Opinion
Errors in Applying the NSD
Concept 1
Colin G. Orton, Ph.D. An analysis of selected papers in which the Nominal Standard Dose (NSD) concept is used reveals that about 50 % of them contain errors. These mistakes are both conceptual and mathematical in nature and average ± 7 %. Typical errors and methods to eliminate them are discussed. A table is provided for interconversion of NSDs, partial tolerances, and time, dose and fractionation (TDF)factors. INDEX TERMS:
Dosimetry, therapeutic. Therapeutic Radiology, treatment planning
Radiology 115:233-235, April 1975
• Use of the Nominal Standard Dose (NSD) concept has increased steadily since its introduction by Ellis in 1967 (1). This is illustrated in Figure 1 which shows the number of papers in which the NSD concept has been utilized in the three journals which publish most of the radiotherapy articles in the United States, viz., Radiology, American Journal of Roentgenology, and Cancer. In 1974, more than 20 such papers were published. This represents 15-20 % of all radiotherapy papers appearing in these journals. Unfortunately, the early articles concerning NSD described its application for the analysis of only relatively simple courses of radiotherapy, whereas in practice. the concept is most useful for the description of complex regimens which do not fall within the realm of the clinical experience of the users. The analysis of such complex applications of NSD is not simple and this has resulted in widespread misuse of the concept since it was not until recently that the full equations for NSD and partial tolerance were published (2). By June 1974, there were 59 articles involving the NSD concept in the three journals cited in Figure 1, and of these articles, 34 were sufficiently detailed for the calculations to be checked. As shown in Table I, it was found that the concept was applied incorrectly (or mathematical mistakes were made) in about 50% of cases. This figure is consistent with the results of a previous survey (3). The average error was ± 7%. Since most of these calculations were for treatments aimed at coming as close as possible to either normal tissue tolerance or cancericidal doses, such 7 % changes in dose are highly significant. There were basically two reasons why these errors were made: Incorrect combinations of regimens. When a course of radiotherapy was divided into several parts, it was not
5no
1969
1970
1971
1972
1973
1974'
YEAR 'BTR~POUHD
FROM
FIRST G MONTHS
Fig. 1. Articles utilizing the NSD concept in the three journals Radiology, American Journal of Roentgenology, and Cancer.
always realized that NSDs calculated using the simple NSD formula (NSD = total dose X ~0.24 X ,0.11) for each separate part of the treatment were meaningless numbers, as they did not represent full tissue tolerance. Furthermore, they could not be added linearly in order to determine the overall NSD. Partial tolerances, which are additive, should have been used (4). However, the calculation of partial tolerance is complicated (2), and authors often took the easy way out (adding NSDs), not appreciating the magnitude of the errors. Inconsistent overall treatment time. When a course of radiotherapy is divided into several parts, the sum of the treatment times used for the calculation of the partial tolerance of each part must equal the overall treatment time. A typical example involves treatment with two fields applied on alternate days. If the overall treatment time is 40 days, the partial tolerance for each field should be calculated using 20 days as the time factor
1 From the Division of Radiation Therapy, Department of Radiology, New York University Medical Center, New York, N. Y. Accepted for publication in November 1974. shan
233
234
OPINION
Table I: Papers Involving Detailed NSD Concept Calculations (to June 1974) in Radiology, American Journal of Roentgenology and Cancer Numberof Papers
No. with Incorrect Application of the NSD Concept
.~--~
34
April 1975
derstand the use of the TDF/NSD/Partial Tolerance Conversion Table.
Number with Mathematical Errors
Problem
--~~-~- ~~~~~~
16(47%)
----
A radiotherapist, having had some success with a treatment regimen consisting of 5 fractions of 300 rads followed by 24 fractions of 200 rads, all at 5 fractions per week, wishes to quote a value for the biologically effective dose in terms of the NSD concept. What number should he quote?
3(9%) ----
----
(T), so that when the partial tolerances of the two regimens are summated, the overall time will still be 40 days. It is very tempting to use T = 40 days for each calculation, thereby doubling the "homeostatic recovery time" of the tissues (1).
