EFTA01201684.pdf

C Timing and heterogeneity of mutations associated with drug resistance in metastatic cancers Ivana Bozic'' and Martin A. Nowa ka'bi 'Program for Evolutionary Dynamics, Department of Mathematics, and °Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138 Edited by Herbert Levine, Rice University, Houston, 1X and approved October 8, 2010 (received for review June 28, 2010) Targeted therapies provide an exciting new approach to combat modeling to investigate the heterogeneity of drug-resistant mu- human cancer. The immediate effect is a dramatic reduction in tations in patients with metastatic cancers. disease burden, but in most cases, the tumor returns as a conse- First mathematical investigations of the evolution of resistance quence of resistance. Various mechanisms for the evolution of to cancer therapy were concerned with calculating the proba- resistance have been implicated, including mutation of target bility that cells resistant to chemotherapy are present in a tumor genes and activation of other drivers. There is increasing evidence of a certain size (16). Later studies expanded these results to that the reason for failure of many targeted treatments is a small include the effects of a fitness advantage or disadvantage preexisting subpopulation of resistant cells; however, little is provided by resistance mutations (17, 18), multiple mutations known about the genetic composition of this resistant subpopu• needed to achieve resistance to several drugs (15, 19-21), and lation. Using the novel approach of ordering the resistant sub• density limitations caused by geometric constraints (22). These dones according to their time of appearance, here we describe studies used generalizations of the famous Luria-Delbrfick model the full spectrum of resistance mutations present in a metastatic for accumulation of resistant cells in exponentially growing bac- lesion. We calculate the expected and median number of cells in terial populations (23). Probability distribution for the number of each resistant subclone. Surprisingly, the ratio of the medians of resistant cells in a population of a certain size in the fully sto- successive resistant clones is independent of any parameter in our chastic formulation of the Luria—Delbrfick model was recently model; for example, the median of the second done divided by the calculated in the large population size limit (24, 25). The focus of median of the first is —1. We find that most radiographically above studies was describing the total number of all resistant detectable lesions harbor at least 10 resistant subclones. Our pre- cells, rather than the composition of the resistant population (26). dictions are in agreement with clinical data on the relative sizes of resistant subclones obtained from liquid biopsies of colorectal can• Results cer patients treated with epidermal growth factor receptor (EGFR) We model the growth of a metastatic lesion as a branching blockade. Our theory quantifies the genetic heterogeneity of re- sistance that exists before treatment and provides information to process (27) that starts from a single cell (the founder cell of the metastasis) that is sensitive to treatment. Sensitive cells divide design treatment strategies that aim to control resistance. with rate b and die with rate d. The net growth rate of sensitive cancer I drug resistance I heterogeneity I mathematical biology cells is r=b —d. During division, one of the daughter cells receives a resistance mutation with probability u. Resistant mutations can be neutral in the absence of treatment, which A d resistance to treatment is a major impediment to successful eradication of cancer. Patients presenting with early-stage cancers can often be cured surgically, but patients means they have the same birth and death rates as sensitive cells, and we initially focus on this case. We also expand our theory to the more general case where resistant cells are nonneutral, which with metastatic disease must be treated with systemic therapies means they have birth and death rates bR and de. respectively. If (1). Traditional treatments such as chemotherapy and radiation 1 that exploit the enhanced sensitivity of cancer cells to DNA damage have serious side effects and, although curative in some Significance cases, often fail due to intrinsic or resistance acquired during treatment Targeted therapies, a new class of drugs, inhibit specific Metastatic dissemination to surgically inaccessible sites is the molecules implicated in tumor development and are typically less major cause of death in cancer patients. Targeted therapies, harmful to normal cells compared with chemotherapy and radiation often initially effective against metastatic disease, invariably fail (2-5). In the case of many targeted treatments, patients initially due to resistance. We use mathematical modeling to study have a dramatic response (6, 7), only to be followed by a regrowth heterogeneity of resistance to treatment and describe for the first time, to our knowledge, the entire ensemble of resistant of most of their lesions several months later (8-10). Acquired re- sistance