Fecal microbiota transplantation is being assessed as a treatment for chronic microbiota-related diseases such as ulcerative colitis. Results from an initial randomized trial suggest that remission rates depend on unobservable features of the fecal donors and observable features of the patients. We use mathematical modeling to assess the efficacy of pooling stools from different donors during multiple rounds of treatment. In the model, there are two types of patients and two types of donors, where the patient type is observable and the donor type (effective or not effective) is not observable. In the model, clinical outcomes from earlier rounds of treatment are used to estimate the current likelihood that each donor is effective, and then each patient in each round is treated by a pool of donors that are currently deemed to be the most effective. Relative to the no-pooling case, pools of size two or three significantly increase the proportion of patients in remission during the first several rounds of treatment. Although based on data from a single randomized trial, our modeling suggests that pooling of stools – via daily cycling of encapsulated stool from several different donors – may be beneficial in fecal microbiota transplantation for chronic microbiota-related diseases.

This unobservable heterogeneity suggests that remission rates might be improved by pooling several different donors’ stools into each patient; this could be achieved via an encapsulated stool protocol [ 3 ] with daily cycling of pills; e.g., if the pool size is two, pills from one donor are given on odd-numbered days and pills from a second donor are administered on even-numbered days. We use mathematical modeling and analysis to propose and assess an adaptive donor allocation policy that statistically infers which donors are effective and then allocates stools (in pool sizes up to five) from the best donors to each patient in each round of treatment.

Fecal microbiota transplantation (FMT), i.e., stool transplanted from a healthy donor that reconstitutes the normal microbiota community in the gut, is an effective treatment for Clostridium difficile infection, achieving a 90% cure rate in recurrent cases [ 1 ]. This treatment modality is now being considered for chronic and difficult-to-treat microbiota-associated diseases such as ulcerative colitis (UC). Although results from a placebo-controlled randomized trial of FMT for UC [ 2 ] show that FMT is only moderately effective (24% remission probability vs. 5% for placebo), they also reveal two interesting phenomena: remission probabilities are highly dependent on the fecal donor (one of five donors achieved seven of the nine remissions) and on the length of time that a patient has had UC (three of four patients with UC for <1 year experienced remission). While this patient heterogeneity is observable prior to treatment (i.e., via the time with UC), heterogeneity in the efficacy of donor material is not observable prior to treatment under the current state of scientific knowledge; in particular, it is not currently possible to predict whether a donor’s stools will lead to remission based on an analysis of 16s rRNA data or a more general metagenomic analysis.

Methods

We describe the mathematical model and then propose an algorithm that dynamically assigns patients to donor pools—or equivalently, allocates donor pools to patients—in a multi-round setting.

