International Journal of Epidemiology © International Epidemiotogical Association 1992
Vol. 21, No. 6 Printed in Great Britain
Pleural Plaques and Exposure to Asbestos: A Mathematical Model BENGT JARVHOLM
Pleural plaques are fibrous lesions, usually situated on the parietal pleura. They are common in people exposed to asbestos but their clinical relevance remains uncertain.1"3 Pleural plaques can also be caused by other fibrous materials, including some zeolites. Even environmental exposure to asbestos has been known to cause an increased prevalence of pleural plaques in some geographical areas, but in most areas occupational exposure to asbestos seems by far the most common cause.3 Other pleural diseases which can be caused by exposure to asbestos are pleural mesothelioma, pleurisy and fibrosis of the visceral pleura, the latter often associated with extension of pleural fibrosis into the lung. A common mathematical model,4 based on epidemiological data, for describing the association between exposure to asbestos and the mortality rate I(t) of mesothelioma is:
KO =
1°
dependent on time since onset of asbestos exposure. Another common observation is that there is a minimal latency period for pleural plaques.2 The aim of this study was to find a mathematical model to describe the occurrence of pleural plaques after exposure to asbestos. A simple model for incidence is assumed. Since the prevalence is empirically observed, it is necessary to transform the model of incidence into a model of prevalence. The fit of the model is then examined in a group of shipyard workers exposed to asbestos. THE MODEL OUT simple model is that incidence depends on cumulative exposure. However, measurements of fibre concentrations are rarely available to make an absolute estimate of cumulative exposure. A crude approximation of cumulative exposure can be taken from the duration of exposure. A model analogous to that proposed for mesothelioma for incidence density in workers still exposed to asbestos would then be:
((t-10) 3 -(t-10-d) 3 )t> 10 + d (t-10) 3 10 + d > t > 1 0 t < 10
I (t) = k(t-w)» t > w (1) I (t) = 0 t< w (2)
where t denotes years since first exposure, d the duration of exposure in years, and f the average concentration of exposure in fibres/cm3. K,^ is a constant depending on type of asbestos. This model emphasizes the time since onset of exposure. The use of a minimal latency period of 10 years in the model is supported in empirical observations.3 Epidemiological data also support the hypothesis that the risk of pleural plaques is highly
where k and a are constants, w is the minimum latency period and t the time since onset of exposure. However, since the prevalence of pleural plaques is measured equation (2) had to be transformed to include the prevalence instead of I (t). Assuming that at some time t since onset of exposure, N, (still employed) people remain to be observed from the original group (Nt0). The number of observable new cases in a short time interval (At) would then be:
Department of Occupational Medicine, Sahlgren's Hospital, St Sigfridsgatan 85, S-412 66 GOteborg, Sweden.
1180
Downloaded from http://ije.oxfordjournals.org/ at University of Winnipeg on August 21, 2015
Jarvholm B (Department of Occupational Medicine, Sahlgren's Hospital, St Sigfridsgatan 85, S-412 66 Gflteborg, Sweden). Pleural plaques and exposure to asbestos: A mathematical model. International Journal of Epidemiology 1992; 21: 1180-1184. The objective of this study was to find a model to describe the relationship between the occurrence of pleural plaques and exposure to asbestos. A simple model based on the cumulative exposure was postulated and empirically tested on shipyard workers occupational^ exposed to asbestos. Exposure time was used to approximate the cumulative dose. It was found that the incidence of pleural plaques could be described as K(t-w)* where ' f is time since onset of exposure; 'K' is a constant that would depend on the level of asbestos exposure; 'W is a latency period and was around 13 years; 'a' is a constant that was 0.4. In subgroups of the workers, i.e. plumbers, fitters and platers, 'a' was 0.4, 0.6 and 0.2 respectively.
