Therapeutic Radiology
Immobilization Error: Some Theoretical Considerations 1 Michael Goitein, Ph.D., and Joel Busse, M.D. 2 The consequences of immobilization error are explored through a model involving the tumor dose-response characteristics. The quantitative effect of such an error depends upon the field size, the shape of the dose-response function, and the position of the nominal dose being delivered on the dose-response function. Squamous-cell carcinoma of the supraglottis and Hodgkin's disease are discussed and values are calculated for the change in tumor control probability which might accompany immobilization error. INDEX TERMS: planning
Therapeutic Radiology, technique. Therapeutic Radioloqy, treatment
Radiology 117:407-412, November 1975
• HE IMMOBILIZATION of patients is a difficult . technical aspect of radiation therapy. Techniques for patient immobilization vary from stable support on the treatment couch without further restraint, to the use of complex formed shells individually designed for the patient. Some theoretical considerations are presented below relating to the importance of patient immobilization. It is important to distinguish between immobilization error and localization error; either can lead to q marginal recurrence, and both are, of course, to be distinguished from failure to control disease in the main body of the treatment volume. Localization error results from the failure to appreciate the full extent of disease or to design adequate treatment fields. Immobilization error, on the other hand, involves displacements of the treatment fields relative to the intended treatment volume with the result that there may be a reduction of the total dose delivered to the periphery of the field. Such errors arise from patient movement during treatment and from inability to reproduce the treatment fields or reposition the patient precisely from treatment to treatment. Several factors influence the consequences of immobilization error; among these are field size, the shape of the dose-response function, and the position of the nominal dose being delivered on the dose-response function. In unfavorable circumstances, the model presented below would predict that quite modest errors in immobilization could result in substantial reductions in control probability. The absolute numbers in the calculations which are presented should not be ascribed any clinical significance, since the model is crude and simplifying assumptions have been made. The calculations are presented with the intention. of suggesting the orders of magnitude involved. It should not be forgotten that immobilization is only one of a number of factors which come into play.
T
(A) ~
SQUAMOUS CELL CA. OF SUPRAGLOTTIS
(B) HODGKIN'S DISEASE
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-
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7000
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I
I
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1000
2000
3000
4000
TOTAL DOSE (
5000
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Fig. 1. Dose-response curves. A. Squamous-cell carcinoma of the supraglottis (stages T 2 and T3 combined) derived from Shukovsky (25). B. Hodgkin's disease, derived from Kaplan (12). The solid lines are approximate fits to the data, giving a linear relationship between probit (of TCP) and dose. Error bars are standard deviations and represent only statistical uncertainty. Other sources of error, including positioning errors, are also present in the data.
METHODS AND MATERIALS
In exploring the effects of field size and of the characteristics of the dose-response function, two diseases have been selected as illustrative examples: locallyadvanced squamous-cell carcinoma of the supraglottis and Hodgkin's disease. These diseases occupy somewhat extreme positions with respect to the effects of interest. The dose-response curve selected for carcinoma of the supraglottis, stages T 2 and T3 combined, is shown in Figure 1, A. It is derived from the data of Shukovsky (26) by converting his NSD values to equivalent total doses of regimens in which fractions of 200 rads are delivered 5 times per week. (Throughout the calcula-
1 From the Department of Radiation Medicine, Massachusetts General Hospital and Harvard Medical School, Boston, Mass. Accepted for publication 'in June 1975. 2 Present address: Department of Radiation Oncology, Rockford Memorial Hospital, Rockford, III. elk
407
408
November 1975
MICHAEL GOITEIN AND JOEL BUSSE
RESULTS
A
B
c
Fig. 2. Treatment fields considered typical for treatment of squamous-cell carcinoma of the supraglottis. A. The tight field used with good immobilization. B. The loose field used with less careful immobilization. The volume of this field is (10 X 8)/(9 X 7) = 27 % larger than that of the tight field. C. A region of reduced dose (II) within the margins of a 9 X 7-cm field due to random ±5 mm immobilization error.
