EFTA01199737.pdf

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 genetic alterations (usually point mutations) in the drug target itself human cancer. The Immediate effect is a dramatic reduction in or in other genes [10. 11. 12. 13. 14 disease burden, but in most cases the tumor returns as a con- Recently. mathematical modeling and clinical data were used to sequence of resistance. Various mechanisms for the evolution show that acquired resistance to an EGFR inhibitor panitumumab in of resistance have been implicated including mutation of target metastatic colorectal cancer patients is a fait accompli. since typi- genes and activation of other drivers. There is increasing evi- dence that the reason for failure of many targeted treatments is a cal detectable metastatic lesions are expected to contain hundreds of small preexisting subpopulation of resistant cells; however, little cells resistant to the drug before the start of treatment [10]. These is known about the genetic composition of this resistant subpopu- cells would then expand during treatment, repopulate the tumor and lation. Using a novel approach of ordering the resistant subclones cause treatment failure. Similar conclusions should hold for targeted according to their time of appearance, here we describe the full treatments of other solid cancers [15]. Successful treatment requires spectrum of resistance mutations present in a metastatic lesion. drugs that are effective against the pre-existing resistant subpopula- We calculate the expected and median number of cells in each tion and must take into account the (possible) heterogeneity of re- resistant subclone. Surprisingly, the ratio of the medians of suc- sistance mutations present in the patient's lesions. In this article we cessive resistant clones Is independent of any parameter in our use mathematical modeling to investigate the heterogeneity of drug- model; for example, the median of the second clone divided by resistant mutations in patients with metastatic cancers. the median of the first is — I. We find that most radiograph- ically detectable lesions harbor at least ten resistant subclones. First mathematical investigations of the evolution of resistance to Our predictions are in agreement with clinical data on the rela- cancer therapy were concerned with calculating the probability that tive sizes of resistant subelones obtained from liquid biopsies of cells resistant to chemotherapy are present in a tumor of a certain size colorectal cancer patients treated with EGFR blockade. Our the- [16]. Later studies expanded these results to include the effects of a ory quantifies the genetic heterogeneity of resistance that exists fitness advantage or disadvantage provided by resistance mutations prior to treatment and provides information to design treatment [IL IS]. multiple mutations needed to achieve resistance to several strategies that aim to control resistance. drugs [15. 19. 20. 21] and density limitations caused by geometric constraints [22]. These studies employed generalizations of the fa- cancer I drug resistance I heterogeneity I math/mask:aft/cagy 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 Significance size in the fully stochastic formulation of the Luria-Delbriick model Metastatic dissemination to surgically inaccessible sites is the major was recently calculated in the large population size limit [24. 25]. cause of death in cancer patients. Targeted therapies, often initially The focus of above studies was describing the total number of all re- effective against metastatic disease. invariably fail due to resistance. sistant cells. rather than the composition of the resistant population We use mathematical modeling to study heterogeneity of resistance [26]. 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 Results subclones of comparable size and compare our predictions to clinical We model the growth of a metastatic lesion as a branching process data on resistance-associated mutations in colorectal cancer patients. [27] that stars from a single cell (the founder cell of the metastasis) Our model provides important information for the development of which is sensitive to treatment. Sensitive cells divide with rate b and second line treatments that aim to inhibit known resistance mutations. 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- 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 tion with probability u. Resistant mutations can be neutral in the ab- metastatic disease must be treated with systemic therapies [I]. Tra- Reserved for Publication Footnotes 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 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 resistant subclones in a metastatic lesion containing Al = 109 cells. rates as sensitive cells, and we initially focus on this case. We also The mean numbers of cells in the first, second and third appearing expand our theory to the more general case where resistant cells are resistant clone are E(Yr) x 2237. E(Y2) x 152 and E(Ya) x 76. non-neutral, which means they have birth and death rates bn and dn. However, the mean for Yi • the size of the first resistant clone. is heav- respectively. If c = (bn— dR)I(b —(1) > 1 then resistance mutations ily influenced by the realizations of the stochastic process in which are advantageous prior to treatment if c < 1 they are deleterious. the first resistance mutation appeared early and is not a good sum- A resistant cell may appear in the population and be lost due to mary of the probability distribution for Y1. Namely. the realizations stochastic drift or it can establish a resistant subclone. We number in which the number of cells in the first clone is greater than the mean the resistant subclones that survive stochastic drift by the order of ap- (2237) account for less than 7% of all cases. The median number of pearance (Fig. IA). A reasonable assumption for the number of point cells in the first resistant clone (Med(Yi)) for the above parameters mutations that can provide resistance to a targeted drug is on the or- is 152, while the medians for Y2 and Y3 are 63 and 40. respectively. der of one hundred [10. 