An optimization framework for measuring spatial access over healthcare networks

Background Measurement of healthcare spatial access over a network involves accounting for demand, supply, and network structure. Popular approaches are based on floating catchment areas; however the methods can overestimate demand over the network and fail to capture cascading effects across the system. Methods Optimization is presented as a framework to measure spatial access. Questions related to when and why optimization should be used are addressed. The accuracy of the optimization models compared to the two-step floating catchment area method and its variations is analytically demonstrated, and a case study of specialty care for Cystic Fibrosis over the continental United States is used to compare these approaches. Results The optimization models capture a patient’s experience rather than their opportunities and avoid overestimating patient demand. They can also capture system effects due to change based on congestion. Furthermore, the optimization models provide more elements of access than traditional catchment methods. Conclusions Optimization models can incorporate user choice and other variations, and they can be useful towards targeting interventions to improve access. They can be easily adapted to measure access for different types of patients, over different provider types, or with capacity constraints in the network. Moreover, optimization models allow differences in access in rural and urban areas. Electronic supplementary material The online version of this article (doi:10.1186/s12913-015-0919-8) contains supplementary material, which is available to authorized users.


Parameter Selection and User Choice
In the optimization models, the congestion weight represents the trade-off between willingness to travel and willingness to wait, and it may be adapted to different applications. To identify a range of values that is reasonable for a given problem setting, we quantify several measures of model performance across the network. Below we describe such performance measures, provide the associated principles if applicable, and give calculation details for the CF case.
 Congestion Difference for Close Facilities: Congestion at facilities that are very close to each other should be similar. We quantify the absolute value of the difference of each pair of facilities within 50 miles of each other, and sum up the differences over the network.  Distance Difference (or Congestion Difference) for Close Patients: Cost experienced by patients who are very close to each other should be similar. We calculate the variance in distance or congestion across individual visits originating in the same county and sum the values over the network.  Variance in Distance (or Congestion) across Network: Heterogeneous networks usually have some disparities in costs experienced; however, very extreme values may not be reasonable. We quantify the mean distance traveled (or congestion experienced) for visits within a county, then we calculate the variance across the counties in the network.  Distance Greater than Shortest Distance: Distance traveled by patients should not be much greater than their shortest possible distances. Distance to closest facility is compared to average distance traveled by each patient.  Total Distance or Total Congestion: Calculated across a network by summing up the distance traveled or congestion experienced for each visit to a facility. These two measures are inversely related. Figure 8 shows the measures for the optimization models under different congestion weights, where the values are normalized [0,1] across results from both models. For patients whose visits are uncovered, we do not include those visits in the calculation of distance or congestion. Note that when the congestion factor is 0, the decentralized optimization is equivalent to the centralized optimization. In this case, the centralized optimization assignment reduces to finding the shortest distance between patients and hospitals. The far right corresponds to splitting congestion evenly among facilities. Thus, the total distance traveled increases with the congestion factor (although not by much), and the total congestion decreases significantly with the congestion factor.
The figure also shows that as the congestion weight increases, the variance of congestion across the network is decreasing, while the variance of distance across the network is increasing. For a very small congestion factor, distance is very important in the assignment to facilities, and thus facilities that are close to each other may have different levels of congestion. Using the principles above, there should be some differences in congestion and distance across the network, but not excessively large gaps, so we view congestion factors of around 10 as the most reasonable for this setting. The results for the centralized model with different congestion factors are also similar.

