Inf. .I. Radiation Oncology Biol. Phys Vol. 21, pp. 123-135 Printed in the U.S.A. All rights reserved.
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0360-3016/91 $3.00 + .oO Q 1991 Pergamon Press plc
0 Original Contribution
FITTING OF NORMAL TISSUE TOLERANCE DATA TO AN ANALYTIC FUNCTION C. BURMAN, PH.D.,’
G. J. KIJTCHER, PH.D.,’
B. EMAMI, M.D.2
AND M.
GOITEIN, PH.D.~
‘Memorial Sloan-Kettering Cancer Center, New York, NY 10021; 2Mallinckrodt Institute of Radiology, Washington University School of Medicine, St. Louis, MO 63110; and 3Massachusetts General Hospital, Department of Radiation Medicine, Boston, MA 02114 and Harvard Medical School During external beam radiotherapy, normal tissues are irradiated along with the tumor. Radiation therapists try to minimize the dose to normal tissues while delivering a high dose to the target volume. Often this is difficult and complications arise due to irradiation of normal tissues. These complications depend not only on the dose but also on volume of the organ irradiated. Lyman (4) has suggested a four-parameter empirical model which can be used to represent normal tissue response under conditions of uniform irradiation to whole and partial volumes as a function of the dose and volume irradiated. In this paper, Lyman’s model has been applied to a compilation of clinical tolerance data developed by Emami et al. (1). The four parameters to characterize the tissue response have been determined and graphical representations of the derived probability distributions are presented. The model may, therefore, be used to interpolate clinical data to provide estimated normal tissue complication probabilities for any combination of dose and irradiated volume for the normal tissues and end points considered. Tissue tolerances, Parameters, Treatment planning, Radiotherapy. from fitting the four-parameter model to the tolerance data of Emami et al. (1). The 4-parameter model estimates the complication probability under conditions of uniform irradiation of a partial organ, but in radiotherapy treatments the dose distribution within an organ is usually nonuniform. Such nonuniform dose distribution can be represented quantitatively by dosevolume histograms. There are methods (2, 5) to reduce a nonuniform dose distribution to an equivalent uniform dose to a partial volume. Once the uniform dose-volume histogram is obtained, the results given in this paper can be used to calculate NTCP. The histogram reduction technique, which uses the results of the model presented here, is discussed in a companion paper (3).
INTRODUCTION In radiation therapy physicians often have to estimate, either implicitly or explicitly, the likelihood of complications due to radiation. These estimates are generally based on published information and the physician’s experience. This paper discusses an attempt to provide a quantitative method for making such estimates based on clinical data available at a few conditions of dose and irradiated volume. Emami et al. (1) have compiled tolerance data for selected normal tissues and organs. These data include estimates of tolerance doses for specific volume and end points, typically TD, (the tolerance dose for a 5% complication) and TD,, (the tolerance dose for a 50% complication) for whole, 5 and 3 uniform organ irradiation. It is necessary to interpolate the data to find the complication probability for an arbitrary uniformly irradiated partial volume and dose. Lyman (4) has suggested a model for this interpolation which uses four parameters to represent the normal tissue complication probability (NTCP) of an organ or tissue for uniform partial volume irradiation to dose D. If the four parameters are known, the response of a tissue to radiation can be calculated. The purpose of this paper is to present probability distributions for uniform irradiation obtained
METHODS
AND MATERIALS
Model The basis of the model is an error function with 4 parameters which connects the three variables of interest, normal tissue complication probability (NTCP), dose (D) and partial volume (v). The following equations describe the interrelation:
Supported in part by NC1 Contracts NO1 CM-47316, NO1 CM-47695, NO1 CM-47696, and NO1 CM-47697. Reprint requests to: C. Burman, Memorial Sloan-Kettering
Cancer Center, Medical Physics, 1275 York Ave., New York, NY 10021.
123
124
I. J. Radiation
NTCP = l/d2P
Oncology
0 Biology 0 Physics
s
exp(- ?/2)dt
(1)
-cc
(2)
v = VN,f t = (D TD(v)
TD,0(v))l(m*TD5c,(v)) = TD(l)
* v-n
(3) (4)
where TD is the tolerance dose and v is the fraction of the organ irradiated, or the volume relative to some reference volume. The above set of equations has 4 parameters, Vrefr TD,,, n and m. TD,, is the dose to the whole organ (or reference volume) which would lead to a complication probability of 50 percent. The volume dependence of the complication probability is determined by the parameter n, the slope of the complication probability vs. dose curve is governed by the value of the parameter, m, and V,, is the reference volume for TD,,.
Curve fitting procedure Emami et al. (1) have compiled tolerance data for many organs as a function of the whole and partial organ volumes. In these cases the parameter V,, is the whole organ volume. For some organs like the spinal cord, the tolerance doses were compiled for lengths of 20, 10, and 5 cm rather than volume. In this case the parameter V,, is a reference length of 20 cm, and V represents length. In the case of skin the tolerance doses were compiled for the areas of 100 cm’, 30 cm2, and 10 cm2. In this case V,, has been selected as a reference area of 100 cm*, and V then refers to the irradiated skin area. The remaining three parameters for the model, TD,,, n and m, were obtained by curve-fitting procedures. Data fitting was done “by eye” rather than by a statistical method since the small and uncertain number of data points did not warrant more sophisticated methods. Emami et al. (1) list six data points each for nine organs-esophagus, heart, brain, liver, lung, larynx, ear, stomach, temporomandibular joint and mandible. For these cases, a set of three curves (probability versus dose for 3 partial volumes, 1, ! and i) were generated for the given value of TD,,( 1) and estimated values of the parameters m and n. If the tit was poor, the parameter n was varied to obtain the best fit to the volume dependence. The next step was to adjust the parameter m to make the probability curve pass through points for TD,,(l) and TD,(l). While adjusting the parameters, more weight was given to the data for a 5% complication probability. For the larynx (end point-cartilage necrosis), temporomandibular joint and mandible, Emami et al. (1) give the same tolerance doses for the whole volume as for i of the volume, but different tolerance doses for i of the volume. Since the model cannot fit this set of data, the parameters have been obtained by using the tolerance data for 3 of the organ and for the whole organ. In this case, therefore, the desired probabilities for g uniform organ irradiation do not
May 15, 1991, Volume 21, Number
1
agree with the input data. For the end point of laryngeal edema the same tolerance doses are given for f and 3 of the volume. For this case the parameters have been obtained using the data for i of the organ and the whole organ volume. For the spinal cord, kidney, brain stem, bladder, lung, small intestine and colon tolerance doses for whole organ irradiation TD,,( 1) and TD,(l) were provided together with at least one data point for partial volume irradiation. The curve-fitting process was similar to the previous case. First, a good fit through the TD,,(l) and TD,(l) data points was obtained by adjusting the parameter m. Then the volume dependence was obtained by adjusting the parameter n, so that the probability curves for the partial volumes for which data is available provide a good fit. For the rectum, cauda equina, lens, retina, femoral head and neck, chiasma, optic nerve, brachial plexus and thyroid, only tolerance doses for whole organ irradiation were provided by Emami et al. (l), whereas for the ribs only tolerance doses for i of the rib cage have been provided. With these data it is possible to estimate only the parameters TD,, and m. To determine the volume dependence parameter, n, for organs with insufficient data, a best clinical estimate was made by a group of investigators. For the parotid gland the clinical data suggested no volume effect between 5 and whole organ irradiation. However, at the same time, the investigators felt