The Model We consider two versions of the model, and begin with the aspects of the model that are common to both versions. There are N 1 patients who have had UC for <1 year and N 2 patients who have had UC for >1 year; they are referred to as type 1 and type 2 patients. We consider T rounds of treatment indexed by t = 1, …, T, and let D t fecal donors be available in treatment round t; as explained later, new donors can be added during each round before allocating treatments. Each donor is either effective (referred to as type 1) or ineffective (type 0). The probability that a donor is effective is p, although a donor’s type is not observable and needs to be estimated from previous treatment results. Treatment results are binary, where patients are either in remission or not in remission after receiving treatment in any given round. In each round, each patient receives 14 days of treatment from a particular donor pool. We assume that treatment is persistent; i.e., the treatment outcome from a particular patient-donor pool pair is the same in all rounds of treatment. In particular, if a patient is in remission after treatment from a particular donor pool, then he remains in remission for as long as he receives treatment from this donor pool. Consequently, we assume that if a patient is in remission after treatment in round t from a particular donor pool, then that patient continues to receive treatment from this donor pool in rounds t + 1, …, T. Moreover, because a patient who is not in remission after treatment from a particular donor pool would not achieve remission for as long as treatment is continued with this donor pool, all patients who are not in remission after treatment in round t are reassigned to a different donor pool in round t + 1. Therefore, the proportion of patients in remission at the end of round t is nondecreasing in t. Turning to the effect of pooled donors, because it takes up to 14 days of treatment to observe whether remission is achieved, we assume that daily cycling of pills from several donors (e.g., ABABAB⋯ if there are two donors and ABCABC⋯ if there are three donors, where A, B and C represent different donors) achieves the same outcome as a pooled stool from these donors. While this assumption is likely to hold for small pool sizes (e.g., two or three), the point at which this assumption breaks down has yet to be assessed empirically. In other words, we implicitly assume that—regardless of pool size (although we only consider pools up to size five)—the probability of engraftment of specific microbes is not lowered due to pooling, and that once the microbes are engrafted, the actual equilibrium and associated clinical outcome would be the same. This assumption is consistent with Lotka-Volterra models of intestinal microbiota [4]-[6], where—after a perturbation due to FMT—the new microbiota equilibrium in the patient will be independent of the initial quantities of microbes from the donor pool beyond these microbes’ absence or presence. While it could take longer to attain the new equilibrium if the initial concentration of certain microbes is lower (i.e., due to pooling), the actual equilibrium—and hence clinical outcome—should be the same. Due to the paucity of data about FMT treatment for UC (and, in particular, the absence of data from pooled stools or from multi-round treatments), we consider two versions of the model that differ in their probabilistic assumptions about treatment outcomes, both within a pool of several donors and across rounds of treatment. In the optimistic (in the sense of achieving remission in more patients) scenario referred to as the independence version of the model, we assume that the treatment results from different donors for a particular patient are independent within a pool and independent across rounds. In the pessimistic scenario referred to as the dependence version of the model, we assume that once a patient receives a treatment outcome from a particular donor type, he experiences that same treatment outcome if he receives treatment from another donor of the same type, whether this other donor is in the same pool as the original donor (i.e., in the same round of treatment) or is in a subsequent round of treatment. Another assumption in the dependence version of the model is that the remissions achieved by type 0 donors are a subset of (i.e., subsumed by) the remissions achieved by type 1 donors; i.e., an ineffective donor is incapable of achieving remission in a patient that did not achieve remission from an effective donor, and hence a patient’s overall remission probability is independent of the number of previous times that he was unsuccessfully treated by an ineffective donor. One way to think about these two versions of the model is that each effective donor possesses an effective unobservable factor (e.g., a microorganism) that achieves a (relatively) high remission probability, and each ineffective donor possesses an ineffective factor that achieves a lower remission probability. In the independence version, there are many of these factors and each donor possesses a different factor (be it effective or ineffective), and the patient is lacking all of these (effective and ineffective) factors; in the dependence version, each effective donor possesses the same effective factor, each ineffective donor possesses the same ineffective factor, each patient is lacking both the effective factor and the ineffective factor, and the effective factor subsumes the ineffective factor. We now formulate and analyze the two versions of the model, culminating in the calculation of the posterior marginal probabilities that each donor is effective at the end of each round of treatment. We then present an allocation strategy that applies to both versions of the model.

The Independence Version of the Model In the absence of donor pooling, the remission probability when a donor of type i first treats a patient of type j is r ij for i = 0, 1 and j = 1, 2, regardless of when (i.e., which round) the treatment occurs. Hence, a patient can fail treatment from a donor in round t and then achieve remission from a different donor of the same type in a later round. If a type j donor, in round t, is treated by a pool comprised of a mix of ineffective donors and effective donors who have never treated this patient before, then the probability of remission is . Thus, the probability of remission after t rounds of treatment for a patient of type j is (1) The key to the analysis is to compute the posterior marginal probability that donor d is of type 1 (i.e., effective) conditioned on all of the treatment results in the first t rounds, which is denoted by ϵ d (t). To compute this posterior probability, we need to introduce additional notation. Let s be the pool size (i.e., the number of donors in a pool), let be the number of pools of size s that can be formed from the D t donors available in round t, index these pools by k = 1, …, K t , and let A k be the set of s donors in pool k. For each patient l, let be the set of all donors from which he has received unsuccessful treatment by the end of round t and be the set of donors who achieved remission in patient l by the end of round t. Note that is either empty or contains the donors in the pool that achieved remission in patient l. Furthermore, is empty for all values of t and l because a patient is never reassigned to a donor from whom he previously received unsuccessful treatment. For d = 1, …, D t , let donor d’s type E d equal i if donor d is of type i, for i = 0, 1. Given the donors’ types E 1 = e 1 , E 2 = e 2 , …, E D t = e D t , define (2) (3) (4) and (5) where e = (e 1 , e 2 , …, e D t ). In words, and are the number of effective and ineffective donors, respectively, who have unsuccessfully treated patient l by the end of round t, and and are the number of effective and ineffective donors, respectively, who have achieved remission in patient l by the end of round t. Note that or 0, depending upon whether or not patient l is in remission at the end of round t. The key to the analysis is the computation of the posterior joint distribution, , of each donor’s type at the end of each round, given all previous treatment results, , where E = (E 1 , E 2 , …, E D t ). The posterior distribution at the end of each round can be updated using only the sufficient statistics and in Eqs (2)–(5). For any integer N, define [N] = {1, 2, …, N}. Then for , we have (6) where Z is the normalization constant defined as (7) A key observation is that the updated posterior probability at the end of the previous round together with the treatment outcomes observed in the current round are sufficient to update the posterior probability at the end of the current round. Specifically, defining and (which are all observable after treatment in round t), and taking (8) we have, for any t = 1, 2, …, T, (9) where the normalization constant Z is (10) Given the posterior joint probability of donors types in Eqs (9) and (10), the posterior marginal probability of each donor d being of type 1 at the end of round t is (11)

