S A Roberts1, J H Hendry. 1. CRC Biomathematics and Computing Unit, Paterson Institute for Cancer Research, Christie Hospital NHS Trust, Manchester, United Kingdom. sroberts@picr.man.ac.uk
Abstract
PURPOSE: To use the time-dependent linear-quadratic model, both in the standard form and in a form modified to incorporate intertumor heterogeneity, in a reanalysis of 4 datasets for larynx tumor control, to provide more representative and direct estimates of the lag period, the time factor (lambda/alpha), and the clonogen population inactivation dose ([lnk]/alpha). METHODS AND MATERIALS: The data comprised 2,225 patients treated in Edinburgh (UK), Glasgow (UK), Manchester (UK), or Toronto (Canada), with tumor control assessed after at least 2 years. Heterogeneity in each series was taken into account using the coefficient of variation (CV) of the clonogen radiosensitivity (alpha). Maximum likelihood techniques were used to provide best estimates of the parameters, and also direct estimation of the more stable parameter ratios of interest. RESULTS: The use of different heterogeneity factors for the different series allowed common dose/time parameters to be fitted across all four series in a way not possible using the standard model, enabling the inherent effect of heterogeneity in flattening dose-response curves and in reducing time factors to be separated from the underlying more-representative values. Radiosensitivity CVs were calculated to be 30% (Edinburgh), 36% (Glasgow), 40% (Manchester), and 71% (Toronto). The lag phase was 32 days (95% CL 20-38 days) which was longer than the value of 23 days (11-36 days) deduced using the standard model without the heterogeneity parameter. The time factor was 1.2 (0.8-2.2) Gy/day, again greater than the value of 0.80 (0.54-1.41) Gy/day derived using the standard model. Similar larger time factors and longer lag periods could be reproduced using the standard model either by using a parameterization based on parameter ratios, or by omitting the discordant Toronto data and refitting just the data from the three UK centers. CONCLUSION: It was concluded that the heterogeneity model provides a better representation of the time factor for tumor control when data are analyzed comprising different stages of disease treated at different centers. The model allows different amounts of heterogeneity in different series, which tend to flatten dose-responses curves and reduce time factors, to be taken in to account. Also, direct maximum likelihood estimates can be made of the lag period, the time factor (lambda/alpha), and the fractionation sensitivity (beta/alpha), as well as the clonogen population inactivation dose (lnk)/alpha. Values of these parameter ratios are more robust and stable than the individual parameter values. The results of the present analysis using a total of 2,225 patients from four centers indicate that the average lag period may be somewhat longer and the average time factor somewhat greater (and the 95% confidence limits of the time factor exclude previous estimates), than the values deduced previously using simpler models and more diverse multi-center datasets.
PURPOSE: To use the time-dependent linear-quadratic model, both in the standard form and in a form modified to incorporate intertumor heterogeneity, in a reanalysis of 4 datasets for larynx tumor control, to provide more representative and direct estimates of the lag period, the time factor (lambda/alpha), and the clonogen population inactivation dose ([lnk]/alpha). METHODS AND MATERIALS: The data comprised 2,225 patients treated in Edinburgh (UK), Glasgow (UK), Manchester (UK), or Toronto (Canada), with tumor control assessed after at least 2 years. Heterogeneity in each series was taken into account using the coefficient of variation (CV) of the clonogen radiosensitivity (alpha). Maximum likelihood techniques were used to provide best estimates of the parameters, and also direct estimation of the more stable parameter ratios of interest. RESULTS: The use of different heterogeneity factors for the different series allowed common dose/time parameters to be fitted across all four series in a way not possible using the standard model, enabling the inherent effect of heterogeneity in flattening dose-response curves and in reducing time factors to be separated from the underlying more-representative values. Radiosensitivity CVs were calculated to be 30% (Edinburgh), 36% (Glasgow), 40% (Manchester), and 71% (Toronto). The lag phase was 32 days (95% CL 20-38 days) which was longer than the value of 23 days (11-36 days) deduced using the standard model without the heterogeneity parameter. The time factor was 1.2 (0.8-2.2) Gy/day, again greater than the value of 0.80 (0.54-1.41) Gy/day derived using the standard model. Similar larger time factors and longer lag periods could be reproduced using the standard model either by using a parameterization based on parameter ratios, or by omitting the discordant Toronto data and refitting just the data from the three UK centers. CONCLUSION: It was concluded that the heterogeneity model provides a better representation of the time factor for tumor control when data are analyzed comprising different stages of disease treated at different centers. The model allows different amounts of heterogeneity in different series, which tend to flatten dose-responses curves and reduce time factors, to be taken in to account. Also, direct maximum likelihood estimates can be made of the lag period, the time factor (lambda/alpha), and the fractionation sensitivity (beta/alpha), as well as the clonogen population inactivation dose (lnk)/alpha. Values of these parameter ratios are more robust and stable than the individual parameter values. The results of the present analysis using a total of 2,225 patients from four centers indicate that the average lag period may be somewhat longer and the average time factor somewhat greater (and the 95% confidence limits of the time factor exclude previous estimates), than the values deduced previously using simpler models and more diverse multi-center datasets.