that there was a threshold volume which implied a large volume effect between i and 5 uniform organ irradiation. A clinical estimate of a large volume dependence was, therefore, provided for the parotid. For the ear (middle/external) tolerance data for two end points, acute serous otitis and chronic serous otitis, were provided. For both these end points the tolerance doses for whole, 3 and 3 of the organ were the same which would lead to no volume dependence (n = 0). This is not believed to be physically correct, since it implies that a very small fraction of the organ irradiated to a dose, D, has the same complication as the irradiation of the whole organ to dose, D. Moreover, NTCP calculations for inhomogeneous irradiation for typical treatment planning examples, for tissues with n = 0, leads to unrealistic probabilities. Therefore, we have assigned, in these cases, a small but finite value of 0.01 to the volume dependence parameter, n. Tolerance doses for skin were given for a reference field size of 10 X 10 cm’. For the end point of necrosis and ulceration a total of four data points are provided, namely TD, and TD,, values for 100 cm2 field size, and TD, values for 30 cm2 and 10 cm2 field sizes. The parameter V,, in this case refers to an area of 100 cm2 and the parameter TD,,, n and m are for this reference area. RESULTS Table 1 lists the 4 parameters for the organs and end points considered by Emami et al. (1). Table 2 lists the
125
Fitting of tolerance data 0 C. BURMAN et al. Table 1. Normal tissue end points and tolerance parameters Fit narameters Organ Bladder Brachial plexus Brain Brain stem Cauda equina Colon Ear (middle/ external Ear (middle/ external Esophagus Femoral head and neck Heart Kidney Larynx Larynx Lens Liver Lung Optic nerve Optic chiasma Parotid Rectum Retina Rib cage Skin Small intestine Spinal cord Stomach Thyroid TM joint and mandible
V ref Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ 100 cm2 Whole organ 20 cm Whole organ Whole organ Whole organ
End point
n
m
0.5
0.11
80
0.03
0.12
75
Symptomatic bladder contracture and volume loss Clinically apparent nerve damage
0.25
0.15
60
Necrosis/infarction
0.16
0.14
65
Necrosis/infarction
0.03
0.12
75
Clinically apparent nerve damage
0.17
0.11
55
Obstruction/perforation/ulceration/fistula
0.01
0.15
40
Acute serous otitis
0.01
0.095
65
Chronic serous otitis
0.06
0.11
68
Clinical stricture/perforation
0.25
0.12
65
Necrosis
0.35
0.10
48
Pericarditis
0.70
0.10
28
Clinical nephritis
0.11
0.075
80
Cartilage necrosis
0.08
0.17
70
Laryngeal edema
0.30
0.27
18
Cataract requiring intervention
0.32
0.15
40
Liver failure
0.87
0.18
24.5
Pneumonitis
0.25
0.14
65
Blindness
0.25
0.14
65
Blindness
0.70
0.18
46
Xerostomia
0.12
0.15
80
Severe proctitis/necrosis
0.20
0.19
65
Blindness
0.10
0.21
68
Pathologic fracture
0.10 0.15
0.12 0.16
70.0 55
Necrosis/ulceration Obstruction/perforation
0.05 0.15
0.175 0.14
66.5 65
Myelitis/necrosis Ulceration/perforation
0.22
0.26
80
Clinical tbyroiditis
0.07
0.10
72
Marked limitation of joint function
T&o
/stenosis/fistula
126
I. .I. Radiation
Oncology
0 Biology 0 Physics
May 15, 1991, Volume 21, Number
Table 2. Tolerance doses of Emami et al. (1) and predictions* TQ,, (GY) volume Organ Bladder Brachial plexus Brain
I
3
2 1
of the 4-parameter
1
model (Eqs. l-4)
Tf%O[%O~)
1
1 3
2 T
1
End point Symptomatic bladder contracture and volume loss Clinically apparent nerve damage Necrosis/infarction
Brain stem
Necrosis/infarction
Cauda equina
Clinically apparent nerve damage
Colon
Obstruction/perforation fistula Acute serous otitis
Ear (middle/ external) Ear (middle/ external) Esophagus
/ulceration/
Chronic serous otitis Clinical stricture/perforation
Femoral head and neck Heart
Necrosis
Kidney
Clinical nephritis
Larynx
Cartilage necrosis
Larynx
Laryngeal edema
Lens
Cataract requiring intervention
Liver
Liver failure
Lung
Pneumonitis
Optic nerve
Blindness
optic chiasma Parotid
Blindness
Pericarditis
Xerostomia
Retina
Severe proctitis/necrosis/stenosis/fistula Blindness
Rib cage
Pathologic fracture
Skin
Necrosis/ulceration
Small intestine
Obstruction/perforation
Rectum
(Continued
on next page)
Fitting of tolerance data ??C. BURMAN et a[.
127
Table 2. (contd)
I
Organ
7
T”,,, (GY)
TD,o,, (GY)
volume
volume
2 5
1
1
1
2 5
1
End point
Spinal cord Myelitis/necrosis Stomach
Ulceration/perforation
Thyroid
Clinical thyroiditis
TM joint and mandible
Marked limitation of joint function
* Predictions
shown in parentheses.
along with the predicted tolerance doses in parentheses. Figures in Apendix A show the calculated curves for the complication probability as a function of dose for 3 partial volumes. The end points along with the parameters used to calculate the curves are indicated. The solid circles are the data compiled by Emami er al. (1).
tolerance
doses
for organs
and end points
DISCUSSION To judge a treatment plan or to compare two plans requires many decisions by a radiotherapist. The physician has to consider the dose delivered to the target volume along with the morbidity associated with the irradiation of surrounding normal tissues. Frequently, the physician has to make these decisions by looking at one or more dose distributions through some selected cross-sections. The process of judging a plan is often subjective depending on the personal preference of the physician, and estimates of cure and complication, which are often qualitative. Fitting of tolerance data to a model is a step in the direction of making the complication estimates quantitative and objective. This model is just a tool to interpolate and extrapolate known or estimated normal tissue tolerances. The accuracy or reasonableness of the estimates depends on the accuracy of the clinical tolerance data which are used to generate the curves. The cases for which a substantial number of data points are provided, for a wide range of partial volumes, leads to some confidence in the calculated values. For those cases where the clinical tolerance data is only for one volume, estimates of the volume dependence have to be made, and there will be much less confidence in the results. And for those cases where there is just one data point available, estimates of 2 parameters have to be made with consequently even less confidence. However, the probability curves that were generated, even under conditions of little data, appeared reasonable to radiation therapists, particularly for plan comparisons. But to put this set
of curves on more solid ground, additional clinical data is needed. The best use of these curves will be obtained if radiation therapists compare the predicted complication probabilities with their own experience; if the curves match with experience, this will suggest the selected values of parameters are reasonable. But if the estimated values consistently differ from their own experience, new curves can be generated to reflect local experience. The above model connects 3 quantities, complication probability, dose and partial volume. The interrelation between these 3 quantities can be represented by a surface as shown in Figure 1. In 2 dimensions we can plot any two variables against one another while keeping the 3rd one constant, as shown by thick lines A, B and C. For type A curves, the partial volume is kept constant and the compli-
Fig. 1. Complication probability for liver, displayed as a threedimensional surface as a function of dose and partial volume for uniform irradiation.
128
I. J. Radiation Oncology 0 Biology 0 Physics
May
15, 1991, Volume 21, Number I
1 .o :
0.s
:
0.9
i E
0.6
i m
0.8
S
0.7
2l
0.7
2 P
0.6
0 E
0.6
5
0.5
;
0.4
z F
0.4
= i %
0.3
5
0.3
0.2
2
0.2
00
0.1
g 0
0.1
0.5
0.0
1 0.
10.
20.
30.
40.
(4
50.
DOSE
60.
70.
60.
90.
0.6
0.7
0.6
0.9
100.
(Gy)
1 .o
0.5
-
0.4
-
0.3
-
0.2
-
0.1
-
0.0
0.1
0.2
0.3
0.4
), -
0.6
-
0.7
-
06-.0.5
-
a E
0.4
-
:
0.3
-
0.2
-
0.1
-
w I
0.5
PARTIAL
(b)
1.0
0.9
? m
0.6
0.6
0.7
0.6
0.9
0.7
5
0.5
c
0.4
-
5 i 5
0.3
-
0.2
-
g
O.l-
0.6
1.0
VOLUME
Esophagus
2 0 :
0.0
0.0
0.9
c
-
0.0
0.1
0.2
@I
0.3
0.4
PARTIAL 1.0
0.5
1 .o
VOLUME
P
Liver
= 9
0.0
1,
I,,
,
I,
I,
I,
I,,
,
I,,
.o. io. io. io. io. io. so.;o. eo.so.
0.
100.
Cc)
DOSE
(Gy)
(cl
10.
20.
30.
40.
DOSE
50.
60.
70.
60.
90.
J
100.
(Gy)
Fig. 2. (a) Complication probability curves fo; liver as a function of dose for uniform irradiation of whole, - and ; organ. (b) Complication probability as a function of uni?orm partial volume irradiation of liver for the doses of 40, 35, and 30 Gy. (c) Uniformly irradiated partial volume of liver as a function of dose for constant complication of 50, 20, and 5 percent.
