Timing and heterogeneity of mutations

Timing and heterogeneity of mutations associated with drug-resistance in metastatic cancers Ivana Bozic • •1 and Martin A. Nowak' 1 .1 • Progam for Evolutionary Dynamics. Department of Mathematics. and I Department of Organismic and Evolutionary Biology. Harvard University. Cambridge. MA 02138. USA 'To whom correspondence may be addressed. E-mail: Submitted to Proceedings of the National Academy of Sciences of the United Slates at America Targeted therapies provide an exciting new approach to combat human cancer. The Immediate effect is a dramatic reduction in disease burden, but in most cases the tumor returns as a con- sequence of resistance. Various mechanisms for the evolution of resistance have been implicated including mutation of target genes and activation of other drivers. There is increasing evi- dence that the reason for failure of many targeted treatments is a small preexisting subpopulation of resistant cells; however, little is known about the genetic composition of this resistant subpopu- lation. Using a novel approach of ordering the resistant subclones according to their time of appearance, here we describe the full spectrum of resistance mutations present in a metastatic lesion. We calculate the expected and median number of cells in each resistant subclone. Surprisingly, the ratio of the medians of suc- cessive resistant clones Is independent of any parameter in our model; for example, the median of the second clone divided by the median of the first is — I. We find that most radiograph- ically detectable lesions harbor at least ten resistant subclones. Our predictions are in agreement with clinical data on the rela- tive sizes of resistant subelones obtained from liquid biopsies of colorectal cancer patients treated with EGFR blockade. Our the- ory quantifies the genetic heterogeneity of resistance that exists prior to treatment and provides information to design treatment strategies that aim to control resistance. cancer I drug resistance I heterogeneity I math/mask:aft/cagy Significance Metastatic dissemination to surgically inaccessible sites is the major cause of death in cancer patients. Targeted therapies, often initially effective against metastatic disease. invariably fail due to resistance. We use mathematical modeling to study heterogeneity of resistance to treatment and describe for the first time the entire ensemble of resistant subclones present in metastatic lesions. We show that ra- diographically detectable metastatic lesions harbor multiple resistant subclones of comparable size and compare our predictions to clinical data on resistance-associated mutations in colorectal cancer patients. Our model provides important information for the development of second line treatments that aim to inhibit known resistance mutations. A cquired 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 with metastatic disease must be treated with systemic therapies [I]. Tra- ditional treatments such as chemotherapy and radiation that exploit the enhanced sensitivity of cancer cells to DNA damage have serious side effects and, although curative in some cases. often fail due to intrinsic or resistance acquired during treatment. Targeted therapies. a new class of drugs. inhibit specific molecules implicated in tumor development and are typically less harmful to normal cells compared to chemotherapy and radiation [2. 3.4. 5]. In the case of many tar- geted treatments, patients initially have a dramatic response [6. 7] only to be followed by a regrowth of most of their lesions several months later [S, 9, 10]. Acquired resistance is often a consequence of genetic alterations (usually point mutations) in the drug target itself or in other genes [10. 11. 12. 13. 14 Recently. mathematical modeling and clinical data were used to show that acquired resistance to an EGFR inhibitor panitumumab in metastatic colorectal cancer patients is a fait accompli. since typi- cal detectable metastatic lesions are expected to contain hundreds of cells resistant to the drug before the start of treatment [10]. These cells would then expand during treatment, repopulate the tumor and cause treatment failure. Similar conclusions should hold for targeted treatments of other solid cancers [15]. Successful treatment requires drugs that are effective against the pre-existing resistant subpopula- tion and must take into account the (possible) heterogeneity of re- sistance mutations present in the patient's lesions. In this article we use mathematical modeling to investigate the heterogeneity of drug- resistant mutations in patients with metastatic cancers. First mathematical investigations of the evolution of resistance to cancer therapy were concerned with calculating the probability that cells resistant to chemotherapy are present in a tumor of a certain size [16]. Later studies expanded these results to include the effects of a fitness advantage or disadvantage provided by resistance mutations [IL IS]. multiple mutations needed to achieve resistance to several drugs [15. 19. 20. 