The Dependence Version of the Model In the dependence version, we assume perfect temporal dependence in the treatment outcome. More specifically, if a type j patient is treated by a pool with at least one effective donor in round 1, then remission is achieved with probability r 1j . But if that patient does not achieve remission, then the patient never achieves remission in any future rounds, regardless of treatment. If a patient is treated by a pool with all ineffective donors in round 1, then remission is achieved with probability r 0j . If remission is not achieved, then the patient never achieves remission in subsequent rounds when treated by a pool with all ineffective donors, and if the patient is subsequently treated by a pool with at least one effective donor, then remission is achieved with probability , so that the patient’s overall remission probability is also r 1j . The first step in deriving the posterior marginal probability, ϵ d (t), that donor d is of type 1 at the end of round t, is to compute the likelihood of the data observed for patient l of type j through the end of round t, conditioned on the vector e of donor states. This likelihood, which is graphically depicted in Fig 1, is (12) where the probability r 1j − r 0j is the product of the initial failure 1 − r 0j and the subsequent conditional probability of remission, . Then, for any , the posterior joint probability of donor types at the end of round t is (13) where Z is the normalization constant defined as (14) Given the posterior joint probability of donors types in Eqs (13) and (14), the posterior marginal probability of each donor d being of type 1 can be computed as in Eq (11). PPT PowerPoint slide

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larger image TIFF original image Download: Fig 1. A graphical depiction of A graphical depiction of Eq (12) https://doi.org/10.1371/journal.pone.0163956.g001

The Allocation Algorithm For a given pool size s and for a given number of initial donors D 1 , we consider the following strategy, which applies to both versions of the model, for assigning patients who are not in remission to donor pools over T treatment rounds. Our approach is to allocate donor pools consisting of the donors who are most likely to be of type 1 (i.e., effective). Let be the number of pools of size s that can be formed from the D t donors available in round t. Step 1: Initially Allocate Patients to Donor Pools in a Balanced Manner: In round t = 1, assign patients of type 1 and patients of type 2 to each of the K 1 donor pools, where ⌊x⌋ is the largest integer less than or equal to x. The remaining type 1 patients are assigned to pools , respectively. If , then assign the remaining type 2 patients to pools , respectively; otherwise, assign them to pools and , respectively. Observe the treatment results in round 1 and let t = 2. Step 2: Compute Posterior Probabilities: Given the observations at the end of round t − 1, compute the posterior marginal probability ϵ d (t − 1) for each donor d = 1, 2, …, D t−1 using Eqs (8)–(10) for the independence version and Eqs (12)–(14) for the dependence version. Step 3: Add Naive Donors: Before allocating treatment, we consider adding naive (i.e., previously unused) donors. At the beginning of round t, identify each donor d for which the following happens for the first time: he has treated more than different patients and his posterior marginal probability is below p (i.e., ϵ d (t − 1) < p), where is a user-defined parameter. For each of these donors, add a naive donor to the system, form the new pools (combining the old and new donors) and update the posterior joint probability of each donor’s type at the end of round t − 1, which is denoted by P t−1 (e 1 , …, e D t−1 ) and derived in Eqs (8)–(10) and (12)–(14) for the two versions of the model, as follows. If u = D t − D t−1 naive donors are added at this step in round t, then (15) Also, for the newly added donors d, define (16) Furthermore, if there is a patient who has been treated by more than D t − s donors and is not in remission (i.e., this patient does not have access to s donors who have not treated him yet), we add as many naive donors as necessary for this patient to have a pool of s unexplored donors. After adding these naive donors, all the possible pools are formed and P t−1 (e) and ϵ d (t − 1) are again updated according to Eqs (15) and (16). Step 4: Reassign Patients who are not in Remission. For each patient who is not in remission and among those donors who have not yet treated him, assign the patient to the pool consisting of the s donors with the highest values of ϵ d (t − 1). Step 5: Iterate. Observe the outcome of the new treatments. Increase t by one and if t ≤ T, return to Step 2. Otherwise, stop.