Fig. 3. (a) Complication probability curves for esophagus as a function of dose for uniform irradiation of whole, i, and 5 organ. (b) Complication probability as a function of uniform partial volume irradiation of esophagus for the doses of 68, 62, and 56 Gy. (c) Uniformly irradiated partial volume of esophagus as a function of dose for constant complication of 50, 20, and 5 percent.
cation probability is represented as a function of dose. For the liver (n = 0.32) curves of this type are shown in Figure 2a for whole, 3 and i uniform volume irradiation. For type B curves the dose is kept constant and complication probability is represented as a function of partial volume.
In Figure 2b, these curves are shown for liver. This organ demonstrates a threshold type behavior; for a given dose, the complication probability does not vary with the partial volume until a certain partial volume is irradiated, and then the probability rises rather rapidly depending upon the
Fitting of tolerance
dose. Type C curves represent the partial volume as a function of dose with the given probability kept constant. Figure 2c shows curves of this type for the liver for complication probabilities of 5, 20, and 50 percent. While Figure 2 showed complication probability curves for a tissue with a large volume dependence (n = 0.32), Figure 3 shows similar curves for the esophagus, a tissue with a small volume dependence (n = 0.06). For type A graphs (Fig. 3a), curves of the complication probability vs. dose for the whole and i of the volume are closely spaced compared to the liver. As the value of the parameter n decreases, the curves for the partial volume irradiation shift towards the curve for the whole volume. As n approaches zero, the complication probability becomes independent of the fraction of the organ irradiated. Type B curves for esophagus are shown in Figure 3b. The shape of the curves, compared to the liver, are different in that there is no threshold type behavior. As mentioned before, the liver shows a threshold behavior; approximately 30 percent of the liver can be treated to TD,, (= 40 Gy) without significant complication whereas for the esophagus, the complication probability rises faster with partial volume at low partial volumes, so that the complication is zero only when no volume is irradiated. For the esophagus a type C curve is shown in Figure 3c. For a constant complication, as the dose is increased, the irradiated partial volume decreases. As the dose is decreased, the curve for esophagus approaches zero partial volume faster than the liver. Uncertainty in parameters The values of the parameters depend upon the tolerance doses. Uncertainties in these doses are reflected in the uncertainties of the parameters. The following are the estimates of these uncertainties. The parameter TD,, is generally determined by the tolerance dose for 50 percent complication for the whole organ and the uncertainty in this parameter is of the same order as the uncertainty in the tolerance dose. It is difficult to assign an uncertainty to this parameter. Emami et al. (1) should be consulted for an indication of the quality of the data. To estimate the uncertainty in the volume dependence parameter, n, equation (4) can be used. If the doses TD,,( 1) and TD,,($ are used to determine the value of the parameter n, then the uncertainty in the parameter (An) is related to uncertainty in the tolerance doses (ATD) through:
For example, in the case of larynx (end point-cartilage necrosis) TD,,(l) = 80 Gy and TDSO($ = 90 Gy, and if the uncertainties in these doses are assumed to be of the
129
data 0 C. BURMANet al.
order of 4 2 Gy, then from equation (5), An = 0.08. Since An is independent of n, this implies that the relative uncertainties in the volume effect will be small for tissues with a large volume effect, and large for tissues with a small volume effect. The parameter m determines the slope of the curve for complication probability as a function of dose. To determine the uncertainty in this parameter, equation (3) can be used. For a 5% complication, the variable t has a value of - 1.647, and from equation (3):
-=ih(l-SYg) or
In the case of larynx TD,(l) = 70 Gy and TD,,(l) = 80 Gy and with an assumed uncertainty of + 2 Gy in the tolerance doses, Am = 0.02. For the larynx the slope parameter m = 0.075, and TD,, = 80 Gy. Suppose the dose to the whole larynx is increased from 80 to 88 Gy which represents a 10% change. Then the complication probability goes from 50% to 91%. If, however, the slope parameter m is changed by 0.02, i.e., m = 0.055, then the same change in dose yields a complication probability of 97% rather than 9 1% . CONCLUSIONS A four-parameter model of complication probability was used which, for the most part, fits a recent compilation of data by Emami et al. Structure of this model implies that the dose and volume are related by power-law relationship. However, the response of tissue to radiation is a complicated process and is not well understood. There have been attempts other than the power-law relationship to understand the process. For example, the model proposed by Schultheiss et aE. (6) suggests a volume dependence which differs from the power-law relationship. Unfortunately, there is not enough clinical data to draw definite conclusion about which, if any, of the models represents the underlying volume dependence for different tissues. As mentioned earlier, in some cases there is insufficient data to determine the parameters so that the derived probability curves represent a substantial extrapolation of the tolerance data. And even when there is sufficient data, the uncertainties in the tolerance doses used to generate the curves can be significant. Therefore, it is prudent to carefully review the probability curves presented here. It is still possible, however, to use these NTCP curves for comparisons between alternative plans, even though the
130
I. J. Radiation Oncology 0 Biology 0 Physics
estimated complication probabilities may have errors associated with them. And it is expected that, as more reliable
May 15, 1991, Volume 21, Number 1
tolerance data becomes available, the estimated complication probabilities will be closer to the actual values.
REFERENCES Emami, B.; Lyman, J.; Brown, A.; Coia, L.; Goiten, M.; Munzenride, J. E.; Shank, B.; Solin, L. J.; Wesson, M. Tolerance of normal tissue to therapeutic radiation. Int. J. Radiat. Oncol. Biol. Phys. 21:109-122; 1991. Kutcher, G. J.; Burman, C. Calculation of complication probability-factors for non-uniform normal tissue irradiation: The effective volume method. Int. J. Radiat. Oncol. Biol. Phys. 16:1623-1630; 1989. Kutcher, G. J.; But-man, C.; Brewster, L.; Goitein, M.; Mohan, R. Histogram reduction method for calculating complication probabilities for three-dimensional treatment plan-
ning evaluations. Int. J. Radiat. Oncol. Biol. Phys. 21:137146; 1991. 4. Lyman, J. T. Complication probability-As assessed from dose-volume histograms. Radiat. Res. 104513-519; 1985. 5. Lyman, J. T.; Wolbarst, A. B. Optimization of radiation therapy, III: A method of assessing complication probability from dose-volume histograms. Int. J. Radiat. Oncol. Biol. Phys. 13:103-109; 1987. 6. Schultheiss, T. E.; Orton, C. G.; Peck, R. A. Models in radiotherapy: Volume effects. Med. Phys. 10:410-415; 1983.
APPENDIX Bladder TD= - 80 ” * 0.5 Ill = 0.11
End Point: Reference
Symptomatic bladder con. lracture and volume loss Volume: Whole organ
End Point: Necrosis/infarction Reference Volume: Whole organ
Brain TDa - 60 n - 025 m - 0.15
1 .o 1 .o 0.9 0.6
:
0.9
i i
0.6
0.7 0.6
;;: 0 E
0.7
g
0.5
;
0.4
5 i 5
0.3
8
0.1
0.6
0.5 0.4 0.3 0.2
0.2
0.1 0.0 0.
10.
20.
30.
40.
50.
60.
70.
60.
90.
0.0
100.
0.
DOSE
Fig. Al. Complication
;r&hi”f
probability
$xxus
Reference
Clinically apparent nerve damage Volume: Whole organ
60.
70.
60.
90.
100.
(Gy)
vs. dose for the brain.
End Point: Necmsitinfarction Reference Volume: Whole organ
” - 0.16 m - 0.14 1.0
i iij
0.6
2
0.7
ZI
g a
0.6
2 0
z
0.5
:
g ~
50.
probability
Brain Stem
1.0
%
40.
Tb-65
0.9
d i
30.
Fig. A3. Complication
c
0 g
20.
DOSE
vs. dose for the bladder.
End Point:
” - 0.w m * 0.12
10.
(Gy)
: i
0.4 0.3 0.2 0.1 i 10.
20.
30.
40.
DOSE
Fig. A2. plexus.
Complication
probability
;o.
so.
;o.
,
ao.
I
,
eo.
0.7 0.6 0.5
;
0.4
2 i %
0.3 0.2 0.1
,
ld0.
0.0 0.
(Gy)
vs. dose
0.6
z
:: I
0.
0.9
10.
20.
30.
40.
DOSE
for the brachial Fig. A4. Complication
probability
50.
60.
70.
60.
ao.
d(I.
1’
(Gy)
vs. dose for the brain stem.
ags
gn
P.
p
z
?
?
.
? ? P ? O-NWCGlmLOlDb I I I I I I I I
?