is often a consequence of genetic alterations (usually point subclones present in metastatic lesions. We show that radio- graphically detectable metastatic lesions harbor multiple re- mutations) in the drug target itself or in other genes (10-14). sistant subclones of comparable size and compare our Recently, mathematical modeling and clinical data were used predictions to clinical data on resistance-associated mutations to show that acquired resistance to an epidermal growth factor in colorectal cancer patients. Our model provides important receptor (EGFR) inhibitor panitumumab in metastatic co- information for the development of second-line treatments that lorectal cancer patients is a fair accompli, because typical aim to inhibit known resistance mutations. detectable metastatic lesions are expected to contain hundreds of cells resistant to the drug before the start of treatment (10). Author contributions: IS. designed research; I.B. and M.A.N. performed research; I.B. and These cells would then expand during treatment, repopulate the M.A.N. analyzed data; and IS. and M.A.N. mote the paper. tumor, and cause treatment failure. Similar conclusions should The authors declare no conflict of Interest. hold for targeted treatments of other solid cancers (15). Suc- This article is a NOS Direct Submissico. cessful treatment requires drugs that are effective against the 'To whom correspondence may be addressed. Email: lbozIcOmath.harvard.edv or preexisting resistant subpopulation and must take into account martinnowakeharvard.edu. the (possible) heterogeneity of resistance mutations present This article contains supporting information online at www.pnas.orgilookup/supplidoi:10. in the patient's lesions. In this article we use mathematical 1073/pnal.141207511INDCSUPplemantal. www.paes.orgegweovio.tozneuistetzozsiti PNAS Early Edition I 1of 5 EFTA01201684  c = (bR —dR)1(b —d)> 1, then resistance mutations are advanta- assume a large M and small u limit and mostly focus on the case geous before treatment; if c < 1, they are deleterious. when Mu >> I. A resistant cell may appear in the population and be lost due Tumor sizes at which successful resistant mutants are pro- to stochastic drift or it can establish a resistant subclone. We duced can be viewed as a Poisson process on [0.,M] with rate u number the resistant subclones that survive stochastic drift by the (SI Tat) (10, 17). The number of successful mutant lineages is order of appearance (Fig. IA). A reasonable assumption for the thus Poisson distributed with mean 2 =Mu. If Mk is the number number of point mutations that can provide resistance to a tar- of cancer cells in the lesion when the kth mutant appeared, which kg geted drug is on the order of 100 (10, 28). Thus, the different survived stochastic drift (Fig. 1A), then Mkt i —Mk is exponen- resistant subclones will typically contain different resistance tially distributed with mean 1/u. Therefore, we expect that the kth clone appeared when the total population size was Mk —klu VA mutations, especially if we only focus on the largest ones. We calculate the number and sizes of resistant subclones in and that roughly the size of the first clone is k times the size of a a metastatic lesion containing Al cells. Typical radiographically the kth clone. The probability that exactly k clones are present in detectable lesions are — I cm in diameter and contain —109 cells. the population of size Al is Ake-A/k1 The mutation rate, u, leading to resistance is the product of the Counting new successful resistant clones in the order of ap- point mutation rate p, which is on the order of —10-9 per base pearance, we calculate the probability distribution for the num- pair per cell division, and the number of point mutations that ber of cells in the kth resistant clone. In particular, if k 4Z Mu, the cumulative distribution function for the number of resistant can confer resistance, which is —100. In our analysis we will cells in the kth clone simplifies to Fk(y) As I (mu dy . MU Ill The excellent agreement between Formula 1 and exact computer simulations of the stochastic process is shown in Fig. lB. The mean number of cells in the kth resistant clone is E(Yi ) Ps [bMulr][log(rIbu)— I] and E(Yk) s /Wu l[r(k — 1)] for k> 2. The median for the number of cells in the kth subclone is given by bMu Med(Yk) (211* —1). [2] —r Interestingly, the ratio of the means of the two subclones k and/ is (i — 1)/(k — I) for k. j> 1. The ratio of their medians is Med(Yk) 21/k — I I31 MeA i'l 2 3— I 1 These ratios are independent of any parameters of the process. In particular, the ratio of the medians of the first and second 0.9 clone is in —1, which implies that they have comparable size 0.8 (same order of magnitude). Cumulative distribution 0.7 Liquid biopsy data were used to obtain estimates for the birth and death rates of cells in metastatic lesions and the number of 0.6 point mutations providing resistance to the EGFR inhibitor 0.5 panitumumab in colorectal cancer (10). The resulting parameter O4 values (b=0.25 and d = 0.181 per day, point mutation rate p =10-9 per base pair per replication, and 42 