Distance to Radiotherapy and Demand – Projection of the Effects of Establishing New Radiotherapy Facilities in Poland by 2025

Relation between distance and the demand for EBRT has been shown by many researchers in various settings, including Poland. On the other hand, utilisation of alternative treatment modalities of patients with cancer, that is, surgery and chemotherapy, does not seem to be highly dependent on distance (Jones et al. 2008). Such difference derives from high costs of establishing new radiotherapy departments. Chemotherapy can be administered in virtually any medical facility, provided the appropriate personnel is available, and for most oncological surgical procedures, a regular operating theatre is sufficient. Meanwhile, the market price for a new linac is around 10M PLN (€2.5M). Moreover, specialised bunkers have to be constructed to house radiotherapy imaging and treatment equipment. That is why the cost of establishing a new radiotherapy facility with two working linacs can be estimated at around 40–50M PLN (€10−12.5M). High setup costs mean that a much smaller number of hospitals provide radiotherapy than chemotherapy or surgery. This phenomenon, combined with long radiotherapy course duration, is the primary cause of the underutilisation

of radiotherapy services in Poland (Więckowska and Czerwiński 2015). Thus, the radiotherapy development plan was introduced, as mentioned (Czerwiński and Więckowska 2018). Fig. 1 presents the planned linac allocation in 2025, together with linac locations in 2012 and 2016 (starting point of the mixed-integer model).

In Poland, all publicly funded health services are recorded by the National Health Fund (NHF) in a centrally managed database. The main purpose of this database is to document all financial transactions between the public payer and healthcare providers. Every publicly funded medical service is registered in this database together with the patient’s national identification number (PESEL), patient’s demographic characteristics (age, sex, place of residence) as well as reported diagnosis-related groups, procedures and diagnoses. In the scope of oncology, the NHF database for 2012 was linked with the National Cancer Registry (Więckowska et al. 2015a). This linked data set was used in this study. In Poland, practically, no EBRT services are being financed

outside of the public system. Private centres without contract with the NHF, who deliver RT pro bono in hopes of receiving such contracts, are the only major exception (e.g. currently a centre located in Otwock operates in this manner). However, in 2012, no such provider operated. Other possible exceptions include financing RT out-of-pocket or through private, voluntary insurance schemes as well as RT courses received abroad. Such events would not be recorded in the NHF data set, and the data on them are not obtainable, which could potentially bias the results of this study. Nevertheless, the fact that every Polish citizen is publicly insured or can obtain insurance in a relatively simple way (e.g. by registering as unemployed) as well as relatively short radiotherapy waiting times (Dziadziuszko 2015) suggest that even if such bias exists, its impact on the results should be minimal. Hence, this oncological data set comprises virtually all EBRT courses delivered in Poland in 2012 and thus can be used to precisely examine EBRT utilisation patterns.

Using the 2012 oncological data set, EBRT utilisation rates were calculated for every Polish county (the overall utilisation rate and 14 cancer-specific rates – Tab. 1). EBRT utilisation rate for cancer group g (URg) is defined as the ratio of all EBRT courses for new group g cases (NCr_g) and retreatments (R_t) divided by the number of new group g cancer cases (NC_g) (Equation 1):

Definition of the overall rate is analogical. Fig. 2a presents the overall EBRT utilisation rate. A considerable number of counties with no cancer cases will cause a limited dependent variable problem. In order to avoid this problem, the following groups were incorporated into the Other group, Melanoma of skin, Upper gastrointestinal tract, Testicle, Bladder, Kidney and Thyroid, and were treated collectively in subsequent analyses.

Ten different distance measures were defined to verify which best describes the relation between county locations and the localisations of EBRT departments. First, the orthodromic (Euclidean) distances between the counties’ capital cities and cities with radiotherapy facilities were calculated. First five distance measures were defined as the mean distance to a given number (1−5) of closest radiotherapy centres. Those measures are further referred to as standard(1)–standard(5), respectively. The use of Euclidean distance is motivated by two reasons. First, it is consistent with the methodology applied to develop the Polish national plan for radiotherapy development and investments (Czerwiński and Więckowska 2018). Second, even though other measures (such as road distance or travel time) might be considered to better reflect a patient’s discomfort, their reliance on other factors, such as road network or commuting network, makes them variant over time and thus problematic when forecasting. Meanwhile, Euclidean distance is constant over time, which means that the same distances can be used for 2012 and 2025. For Poland, the following approximations were suggested: 1 km of