21] and density limitations caused by geometric constraints [22]. These studies employed generalizations of the fa- mous Luria-Delbruck model for accumulation of resistant cells in exponentially growing bacterial populations [23[. Probability distri- bution for the number of resistant cells in a population of a certain size in the fully stochastic formulation of the Luria-Delbriick model was recently calculated in the large population size limit [24. 25]. The focus of above studies was describing the total number of all re- sistant cells. rather than the composition of the resistant population [26]. Results We model the growth of a metastatic lesion as a branching process [27] that stars from a single cell (the founder cell of the metastasis) which is sensitive to treatment. Sensitive cells divide with rate b and die with rate d. The net growth rate of sensitive cells is r = b — d. During division one of the daughter cells receives a resistance muta- tion with probability u. Resistant mutations can be neutral in the ab- Reserved for Publication Footnotes www.pnas.orgtgildoif10.1073/pnas.0709640104 PNAS I IssueDate I Volume I Issue Number I I-5 EFTA01199737 sence of treatment, which 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 non-neutral, which means they have birth and death rates bn and dn. respectively. If c = (bn— dR)I(b —(1) > 1 then resistance mutations are advantageous prior to treatment if c < 1 they are deleterious. A resistant cell may appear in the population and be lost due to stochastic drift or it can establish a resistant subclone. We number the resistant subclones that survive stochastic drift by the order of ap- pearance (Fig. IA). A reasonable assumption for the number of point mutations that can provide resistance to a targeted drug is on the or- der of one hundred [10. 28]. Thus, the different resistant subclones will typically contain different resistance mutations, especially if we only focus on the largest ones. We calculate the number and sizes of resistant subclones in a metastatic lesion containing Al cells. Typical radiographically de- tectable lesions are 1 cm in diameter and contain 109 cells. The mutation rate. u. leading to resistance is the product of the point mutation rate p. which is on the order of ti 10-9 per base pair per cell division, and the number of point mutations that can confer re- sistance, which is ••••• 100. In our analysis we will assume a large Al and small u limit and mostly focus on the case when Mu > 1. Tumor sizes at which successful resistant mutations are produced can be viewed as a Poisson process on 10. MI with rate u (see Sup- plementary Information) [17. 10]. The number of successful mutant lineages is thus Poisson-distributed with mean A = Mu. If Afk is the number of cancer cells in the lesion when the k-th mutant appeared. which survived stochastic drift (Fig. IA). then .44 — Afk is expo- nentially distributed with mean 1/u. Therefore, we expect that the k-th clone appeared when the total population size was .41k k 1 u and that roughly. the size of the first clone is k times the size of the k-th clone. The probability that exactly k clones are present in the population of size Al is Ake- A/k!. Counting new successful resistant clones in the order of appear- ance, we calculate the probability distribution for the number of cells in the k-th resistant clone. In particular, if k Mu the cumulative distribution function for the number of resistant cells in the k-th clone simplifies to Afu o, \ 14(0 1 ( Mu + y — dylb ) " The excellent agreement between formula [1 1 and exact computer simulations of the stochastic process is shown in Fig. I B. The mean number of cells in the k-th resistant clone is E(Yr) tt EbA/u/rillog(r/bu) — 1] and E(Yk) x bA/ ul(r(k — 1)1 fork ≥ 2. The median for the number of cells in the k-th subclone is given by Interestingly, the ratio of the means of the two subclones k and j is (j — 1)/(k — 1) for k. j > 1. The ratio of their medians is Med(Yk) 21Th — 1 Mecl(y,) 21/3 — 1 Note that the these ratios are independent of any parameters of the process. In particular, the ratio of the medians of the first and second clone is 1,./2 — 1, which implies that they have comparable size (same order of magnitude). Liquid biopsy data were used to obtain estimates for the birth and death rates of cells in metastatic lesions and the number of point mutations providing resistance to the EGFR inhibitor panitu- mumab in colorectal cancer [10]. The resulting parameter values (b = 0.25,d = 0.181 per day. point mutation rate p = 10-9 per base pair per replication and 42 point mutations conferring resis- tance) can be used to calculate the mean and median sizes of the Ill 131 resistant subclones in a metastatic lesion containing Al = 109 cells. The mean numbers of cells in the first, second and third appearing resistant clone are E(Yr) x 2237. E(Y2) x 152 and E(Ya) x 76. However, the mean for Yi • the size of the first resistant clone. is heav- ily influenced by the realizations of the stochastic process in which the first resistance mutation appeared early and is not a good sum- mary of the probability distribution for Y1. Namely. the realizations in which the number of cells in the first clone is greater than the mean (2237) account for less than 7% of all cases. The median number