tions. dose-response curves have been used without reference to the details of the fractionation schedule. Such considerations would lead to slight quantitative changes but are not qualitatively important.) The data represented in Figure 1, A show a fairly steep slope of the dose-response curve. The steepness of the curve is, in large part, forced by the point at the lowest dose which corresponds to 1 recurrence out of a total of only 6 cases. The statistical uncertainty (in the figures, error bars indicate standard deviations) clearly would permit a less steep fit. Different groups of the data presented by Shukovsky, such as into stages T 1 + T 2 and T 3 + T 4, have less steep dose-response curves. Other data (5, 19, 28) also suggest a shallower doseresponse curve, as do some theoretical considerations (3, 4). On the other hand, an estimate (relating to squamous-ceil carcinoma at various sites) by Paterson (24) supports the slope of the curve suggested by the T 2 + T 3 grouping of Shukovsky. Thus the curve selected represents the steepest curve which can be supported by the data and is chosen to represent a fairly extreme but nevertheless possible situation. It is assumed that the treatments for squamous-cell carcinoma of the supraglottis would be delivered through parallel opposed 9 X 7-crn fields, and the fact that the fields would generally be reduced towards the end of treatment is ignored. A typical field might be as indicated in Figure 2, A. The dose-response curve for Hodgkin's disease is presented in Figure 1, B. This is derived from Kaplan's data (12) and represents the probability of true local control as a function of dose. The mantle field is a very complex one with an unusually high ratio of perimeter to area. For purposes of the present work, both the anterior and posterior mantle fields are crudely characterized as pairs of 30 X 12-cm fields, i.e., left- and right-sided fields. The ratio of perimeter to area of these simplified fields is comparable to that of conventional mantle fields.
Immobilization errors may act to reduce local tumor control probability (TCP) through two effects. The first is associated with a dose reduction necessitated by the decreased tolerance of normal tissues treated to high doses which may accompany unnecessarily enlarged treatment fields. The second is related to a dose reduction to potential tumor in the margin of the treatment field. These are now considered with calculations for squamous-cell carcinoma of the supraglottis and for Hodgkin's disease. I. Enlarged Treatment Fields-Volume Effect
The first effect involves the therapist's strategy concerning the field size used. If stringent measures are taken to achieve patient immobility they will generally be advantageous in reducing the treatment field area. Conversely, where there is a lack of confidence in patient immobilization, fields are likely to be enlarged to allow for some degree of movement. There is general agreement that the larger the treatment volume, the smaller the dose which can be delivered before normal tissue tolerance is reached. This may be because sensitive critical structures may be included in the larger field or because of a nonspecific volume effect. One might therefore expect that a therapist using a more generous field would be led, perhaps more by empirical observation than conscious calculation, to use lower doses than would be employed with smaller fields. The consequence of the use of a lower dose, as has been well documented for animal tumors and is implicit in the dose-response curves obtained clinically, would be a reduction in the TCP. The extent (and therefore the importance) of this reduction is the subject of the following calculations. A. Supraglottic lesions: Consider the treatment through parallel-opposed fields of a lesion with the dose-response characteristics of Figure 1, A. Suppose that with careful immobilization, fields of 9 X 7 cm adequately cover the lesion (Fig. 2, A) and that the tolerance dose which could be delivered is 6,600 rads (delivered in 200 rads/fraction, 5 times/week). From the dose-response curve of Figure 1, A, this can be seen to correspond to a TCP of 59 % . If less careful immobilization were involved, allowing 5 mm of additional patient motion, the wary therapist might employ fields enlarged by 5 mm around the margin, i.e., 10 X 8-cm fields as indicated in Figure 2, B. This would increase the treated volume by 27 % (10 X 8 X depth/9 X 7 X depth) = 1.27. Several investigators have made quantitative estimates of the relationship between tolerance dose and volume treated. All have developed a power law relationship of the form
Vol. 117
IMMOBILIZATION ERROR: SOME THEORETICAL CONSIDERATIONS