COMPLICATION
!J
I
I
I
?
II
P
II
0
I I I,
P P
I I I I I
PROBABILITY
I
PROBABILITY
I I I
I I I I I I I I I I I
COMPLICATION 2 cm
?
COMPLICATION
0 ? ? ? o~Nwcba.Jolnb
COMPLICATION
PROBABILITY
PROBABILITY
*pp?p-
1
o-i.aucuo~biDb I I I I I I I,
?
oopppp~oo-
PROBABILITY
PROBABILITY
I I I I I I I I I I I,
COMPLICATION
COMPLICATION
-I
132
I. J. Radiation Oncology 0 Biology 0 Physics
May 15, 1991, Volume 21, Number 1
End Point: Pericardltis reference Volume: Whole organ
HeaR - 46 TDm
End Point: Laryngeal edema Reference Volume: Whole organ
Larynx TD* 70.0 n = 0.08 m - 0.17
n = 0.35 Ill = 0.10
1 .o g
0.9
i m
0.6
2 0 :
0.7
z
0.5
5 0 i %
o.4 0.3
0 0
0.8
0.2 0.1 0.0 0.
10.
20.
30.
probability
50.
60.
DOSE
DOSE (Gy) Fig. Al 1. Complication
40.
70.
60.
90.
100.
(Gy)
Fig, A14. Complication probability vs. dose for the larynx for the end point of laryngeal edema.
vs. dose for the heart.
End Point: Clinical nephritis Reference Volume: Whole otgan
End Point: Cataract reqUirlng intewention Reference Volume: Whole organ
Lens TD6O - 16 ” = 0.30 m = 0.27 1.0 :
0.9
i G
0.6
a al
0.7
0 g
0.6
z
0.5
0 c
0.4
2 i I”
0.3
00
0.1
0.2
0.0 0.
10.
20.
30.
40.
50.
DOSE
Fig. A12. Complication
60.
70.
60.
90.
100.
DOSE
(Gy)
probability
vs. dose for the kidney.
End Point: Cartilage necros~e Reference Volume: Whole organ
Fig. AH.
Complication
(Gy)
probability
Liver TD!jr) - 40 n = 0.32 m = 0.15
vs. dose for the lens.
End Point: Liver failure Reference Volume: Whole organ
1.0 :
0.9
i m
0.0
$
0.5
s 0 i a I 00
o.4 0.3
-1
Ii
I
0.2 0.1
DOSE
(Gy)
Fig. A13. Complication probability vs. dose for the larynx for the end point of cartilage necrosis.
DOSE Fig. A16. Complication
probability
(Gy)
vs. dose for the liver.
133
Fitting of tolerance data 0 C. BURMAN ef a[. End Point: Pneumomtls Reference Volume: Whole
= 24.5 n - 0.67 m = 0.16 1.0 z
0.9
5 G
0.6
End Point: Xerostomia Reference Volume: Whole organ
Parotid TDw - 46 n = 0.70 m = 0.16
organ
1.0 0.9 0.6
2 0 @z a
0.7
5
0.5
F
0.4
= 5 2
0.3 0.2
0.2
z
0.1
0.1
0.7
0.6
0.6 0.5 0.4 0.3
0.0
I 0.
10.
20.
30.
40.
50.
DOSE
Fig. A17. Complication
’ II 60.
1’1’1 70.
0.0
1 60.
90.
100.
0.
10.
20.
30.
(Gy)
probability
40.
50.
DOSE
vs. dose for the lung.
End Point: Blindness Reference Volume: Whole
Fig. A20. Complication
probability
Rectum TD50 = 60
organ
70.
60.
90.
1
vs. dose for the parotid.
End Palm: Reference
n =025 m-014
Severe proct~t~s/necros~sJ stenosis/fistula Volume: Whole organ
1.0
1.0
0.9
0.9
0.6
0.6
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0 0.
10.
20.
30.
40.
50.
DOSE
Fig. A18. Complication
60.
70.
60.
90.
0.
100.
10.
20.
30.
40.
50.
DOSE
(Gy)
probability
Fig. A21. Complication
vs. dose for the optic nerve.
End Point: Blindness Reference Volume: Whole
0 tic chiasma T B 50 = 65 ” = 0.25 m = 0.14 1.0
60.
70.
60.
90.
100.
(Gy)
probability
Retina TD50 - 65 ” = 0.20 m = 0.19
organ
vs. dose for the rectum.
End Point: Blindness Reference Volume: Whole
organ
1.0
:
0.9
t?
0.9
i =
0.6
i 6
0.6
2 0 a a
0.7
tz
0.7
0.6
z a
0.6
z
0.5
z
0.5
i=
0.4
F
0.4
= i s
0.3
5 i :
0.3
0.2 0.1
00
0 0
60.
(Gy)
0.0
0.2 0.1 0.0
0.
10.
20.
30.
40.
DOSE
Fig. A19. Complication asma.
probability
50.
60.
70.
60.
90.
100.
0.
10.
20.
(Gy)
vs. dose for the optic chi-
30.
40.
DOSE
Fig. A22. Complication
probability
50.
60.
70.
80.
90.
(Gy)
vs. dose for the retina.
100.
134
I. J. Radiation Oncology 0 Biology 0 Physics
May 15, 1991, Volume 21, Number 1
End Point: Pathologic fracture Reference Volume: Whole organ
Small TDs0
n = 0.10 m = 0.21 1.0 c
0.9
i E
0.6
0.9
i
m
0.6 0.7
0.6
0.6
0.5
5
0.5
0.4
F
0.4
5 i $
0.3
0.3
2
0.1
2 i a I 00
0.7
z
6 t=
End Point: Obstructionlperioration RaferenCa Volume: Whole organ
1.0
: 2 0 :
; 0 :
intestine - 55
” = 0.15 m - 0.16
0.2
0.2 0.1 0.0
0.0 0.
10.
20.
30.
40.
50.
DOSE
Fig. A23. Complication
70.
60.
90.
0.
20.
30.
40.
50.
DOSE
Fig. A25. Complication tine.
vs. dose for the rib cage.
probability
S mal cord T&0 = 66.5 ” = 0.05 m = 0.175
End Point: NecroslSiulceraton Reference Area: 100 cm2
= 70.0 n = 0.10 m = 0.12
10.
(Gy)
probability
Skm TD50
60.
1 .o
60.
70.
60.
90.
1
0.
(Gy)
vs. dose for the small intes
End Point: Myelltislnecrosls Reference Length: 20 cm
1.0
t:
0.9
:
0.9
i E
0.8
i m
0.6
2
0.7
4 :
0.7
z a
0.6
:
0.6
g
0.5
5
0.5
;
0.4
F
0.4
2 i &
0.3 0.2
s i %
0.2
g
0.1
$
0.1
0.0
0.3
0.0 0.
10.
20.
so.
40.
50.
DOSE
Fig. A24. Complication
60.
70.
60.
90.
100.
0.
10.
20.
30.
(Gy)
probability
vs. dose for the skin.
Fig. A26. Complication
probability
End Pomt: Ukeratuw edoration d hole organ Reference Volume:
Stomach TD50 - 65 ” - 0.15 m = 0.14 1 .o t:
0.9
i m
0.6
2
0.7
g a
0.6
5
0.5
c
0.4
d
0.3
2
0.2
; 0
40.
DOSE
0.1 0.0 0.
10.
20.
30.
40.
DOSE
Fig. A27. Complication
probability
50.
60.
70.
00.
90.
100.
(Gy)
vs. dose for the stomach.
50.
60.
70.
60.
90.
10’ 0.
(Gy)
vs. dose for the spinal cord.
Fitting of tolerance data ??C. BLRMAN et al.
1.0
TM Joint TD5o ” = Ill =
End Point: Clinical thyroidilis Reference Volume: Whole organ
Thyrold TD50 i 80 ” - 0.22 m=OZ6
135 End Point:
and mandible 72 0.07 0.10
Reference
Marked llmltation of joint function Volume: Whole organ
,
5 E
0.0
2
0.7
0 :
0.6
z
0.5
0 c
0.4
= i % g
0.3 0.2 0 0
0.1 0.0 0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
DOSE (Gy) Fig. A28. Complication
probability
vs. dose for the thyroid.
100.
0.1
-
0.0
0.
10.
20.
30.
40.
SO.
60.
70.
80.
90.
1
0.
