A fundamental goal of research into carcinogenesis, with implications for cancer prevention, is to determine the order of mutations that occur in a neoplasm as it progresses from normal tissue to cancer. This has been explored in various cancer types, including breast cancer (3), lung cancer (4), and melanoma (5). In a canonical paper (6), the order of mutations for colorectal cancer was reconstructed from 172 colorectal tumor specimens (7). The specimens were classified according to tumor size and grade, and a small set of genetic alterations was characterized in these same samples. The order of mutations leading to colon cancer was constructed from these data by identifying the mutations whose frequency across tumors increased in conjunction with increases in the tumor size and grade (6). This analysis has been widely influential and used as a model for other cancer systems (4, 8, 9). Even at its conception, this model was not meant to provide a fixed roadmap for the accumulation of genetic alterations. In fact, in observations by the original authors of the linear genetic model, 2 of 7 tumors were inconsistent with the linear model, as determined through experiments in which the genetic alterations were identified in different stages within the same tumor. This was independently verified through experiments identifying the mutational state of colorectal tumors from over 100 patients (10). We analyzed the data from (10) and found that just 26.2% of the neoplasms had a mutational state consistent with the canonical genetic pathway inferred from cross-sectional data.

We begin by discussing two types of models of progression: the canonical path model and its extension, the tree model. A path model for a type of cancer is a linear sequence of mutations that must occur in order, beginning with wild-type (Fig. 1A and B). Because this is a path, each mutation has at most one following mutation, and a given mutation increases the chances of finding the next mutation. Oncogenetic tree models were developed to capture multiple independent mutational events, in effect enumerating several possible paths to cancer (11). Here, the mutational order can be a tree rather than a path (Fig. 1C and D). This is not an evolutionary model because the oncogenetic tree does not represent ancestral relationships within a neoplasm, but rather a summary of the observed co-occurrences of mutations across independent neoplasms. In this branching tree model, each mutation occurs just once in the tree, and can have multiple independently occurring following mutations. As before, the interpretation is that a given mutation increases the chances of obtaining the following mutation. A number of studies have constructed oncogenetic trees from cross-sectional data, including for renal cell carcinoma (12), endometrial adenocarcinoma (13), gastrointestinal stromal tumors (14), oral cancer (15), and colon cancer (16).

First, we characterized the simulated neoplasms that progressed to cancer. We found that the necessary and sufficient mutant phenotypes for progression to cancer were loss of differentiation, evasion of apoptosis, and sustained angiogenesis (Fig. S1). We observed that tumors are heterogeneous and are comprised of cells from multiple different clones, which agrees with recent experimental evidence (27, 28). There were 327 (mean, s.e.m. = 17) clones generated over the lifetime of a tumor. When the tumor was large enough to be detected, it was made up of 256 distinct clones (mean, s.e.m. = 17), the largest clone of which comprised 67% (mean, s.e.m. = 2%) of the neoplasm. The Shannon Index of the tumors was 1.7 (mean, s.e.m. = 0.1). The temporal order of phenotypic mutations in the neoplasms that progressed to cancer was different between tumors; there were 50 different temporal orders found in the 91 cancers (Table S2). This agrees with recent experimental evidence suggesting that the same types of tumors can have different sets of mutations (29, 30).

In order to characterize the actual, temporal order of mutations within a neoplasm, we extracted the temporal order of mutations for every cell that survived to become part of a cancer from the summary data. For our analyses, we focused on characterizing the ordering of mutant phenotypes. Of the 14,540^* possible temporal paths to cancer, 381 were observed in cancer and 26 were found in more than 1% of the cells. Sixty-four percent of the cells had one of these 26 evolutionary paths (Fig. 3). Loss of differentiation was by far the most common first step in progression and occurred in 72% of the cells. For the subsequent mutations, there is at best a weak tendency for one mutation to occur as opposed to another (Fig. S2).