28]. Thus, the different resistant subclones In Supplementary Information we calculate the probability dis- will typically contain different resistance mutations, especially if we tribution for the ratio of resistant clone sizes Y1/ Yk and show that only focus on the largest ones. it is also independent of the parameters of the process. Even though We calculate the number and sizes of resistant subclones in a the first appearing clone is expected to be the largest. followed by the metastatic lesion containing Al cells. Typical radiographically de- second clone and so on, we show that this ordering is often violated. tectable lesions are 1 cm in diameter and contain 109 cells. In 31% of lesions the first successful subclone is smaller than the sec- The mutation rate. u. leading to resistance is the product of the point ond one: on the other hand. in 24% of lesions the first subclone is at mutation rate p. which is on the order of ti 10-9 per base pair per least 10 times larger than the second one. cell division, and the number of point mutations that can confer re- Fig. 2 shows different realizations of the stochastic process of sistance, which is ••••• 100. In our analysis we will assume a large Al evolution of resistance in metastatic lesions containing 108 and 109 and small u limit and mostly focus on the case when Mu > 1. cancer cells. The same parameters were used to generate all lesions. Tumor sizes at which successful resistant mutations are produced The size of each subclone is shown (in number of cells), and the can be viewed as a Poisson process on 10. MI with rate u (see Sup- subclones are ordered by their time of appearance. In lesion LI the plementary Information) [17. 10]. The number of successful mutant first three subclones are the largest and each have around 100 cells. lineages is thus Poisson-distributed with mean A = Mu. If Afk is the Lesion L5 contains only two subclones. while L6 contains seven sub- number of cancer cells in the lesion when the k-th mutant appeared. clones but none has more than 10 cells. In each lesion of total size which survived stochastic drift (Fig. IA). then .44 — Afk is expo- 109 cells there are more than 10 resistant subclones. In L7 the two nentially distributed with mean 1/u. Therefore, we expect that the largest subclones contain 1500 and 460 cells. In L8 there are five k-th clone appeared when the total population size was .41k k1u subclones of about 100 cells. and that roughly. the size of the first clone is k times the size of the In Table I we show clinical data for the number of circulating k-th clone. The probability that exactly k clones are present in the tumor DNA (ctDNA) fragments harboring mutations in five genes population of size Al is Ake- A/k!. associated with resistance to anti-EGFR treatment in IS colorectal Counting new successful resistant clones in the order of appear- cancer patients who developed more than one mutation in those genes ance, we calculate the probability distribution for the number of cells [291. These mutations were not detectable in patients' serum prior to in the k-th resistant clone. In particular, if k Mu the cumulative therapy. but became detectable during the course of anti-EGFR treat- distribution function for the number of resistant cells in the k-th clone ment. The number of ctDNA fragments correlates with the number simplifies to of tumor cells harboring that mutation - it was previously estimated (using the tumor burdens and pre-treatment ctDNA levels measured \ o, Afu in patients who had KRAS mutations in their tumors before therapy) 14(0 1 ( Mu + y — dylb ) " Ill 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 The excellent agreement between formula [11 and exact computer resistant clone sizes can be obtained from the ratios of the numbers of simulations of the stochastic process is shown in Fig. I B. ctDNA fragments harboring resistance-associated mutations. These The mean number of cells in the k-th resistant clone is E(Yr) tt data provide a unique opportunity to test our theory and compare the EbA/u/rillog(r/bu) — 1] and E(Yk) x bA/ ul(r(k — 1)1 fork ≥ 2. relative sizes of resistant clones inferred from the data with those The median for the number of cells in the k-th subclone is given by 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 Interestingly, the ratio of the means of the two subclones k and j is our model predicts 2.51. The median ratio of the sizes of the first and (j — 1)/(k — 1) for k. j > 1. The ratio of their medians is 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 Med(Yk) 21Th — 1 131 that the ratio of resistant clone sizes is independent of parameters. Mecl(y,) 21/3 — 1 Our mathematical results describe the relative sizes of resistant Note that the these ratios are independent of any parameters of the 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 process. In particular, the ratio of the medians of the first and second age is unknown. We quantify the extent to which this difference in clone is 1,./2 — 1, which implies that they have comparable size (same clonal ordering by size versus age influences our statistics using ex- order of magnitude). act computer simulations (Table I). In the relevant parameter regime Liquid biopsy data were used to obtain estimates for the birth of large lesion size, Af, and small mutation rate. u. with Mu > 1, and death rates of cells in metastatic lesions and the number of the results are largely independent of parameters (median ratios of point mutations providing resistance to the EGFR inhibitor panitu- clone sizes vary by < 10% for different parameter combinations). mumab in colorectal cancer [10]. The resulting parameter values We show simulation results for median ratios of clone sizes when (b = 0.25,d = 0.181 per day. point mutation rate p = 10-9 per clones are ordered by size for typical parameter values (from Ref. 9). base pair per replication and 42 point mutations conferring resis- tance) can be used to calculate the mean and median sizes of the 2 I vmetprias.orgicgitbVt0.1073/pnes.0709640104 Foottine Author EFTA01199738 As we see in Table I. the ordering of experimental data by size does employed a generalization of the Luria-Delbruck model in which sen- not significantly change the results of our analysis. sitive cells grow deterministically and