point mutations 0.3 conferring resistance) can be used to calculate the mean and 0.2 median sizes of the resistant subclones in a metastatic lesion containing M = 109 cells. The mean numbers of cells in the first, 0.1 second, and third appearing resistant clone are E(Y1)x2237, 0 E(Y2) PS 152, and E(Y3) PS76, respectively. However, the mean for 1 10 100 1000 10000 100000 Yi, the size of the first resistant clone, is heavily influenced by the Clone size realizations of the stochastic process in which the first resistance Ng. 1. Evolution of resistance in a metastatic lesion. (A) As the lesion mutation appeared early and is not a good summary of the (green) grows from one cell to detectable size, new resistant subclones ap- probability distribution for Yi. Namely, the realizations in which pear. Some of them are lost to stochastic drift (yellow and pink), while others the number of cells in the first clone is greater than the mean survive (purple, red and orange triangle). Instead of looking at the time of (2,237) account for less than 7% of all cases. The median appearance of new clones, our approach takes into account the total size of number of cells in the first resistant clone [Med(Yi )] for the the lesion when the resistance mutation first occurred. (8) Agreement be- above parameters is 152, whereas the medians for Y2 and Y3 are tween computer simulations and formula (I) for the cumulative distribution 63 and 40, respectively. function for the number of cells in the first four resistant clones. The first subclone contains 10 or fewer cells with probability 0.06, between 10 and In SI Tar, we calculate the probability distribution for the 100 cells with probability 0.30, between 100 and 1000 cells with probability ratio of resistant clone sizes YI /Yk and show that it is also in- 0.07 and more than 1000 cells with probability 0.13. The second subclone dependent of the parameters of the process. Even though the contains more than 100 cells with probability 0.36. Parameters b= 0 25, first appearing clone is expected to be the largest, followed by the d=0.181, M=109, u=42.104 . second clone and so on, we show that this ordering is often 2 of 5 I www.poes.cregfCgilda/10.107342nat.1412075111 Bozic and Nowak EFTA01201685 violated. In 31% of lesions, the first successful subclone is smaller that the ratio of resistant clone sizes is independent of than the second one: on the other hand, in 24% of lesions the first parameters. subclone is at least 10 times larger than the second one. Our mathematical results describe the relative sizes of re- Fig. 2 shows different realizations of the stochastic process of sistant clones ordered by age, whereas the experimental data in evolution of resistance in metastatic lesions containing 109 and Table I are ordered by size, which serves as a proxy for age, 109 cancer cells. The same parameters were used to generate all because exact clonal age is unknown. We quantify the extent to lesions. The size of each subclone is shown (in number of cells), which this difference in clonal ordering by size vs. age influences and the subclones are ordered by their time of appearance. In our statistics using exact computer simulations (Table 1). In the lesion LI, the first three subclones are the largest, and each have relevant parameter regime of large lesion size, At, and small around 100 cells. Lesion L.5 contains only two subclones, whereas mutation rate, a, with Mu >, 1, the results are largely in- L6 contains seven subclones, but none has more than 10 cells. In dependent of parameters (median ratios of clone sizes vary by each lesion of total size 109 cells, there are more than 10 resistant <10% for different parameter combinations). We show simula- subclones. In L7, the two largest subclones contain 1,500 and 460 tion results for median ratios of clone sizes when clones are cells. In IS, there are five subclones of about 100 cells. ordered by size for typical parameter values (10). As we see in In Table 1, we show clinical data for the number of circulating Table 1, the ordering of experimental data by size does not sig- tumor DNA (ctDNA) fragments harboring mutations in five nificantly change the results of our analysis. genes associated with resistance to anti-EGFR treatment in 18 We can generalize our approach to the case when resistance colorectal cancer patients who developed more than one muta- mutations are not neutral, but provide a fitness effect already tion in those genes (29). These mutations were not detectable in before treatment (formulas shown in Si Text). In Table 2, we patients' serum before therapy, but became detectable during the compare the predicted medians for the first five resistant clones course of anti-EGFR treatment. The number of ctDNA frag- in a metastatic lesion containing M= 109 cells when resistance is deleterious, neutral, or advantageous. We see from Table 2 that ments correlates with the number of tumor cells harboring that even if resistant cells are only 10% as fit as sensitive cells, they mutation: it was previously estimated (using the tumor burdens will still be present in typical lesions. The average number of