DOSE (Gy) Fig. A29. Complication probability vs. dose for the temporomandibular joint and mandible.
Copyrrght
0360-3016/91 $3.00 + .oO Q 1991 Pergamon Press plc
0 Original Contribution
FITTING OF NORMAL TISSUE TOLERANCE DATA TO AN ANALYTIC FUNCTION C. BURMAN, PH.D.,’
G. J. KIJTCHER, PH.D.,’
B. EMAMI, M.D.2
AND M.
GOITEIN, PH.D.~
‘Memorial Sloan-Kettering Cancer Center, New York, NY 10021; 2Mallinckrodt Institute of Radiology, Washington University School of Medicine, St. Louis, MO 63110; and 3Massachusetts General Hospital, Department of Radiation Medicine, Boston, MA 02114 and Harvard Medical School During external beam radiotherapy, normal tissues are irradiated along with the tumor. Radiation therapists try to minimize the dose to normal tissues while delivering a high dose to the target volume. Often this is difficult and complications arise due to irradiation of normal tissues. These complications depend not only on the dose but also on volume of the organ irradiated. Lyman (4) has suggested a four-parameter empirical model which can be used to represent normal tissue response under conditions of uniform irradiation to whole and partial volumes as a function of the dose and volume irradiated. In this paper, Lyman’s model has been applied to a compilation of clinical tolerance data developed by Emami et al. (1). The four parameters to characterize the tissue response have been determined and graphical representations of the derived probability distributions are presented. The model may, therefore, be used to interpolate clinical data to provide estimated normal tissue complication probabilities for any combination of dose and irradiated volume for the normal tissues and end points considered. Tissue tolerances, Parameters, Treatment planning, Radiotherapy. from fitting the four-parameter model to the tolerance data of Emami et al. (1). The 4-parameter model estimates the complication probability under conditions of uniform irradiation of a partial organ, but in radiotherapy treatments the dose distribution within an organ is usually nonuniform. Such nonuniform dose distribution can be represented quantitatively by dosevolume histograms. There are methods (2, 5) to reduce a nonuniform dose distribution to an equivalent uniform dose to a partial volume. Once the uniform dose-volume histogram is obtained, the results given in this paper can be used to calculate NTCP. The histogram reduction technique, which uses the results of the model presented here, is discussed in a companion paper (3).
INTRODUCTION In radiation therapy physicians often have to estimate, either implicitly or explicitly, the likelihood of complications due to radiation. These estimates are generally based on published information and the physician’s experience. This paper discusses an attempt to provide a quantitative method for making such estimates based on clinical data available at a few conditions of dose and irradiated volume. Emami et al. (1) have compiled tolerance data for selected normal tissues and organs. These data include estimates of tolerance doses for specific volume and end points, typically TD, (the tolerance dose for a 5% complication) and TD,, (the tolerance dose for a 50% complication) for whole, 5 and 3 uniform organ irradiation. It is necessary to interpolate the data to find the complication probability for an arbitrary uniformly irradiated partial volume and dose. Lyman (4) has suggested a model for this interpolation which uses four parameters to represent the normal tissue complication probability (NTCP) of an organ or tissue for uniform partial volume irradiation to dose D. If the four parameters are known, the response of a tissue to radiation can be calculated. The purpose of this paper is to present probability distributions for uniform irradiation obtained
METHODS
AND MATERIALS
Model The basis of the model is an error function with 4 parameters which connects the three variables of interest, normal tissue complication probability (NTCP), dose (D) and partial volume (v). The following equations describe the interrelation:
Supported in part by NC1 Contracts NO1 CM-47316, NO1 CM-47695, NO1 CM-47696, and NO1 CM-47697. Reprint requests to: C. Burman, Memorial Sloan-Kettering
Cancer Center, Medical Physics, 1275 York Ave., New York, NY 10021.
123
124
I. J. Radiation
NTCP = l/d2P
Oncology
0 Biology 0 Physics
s
exp(- ?/2)dt
(1)
-cc
(2)
v = VN,f t = (D TD(v)
TD,0(v))l(m*TD5c,(v)) = TD(l)
* v-n
(3) (4)
where TD is the tolerance dose and v is the fraction of the organ irradiated, or the volume relative to some reference volume. The above set of equations has 4 parameters, Vrefr TD,,, n and m. TD,, is the dose to the whole organ (or reference volume) which would lead to a complication probability of 50 percent. The volume dependence of the complication probability is determined by the parameter n, the slope of the complication probability vs. dose curve is governed by the value of the parameter, m, and V,, is the reference volume for TD,,.
Curve fitting procedure Emami et al. (1) have compiled tolerance data for many organs as a function of the whole and partial organ volumes. In these cases the parameter V,, is the whole organ volume. For some organs like the spinal cord, the tolerance doses were compiled for lengths of 20, 10, and 5 cm rather than volume. In this case the parameter V,, is a reference length of 20 cm, and V represents length. In the case of skin the tolerance doses were compiled for the areas of 100 cm’, 30 cm2, and 10 cm2. In this case V,, has been selected as a reference area of 100 cm*, and V then refers to the irradiated skin area. The remaining three parameters for the model, TD,,, n and m, were obtained by curve-fitting procedures. Data fitting was done “by eye” rather than by a statistical method since the small and uncertain number of data points did not warrant more sophisticated methods. Emami et al. (1) list six data points each for nine organs-esophagus, heart, brain, liver, lung, larynx, ear, stomach, temporomandibular joint and mandible. For these cases, a set of three curves (probability versus dose for 3 partial volumes, 1, ! and i) were generated for the given value of TD,,( 1) and estimated values of the parameters m and n. If the tit was poor, the parameter n was varied to obtain the best fit to the volume dependence. The next step was to adjust the parameter m to make the probability curve pass through points for TD,,(l) and TD,(l). While adjusting the parameters, more weight was given to the data for a 5% complication probability. For the larynx (end point-cartilage necrosis), temporomandibular joint and mandible, Emami et al. (1) give the same tolerance doses for the whole volume as for i of the volume, but different tolerance doses for i of the volume. Since the model cannot fit this set of data, the parameters have been obtained by using the tolerance data for 3 of the organ and for the whole organ. In this case, therefore, the desired probabilities for g uniform organ irradiation do not
May 15, 1991, Volume 21, Number
1
agree with the input data. For the end point of laryngeal edema the same tolerance doses are given for f and 3 of the volume. For this case the parameters have been obtained using the data for i of the organ and the whole organ volume. For the spinal cord, kidney, brain stem, bladder, lung, small intestine and colon tolerance doses for whole organ irradiation TD,,( 1) and TD,(l) were provided together with at least one data point for partial volume irradiation. The curve-fitting process was similar to the previous case. First, a good fit through the TD,,(l) and TD,(l) data points was obtained by adjusting the parameter m. Then the volume dependence was obtained by adjusting the parameter n, so that the probability curves for the partial volumes for which data is available provide a good fit. For the rectum, cauda equina, lens, retina, femoral head and neck, chiasma, optic nerve, brachial plexus and thyroid, only tolerance doses for whole organ irradiation were provided by Emami et al. (l), whereas for the ribs only tolerance doses for i of the rib cage have been provided. With these data it is possible to estimate only the parameters TD,, and m. To determine the volume dependence parameter, n, for organs with insufficient data, a best clinical estimate was made by a group of investigators. For the parotid gland the clinical data suggested no volume effect between 5 and whole organ irradiation. However, at the same time, the investigators felt that there was a threshold