Search results

In review step 1, 4059 citations were retrieved. After exclusion of 1727 duplicates, 1173 titles were excluded because they were out of the scope of this study. Upon abstract and full text-screening, 1116 citations were excluded because they either did not include a gravity model, were not in the field of healthcare, did not fit the journal/language criteria, or did not constitute a methodological development. We resulted with a final set of 43 studies that presented gravity model-type methodological developments for measuring spatial healthcare access (Additional file 1: Appendix B).

For review step 2, we retrieved 2732 citations via forward citation search and amended them with 315 citations that had been excluded in review step 1 solely because they did not constitute a methodological development. Next to the exclusion of 1517 duplicates, we additionally automatically excluded 203 citations that had been excluded in review step 1 because either the title was out of scope, the study showed no gravity model, or was not in the healthcare field. We further excluded the 43 studies that had been included in step 1, since they were not application studies. Upon title screening, an additional 287 studies were excluded. Upon abstract and full text-screening, we excluded 688 studies that were lacking either a gravity model or a healthcare focus. Finally, 309 studies were included in review step 2. These 309 studies contained a total of 346 individual measures of spatial access to healthcare (Additional file 2).

Review step 1: methodological developments

Starting from the base gravity model, two major conceptual methodological developments were proposed: The Two-Step Floating Catchment Area (2SFCA) method [11], and the Kernel Density (KD) method [30]. From 2004 onward, methodological developments almost exclusively built on the 2SFCA method and proposed developments within this family of methods, thus representing a second generation of gravity model-type methodological developments. First-generation developments (= Gravity Model Developments) and second-generation developments (= 2SFCA Developments) are discussed separately.

1st Generation developments

Two major conceptual developments of the base gravity model were put forward. The 2SFCA and the KD method are conceptual developments in that 2SFCA employs predefined catchment areas that are floated over population/provider locations instead of considering all populations/providers; and KD overlays a provider and a population density layer. Mathematically however, both methods are simply special forms of the base gravity model. In addition to these conceptual developments, a small number (N = 2) of methodological papers [7, 31] proposed adaptations to the base gravity model while staying conceptually similar to it. Key characteristics of these adaptation studies are presented in Additional file 1: Appendix B.

Two-step floating catchment area method

Advancement: Floating catchment areas for determining access.

Method: Luo and Wang [11] introduced the Two-Step Floating Catchment Area (2SFCA) method. In the first step, all population locations within a predefined catchment area around each provider location are searched. The provider capacity is divided by the total population within the catchment area to create location-specific provider-to-population ratios. In the second step, all provider locations within the same predefined catchment area around each population location are searched. The provider-to-population ratios of all provider locations within the catchment area are summed up to build the spatial access measure. Technically speaking, the 2SFCA is a special case of the gravity model proposed by Joseph and Bantock [10] using a binary distance decay function. The function takes on a value of 1 if the provider/population point lies within the predefined catchment area (complete access assumed), and a value of 0 if it does not (no access assumed). While the base gravity model considers all providers/populations within the system, the 2SFCA method introduces the notion that beyond a certain threshold, providers are not relevant to the population anymore. Hence those providers are deemed not accessible. The reduction of providers/populations considered to only those within a catchment area implies a reduction of methodological and computational complexity. This in turn allows for more complexity and precision in the distance computation. Instead of the Euclidean distance, Luo & Wang use the estimated shortest path travel time along a road network. Interpretation of the 2SFCA access measure is in terms of providers per population, making the measure easily understandable. Adding up all 2SFCA values in the system results in the total number of providers. However, the 2SFCA measure also has drawbacks. Namely, the catchment area delineation is arbitrary. Within the catchment area, all providers are treated as equally accessible. The binary distance decay implies sharp edges where providers that are very close to each other can be treated as very differently accessible.

Purpose: Approximating reality.

Methodological complexity: Introducing catchment areas and considering only those locations within a catchment area implies a reduction of methodological complexity.

Data requirements: No additional data requirements.

Kernel density method

Advancement: Overlying provider and population density layers.

