mathematical modeling links pregnancy-...

Integrated Systems and Technologies: Mathematical Oncology

Mathematical Modeling Links Pregnancy-Associated Changes and Breast Cancer RiskDaniel Temko1,2,3,4, Yu-Kang Cheng1, Kornelia Polyak5, and Franziska Michor1

Abstract

Recent debate has concentrated on the contribution of badluck to cancer development. The tight correlation between thenumber of tissue-specific stem cell divisions and cancer risk ofthe same tissue suggests that bad luck has an important role toplay in tumor development, but the full extent of this contri-bution remains an open question. Improved understanding ofthe interplay between extrinsic and intrinsic factors at themolecular level is one promising route to identifying the limitson extrinsic control of tumor initiation, which is highly relevantto cancer prevention. Here, we use a simple mathematical

model to show that recent data on the variation in numbersof breast epithelial cells with progenitor features due to preg-nancy are sufficient to explain the known protective effect offull-term pregnancy in early adulthood for estrogen receptor–positive (ERþ) breast cancer later in life. Our work provides amechanism for this previously ill-understood effect and illumi-nates the complex influence of extrinsic factors at the molecularlevel in breast cancer. These findings represent an importantcontribution to the ongoing research into the role of bad luck inhuman tumorigenesis. Cancer Res; 77(11); 2800–9. �2017 AACR.

IntroductionA recent study (1) by Tomasetti and Vogelstein analyzed the

relationship between the number of stem cell divisions andcancer risk across tissues to investigate the role of "bad luck" incarcinogenesis. The authors demonstrated that the logarithm oflifetime cancer incidence in a tissue is correlated with thelogarithm of the cumulative number of stem cell divisions in

the same tissue (R2 ¼ 0.64). As a result, the authors claimed thatthe majority of the variance in cancer risk among tissues is dueto bad luck (Fig. 1A).

In the reporting of the study and ensuing debate, some com-mentators drew broader conclusions from the correlation foundby Tomasetti and Vogelstein. While the initial study claimed thattwo thirds of the variation in cancer risk between tissues is due tobad luck, an accompanying commentary suggested that two thirdsof all cancers, rather than two thirds of the variation, are due torandommutations in healthy cells (2). Subsequent analyses haveshown that the initial correlation is not sufficient to imply a lowerbound on the proportion of all cancers that are due to bad luck at64%. Todraw this conclusion from the studywould require strongassumptions about the possible effects of controllable factors inthe data set considered (3).

Importantly, the regression analysis used by Tomasetti andVogelstein cannot quantify the possible effects of extrinsic factorsthat do not already vary within the data set used, which notablydid not include breast cancer (4). Therefore, the regression cannotbe used to draw conclusions about unavoidable bad luck, takinginto account the variation of all possible extrinsic factors. Toillustrate this point, consider the (perhaps unlikely) possibilitythat it is possible to safely alter the fitness advantage of mutationsthat can lead to cancer. The correlation analysis presented cannottell us about the impact such variation could have on cancer risk.

The insufficiency of the current evidence to draw conclusionsabout the contribution of unavoidable bad luck to cancer demon-strates the important potential role of mechanistic models indetermining the contribution of controllable factors to differentcancer types, andwhether these factors canbeharnessed for cancerprevention. The changes that lead to cancer are thought to developin a complex molecular setting, which defies simple characteri-zation. In this setting, variation of any number of parameters mayaffect lifetime risk of cancer; these include but are not limited tothe number of cells susceptible to transformation, the mutationrate of cells, and the fitness advantage conferred by those muta-tions when they occur (Fig. 1B).

1Department of Biostatistics and Computational Biology, Dana-Farber CancerInstitute, and Department of Biostatistics, Harvard T. H. Chan School of PublicHealth, Boston, Massachusetts. 2Centre for Mathematics and Physics in the LifeSciences and Experimental Biology (CoMPLEX), University College London,London, United Kingdom. 3Department of Computer Science, University CollegeLondon, London, United Kingdom. 4Barts Cancer Institute, Queen Mary Univer-sity of London, London, United Kingdom. 5Department of Medical Oncology,Dana-Farber Cancer Institute, and Department of Medicine, Harvard MedicalSchool, Boston, Massachusetts.

Notes: Supplementary data for this article are available at Cancer ResearchOnline (http://cancerres.aacrjournals.org/).