Solutions (a) If he considers this to represent a tolerance level, he can quote the overall NSD calculated using partial tolerances. However, such calculations are quite complicated, thus risking computational error (2). A simpler solution is to use TDFs. From Table V in Orton and Ellis (5), the TDFs for the first and second parts of the treatment are 31 and 79, respectively. The overall TDF is 31 + 79 = 110. The therapist could, if he wished, use this TDF number to represent the biologically effective dose, or alternatively, he could convert this to an NSD using Table II, which shows that a TDF of 110 is equivalent to an NSD of 1892 rets. (b) If the therapist considers an NSD of, say, 1800 rets to represent tolerance, then he could calculate the sum of the PTs for the 2 parts of the treatment. Such a calculation is again quite complicated (2). More simply, however, he could refer to Table II, which shows that a
DISCUSSION These misunderstandings of the NSD concept were first realized as a result of a survey conducted in 1972 (3). This led to the introduction of time, dose, and fractionation (TDF) factors (3, 5, 6, 8) in order to simplify the practical utilization of the NSD concept, and to reduce errors in the interpretation of the "rules of application" of NSDs and PTs. The inter-relationship between TDFs, NSDs, and PTs is shown in Table II, which has been derived from the fundamental equations for these quantities (see ApPENDIX).
USE OF TABLE II The following example should help the reader to unTable II:
TDF/NSD/Partial Tolerance Conversion Table Chosen NSD Value
TDF
(NSD)
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
10 20 25 30 35
(398) (625)
257 513 641 770 898
242 485 606 727 849
230 461 576 691 806
220 440 549 659 769
211
(813) (899)
273 547 683 820 957
421 526 632 737
202 405 506 607 708
195 390 487 585 682
188 376 471 565 659
182 364 455 547 638
177 353 442 530 618
172 343 429 515 601
167 334 417 501 584
40 45 50 55 60
(980) (1058) (1133) (1206) (1276)
1094 1230 1367 1504 1640
1026 1155 1283 1411 1540
970 1091 1212 1333 1455
921 1036 1152 1267 1382
879 989 1099 1209 1319
842 947 1053 1158 1263
809 910 1011 1112 1214
780 877 974 1072 1169
753 847 941 1035 1129
729 820 911 1002 1093
707 795 883 972 1060
686 772 858 944 1030
668 751 835 918 1002
65 70 75 80 85
(1344) (1410) (1475) (1538) (1600)
1668 1796 1924
1576 1697 1818 1940 2061
1497 1612 1727 1843 1958
1429 1538 1648 1758 1868
1368 1474 1579 1684 1789
1315 1416 1517 1618 1719
1267 1364 1462 1559 1657
1224 1318 1412 1506 1600
1184 1275 1366 1458 1549
1148 1237 1325 1413 1502
1115 1201 1287 1373 1459
1085 1168 1252 1335 1419
90 95 100 105 110
(1661) (1720) (1778) (1836) (1892)
2073 2188 2303
1978 2088 2198 2308 2418
1895 2000 2105 2210 2316
1820 1922 2023 2124 2225
1754 1851 1949 2046 2144
1694 1788 1882 1976 2071
1640 1731 1822 1913 2004
1590 1678 1767 1855 1943
1544 1630 1716 1802 1888
1502 1586 1669 1753 1836
115 120 125 130 135
(1947) (2002) (2056) (2109) (2161)
2421 2526 2631
2326 2427 2528 2630 2731
2241 2339 2436 2534 2631
2165 2259 2353 2447 2541
2095 2186 2277 2368 2460
2032 2120 2208 2297 2385
1973 2059 2145 2231 2317
1920 2003 2087 2170 2253
140 160 180 200
(2213) (2414) (2606) (2790)
2832
2729 3118
2635 3012 3388
2551 2915 3279 3644
2473 2827 3180 3533
2402 2746 3089 3432
2337 2671 3005 3339
(722)
235
OPINION
Vol. 115
TDF of 110 is equivalent to a PT of 1943 rets (this numbers appears in row 110, column 1800). The biologically effective dose can be quoted as represented by
Opinion
the NSD or PT solutions. For brachytherapy, TDF solutions (6) will be close to those using the PT or NSQ methods (9), differences being due to variations in the clinical data on which the equations are based. CONCLUSIONS
and
It is clear from this example that Table II can be used both by authors who wish to express their results in terms of biologically effective doses with minimal chance of error, and by readers who wish to intercompare published data. In order to avoid further confusion, the important features of some of the terms used in connection with the NSD concept are described below.
Calculations based on the NSD concept are complicated and quite readily lead to errors. Some of these errors are mathematical and some conceptual, but most are avoidable. TDF tables (5, 6), together with Table II, provide a rapid and accurate method for "checking" such complex calculations. Department of Radiology Division of RadiationTherapy New York University Medical Center 560 First Avenue New York, N. Y. 10016
Nominal Standard Dose (NSD)
(1)
(iI)
(iii)
NSD should not be used to describe a course of radiotherapy which does not result in full normal tissue tolerance (1). NSD values (calculated using the basic NSD equation) are not linearly additive (3). The "decay factor" cannot be applied directly to NSDs (2). Partial Tolerance (PT)
(1)
(ii)
PTs are linearly additive, and for complex treatment regimens may be calculated using the original equation (1), Dixon's equations (2), or by the application of TDFs, which can be converted to PTs by the use of Table II. The calculation of PTs requires the assumption of a "known" NSD tolerance value (4).
