Optimal screening and donor management in a public stool bank

Background Fecal microbiota transplantation is an effective treatment for recurrent Clostridium difficile infection and is being investigated as a treatment for other microbiota-associated diseases. To facilitate these activities, an international public stool bank has been created, which screens donors and processes stools in a standardized manner. The goal of this research is to use mathematical modeling and analysis to optimize screening and donor management at the stool bank. Results Compared to the current policy of screening active donors every 60 days before releasing their quarantined stools for sale, costs can be reduced by 10.3 % by increasing the screening frequency to every 36 days. In addition, the stool production rate varies widely across donors, and using donor-specific screening, where higher producers are screened more frequently, also reduces costs, as does introducing an interim (i.e., between consecutive regular tests) stool test for just rotavirus and C. difficile. We also derive a donor release (i.e., into the system) policy that allows the supply to approximately match an exponentially increasing deterministic demand. Conclusions More frequent screening, interim screening for rotavirus and C. difficile, and donor-specific screening, where higher stool producers are screened more frequently, are all cost-reducing measures. If screening costs decrease in the future (e.g., as a result of bringing screening in house), a bottleneck for implementing some of these recommendations may be the reluctance of donors to undergo serum screening more frequently than monthly. Electronic supplementary material The online version of this article (doi:10.1186/s40168-015-0140-3) contains supplementary material, which is available to authorized users.


Parameter Estimation
The PMF of the stool production rate is derived in §1.1, and the test failure rates η and γ are derived in §1.2.

Stool Production Rate
For each of 30 donors, we compute the donor stool production rate in grams per day, which is the donor's total stool production in grams divided by the total time between the day the donor begins donating and the day the donor exits the system (or until the end of our data collection period if the donor has not exited). We find that the mean production rate is 87.2 grams per day and the coefficient of variation (standard deviation divided by the mean) is 0.74. We use n = 9 classes for the intervals (4.7 − 25. 3 In addition, we examine how the number of weekly visits by a donor varies over time. For donor i = 1, . . . , 30, let v iτ be the number of visits by donor i in week τ of the donor's donation period, and let T i be donor i's lifetime (in weeks) as a donor. The mean (averaged over the number of active donors at each point in time) visit rate in week τ is which is plotted vs. τ in Fig. 2

Test Failure Rates
The small amount of data precludes us from estimating the exact nature of the time-to-failure distribution. Hence, we assume that the time-to-failure has an exponential distribution, where η is the exponential rate associated with rotavirus and CDI, and γ is the exponential rate associated with the other 25 infectious agents.
Group 2: Donors who failed their first test because of the other 25 agents, which occurs with Group 3: Donors who passed their first test but failed their second test due to rotavirus or CDI, which occurs with probability Group 4: Donors who are still active after passing a single test, which occurs with probability 2 Letting i ∈ G j denote that donor i is in group j, we find that the negative log-likelihood associated with our dataset is The function L(η, γ) in (6) is convex in η and γ, and minimizing this function yields the maximum likelihood values, η = 0.0066/day and γ = 0.0040./day

Problem Analysis
The problem is formulated in §2.1 in the case of deterministic stationary demand (i.e., β = 0). In §2.2, we show that the optimal screening strategy in §2.1 also holds for the case of nonstationary demand (i.e., β > 0).

Stationary Demand
In the stationary demand case, β = 0 and the demand rate equals α for all t. Because demand is stationary, we drop the dependence on time t in this subsection. The analysis is performed in two steps: to derive the mean number of donors and the release rate in equilibrium, and then to derive the cost function.
We can think of each donation cycle for a class k donor as having two stages: the first stage consists of the d k days before the interim test, and the second stage consists of the D k − d k days between the interim test and the regular test. Referring to Fig. 1 in the main text, new donors of class k arrive at an average rate of rf k p 0 p s p b . After d k days, class k donors exit the first stage after interim testing, and they fail the interim test (and exit the system) with probability 1 − e −ηd k and move to the second stage with probability e −ηd k .
Upon entering the second stage, class k donors wait D k − d k days and then undergo regular 3 testing, and they fail the regular test (and exit the system) with probability 1 − e ηd k −(η+γ)D k and return to the first stage with probability e ηd k −(η+γ)D k . Let x k and y k be the mean number of class k donors in the first stage and second stage, respectively. Then the above reasoning leads to the system of ordinary differential equations, Setting the left sides of (7)-(8) to zero and solving yields With x k and y k in hand, we can compute the release rate r such that the rate at which salable stool is produced equals the demand rate α. Because s k D k grams of stool are released from quarantine and made available for sale whenever a class k donor passes a regular test, we have the equation Solving for r in (11) gives the input rate 4 Referring again to Fig. 1 in the main text, we can express the total cost per day by where, in the last term in (13), we assume that those failing the regular test will fail the stool test. Substituting (9), (10) and (12) into (13) gives The optimization problem is given by The cost function in (14) is convex in the domain of interest, and (14)-(16) can be solved via standard convex optimization methods such as projected gradient descent [2]. We round off the solution to the nearest integer to get the results stated in the main text.

Nonstationary Demand
Turning to the nonstationary case where the demand rate at time t is αe βt with β > 0, we divide the donation cycle into two stages as in §2.1, but we define x k (t) and y k (t) differently.
Let Taken together, it follows that the differential equations governing x k (t) and y k (t) arė where (19) ensures that the nonstationary demand is satisfied.
The above 2n + 1 equations can be solved to derive the 2n + 1 unknown functions (x 1 (t), . . . , x n (t), y 1 (t), . . . , y n (t), r(t)). We can represent and solve the above equations in the Laplace domain as follows. Let X k (z), Y k (z) and R(z) be the Laplace transform of x k (t), y k (t) and r(t), respectively. Then equations (17)-(19) are equivalent to Solving (20)-(21) for X k and Y k gives and substituting (24) into (22) yields Now that we have the solution (23)-(25), there are two remaining tasks: attempt to invert equation (25) to obtain the optimal release rate r(t), and derive the cost function and optimal inter-testing times; we begin with the latter task. The total cost rate at time t iŝ Note that the total cost rate in (26) grows exponentially with time. We re-express our objective as minimizing the total cost per gram of demanded stool, and let the time horizon go to infinity, i.e., LetĈ(z) be the Laplace transform ofĉ(t), which from (26) is given bŷ Substituting (23)-(25) into (28) givesĈ where Letting g(t) be the inverse Laplace transform of G(z), we re-express (29) in the time domain where * denotes the convolution operation. Substituting  Fig. 1: The PMF f k of the donor stool production rate (in grams/day).