and pretreatment ctDNA levels measured in patients who had resistant cells produced until the lesion reaches size M is Mu/s. KRAS mutations in their tumors before therapy) that one mu- 17 tant DNA fragment per milliliter of serum corresponds to 44 Here s= 1 —d/b is the survival probability of sensitive cells, e which is the probability that the lineage of a single sensitive cell million mutant cells in the patient's tumor (10). Thus, the ratios will not die out. For typical parameter values (i.e., those used in of the resistant clone sizes can be obtained from the ratios of the Table 2), the number of resistant cells produced by sensitive cells numbers of ctDNA fragments harboring resistance-associated in a single lesion is — 150. Resistant cells that are 10% as fit as mutations. These data provide a unique opportunity to test our sensitive cells have a survival probability of 4%; so on average, theory and compare the relative sizes of resistant clones inferred from the data with those predicted using our model. Assuming that resistance-associated mutations with higher ctDNA counts six of them will form surviving clones. The effect that mutations can cause treatment failure, although they have high fitness cost is a consequence of the high number of resistant mutants pro- BR appeared before those with lower ctDNA counts, we find ex- duced by billion(s) of sensitive cells in a lesion and the specific cellent agreement between the data and our model predictions. properties of the branching process, namely the independence 1 For example, the median ratio of the sizes of the first two re- of lineages. sistant clones inferred from clinical data (29) is 2.21, whereas our model predicts 2.51. The median ratio of the sizes of the first and Discussion third clones from clinical data are 4.3, and our model predicts In this paper we describe the heterogeneity of mutations pro- 4.12 (Table 1). This comparison is parameter free, as we showed viding resistance to cancer therapy that can be found in any one A Lesion size M-108 cells in two LI moo L2 'coo L3 woo woo 1.3 moo L6 •0 too too 100 too too too to II to 10 10 to I0 Z • I II VIII_ 1 2 3 4 5 6 7 8 9 ID 2 3 4 5 6 7 8 9 10 1 2 1 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 910 1 2 3 4 3 6 7 8 910 1 2 3 4 5 6 7 8 9 10 B Lesion size M109 cells L7 U L9 LI0 LII 1000 L12 4 t000 I 1 1000 1000 I 100 (.... 100 100 100 0 10 10, to io 10 hid Z I/1 2 3 4 3 6 7 8 9 IS 1 2 3 4 3 6 7 8 9 10 1 2345678910 1234 567 9 10 1 2 3 4 5 6 7 8 9 10 1 2 34 567 9 10 Resistant clones (order of appearance) Fig. 2. Resistant subclones in metastatic lesions. Different realizations of the same stochastic process are shown in each panel. (A) Six lesions of size 10° and (8) six lesions of size le cells. The first ten resistant clones are shown, which survived until time of detection. They are ordered according to their time of appearance. Parameter values for all simulations: b =0.25, d=0.181, u=42.104. Bozic and Nowak PNAS Batty Edition I 3 of 5 EFTA01201686  Table 1. Comparison of predicted ratios of resistant clone sizes and ratios obtained from clinical data Patient Yi* Y2 1 12 1 14 YI/Y2 YI/Y3 YI/Y4 1 168 90 1.87 2 129 120 1.08 3 82 80 30 1.03 2.73 4 948 120 104 100 7.9 9.12 9.48 5 28 15 1.87 FA 6 114 40 2.85 a 7 8 9 6,760 220 848 4,940 30 374 4,100 135 3,900 133 1.37 7.33 2.27 1.65 6.28 1.73 6.38 10 61 25 2.44 11 244 83 57 2.94 4.28 12 429 400 100 1.07 4.29 13 394 13 4 30.31 98.5 14 308 265 208 139 1.16 1.48 2.22 15 130 13 10 16 28 13 2.15 17 131 45 12 11 2.91 10.92 11.91 18 250 173 58 3t 1.45 431 8.06 Median from patients 2.21 43 7.22 Predicted median 2.51 4.12 5.74 Predicted median 2.05 3.63 5.25 (order by size) •Number of circulating tumor DNA (ctDNA) fragments per milliliter (VI to Y4) harboring different mutations associated with resistance to anti-EGFR agents in colorectal cancer patients treated with EGFR blockade (29). Ratio of resistant clone sizes is given by the ratio of the ctDNA counts for any two resistance-associated muta- tions. We assumed that mutations with higher ctDNA counts in the patient data appeared before mutations with smaller ctDNA counts. We also report predicted median ratios obtained from computer simulations when clones are ordered by size (parameters: b =0.25, d=0.181, M=109, u =42 x 10-9). metastatic lesion. Our results can be generalized to take into rate u, the statistics of the relevant clones in a branching process account all of the patient's lesions, assuming that they evolve with death remain approximately Yule (31). Similarly, it can be according to the same branching proetts and that the number of shown that in the Yule process, in the above limits, the mean size lesions is much smaller than l/u. In that case, the probability of the kth largest clone is —Mu/(k-1), and the ratio of the distribution for the size of the kth appearing resistant clone in mean sizes of the kth and jth largest clones is —(f —1)/(k — 1) the patient's cancer is given by Formula 1 if we let M be the (31, 32). This formula is exactly the result we obtain for the ratio number of cancer cells in all of the patient's lesions. All our of mean