where D is the maximum dose which can be delivered to volume V and Do is the maximum dose which may be delivered to a reference volume Vo (usually a volume of 1,000 ern"). The value for the exponent P has been variously estimated as 0.16 (14), 0.12 (15), 0.12 (2), and 0.085 (1). We chose the value of 0.12. Such a quotient would lead to a 2.9 % dose reduction if the irradiated volume were enlarged by 27 % in order to reach the same level of normal tissue damage. The consequence of a 2.9 % (191 rads) reduction in the total dose delivered can be determined by inspection of Figure 1, A to be a reduction in TCP from 59 % to 44 % . If the dose which could be delivered to the 9 X 7-crn field were 7,400 rads, corresponding to a 95 % TCP, then a 5-mm enlargement and the consequent 2.9 % dose reduction would lower the TCP to 90 %. Thus the reduction in TCP depends on the position of the nominal dose on the dose-response curve. Control rates in the neighborhood of 50 % lie on the steepest portion of the curve and are most sensitive to this effect. The initial size of the irradiation field plays an important role in determining the reduction in TCP brought about by adding 5 mm to the margins. Assuming a dose-response curve similar to that shown in Figure 1, A, the addition of 5 mm to the margins of a 20 X 15-cm field results in a 12 % increase. This corresponds to a 1.4 % dose reduction using the above equation, which in turn corresponds to a drop in TCP of 7 % from 59 % to 52%. B. Hodgkin's Disease: The same calculations can be performed for this second illustrative disease category. With the larger fields and greater difficulty in patient immobilization and position reproducibility, it would be more realistic to consider a 1-cm enlargement around the field periphery as a response to less adequate technique. The resulting field enlargement would be by a factor of 1.24 (14 X 32 X depth/12 X 30 X depth). The reduction in dose suggested by the volumedose relationship in the above equation would then be 2.7%. Assuming the tight immobilization dose would be 4,400 rads, it can be seen by inspection of Figure 1, B that a 2.7 % dose reduction leads to a 0.4 % reduction in TCP from 98.7% to 98.3 %. II. Reduced Dose to Tumor in the Field Margin The second and perhaps more obvious consequence of patient motion is the loss of the sharp dose falloff at the field periphery. An increased dose may be delivered to tissues which lie immediately outside the intended treatment volume, or a reduced dose may be delivered to tissues just within the field margin, depending upon the direction of patient motion. The latter effect may, if the tumor extends into the reduced dose region, lead to an increased probability of recurrence. If motion were random, the loss of sharpness of the field edge would be smooth, with the dose profile resembling an error function with width determined by the root-mean-square positioning error. While calculations
409
Therapeutic Radiology
can readily be performed by integration over such a profile, a simpler and more transparent model is adopted here. It is assumed that the intended dose is delivered for all but a certain fraction of treatments and that, for those treatments, a fixed positioning error is introduced with random sign. Marks et al. (17) have reported an incidence of significant localization error (sufficient to exclude known or suspected sites of disease from the treatment volume) in 36 % (330/902) of treatments studied. We adopt this figure as characteriizing the error frequency in both mantle treatments of Hodgkin's disease and in the treatment of supraglottic disease. That is, a 36 % error rate (18 % left, 18 % superior, 18 % right and 18 % inferior) is selected for the model. In addition, results for 100 % of treatments in error are given. The model, therefore, comprises a central portion of the field (smaller than the intended treatment volume) which will have received a full dose and a strip around the margin which will have received 18 % less dose (50 % in the alternative calculation). The width of this strip is the mean immobilization error, which is assumed to be 5 mm in the example of supraglottic disease (Fig. 2, C) and 1 cm in the case of the less easily immobilized patient with Hodgkin's disease. In order to estimate the TCP, it is necessary to know the relative number of malignant cells in the low-dose strip. It is not likely that there will be the same density of malignant cells in the peripheral region as in the center. For purposes of calculation, it is assumed that the mean density of malignant cells in the peripheral strip will be 25 % of that in the central high-dose region. Some such assumption is needed to allow one to construct doseresponse curves for the low-dose region. A. Supraglottic lesions: Consider the situation in which a dose of 6,600 rads is delivered through 9 X 7-cm parallel-opposed fields. Assume that the dose-response curve for such an irradiated volume is that of Figure 1, A and thus would predict a 59 % TCP. The TCP for the entire field (Fig. 2, C) is the product of the probabilities for inactivating all malignant cells in the central region (Region I of Fig. 2, C) and in the peripheral strip (Region II of Fig. 2, C). The dose-response curve for these smaller volumes may be derived from the dose-response curve for the entire volume using the proportional relationship between the logarithm of the TCP and the total number of malignant cells (3, 4). This derivation implicitly assumes that the tumor cell kinetics are the same in the periphery as in the central tumor volume. Such a simplification is certainly suspect; peripheral cells would be likely to have a smaller hypoxic component and might therefore be more radiosensitive. The dose-response curves for the two regions mentioned are shown in Figure 3, A. The control probabilities for these two regions considered separately are, at the 6,600 rad level, 61.4 % and 96.0 %, respectively. The product of these two probabilities is, of course, 59 % which is the TCP for the entire volume. If the outer margin of the field (Region II) receives 18 % less dose, the TCP drops to 76.4 % in the