volume which implied a large volume effect between i and 5 uniform organ irradiation. A clinical estimate of a large volume dependence was, therefore, provided for the parotid. For the ear (middle/external) tolerance data for two end points, acute serous otitis and chronic serous otitis, were provided. For both these end points the tolerance doses for whole, 3 and 3 of the organ were the same which would lead to no volume dependence (n = 0). This is not believed to be physically correct, since it implies that a very small fraction of the organ irradiated to a dose, D, has the same complication as the irradiation of the whole organ to dose, D. Moreover, NTCP calculations for inhomogeneous irradiation for typical treatment planning examples, for tissues with n = 0, leads to unrealistic probabilities. Therefore, we have assigned, in these cases, a small but finite value of 0.01 to the volume dependence parameter, n. Tolerance doses for skin were given for a reference field size of 10 X 10 cm’. For the end point of necrosis and ulceration a total of four data points are provided, namely TD, and TD,, values for 100 cm2 field size, and TD, values for 30 cm2 and 10 cm2 field sizes. The parameter V,, in this case refers to an area of 100 cm2 and the parameter TD,,, n and m are for this reference area. RESULTS Table 1 lists the 4 parameters for the organs and end points considered by Emami et al. (1). Table 2 lists the
125
Fitting of tolerance data 0 C. BURMAN et al. Table 1. Normal tissue end points and tolerance parameters Fit narameters Organ Bladder Brachial plexus Brain Brain stem Cauda equina Colon Ear (middle/ external Ear (middle/ external Esophagus Femoral head and neck Heart Kidney Larynx Larynx Lens Liver Lung Optic nerve Optic chiasma Parotid Rectum Retina Rib cage Skin Small intestine Spinal cord Stomach Thyroid TM joint and mandible
V ref Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ Whole organ 100 cm2 Whole organ 20 cm Whole organ Whole organ Whole organ
End point
n
m
0.5
0.11
80
0.03
0.12
75
Symptomatic bladder contracture and volume loss Clinically apparent nerve damage
0.25
0.15
60
Necrosis/infarction
0.16
0.14
65
Necrosis/infarction
0.03
0.12
75
Clinically apparent nerve damage
0.17
0.11
55
Obstruction/perforation/ulceration/fistula
0.01
0.15
40
Acute serous otitis
0.01
0.095
65
Chronic serous otitis
0.06
0.11
68
Clinical stricture/perforation
0.25
0.12
65
Necrosis
0.35
0.10
48
Pericarditis
0.70
0.10
28
Clinical nephritis
0.11
0.075
80
Cartilage necrosis
0.08
0.17
70
Laryngeal edema
0.30
0.27
18
Cataract requiring intervention
0.32
0.15
40
Liver failure
0.87
0.18
24.5
Pneumonitis
0.25
0.14
65
Blindness
0.25
0.14
65
Blindness
0.70
0.18
46
Xerostomia
0.12
0.15
80
Severe proctitis/necrosis
0.20
0.19
65
Blindness
0.10
0.21
68
Pathologic fracture
0.10 0.15
0.12 0.16
70.0 55
Necrosis/ulceration Obstruction/perforation
0.05 0.15
0.175 0.14
66.5 65
Myelitis/necrosis Ulceration/perforation
0.22
0.26
80
Clinical tbyroiditis
0.07
0.10
72
Marked limitation of joint function
T&o
/stenosis/fistula
126
I. .I. Radiation
Oncology
0 Biology 0 Physics
May 15, 1991, Volume 21, Number
Table 2. Tolerance doses of Emami et al. (1) and predictions* TQ,, (GY) volume Organ Bladder Brachial plexus Brain
I
3
2 1
of the 4-parameter
1
model (Eqs. l-4)
Tf%O[%O~)
1
1 3
2 T
1
End point Symptomatic bladder contracture and volume loss Clinically apparent nerve damage Necrosis/infarction
Brain stem
Necrosis/infarction
Cauda equina
Clinically apparent nerve damage
Colon
Obstruction/perforation fistula Acute serous otitis
Ear (middle/ external) Ear (middle/ external) Esophagus
/ulceration/
Chronic serous otitis Clinical stricture/perforation
Femoral head and neck Heart
Necrosis
Kidney
Clinical nephritis
Larynx
Cartilage necrosis
Larynx
Laryngeal edema
Lens
Cataract requiring intervention
Liver
Liver failure
Lung
Pneumonitis
Optic nerve
Blindness
optic chiasma Parotid
Blindness
Pericarditis
Xerostomia
Retina
Severe proctitis/necrosis/stenosis/fistula Blindness
Rib cage
Pathologic fracture
Skin
Necrosis/ulceration
Small intestine
Obstruction/perforation
Rectum
(Continued
on next page)
Fitting of tolerance data ??C. BURMAN et a[.
127
Table 2. (contd)
I
Organ
7
T”,,, (GY)
TD,o,, (GY)
volume
volume
2 5
1
1
1
2 5
1
End point
Spinal cord Myelitis/necrosis Stomach
Ulceration/perforation
Thyroid
Clinical thyroiditis
TM joint and mandible
Marked limitation of joint function
* Predictions
shown in parentheses.
along with the predicted tolerance doses in parentheses. Figures in Apendix A show the calculated curves for the complication probability as a function of dose for 3 partial volumes. The end points along with the parameters used to calculate the curves are indicated. The solid circles are the data compiled by Emami er al. (1).
tolerance
doses
for organs
and end points
DISCUSSION To judge a treatment plan or to compare two plans requires many decisions by a radiotherapist. The physician has to consider the dose delivered to the target volume along with the morbidity associated with the irradiation of surrounding normal tissues. Frequently, the physician has to make these decisions by looking at one or more dose distributions through some selected cross-sections. The process of judging a plan is often subjective depending on the personal preference of the physician, and estimates of cure and complication, which are often qualitative. Fitting of tolerance data to a model is a step in the direction of making the complication estimates quantitative and objective. This model is just a tool to interpolate and extrapolate known or estimated normal tissue tolerances. The accuracy or reasonableness of the estimates depends on the accuracy of the clinical tolerance data which are used to generate the curves. The cases for which a substantial number of data points are provided, for a wide range of partial volumes, leads to some confidence in the calculated values. For those cases where the clinical tolerance data is only for one volume, estimates of the volume dependence have to be made, and there will be much less confidence in the results. And for those cases where there is just one data point available, estimates of 2 parameters have to be made with consequently even less confidence. However, the probability curves that were generated, even under conditions of little data, appeared reasonable to radiation therapists, particularly for plan comparisons. But to put this set
of curves on more solid ground, additional clinical data is needed. The best use of these curves will be obtained if radiation therapists compare the predicted complication probabilities with their own experience; if the curves match with experience, this will suggest the selected values of parameters are reasonable. But if the estimated values consistently differ from their own experience, new curves can be generated to reflect local experience. The above model connects 3 quantities, complication probability, dose and partial volume. The interrelation between these 3 quantities can be represented by a surface as shown in Figure 1. In 2 dimensions we can plot any two variables against one another while keeping the 3rd one constant, as shown by thick lines A, B and C. For type A curves, the partial volume is kept constant and the compli-
Fig. 1. Complication probability for liver, displayed as a threedimensional surface as a function of dose and partial volume for uniform irradiation.
128
I. J. Radiation Oncology 0 Biology 0 Physics
May
15, 1991, Volume 21, Number I
1 .o :
0.s
:
0.9
i E
0.6
i m
0.8
S
0.7
2l
0.7
2 P
0.6
0 E
0.6
5
0.5
;
0.4
z F
0.4
= i %
0.3
5
0.3
0.2
2
0.2
00
0.1
g 0
0.1
0.5
0.0
1 0.
10.
20.
30.
40.
(4
50.
DOSE
60.
70.
60.
90.
0.6
0.7
0.6
0.9
100.
(Gy)
1 .o
0.5
-
0.4
-
0.3
-
0.2
-
0.1
-
0.0
0.1
0.2
0.3
0.4
), -
0.6
-
0.7
-
06-.0.5
-
a E
0.4
-
:
0.3
-
0.2
-
0.1
-
w I
0.5
PARTIAL
(b)
1.0
0.9
? m
0.6
0.6
0.7
0.6
0.9
0.7
5
0.5
c
0.4
-
5 i 5
0.3
-
0.2
-
g
O.l-
0.6
1.0
VOLUME
Esophagus
2 0 :
0.0
0.0
0.9
c
-
0.0
0.1
0.2
@I
0.3
0.4
PARTIAL 1.0
0.5
1 .o
VOLUME
P
Liver
= 9
0.0
1,
I,,
,
I,
I,
I,
I,,
,
I,,
.o. io. io. io. io. io. so.;o. eo.so.
0.
100.
Cc)
DOSE
(Gy)
(cl
10.
20.
30.
40.
DOSE
50.
60.
70.
60.
90.
J
100.
(Gy)
Fig. 2. (a) Complication probability curves fo; liver as a function of dose for uniform irradiation of whole, - and ; organ. (b) Complication probability as a function of uni?orm partial volume irradiation of liver for the doses of 40, 35, and 30 Gy. (c) Uniformly irradiated partial volume of liver as a function of dose for constant complication of 50, 20, and 5 percent.