of cells in the first resistant clone (Med(Yi)) for the above parameters is 152, while the medians for Y2 and Y3 are 63 and 40. respectively. In Supplementary Information we calculate the probability dis- tribution for the ratio of resistant clone sizes Y1/ Yk and show that it is also independent of the parameters of the process. Even though the first appearing clone is expected to be the largest. followed by the second clone and so on, we show that this ordering is often violated. In 31% of lesions the first successful subclone is smaller than the sec- ond one: on the other hand. in 24% of lesions the first subclone is at least 10 times larger than the second one. Fig. 2 shows different realizations of the stochastic process of evolution of resistance in metastatic lesions containing 108 and 109 cancer cells. The same parameters were used to generate all lesions. The size of each subclone is shown (in number of cells), and the subclones are ordered by their time of appearance. In lesion LI the first three subclones are the largest and each have around 100 cells. Lesion L5 contains only two subclones. while L6 contains seven sub- clones but none has more than 10 cells. In each lesion of total size 109 cells there are more than 10 resistant subclones. In L7 the two largest subclones contain 1500 and 460 cells. In L8 there are five subclones of about 100 cells. In Table I we show clinical data for the number of circulating tumor DNA (ctDNA) fragments harboring mutations in five genes associated with resistance to anti-EGFR treatment in IS colorectal cancer patients who developed more than one mutation in those genes [291. These mutations were not detectable in patients' serum prior to therapy. but became detectable during the course of anti-EGFR treat- ment. The number of ctDNA fragments correlates with the number of tumor cells harboring that mutation - it was previously estimated (using the tumor burdens and pre-treatment ctDNA levels measured in patients who had KRAS mutations in their tumors before therapy) that one mutant DNA fragment per ml of serum corresponds to 44 million mutant cells in the patient's tumor 1101. Thus the ratios of the resistant clone sizes can be obtained from the ratios of the numbers of ctDNA fragments harboring resistance-associated mutations. These data provide a unique opportunity to test our theory and compare the relative sizes of resistant clones inferred from the data with those predicted using our model. Assuming that resistance-associated mu- tations with higher ctDNA counts appeared prior to those with lower ctDNA counts, we find excellent agreement between the data and our model predictions. For example. the median ratio of the sizes of the first two resistant clones inferred from clinical data [29] is 2.21. while our model predicts 2.51. The median ratio of the sizes of the first and third clones from clinical data is 4.3 and our model predicts 4.12 (Ta- ble 1). Note that this comparison is parameter-free. as we showed that the ratio of resistant clone sizes is independent of parameters. Our mathematical results describe the relative sizes of resistant clones ordered by age. while the experimental data in Table I are or- dered by size, which serves as a proxy for age. because exact clonal age is unknown. We quantify the extent to which this difference in clonal ordering by size versus age influences our statistics using ex- act computer simulations (Table I). In the relevant parameter regime of large lesion size, Af, and small mutation rate. u. with Mu > 1, the results are largely independent of parameters (median ratios of clone sizes vary by < 10% for different parameter combinations). We show simulation results for median ratios of clone sizes when clones are ordered by size for typical parameter values (from Ref. 9). 2 I vmetprias.orgicgitbVt0.1073/pnes.0709640104 Foottine Author EFTA01199738 As we see in Table I. the ordering of experimental data by size does not significantly change the results of our analysis. We can generalize our approach to the case when resistance mu- tations are not neutral, but provide a fitness effect already before treatment (formulas shown in Supplementary Information). In Ta- ble 2 we compare the predicted medians for the first five resistant clones in a metastatic lesion containing Al = 109 cells when resis- tance is deleterious, neutral or advantageous. We see from Table 2 that even if resistant cells are only 10% as fit as sensitive cells, they will still be present in typical lesions. The average number of resis- tant cells produced until the lesion reaches size Al is AI Ws. Here s = 1 — di!, is the survival probability of sensitive cells, which is the probability that the lineage of a single sensitive cell will not die out. For typical parameter values (i.e. those used in Table 2) the number of resistant cells produced by sensitive cells in a single le- sion is •••••• 150. Resistant cells that are 10% as fit as sensitive cells have a survival probability of 4%: so on average 6 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 produced by billion(s) of sensitive cells in a lesion and the specific properties of the branching process, namely the