D. Temko and Y.-K. Cheng contributed equally to this article.

Corresponding Authors: Franziska Michor, Dana-Farber Cancer Institute, 450Brookline Avenue, CLSB-11029, Boston, MA 02215. Phone: 617-632-5045; Fax:617-632-2444; E-mail: [email protected]; Kornelia Polyak, Phone: 617-632-2106; Fax: 617-582-8490; E-mail: [email protected]

doi: 10.1158/0008-5472.CAN-16-2504

�2017 American Association for Cancer Research.

Major FindingsA mathematical model demonstrates that the pregnancy-

associated reduction in Ki67þ and p27þ cell numbers in thehuman breast can explain the protective effect of pregnancyagainst ERþ breast cancer.

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Full-term pregnancy in young adulthood is a well-documentednatural protective factor for breast cancer (5, 6). Estimates suggestthat risk increases by 5% for every 5-year increase in the age at firstbirth for women with one birth (6). The specific effects of parityvary by hormone-receptor status of the resulting tumors (7).Analysis of the Nurses Health Study (NHS) cohort showed thatthe risk for ERþ breast cancer decreases with the number ofpremenopausal years accumulated since first birth (7). Hence,early first birth confers the greatest protective effect; a womanwithfour births at age 20, 23, 26, and 29 years old has an estimated29% reduced risk of ERþ/progesterone receptor–positive (PRþ)breast cancer between the ages of 30 and 70, compared with anulliparous woman during the same time period. The same studyfound thatfirst birth causes a one-off increase in risk for PR– cancercompared with nulliparous women, with an effect size that

increases with age at first birth. As a result, women with a firstbirth over the age of 35 can be at an increased risk of breast cancer.

In the absence of high-resolution single-cell data, which arenearly impossible to obtain in large cohorts of humans, mathe-matical models have demonstrated the plausibility of generalmolecular explanations for the protective effects of pregnancy. Animportant study by Moolgavkar and colleagues explored a frame-work where breast cancer is caused by two cellular transitionsoccurring in normal cells (8). In this model, pregnancy increasesthe rate of differentiation of normal and partially transformedcells, decreasing the pool of cells susceptible to the cellulartransitions leading to cancer. The study leads to a good fit to thedata of MacMahon and colleagues. (5). The model of Pike andcolleagues (9) uses a concept of breast tissue age: breast cancerincidence is modeled as a linear function of the logarithm of

Quick Guide to Equations and AssumptionsCellular dynamics of the stem cell and proliferative progenitor cell populations

There are N stem cells per terminal end duct. The stem cells follow a stochastic process known as the Moran model. One celldivision occurs during each time step of length tcycle/N. In each time step, a single stemcell is randomly chosen to divideproportionalto the fitness of the cell, with the two daughter cells replacing the divided cell and another randomly chosen cell.

With probability P, stem cell divisions are asymmetric, giving rise to one stem cell that replaces the divided cell and one progenitorcell that forms the founder in a new cascade of progenitors. All cells in a progenitor cascade divide during every time step. Innonpregnant women, wild-type progenitor cells can divide a total of z times before becoming terminally differentiated (see belowfor the effects of mutations and effects of pregnancy). Cells that are terminally differentiated exit the simulation.

Cancer initiationDuring each cell division, one of the two daughter cells in a division attains a new (epi)genetic mutation with probability m.

In stem cells, mutations increase the relative fitness of the cell by a factor of fmut. In progenitor cells, mutations increase the numberof levels in the differentiation hierarchy by zmut levels. Thus, a stem cell with n mutations has relative fitness in the Moranmodel given by Equation A, and a progenitor cell with n mutations is able to divide a total number of times given by Equation Bbefore terminal differentiation:

fmutð Þn ðAÞ

zþ n � zmut ðBÞ

In addition, progenitor cells must acquire the ability to self-renew to become cancer cells. We assumed that the probability of aprogenitor cell at differentiation level 0 � i � z þ n � zmut attaining self-renewal is given by Equation C:

g ¼ gbase � i � gbaseð Þ=ð2 � zÞ ðCÞ

We assumed that cancer initiation occurs when a cell has accumulated a total of nmutmutations and either retained (through being astem cell) or attained (through a self-renewal event) the ability to self-renew.

Effect of pregnancyOur model simulates an entire life-course over ttotal years. The model takes into account possible changes to cellular dynamics

during pregnancy, after pregnancy, and after menopause. During pregnancy, we assumed that the stem cell cycle length decreases totcycle,preg, whereas the number of levels in the differentiation hierarchy of progenitor cells increases by zpreg levels. After menopause,the stem cell cycle length increases to tcycle,menopause.