Method: Guagliardo et al. [8] introduced the Kernel Density (KD) method, which overlays provider and population density layers to compute the spatial access measure. Conceptually, this measure is thus closer to a provider-to-population ratio than the base gravity model or the 2SFCA. The KD method employs the Kernel Density function to create a density layer of providers out of the (in reality) discretely allocated provider points. This can be understood as a smoothing of provider points over a raster grid. Thus, the reach of providers, and hence the distance decay, is considered via the Kernel Density function. Similarly, a population density layer is created, smoothing out population data to mimic the real population distribution. Overlaying those layers, i.e., dividing the provider by the population layer results in a provider-to-population ratio for the grid cells. In the final access index, the provider-to-population ratios of all grid cells within the target area are averaged. Thus, the KD spatial access measure represents a localized provider-to-population ratio. The methodological simplicity and interpretability are clear advantages of the KD method. One downside of the KD method is that for creating the density layers, it requires somewhat arbitrarily choosing a cone radius. Another one is that the KD method does not integrate real travel behavior as travel friction is only considered in terms of Euclidean distances. However, compared to the base gravity model and the 2SFCA method, the KD method has the clear advantage of avoiding the circularity issue of other gravity models. Provider availability in Joseph and Bantock [10] and Luo and Wang [11] are adjusted according to the potential demand (i.e., size) of the surrounding population. One might argue, that for the estimation of this potential demand of the surrounding population, one would have to adjust for the potential availability of the providers surrounding those population points. Taking this line of argument further, it becomes circular. By overlaying density layers, the KD method does not suffer from this circularity issue.

Purpose: Approximating reality.

Methodological complexity: Introducing provider and population density layers implies a reduction of methodological complexity as it is conceptually close to a standard provider-to-population ratio.

Data requirements: No additional data requirements.

2nd Generation developments (2SFCA family)

There was a plethora (N = 39) of methodological developments within the 2SFCA family. We classified them into novelty categories for which we present intuition, first study introducing the type of development, following adaptations within the category, purpose, methodological complexity, and data requirements. Additionally, all individual studies presenting a methodological development within the 2SFCA method family are given in Additional file 1: Appendix B.

Distance decay within catchment area

Advancement: Introducing a distance decay function within the catchment area.

Intuition: A major drawback of the 2SFCA method by Luo and Wang [11] is the binary distance decay function treating all providers within the catchment area as completely accessible and all providers outside the catchment area as not accessible. Realistically, farther away providers are less likely to be visited. To mitigate this problem and to incorporate the intuition that within a catchment area, not all providers are equally accessible, distance decay within the catchment area was introduced.

First study: Luo and Qi [12] built on the 2SFCA method and introduced distance decay within the catchment area. This was done to depict travel impedance more realistically. Specifically, the catchment area was split into several catchment radii, with each section being assigned a different weight. Thus, population points distant from the provider are weighted less and thus contribute less to the potential demand. Provider points distant from the population location equally are weighted less and hence contribute less to the spatial access measure. As a discrete average weight (computed by a Gaussian function) is assigned to all points lying within a sub-radius, this type of distance decay is considered a discrete stepwise distance decay.

Development within category: Several studies have suggested different functional forms of the decay function. This includes continuous functions (e.g., power function, Gaussian function) and a combination of a stepwise and a continuous function. Schuurman et al. [32] for example suggested no distance decay for an initial catchment radius, linear decay for the next catchment radius, and full decay (= no access) after the third threshold. Jin et al. [33] further introduced a decay function that is dependent on the provider specialization degree (i.e., they assume slower distance decay for more specialized providers). Tao et al. [34] assumed that the distance decay function varies by rurality (i.e., they assume slower distance decay in rural areas). The developments concerning distance decay have been synthesized by Wang [21], who suggested a generalized 2SFCA. Instead of specifying a particular distance decay function, the generalized 2SFCA introduced a general decay function, leaving it up to the researcher to decide which empirical functional form fits best their context and assumptions about distance decay. The generalized 2SFCA consolidated previous measures, formalizing the general principle used by other authors.

Purpose: Approximating reality.

Methodological complexity: Introducing a (stepwise, continuous, or hybrid) distance decay function within the catchment area moderately increases modeling complexity. In addition to determining for each provider-population pair whether they are within the catchment area threshold distance, a distance decay function must be applied for each provider-population pair within the catchment area.

Data requirements: No additional data requirements.

Variable catchment area sizes

Advancement: Variable catchment size definition.