410
MICHAEL GOITEIN AND JOEL BUSSE
(A) SQUAMOUS CELL CA. OF SUPRAGLOTTIS
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100
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::::! Cl:l
Immobilization Error: Some Theoretical Considerations 1 Michael Goitein, Ph.D., and Joel Busse, M.D. 2 The consequences of immobilization error are explored through a model involving the tumor dose-response characteristics. The quantitative effect of such an error depends upon the field size, the shape of the dose-response function, and the position of the nominal dose being delivered on the dose-response function. Squamous-cell carcinoma of the supraglottis and Hodgkin's disease are discussed and values are calculated for the change in tumor control probability which might accompany immobilization error. INDEX TERMS: planning
Therapeutic Radiology, technique. Therapeutic Radioloqy, treatment
Radiology 117:407-412, November 1975
• HE IMMOBILIZATION of patients is a difficult . technical aspect of radiation therapy. Techniques for patient immobilization vary from stable support on the treatment couch without further restraint, to the use of complex formed shells individually designed for the patient. Some theoretical considerations are presented below relating to the importance of patient immobilization. It is important to distinguish between immobilization error and localization error; either can lead to q marginal recurrence, and both are, of course, to be distinguished from failure to control disease in the main body of the treatment volume. Localization error results from the failure to appreciate the full extent of disease or to design adequate treatment fields. Immobilization error, on the other hand, involves displacements of the treatment fields relative to the intended treatment volume with the result that there may be a reduction of the total dose delivered to the periphery of the field. Such errors arise from patient movement during treatment and from inability to reproduce the treatment fields or reposition the patient precisely from treatment to treatment. Several factors influence the consequences of immobilization error; among these are field size, the shape of the dose-response function, and the position of the nominal dose being delivered on the dose-response function. In unfavorable circumstances, the model presented below would predict that quite modest errors in immobilization could result in substantial reductions in control probability. The absolute numbers in the calculations which are presented should not be ascribed any clinical significance, since the model is crude and simplifying assumptions have been made. The calculations are presented with the intention. of suggesting the orders of magnitude involved. It should not be forgotten that immobilization is only one of a number of factors which come into play.
T
(A) ~
SQUAMOUS CELL CA. OF SUPRAGLOTTIS
(B) HODGKIN'S DISEASE
::....~ 1 00 r------~.-
I-... ...... ~ ~
~ C)
&:
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-
80
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-
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7000
8000
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-
I
I
I
I
1000
2000
3000
4000
TOTAL DOSE (
5000
raas )
Fig. 1. Dose-response curves. A. Squamous-cell carcinoma of the supraglottis (stages T 2 and T3 combined) derived from Shukovsky (25). B. Hodgkin's disease, derived from Kaplan (12). The solid lines are approximate fits to the data, giving a linear relationship between probit (of TCP) and dose. Error bars are standard deviations and represent only statistical uncertainty. Other sources of error, including positioning errors, are also present in the data.
METHODS AND MATERIALS
In exploring the effects of field size and of the characteristics of the dose-response function, two diseases have been selected as illustrative examples: locallyadvanced squamous-cell carcinoma of the supraglottis and Hodgkin's disease. These diseases occupy somewhat extreme positions with respect to the effects of interest. The dose-response curve selected for carcinoma of the supraglottis, stages T 2 and T3 combined, is shown in Figure 1, A. It is derived from the data of Shukovsky (26) by converting his NSD values to equivalent total doses of regimens in which fractions of 200 rads are delivered 5 times per week. (Throughout the calcula-
1 From the Department of Radiation Medicine, Massachusetts General Hospital and Harvard Medical School, Boston, Mass. Accepted for publication 'in June 1975. 2 Present address: Department of Radiation Oncology, Rockford Memorial Hospital, Rockford, III. elk
407
408
November 1975
MICHAEL GOITEIN AND JOEL BUSSE
RESULTS
A
B
c
Fig. 2. Treatment fields considered typical for treatment of squamous-cell carcinoma of the supraglottis. A. The tight field used with good immobilization. B. The loose field used with less careful immobilization. The volume of this field is (10 X 8)/(9 X 7) = 27 % larger than that of the tight field. C. A region of reduced dose (II) within the margins of a 9 X 7-cm field due to random ±5 mm immobilization error.