In parous scenarios, after the first birth, the probability of asymmetric stem cell division changes by amultiplicative factor ppost,init(0 < ppost,init<1). After the secondbirth and subsequent births, the probability of asymmetric stem cell division changes by a factor ofppost,subs (ppost,init < ppost,subs < 1).

Mammary Progenitors and Breast Cancer Risk

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breast tissue age, and risk factors for breast cancer alter the rate ofbreast tissue aging. First full-term pregnancy causes a one-offincrease in breast tissue age, but decreases its subsequent rate ofincrease. This study also demonstrated a good fit to the Mool-gavkar and colleagues (8) data. Rosner and Colditz then adaptedand extended the model developed by Pike and colleagues. (9),including changes to further improve the fit and accommodatemultiple births, and applied the adapted model to data from theNHS cohort (7, 10, 11). The fit of these models to epidemiologicdata provides support for the theory that pregnancy alters thenumber of cells that are at risk for accumulating changes leading tobreast cancer. However, they do not identify the molecularmechanisms responsible, nor do they accommodate the effectsof a cellular hierarchy of stem and progenitor cells.

Recently, single-cell technology has made it possible to collectquantitative data on changes in individual mammary subpopula-tions, presenting the possibility to quantitatively assess themolec-ular-level changes, as well as the epidemiologic incidence curves,associated with pregnancy. Studies in mice and humans provideevidence that p27þ mammary epithelial cells with progenitorfeatures decrease in number with pregnancy and are present inhigh numbers in BRCA1 and BRCA2 germline mutation carriers(12, 13). Evidence was presented that a subset of p27þ cells withprogenitor features are hormone-responsive quiescent luminalprogenitors with proliferative potential, and that their variationcould relate to breast cancer risk (12). Here, we use a simplemathematical model to test whether, given a role for p27þ

progenitor cells as proliferative progenitors, which can accumu-late changes leading tobreast cancer, theobserved reduction in thepopulations of p27þ progenitor cells with pregnancy is sufficientto explain the protective effect of pregnancy.

Materials and MethodsWe aimed to test the hypothesis that a decreasing cell number

and proliferative capacity of luminal progenitor cells after preg-nancy can result in aprotective effect against breast cancer and thatthe effect decreases with increasing age of pregnancy. To this end,we designed a mathematical model of the dynamics of prolifer-ating cells in the breast tissue that can accumulate the changesleading to cancer initiation. We considered two types of cells: aself-renewing population of stem cells and a population ofproliferating luminal progenitor cells that result from differenti-ationof these stem cells and respond tohormonal stimuli.Wefirst

tested whether we could identify a biologically plausible param-eter setting in our model, under which the variation in progenitorcell numbers results in a risk decrease that fits the quantitative riskdecreases observedwith pregnancy.We then tested the robustnessof the fit of our model in the surrounding parameter space.

We first studied the dynamics of stem cells in the breast ductalsystem. Given the population structure inherent to breast ducts,we considered the stem cells in each duct to act independently. Assuch, we investigated the dynamics of a single duct within thebreast since the total probability of cancer initiation is given by theprobability per niche times thenumber of niches; thus, the relativelikelihood of cancer initiation is not altered by considering onlyone niche. The overall number of stem cells in the breast isestimated to be on the order of 5 to 10 cells per duct (14, 15),and we denoted this number by N, although there is someuncertainty in these estimates. We defined a fundamental timeunit of our system to be dictated by the division time of stem cells,tcycle, which varies during pregnancy. In in vivo experiments, themean cell-cycle length of benign breast cancer cells was approx-imately 162 hours per cell (16). We assumed that even precan-cerous cells divide faster than stem cells; thus, using tcycle ¼ 162hours as the average premenopausal stem cell cycle length whennot pregnantmay be an overestimation of the number of stem celldivisions that occur in the normal breast, and we verified that ourresults were unaffected at higher stem cell cycle lengths. Further,previous data by our lab (12) and several others (17–22) suggestthat the percentage of cells in normal breast that stain positive forKi67 are approximately 3% and 12% in the follicular and lutealphases of the menstrual cycle, respectively. Assuming that theduration of these twomenstrual cycle phases is roughly the same,at 2 weeks per cycle, leads to an average Ki67 value of 7.5%.Considering that Ki67 is detectable for 24 hours during the activephases of the cell cycle (23, 24), this translates to an estimate of320 hours (24/0.075) for the average cell-cycle length, which isalso within the range tested (162 hours to 324 hours). Otherstudies have shown a broadly consistent range of Ki67/KiS5values (20) or lower values consistent with still longer cell cycletimes (18, 19).

