M constraint are defined below: xijk = decision variable is 1 if patient

See Additional file 1 section 3 for more details.Review of catchment modelsGravity AcadesineMedChemExpress AICA Riboside models use the following general form to calculate an "attraction" measure for each patient i: ??Xm S j w d ij AG ???Xk ?? i j? Pi w d ij i? where Sj is the supply at provider j, Pi is the population at location i, w(dij) is the decay function based on distance of each patient-provider pair (i,j). In the second step of E2SFCA, the method defines the accessibility of each patient or community i based on the ratios at each provider and the zone weights: Ai ?XXrThe Institutional Review Board of the Georgia Institute of Technology approved the overall research project using data from the Cystic Fibrosis Founda.M constraint are defined below: xijk = decision variable is 1 if patient i chooses facility j for visit k, or 0 otherwise; Xn Xvp d ij ?j p? k? xpjk d iq ??Xn Xv p �q x ?1 ; q j; i; k k? pqk p? The equilibrium condition includes a separate constraint for each patient's visit and each location when there is no distance decay function. The left-hand side is the distance and congestion associated with current facility choice j for a visit k, and the right-hand side is the distance and congestion at any location other than j. See Additional file 1 section 3 for more details.Review of catchment modelsGravity models use the following general form to calculate an "attraction" measure for each patient i: ??Xm S j w d ij AG ???Xk ?? i j? Pi w d ij i? where Sj is the supply at provider j, Pi is the population at location i, w(dij) is the decay function based on distance of each patient-provider pair (i,j). The original 2SFCA method was introduced by Luo and Wang [7]; it allows the catchment title= jir.2011.0094 of each provider and patient to float based on the distances between each pair. E2SFCA is a variation that suggests applying different weights within travel time zones to account for decaying of the willingness to travel as distance increases [8]. Under the E2SFCA model, in the first step the "physician-to-population ratio" at each provider is calculated. Although the E2SFCA aims to estimate the number of patients that may potentially use a facility, it is easy to extend the metrics to estimate the number ofWith optimization models, many variations are possible, including through the addition of constraints, the use of different objective function values, or by differentiating decision variables by type. Here we describe a major variation in our model, optimization with user choice ("Decentralized"), and include many others title= fnins.2013.00251 such asLi et al. BMC Health Services Research (2015) 15:Page 4 ofvisits by replicating each patient using visits demanded (e.g., a patient demanding 10 visits can be viewed as 10 patients) [25, 26]. We make a minor adjustment to allow for each patient to have multiple visits to a provider, so we use physician-to-visits ratio instead. Thus we obtain: Rj ?X XrE2SFCA method. For the M2SFCA method, a similar calculation can be made, where the composite patientcoverage accessibility measure is AM ?congestion. iHuman subject study approvalSj V iW r;??ifdij