independence of lineages. Discussion In this paper we describe the heterogeneity of mutations providing resistance to cancer therapy that can be found in any one metastatic lesion. Our results can be generalized to take into account all of the patient's lesions, assuming that they evolve according to the same branching process and that the number of lesions is much smaller than 1/u. In that case, the probability distribution for the size of the k-th appearing resistant clone in the patient's cancer is given by formula II ] if we let Al be the number of cancer cells in all of the patient's lesions. All our results generalize similarly. While the mean and median clone sizes in our model depend on the parameters of the process. their ratios are generally parameter- free. The universality of the clone ratio statistics follows from the fact that the skeleton of our branching process. which includes only cells with infinite line of descent, can be approximated by a Yule (pure birth) process 1301. It has been shown that in the limit of large lesion size M and small mutation rate u. the statistics of the relevant clones in a branching process with death remain approximately Yule 131]. Similarly. it can be shown that in the Yule process. in the above limits, the mean size of the k-th largest clone is •••-• A/ u/ (k — 1) and the ratio of the mean sizes of the k-th and j-th largest clones is (j — 1)/(k — 1) 131. 32]. This is exactly the result we obtain for the ratio of mean clone sizes even though we order clones by age. A few recent investigations studied the dynamics of single clones resistant to therapy 128, 33]. In one of the studies 133]. the authors 1. Vogelsbein 8. el at (2013) Cancer genome landscapes. Science 339:1546-1558. 2- Strayer* C (2009) Targeted cancer therapy. Nature 432:294297. 3. 'acne, F. el at (2005) Dynamics of chronic myeloid leukaemia. Nature 035 (7046): 1267.1270. 4. 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Nature 486:537440. employed a generalization of the Luria-Delbruck model in which sen- sitive cells grow deterministically and calculated the number of indi- vidual resistant clones and the probability distribution for the num- ber of cells in a single resistant clone after time 1. In another study 1281. mathematical modeling along with in vitro growth rates of cells harboring 12 point mutations providing resistance to BCR-ABL in- hibitor imatinib were used to calculate the number of resistant clones and the expected number of resistant cells with a particular resis- tance mutation at the time of diagnosis of chronic myeloid leukemia (CML). The authors found that at most one resistant clone is ex- pected to be present. as the total number of CML stem cells at di- agnosis is estimated to be approximately hi •-•.• 100.000 cells and is much smaller than the billions of cells typically present in a single detectable lesion of a solid tumor. In this paper we break new ground by using a different mathematical technique and the novel approach of ordering the resistant clones according to their time of appearance, which allows us for the first time to describe the full spectrum of resistance mutations present in a lesion. Our study is challenging the conventional view of the evolution of resistance in cancer. For every therapy that is opposed by mul- tiple potential resistance mutations, which is the case for every tar- geted drug developed so far, we can expect multiple resistant clones of comparable size in every lesion. Our theory provides a precise quantification of the relative sizes of those resistant subclones. The heterogeneity of resistance mutations is further amplified when tak- ing into account multiple metastatic lesions in a patient. This infor- mation is pertinent to the development of second line treatments that aim to inhibit known resistance mutations. Materials and Methods Model. We model the growth and evolution of a metastatic lesion as a contin- uous lime multitype branching process I34I. The growth of a lesion is initiated by a single cell sensitive to the drug. Sensitive cells produce a resistant cell at each division with probability u and each resistant cell produced by sensitive cells starts a new resistant type. Analysis. In our analysis we use the approximation that resistant cells produced by sensitive cells appear as a Poisson process on the number of sensitive cells 0 7j. For more detais and derivations of our results please see Supplementary Information. Simulations. We perform Monte Carlo simulations of the multdype branching process using the Gillespie algorithm [34 Between 5000 and 10000 surviving runs are used for each parameter combination. ACKNOWLEDGMENTS. We thank Berl Vogeistein for critical reading of the manuscript and Rick Durrett for discussion cawing the conception of this work. We are grateful for the support from the Foundational Cuestions in Evolutionary Biology Grant IRFP-12-17 and the John Templeton Foundation. 11. Pao W. et at (2005) Acquired resistance of lung adenocarcinomes to gelitinib or edolinib is associated with a second mutation in the EOFR klnase domain. PLOS Med 2:673. 