Issue 
Acta Acust.
Volume 8, 2024



Article Number  19  
Number of page(s)  17  
Section  Hearing, Audiology and Psychoacoustics  
DOI  https://doi.org/10.1051/aacus/2024003  
Published online  19 April 2024 
Scientific Article
Modelling prevalence development in a population group exposed to vibration, and noise: Application to handtransmitted vibration
^{1}
School of Engineering and Design, Technical University of Munich, Boltzmannstraße 15, 85748 Garching, Germany
^{2}
Department of Medicine, University of Connecticut Health, 263 Farmington Avenue, Farmington, CT 06030, USA
^{*} Corresponding author: magdalena.scholz@tum.de
Received:
21
March
2023
Accepted:
27
January
2024
An invertible, generalisable population model for the time course of the prevalence of a health effect resulting from habitual exposure to a physical agent, such as vibration, and noise, is proposed. The model includes a timedependent factor representing the daily exposure, expfac(t), applied to a polynomial fit of prevalencetime data recorded in a population group and one additional numerical parameter, a^{'}_{1}, to adjust the model for exposurespecific conditions. A model is constructed for the prevalence of vibrationinduced white finger (VWF) to confirm and validate its performance. A 4thorder polynomial fits representative data recorded in a population group from the commencement of exposure. Using the same polynomial coefficients and solely adjusting expfac(t) enables the model to fit period prevalence data for VWF from all available population groups as a function of time. Adjusting a^{'}_{1} enables the model to predict the point prevalence. If expfac(t) is specified in terms of the daily 8h, energyequivalent, frequencyweighted triaxial acceleration, the prevalences observed in different populations groups can be interrelated and interpolated to a common value (e.g., 10%) by inverting the model. This will enable tolerable daily exposures suitable for occupational environments to be defined for handtransmitted vibration.
Key words: Modelling prevalence / Handtransmitted vibration / Vibrationinduced white finger / Noise
© The Author(s), Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Habitual exposure to vibration or noise has long been associated with chronic health effects that accumulate over time. For vibration entering the hands, the outcome is commonly sensory loss in the hands and episodic white fingers [1], while for wholebody vibration, the outcome is commonly back pain and, from extreme exposures, spinal injury [2, 3]. For noise, the outcome is commonly sensorineural hearing loss and hearing impairment, though there are numerous other symptoms [4, 5]. A continuing concern is the appropriateness of occupational health guidelines for limiting exposures to these physical agents. This is especially so for vibration entering the hands, where the accuracy of guidelines contained in the international standard for handtransmitted vibration, ISO 53491:2001, has been repeatedly questioned [6].
It is often overlooked that almost all handheld vibrating power tools and machines produce noise so that exposure to handtransmitted vibration usually also involves exposure to noise. An excess hearing loss has been documented in some occupations in which exposure occurs simultaneously to these two physical agents [7]. The excess hearing loss in power tool operators suffering from vibrationinduced white finger (VWF) has been suggested to result from a common mechanism associated with exposures to vibration and noise: vasoconstriction in both cochlear and peripheral blood vessels meted by the sympathetic nervous system [8, 9]. This is supported by recent work in which a robust association between hearing loss and episodic white fingers has been found [10]. Recently, a combined noise and vibration exposure index has been proposed for occupational exposures [11]. The index constructs the sum of energylike daily exposures to each physical agent, which together form the daily exposure, divided by their respective exposure limits. While it may not be possible to establish retrospectively the relative magnitude of the hazards posed by vibration or noise and hence construct a combined index, it is plausible to model such a daily exposure by an “exposure factor” without initially specifying the contributions from different physical agents. This is the approach adopted here.
The development of a health effect amongst workers in a given occupation can be described for an individual or for a population: the latter description is the subject of this investigation. The focus is on population groups, each of which is taken to consist of members who are engaged neardaily in the same activity and are assumed to experience essentially the same daily exposure (e.g., operators of chain saws felling and processing trees in a given forest, or factory workers operating essentially the same machines to fabricate similar parts or products). This definition enunciates the assumptions underlying most studies of working populations, although it is not commonly mentioned in the epidemiologic reports that are available as data sources for modelling. The extent of a specific health effect in such a population group resulting from ongoing, quantifiable exposure to vibration, and noise, is described here by the prevalence, which needs to be expressed as either a point or period prevalence. The former is the prevalence in a population group at a given point in time, while the latter is the prevalence integrated over a time period [12].
Empirical populationbased exposureresponse data are sometimes reported for the time course of the prevalence observed in a single population group. More commonly, studies of working populations report a single prevalence, observed when the study was performed. Mathematical models have been developed for predicting disease progression such as DisMod, which was described by the World Health Organization [13]. A variant of this model has been reported for hearing loss but not specifically for impairment resulting from noise exposure [14]. Thus, the role of physical agents in the development of the health effect is not explicitly considered and the adequacy of occupational health guidelines for limiting exposures to the agent(s) cannot be assessed. In view of the continuing debate concerning the appropriateness of the guidelines for exposure to handheld vibrating tools and machines, establishing a model for the development of VWF in occupations involving handtransmitted vibration was chosen for the present investigation. The guidelines are provided in ISO 53491:2001 as a relation between the total time exposed at a given daily exposure for the prevalence of VWF to reach 10% in an exposed population group. For models based on data from different populations, this formulation requires the exposure time at which the prevalences were recorded to be transformed to those corresponding to when a 10% prevalence would have occurred in each population group. Thus, a model between prevalence and total exposure time must be established before the exposureresponse relation in the standard can be constructed. The formulation of mathematical models relating VWF prevalence to exposure time applicable to all data sets that can be inverted to predict the exposure time corresponding to 10% prevalence does not appear to have been treated extensively in the literature. In previous work, the growth of prevalence with continuing exposure to vibration has been modelled both for individuals engaged in a single occupation and for population groups from different occupations.
The development of VWF was modelled as almost a linear increase in prevalence with exposure time in forestry workers and proportional to the square root of exposure time in stone workers [15–17]. Neither of these functions can satisfy the need for a generalisable model, which must possess the same functional form for all population groups and produce a curvilinear relation between prevalence and exposure time as prevalence will ultimately reach a maximum value (e.g., at or less than 100%). A populationbased model that generates a curvilinear function61 has been proposed and was selected by ISO as the basis for their exposureresponse relation in ISO 5349 [18, 19]. This models the growth of prevalence by a cumulative normal distribution. However, while derived from, and applicable to, all population groups selected for the analysis, it suffers from the limitation that all exposures will ultimately result in a VWF prevalence of 100%. In contrast, the maximum prevalence observed in population groups can be expected to be related to the hazard presented by the activity and is commonly found to peak at values other than 100%. A populationbased model is thus required that more closely replicates the epidemiologic findings.
