutations emerge during bacterial growth.Their analytical results describing the distri-bution of the number of mutants in an expo-nentially growing population became known asthe Luria–Delbruck distribution. This distribu-tion is also useful for situations arising in cancerand has been studied for more than half a cen-tury (Skipper 1983; Sarkar 1991; Zheng 1999;Angerer 2001; Frank 2003; Dewanji et al. 2005;Iwasa et al. 2006; Haeno et al. 2007; Komarovaet al. 2007; Zheng 2008). Several different mod-els have been suggested that are based on purebirth processes and do not include the possibil-ity of cell death (Zheng 1999). However, in mostexamples of cancer growth, cancer cell deathcannot be neglected.

In 2003, Frank developed a branching pro-cess model to investigate the distribution ofcells with different numbers of mutations (Frank2003). He found that the distribution of cellswith two mutations provides information aboutthe effective time of occurrence of the first mu-tation; the latter—depending on its frequencyin different tissues in the body—could evenhave arisen during embryonal development.The time at which the first mutation emergescontains information about the number andtypes of tissues harboring this mutation, whichin turn influences cancer prognoses and pro-vides information about tissue specificity ef-fects. In 2005, Dewanji and colleagues de-veloped an extension of the Luria–Delbruckmodel that considered non-exponential growthdynamics (Dewanji et al. 2005). The authorsfound that including the possibility of cell death,as well as the assumption of Gompertziangrowth of mutant cells, leads to larger varia-tions in the number of cells harboring resistancemutations.

Iwasa and co-workers (2006) designed abranching process model to calculate the prob-ability of resistance mutations existing at thetime of diagnosis, as well as the expected num-ber of resistant cells, as a function of detectionsize and mutation rate. These authors also ex-plicitly considered cell death. They concludedthat the probability of resistance is an increasing

function of the detection size times the muta-tion rate, and that a tumor with larger apop-tosis rates has a higher incidence of resistance.Haeno and collaborators (2007) extended thisstochastic model to incorporate resistance dueto the accumulation of two mutations and ar-rived at similar conclusions. Later Komarovaand colleagues (2007) presented a variation ofthe Luria–Delbruck model for a stochasticallygrowing population of cancer cells. As in priorpapers, it was shown that a higher cell-deathrate leads to a larger number of mutants. Fur-thermore, irreversible mutations were found toact as a selective force in large cell populationseven if they do not confer a fitness advantageto the cell; this effect results from consideringunidirectional mutations, which continuouslyreduce the pool of unmutated cells.

These investigations of resistance mutationsexisting prior to diagnosis are important for aquantification of the risk of resistance as well asfor deciding on appropriate treatment choicesat diagnosis. Unfortunately, cell-specific param-eters such as growth and death rates and muta-tion rates are unknown for most cancer types.To adequately predict the risk of pre-existingresistance for a given tumor type, quantitativemeasurements of these parameters are essen-tial. It is therefore import to perform in vitro

experiments to determine these rates.In 2005, Komarova and Wodarz developed

a stochastic mathematical model to investigatetreatment strategies involving multiple drugs(Komarova & Wodarz 2005). The effect of ther-apy was modeled by increasing the ratio of celldeath and cell birth in cancer cells. The authorsassumed that a single mutation is sufficient toconfer resistance to one particular drug; hence,resistance against therapies consisting of n dif-ferent drugs requires the accumulation of n mu-tations. The authors investigated the stochas-tic process model for situations with differentmagnitudes of cell turnover and evaluated theimportance of resistance mutations emergingduring treatment with multiple drugs. Theyfound that the treatment phase is unimportantfor the production of resistant cells and can


be neglected when determining the total riskof resistance—the majority of resistance muta-tions arise before the tumor is diagnosed. Thechance of resistance increases with enhancedturnover rates independently of the number ofdrugs administered. The model was also usedto find the optimum number of drugs such thatboth toxicity and the evolution of resistance areminimized; this number depends on the growthand death rates of cells, the mutation rate, andthe population size of cancer cells at the start oftherapy (Fig. 6C). When the model was appliedto chronic myeloid leukemia (CML) to investi-gate the optimum number of drugs, it was foundthat administration of three chemotherapeu-tics minimizes the risk of resistance. Later on,Komarova and Wodarz studied the importanceof stem cell quiescence in the evolution of resis-tance against anti-cancer treatment (Komarova& Wodarz 2007). The authors used a stochasticprocess model to investigate the dynamics of apopulation of cancer stem cells that cycle be-tween an active, proliferative state and an inac-tive, quiescent state. The model was applied toCML and could reproduce the biphasic declinein blood counts seen in CML patients (Michoret al. 2005a), which in the context of this modelwas explained by the differential response ofcycling and dormant stem cells to treatmentwith the targeted agent Gleevec. The authorsalso predicted that the probability of resistancewould be independent of the presence and ex-tent of quiescence if therapy involves only a sin-gle drug. When two or more drugs are used, therisk of resistance increases with the percentageof quiescent stem cells. In line with previousresults obtained by these authors, the risk ofresistance emerging during therapy is negligi-ble when compared with resistance arising be-fore the start of treatment. These findings haveimportant implications for the design of treat-ment strategies since that showed that the useof drugs that reduce the number of quiescentcancer stem cells cannot decrease the chanceof resistance effectively.