Experimental data suggest that proliferation decreases 4- to 5-fold after menopause, irrespective of parity (12, 25). To take thiseffect into account, we assumed that the cell-cycle length increasesby a factor of amenopause ¼ 4 after menopause. In our model, asingle stem cell in each duct is randomly chosen to divide duringeach time step, proportional to the fitness of the cell, following a

Figure 1.

Multiple factors can affect cancer riskin a complex setting. A, An analysis byTomasetti and Vogelstein (1)demonstrated a close correlationbetween the log of lifetime cancerincidence in a tissue and thecumulative number of stem celldivisions in the same tissue. Plot shownis a schematic using simulated data. B,Variation in multiple molecular factorsmay affect cancer risk when theychange from the homeostatic state(top left), including the number ofprogenitor cells (top right), themutation rate (bottom left), and thefitness effect conferred by mutations(bottom right).

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stochastic process known as the Moran model (26). According tothismodel, the divided cell is replaced by one of the daughter cellsof the division, whereas the other daughter replaces another stemcell that was randomly selected from the population to die. Theuse of this model ensures preservation of homeostasis in thenormal breast epithelial cell population. Because the specificdynamics of stem cells in the breast are not known, we chose theMoran model as it has been used to model stem cell populationsin other tissues (27–29). For each cell division, we allowed for asingle mutation to arise in one of the two daughter cells of thedivision with a certain probability.

In the mature breast, stem cells divide primarily to maintaincellular integrity. However, differentiating events do occur,although rarely (30–32). In our model, with probability P, weallowed the cell division in the current time step to be asym-metric, producing one daughter stem cell to maintain the stemcell population and one progenitor daughter. Because the exactrate of differentiation is unknown, we tested P ¼ 10–1 to 10–3.With the remaining 1 – P probability, the stem cell division is

symmetric and follows the usual Moran division dynamics. Ineach time step thereafter, all cells resulting from the progenitordaughter divide and differentiate further until a total of z celldivisions are accumulated. The number of luminal epithelialprogenitors in humans is unknown. As a result, we set z ¼ 10 tofit data from mouse mammary fat pad transplantation experi-ments (33) and tested a wide range of alternate values for thisparameter. After z divisions, we considered the cells differenti-ated, and at this point, they are no longer considered in ourmathematical model. Thus, in the wild-type system, there are Nstem cells per duct and 2zþ1 � 1 progenitor cells per differen-tiation cascade. Because the dynamics of progenitor cells in thehuman breast are not known, we have adopted the assumptionthat progenitor cells undergo a limited number of divisions,similar to what has been observed for transit-amplifying cells inthe colon and other tissues. Figure 2A describes the temporaldynamics of the system.

During each cell division, genetic alterations contributing tocancer initiation may arise with a small probability. We

Figure 2.

Schematic representation of the mathematical model. A, Initially, there are N wild-type stem cells (blue), which give rise to a differentiation cascadeof 2zþ1 � 1 wild-type luminal progenitor cells (purple). At each time step, all progenitor cells as well as one randomly selected stem cell divide. Withprobability 1 – P, the stem cell divides symmetrically and one daughter cell replaces another randomly chosen stem cell. With probability P, the stem celldivides asymmetrically, and one daughter cell remains a stem cell, whereas the other daughter cell becomes committed to the progenitor population (light pink).Regardless of the dividing stem cell's fate, all existing progenitor cells divide symmetrically for a total of z times to give rise to successively moredifferentiated cells (progressively darker shades of purple) before becoming terminally differentiated. In the figure, the darkening purple gradations refer tosuccessively more differentiated cells and serve to clarify a single time step of the stochastic process. B, The acquisition of mutations leading to breastcancer initiation all result in an increased relative fitness (i.e., growth rate) fmut in stem cells (red) as compared with wild-type cells (blue), and an additionalnumber of divisions zmut that progenitor cells can undergo before terminally differentiating. C, During pregnancy, progenitor cells experience an expansion inproliferative capacity through an additional number of divisions zpreg in order to form terminally differentiated milk-producing cells (dotted triangle) and adecrease in cell-cycle length.






considered a number nmut of mutations that, when combined,result in a single cell leading to cancer initiation. These mutationscould each be any of the many mutations commonly found inbreast cancer with initiation potential. As a simplifying assump-tion, we considered a mutation rate on the order of 10�5 muta-tions per oncogenic mutation per cell division to limit therequired number of simulations for detection to a reasonablenumber.