Now, sensorineural hearing loss and hearing impairment have long been quantifiable by audiometric measurements. While equivalent metrics have been demonstrated to record small sensory changes related to tactile acuity in the hands of persons occupationally exposed to vibration [20], and limits suggested for their use as a sensory test [21], they have not generally been employed in population studies. The same can be said, though to a somewhat lesser extent, of vascular function tests for quantifying the prevalence of VWF [22]. In consequence, the available health data are commonly in the form of responses to questionnaires and diagrams showing the location of signs and symptoms as well as colour changes in the hands. The signs and symptoms are evaluated against descriptive numerical stages for the severity of VWF (e.g., Stage 1, Stage 2, etc.). Fortunately, only one such staging is in common use, the Stockholm Workshop scale (SWS) for the vascular component of the syndrome [23], with an earlier classification of signs and symptoms by Taylor and Pelmear (TPS) being used prior to its development [24].
The purpose of this contribution is thus to develop an invertible, generalisable mathematical model that accurately represents the growth in prevalence of a health effect in a group of persons all engaged neardaily in the same activity, which is assumed to result in essentially the same daily exposure to vibration, and noise, for low prevalences and up to at least 50%. This model is to be adaptable to different exposures and changes in a population to fit period and point prevalences of different populations, and to allow the prediction of lower prevalences at earlier times from a provided prevalence. A successful model for the growth of the prevalence of a health effect will find immediate application to the establishment of populationbased exposureresponse relations for this health hazard. Furthermore, it will enable the development of occupational exposure limits, which require the prevalences recorded in studies of different population groups to be transformed to a common point prevalence, e.g., 10% [6]. Clearly, it is only possible to create work environments and schedule exposures that balance productivity with the health and safety of the persons involved if the prevalence development of a significant health effect under actual working conditions can be predicted with confidence.
In this contribution, models are constructed from published period and point prevalence data in which the duration of individuals’ exposure has been expressed in years (typically years of employment in the occupation). All members of a population group have performed basically the same tasks neardaily and hence are assumed to have experienced essentially the same daily exposure. Variations amongst individuals are hence subsumed by other model parameters. These conditions permit functional relations between prevalence and exposure duration to be derived from published occupational studies of population groups. At first, generalised models are described for the period prevalence and point prevalence in Section 2. Then, these are applied to construct models for the development of the period prevalence and point prevalence of VWF in Section 3. Unfortunately, there are only a few studies of VWF that report data in the form needed to construct and evaluate the models; all those known to the authors are included in the analyses here. A period prevalence model is initially constructed in Section 3.1 using representative data sets and the adjustment to the exposure factor necessary to accommodate different descriptions of the symptoms established. The potential for applying a model constructed from one data set to another independently obtained data set with a different exposure factor is then considered in Section 3.2, followed by transforming the period prevalence model into a point prevalence model (Sect. 3.3). A method for estimating a chosen common prevalence (e.g., 10%) for studies in which only a single value was reported at a higher prevalence is proposed and evaluated in Section 3.4. Additionally, in the absence of sufficient information on noise exposure, the exposure factor for developing VWF is defined in terms of the available vibration metrics and the resulting model shown to adequately predict the prevalence of episodic white fingers in Section 3.5. The models and their implications are discussed in Section 4 with conclusions in Section 5.
2 Generalised models
The following hypotheses summarise the requirements for a generalised model. For all persons engaged neardaily in the same activity resulting in essentially the same quantifiable exposure factor, so defining a population group:
The period prevalence of a health effect common to the population group and its growth in time can be predicted by an mathematical model at prevalences from 0% to at least 50%.
For population groups engaged in different occupational activities, the period prevalence of a health effect common to all groups and its growth in time can be predicted by the same mathematical model after adjustment for the differences in exposure factors between the population groups.
For population groups engaged in different occupational activities, the point prevalence of a health effect common to all groups and its change in time can be predicted from the period prevalence model after adjustment for the differences in exposure factors between the population groups and an estimate of the rates at which persons join and leave the groups.
For a population group, the point prevalence of a health effect for an exposure duration at which no data are available can be predicted by the point prevalence model for the population group from a single value observed at a later time and higher prevalence.
Data linking the prevalence and exposure to the physical agent(s) on an ongoing basis are employed to establish the development of responses such as hearing loss or VWF over time in an exposed population group in evaluating the risk of health effects due to exposures to vibration, and noise and in creating a generalisable model. In these studies, workers have been grouped by the number of years each person was exposed, and the prevalence (and/or incidence) of the investigated health effect has been tracked.
For vibration, and noise, the resulting growth of prevalence can be expected to form a curvilinear function of the time exposed, the shape of which depends on whether period or point prevalences are analysed. If a study follows a group of workers over a period of time and accumulates the prevalence, calculated from the commencement of exposure, from all persons irrespective of whether they leave the group resulting in a constant population size, the determined prevalence is the period prevalence modelled here. If, at any point in time, all workers are being investigated and, therefore, the population size and its membership are dependent on the date chosen for conducting the crosssectional study, the resulting prevalence is considered a point prevalence.
2.1 Period prevalence model
For a population group commencing an activity in which all members are performing the same work involving essentially the same exposure to vibration, and noise, at first, those most susceptible will show lasting health effects. Over time, this number will increase and at some point reach a maximum value. This results in the prevalence change with time possessing a curvilinear shape, as is evident in all data sets for VWF available to the authors [25, 26], and for noise when age is a surrogate for exposure time [27]. Thus, the period prevalence as a function of exposure time can be modelled using a polynomial of the general form:(1)where t is the time each member of the population group has been exposed, typically provided as the years of employment in an occupation involving the usage of handheld vibrating tools, and a_{0} through a_{n} are numerical constants. a_{0} represents the background prevalence of the health effect in the population that has not been caused by the physical agent(s). In the present study, the background (i.e., nonvibration related) prevalence is excluded from the data and therefore a_{0} is removed from this equation. As the general shape of the period prevalence can be achieved with polynomials of different orders, several orders are compared here regarding their quality of fit to the data. By choosing a data set ranked highly in terms of epidemiologic “quality” (e.g., see Tab. 1 of Ref. [28]) that contains the information necessary to fulfil the purposes of this investigation and by selecting the best polynomial fit, a potentially generalisable model is obtained.
Coefficients of different order polynomial fits to the combined data set for SWS (Stockholm Workshop Scale) left and right hands of Nilsson et al. [25], and the respective r^{2}values. The respective 95percentile confidence intervals are shown in brackets (N.B. The dimensions of the coefficients are given for a dimensionless exposure factor). The bold coefficients are the ones used in the following analyses.
This polynomial is adapted to other population groups by introducing an exposure factor that represents the different daily exposures. Conceptually, the exposure factor is modelled as a timedependent measure of the exposure to include magnitudes that can change during the lifetime exposure, expfac(t). This yields:(2)
2.2 Point prevalence model
Point prevalence is distinguished from period prevalence by the possibility for people to join the population group and for individuals, independent of whether they are affected by health effects from exposure to the physical agent(s), to leave the group. This naturally influences the observed group prevalence and thus the shape of the growth curve represented by the polynomial in equation (2). Hence, as a first approach, this change can be represented by introducing to into equation (2):(3)where to are numerical parameters. The practical application of this model will be limited by the large number of unknown parameters that will need to be determined for highorder polynomials. Hence, in this study, it is assumed that the net change in prevalence from people joining and leaving the group during a given time period (e.g., per year) can be approximately represented by a single parameter, . This would allow the fit to the period prevalence to be adapted to predict point prevalence data by(4)where has been inserted into equation (2). Provided the polynomial coefficients can first be satisfactorily fitted to period prevalence growth data typical for exposure to a given physical agent or agents, the simple model in equation (4) will contain only two unknowns (expfac(t) and ). These can be derived, in principle, in many crosssectional studies of population groups exposed to physical agents such as vibration, and noise, as zero prevalence at zero exposure time can be assumed. In this way, the point prevalence to be expected in the group after any given exposure time may be predicted.