tions. dose-response curves have been used without reference to the details of the fractionation schedule. Such considerations would lead to slight quantitative changes but are not qualitatively important.) The data represented in Figure 1, A show a fairly steep slope of the dose-response curve. The steepness of the curve is, in large part, forced by the point at the lowest dose which corresponds to 1 recurrence out of a total of only 6 cases. The statistical uncertainty (in the figures, error bars indicate standard deviations) clearly would permit a less steep fit. Different groups of the data presented by Shukovsky, such as into stages T 1 + T 2 and T 3 + T 4, have less steep dose-response curves. Other data (5, 19, 28) also suggest a shallower doseresponse curve, as do some theoretical considerations (3, 4). On the other hand, an estimate (relating to squamous-ceil carcinoma at various sites) by Paterson (24) supports the slope of the curve suggested by the T 2 + T 3 grouping of Shukovsky. Thus the curve selected represents the steepest curve which can be supported by the data and is chosen to represent a fairly extreme but nevertheless possible situation. It is assumed that the treatments for squamous-cell carcinoma of the supraglottis would be delivered through parallel opposed 9 X 7-crn fields, and the fact that the fields would generally be reduced towards the end of treatment is ignored. A typical field might be as indicated in Figure 2, A. The dose-response curve for Hodgkin's disease is presented in Figure 1, B. This is derived from Kaplan's data (12) and represents the probability of true local control as a function of dose. The mantle field is a very complex one with an unusually high ratio of perimeter to area. For purposes of the present work, both the anterior and posterior mantle fields are crudely characterized as pairs of 30 X 12-cm fields, i.e., left- and right-sided fields. The ratio of perimeter to area of these simplified fields is comparable to that of conventional mantle fields.
Immobilization errors may act to reduce local tumor control probability (TCP) through two effects. The first is associated with a dose reduction necessitated by the decreased tolerance of normal tissues treated to high doses which may accompany unnecessarily enlarged treatment fields. The second is related to a dose reduction to potential tumor in the margin of the treatment field. These are now considered with calculations for squamous-cell carcinoma of the supraglottis and for Hodgkin's disease. I. Enlarged Treatment Fields-Volume Effect
The first effect involves the therapist's strategy concerning the field size used. If stringent measures are taken to achieve patient immobility they will generally be advantageous in reducing the treatment field area. Conversely, where there is a lack of confidence in patient immobilization, fields are likely to be enlarged to allow for some degree of movement. There is general agreement that the larger the treatment volume, the smaller the dose which can be delivered before normal tissue tolerance is reached. This may be because sensitive critical structures may be included in the larger field or because of a nonspecific volume effect. One might therefore expect that a therapist using a more generous field would be led, perhaps more by empirical observation than conscious calculation, to use lower doses than would be employed with smaller fields. The consequence of the use of a lower dose, as has been well documented for animal tumors and is implicit in the dose-response curves obtained clinically, would be a reduction in the TCP. The extent (and therefore the importance) of this reduction is the subject of the following calculations. A. Supraglottic lesions: Consider the treatment through parallel-opposed fields of a lesion with the dose-response characteristics of Figure 1, A. Suppose that with careful immobilization, fields of 9 X 7 cm adequately cover the lesion (Fig. 2, A) and that the tolerance dose which could be delivered is 6,600 rads (delivered in 200 rads/fraction, 5 times/week). From the dose-response curve of Figure 1, A, this can be seen to correspond to a TCP of 59 % . If less careful immobilization were involved, allowing 5 mm of additional patient motion, the wary therapist might employ fields enlarged by 5 mm around the margin, i.e., 10 X 8-cm fields as indicated in Figure 2, B. This would increase the treated volume by 27 % (10 X 8 X depth/9 X 7 X depth) = 1.27. Several investigators have made quantitative estimates of the relationship between tolerance dose and volume treated. All have developed a power law relationship of the form
Vol. 117
IMMOBILIZATION ERROR: SOME THEORETICAL CONSIDERATIONS