1181
PLEURAL PLAQUES—A MODEL
N, (1-P) I(t)At = N,Ap
where P is the prevalence at t, I(t) the incidence at t and Ap is the change in prevalence between t and t +At. This formula assumes that there is no selective withdrawal from the observable group, i.e. that employees with or without plaques terminate employment (hence the possibility to be observed) at different rates. Given that pleural plaques are benign lesions with no pain, and other asbestos-associated conditions such as tumours and fibrosis are far less common than pleural plaques in most data sets, this seems to be a reasonable assumption. Including (1) in (2) gives (if P
Vol. 21, No. 6 Printed in Great Britain
Pleural Plaques and Exposure to Asbestos: A Mathematical Model BENGT JARVHOLM
Pleural plaques are fibrous lesions, usually situated on the parietal pleura. They are common in people exposed to asbestos but their clinical relevance remains uncertain.1"3 Pleural plaques can also be caused by other fibrous materials, including some zeolites. Even environmental exposure to asbestos has been known to cause an increased prevalence of pleural plaques in some geographical areas, but in most areas occupational exposure to asbestos seems by far the most common cause.3 Other pleural diseases which can be caused by exposure to asbestos are pleural mesothelioma, pleurisy and fibrosis of the visceral pleura, the latter often associated with extension of pleural fibrosis into the lung. A common mathematical model,4 based on epidemiological data, for describing the association between exposure to asbestos and the mortality rate I(t) of mesothelioma is:
KO =
1°
dependent on time since onset of asbestos exposure. Another common observation is that there is a minimal latency period for pleural plaques.2 The aim of this study was to find a mathematical model to describe the occurrence of pleural plaques after exposure to asbestos. A simple model for incidence is assumed. Since the prevalence is empirically observed, it is necessary to transform the model of incidence into a model of prevalence. The fit of the model is then examined in a group of shipyard workers exposed to asbestos. THE MODEL OUT simple model is that incidence depends on cumulative exposure. However, measurements of fibre concentrations are rarely available to make an absolute estimate of cumulative exposure. A crude approximation of cumulative exposure can be taken from the duration of exposure. A model analogous to that proposed for mesothelioma for incidence density in workers still exposed to asbestos would then be:
((t-10) 3 -(t-10-d) 3 )t> 10 + d (t-10) 3 10 + d > t > 1 0 t < 10
I (t) = k(t-w)» t > w (1) I (t) = 0 t< w (2)
where t denotes years since first exposure, d the duration of exposure in years, and f the average concentration of exposure in fibres/cm3. K,^ is a constant depending on type of asbestos. This model emphasizes the time since onset of exposure. The use of a minimal latency period of 10 years in the model is supported in empirical observations.3 Epidemiological data also support the hypothesis that the risk of pleural plaques is highly
where k and a are constants, w is the minimum latency period and t the time since onset of exposure. However, since the prevalence of pleural plaques is measured equation (2) had to be transformed to include the prevalence instead of I (t). Assuming that at some time t since onset of exposure, N, (still employed) people remain to be observed from the original group (Nt0). The number of observable new cases in a short time interval (At) would then be:
Department of Occupational Medicine, Sahlgren's Hospital, St Sigfridsgatan 85, S-412 66 GOteborg, Sweden.
1180
Downloaded from http://ije.oxfordjournals.org/ at University of Winnipeg on August 21, 2015
Jarvholm B (Department of Occupational Medicine, Sahlgren's Hospital, St Sigfridsgatan 85, S-412 66 Gflteborg, Sweden). Pleural plaques and exposure to asbestos: A mathematical model. International Journal of Epidemiology 1992; 21: 1180-1184. The objective of this study was to find a model to describe the relationship between the occurrence of pleural plaques and exposure to asbestos. A simple model based on the cumulative exposure was postulated and empirically tested on shipyard workers occupational^ exposed to asbestos. Exposure time was used to approximate the cumulative dose. It was found that the incidence of pleural plaques could be described as K(t-w)* where ' f is time since onset of exposure; 'K' is a constant that would depend on the level of asbestos exposure; 'W is a latency period and was around 13 years; 'a' is a constant that was 0.4. In subgroups of the workers, i.e. plumbers, fitters and platers, 'a' was 0.4, 0.6 and 0.2 respectively.
1181
PLEURAL PLAQUES—A MODEL
N, (1-P) I(t)At = N,Ap
where P is the prevalence at t, I(t) the incidence at t and Ap is the change in prevalence between t and t +At. This formula assumes that there is no selective withdrawal from the observable group, i.e. that employees with or without plaques terminate employment (hence the possibility to be observed) at different rates. Given that pleural plaques are benign lesions with no pain, and other asbestos-associated conditions such as tumours and fibrosis are far less common than pleural plaques in most data sets, this seems to be a reasonable assumption. Including (1) in (2) gives (if P