While the general approach includes a timedependent exposure factor, expfac(t), in many cases the work and the tools will not change from day to day and therefore the daily exposure will remain virtually constant over time.
3 Models for vibrationinduced white finger (VWF)
3.1 Period prevalence model
In this study, the previously described approach is applied to the development of VWF in population groups as defined here. The longitudinal study from which the period prevalence data are sourced was reported by Nilsson et al. [25]. The prevalence of VWF in this population group over a period of 36 years is given. It is used here to create a generalisable model using equation (2) in which the exposure factor is assumed to be unity.
Reference [25] provides three sets of data for the period prevalence. The authors have identified VWF in the observed group of workers using the TPS and the SWS clinical staging protocols. The latter is done separately for the left and the right hands. In order to create a model, the data sets obtained by applying the SWS protocol were combined. For this combined data set, regression analyses were carried out with 3rd to 6thorder polynomials to find the lowest order that best represents the general shape of the prevalencetime curve.
To avoid overfitting that creates unwanted deviations from the Sshaped progression evident from the data, three additional “data points” are used to extend the data set to an exposure time of 50 years. These “data points” follow the tendency of the last four points of the data set. The regression analyses are conducted with this extended data set. The results are shown in Figure 1. The values of the polynomial coefficients a_{1}…a_{n} and the respective coefficients of determination, r^{2}, are given in Table 1.
Figure 1 Period prevalence of vibrationinduced white finger (VWF) as a function of exposure time. The SWS (Stockholm Workshop Scale) left and right hand data sets combined are shown as asterisks, with the 3rdorder fit (dotted line), 4thorder fit (thick line), and 6thorder fit (dashed line). 5thorder fit is not shown, as it lies on the 4thorder fit. 
In the regression analyses, both the 3rd and 6thorder fits give good results but showed noticeable deviations from the trends evident from the data for up to 10 years and for more than 30 years of exposure (see Fig. 1). The 4th and 5thorder fits delivered the best fit, judging by the r^{2}values, the difference between the fits being hardly visible. In order to keep the model as simple as possible, the 4thorder polynomial is chosen for further analyses in this study (thick line in Fig. 1).
This 4thorder fit will be considered the presumptive generalisable model for VWF in what follows. It was first used with an exposure factor to fit each data set provided in Ref. [25].
Hence, equation (2) with i up to i = 4 will be employed with the values for the coefficients a_{1} to a_{4} provided for the 4thorder in Table 1 and in which the prevalence, P, is given in percent and the time exposed, t, is in years. No information was provided on the noise exposure of the population group by Nilsson et al. [25]. A value for the fourhour, frequencyweighted, energyequivalent acceleration, A(4) [29], was provided in the study, and there is information on it not having changed during the exposure; hence, the exposure factor is assumed to be constant. In consequence, a constant exposure factor is employed in the generalisable model as a fit parameter in the regression analyses to fit it to each data set provided by Nilsson et al.[25]. The resulting exposure factors and r^{2}values are listed in Table 2 for the data sets reported separately for the right, and left, hands using the SWS classification and for each subject evaluated by the TPS classification (i.e., VWF counted when reported in either or both hands). The TPS data set with its fit is shown in Figure 2 by the continuous line as well as the 4thorder model constructed using the combined data for left and right hands from Figure 1 (dashdotted line).
Figure 2 Period prevalence of vibrationinduced white finger (VWF) as a function of exposure time. The TPS (TaylorPelmear Scale) data set is shown as asterisks, the generalised fit to this data set adapted with an exposure factor is shown by the continuous line, and the 4thorder fit to the data for the left and right hands from Figure 1 by the dashdotted line. 
Exposure factors obtained by fitting the generalisable model to each data set (SWS = Stockholm Workshop Scale, TPS = TaylorPelmear Scale) of Nilsson et al. (1989) [25] by adjusting expfac in the generalisable model and the respective r^{2}values. The 95percentile confidence interval is shown in brackets. The exposure factor is dimensionless.
Figure 2 shows that by introducing a nonunity exposure factor, the model fits the TPS data set very well. The changes in the curve introduced by the exposure factor compared to the generalisable model in Figure 1 are equally evident.
Table 2 shows that taking the data set for the left hand created using the SWS staging as a reference, the exposure factors of the other two data sets vary by plus and minus 10%, respectively. With these exposure factors, the fits achieve r^{2}values of 0.963 and higher. The results for the SWS staging suggest that the hands experienced different exposures, while the difference between the exposure factors necessary to fit the TPS staging and SWS staging for the left hand is reflective of the consequences of combining the data for the two hands as well as uncertainties in the models.
3.2 Application of the period prevalence model to data from another study
The generalisable model is now tested by applying it to data from another study. Futatsuka and coworkers have analysed the prevalence of VWF in forestry workers in Japan [26]. They give results for three groups of workers that were created based on the year in which their exposure began. There is a data set obtained from a group that started in 1958, another that started in 1962, and one in which all workers started in 1966 (subsequently referred to as 1958, 1962 and 1966, respectively). The authors have supplied both period and point prevalence data for each group of workers and tracked them for 12–20 years. The benefit of this study is that workers only used one kind of power tool and had a limited number of work processes (in contrast to the study by Nilsson et al. [25] in the previous section in which workers used several different power tools). The tools used, chain saws, underwent drastic changes in design during this period and consequently in the acceleration at the handles. While a measure of noise exposure was provided, no information on any change in it with the evolution of chain saws during the course of exposure was given. It is possible that there were only insignificant changes in noise exposure during the twenty years of the study. Hence, an attempt was first made to fit the period prevalence data with a timeindependent exposure factor. Under these conditions, a poor fit was obtained by the generalisable model, with the r^{2} values typically in the range 0.50.7 and especially so to the prevalencetime data in the region around 10% prevalence. Therefore, a timedependent exposure factor based on the handle vibration has been used to fit the generalisable model to the three data sets. In Futatsuka and Ueno [26], a measure of the acceleration at the chain saw handles is depicted from 1965 until 1978 (see Fig. 2 of Ref. [26]). This is used to find the appropriate form for a timedependent exposure factor by fitting the data with a regression analysis as shown in Figure 3.
Figure 3 Reduction in the acceleration at chain saw handles over time where time 0 represents the year 1965, the measured accelerations is shown as asterisks, and the fit of the curve is based on equation (5). 
The analysis shows that the exponential function(5)provides an approximate fit to the change in acceleration at the chain saw handles over time. This expression is used as the form for the exposure factor in the analyses of Futatsuka and Ueno’s data, where k, c and b are adapted to fit the generalisable model to each data set. Adjusting these parameters allows to account for the different times in which the respective populations started using chain saws and the corresponding differences in their daily exposure.