where D is the maximum dose which can be delivered to volume V and Do is the maximum dose which may be delivered to a reference volume Vo (usually a volume of 1,000 ern"). The value for the exponent P has been variously estimated as 0.16 (14), 0.12 (15), 0.12 (2), and 0.085 (1). We chose the value of 0.12. Such a quotient would lead to a 2.9 % dose reduction if the irradiated volume were enlarged by 27 % in order to reach the same level of normal tissue damage. The consequence of a 2.9 % (191 rads) reduction in the total dose delivered can be determined by inspection of Figure 1, A to be a reduction in TCP from 59 % to 44 % . If the dose which could be delivered to the 9 X 7-crn field were 7,400 rads, corresponding to a 95 % TCP, then a 5-mm enlargement and the consequent 2.9 % dose reduction would lower the TCP to 90 %. Thus the reduction in TCP depends on the position of the nominal dose on the dose-response curve. Control rates in the neighborhood of 50 % lie on the steepest portion of the curve and are most sensitive to this effect. The initial size of the irradiation field plays an important role in determining the reduction in TCP brought about by adding 5 mm to the margins. Assuming a dose-response curve similar to that shown in Figure 1, A, the addition of 5 mm to the margins of a 20 X 15-cm field results in a 12 % increase. This corresponds to a 1.4 % dose reduction using the above equation, which in turn corresponds to a drop in TCP of 7 % from 59 % to 52%. B. Hodgkin's Disease: The same calculations can be performed for this second illustrative disease category. With the larger fields and greater difficulty in patient immobilization and position reproducibility, it would be more realistic to consider a 1-cm enlargement around the field periphery as a response to less adequate technique. The resulting field enlargement would be by a factor of 1.24 (14 X 32 X depth/12 X 30 X depth). The reduction in dose suggested by the volumedose relationship in the above equation would then be 2.7%. Assuming the tight immobilization dose would be 4,400 rads, it can be seen by inspection of Figure 1, B that a 2.7 % dose reduction leads to a 0.4 % reduction in TCP from 98.7% to 98.3 %. II. Reduced Dose to Tumor in the Field Margin The second and perhaps more obvious consequence of patient motion is the loss of the sharp dose falloff at the field periphery. An increased dose may be delivered to tissues which lie immediately outside the intended treatment volume, or a reduced dose may be delivered to tissues just within the field margin, depending upon the direction of patient motion. The latter effect may, if the tumor extends into the reduced dose region, lead to an increased probability of recurrence. If motion were random, the loss of sharpness of the field edge would be smooth, with the dose profile resembling an error function with width determined by the root-mean-square positioning error. While calculations
409
Therapeutic Radiology
can readily be performed by integration over such a profile, a simpler and more transparent model is adopted here. It is assumed that the intended dose is delivered for all but a certain fraction of treatments and that, for those treatments, a fixed positioning error is introduced with random sign. Marks et al. (17) have reported an incidence of significant localization error (sufficient to exclude known or suspected sites of disease from the treatment volume) in 36 % (330/902) of treatments studied. We adopt this figure as characteriizing the error frequency in both mantle treatments of Hodgkin's disease and in the treatment of supraglottic disease. That is, a 36 % error rate (18 % left, 18 % superior, 18 % right and 18 % inferior) is selected for the model. In addition, results for 100 % of treatments in error are given. The model, therefore, comprises a central portion of the field (smaller than the intended treatment volume) which will have received a full dose and a strip around the margin which will have received 18 % less dose (50 % in the alternative calculation). The width of this strip is the mean immobilization error, which is assumed to be 5 mm in the example of supraglottic disease (Fig. 2, C) and 1 cm in the case of the less easily immobilized patient with Hodgkin's disease. In order to estimate the TCP, it is necessary to know the relative number of malignant cells in the low-dose strip. It is not likely that there will be the same density of malignant cells in the peripheral region as in the center. For purposes of calculation, it is assumed that the mean density of malignant cells in the peripheral strip will be 25 % of that in the central high-dose region. Some such assumption is needed to allow one to construct doseresponse curves for the low-dose region. A. Supraglottic lesions: Consider the situation in which a dose of 6,600 rads is delivered through 9 X 7-cm parallel-opposed fields. Assume that the dose-response curve for such an irradiated volume is that of Figure 1, A and thus would predict a 59 % TCP. The TCP for the entire field (Fig. 2, C) is the product of the probabilities for inactivating all malignant cells in the central region (Region I of Fig. 2, C) and in the peripheral strip (Region II of Fig. 2, C). The dose-response curve for these smaller volumes may be derived from the dose-response curve for the entire volume using the proportional relationship between the logarithm of the TCP and the total number of malignant cells (3, 4). This derivation implicitly assumes that the tumor cell kinetics are the same in the periphery as in the central tumor volume. Such a simplification is certainly suspect; peripheral cells would be likely to have a smaller hypoxic component and might therefore be more radiosensitive. The dose-response curves for the two regions mentioned are shown in Figure 3, A. The control probabilities for these two regions considered separately are, at the 6,600 rad level, 61.4 % and 96.0 %, respectively. The product of these two probabilities is, of course, 59 % which is the TCP for the entire volume. If the outer margin of the field (Region II) receives 18 % less dose, the TCP drops to 76.4 % in the
410
MICHAEL GOITEIN AND JOEL BUSSE
(A) SQUAMOUS CELL CA. OF SUPRAGLOTTIS
~
-:
I--. .....
100
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::::! Cl:l