REFERENCES 1. Ellis F: Fractionation in radiotherapy. (In] Modern Trends in Radiotherapy, Deeley TJ, Wood CAP, ed. New York, Appleton-Century-Crofts, 1967, Vol 1, pp 34-51 2. Dixon RL: General equation for the calculation of nominal standard dose. Acta Radiol [Ther] 11:305-311, Aug 1972 3. Orton CG: Analysis and discussion of the time/doselfractionation problem. AAPM Quart Bull 6: 173-175, 1972 4. Ellis F: Dose, time and fractionation: A clinical hypothesis. Clin RadioI20:1-7, 1969 5. Orton CG, Ellis F: A simplification in the use of the NSD concept in practical radiotherapy. Br J RadioI46:529-537, Jul 1973 6. Orton CG: Time-dose factors (TDFs) in brachytherapy. Br J RadioI47:603-607, Sep 1974 7. Orton CG: The split-course radiotherapy problem. AAPM Quart Bull 7:22-23, 1973 8. Hall EJ: Radiobiology for the Radiologist. Hagerstown, Md., Harper and Row, 1973 9. Ellis F, Sorensen A: A method of estimating biological effect of combined intracavitary low dose rate radiation with external radiation in carcinoma of the cervix uteri. Radiology 110:681-686, Mar 1974
Time, Dose, and Fractionation (TDF) Factors (5, 6, 8) (1)
(iI) (iii)
TDFs have all the properties of PTs but do not require the assumption of a known NSD value. They can readily be converted to NSDs or PTs, using the conversion table in this paper (Table II). For fractionated radiotherapy, the solutions to practical problems using TDFs will be identical to
APPENDIX Table II was generated using the following two equations, which come from equation 5 in Orton and Ellis (5):
TDF=110 or NSD 1892 rets 1943 rets (if the NSD is taken or PT as 1800)
= =
Errors in Applying the NSD
Concept 1
Colin G. Orton, Ph.D. An analysis of selected papers in which the Nominal Standard Dose (NSD) concept is used reveals that about 50 % of them contain errors. These mistakes are both conceptual and mathematical in nature and average ± 7 %. Typical errors and methods to eliminate them are discussed. A table is provided for interconversion of NSDs, partial tolerances, and time, dose and fractionation (TDF)factors. INDEX TERMS:
Dosimetry, therapeutic. Therapeutic Radiology, treatment planning
Radiology 115:233-235, April 1975
• Use of the Nominal Standard Dose (NSD) concept has increased steadily since its introduction by Ellis in 1967 (1). This is illustrated in Figure 1 which shows the number of papers in which the NSD concept has been utilized in the three journals which publish most of the radiotherapy articles in the United States, viz., Radiology, American Journal of Roentgenology, and Cancer. In 1974, more than 20 such papers were published. This represents 15-20 % of all radiotherapy papers appearing in these journals. Unfortunately, the early articles concerning NSD described its application for the analysis of only relatively simple courses of radiotherapy, whereas in practice. the concept is most useful for the description of complex regimens which do not fall within the realm of the clinical experience of the users. The analysis of such complex applications of NSD is not simple and this has resulted in widespread misuse of the concept since it was not until recently that the full equations for NSD and partial tolerance were published (2). By June 1974, there were 59 articles involving the NSD concept in the three journals cited in Figure 1, and of these articles, 34 were sufficiently detailed for the calculations to be checked. As shown in Table I, it was found that the concept was applied incorrectly (or mathematical mistakes were made) in about 50% of cases. This figure is consistent with the results of a previous survey (3). The average error was ± 7%. Since most of these calculations were for treatments aimed at coming as close as possible to either normal tissue tolerance or cancericidal doses, such 7 % changes in dose are highly significant. There were basically two reasons why these errors were made: Incorrect combinations of regimens. When a course of radiotherapy was divided into several parts, it was not
5no
1969
1970
1971
1972
1973
1974'
YEAR 'BTR~POUHD
FROM
FIRST G MONTHS
Fig. 1. Articles utilizing the NSD concept in the three journals Radiology, American Journal of Roentgenology, and Cancer.