The period prevalence data sets and the resulting fits with the exponential exposure factors are shown in Figure 4. The parameters k, c, b and the respective r^{2}values are given in Table 3.
Figure 4 Period prevalence of vibrationinduced white finger (VWF) as a function of exposure time. The data sets are shown as asterisks, and the generalisable model fitted with an exponential exposure factor as a continuous line for: a) the 1958 data, b) the 1962 data, and c) the 1966 data. 
The generalisable period prevalence model fits these data very well using the timedependent exposure factor. This is evident from Figure 4 as well as from the r^{2}values, which are 0.995 and higher. Thus, the generalisable model constructed in Section 3.1 has been shown to fit period prevalence data for VWF from another completely unrelated study of workers using different power tools.
The 1958 data set has the lowest r^{2}value. Further, this is the data set that shows the largest effect of two events during the exposure reported by Futatsuka and coworkers. In 1970, the maximum daily usage time of chain saws in Japan was reduced to 2 hours. Before this, workers had used chain saws for 5 hours per day on average. The second change that shows in the data set occurred in 19731974 when the possibility of changing jobs within the forestry industry was introduced to allow workers to avoid further vibration exposure. Both events occurred at times coinciding with plateaus in the 1958 data after about 10 and 16 years of exposure, as can be seen in Figure 4a.
3.3 Point prevalence model
As already noted, Futatsuka and Ueno [26] provided both point and period prevalence for the same groups of workers. Despite the point prevalence having been recorded for the same time span as the period prevalence, the data are only used until the year 1970 in the present analyses. This cutoff was chosen due to the previously mentioned events that resulted in major changes to the incidence of VWF. The consequent changes in the prevalence are not considered to be representative of the development of VWF, and the generalisable model cannot account for these by introducing only a single adaptive parameter .
The point prevalences of the groups for the restricted time ranges are plotted in Figures 5a–5c together with the respective model fits. The fits were obtained by using the generalisable model modified to include , so employing equation (4) up to i = 4 with the values for a_{1}–a_{4} in Table 1 and with the timedependent exposure factor that was determined previously for the corresponding period prevalence data set and adapting only the parameter . Here, has been added to the a_{1} of the model in equation (4). Values for and the resulting r^{2}values are given in Table 4.
Figure 5 Point and period prevalence of vibrationinduced white finger (VWF) as a function of exposure time. The data sets are shown as asterisks and capped at the year 1970. The point prevalence model obtained from equation (4) up to the 4thorder with the respective parameter values from Table 1 with the timedependent exposure factor of the respective period prevalence data set fitted by introducing is shown by the continuous line for: a) the 1958 data, b) the 1962 data and c) the 1966 data. The dashdotted line shows the model fitted to the respective period prevalence data, i.e., equation (2) up to i = 4 with the values for a_{1}–a_{4} from Table 1 without , to show its influence on the relation between prevalence and exposure time. 
 and r^{2}values for the models obtained from fitting the generalisable model to the point prevalence data sets from Futatsuka and Ueno [26] using equation (4) with the parameter values from Table 1 for the 4thorder and the timedependent exposure factor. The timedependent exposure factors are determined from the corresponding period prevalence data (see Tab. 3). The 95percentile confidence intervals for are given in brackets. The bold coefficients are the ones used in the following analyses.
Both Figures 5a–5c and the r^{2}values in Table 4 show that the achieved fits represent the data well, but also bigger deviations than for the period prevalences. Furthermore, it can be seen that the time until the first workers are affected by VWF can only be accounted for by the model taking on negative values at small values of t. While prevalence cannot be negative, the artefact allows the model to follow the progression of the data points. The effect of including the parameter becomes evident by comparing the point prevalence fits (continuous lines) to the corresponding period prevalence fits shown as dashdotted lines in the figure.
3.4 Linear and polynomial interpolation of point prevalences from a single value
Most studies of health effects from exposure to physical agents are crosssectional in nature and hence report only the point prevalence at the time of the study. In order to compare such data obtained in different studies, it is often necessary to predict the exposure time to reach a common prevalence in each study. In the metaanalysis undertaken by Nilsson et al. [28], the observed prevalence in a study at the respective exposure time was interpolated linearly to estimate the exposure time at the prevalence that was being investigated, which was 10%. This will lead to interpolated exposure times with varying error. Figures 6a–6c show examples of two methods of interpolating: linear and polynomial, the latter using the model for predicting point prevalence described here.
Figure 6 Point prevalence of vibrationinduced white finger (VWF) as a function of exposure time. The data sets from Futatsuka and Ueno [26] are shown as small asterisks, the fits to the respective data sets from Section 3.3 as a thin line and the assumed known data point from which to estimate the exposure time to reach 10% prevalence as a large asterisk. Linear interpolation from the assumed known data point is shown by the dashed line and polynomial interpolation using equation (4) up to the 4thorder with the respective parameter values from Table 1 by the thick line. The prevalence of interest (10%) is shown by the horizontal dotted line. 
In these examples using the data sets from Futatsuka and Ueno [26], the assumed (single) observed point prevalence reported in the notional crosssectional study is shown by the large asterisk. Both methods of interpolation assume that the prevalence of VWF is zero at zero exposure time. Linear interpolation is shown by the dashed line. Polynomial interpolation is performed by using the method described in Section 3.3 to determine the appropriate value for from the observed point prevalence and exposure time, and zero prevalence and exposure time. As before, the fits were obtained by using the point prevalence model in equation (4) up to the 4thorder with the respective parameter values from Table 1 with the timedependent exposure factor that was determined previously for the corresponding period prevalence data set and adapting only the parameter . While the polynomial interpolation of the data (thick line) shows some deviation from the fit of the point prevalence model to the entire point prevalence data set (thin line), it follows its trend closely. Depending on the assumed known data point, it varies which of the two interpolation methods estimates the exposure time for 10% prevalence with the least error. It is evident from Figures 6a–6c that the accuracy of estimating the exposure time at which a given prevalence occurs will depend on multiple factors: the rate of development of the prevalence over time and where on the progression the assumed known data point lies, as well as what percentage prevalence is to be estimated.
3.5 Exposure factor
3.5.1 Specification
The study initially employed here to develop prevalence models did not include the noise exposure of the population group, as already noted. Therefore, constructing an exposure factor or occupational index that includes both noise and vibration is not possible from these data. However, the study did provide separate results for the left and right hands. A twosided ttest of the exposure factors for the period prevalence models of these data reveals that expfac for the left hands differs significantly from that for the right hands (Tab. 2) (p < 0.001). The difference between the prevalences of VWF experienced by the left and right hands in this population group suggests that the dominant exposure factor was localised to the hands and thus is unlikely to have been noise, as this would be expected to influence the development of symptoms equally in all extremities. Futatsuka and Ueno [26] reported that the populations they studied were exposed to sound levels between 105 and 115 dBA for, on average, 25 hours per week with a range of 10–30 h. While this information would enable calculation of the combined noise and vibration index, it has not been attempted in view of the changes in vibration experienced by the workers during the course of exposure (see Fig. 3) and the unknown correlation between changes in the noise and vibration produced by the chain saws. A primary change in the technology during this period was an increase in the engine speed for optimal performance. This commonly led to reduced vibration and often also reduced noise [8]. Additionally, vibrationisolation systems for the handles, which reduced vibration but not noise, were introduced in the 1970s.