Next, we repeated the experiments to obtain the cross-sectional order of mutations, as was done in colon cancer (6), using our data. First, we obtained the mutational state of the simulated neoplasms at increasing tumor grades that we approximated by the neoplasm size, defined as the number of cells comprising a neoplasm. To do so, we biopsied neoplasms that progressed to cancer as they crossed various size thresholds and genotyped them for the presence of mutant alleles in the majority of cells. The frequency-based ordering of mutations inferred from this analysis was loss of differentiation, evasion of apoptosis, limitless replicative potential, sustained angiogenesis, genomic instability, self-sufficiency in growth signals, and insensitivity to antigrowth signals (Fig. 4A and B).

In order to quantify how well the temporal order matched the path order, we determined the percentage of tumor cells when cancer was detected whose evolutionary order over time matched the path order obtained from cross-sectional data. We found that most simulations had few cells in their tumors with consistent temporal and cross-sectional path orders (mean 7.3%, s.e.m. 1.0%; Fig. 4C). In fact, 41% of tumors had no cells with consistent orders and 68% had at most 1% of cells with consistent order. Thus, we saw that the order inferred from cross-sectional data was inconsistent with the temporal order of mutations for most cells. This is true across a range of parameter, or fitness, values for the cancer phenotypes, including telomere length, neoplasm size, mutation rate, and the probability of cell division. The sensitivity analyses are summarized in Table S3.

Then, we examined whether cells’ temporal orders were consistent with the order reconstructed from an oncogenetic tree model. We obtained the oncogenetic branching tree model from our cross-sectional cancer data (Fig. S3) and compared it to the temporal order from individual cells in cancers. We found that more temporal paths were consistent with the oncogenetic tree models than the simple path model, but most tumors still had relatively low fractions of temporal paths consistent with the oncogenetic tree (mean 11.5%; s.e.m. 2.2%; Fig. S3). Like for the path model comparison, 41% of tumors have no cells with consistent temporal orders to the reconstructed oncogenetic tree. However, the cells comprising the rest of the tumors are more consistent with the oncogenetic tree order; 54% of tumors have more than 1% of cells with consistent temporal orders. Like path models, these models do not fully specify the mutational spectrum observed in experimental cross-sectional studies of neoplasms. In one study in colon cancer, 11% of the neoplasms had combinations of mutations that were inconsistent with the oncogenetic tree identified (16). Of course, characteristics of neoplasms like mutation rate, number of cells, cell motility, and the relative fitness of new mutations affects the probability that a clone can reach fixation. In turn, this affects the percent of clones with consistent temporal and cross-sectional mutational orders.

Path models implicitly assume that neoplasms pass through a series of selective sweeps, each of which homogenizes a tumor’s genotype. With our model, we have shown that there are many possible temporal paths to cancer and that, at detection, neoplasms are comprised of many clones. This heterogeneity has been observed experimentally (35–37). Exactly how much heterogeneity exists depends on the evolutionary parameters of the tumor, including the mutation rate, fitness advantage of the new mutations, and the level of cell motility within the neoplasm. Cancer cells are an evolving population of asexually-reproducing cells, and the rate and dynamics of adaptation in asexually-evolving populations has been studied experimentally and theoretically in evolutionary biology (reviewed in (2)). Homogenous and linear clonal evolution occurs when each clone outcompetes others and reaches fixation before the next mutation occurs. This tends to occur when mutations are infrequent, have strong selective advantages, and there is a high level of cell turnover in the neoplasm. In this case, the clear temporal path to cancer is simply the order of clonal selective sweeps and heterogeneity is likely to decrease as the neoplasm homogenizes, especially with low mutation rates. Clonal interference tends to occur when new mutant clones with relatively high fitness advantages occur frequently enough to interfere with fixation of other clones. Because there are many competing clones, determining the single temporal path to cancer for a neoplasm from its heterogeneous subclones becomes more complicated. Clonal interference, and the heterogeneity that accompanies it, can occur when mutation rates are high but the fitness effects of new mutations are small. Another condition generating exceptionally high intra-tumor heterogeneity occurs when mutation rates are high and there is geographical isolation of subpopulations of cells. It is currently unknown which of these clonal evolution dynamics are found in real tumors, though there is evidence for high mutation rates and small fitness effects in colon cancer (38), and evidence for small fitness effects in pancreatic cancer and glioblastoma multiforme (39). We observed dynamics consistent with both clonal evolution and clonal interference in our simulations, even within the same neoplasm (Fig. 5C). For example, we observed that heterogeneity decreased when a clone that acquired the loss of differentiation mutation fixed in the population, as might occur during a clonal selective sweep (Fig. 5B). Subsequently, heterogeneity increased as clonal interference dominated the rest of the neoplastic dynamics. In general, heterogeneity increased over the course of progression in the simulations, with occasional drops that were immediately followed by increases. Thus, one of the explanations for the low percent of clones within a tumor with matching temporal and path orders is that there is not a single evolutionary path for tumor. Instead, there are many clones, each of which can independently acquire genetic alterations as has been suggested in Barrett’s esophagus (21).

