rnt J. &diatron
Oncology
Bid
Phys
, 1977,
Vol. 2, p. 823.
Pergamon Press.
Printed in U S.A
??Editorial MATHEMATICAL BOWEN University
MODELING
E. KELLER,
M.S.
of Rochester Cancer Center, Division of Radiation Oncology, 601 Elmwood Avenue, Rochester, New York 14642, U.S.A.
The analysis of time-dose relationships directed toward finding an optimal treatment regimen in radiation oncology is multifarious. Methods most commonly used for the analysis are often unsatisfactory. A satisfactory mathematical model for the clinical situation must recognize that the results of treatment are probabilities. Some measure of the variability ofresponse among patients must be specified. Estimated parameters should measure the characteristics ofthe tumor and normal tissue, and a method should provide confidence limits for these estimated parameters. Ideally, the model should be flexible enough to take into account unusual treatment parameters such as unequal fractionation, gaps in treatment, treatment volume, and biological variables such as tumor size and growth rate, oxygenation, etc., within the sampled population. In such a model, the physical and biological concepts are identified with mathematical concepts, which are given the same names as their physical and biological counterparts. The mathematical concepts are postulated to fulfill certain axioms, which are idealized properties of their physical and biological counterparts. The model will be “successful” when it describes the physical and biological system and predicts the outcome of treatment within predetermined confidence limits. We appear to be at a new threshold in our understanding of the complex interrelationships between all of these factors and of the probability of what the outcome of treatment will be. Included in this issue are two papers presenting methodologies for the assessment of the clinical significance of non-compliance with prescribed schedules of irradiation. The reduction of clinical data is largely anecdotal. Herbert’ has pointed out that radiation biology has not yet demonstrated that its findings can be extrapolated from mouse to man. Therefore, we have developed “isoeffect” methods to re-
Box 647
duce clinical data to predictive forms. Herbert has described the dependence of the probability of outcome upon the change in the prescribed dose and time. This dependence is represented by a transform, and he describes two alternatives: the multivariate probit and the multivariate logistic. The multivariate probit model is preferred for small clinical samples, and the multivariate logistic model for larger numbers of patients. Using these models, he is able to predict that the clinical effects of deviations from the prescribed time and dose depend upon the design value of the probability of control. The logistic regression method described by Fischer’ avoids the problem of assuming the shape of the time-dose isoeffect curve prior to analysis. His method provides confidence limits for the estimates of the isoeffect curves and a measure of the “goodness of fit”. It thus serves the useful function of indicating to the oncologist when his data are inadequate to test his hypothesis. Both Herbert and Fischer draw conclusions that are at variance with Ellis. Herbert suggests that one reason for the discrepancy is that the form and the location and shape parameters of the response surface for a given effect are local functions of the prognostic factors. The utilization of these mathematical models to extract better treatment regimes is the goal of these reports. However, the value of any new treatment regimes will be determined in clinical practice. REFERENCES 1.
2.
823
Fischer, D. B., Fischer, J. J.: Dose response relationships in radiotherapy: Applications of a logistic regression model. Intern. J. Radiat. Oncol. Biol. Phvs. 2773-781, 1977. Herbert, D.: The assessment of the clinical significance of non-compliance with prescribed schedules of irradiation. Intern. J. Radik Oncol. Biol. Phys. 2:763-772. 1977.
Oncology
Bid
Phys
, 1977,
Vol. 2, p. 823.
Pergamon Press.
Printed in U S.A
??Editorial MATHEMATICAL BOWEN University
MODELING
E. KELLER,
M.S.
of Rochester Cancer Center, Division of Radiation Oncology, 601 Elmwood Avenue, Rochester, New York 14642, U.S.A.
The analysis of time-dose relationships directed toward finding an optimal treatment regimen in radiation oncology is multifarious. Methods most commonly used for the analysis are often unsatisfactory. A satisfactory mathematical model for the clinical situation must recognize that the results of treatment are probabilities. Some measure of the variability ofresponse among patients must be specified. Estimated parameters should measure the characteristics ofthe tumor and normal tissue, and a method should provide confidence limits for these estimated parameters. Ideally, the model should be flexible enough to take into account unusual treatment parameters such as unequal fractionation, gaps in treatment, treatment volume, and biological variables such as tumor size and growth rate, oxygenation, etc., within the sampled population. In such a model, the physical and biological concepts are identified with mathematical concepts, which are given the same names as their physical and biological counterparts. The mathematical concepts are postulated to fulfill certain axioms, which are idealized properties of their physical and biological counterparts. The model will be “successful” when it describes the physical and biological system and predicts the outcome of treatment within predetermined confidence limits. We appear to be at a new threshold in our understanding of the complex interrelationships between all of these factors and of the probability of what the outcome of treatment will be. Included in this issue are two papers presenting methodologies for the assessment of the clinical significance of non-compliance with prescribed schedules of irradiation. The reduction of clinical data is largely anecdotal. Herbert’ has pointed out that radiation biology has not yet demonstrated that its findings can be extrapolated from mouse to man. Therefore, we have developed “isoeffect” methods to re-
Box 647
duce clinical data to predictive forms. Herbert has described the dependence of the probability of outcome upon the change in the prescribed dose and time. This dependence is represented by a transform, and he describes two alternatives: the multivariate probit and the multivariate logistic. The multivariate probit model is preferred for small clinical samples, and the multivariate logistic model for larger numbers of patients. Using these models, he is able to predict that the clinical effects of deviations from the prescribed time and dose depend upon the design value of the probability of control. The logistic regression method described by Fischer’ avoids the problem of assuming the shape of the time-dose isoeffect curve prior to analysis. His method provides confidence limits for the estimates of the isoeffect curves and a measure of the “goodness of fit”. It thus serves the useful function of indicating to the oncologist when his data are inadequate to test his hypothesis. Both Herbert and Fischer draw conclusions that are at variance with Ellis. Herbert suggests that one reason for the discrepancy is that the form and the location and shape parameters of the response surface for a given effect are local functions of the prognostic factors. The utilization of these mathematical models to extract better treatment regimes is the goal of these reports. However, the value of any new treatment regimes will be determined in clinical practice. REFERENCES 1.
2.
823
Fischer, D. B., Fischer, J. J.: Dose response relationships in radiotherapy: Applications of a logistic regression model. Intern. J. Radiat. Oncol. Biol. Phvs. 2773-781, 1977. Herbert, D.: The assessment of the clinical significance of non-compliance with prescribed schedules of irradiation. Intern. J. Radik Oncol. Biol. Phys. 2:763-772. 1977.