Fig. 3. (a) Complication probability curves for esophagus as a function of dose for uniform irradiation of whole, i, and 5 organ. (b) Complication probability as a function of uniform partial volume irradiation of esophagus for the doses of 68, 62, and 56 Gy. (c) Uniformly irradiated partial volume of esophagus as a function of dose for constant complication of 50, 20, and 5 percent.
cation probability is represented as a function of dose. For the liver (n = 0.32) curves of this type are shown in Figure 2a for whole, 3 and i uniform volume irradiation. For type B curves the dose is kept constant and complication probability is represented as a function of partial volume.
In Figure 2b, these curves are shown for liver. This organ demonstrates a threshold type behavior; for a given dose, the complication probability does not vary with the partial volume until a certain partial volume is irradiated, and then the probability rises rather rapidly depending upon the
Fitting of tolerance
dose. Type C curves represent the partial volume as a function of dose with the given probability kept constant. Figure 2c shows curves of this type for the liver for complication probabilities of 5, 20, and 50 percent. While Figure 2 showed complication probability curves for a tissue with a large volume dependence (n = 0.32), Figure 3 shows similar curves for the esophagus, a tissue with a small volume dependence (n = 0.06). For type A graphs (Fig. 3a), curves of the complication probability vs. dose for the whole and i of the volume are closely spaced compared to the liver. As the value of the parameter n decreases, the curves for the partial volume irradiation shift towards the curve for the whole volume. As n approaches zero, the complication probability becomes independent of the fraction of the organ irradiated. Type B curves for esophagus are shown in Figure 3b. The shape of the curves, compared to the liver, are different in that there is no threshold type behavior. As mentioned before, the liver shows a threshold behavior; approximately 30 percent of the liver can be treated to TD,, (= 40 Gy) without significant complication whereas for the esophagus, the complication probability rises faster with partial volume at low partial volumes, so that the complication is zero only when no volume is irradiated. For the esophagus a type C curve is shown in Figure 3c. For a constant complication, as the dose is increased, the irradiated partial volume decreases. As the dose is decreased, the curve for esophagus approaches zero partial volume faster than the liver. Uncertainty in parameters The values of the parameters depend upon the tolerance doses. Uncertainties in these doses are reflected in the uncertainties of the parameters. The following are the estimates of these uncertainties. The parameter TD,, is generally determined by the tolerance dose for 50 percent complication for the whole organ and the uncertainty in this parameter is of the same order as the uncertainty in the tolerance dose. It is difficult to assign an uncertainty to this parameter. Emami et al. (1) should be consulted for an indication of the quality of the data. To estimate the uncertainty in the volume dependence parameter, n, equation (4) can be used. If the doses TD,,( 1) and TD,,($ are used to determine the value of the parameter n, then the uncertainty in the parameter (An) is related to uncertainty in the tolerance doses (ATD) through:
For example, in the case of larynx (end point-cartilage necrosis) TD,,(l) = 80 Gy and TDSO($ = 90 Gy, and if the uncertainties in these doses are assumed to be of the
129
data 0 C. BURMANet al.
order of 4 2 Gy, then from equation (5), An = 0.08. Since An is independent of n, this implies that the relative uncertainties in the volume effect will be small for tissues with a large volume effect, and large for tissues with a small volume effect. The parameter m determines the slope of the curve for complication probability as a function of dose. To determine the uncertainty in this parameter, equation (3) can be used. For a 5% complication, the variable t has a value of - 1.647, and from equation (3):
-=ih(l-SYg) or
In the case of larynx TD,(l) = 70 Gy and TD,,(l) = 80 Gy and with an assumed uncertainty of + 2 Gy in the tolerance doses, Am = 0.02. For the larynx the slope parameter m = 0.075, and TD,, = 80 Gy. Suppose the dose to the whole larynx is increased from 80 to 88 Gy which represents a 10% change. Then the complication probability goes from 50% to 91%. If, however, the slope parameter m is changed by 0.02, i.e., m = 0.055, then the same change in dose yields a complication probability of 97% rather than 9 1% . CONCLUSIONS A four-parameter model of complication probability was used which, for the most part, fits a recent compilation of data by Emami et al. Structure of this model implies that the dose and volume are related by power-law relationship. However, the response of tissue to radiation is a complicated process and is not well understood. There have been attempts other than the power-law relationship to understand the process. For example, the model proposed by Schultheiss et aE. (6) suggests a volume dependence which differs from the power-law relationship. Unfortunately, there is not enough clinical data to draw definite conclusion about which, if any, of the models represents the underlying volume dependence for different tissues. As mentioned earlier, in some cases there is insufficient data to determine the parameters so that the derived probability curves represent a substantial extrapolation of the tolerance data. And even when there is sufficient data, the uncertainties in the tolerance doses used to generate the curves can be significant. Therefore, it is prudent to carefully review the probability curves presented here. It is still possible, however, to use these NTCP curves for comparisons between alternative plans, even though the
130
I. J. Radiation Oncology 0 Biology 0 Physics
estimated complication probabilities may have errors associated with them. And it is expected that, as more reliable
May 15, 1991, Volume 21, Number 1
tolerance data becomes available, the estimated complication probabilities will be closer to the actual values.
REFERENCES Emami, B.; Lyman, J.; Brown, A.; Coia, L.; Goiten, M.; Munzenride, J. E.; Shank, B.; Solin, L. J.; Wesson, M. Tolerance of normal tissue to therapeutic radiation. Int. J. Radiat. Oncol. Biol. Phys. 21:109-122; 1991. Kutcher, G. J.; Burman, C. Calculation of complication probability-factors for non-uniform normal tissue irradiation: The effective volume method. Int. J. Radiat. Oncol. Biol. Phys. 16:1623-1630; 1989. Kutcher, G. J.; But-man, C.; Brewster, L.; Goitein, M.; Mohan, R. Histogram reduction method for calculating complication probabilities for three-dimensional treatment plan-
ning evaluations. Int. J. Radiat. Oncol. Biol. Phys. 21:137146; 1991. 4. Lyman, J. T. Complication probability-As assessed from dose-volume histograms. Radiat. Res. 104513-519; 1985. 5. Lyman, J. T.; Wolbarst, A. B. Optimization of radiation therapy, III: A method of assessing complication probability from dose-volume histograms. Int. J. Radiat. Oncol. Biol. Phys. 13:103-109; 1987. 6. Schultheiss, T. E.; Orton, C. G.; Peck, R. A. Models in radiotherapy: Volume effects. Med. Phys. 10:410-415; 1983.
APPENDIX Bladder TD= - 80 ” * 0.5 Ill = 0.11
End Point: Reference
Symptomatic bladder con. lracture and volume loss Volume: Whole organ
End Point: Necrosis/infarction Reference Volume: Whole organ
Brain TDa - 60 n - 025 m - 0.15
1 .o 1 .o 0.9 0.6
:
0.9
i i
0.6
0.7 0.6
;;: 0 E
0.7
g
0.5
;
0.4
5 i 5
0.3
8
0.1
0.6
0.5 0.4 0.3 0.2
0.2
0.1 0.0 0.
10.
20.
30.
40.
50.
60.
70.
60.
90.
0.0
100.
0.
DOSE
Fig. Al. Complication
;r&hi”f
probability
$xxus
Reference
Clinically apparent nerve damage Volume: Whole organ
60.
70.
60.
90.
100.
(Gy)
vs. dose for the brain.
End Point: Necmsitinfarction Reference Volume: Whole organ
” - 0.16 m - 0.14 1.0
i iij
0.6
2
0.7
ZI
g a
0.6
2 0
z
0.5
:
g ~
50.
probability
Brain Stem
1.0
%
40.
Tb-65
0.9
d i
30.
Fig. A3. Complication
c
0 g
20.
DOSE
vs. dose for the bladder.
End Point:
” - 0.w m * 0.12
10.
(Gy)
: i
0.4 0.3 0.2 0.1 i 10.
20.
30.
40.
DOSE
Fig. A2. plexus.
Complication
probability
;o.
so.
;o.
,
ao.
I
,
eo.
0.7 0.6 0.5
;
0.4
2 i %
0.3 0.2 0.1
,
ld0.
0.0 0.
(Gy)
vs. dose
0.6
z
:: I
0.
0.9
10.
20.
30.
40.
DOSE
for the brachial Fig. A4. Complication
probability
50.
60.
70.
60.
ao.
d(I.
1’
(Gy)
vs. dose for the brain stem.
ags
gn
P.
p
z
?
?
.
? ? P ? O-NWCGlmLOlDb I I I I I I I I
?
COMPLICATION
!J
I
I
I
?