Another reason the temporal order does not match the path order in cross-sectional studies is due to the detection of transient clones. Transient clones increase in size early in progression only to go extinct which may occur if a clone is outcompeted by another clone or if it fails to stabilize its telomeres. We observed both events (Fig. 5). For example, we observed a clone that acquired the insensitivity to antigrowth signals mutation early on which allowed the clone to expand to a detectable size. Then, a loss of differentiation mutation occurred independently in a wild-type cell. This new clone quickly expanded and drove the original clone extinct. Later in progression of this same neoplasm, a large clone eventually went extinct due to failure to stabilize its telomeres.

There is a further problem specifically with the construction of path models. Building path models requires the characterization of several different neoplasms at different stages. The stages are then ordered according to increasing size and grade, which is assumed to correspond to a single, linear order of changes during progression to cancer. This is how the in the path model in Fig. 4 was constructed. It may be an obvious point, but by basing the path model on mutations associated with increasing size and grade, we are identifying those mutations involved in increasing the neoplasm’s size and grade. That these mutations are involved in progression to cancer is an assumption (40). Thus, if histological grading does not reflect the necessary temporal sequence during progression, then studies based on that ordering will of course be invalid (41).

Previous work has identified other concerns with the cross-sectional approach. Using a probabilistic framework, Szabo and Yokovlev (42) showed that there are technical limitations in inferring the ordering of genetic events from frequency and correlation data, regardless if the cross-sectional order obtained was a path or a tree. In particular, small sample sizes, inherent undercounting of mutations associated with early tumor grades, and current methods that assume that the mutations are independent are problematic.

Oncogenetic tree models (11) accommodated more of the heterogeneity between tumors than path models because they do not impose a strict order on every mutation in a tumor. They also relax the assumption that the mutations that lead to neoplasms of increasing size and grade are the same mutations that lead to cancer. However, we have shown that even the oncogenetic tree order of mutations does not match the true evolutionary path. Oncogenetic tree models have already been extended. Distance-based methods have also been used to reconstruct oncogenetic tree models (43) and conjunctive Bayesian network models have used directed acyclic graphs to represent mutation ordering (44) and have been applied to cross-sectional data (45). These models still suffer from the weakness of cross-sectional data. Recently, a computational approach was developed to identify the most likely paths through a mutational network for colorectal cancer and glioblastoma using cross-sectional data; the authors found that not all evolutionary paths are accounted for in the mutational networks, perhaps due to heterogeneity of temporal orders within cancer types (46).

While understanding the dynamics of mutation accumulation has important implications for cancer prevention and risk stratification, it is difficult to reconstruct temporal order from cross-sectional data. A fundamental problem in the use of cross-sectional data to infer the temporal order of events is the assumption that the state of one tumor is informative for the history of a tumor in a different patient. Both our model and recent cancer resequencing efforts (30, 47, 48) show that there are likely many possible evolutionary paths to cancer, not just between types of cancers, but even within a given type of cancer. Each tumor is a unique evolutionary trajectory with occasional necessary and sufficient phenotypic mutations that can be acquired differently in different tissues. Further, tumors are populations of heterogeneous clones, each of which is evolving along a distinct path. Thus, identifying a single path or oncogenetic tree of mutational events is insufficient to describe this process. Reconstructing cell lineages within individual tumors should reveal the true temporal order of events for the different clones within a tumor (49).

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