The baselinemutation rate is roughly 5� 10�9 per base pair percell division (34, 35). It is estimated that there are roughly 34,000possible driver base pairs in the genome (36), thus it may bereasonable to assume that there are on the order of 10,000possible ways to achieve each oncogenic mutation, which wouldlead to the above rates on the order of 10�5 mutations peroncogenic mutation. However, it is important to note that notall driver loci are relevant in breast cancer, and in particular, theexact combinations of driver loci that could causebreast cancer areunknown, thus the 10�5

figure can only be a broad approxima-tion. For this reason, we also tested our model at other mutationrates and found that our main conclusions were also consistent atlower mutation rates.

We studied the followingmutational effects for eachmutation:under the default assumptions in stem cells, mutant cells had arelativefitness of fmut¼1.1, i.e., afitness increase of 10%, resultingin an increased probability of dividing, while mutant progenitorcells divided an additional zmut ¼ 1 times (Fig. 2B). Because thenumber of stem cells per duct is small, the fitness ofmutant alleleshas little effect on cancer initiation probabilities, as the fixationtime of mutations is much smaller than the mutation accumu-lation time (27); we also tested our results at other values of fmut

and zmut. In addition, progenitor cells must accumulate somepropensity toward self-renewal:wedefined aparameter g ¼ gbase–

(i � gbase)/(2 � z) as the probability of a progenitor cell atdifferentiation level 0 � i � z þ n � zmut to acquire self-renewal.We chose this functional form to capture a decrease in theprobability of attaining self-renewal as progenitor cells differen-tiate, and explored different values of gbase within this framework.We defined cancer initiation as a single cell that accumulated allrequired mutations and either retained or acquired the ability toself-renew, either through being a stem cell or through acquiring agenetic or epigenetic self-renewal event.

As we were interested in the effects of the timing of pregnancy,we considered the phenotypic alterations that occur in the breastduring pregnancy and as a result of pregnancy. For the purposes ofthis simulation, we considered the 280-day period of time for thepregnancy itself as the time period during which parameters arealtered by pregnancy. Evidence suggests that pregnancy results inthe differentiation of mammary epithelial cells (37, 38) as well astheir increased proliferation (19, 39). To model these effects, weallowed further differentiation of progenitor cells during preg-nancyby an additional zpreg differentiation levels and adecrease inthe cell-cycle length of stem cells (Fig. 2C). There is a 4.5- to 8.5-fold increase in the number of Ki67þ cells during pregnancy (19,39). Thus, we allowed a 4-fold to 8-fold increase in progenitorcells during pregnancy, corresponding to zpreg ¼ 2 to 3. Theremaining approximately 1.1-fold increase in proliferation wasmodeled as a decrease in stem cell cycle length, specifically achange by a factor of apreg ¼ (1/1.1). Importantly, we consideredthat pregnancy reduces the progenitor population in our model.We simulated this change in population structure by decreasingthe rate of asymmetric division of stem cells giving rise to

progenitor cells by a factor of ppost,init after an initial pregnancy.Our experiments suggested a 2- to 3-fold drop in p27þ expressingprogenitor cells, which suggests a value of ppost,init ¼ 0.5 (12).

We also modeled the effects of later pregnancies. In runs of themodel with more than one birth, we considered the effect of theperiod of subsequent pregnancies to be the same as for the firstbirth. That is, the number of levels in the differentiation hierarchyof progenitor cells increases by zpreg levels, and the cell-cyclelength of stem cells decreases to tcycle,preg ¼ 147 hours. Regardingthe lasting effects of pregnancy on the structure of the breastepithelium, we allowed for the possibility of a smaller decrease inthe probability of asymmetric stem cell division after later birthscompared with the decrease after the first birth, and defined aseparate parameter, ppost,subs, for the decrease in asymmetricdivisions after subsequent births.

Our simulation spanned from menarche to death or initiationof cancer within the duct. Our total simulation time was calcu-lated from the average woman's life expectancy in the UnitedStates, which was 81.2 years in 2014 (40), and the average age ofmenarche, which ranged between 12.2 and 12.8 years of age fordifferent ethnic groups in 2007 (41). We used the mean age ofmenarche between the groups, which was 12.5 years, and thusresulted in a total of 68.7 years of simulation time.