Intuition: Luo and Wang’s [11] 2SFCA formulation assumed fixed and uniform catchment area sizes for all provider and population locations. However, urban and rural populations might have a differing willingness to travel due to the presence of nearby providers. In other words, when providers are far away, patients will travel farther to get access to healthcare; when many providers are nearby, the willingness to travel will be lower.

First study: Luo and Whippo [13] introduced variable catchment sizes to the 2SFCA method (= V2SFCA). This was done to reflect differences in surrounding provider density leading to differences in accepted initial travel time. They proposed to dynamically determine catchment sizes by increasing a catchment size until a predefined provider-to-population threshold is met. This specification cannot result in population locations with zero spatial access. This fully variable catchment size depends both on the provider capacity and the population density and results in unique catchment sizes for each provider/population location. More generally speaking, in the variable 2SFCA the catchment threshold is a function of certain characteristics of the provider/population location.

Development within category: Reflecting differences in catchment thresholds for urban and rural populations has been conducted in different ways. While Luo and Whippo [13] defined catchment sizes dependent on a base provider-to-population threshold, McGrail and Humphreys [35] chose to define discrete catchment sizes. They suggested to assign one of five catchment area sizes to a provider/population location based on the rurality classification of that location. Variable catchment sizes have not only been implemented to depict urban and rural differences, but also to account for varying reach depending on provider specialization degrees. Such approaches reflect that more specialized or larger providers serve a larger population. Kim et al. [36] for example used the number of physicians per hospital as determinant for the catchment size, while Tao et al. [37] defined the catchment area size as dependent on the level of hospital specialization.

Purpose: Approximating reality.

Methodological complexity: Introducing variable catchment sizes only slightly increases modeling complexity. The criterion for determining whether a provider-population pair is within a catchment area shifts from a single condition (e.g., less than 30 min driving time) to combined conditions (e.g., less than 30 min driving time for population in urban area and less than 60 min driving time for population in rural area).

Data requirements: Additional data may be required in some cases, e.g., a rurality classification for all provider and population locations. Additional data is not required when catchment sizes are determined conditional on a base provider-to-population threshold.

Outcome unit modification

Advancement: Outcome unit modification to interpretation in relative terms.

Intuition: Accounting for distance decay within a catchment area (like in the generalized 2SFCA) requires the definition of a distance impedance parameter. Since empirical verification of distance disutility is difficult, the choice of the impedance parameter is somewhat arbitrary. Yet, different values of the distance impedance parameter can vastly alter the magnitude of the spatial access measure. To account for this uncertainty, an outcome unit modification was proposed.

First study: Wan, Zhan et al. [38] proposed to compute a spatial access ratio (SPAR) from the 2SFCA measure. The SPAR is defined as the ratio between a population location’s spatial access measure and the mean spatial access measure of all population locations. SPAR thus represents a dimensionless relative outcome measure rather than an absolute provider-to-population-like measure. SPAR is stable to different distance impedance parameter specifications and useful for mapping when no absolute outcome values are needed. This development comes at no cost in terms of data requirements or computation.

Development within category: No further studies suggested an outcome unit modification.

Purpose: Correcting methodology.

Methodological complexity: Introducing an outcome unit modification does not increase modeling complexity. An additional step is required to compute the Spatial Access Ratio. However, it requires simply dividing the population location’s access index by the mean access measure of all population locations, thus not noticeably adding complexity.

Data requirements: No additional data requirements.

Provider competition

Advancement: Correcting demand overestimation by introducing a provider competition-based selection weight.

Intuition: In the first step of 2SFCA, all population locations within a provider’s catchment area are considered to contribute to the potential demand at that provider location. Thus, population points may be counted fully as contributing to the potential demand at several provider locations, implying an overestimation of demand. However, individuals do not visit all accessible providers, but rather choose one provider. To correct the implicit demand overestimation, a provider competition scheme was suggested. This scheme assigns populations based on a choice probability, reflecting that populations face several provider options.

First study: Wan, Zou, and Sternberg [39] proposed the three-step floating catchment area method (3SFCA) which computes a distance impedance-based selection weight for each provider-population pair in an additional first step. This approach considers all surrounding providers (and distances to them) a population can choose from in its catchment area. As such, the potential demand more accurately depicts the actual demand on a provider. The closer a provider is to a population location, the more likely the population will be to visit the provider; hence the selection weight assigns a larger share of the population to closer providers. The additional step is computed using a Huff-model based selection weight.