Information on these and other potential influencing factors is unavailable. Moreover, the observation that the exposure factors for Futatsuka and Ueno’s [26] data sets provide good fits to the health effect, when following approximately the changing acceleration of the tools, suggests that the previously unspecified exposure factor may be linked to the vibration exposure by a relation of the form(6)in which X is either 4 or 8, A(4, t) is the daily 4h frequencyweighted, energyequivalent, dominant component acceleration and A(8, t) is the daily 8h frequencyweighted, energyequivalent, triaxial accelerationat time t [29, 30] experienced by a population group of interest and A(X)_{ref} is the corresponding daily frequencyweighted, energyequivalent, triaxial acceleration experienced by a reference group, which is yet to be selected. The index q also remains to be determined.
3.5.2 Numerical values for expfac(t)
In order to apply the models to different population groups, numerical values will need to be established for expfac(t), which from equation (6) requires numerical values for q and A(8)_{ref}. As the generalisable model is based on the data from Nilsson et al. [25], the A(4)value given in that study needs to be converted into an A(8)value in order to be used as A(X)_{ref} in studies with vibration measurements complying with ISO 53491:2001. For this purpose, the given frequencyweighted, singleaxis accelerations of the tools used were converted into triaxial values using the respective mean, minimum and maximum conversion factors for the populations of chipping hammers (N = 186), straight grinders (N = 287) and die grinders (N = 194) extracted on 11 January 2021 from a database of field measurements operated by the Health and Safety Executive UK (P. Pitts, private communication). These were constructed from the mean values of the dominant singleaxis and triaxial accelerations for populations of the different tool types. Using these triaxial values and the given usage times of each tool, the total mean, minimum and maximum A(8)values were calculated. The A(8)value was assumed to be constant, just like the A(4)values reported by Nilsson et al. [25], who stated that there were “no major changes in the vibration exposure”.
In a study by Bovenzi [31], the signs and symptoms experienced by forestry and stone workers were established over the course of three years and form the only sources of data known to the authors that are suitable for the present analysis to determine q. Thus, for each population group, there were two prevalences observed at two different exposure times. It should be noted the prevalences were determined for exposure times that were the average times workers were employed using power tools at the time of the study, and not the prevalences at equal exposure times of individual group members as used in the models of Sections 3.1–3.4. There is a detailed description of the vibration measurements undertaken in this study [31], but no A(8)values were reported. There is also no mention of the vibration exposures changing over time. Hence, the A(8)values reported elsewhere by the same research group within the same time period on the same groups of workers are used here [32].
To determine q, the point prevalence model is applied to the two data sets from Bovenzi [31] using the exposure factor as described in equation (6) and the converted A(4)_{ref}values from Nilsson et al. [25], henceforth referred to as A(8)_{ref}. Thus,(7)predicts the point prevalence of VWF for these data sets.
This equation is used in a regression analysis with the data from Bovenzi [31] and the A(8)values established for the two analysed groups. In view of the range of conversion factors from singleaxis A(4)_{ref} to triaxial A(8)_{ref}, the analysis has been performed for three values of A(8)_{ref}: maximum, mean and minimum. The A(8)value for the forestry workers is almost equal to the converted mean A(8)_{ref} and also very close to the minimum value, and thus the exposure factor is almost equal to 1 for two of the three cases. This does not allow for a determination of q as a value close to 1 requires large changes in the exponent to adjust the magnitude of the exposure factor (i.e., from −22 to 0.12). Hence, the data for the stone workers are used to determine a value for q (i.e., two nonzero prevalences and zero prevalence at zero exposure). The results of these analyses using equation (7) in which q and are fitting parameters are shown in Table 5.
It should be noted that the values for q from the regression analyses with the respective mean, minimum or maximum A(8)_{ref} result in the same exposure factor for the stone workers in all three cases (expfac_{stone} = 1.318). While the maximum A(8)_{ref} is roughly 1.7 times the mean A(8)_{ref}, the maximum q is roughly 2.3 times the mean value. The minimum A(8)_{ref} as well as the minimum q are both about 10% smaller than their respective mean values.
3.5.3 Tests of models
In order to analyse the stability of the results in Table 5 and how applicable the respective qvalues are to other groups of workers, the qvalues in Table 5 together with the respective A(8)_{ref}values are now used to fit the forestry data sets from Bovenzi [31] with as a fitting parameter in equation (7) (i.e., using data for two nonzero prevalences and zero prevalence at zero exposure for each data set). The results are shown in Table 6.
Fitted values for forestry workers using both nonzero data points and zero prevalence at zero exposure time when q set to the mean, minimum and maximum values of the stone workers in Table 5 and the respective A(8)_{ref}values are used. The fit employs equation (7) with the q in the respective column, the corresponding A(8)_{ref} and the A(8) of the respective group of workers.
For the forestry workers, the resulting exposure factor and in Table 6 depend on the qvalue selected for the analysis. The r^{2}values of all fits are unity. The exposure factor calculated for the forestry workers using the mean A(8)_{ref} and the corresponding qvalue deviate least from that obtained from an analysis in which both q and have been fitted (expfac_{forestry} = 0.941). Therefore, it appears appropriate for further analyses to employ the mean A(8)_{ref} and qvalues.
The results of the analysis of all values in the data set for the stone workers (i.e., two observed prevalences and an assumed zero prevalence at zero exposure time) are shown by continuous lines in Figure 7. The graphs show both types of analysis that have been conducted in which all the available data points, depicted by asterisks, are fitted with the models previously developed and summarised by Eqs. (7) and (4) for i up to 4 and the respective values for a_{1}–a_{4}. The parameter adjusted to obtain a fit to the data is in both cases. In Figure 7a, the fit to the data is obtained using equation (7), in which the exposure factor is a fixed value calculated using equation (6), and in Figure 7b the exposure factor is an additional timeindependent fit parameter and the fit is obtained using equation (4) with the 4thorder parameters.
Figure 7 Vibrationinduced white finger (VWF) point prevalence data as a function of exposure time from Bovenzi [31] for stone workers shown as asterisks. In (a) and (b), the fit to all known data points is shown by the continuous line. In (a), the exposure factor is a fixed value calculated using equation (6) as included in equation (7) that is used for the prevalence model, and in (b) the exposure factor is a timeindependent additional fit parameter and the prevalence model uses equation (3) with the 4thorder parameters. is an adjustable fit parameter in both models. In (a), the model with the fit parameter set to zero (dashed line) shows the effect of when the same calculated exposure factor is used. 
In summary, it is evident from inspection of the modelled prevalences in Figure 7 that the models represented by equation (7) can fit the data well (numerical values are shown in Table 7, top row), thus implicitly confirming the appropriateness of approximating expfac for VWF by equation (6) for this data set. The analysis also suggests that the models are insensitive to specifying exposure duration either as that for individual group members or as the average duration of employment of group members.