always realized that NSDs calculated using the simple NSD formula (NSD = total dose X ~0.24 X ,0.11) for each separate part of the treatment were meaningless numbers, as they did not represent full tissue tolerance. Furthermore, they could not be added linearly in order to determine the overall NSD. Partial tolerances, which are additive, should have been used (4). However, the calculation of partial tolerance is complicated (2), and authors often took the easy way out (adding NSDs), not appreciating the magnitude of the errors. Inconsistent overall treatment time. When a course of radiotherapy is divided into several parts, the sum of the treatment times used for the calculation of the partial tolerance of each part must equal the overall treatment time. A typical example involves treatment with two fields applied on alternate days. If the overall treatment time is 40 days, the partial tolerance for each field should be calculated using 20 days as the time factor
1 From the Division of Radiation Therapy, Department of Radiology, New York University Medical Center, New York, N. Y. Accepted for publication in November 1974. shan
233
234
OPINION
Table I: Papers Involving Detailed NSD Concept Calculations (to June 1974) in Radiology, American Journal of Roentgenology and Cancer Numberof Papers
No. with Incorrect Application of the NSD Concept
.~--~
34
April 1975
derstand the use of the TDF/NSD/Partial Tolerance Conversion Table.
Number with Mathematical Errors
Problem
--~~-~- ~~~~~~
16(47%)
----
A radiotherapist, having had some success with a treatment regimen consisting of 5 fractions of 300 rads followed by 24 fractions of 200 rads, all at 5 fractions per week, wishes to quote a value for the biologically effective dose in terms of the NSD concept. What number should he quote?
3(9%) ----
----
(T), so that when the partial tolerances of the two regimens are summated, the overall time will still be 40 days. It is very tempting to use T = 40 days for each calculation, thereby doubling the "homeostatic recovery time" of the tissues (1).
Solutions (a) If he considers this to represent a tolerance level, he can quote the overall NSD calculated using partial tolerances. However, such calculations are quite complicated, thus risking computational error (2). A simpler solution is to use TDFs. From Table V in Orton and Ellis (5), the TDFs for the first and second parts of the treatment are 31 and 79, respectively. The overall TDF is 31 + 79 = 110. The therapist could, if he wished, use this TDF number to represent the biologically effective dose, or alternatively, he could convert this to an NSD using Table II, which shows that a TDF of 110 is equivalent to an NSD of 1892 rets. (b) If the therapist considers an NSD of, say, 1800 rets to represent tolerance, then he could calculate the sum of the PTs for the 2 parts of the treatment. Such a calculation is again quite complicated (2). More simply, however, he could refer to Table II, which shows that a
DISCUSSION These misunderstandings of the NSD concept were first realized as a result of a survey conducted in 1972 (3). This led to the introduction of time, dose, and fractionation (TDF) factors (3, 5, 6, 8) in order to simplify the practical utilization of the NSD concept, and to reduce errors in the interpretation of the "rules of application" of NSDs and PTs. The inter-relationship between TDFs, NSDs, and PTs is shown in Table II, which has been derived from the fundamental equations for these quantities (see ApPENDIX).
USE OF TABLE II The following example should help the reader to unTable II:
TDF/NSD/Partial Tolerance Conversion Table Chosen NSD Value
TDF
(NSD)
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
10 20 25 30 35
(398) (625)
257 513 641 770 898
242 485 606 727 849
230 461 576 691 806
220 440 549 659 769
211
(813) (899)
273 547 683 820 957
421 526 632 737
202 405 506 607 708
195 390 487 585 682
188 376 471 565 659
182 364 455 547 638
177 353 442 530 618
172 343 429 515 601
167 334 417 501 584
40 45 50 55 60
(980) (1058) (1133) (1206) (1276)
1094 1230 1367 1504 1640
1026 1155 1283 1411 1540
970 1091 1212 1333 1455
921 1036 1152 1267 1382
879 989 1099 1209 1319
842 947 1053 1158 1263
809 910 1011 1112 1214
780 877 974 1072 1169
753 847 941 1035 1129
729 820 911 1002 1093
707 795 883 972 1060
686 772 858 944 1030
668 751 835 918 1002
65 70 75 80 85
(1344) (1410) (1475) (1538) (1600)
1668 1796 1924