Development within category: While the initial 3SFCA proposed by Wan, Zou, and Sternberg [39] considered only distance impedance as determinant of the selection probability, other developments incorporated additional factors into the provider competition scheme. Luo [40] amended the selection weight computed in the first step by including provider capacity in addition to the distance impedance function. Other functions for the computation of the selection probability have also been proposed, incorporating a plethora of factors determining the selection weight [e.g., 41,42,43]. Notably, Paez, Higgins, and Vivona [44] showed that standard 2SFCA methods not only overestimate demand, but also overestimate supply as the provider capacity computed in the provider-to-population ratio may be counted to the final access score at several population locations. Thus, instead of computing a selection probability to be applied in the provider-to-population ratio, Paez et al. [44] fully proportionally allocated population and providers to account for potential demand and supply inflation. This is done by standardizing the distance decay weights within the distance decay function in each catchment area to 1. Hence the total level of demand and supply within the system are preserved in this measure.

Purpose: Correcting methodology.

Methodological complexity: Introducing a selection weight heavily increases modeling complexity. An additional first step is required to compute the selection weight. In addition to computing a provider-to-population ratio at each provider location considering all surrounding population locations, a selection weight for each provider-population pair considering all surrounding provider locations must be computed. This almost doubles computational intensity.

Data requirements: No additional data requirements.

Local and global distance decay

Advancement: Correcting for sub-optimally configured healthcare system by modeling local and global distance decay.

Intuition: Delamater [45] showed that the standard configuration of 2SFCA methods only considers relative distances within a catchment area and ignores absolute distances. That is, only the relative distance differences among providers within the population’s catchment area impact the final access score, not the absolute distance differences across catchment areas. In a simple example, population locations with only one provider within the catchment area would receive the exact same spatial access score, irrespective of whether the provider is close by or far away. This implicitly assumes optimally configured healthcare systems, where population locations do not face different absolute distances to providers. Arguably, even if two population locations have the same provider availability within their catchment area, we might consider a population location from which those providers are farther away to have poorer accessibility.

First study: Delamater [45] put forward a methodological development to correct for sub-optimally configured healthcare systems as they are found in reality. By introducing an additional distance decay function in the first step, the method produces specific pairwise supply ratios rather than single supply ratios per provider and thus accounts for both relative and absolute distances. Conceptually, this can be understood as having a local distance decay function (= relative distances) and a global distance decay function (= absolute distances).

Development within category: Bauer and Groneberg [46] further advanced this method by using two different functional forms for the local and the global distance decay function. Assumptions about the nature of the relative distance decay within the catchment area are formalized in the local decay function, while assumptions about the role of absolute distance are captured in the global distance decay function. This allows to differentially conceptualize the role of relative and absolute distances.

Purpose: Correcting methodology.

Methodological complexity: Introducing an additional distance decay function slightly increases modeling complexity, more so, if different functional forms are used for the local and global distance decay functions.

Data requirements: No additional data requirements.

Subgroup-specific access

Advancement: Subgroup-specific access measure for selective population-provider pairing.

Intuition: Factors other than spatial impedance and provider capacity have an impact on whether an individual can access a provider, i.e., not all providers are available to all populations equally. This might be due to e.g., language restrictions, insurance plans, or referral systems. In turn, subgroups of individuals compete for a subgroup of providers, both of which may exhibit spatial variation. Standard 2SFCA methods, however, assume equal non-spatial access for all populations to all providers.

First study: Wang [47] proposed an additional step after computing the access measure for quantifying group-specific access of Chinese immigrants to ethnic Chinese physicians. This was done by considering the relative abundance of ethnic Chinese physicians as well as the share of Chinese immigrants within the catchment area of a population location. Thus, demand competition of a population subgroup for a provider subgroup is reflected.

Development within category: Other context-specific methodological developments also reflected that not all providers are available to all populations equally. Xiao et al. [48] integrate referrals within a hierarchical healthcare system where higher specialized providers can only be accessed upon referral. Shao and Luo [49] incorporated group-specific healthcare access resulting from different health insurance plans. In their method, a provider’s resources are only counted toward the accepted insurance plan of a population subscribed to that same plan. The resulting measure yields access scores by insurance plan.