Observed and calculated prevalences at the time of the first survey by Bovenzi [31] predicted by models when fitting both nonzero data points and zero prevalence at zero time (top row, using calculated expfac or fitted expfac), and when using one nonzero data point and zero prevalence at zero time (second and third rows using calculated expfac). is a fitted parameter in both models. expfac is also a fitted parameter in the model using equation (4) with the 4thorder parameters.
3.5.4 Application to interpolation
To interrelate prevalence data obtained in different population studies, which would have usually been obtained after different exposure times and possess different prevalences, it is necessary to establish the exposure time to reach a common prevalence (e.g., 10%) as in Section 3.4 in order to estimate the limits of “safe” exposures.
It is now appropriate to establish whether the exposure factor defined in Section 3.5.1 can be used for this purpose for population studies reporting only a single prevalence, mean exposure time and A(8). This information is commonly all that is available. For this reason, and because the mean A(8)_{ref}value is likely to be closest to the true A(8)value of the reference population, further analyses are based on this value. Converting the single axis A(4)value provided by Nilsson et al. [25] into a triaxial A(8)value is bound to introduce uncertainty. Therefore, we approximate the mean qvalue to q = 0.3 for use in future analyses.
In the following application of the point prevalence model, a comparison is made between interpolation using the calculated exposure factors in equation (7) and linear interpolation. The model is evaluated for both groups of workers in Bovenzi [31] assuming only one nonzero data point as well as zero prevalence at zero exposure time.
The results of the analyses are shown in Figure 8 and in the rows labelled “only one data point” in Table 7. The assumed known prevalence is shown in Figure 8 by the asterisk for the stone workers and open circles for the forestry workers. The observed prevalences at 15 years’ exposure (cross and filled circle for the stone and forest workers, respectively) are assumed “unknown” for the purposes of evaluating the performance of the models. The predicted relations between prevalence and exposure time from which to estimate the prevalence after 15 years exposure are shown by the continuous lines. The calculated exposure factors are expfac_{stone} = 1.31 and expfac_{forestry} = 1.00 when q = 0.3 (not shown), and the values of are listed under “calculated expfac”. The observed and predicted prevalences for the “unknown” data point are presented in Table 7 for the two groups of workers as well. In both cases, has a positive value, indicating an increase in curvature of the model fit relative to a period prevalence model (as can also be seen by comparing the continuous and dashdotted lines in Figure 7).
Figure 8 Point prevalence of vibrationinduced white finger (VWF) as a function of exposure time. Here, only the higher prevalence data given in Bovenzi [31] for the stone and forestry workers are assumed to be known, shown by an asterisk and open circle, respectively, and the prevalence is interpolated from that value using linear interpolation (dashed lines) and polynomial interpolation (continuous lines). The second (lower) prevalence point is assumed to be unknown and is shown by a plus sign (stone workers) and a filled circle (forest workers). The prevalence model uses equation (7) that includes the calculated exposure factor by means of equation (6). A horizontal (dotted) line at 10% prevalence shows the target prevalence of some studies [6, 28]. 
Using the values from Table 7 in equation (7) to predict the prevalence of the second data point for both groups of workers at an exposure time of t = 15 years shows that the calculated exposure factor allows for the prevalence of the “unknown” data point to be estimated precisely. Thus, the polynomial method described in Section 3.4 for interpolating point prevalences from a single observed prevalence has been confirmed when the exposure factor is calculated by means of equation (6). The only information available with which to predict the point prevalence was the prevalence of VWF reported in the study, the A(8)value that characterised the vibration exposure of the population group and the mean time of exposure at the time of the study.
Aside from the data points and the respective fits, Figure 8 also shows linear interpolation as a dashed line and a target prevalence for which the exposure time is to be estimated by a horizontal dotted line (at 10% prevalence). In both cases, linear interpolation from the “known” data point does not go through the second (“unknown”) prevalence point. In addition, it is evident that linear interpolation, as applied in Nilsson et al. [28] estimates a smaller number of years for 10% prevalence to occur in these population groups than the polynomial models.
4 Discussion
In a recent metaanalysis of 4335 studies of VWF published since 1945, Nilsson et al. found 41 that adhered to strict criteria for scientific “quality” and hence are believed reliable [28]. Of these, the study employed to construct the generalisable model (Eq. (2)) up to the 4thorder with the parameter values from Table 1) was ranked #18 (#1 most reliable), and that used in Section 3.5 was ranked #5. The study employed in Sections 3.2–3.4 was not ranked by Nilsson et al. [28].
The present analyses show that a polynomial model can be used to fit and hence represent period prevalence data for VWF resulting from exposure to one or more physical agents described initially by an exposure factor set to unity. This becomes evident from the goodness of the fit achieved for the data from Nilsson et al. [25]. r^{2}values of 0.963–0.985 for the three data sets (SWS left, SWS right, and TPS both hands), together with the close relation between the reported data and the predicted prevalence curve in Figure 1, provide evidence that a polynomial model of the 4th order can achieve a near perfect fit to the mean period prevalence for the left and right hand data sets (Table 1). Thus, the first hypothesis has been satisfied.
Applying this model (Eq. (2)) up to the 4thorder with the parameter values from Table 1) to period prevalence data from a second longitudinal study by Futatsuka and Ueno [26] and introducing a nonunity exposure factor to account for the differences in exposure between the observed groups in the two studies suggests that the modelling approach is generalisable as required by the second hypothesis. With a timedependent exposure factor that reflected the known changes in tool vibration, the model based on the data from Nilsson et al. [25] achieved even higher r^{2}values for the data from the Futatsuka and Ueno (1985) study than for the data sets for individual hands in Nilsson et al., with the lowest being 0.995 (Table 3).
Fitting the generalisable model to the data of Futatsuka and Ueno [26] showed that the quality of the fit depends on knowledge of the time course of the exposure that the respective population groups experienced. If the exposure and, therefore, the exposure factor is constant or nearly constant over time, the period prevalence model is likely to work well. In cases of a noticeably changing exposure, it may fail, especially if the changes are not known well enough to incorporate them adequately into the exposure factor. With a constant exposure factor, it was not possible to fit the data from Futatsuka and Ueno [26]. Only with approximate knowledge of the change in tool vibrations over time was it possible to fit the period prevalence data.
Applying the generalisable model created from the combined SWS data set for the left and right hands in Nilsson et al. [25] to each of the three data sets from that paper and adapting the model with an exposure factor also demonstrated the uncertainty in assessing the exposure. The differences in the exposure factor are shown in Table 2. Classifying VWF in the same population with two different staging protocols created a difference of 10% in the exposure factor, which was comparable to that obtained when comparing the results for the left and the right hands. Yet, the difference between the models for the left and the right hands, which led to the exposure factor for the left hands being ~10% greater than that for the right hands, does not reflect the difference in A(4) between the left and the right hands that was reported by Nilsson et al. [25]. Their vibration measurements found that the vibration exposure of the right hands exceeded that of the left by about 25%. The disagreement between the growth of VWF in the two hands and vibration exposure opens the possibility of physical agents or factors other than vibration influencing the exposure factor in this population. Noise exposure is considered unlikely to be a dominant contributor to the exposure factor owing to the difference in signs and symptoms between left and right hands.