The parameters in Supplementary Table S1 were set at fixedvalues from the literature. Theparameters in Supplementary TableS2 were set at values that fit to epidemiologic data, as describedbelow. We tested the robustness of the fit by varying each of theseparameters individually.

ResultsWe first investigated whether our model could quantitatively

match the epidemiologic data available on the protective effect ofearly pregnancy on breast cancer risk, within the space of biolog-ically plausible parameters. From the literature, a woman withonebirth at age20has a cumulative relative risk of ERþ/PRþbreastcancer of 0.88 [confidence interval (CI), 0.81–0.96] between theages of 30 and 70, compared with a nulliparous woman, whereasa woman with four births at ages 20, 23, 26, and 29 has acumulative relative risk of 0.71 (CI, 0.60–0.84) over the sameage range (7). Tomatch these rates,we varied theprobability that aprogenitor cell acquires the ability to self-renew, gbase, and thereduction of the size of the p27þ progenitor cell population afterthe second pregnancy and later pregnancies, ppost,subs. We foundthat with g ¼ 3.2 � 10�3 and ppost,subs ¼ ppost,init ¼ 0.5, themodeled relative risks were within the CIs reported in the liter-ature for these two data points, at 0.86 and 0.73, respectively (Fig.3A). Due to binning ofmodeled incidence into annual groups, weconsidered risk during the 40-year period from ages 30.5 to 70.5.Note that there are likely other parameter settings that couldfit thedata, in addition to those that we used; the ones presented hereserve as an example of how ourmodel can explain the data, ratherthan as an exact parameter estimation approach.

Using the fitted model, we first tested the effects of varyingmodel parameters in the nulliparous simulations to test thebehavior of the model. As expected, we found that the rate ofcancer initiation per duct was increased by increasing thenumber of stem and progenitor cells per duct, the rate ofasymmetric stem cell division, the mutation rate, the proba-bility of progenitor cells attaining self-renewal capacity, and thefitness advantage of mutated progenitor cells compared with

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wild-type cells. By contrast, the rate of cancer initiation per ductwas increased by decreasing the number of mutations requiredfor cancer initiation. Also, as expected, changes in the prolif-erative capacity of progenitor cells during pregnancy, and theeffects of subsequent pregnancies, have no effect in the nul-liparous state (Fig. 3B and C).

We then tested the robustness of the fit of our model to theresult that early pregnancy protects against breast cancer in thesurrounding parameter space. We compared the relative likeli-hood of cancer initiation with pregnancy occurring at 5-year

intervals during a woman's childbearing years as compared withthe nulliparous simulations. We tested for the effects of preg-nancy occurring from the age of menarche until immediatelybefore menopause at the average age of 51.3 in 1998 (42). Wetested the effects of varying the simulation parameters indepen-dently for each pregnancy age tpreg. All fixed value parameters arelisted in Supplementary Table S1, whereas Supplementary TableS2 lists the values of all other parameters. We found that theprobability of cancer initiation in a duct increases as the age offirst pregnancy increases within the range of all simulated

Figure 3.

Model fitting and effect of parameter variation on cancer initiation in nulliparous simulations. A, Evolution of initiation-free ducts with age under the defaultparameter settings for three birth scenarios (nulliparous, a single birth at age 20, and four births at ages 20, 23, 26, and 29). B, Effects of varying individualparameters of the model on nulliparous cancer initiation. C, Evolution of initiation-free ducts with age under the nulliparous scenario for different settingsof the probability of asymmetric division (top), the mutation rate (middle), and the effect of mutation in progenitor cells (bottom). Default values were tcycle ¼ 162,N ¼ 8, z ¼ 10, P ¼ 10�2, m ¼ 2 � 10�5, fmut ¼ 1.1, zmut ¼ 1, nmut ¼ 2, zpreg ¼ 2, rpost,subs ¼ 0.5.






parameters (Fig. 4A). In addition, the average probability ofcancer initiation across birth ageswas lower than the nulliparousrisk for all parameter settings. Both of these effects were lessmarked under parameter settings in which most of the cancersresulted from the stem cells under the nulliparous scenario(P ¼ < 4 � 10�4 Spearman's rank correlation coefficient, inboth cases). Indeed, the z¼ 6 setting had the highest proportionof stem cell cancers under the nulliparous setting.

We also investigated the effects ofmultiple births on cancer risk.We tested model runs with one, two, three, and four total births.