Euclidean distance = 1.2 km of road distance = 1.27 min of travel time (Kopczewska 2013). Second five distance measures were gravitational. A number of working linear accelerators in 2012 was used as a proxy of the size of a location. Gravitational measures were examined, as larger centres may have influence over larger areas, because of, for example, reputation, experience, connections within the medical community or the capability to provide more advanced treatment. First, gravitational indices for the pairs of counties and EBRT facility locations were defined as a quotient of the orthodromic distance and the number of linacs in a location (as the analysis is discrete, no distinction was made between different providers located in the same county). Proposed definition is an inversion of the standard definition of gravitational indices (quotient of mass and distance). However, such construction should facilitate the interpretation of results. EBRT utilisation is negatively correlated with distance, thus the proposed definition allows to expect that gravitational measures are negatively correlated with utilisation as well. Subsequently, five gravitational measures were defined as the mean of a given number of gravitational indices (1−5) for the respective number of closest radiotherapy centre locations (in terms of gravitational indices). These measures are further referred to as gravitational(1)–gravitational(5). Formally, proposed distance measures can be denoted as follows. Let

I be the set of counties,

J be the set of radiotherapy facility locations,

d _ij be the orthodromic distance between county i ∈ I and location j ∈ J,

x_j be the number of linear accelerators in location j ∈ J

If the sets of distances and gravitational indices are denoted as

D={dij:i∈I∧j∈J}$$D=\left\{ {{d}_{ij}}:\,\,i\,\in I\wedge j\in J \right\}$$the set of orthodromic distances,

X={dijxj:i∈I∧j∈J}$$X=\left\{ \frac{{{d}_{ij}}}{{{x}_{j}}}:i\in I\wedge j\in J \right\}$$the set of gravitational indices, then the measures standard(1)–standard(5) and gravitational(1)–gravitational(5) for the county î∈I$\hat{i}\in I$can be expressed as shown by Equations (2) and (3).

One-way spatial interaction models (Fortheringham and O’Kelly 1989) and linear regression models were used to assess the relation between distance and EBRT utilisation, as well as to forecast the demand for EBRT in 2025. For every utilisation rate (overall and cancer-specific), 280 models with different functional forms and different independent variables were estimated. Afterwards, the Akaike information criterion (AIC; Akaike 1998) was used to select the model with the best fit. For every distance measure (10 in total), 14 linear models and 14 spatial error models were estimated. Spatial error models evaluate spatial autocorrelation in the residuals. In terms of EBRT utilisation, this autocorrelation can be caused by a plethora of factors, including communication (roads, railways, urban transport), behaviour patterns of healthcare providers in the area (in particular cancer detection patterns), demographic and socio-economic structure of the region and so on. The main purpose of the estimated models was to forecast EBRT utilisation. It is hardly possible to project all factors associated with the above-mentioned possible reasons for spatial autocorrelation as their impact may change rapidly over time, especially given the relatively long time horizon of the analysis. Therefore, it was assumed that the functional forms should include only those geographical and administrative variables, which are known for 2025. As the main purpose of modelling was to estimate the influence of distance on the demand for EBRT services, the spatial interaction models were considered in a one-way version. Counties were connected neighbours. Functional forms of the models comprised two components, a function of a distance measure and a binary variable indicating, whether a radiotherapy facility was present in the county. The binary indicator was included, because it was observed that Polish counties with a working radiotherapy facility had lower EBRT utilisation rates than their neighbours (Więckowska and Czerwinski 2015).

Seven distance measure functions were considered: exponential, power and polynomial (degrees of 1–5). Formally, each of the 280 functional forms can be represented by the following equation:

where ⍰(UR) is a function of the utilisation rate; f(d) is a function of the distance measure; ind is a dichotomous variable equal to 1, if a radiotherapy facility was located

in the county (0 otherwise); and e is the error term. Models are clearly defined by four features:

where u~N(0, σ²), and W is a row-standardised spatial weight matrix.

Models with the best fit according to AIC were used to calculate the expected utilisation rates in 2025. It was assumed that linac distribution will be consistent with the radiotherapy development plan (Fig. 1). National utilisation rates were calculated using the estimated utilisation rates for counties and the cumulative cancer incidence forecast for 2025 (Więckowska et al. 2015b).

Fig. 1

Fig. 2

In case of spatial error models, in contrast to linear regression, forecasting for the new set of explanatory variables (2025) will differ from calculating predicted values for the present (2012) (Haining 1990; Bivand 2002). For the linear regression model,

Predicted values ŷ are calculated as follows:

where β^$$\hat{\beta }$$is the vector of coefficients. Equation (10) will be correct for the estimation data set and for every new set of observations. In case of spatial error models,

y=Xβ+u,whereu=λWu+eande∼N(0,σ2).$$y=X\beta +u,\,\,\text{where}\,\,u=\lambda Wu+e\,\,\text{and}\,\,e\sim N\left( 0,\,{{\sigma }^{2}} \right).$$

Equation (11) can be rewritten as

And y can be presented as a sum of three components: trend, signal and noise (Bivand 2002).

Therefore, predicted values for the estimation data set can comprise either the trend alone or the trend and signal components. However, in case of the new data set (forecasting), y is not known, and thus, the predicted values can only comprise the trend. Despite not differing much from the linear regression, forecasting with spatial error models (in the presence of spatial autocorrelation) ensures that the estimators are not only unbiased but also efficient (Anselin 2001).

