An EM algorithm to improve the estimation of the probability of clonal relatedness of pairs of tumors in cancer patients

摘要

Many studies have been published over the past 20 years that involved examining pairs of tumors at the molecular level from a set of patients to determine if, for some patients, the tumors are clonal, i.e. one of the tumors is a metastasis of the other tumor. We focus in this article on the setting where the data comprise somatic mutations from a panel of genes. Various statistical methods have been proposed in the literature. One approach has been to characterize the evidence for clonality using an index of clonal relatedness (see [ ] and [ ]). However in constructing the index these authors have focused solely on mutations that are shared between the two tumors, ignoring the information from mutations that occur in one tumor but not the other, evidence that argues against clonal relatedness. Other authors have used the proportion of observed mutations that are shared as the index [ , ], while Bao et al. [ ] formalized this idea by assuming that the matched mutations follow a binomial distribution. All of these approaches analyze each case independently. To our knowledge, the approach we discuss in this article, improving upon Mauguen et al. [ ], is the only available method that models the data from all cases collectively to obtain parametric estimates of the proportion of cases in the population that are clonal. Also our method relies heavily on the recognition of the fact that the probabilities of occurrence of the observed mutations are crucially informative,especially for shared mutations. Motivated by a study of contralateral breast cancer that will be described in more detail in the next section, we developed a random-effects model to simultaneously analyze each case for clonal relatedness and to obtain an estimate of how frequently this occurs [ ]. The corresponding function mutation.rem has been added to the R package , originally described in Ostrovnaya et al. [ ]. Overall, the properties of this model were demonstrated to be quite good, in the sense that the parameter estimation has generally low bias except in small samples, where only a few cases from the population are available [ ]. Recently, in applying the model anecdotally, we noticed that in such small datasets, examples can arise where the maximum likelihood estimator of the proportion of clonal cases is zero, even when mutational matches have been observed in some cases. This tends to occur if the absolute number of cases with matches is small, either because the overall number of cases is small, or the proportion of cases that are clonal is small, or in clonal cases the proportion of mutations that are matches is small. This is problematic because it renders the probabilities of clonal relatedness to be exactly zero for all individual cases, an estimate that seems unreasonable, especially if matches on rare mutations have been observed. We thus became interested in alternate estimation methods. In this article we compare estimates obtained by the EM algorithm versus our first approach using a one-step estimate of the conditional likelihood.