Purpose: Fitting context.

Methodological complexity: Introducing a subgroup-specific access index heavily increases modeling complexity. An additional third step is required to compute the subgroup-specific index. In addition to computing the standard 2SFCA measure, a weight considering all surrounding subgroup-specific provider and population locations must be computed. This almost doubles computational intensity.

Data requirements: Additional data are required for information on subgroup-specific provider and population shares at each location.

Multiple transportation modes

Advancement: Multiple transportation modes.

Intuition: Standard 2SFCA methods considered one transportation mode only—typically, travel time along a road network by private vehicle was used to model distance impedance. This is a simplified assumption. For one, traveling by public transport or by foot implies different travel times and thus accessibility. For another, population locations differ in terms of mobility, i.e., the share of individuals using each mode of transport.

First study: Mao and Nekorchuk [14] suggested a multi-mode 2SFCA which incorporates multiple transportation modes that individuals might use to reach a provider. The method captures several transport mode options as well as varying mobility by population location. Specifically, transport-mode specific catchment sizes are defined (e.g., 30-min travel time by private vehicle and 60-min travel time by public transit). In the first step, for each transport mode, only population points within the mode-specific catchment are considered, and only the assumed mode-using share of the population is counted to build a mode-specific provider-to-population ratio. These mode-specific ratios are summed up to form the full provider-to-population ratio. The same approach is used for the second step, additionally weighting the mode-specific reachable provider ratios by the corresponding mode-using share of the population at the location. As such, this measure can also account for competition among populations with differing mobility.

Development within category: Producing a more individualized measure, Langford, Higgs, and Fry [50] suggested computing separate access measures for each transport mode. When computing provider-to-population ratios in the first step, all transportation modes are considered (just like in the Mao & Nekorchuk [14] method). In the second step, only the mode-specific reachable provider ratios are summed up. This allows to incorporate demand competition for providers, while producing travel mode-specific scores. Zhou et al. [51] incorporated different travel mode choice probabilities for each provider-to-population pair instead of simply using different fractions of mode users by transport type per population location, thus producing a more detailed measure.

Purpose: Approximating reality.

Methodological complexity: Introducing multiple transportation modes heavily increases modeling complexity. For each provider-population pair, travel distance/time must be computed for at least two modes of transportation. Additionally, demand intensity is weighted by the mode-specific user shares at each population location. This at least doubles computational intensity.

Data requirements: Additional data are required for information on travel distance/time by transport mode as well as data on mode-specific user shares for each population location.

Time-dependent access

Advancement: Spatio-temporal access measure using time-varying parameters.

Intuition: Previous 2SFCA methods were static in the sense that no temporal variability of the input parameters was considered. Individuals might face greatly varying spatial access depending on the time of day or year due to e.g., traffic congestion, opening hours, or road conditions. Spatio-temporal access measures explicitly model temporal variability, producing a range of access measures rather than relying on one static measure which reflects spatial access only at a specific time point.

First study: Ma et al. [52] introduced a temporal dimension in their 2SFCA measure by incorporating time-varying travel times. Using real-time traffic data at different time points within the same day, they captured varying traffic congestion. Thus, the travel impedance becomes a time-dependent parameter in this method. While previous 2SFCA measures frequently assumed average travel speeds for road segments to compute travel times, this method relies on routing algorithms using empirical travel speeds. Importantly, this approach does not produce one combined static access measure. Instead, it gives a range of access measures that depict spatial access at different time points.

Development within category: Instead of looking at within-day variability, Song et al. [53] incorporated seasonal differences in average travel times due to varying precipitation levels. Further developments incorporated time-varying demand-size [54] and time-varying supply size [55]. The former aims to reflect variable demand intensity due to commuting behavior by using mobile phone GPS data [54]. The latter reflects variable provider availability throughout the day by incorporating provider opening hours [55].

Purpose: Approximating reality.

Methodological complexity: Methodological complexity is not affected by this development as the basic configurations of the 2SFCA method remain unchanged. However, as this development requires generating a range of access measures with different parameter inputs computational intensity will be heavily affected.

Data requirements: Additional data are required for information on time-varying travel times, population size, or provider opening hours (Table 1).