1576 1697 1818 1940 2061
1497 1612 1727 1843 1958
1429 1538 1648 1758 1868
1368 1474 1579 1684 1789
1315 1416 1517 1618 1719
1267 1364 1462 1559 1657
1224 1318 1412 1506 1600
1184 1275 1366 1458 1549
1148 1237 1325 1413 1502
1115 1201 1287 1373 1459
1085 1168 1252 1335 1419
90 95 100 105 110
(1661) (1720) (1778) (1836) (1892)
2073 2188 2303
1978 2088 2198 2308 2418
1895 2000 2105 2210 2316
1820 1922 2023 2124 2225
1754 1851 1949 2046 2144
1694 1788 1882 1976 2071
1640 1731 1822 1913 2004
1590 1678 1767 1855 1943
1544 1630 1716 1802 1888
1502 1586 1669 1753 1836
115 120 125 130 135
(1947) (2002) (2056) (2109) (2161)
2421 2526 2631
2326 2427 2528 2630 2731
2241 2339 2436 2534 2631
2165 2259 2353 2447 2541
2095 2186 2277 2368 2460
2032 2120 2208 2297 2385
1973 2059 2145 2231 2317
1920 2003 2087 2170 2253
140 160 180 200
(2213) (2414) (2606) (2790)
2832
2729 3118
2635 3012 3388
2551 2915 3279 3644
2473 2827 3180 3533
2402 2746 3089 3432
2337 2671 3005 3339
(722)
235
OPINION
Vol. 115
TDF of 110 is equivalent to a PT of 1943 rets (this numbers appears in row 110, column 1800). The biologically effective dose can be quoted as represented by
Opinion
the NSD or PT solutions. For brachytherapy, TDF solutions (6) will be close to those using the PT or NSQ methods (9), differences being due to variations in the clinical data on which the equations are based. CONCLUSIONS
and
It is clear from this example that Table II can be used both by authors who wish to express their results in terms of biologically effective doses with minimal chance of error, and by readers who wish to intercompare published data. In order to avoid further confusion, the important features of some of the terms used in connection with the NSD concept are described below.
Calculations based on the NSD concept are complicated and quite readily lead to errors. Some of these errors are mathematical and some conceptual, but most are avoidable. TDF tables (5, 6), together with Table II, provide a rapid and accurate method for "checking" such complex calculations. Department of Radiology Division of RadiationTherapy New York University Medical Center 560 First Avenue New York, N. Y. 10016
Nominal Standard Dose (NSD)
(1)
(iI)
(iii)
NSD should not be used to describe a course of radiotherapy which does not result in full normal tissue tolerance (1). NSD values (calculated using the basic NSD equation) are not linearly additive (3). The "decay factor" cannot be applied directly to NSDs (2). Partial Tolerance (PT)
(1)
(ii)
PTs are linearly additive, and for complex treatment regimens may be calculated using the original equation (1), Dixon's equations (2), or by the application of TDFs, which can be converted to PTs by the use of Table II. The calculation of PTs requires the assumption of a "known" NSD tolerance value (4).
REFERENCES 1. Ellis F: Fractionation in radiotherapy. (In] Modern Trends in Radiotherapy, Deeley TJ, Wood CAP, ed. New York, Appleton-Century-Crofts, 1967, Vol 1, pp 34-51 2. Dixon RL: General equation for the calculation of nominal standard dose. Acta Radiol [Ther] 11:305-311, Aug 1972 3. Orton CG: Analysis and discussion of the time/doselfractionation problem. AAPM Quart Bull 6: 173-175, 1972 4. Ellis F: Dose, time and fractionation: A clinical hypothesis. Clin RadioI20:1-7, 1969 5. Orton CG, Ellis F: A simplification in the use of the NSD concept in practical radiotherapy. Br J RadioI46:529-537, Jul 1973 6. Orton CG: Time-dose factors (TDFs) in brachytherapy. Br J RadioI47:603-607, Sep 1974 7. Orton CG: The split-course radiotherapy problem. AAPM Quart Bull 7:22-23, 1973 8. Hall EJ: Radiobiology for the Radiologist. Hagerstown, Md., Harper and Row, 1973 9. Ellis F, Sorensen A: A method of estimating biological effect of combined intracavitary low dose rate radiation with external radiation in carcinoma of the cervix uteri. Radiology 110:681-686, Mar 1974
Time, Dose, and Fractionation (TDF) Factors (5, 6, 8) (1)
(iI) (iii)
TDFs have all the properties of PTs but do not require the assumption of a known NSD value. They can readily be converted to NSDs or PTs, using the conversion table in this paper (Table II). For fractionated radiotherapy, the solutions to practical problems using TDFs will be identical to
APPENDIX Table II was generated using the following two equations, which come from equation 5 in Orton and Ellis (5):
TDF=110 or NSD 1892 rets 1943 rets (if the NSD is taken or PT as 1800)
= =