The role of other cofactors (e.g., ergonomic, biodynamic, lifestyle and environmental factors) in the signs and symptoms experienced by this population group remains unclear. Nevertheless, the ability of the model to fit data involving exposure both to single power tools (chain saws) and multiple power tools (chipping hammers, straight grinders and die grinders) is noteworthy. A further complication for the models is interventions, such as a mandatory reduction of permitted tool usage time per day or the wearing of hearing protectors for exposures involving noise, as these may cause major deviations in the development of the prevalence within the group. The effects of such interventions are hard to account for in the models.
By introducing a single parameter intended to account for changes in the number of group members, it was possible to adapt the period prevalence model to predict the point prevalence of Futatsuka and Ueno’s data as a function of the duration of exposure. While the model produced a negative prevalence during the initial period of exposure when the observed point prevalence was zero, the goodness of the fit was high overall (Table 4). The model fits the growth of prevalence in these data sets, suggesting that the third hypothesis is achievable.
The parameter was initially intended to allow the point prevalence model to differ from the period prevalence model, as can be seen from Figure 5. Table 4 shows the range of for the Futatsuka and Ueno [26] data. Predominantly, people suffering from the health effect are likely to leave the group (known as the “healthy worker effect” [12]), which is thought to be the reason for being negative for all three data sets. Positive values of , however, as found in the model fits to the data from Bovenzi [31] in Tables 5–7 and shown in Figure 7a, do not necessarily reflect changes in group membership and necessitate rethinking the interpretation of . Reference to Figure 2 shows that changing the exposure factor produces changes in the predicted prevalence in which the general curve shape is preserved. These can be considered to result from the exposure factor acting as a form of amplifier. With this interpretation, serves to adjust the curve shape to accommodate factors that render the exposure of one population group different from that experienced by another. They could involve physical agents other than vibration and factors peculiar to a particular work activity in addition to people entering and leaving the group. Such factors could include exposures involving overhead work, for example, which is known with other postures to influence the vibration transmitted within the hand and arm [33]; the wearing of antivibration gloves that influence the biodynamic response of the handarm system and hence their effectiveness in attenuating the vibration entering the hand [34, 35]; and the effect of climate, exposure in a tropical environment being known to precipitate fewer vasospasms [36]. While these factors affect exposure and could possibly be included in the models by means of additional parameters in equation (6), this has not been attempted here. An excellent fit to the available point prevalence data has been obtained by adjusting , as can be seen from Figure 7 and Table 7 (“both data points”). This result, obtained by models with calculated and fitted exposure factors that yield almost identical values for expfac and , suggests that the sources of variability can provisionally be subsumed in , hence allowing the simplicity of twoparameter models to be retained while accepting the limitations inherent in using equation (6).
Being able to model the development of the prevalence of a health effect resulting from exposure to a physical agent enables estimation of the prevalence in a population group at a different exposure time from that at which a single point prevalence value was observed. Hence, analyses have been conducted with the goal of being able to interpolate to a chosen prevalence from any higher point prevalence.
This has been done for handarm vibration by linear interpolation, for example, in Nilsson et al. [28] and Scholz et al. [6]. Linear interpolation cannot account for the latency, i.e., the time until the first case of the health effect occurs in the group, or for the saturation in prevalence that occurs after a certain exposure time.
Examples of comparisons of the two interpolation methods are given in Sections 3.4 and 3.5. In Figure 6, it is evident that linear interpolation does not replicate the progression of point prevalence with time. Interpolation with the polynomial model follows the trend of the data points and hence is close to the assumed “true” exposure time (shown by the thin line, which is the best fit to all data points) at 10% prevalence. Thus, the fourth hypothesis has been achieved. However, depending on where on the progression of the prevalence curve the (single) observed data point lies, linear interpolation may get closer to the assumed true exposure time at a given prevalence (e.g., see Fig. 6c).
As in most cases only a single data point will be provided by a population study conducted at a single point in time, the exposure factor cannot be derived from fitting the model to a progression of period or point prevalences as in Sections 3.1–3.3. Therefore, it is essential to create a link between the exposure factor and a quantitative measure of exposure, which, in view of the lack of information on the contribution of noise to the exposure factor in the studies, has become, by default, a metric of vibration exposure, here A(4) or A(8). In Section 3.5, the exposure factor is defined as the ratio between the A(8)value of the respective study and the overall A(8)value estimated from Nilsson et al.’s [25] measurements to the power of q. This approach is applied to the data sets from a study by Bovenzi [31]. When models are fitted to the two prevalences observed in the stone workers population group three years apart, the model constructed using the quantitative measure of exposure (Eq. (7)) and the model using the fitted exposure factor (Eq. (4) with the 4thorder parameters) produce excellent fits to the data (see Fig. 7). This suggests that calculating the exposure factor using equation (6) produces a stable point prevalence model and yields acceptable results even though the magnitude of q cannot be determined with precision and positive values of are required. Furthermore, when only one observed prevalence is available, the applicable model, equation (7), can accurately predict the “unknown” observed prevalence (e.g., see second and third rows in Table 7). These results suggest that for VWF the exposure factor can be quantified by means of the A(8)value alone, provided both positive and negative values of are accepted. This is not without reservation as uncertainties in the frequencies to include in, and the construction of, A(4) and A(8) are well documented [6, 37–39], as well as the need to consider alternate measures of vibration magnitude [40, 41].
Nevertheless, the lack of information in the studies employed in this contribution on the role of different cofactors in the development of VWF does not exclude their involvement. In particular, the vasoconstriction in response to exposure to noise, and to vibration entering the hands, suggests that a metric constructed to combine the effects of these two physical agents may prove essential for quantifying the hazard of developing VWF from some exposures.
It remains to be seen whether the generalised models (Sect. 2) proposed here will find application to predict health effects from exposure to other physical agents that have a binary outcome, such as hearing impairment and back pain.
5 Conclusions
A method for modelling the prevalence of a health effect in a population group resulting from habitual exposure to physical agents such as vibration, and noise, has been proposed. The method involves applying a factor representing the daily exposure, expfac(t) to a polynomial fit to observed prevalencetime data. A suitable population group for this modelling is one in which all members are engaged neardaily in the same activity that involves exposure to the physical agent(s).
For VWF, the period prevalence from the onset of exposure and its growth in time resulting from vibration entering the hands and noise can be predicted by a 4thorder polynomial. If an adjustment is made for the difference in exposure between population groups, the same polynomial model can be applied to predict the period prevalence of VWF in all the considered groups, leading to its characterisation as a generalisable model.
This model can be transformed to fit the point prevalence of VWF in a population group by introducing a single numerical parameter, , if the exposure and therefore the exposure factor, expfac(t), of the group are adequately known.