clone sizes even though we order clones by age. results generalize similarly. A few recent investigations studied the dynamics of single Although the mean and median clone sizes in our model de- clones resistant to therapy (28, 33). In one of the studies (33), the pend on the parameters of the process, their ratios are generally authors used a generalization of the Luria-Delbriick model in 1 parameter free. The universality of the clone ratio statistics fol- which sensitive cells grow deterministically and calculated the lows from the fact that the skeleton of our branching process, number of individual resistant clones and the probability distri- which includes only cells with infinite line of descent, can be bution for the number of cells in a single resistant clone after approximated by a Yule (pure birth) process (30). It has been time I. In another study (28), mathematical modeling along shown that in the limit of large lesion size M and small mutation with in vitro growth rates of cells harboring 12 point mutations Table 2. Sizes of resistant clones when resistance Is deleterious, neutral, or advantageous c.(bR -dR)/(b-d) First done* Second clone Third done Fourth clone Fifth done 0.01 0 0 0 0 0 0.1 10 6 4 2 0.5 27 17 13 11 10 0.7 50 26 19 15 13 0.9 103 46 30 23 18 0.95 125 54 35 26 20 1 152 63 40 29 23 1.05 186 74 45 33 25 t.1 229 87 52 37 28 •Median number of cells in the first five successful resistant clones in a metastatic lesion with M=109 cells when resistant cells are less fit than sensitive cells (c < 1), neutral (c =1), and more fit than sensitive cells (c> 1). We fix the birth and death rate of sensitive cells, b= a 25 and d=0.181, and the death rate of resistant cells dR =d. We vary the relative fitness of resistant cells, G and let the birth rate of resistant cells be bit =dR +c(b—d). Mutation rate u =42 x 104. For c =0.1 we report simulation results, and for c> 0 1, we use Eq. S13; see SI Text for details. 4 of 5 I wwwonas.orgfcgildoi/10.10734ffias.14I2075111 Bozic and Nowak EFTA01201687 providing resistance to BCR-ABL (fusion of breakpoint cluster re- metastatic lesions in a patient. This information is pertinent to gion gene and Abelson murine leukemia viral oncogene homolog the development of second line treatments that aim to inhibit inhibitor imatinib were used to calculate the number of resistant known resistance mutations. clones and the expected number of resistant cells with a particular resistance mutation at the time of diagnosis of chronic myeloid Materials and Methods leukemia. The authors found that at most one resistant clone is Model. We model the growth and evolution of a metastatic lesion as a con- expected to be present, as the total number of CML stem cells at tinuous time multitype branching process (34). The growth of a lesion is diagnosis is estimated to be approximately M— 100.000 cells and is initiated by a single cell sensitive to the drug. Sensitive cells produce a re- much smaller than the billions of cells typically present in a single sistant cell at each division with probability ta and each resistant cell pro- detectable lesion of a solid tumor. In this paper, we use a different duced by sensitive cells starts a new resistant type. mathematical technique and the novel approach of ordering the resistant clones according to their time of appearance, which Analysis. In ow analysis, we use the approximation that resistant cells pro- allows us for the first time, to our knowledge, to describe the full duced bysensitive cells appear as a Poisson process on the number of sensitive spectrum of resistance mutations present in a lesion. cells (17). For more details and derivations of our results, please see SI Ten. Our study is challenging the conventional view of the evolution of resistance in cancer. For every therapy that is opposed by Simulations. We perform Monte Carlo simulations of the multitype branching multiple potential resistance mutations, which is the case for process using the Gillespie algorithm (35). Between 5,000 and 10,000 sur- every targeted drug developed thus far, we can expect multiple viving runs are used for each parameter combination. resistant clones of comparable size in every lesion. Our theory ACKNOWLEDGMENTS. We thank Bert Vogelstein for critical reading of the provides a precise quantification of the relative sizes of those manuscript and Rick Durrett for discussion during the conception of this work. resistant subclones. The heterogeneity of resistance muta- We are grateful for the support from Foundational Questions in Evolutionary tions is further amplified when taking into account multiple Biology Grant RFP-12-17 and the John Templeton Foundation. 1. 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Bozic and Nowak PNAS Early Edition I 5 of 5 EFTA01201688  Supporting Information Bozic and Nowak 10.1073/pnas.1412075111 Si Text population that survives stochastic drift when there are Al- IA The Model. We model the growth of a metastatic lesion as a branching process (1) that starts from a single cell sensitive to sensitive cells, conditioned on A4 <M. By the time the sensitive population reache