12. Antonescu CR. of at (2005) Acquired resistance to Waling in gastrointestinal *no- m.' tumor occurs through secondarygene mutation. CM Cancer Res 11:4182.4190. 13. O'Hare T. et at (2007) Bcr.Abl kinase domain mutations, drug resistance. and the road to a cure for chronic myeloid leukemia. Blood 1102242.2249. 14. kabala S. el at (2012) Emergence of KRAS mutations and acquired resistance to enti.EGFR therapy in colorectal cancer. Nature486:532-536. 15. Bozic 1. et al. (2013) Evolutionary dynamics al cancer In response to targeted com- bination therapy. °We 2s)00747. http:,',0x.dolorgi0.75541eLlle.00797 IS. 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Winne Med 6(224):22ara24. DOI: 10.11265cl. transtmed.3007094 30. O'Connell N (1993) Yule Process Approximation for the Skeleton of a Branching Process. J Apia Prob 30:725.729. 31. Manrubia SC. Lanett, OH (2002) At the Boundary between Biological and Cultural Evolution: The Origin of Surname Distributions. A Thor 8/01 215:451-477. 32. Maruvka YE. Shnerb NM. Kessler DA (2010) Universal features of surname distribu- tion in a subsample of a growing population../. Theor. Slot 262:2457256. 33. Dewan)IA. Luebeck EG,IAooMavkar SH (2005)A generalized Lurla-Delbitlek model. Math Mosel 197:140-152. 34. Athreya KB. Hey PE (1972) Branching Processes (Springer•Yerlag. Berlin). 35. Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem91:234041. 4 I www.prias.org/cgitbitt0.1073/pnes.0709840104 Footline Author EFTA01199740 Figure Legends Fig. I. Evolution of resistance in a metastatic lesion. (A) As the lesion (green) grows from one cell to detectable size, new resistant subclones appear. Some of them are lost to stochastic drift (yellow and pink), while others survive (purple. red and orange triangle). Instead of looking at the time of appearance of new clones, our approach takes into account the total size of the lesion when the resistance mutation first occurred. (B) Agreement between computer simulations and formula 111 for the cumulative distribution 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 100 cells with probability 0.34. between 100 and 1000 cells with probability 0.47 and more than 1000 cells with probability 0.13. The second subclone contains more than 100 cells with probability 036. Parameters b = 0.25, d = 0.181, M = 109, u = 42 • 10-9. Fig. 2. Resistant subclones in metastatic lesions. Different realizations of the same stochastic process are shown in each panel. (A) Six lesions of size le and (B) six lesions of size 109 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 • 10-9. Footline Author PNAS I Issue Date I Volume I Issue Number I S EFTA01199741 A B 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Cumulative distribution 10 100 1000 Clone size 10000 100000 EFTA01199742 A Lesion size M=108 cells <a moo Ll woo L2 moo L3 two L4 moo "tri loo too 100 loo 100 10 1 10 10 10 10 § , 1 111i.. . . z 1 2 3 4 5 6 7 8 910 1 2 3 4 5 6 7 8 910 1 2 3 4 5 6 7 8 910 1 2 3 4 5 6 7 8 910 1 B Lesion size M=109 cells L7 ti woo I moo 14) 100 100 io III '1II10 L8 1000 100 10 II 1 L9 moo 100 LIO 1000 100 L5 1000 100 10 L6 2 3 4 5 6 7 8 910 1 2 3 4 5 6 7 8 910 LI I 1000 100 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 L12 1 2 3 4 5 6 7 8 9 10 Resistant clones (order of appearance) EFTA01199743 Table 1. Comparison of predicted ratios of resistant clone sizes and ratios obtained from clinical data. -'a;IE .il Yi. Y2 i ts Y.4 Y. /Y2 Yi/Yz 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 6 114 40 2.85 7 6760 4940 4100 3900 1.37 1.65 1.73 8 220 30 7.33 9 848 374 135 133 227 6.28 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 31 1.45 4.31 8.06 Median from patients 221 4.3 7.22 Predicted median 2.51 4.12 5.74 Predicted median (order by size) 2.05 3.63 5.25 'Number of cioula6ng tumor DNA (ctDNA) fragments per ml to 1'.0 harboring Efferent mutations associated with resistance to anti•EGFR agents in colorectal cancer patients treated with EGFR blockade (283. Ratio of resistant clone sizes is given by the ratio of the aDNA counts for any two resistance-associated mutations. We assumed that mutations with higher clDNA cants in the patient data appeared prior to 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. Al = 109, u = 42. 10-9). Foottine Author PNAS I Issue Date I Volume I Issue Number r 1 EFTA01199744 Table 2. Sizes of resistant clones when resistance is deleterious, neutral or advantageous. o = (bR - dR)/(b - d) 1st clone" 2nd clone 3rd clone 4th clone 5th clone 0.01 0 0 0 0 0 0.1 10 6 4 2 1 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 1.1 229 87 52 37 28 • Median number of cells m the first five successful resistant clones it a metastatic lesion with Af = 109 cells when resistant cells are less lit than sensitive cells (c < 1), neutral (e = I) and more lit than sensitive cets Ie > I). We fix the birth and death rate of sensitive cells. b = 0.25. d = 0.181 and the death rate of resistant cells dR = d. We vary the relative fitness of resistant cells. c. and let thebirth rate of resistant cells be bit = dR+c(b-d). Mutation rate u = 42.10-9. For e = 0.1 we report simulation results and for c > 0.1 we use brmula (613) from the SI: see SI for details. Footline Author PNAS I Issue Date I Volume I Issue Number I 1 EFTA01199745