The effect of quiescence on the risk of re-sistance is an important phenomenon to con-

sider, and there are other situations in cancerthat should also be addressed. For instance,it is of interest to study the evolutionary dy-namics of cancer cell resistance when migra-tion and adaptation to foreign environmentsare included; a change in microenvironmentmight lead to modifications in gene expression,which can in turn modulate the risk of resis-tance. Furthermore, the interaction betweentumor cells, stroma, immune system cells, andvascular endothelial cells may alter the riskof resistance. Future studies of resistance dy-namics should address these more complicatedsituations.

While most mathematical models have fo-cused on the action of cytotoxic or targeteddrugs whose objective is to kill cancer cells,Maley and colleagues recently studied the pos-sibility of using benign cell “boosters” to in-crease the fitness of healthy cells (Maley et al.2004b). The authors developed a computa-tional model to investigate the dynamics ofmutations emerging in cells that proliferate ona two-dimensional lattice. In this model, thegenetic information of cells is represented bya small number of loci acting as oncogenes,tumor suppressor genes, genes preventingmutator phenotypes, and genes conferring sen-sitivity to drugs. The population size is as-sumed to be constant, thus exploiting com-petition among different clones. The authorsshowed that therapies that increase the fitnessof benign cells, so-called boosters, are effec-tive in all stages of the disease when com-bined with traditional chemotherapy. Theyalso suggested that increasing the fitness ofchemosensitive cells prior to treatment wouldimprove treatment outcomes. In a different ap-proach to cancer therapy, Wodarz designed amathematical model to investigate the use ofviruses as anti-tumor agents (Wodarz 2001).He studied a system of differential equationsfor three possible scenarios of the interactionbetween virus particles and tumor cells, arriv-ing at predictions about the necessary con-ditions for successful therapy. Such alterna-tive approaches to treating cancer are worth


exploring and should be investigated withquantitative techniques.

Conclusions and Outlook

The application of evolutionary thinking tothe study of cancer initiation, progression, het-erogeneity, and resistance has produced manysuccessful studies. However, several aspects ofcancer remain underrepresented in the evo-lutionary literature and could benefit tremen-dously from more intensive quantitative inves-tigations. The putative cancer stem cells andtheir dynamics during cancer progression andtherapy, for example, have not received muchattention. This fact may in part be due tothe relative novelty of experimental studies ofcancer stem cells and the comparatively smallamount of biological information available tomodelers; however, quantitative and evolution-ary approaches to cancer stem cells will benecessary for driving the investigations prov-ing (or disproving) their existence, their mech-anisms of treatment insensitivity, and relatedaspects. Similarly, a full understanding of re-sistance from an evolutionary viewpoint stillrequires considerable work. Many cancers re-lapse despite therapy shown to be effectivein vitro, and the causes of refractory disease re-main to be identified in many cases. Pharma-cological modeling such as the investigation ofpharmacokinetics and pharmacodynamics incombination with the emergence of resistancesimilarly requires further input from evolution-ary theorists. The questions about the optimumdosing strategies that utilize small therapeuticwindows need to be answered with sophisti-cated mathematical models that push the fieldbeyond the current state.

The main barriers to theoretical investiga-tions of cancer remain a lack of generally acces-sible and quality-controlled databases of treat-ment response data, mutation status of cancersfor resistance investigations, heterogeneity in-formation in the form of genomic data such asSNP arrays, gene expression data and sequence

information, and other data. Such data shouldnot only provide spatial, but also temporal in-formation about tumors. The development ofinexpensive, high-throughput single-cell assays,as well as next-generation sequencing meth-ods, will be beneficial for theoretical endeav-ors. Furthermore, most evolutionary parame-ters such as mutation rates, fitness landscapesof neoplasms, and population sizes and struc-tures are virtually unknown. These parame-ters can be used to validate evolutionary mod-els; for example, the predictions of the Luria–Delbruck distribution of resistance mutationspre-existing to therapy (Skipper 1983; Zheng1999; Sarkar 1991; Angerer 2001; Frank 2003;Dewanji et al. 2005; Iwasa et al. 2006; Haenoet al. 2007; Komarova et al. 2007; Zheng 2008)should be tested in cell line experiments thatcan directly assay the frequency of apoptosisas well as the growth and mutation rates ofcells. Such studies of experimental evolutionremain rare but would provide valuable infor-mation to modelers. Once more data are acces-sible to theoreticians, the evolution of the fieldwill be accelerated. Interdisciplinary investiga-tions are furthermore hindered by the compli-cations of securing funding. Most major medi-cal funding sources prefer traditional molecularbiology or biochemical studies to mathemati-cal investigations, and the field would benefittremendously from a funding source for inter-disciplinary work. Also, the training of the nextgeneration of evolutionary modelers of can-cer requires the design of a useful curriculum,since the sheer amount of knowledge necessary(cancer biology, evolution, applied mathemat-ics, bioinformatics, and statistics) renders moststudy courses superficial. Along the same line,evolutionary training of oncologists and clini-cians would be beneficial for the understandingof cancer.

In closing, we believe that the application ofevolutionary modeling to cancer will continueto increase our understanding of neoplasmsand will eventually contribute to the design ofsystems biology-based strategies to detect andmanage the disease.


Acknowledgments

We would like to thank Steven Frank, CarloMaley, Robert Downey, and the Michor lab forcritical reading and comments.

Conflicts of Interest

The authors declare no conflicts of interest.

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09 Evolutionary Theory of Cancer

Documents

cancer cell

cancer research

cancer results

cancer progression tomlinson

cancer initiation purple

genetic cause of cancer

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