II
P
II
0
I I I,
P P
I I I I I
PROBABILITY
I
PROBABILITY
I I I
I I I I I I I I I I I
COMPLICATION 2 cm
?
COMPLICATION
0 ? ? ? o~Nwcba.Jolnb
COMPLICATION
PROBABILITY
PROBABILITY
*pp?p-
1
o-i.aucuo~biDb I I I I I I I,
?
oopppp~oo-
PROBABILITY
PROBABILITY
I I I I I I I I I I I,
COMPLICATION
COMPLICATION
-I
132
I. J. Radiation Oncology 0 Biology 0 Physics
May 15, 1991, Volume 21, Number 1
End Point: Pericardltis reference Volume: Whole organ
HeaR - 46 TDm
End Point: Laryngeal edema Reference Volume: Whole organ
Larynx TD* 70.0 n = 0.08 m - 0.17
n = 0.35 Ill = 0.10
1 .o g
0.9
i m
0.6
2 0 :
0.7
z
0.5
5 0 i %
o.4 0.3
0 0
0.8
0.2 0.1 0.0 0.
10.
20.
30.
probability
50.
60.
DOSE
DOSE (Gy) Fig. Al 1. Complication
40.
70.
60.
90.
100.
(Gy)
Fig, A14. Complication probability vs. dose for the larynx for the end point of laryngeal edema.
vs. dose for the heart.
End Point: Clinical nephritis Reference Volume: Whole otgan
End Point: Cataract reqUirlng intewention Reference Volume: Whole organ
Lens TD6O - 16 ” = 0.30 m = 0.27 1.0 :
0.9
i G
0.6
a al
0.7
0 g
0.6
z
0.5
0 c
0.4
2 i I”
0.3
00
0.1
0.2
0.0 0.
10.
20.
30.
40.
50.
DOSE
Fig. A12. Complication
60.
70.
60.
90.
100.
DOSE
(Gy)
probability
vs. dose for the kidney.
End Point: Cartilage necros~e Reference Volume: Whole organ
Fig. AH.
Complication
(Gy)
probability
Liver TD!jr) - 40 n = 0.32 m = 0.15
vs. dose for the lens.
End Point: Liver failure Reference Volume: Whole organ
1.0 :
0.9
i m
0.0
$
0.5
s 0 i a I 00
o.4 0.3
-1
Ii
I
0.2 0.1
DOSE
(Gy)
Fig. A13. Complication probability vs. dose for the larynx for the end point of cartilage necrosis.
DOSE Fig. A16. Complication
probability
(Gy)
vs. dose for the liver.
133
Fitting of tolerance data 0 C. BURMAN ef a[. End Point: Pneumomtls Reference Volume: Whole
= 24.5 n - 0.67 m = 0.16 1.0 z
0.9
5 G
0.6
End Point: Xerostomia Reference Volume: Whole organ
Parotid TDw - 46 n = 0.70 m = 0.16
organ
1.0 0.9 0.6
2 0 @z a
0.7
5
0.5
F
0.4
= 5 2
0.3 0.2
0.2
z
0.1
0.1
0.7
0.6
0.6 0.5 0.4 0.3
0.0
I 0.
10.
20.
30.
40.
50.
DOSE
Fig. A17. Complication
’ II 60.
1’1’1 70.
0.0
1 60.
90.
100.
0.
10.
20.
30.
(Gy)
probability
40.
50.
DOSE
vs. dose for the lung.
End Point: Blindness Reference Volume: Whole
Fig. A20. Complication
probability
Rectum TD50 = 60
organ
70.
60.
90.
1
vs. dose for the parotid.
End Palm: Reference
n =025 m-014
Severe proct~t~s/necros~sJ stenosis/fistula Volume: Whole organ
1.0
1.0
0.9
0.9
0.6
0.6
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0 0.
10.
20.
30.
40.
50.
DOSE
Fig. A18. Complication
60.
70.
60.
90.
0.
100.
10.
20.
30.
40.
50.
DOSE
(Gy)
probability
Fig. A21. Complication
vs. dose for the optic nerve.
End Point: Blindness Reference Volume: Whole
0 tic chiasma T B 50 = 65 ” = 0.25 m = 0.14 1.0
60.
70.
60.
90.
100.
(Gy)
probability
Retina TD50 - 65 ” = 0.20 m = 0.19
organ
vs. dose for the rectum.
End Point: Blindness Reference Volume: Whole
organ
1.0
:
0.9
t?
0.9
i =
0.6
i 6
0.6
2 0 a a
0.7
tz
0.7
0.6
z a
0.6
z
0.5
z
0.5
i=
0.4
F
0.4
= i s
0.3
5 i :
0.3
0.2 0.1
00
0 0
60.
(Gy)
0.0
0.2 0.1 0.0
0.
10.
20.
30.
40.
DOSE
Fig. A19. Complication asma.
probability
50.
60.
70.
60.
90.
100.
0.
10.
20.
(Gy)
vs. dose for the optic chi-
30.
40.
DOSE
Fig. A22. Complication
probability
50.
60.
70.
80.
90.
(Gy)
vs. dose for the retina.
100.
134
I. J. Radiation Oncology 0 Biology 0 Physics
May 15, 1991, Volume 21, Number 1
End Point: Pathologic fracture Reference Volume: Whole organ
Small TDs0
n = 0.10 m = 0.21 1.0 c
0.9
i E
0.6
0.9
i
m
0.6 0.7
0.6
0.6
0.5
5
0.5
0.4
F
0.4
5 i $
0.3
0.3
2
0.1
2 i a I 00
0.7
z
6 t=
End Point: Obstructionlperioration RaferenCa Volume: Whole organ
1.0
: 2 0 :
; 0 :
intestine - 55
” = 0.15 m - 0.16
0.2
0.2 0.1 0.0
0.0 0.
10.
20.
30.
40.
50.
DOSE
Fig. A23. Complication
70.
60.
90.
0.
20.
30.
40.
50.
DOSE
Fig. A25. Complication tine.
vs. dose for the rib cage.
probability
S mal cord T&0 = 66.5 ” = 0.05 m = 0.175
End Point: NecroslSiulceraton Reference Area: 100 cm2
= 70.0 n = 0.10 m = 0.12
10.
(Gy)
probability
Skm TD50
60.
1 .o
60.
70.
60.
90.
1
0.
(Gy)
vs. dose for the small intes
End Point: Myelltislnecrosls Reference Length: 20 cm
1.0
t:
0.9
:
0.9
i E
0.8
i m
0.6
2
0.7
4 :
0.7
z a
0.6
:
0.6
g
0.5
5
0.5
;
0.4
F
0.4
2 i &
0.3 0.2
s i %
0.2
g
0.1
$
0.1
0.0
0.3
0.0 0.
10.
20.
so.
40.
50.
DOSE
Fig. A24. Complication
60.
70.
60.
90.
100.
0.
10.
20.
30.
(Gy)
probability
vs. dose for the skin.
Fig. A26. Complication
probability
End Pomt: Ukeratuw edoration d hole organ Reference Volume:
Stomach TD50 - 65 ” - 0.15 m = 0.14 1 .o t:
0.9
i m
0.6
2
0.7
g a
0.6
5
0.5
c
0.4
d
0.3
2
0.2
; 0
40.
DOSE
0.1 0.0 0.
10.
20.
30.
40.
DOSE
Fig. A27. Complication
probability
50.
60.
70.
00.
90.
100.
(Gy)
vs. dose for the stomach.
50.
60.
70.
60.
90.
10’ 0.
(Gy)
vs. dose for the spinal cord.
Fitting of tolerance data ??C. BLRMAN et al.
1.0
TM Joint TD5o ” = Ill =
End Point: Clinical thyroidilis Reference Volume: Whole organ
Thyrold TD50 i 80 ” - 0.22 m=OZ6
135 End Point:
and mandible 72 0.07 0.10
Reference
Marked llmltation of joint function Volume: Whole organ
,
5 E
0.0
2
0.7
0 :
0.6
z
0.5
0 c
0.4
= i % g
0.3 0.2 0 0
0.1 0.0 0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
DOSE (Gy) Fig. A28. Complication
probability
vs. dose for the thyroid.
100.
0.1
-
0.0
0.
10.
20.
30.
40.
SO.
60.
70.
80.
90.
1
0.
DOSE (Gy) Fig. A29. Complication probability vs. dose for the temporomandibular joint and mandible.