For each of these cases, we investigated varying the age atfirst birthin 5-year intervals as above from the age ofmenarche to the age ofmenopause, assuming that all subsequent births were distributedevenly across the intervening years between the first birth and theage of menopause. For all numbers of total births, risk increasedwith increasing age atfirst birth. In addition, as expected, scenarioswith a larger total number of births were at a lower risk comparedwith scenarios with fewer births (Fig. 4B).

We also tested for robustness of the quantitative fit to the twodata points considered. As expected, we found that for some

Figure 4.

Relative probability of cancer initiation per duct as compared with nulliparous simulations. A, Variation in cancer initiation relative to nulliparous for differentages at first birth under default parameter settings (green lines), and when varying individual model parameters upward (red lines) or downward (blue lines).Left to right from top left, effects of varying stem cell cycle time, number of stem cells, number of progenitors, probabilities of stem cell differentiation,mutation rate, probability of progenitor cells attaining ability to self-renew, fitness effects of mutations, number of mutations required for cancer initiation,and additional pregnancy divisions are shown. B, Variation in cancer initiation relative to nulliparous for different ages at first birth and different numbers oftotal births.

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parameters, the decrease in risk in the two modeled scenariosremained within the bounds of the CIs for all settings tested,whereas for other parameters, there were some settings wherethe risk decrease did not match the literature values (Supple-mentary Fig. S1). In particular, the quantitative fit to both datapoints was robust to changes in the cell-cycle time of stem cells,the number of stem cells per duct, the fitness effects of muta-tions in stem cells, the number of additional progenitor celldivisions during pregnancy, and the reduction in numbers ofprogenitors with subsequent births, within the range of valuestested. The quantitative fit was also robust to decrease in themutation rate in the range of values tested. Thus, our analysisdemonstrates that the hypothesis can explain these twoobserved quantitative decreases in breast cancer risk, undersome, but not all, plausible biological settings. Our hypothesisis thus one possible explanation for the observed protectiveeffect of parity. However, we cannot rule out other possibleexplanations for the relatively limited amount of available dataon the quantitative risk reduction.

Another interesting result is the specificity of the effect of thedecrease in theprogenitor poolwithpregnancy todecrease the riskof cancers initiating from the progenitor compartment. We notedthat the risk of cancers initiating from the progenitor cell com-partment increased with age at first birth, whereas the risk ofcancers where the final mutation occurs in the stem cell compart-ment showed a (smaller) decrease (P < 4 � 10�5 in both casesunder linear regression). Similarly, under the default parametersettings, whereas the risk of cancers initiated from the progenitorcompartment was lower under all parous scenarios comparedwith the nulliparous scenarios, the risk of cancers initiated fromthe stem compartment was slightly higher under all parousscenarios.

This result raises one possible explanation for the specificity ofthe protective effect of early pregnancy to ERþ/PRþ cancer (7).Mounting experimental evidence suggests that the typical cell oforigin of breast carcinomas is a stem or progenitor cell (43). Thespecificity of the protective effects in ourmodel to a single cellularcompartment poses the question of whether other breast cancermolecular subtypesmayhave adifferent cell of origin as a possibleexplanation for the observed specificity of protective effects.Relatedly it is also possible that changes during carcinogenesisrender other breast cancer subtypes insensitive tohormone-drivengrowthor that someof themolecular parameters considered differbetween breast cancer subtypes. By the same token, our model isagnostic on whether the pregnancy should protect against otherhistologic breast cancer types, such as lobular cancers. Whether ornot protective effects would be expected for these subtypesdepend on the extent to which the etiology of these cancer types,in terms of cell of origin and other molecular parameters, corre-sponds to ERþ cancers.

As a further test of our framework, we investigated whether ourmodel reproduced the known effect that breast cancer risk isincreased for a short period immediately following pregnancy(6). For these purposes, we investigated an extended modelincluding a variable delay between initiation of cancer withinthe duct and clinical presentation. We investigated two scenarios,first birth at age 20, and first birth at age 40, and calculated therelative risk comparedwith nulliparouswomen ofmatched age inthe years following pregnancy for varying averagewaiting times toclinical presentationbetween0 and5 years.We found thatwith anaverage waiting time of 1 year, relative risk in both parous

scenarios was greater than one during the 2 years following thepregnancy (Supplementary Fig. S2)