From the studies available for the present analyses, it has been shown possible to construct the exposure factor for VWF on the basis of a ratio of the observed A(8)value to the estimated A(8)value for the data that led to the generalisable polynomial, A(8)_{ref}. This makes the model easily adaptable to data from any group of workers for which the prevalence and their daily vibration exposure in the form of an A(8)value is given at one group exposure time. The index of the ratio is not known with precision, and its definition will need to be improved in future work.
A comparison of interpolation methods for predicting the development of VWF in a population group from a single observed prevalence has shown that the point prevalence at an earlier time and lower prevalence can be inferred from the point prevalence model. The prediction involves a measure of the exposure factor for the group and a fitted value for . The method has been validated in population groups in which the prevalence was recorded on at least two occasions separated in time.
The analyses have shown that the first two hypotheses can be confirmed. They also show that the third and fourth hypotheses can be confirmed if the interpretation of the adjustable parameter is broadened to include factors that render the exposure of one population group different from that experienced by another.
It is believed that the time course of the period and point prevalences of other health effects resulting from habitual exposure to a physical agent, such as noise or wholebody vibration, could be treated by equivalent models.
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or notforprofit sectors.
Conflict of interest
The authors declare that they have no known competing interests that could have appeared to influence the work reported in this paper.
Data availability statement
The research data associated with this article are included within the article.
Authorship contribution statement
Magdalena F. Scholz: Conceptualization Ideas, Methodology, Software Programming, Validation, Formal Analysis, Investigation, Data Curation, Writing – Original Draft, Visualization. Anthony J. Brammer: Conceptualization Ideas, Methodology, Investigation, Writing – Original Draft, Writing – Review & Editing, Supervision. Steffen Marburg: Resources, Writing – Review & Editing, Supervision.
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All Tables
Coefficients of different order polynomial fits to the combined data set for SWS (Stockholm Workshop Scale) left and right hands of Nilsson et al. [25], and the respective r^{2}values. The respective 95percentile confidence intervals are shown in brackets (N.B. The dimensions of the coefficients are given for a dimensionless exposure factor). The bold coefficients are the ones used in the following analyses.
Exposure factors obtained by fitting the generalisable model to each data set (SWS = Stockholm Workshop Scale, TPS = TaylorPelmear Scale) of Nilsson et al. (1989) [25] by adjusting expfac in the generalisable model and the respective r^{2}values. The 95percentile confidence interval is shown in brackets. The exposure factor is dimensionless.
 and r^{2}values for the models obtained from fitting the generalisable model to the point prevalence data sets from Futatsuka and Ueno [26] using equation (4) with the parameter values from Table 1 for the 4thorder and the timedependent exposure factor. The timedependent exposure factors are determined from the corresponding period prevalence data (see Tab. 3). The 95percentile confidence intervals for are given in brackets. The bold coefficients are the ones used in the following analyses.
Fitted values for forestry workers using both nonzero data points and zero prevalence at zero exposure time when q set to the mean, minimum and maximum values of the stone workers in Table 5 and the respective A(8)_{ref}values are used. The fit employs equation (7) with the q in the respective column, the corresponding A(8)_{ref} and the A(8) of the respective group of workers.
Observed and calculated prevalences at the time of the first survey by Bovenzi [31] predicted by models when fitting both nonzero data points and zero prevalence at zero time (top row, using calculated expfac or fitted expfac), and when using one nonzero data point and zero prevalence at zero time (second and third rows using calculated expfac). is a fitted parameter in both models. expfac is also a fitted parameter in the model using equation (4) with the 4thorder parameters.
All Figures
Figure 1 Period prevalence of vibrationinduced white finger (VWF) as a function of exposure time. The SWS (Stockholm Workshop Scale) left and right hand data sets combined are shown as asterisks, with the 3rdorder fit (dotted line), 4thorder fit (thick line), and 6thorder fit (dashed line). 5thorder fit is not shown, as it lies on the 4thorder fit. 

In the text 
Figure 2 Period prevalence of vibrationinduced white finger (VWF) as a function of exposure time. The TPS (TaylorPelmear Scale) data set is shown as asterisks, the generalised fit to this data set adapted with an exposure factor is shown by the continuous line, and the 4thorder fit to the data for the left and right hands from Figure 1 by the dashdotted line. 

In the text 
Figure 3 Reduction in the acceleration at chain saw handles over time where time 0 represents the year 1965, the measured accelerations is shown as asterisks, and the fit of the curve is based on equation (5). 

In the text 
Figure 4 Period prevalence of vibrationinduced white finger (VWF) as a function of exposure time. The data sets are shown as asterisks, and the generalisable model fitted with an exponential exposure factor as a continuous line for: a) the 1958 data, b) the 1962 data, and c) the 1966 data. 

In the text 
Figure 5 Point and period prevalence of vibrationinduced white finger (VWF) as a function of exposure time. The data sets are shown as asterisks and capped at the year 1970. The point prevalence model obtained from equation (4) up to the 4thorder with the respective parameter values from Table 1 with the timedependent exposure factor of the respective period prevalence data set fitted by introducing is shown by the continuous line for: a) the 1958 data, b) the 1962 data and c) the 1966 data. The dashdotted line shows the model fitted to the respective period prevalence data, i.e., equation (2) up to i = 4 with the values for a_{1}–a_{4} from Table 1 without , to show its influence on the relation between prevalence and exposure time. 

In the text 
Figure 6 Point prevalence of vibrationinduced white finger (VWF) as a function of exposure time. The data sets from Futatsuka and Ueno [26] are shown as small asterisks, the fits to the respective data sets from Section 3.3 as a thin line and the assumed known data point from which to estimate the exposure time to reach 10% prevalence as a large asterisk. Linear interpolation from the assumed known data point is shown by the dashed line and polynomial interpolation using equation (4) up to the 4thorder with the respective parameter values from Table 1 by the thick line. The prevalence of interest (10%) is shown by the horizontal dotted line. 

In the text 
Figure 7 Vibrationinduced white finger (VWF) point prevalence data as a function of exposure time from Bovenzi [31] for stone workers shown as asterisks. In (a) and (b), the fit to all known data points is shown by the continuous line. In (a), the exposure factor is a fixed value calculated using equation (6) as included in equation (7) that is used for the prevalence model, and in (b) the exposure factor is a timeindependent additional fit parameter and the prevalence model uses equation (3) with the 4thorder parameters. is an adjustable fit parameter in both models. In (a), the model with the fit parameter set to zero (dashed line) shows the effect of when the same calculated exposure factor is used. 

In the text 
Figure 8 Point prevalence of vibrationinduced white finger (VWF) as a function of exposure time. Here, only the higher prevalence data given in Bovenzi [31] for the stone and forestry workers are assumed to be known, shown by an asterisk and open circle, respectively, and the prevalence is interpolated from that value using linear interpolation (dashed lines) and polynomial interpolation (continuous lines). The second (lower) prevalence point is assumed to be unknown and is shown by a plus sign (stone workers) and a filled circle (forest workers). The prevalence model uses equation (7) that includes the calculated exposure factor by means of equation (6). A horizontal (dotted) line at 10% prevalence shows the target prevalence of some studies [6, 28]. 

In the text 
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