In order to establish the impact of the implementation of the radiotherapy development plan on the demand for EBRT, a base-case scenario was developed. This scenario assumed that in 2025, utilisation rates (overall or cancer-specific) in every county will be equal to the utilisation rates observed in this county in 2012. Subsequently, base-case national utilisation rates were calculated using the cumulative cancer incidence forecast for 2025.

Tab. 2 presents the overall utilisation models with the best fit (linear and spatial). According to the AIC, spatial models were characterised by a much better fit. For the best model (standard(1), power*), the AIC value for the spatial version was equal to −1,247.33, whilst the AIC value for the linear version was −1,062.53 (lowest value amongst linear models). For every pair of linear and spatial models with the same functional form and the same distance measure, the spatial version was characterised by a lower AIC value. Spatial coefficient λ was significant at 1% and positive for every specification, which confirms the validity of the use of spatial models. Relatively high differences between the coefficient estimates of the best linear model and the best spatial model, as well as the discrepancy between the projected national EBRT utilisation rates in 2025 (Tab. 2), highlight the importance of using spatial models to healthcare services utilisation studies in the presence of spatial autocorrelation. For every model, the relation between EBRT utilisation rates and a given distance measure was negative, which confirms that as the distance to the EBRT facility increases, the EBRT utilisation diminishes. Fig. 2b and 2c compares the fitted values from the best spatial model (standard(1), power*) depending on whether the signal component is included.

Similar to the overall utilisation modelling, cancer-specific spatial models were characterised by lower AIC values than their linear counterparts (Tab. 3).

For all cancer-specific cases, excluding the central nervous system (where AIC of the spatial specification was still lower than the AIC of the linear specification), the parameter was positive and significant at p < 0.001. According to the adopted modelling approach, the pace at which the EBRT utilisation rate decreases with distance differs between cancer groups (Fig. 3).

Fig. 3

Utilisation rates for two cancer groups (prostate and central nervous system) are exponentially related to the distance. In other words, counties located closer to the radiotherapy centres suffer higher decrements in utilisation per kilometre than the more remote counties. This has a simple quantitative interpretation in case of prostate cancer, as distance to the nearest radiotherapy centre, standard(1), is used in the best prostate cancer model. For prostate cancer, a 10-km distance

increase near radiotherapy centres is associated with a 0.09 decrease in the utilisation rate, whilst at around 100 km, the drop is more than 20 times smaller (0.004). To put it differently, at the distance equal to zero (cities where radiotherapy centres are located), on an average, 860 radiotherapy courses are administered per 1,000 new patients with prostate cancer, whilst this number declines to 771 at 10 km, to 694 at 100 km and to 690 at 110 km. Utilisation rates for the remainder of cancer groups decline at a pace close to linear (with respect to the appropriate distance measure). Again, simple quantitative interpretation can be done for the models with standard(1) distance measure. For example, breast cancer EBRT utilisation rate decreases at the rate of approximately 0.019 per 10 km in the first 50 km and at approximately 0.018/10 km between 50 and 100 km from the nearest radiotherapy centre, which corresponds to 19 and 18 less courses per 1,000 new patients with breast cancer. The overall utilisation is given by the exponential function, similar to prostate and central nervous system cancers; however, because of a significant

Lines are restricted to the maximum and minimum values of the appropriate distance measure observed in 2012. Numbers in brackets, together with the x-axis, indicate the distance measure of the respective model with the best fit.

decline in utilisation in cities with radiotherapy centres (relative to the utilisation curve – Fig. 3), the per kilometre decrease is less steep. At the distance of zero, 500 courses are given per 1,000 new cancer cases; this number falls to 432 at the distance of 50 km and to 409 at the distance of 100 km.

Prognosis results are presented in Tab. 4. Fig. 2 allows to compare fitted values for 2025 (d), fitted values for 2012 (b, c) and observed utilisation in 2012 (a).

Prognosis results for the overall utilisation rates show that the Polish national plan for radiotherapy development and investment should fulfil its main goal, that is, assuring better equity in access to this treatment modality for the entire population. As shown in Fig. 4, the highest utilisation increase is projected in the most remote counties, where currently the uptake is smallest because of geographical and transportation barriers. The expected increase in utilisation for counties that were located closer than 50 km from the nearest radiotherapy centre in 2012 is equal to 4 courses per 1,000 new cancer cases (simple county average, not accounting for population), whilst this value is equal to 23 for counties with the distance between 50 and 100 km and 49 for counties with the distance more than 100 km.

Fig. 4

Using the estimated 2025 utilisation rates and the cumulative cancer incidence forecast, the projected number of EBRT courses was calculated for every county. In total, the expected number of EBRT courses in 2025 in the overall utilisation model was 95,312, and the cancer-

specific models amounted to 97,932 courses, compared to 64,803 courses delivered in 2012, 92,483 courses in the overall base-case scenario and 94,940 courses in the cancer-specific base-case scenario. Hence, the overall (cancer-specific) utilisation modelling indicates that the full implementation of the EBRT development plan by 2025 will increase the EBRT utilisation by 3.05% (3.15%).