Discussion

Here, we investigated whether variation in the size of theprogenitor cell population is sufficient to explain the protectiveeffects of pregnancy.We used a simplemathematicalmodel of thesteps leading to cancer initiation, which included both stem cellsand progenitor cells. We found that within the range of biolog-ically plausible parameters, our model matches the observeddecrease in ERþ/PRþ cancer risk for a woman with a birth at age20 and a woman with four births in her 20s compared with anulliparous woman. Using these parameter settings, we foundthat the risk of cancer in our model decreased with increasing ageof first birth in scenarios with one birth. Moreover, the risk ofcancerwas lower in all scenarioswithonebirth comparedwith thenulliparous case. This behavior was robust to variation in keymodel parameters. The ability of our model to robustly recreatethe effect on cancer risk when varying the progenitor populationsizewith pregnancy is striking given themodeled assumption thatprogenitor cells terminally differentiate after a finite number ofdivisions, so thatmutations arising in progenitor cells are liable toleave the population without any functional impact. Takentogether, these results support the hypothesis that a subset ofp27þ cells represents quiescent hormone-responsive luminalprogenitor cells with proliferative potential.

Our mathematical modeling approach for breast cancer canbe useful in understanding the contribution of unavoidablebad luck to cancer risk. We have presented evidence that, in thesetting of breast cancer, the size of a sub-population ofprogenitor cells may vary safely over the course of a life toalter breast cancer risk, independent of the probability ofmutations. Although it is possible that the mechanismsexplored here are specific to the breast cancer setting, ourresults highlight the possibility that extrinsic factors can inter-act with molecular parameters to affect cancer risk in ways thatare not yet fully mapped out. These results therefore furthermotivate the use of complementary approaches to assess thecontribution of bad luck to cancer risk that do not rely onstrong assumptions about the effects of extrinsic factors, whichmay still be subject to revision. The modeling approachdeveloped here is one such possible complementary approach.Therefore, the main implications of our study are support for amechanism in the breast cancer setting, with potential impli-cations for other cancers with an important role for hormone-driven growth, including endometrial and ovarian cancers.And, in addition, the current approach may be usefully appliedin a range of cancer types.

In conclusion, our results demonstrate that variation in the sizeof the pool of progenitor cells with proliferative potential iscapable of explaining the protective effect of early pregnancyagainst breast cancer. We obtained good agreement between oursimple model's predictions and specific epidemiological datapoints within the range of plausible parameters. Intense recentdebate, prompted by the work of Tomasetti and Vogelstein (1),has indicated the limits of regression techniques for determiningthe ultimate contribution of bad luck to cancer incidence. Con-tinuing improvements in our mechanistic understanding of theetiology of different cancers can help elucidate the contribution ofbad luck to cancer risk and the limits of cancer prevention






strategies. Given the complexity of the molecular setting in whichcancer develops, mathematical models can be a useful tool indeveloping such a mechanistic understanding. Our work hasdeveloped this approach for the case of breast cancer to provideevidence for a possiblemechanism for the protective effect of earlypregnancy against the disease.

Disclosure of Potential Conflicts of InterestNo potential conflicts of interest were disclosed.

Authors' ContributionsConception and design: K. Polyak, F. MichorDevelopment of methodology: D. Temko, Y.-K. Cheng, F. MichorAnalysis and interpretation of data (e.g., statistical analysis, biostatistics,computational analysis): D. Temko, Y.-K. Cheng, F. MichorWriting, review, and/or revision of the manuscript: D. Temko, K. Polyak,F. Michor

Administrative, technical, or material support (i.e., reporting or organizingdata, constructing databases): D. TemkoStudy supervision: K. Polyak, F. Michor

AcknowledgmentsWe thank members of the Michor and Polyak labs for their critical reading

of our article.

Grant SupportThis work was supported by NCI U54CA193461 (F. Michor and K. Polyak),

by EPSRC doctoral training grant EP/F500351/1 (D. Temko), and byP01CA080111 and a Susan G. Komen Foundation grant, both to K. Polyak.

The costs of publication of this articlewere defrayed inpart by the payment ofpage charges. This article must therefore be hereby marked advertisement inaccordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Received September 12, 2016; revised October 24, 2016; accepted March 24,2017; published OnlineFirst March 30, 2017.

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2017;77:2800-2809. Published OnlineFirst March 30, 2017.Cancer Res Daniel Temko, Yu-Kang Cheng, Kornelia Polyak, et al. Breast Cancer RiskMathematical Modeling Links Pregnancy-Associated Changes and

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