Table 1 Methodological developments in the 2SFCA familyReview step 2: application studiesStudy characteristics

The 309 application studies entailed a total of 346 individual measures that constituted applications of the earlier identified methodological developments. Since the application of specific methods is of interest here, the following numbers refer to individual measures rather than studies. Frequency statistics on the characteristics of access measures are given in Table 2. An overview of the measures applied by methodological development type is given in Fig. 2. Most healthcare access measures were applied either in an Asian (N = 160; 46%) or a North American (N = 136; 39%) healthcare setting, with USA (N = 109), China (N = 91), and Canada (N = 27) being the most frequently investigated countries. In 45% of the application cases (N = 157), spatial access to providers within the primary care sector was investigated, 39% of the measures (N = 135) were applied to secondary care providers and 16% (N = 54) to both primary and secondary care providers. Provider capacity was measured in a variety of ways, including physician head counts (e.g., [56]), physician full-time equivalents (e.g., [15]), number of hospital beds (e.g., [57]), number of sites (e.g., [29]), or number of services (e.g., [58]) available. Most spatial access measures (N = 186; 54%) were applied to a geographic scope covering both urban and rural regions, while 41% (N = 142) were applied to an exclusively urban, and 5% (N = 18) to an exclusively rural context. To operationalize travel friction in the gravity model, most applications (72%; N = 249) used the estimated shortest travel time along a road network between population and provider location. Another 16% (N = 56) of the measures incorporated travel friction by means of estimated shortest travel distance, and 12% (N = 41) used Euclidean distance. Almost all spatial access measures (98%, N = 338) were computed within the scope of the respective studies. However, a small share of studies relied on secondary data (2%; N = 8), using access indices that had been previously computed by other researchers.Footnote 1 There was a range of purposes that the healthcare access measures were applied for: 55% (N = 189) of the measures served the purpose of describing the spatial access in a particular healthcare system. In 19% of the cases (N = 67), the spatial access measure was used as a covariate to explain health outcomes (e.g., [59]). Another 15% of the measures (N = 51) were applied in a methodologically exploratory way; that is, investigating model properties by applying different parameters (e.g., [60]). Lastly, 11% (N = 39) of the measures were applied to conduct a comparison of methods, either comparing gravity model indicators to non-gravity type accessibility or availability measure (e.g., [61]), or comparing several different gravity model methodologies (e.g., [62]).

Table 2 Characteristics of measures applying gravity model developmentsFig. 2

Lineage and application frequency of gravity model developments. Displayed in light blue is the base gravity model, displayed in blue are the conceptual developments derived from it (2SFCA method, Kernel Density method), displayed in dark blue are categories of methodological developments derived from the 2SFCA method. Number of access measures that applied the respective methodological development is given in gray boxes, bubble sizes are weighted by frequency of application. Each applied measure was assigned to either the base gravity model, the 2SFCA method family, or the Kernel Density method (= mutually exclusive). In addition, for all 324 2SFCA-type measures, the methodological development(s) applied were identified. One 2SFCA-type measure can implement more than one methodological development (= not mutually exclusive). Therefore, the sum of 2SFCA-type methodological developments applied exceeds the number of measures classified as 2SFCA-type. 2SFCA Two-Step Floating Catchment Area, E2SFCA Enhanced Two-Step Floating Catchment Area, V2SFCA Variable Two-Step Floating Catchment Area, SPAR Spatial Access Ratio, 3SFCA Three-Step Floating Catchment Area, M2SFCA Modified Two-Step Floating Catchment Area, MM-2SFCA Multi-Mode Two-Step Floating Catchment Area

Application frequency: 1st generation developments

Applications of gravity models for spatial healthcare access measurement were almost exclusively restricted to methods from within the 2SFCA family (94%, N = 324). Only few measures applied a base gravity model or the KD method: 4% (N = 14) of the applications represented measures derived from the base gravity model method and 2% (N = 8) applied the KD method to measure potential spatial access to healthcare.Footnote 2

Application frequency: 2nd generation developments (2SFCA family)

Out of the 324 measures implementing a 2SFCA-type methodology, 25% (N = 82) computed the measure exactly like Luo and Wang [11] had first suggested it. That is, no distance decay within the catchment area, fixed catchment size