Other Variations on Optimization Model
Capacity: Some providers or facilities may have limited resources. This can be introduced by adding a capacity constraint to the basic model. Define = capacity for provider . The corresponding constraint is Unmet Demand: If resources in the network are limited, it may not be possible to meet all demand. In this case, the assignment constraint should be modified to ≤. In addition, to ensure that as much demand is met as possible, one can add constraints to ensure that for community , the minimum service level requirement is met, that is,  . [9] Alternatively, one can add a penalty to the objective function for all visits not assigned.
Willingness to Travel: If patients are located too far from providers, they may not be as willing to travel to that provider. In the basic optimization model, the cost to travel is linear with distance. By adjusting the distance values, one can make the cost to travel nonlinear with distance, which represents a patient's higher willingness to travel to close distances. Particular adjustments can be chosen to match the weights of zones as used in the catchment models.
Patient or Provider Types: Some providers choose not to accept Medicaid patients (or limit how many they will accept), which can reduce the spatial access for those patients.
One way to represent this in the model is by creating separate assignment variables for each patient type, and adding constraints to limit their assignment to providers with those preferences (33).This allows the optimization approach to incorporate the link between affordability and spatial accessibility.
On the demand side, patients may have preferences for providers with certain characteristics, e.g., children and their caregivers may desire providers focused on pediatric care. One way to incorporate this is to adjust the travel cost to be relatively lower for providers of the preferred characteristics. This example shows how the optimization model can incorporate acceptability (1) in the measurement of access. A similar approach (adjusting distances) can be used to capture differences in patient mobility, e.g., for families with automotive vehicles or not.
Objective Function: In Section 2.1 we describe a model with an objective function that has a particular congestion cost. Many other variations on congestion are possible, including linear with the number of visits at a facility, exponential with the number of visits, or others. More generally, many variations on the objective function are possible.
Interventions: Decision variables can be added to optimization models to represent whether or not a new facility should be located in a network at particular locations, whether or by how much to increase capacity, or other interventions. The interventions can be designed to optimize the overall system performance or to reduce the disparities among subpopulations.

Minimum cost network flow transformation for decentralized model
Decision Variables: = the percentage of time that patients in location visit facility = 1 if the ℎ visit is selected for facility Parameters: = distance between patient location and facility ( ) = decay function value for distance = demand of patient location = capacity at facility = , the cost of marginal congestion for the ℎ visit = congestion weight Model: Constraints: ≥ 0, ∀ = 1, … , and ∀ = 1, … , . 0 ≤ ≤ 1, ∀ = 1, … , and ∀ = 1, … , .

Analytical Results: Networks with Overlapping Service Areas
Result 4: Optimization models show higher accessibility in non-overlapping service areas.
It can be difficult to understand model differences across complex networks like we study in Section 4. Thus we analyze one more simulated system that can assist in making comparisons between the 2SFCA approaches and optimization models.
1. When the population density is homogenous over the network: Consider System I with two facilities, each with a population surrounding them in a circle of radius R. The distance between the two facilities is also R, so some population resides between both facilities. Define the decay function ( ) = − , where ≤ ≤ . The density of the areas is 1 unit per square mile and the supply C is the same in each facility. We will compare composite measures across the network in Figure 9. the distance between the patient and the facility. For the population in overlapping catchment areas, a patient's accessibility can be calculated as 1 , 2 = ( − 1 + − 2 ) . Where 1 is the distance to the first facility and 2 is the distance to the second facility. We also have 1 + 2 ≤ , 1 , 2 ≤ , ⇒ 2 − ≤ − 1 + − 2 ≤ 2 − 2 .
For the optimization models, we initially use a congestion weight such that patients will visit their closest facility. The congestion at each facility is = For the population inside the catchment of only one facility, the patient's congestion is = . For the population in the overlapping catchment areas the congestion experienced by each patient is = .
If a patient is inside a single circle, then the optimization model shows higher accessibility than the 2SFCA approaches since = < 1 . This is true for larger congestion weights if there is no decay function or if the congestion weights are extreme. The result occurs because visits are over-counted in the 2SFCA methods, while the optimization model is capturing the cost associated with user experience. For patients in the overlapping areas, we find that which method estimates higher accessibility depends on the value of radius R. If < 4 3 , then the accessibility for patients in the middle is: ≥ 2 − > 3 2 > 1 . This implies that the overall range of accessibility in the optimization model is smaller than the 2SFCA methods, so the access appears smoother. R values that are small represent dense areas. 2. When the population density is non-homogenous over the network Consider system II. The E2SFCA facility and patient level accessibility measures are: For a patient inside the catchment of facility 1 only: 1 = − 1 .
For a patient inside the catchment of facility 2 only: 2 = − 2 .
The facility and patient level congestion measures for Shortest Distance are: .
For a patient inside the catchment of facility 1 only: 1 = 1 .
For a patient inside the catchment of facility 2 only: 2 = 2 .
For a patient inside the overlapping area: 1 ≤ ≤ 2 .
At the patient level, it is obvious that 1 < 1 and 2 < 2 .

5
Incidence matrix for race/ethnicity   We estimate the number of visits for each patient-CF center pair i,j based on an exponentially decaying function = 10 −0.02 , ≤ 150 miles, and = 0 , > 150 miles. For all patients with family income below two times the FPL, = 0 for all centers located in other states.