calculated the number of indi- We can generalize our approach to the case when resistance mu- vidual resistant clones and the probability distribution for the num- tations are not neutral, but provide a fitness effect already before ber of cells in a single resistant clone after time 1. In another study treatment (formulas shown in Supplementary Information). In Ta- 1281. mathematical modeling along with in vitro growth rates of cells ble 2 we compare the predicted medians for the first five resistant harboring 12 point mutations providing resistance to BCR-ABL in- clones in a metastatic lesion containing Al = 109 cells when resis- hibitor imatinib were used to calculate the number of resistant clones tance is deleterious, neutral or advantageous. We see from Table 2 and the expected number of resistant cells with a particular resis- that even if resistant cells are only 10% as fit as sensitive cells, they tance mutation at the time of diagnosis of chronic myeloid leukemia will still be present in typical lesions. The average number of resis- (CML). The authors found that at most one resistant clone is ex- tant cells produced until the lesion reaches size Al is AIWs. Here pected to be present. as the total number of CML stem cells at di- s = 1 — di!, is the survival probability of sensitive cells, which is agnosis is estimated to be approximately hi •-•.• 100.000 cells and is the probability that the lineage of a single sensitive cell will not die much smaller than the billions of cells typically present in a single out. For typical parameter values (i.e. those used in Table 2) the detectable lesion of a solid tumor. In this paper we break new ground number of resistant cells produced by sensitive cells in a single le- by using a different mathematical technique and the novel approach sion is •••••• 150. Resistant cells that are 10% as fit as sensitive cells of ordering the resistant clones according to their time of appearance, have a survival probability of 4%: so on average 6 of them will which allows us for the first time to describe the full spectrum of form surviving clones. The effect that mutations can cause treatment resistance mutations present in a lesion. failure although they have high fitness cost is a consequence of the Our study is challenging the conventional view of the evolution high number of resistant mutants produced by billion(s) of sensitive of resistance in cancer. For every therapy that is opposed by mul- cells in a lesion and the specific properties of the branching process, tiple potential resistance mutations, which is the case for every tar- namely the independence of lineages. 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 Discussion heterogeneity of resistance mutations is further amplified when tak- In this paper we describe the heterogeneity of mutations providing ing into account multiple metastatic lesions in a patient. This infor- resistance to cancer therapy that can be found in any one metastatic mation is pertinent to the development of second line treatments that lesion. Our results can be generalized to take into account all of the aim to inhibit known resistance mutations. patient's lesions, assuming that they evolve according to the same branching process and that the number of lesions is much smaller Materials and Methods than 1/u. In that case, the probability distribution for the size of Model. We model the growth and evolution of a metastatic lesion as a contin- the k-th appearing resistant clone in the patient's cancer is given by uous lime multitype branching process I34I. The growth of a lesion is initiated formula II ] if we let Al be the number of cancer cells in all of the by a single cell sensitive to the drug. Sensitive cells produce a resistant cell at patient's lesions. All our results generalize similarly. each division with probability u and each resistant cell produced by sensitive While the mean and median clone sizes in our model depend on cells starts a new resistant type. the parameters of the process. their ratios are generally parameter- free. The universality of the clone ratio statistics follows from the Analysis. In our analysis we use the approximation that resistant cells produced fact that the skeleton of our branching process. which includes only by sensitive cells appear as a Poisson process on the number of sensitive cells cells with infinite line of descent, can be approximated by a Yule 0 7j. For more detais and derivations of our results please see Supplementary Information. (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 Simulations. We perform Monte Carlo simulations of the multdype branching clones in a branching process with death remain approximately Yule process using the Gillespie algorithm [34 Between 5000 and 10000 surviving 131]. Similarly. it can be shown that in the Yule process. in the above runs are used for each parameter combination. 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 ACKNOWLEDGMENTS. We thank Berl Vogeistein for critical reading of the (j — 1)/(k — 1) 131. 32]. This is exactly the result we obtain for manuscript and Rick Durrett for discussion cawing the conception of this work. the ratio of mean clone sizes even though we order clones by age. We are grateful for the support from the Foundational Cuestions in Evolutionary Biology Grant IRFP-12-17 and the John Templeton Foundation. A few recent investigations studied the dynamics of single clones resistant to therapy 128, 33]. 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(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 Cumulative distribution 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 100 1000 10000 100000 Clone size EFTA01199742 A Lesion size M=108 cells <a moo Ll woo L2 moo L3 two L4 moo L5 1000 L6 "tri loo too 100 loo 100 100 10 1 10 10 10 10 10 z§ , 1 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 111i.. . . 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 B Lesion size M=109 cells L7 L8 L9 LI I 1000 ti woo I moo 1000 moo LIO 1000 L12 III '1II10 100 100 100 100 100 100 10 14) io 10 1 2 3 4 5 6 7 8 9 10 II 1 2 3 4 5 6 7 8 9 10 1 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 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