

ORIGINAL ARTICLE 

Year : 2005  Volume
: 1
 Issue : 3  Page : 168179 

Analytical approach to estimate normal tissue complication probability using best fit of normal tissue tolerance doses into the NTCP equation of the linear quadratic model.
TS Kehwar
Department of Radiotherapy, Postgraduate Institute of Medical Education and Research, Chandigarh – 160 012, India
Correspondence Address: T S Kehwar Department of Radiation Oncology, Postgraduate Institute of Medical Education and Research, Chandigarh – 160012 India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09731482.19597
Aims and Objectives: Aims and objectives of this study are to get the best fit of the normal tissue tolerance doses to the NTCP model of the linear quadratic model. Methods and Materials: To compute the NTCP, the modified form of the Poisson cell kill model of NTCP, based on linearquadratic model, is used. The model has been applied to compute the parameters of the NTCP model using clinical tolerance doses of various normal tissues / organs extracted from published reports of various authors. The normal tissue tolerance doses are calculated for partial volumes of the organs using the values of abovesaid parameters for published data on normal tissue tolerance doses. In this article, a graphical representation of the computed NTCP for bladder, brain, heart and rectum is presented. Results and Conclusion: A fairly good correspondence is found between the curves of 2 sets of data for brain, heart and rectum. Hence the model may, therefore, be used to interpolate clinical data to provide an estimate of NTCP for these organs for any altered fractionated treatment schedule. Keywords: Normal tissue complication probability, Normal tissue tolerance dose, Linearquadratic model, External beam radiotherapy, volume effect
How to cite this article: Kehwar T S. Analytical approach to estimate normal tissue complication probability using best fit of normal tissue tolerance doses into the NTCP equation of the linear quadratic model. J Can Res Ther 2005;1:16879 
How to cite this URL: Kehwar T S. Analytical approach to estimate normal tissue complication probability using best fit of normal tissue tolerance doses into the NTCP equation of the linear quadratic model. J Can Res Ther [serial online] 2005 [cited 2018 Jan 19];1:16879. Available from: http://www.cancerjournal.net/text.asp?2005/1/3/168/19597 
> Introduction   
Estimation of the normal tissue complication probability (NTCP) of critical organs is an essential factor prior to the delivery of external beam radiotherapy (EBRT), because very often critical organs, within the vicinity of the tumour, receive a radiation dose equal to that of the tumour, which is generally based on the published data on normal tissue complications and clinical experience of the radiation oncologist.[1], [2] First set of normal tissue tolerance doses was published by Rubin and Cassarett,[3] in terms of TD[5]/[5] and TD[50]/[5] (the NTCP at 5% and 50%, respectively, within 5 years after radiotherapy) for a large number of normal tissues and organs. Some other investigators had also done in this direction but their work was little comprehensive and systematic.[4], [5], [6], [7]
Similar concept of TD[5]/[5] and TD[50]/[5] has been adopted by Emami et al [8] to report the normal tissue tolerance doses for selected organs. The normal tissue tolerance doses were defined for uniformly irradiated 1/3, 2/3 and 3/3 partial volumes of the organs only for conventional fractionation schedules of 1.8 to 2 Gy per fraction, 5 fractions a week. The work of many other researchers is sparsely scattered in the literature and are for limited organs with varied end points. [9],[10],[11],[12],[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[25],[26],[27],[28],[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40],[41],[42],[43],[44],[45],[46],[47],[48],[49],[50],[51]
Because of radiobiological bearings of the empirical model based on linearquadratic (LQ) model, proposed by Kallman et al [52] and modified by Zaider & Amols,[53] has been used, in this study, to fit these data with consideration of quadratic term. A method of least square fit was used to compute the values of the parameters of the model for the normal tissue tolerance doses. The values of the tissue specific LQ parameters, a and b, are determined using the value of a factor, aG, of the NTCP equation obtained from the above said least square fit and other researches using the published values of the a/b ratio for different normal tissue and organs extracted from the literature, where G = [1 + d/(a/b)]. A set of representative curves have also been plotted between dose and computed NTCP to demonstrate the applicability of the NTCP model.
> Methods and Materials   
NTCP Model
The proposed equation of the NTCP model has radiobiological bearings and is similar to that proposed by Zaider & Amols.[53] The expression of the equation of the NTCP model may be written as
NTCP(D, v) = exp[N0 v k exp{aDG}] (1)
Where G = [1+d/(a/b)], a is the coefficient of lethal damage and a/b is the ratio of the coefficients of lethal and sublethal damages. The N0 and k are nonnegative adjustable parameters, v is the uniformly irradiated partial volume of the tissue/organ (i.e. v = V/Vref, where V is uniformly irradiated volume of the normal tissue/organ and Vref is the reference volume of the normal tissue/organ). D is the normal tissue dose in terms of TD[5]/[5] or TD[50]/[5], delivered with d dose per fraction. The expression in the exponent, exp (aDG), is the reminiscent of the LQ model for cellular survival.
The expression of the relative effectiveness (RE) per unit dose can be written as
RE = G (2)
Using equation (2) into equation (1) the expression of NTCP may be written as
NTCP (D, v) = exp[N0 v k exp{aD*RE}]
Or NTCP (D, v) = exp [N0 v k exp {a BED}] (3)
Where BED = D*RE. In equation (1), if N0 is considered to be the clonogenic cell density of the tumour cells, and the exponent of the partial volume v is taken as k = 1, then the product of N0v represents the total number of the clonogenic cells in the tumour volume and the expression will be of the tumour control probability (TCP) model. But here in equation (1) the N0 and k are assumed to be nonnegative adjustable parameters and are allowed to vary depending on the type of the tissue / organ. To get the best fit of normal tissue tolerance doses, it is required that parameter k should be greater than zero, i.e. k > 0, and as the volume of the irradiated tissue / organ increases, the NTCP of the tissue must also increase.
Normal Tissue Tolerance Doses
To get the best fit of equation (1) the published normal tissue tolerance doses of Emami et al [8] and other investigators [9],[10],[11],[12],[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[25],[26],[27],[28],[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40],[41],[42],[43],[44],[45],[46],[47],[48],[49],[50],[51] have been used. The Emami et al's data are in the form of TD5/5 and TD50/5 defined for 1/3, 2/3, and 3/3 partial volumes or a reference volume (length or area) of the organs. The partial volume of a organ is presented in terms of fraction of the reference volume Vref. In many cases the reference volume of the organ is considered to be the whole volume of the organ while in some it is assumed to be a part of the organ or length of the organ, such as spinal cord.
Many other workers have also reported normal tissue tolerance doses for different organs / tissue, but these are widely scattered in the literature and is very difficult to extract from all reports. [9],[10],[11],[12],[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[25],[26],[27],[28],[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40],[41],[42],[43],[44],[45],[46],[47],[48],[49],[50],[51] Hence, in this study, an attempt is made to collect normal tissue tolerance doses from published reports for the organs for which Emami et al [8] have compiled. I have chosen only those reports which have tolerance doses at different NTCP levels for fractional (partial) volumes of the organs or at different NTCP levels for whole organ or at same NTCP level for fractional volumes. The references of the reports from where data have been extracted, other than Emami et al,[8] are listed in [Table  1]. There has not been any control on the tolerance data and these may be of less severe endpoints.
> Results   
Normal tissue tolerance data of Emami et al [8] used to fit into the equation (1) to obtain the values of aG, k and N0. The method of least square fit is used to compute the parameters using transformed linear expression of the equation (1). The values of aG, k and N0 for Emami et al [8] are listed in [Table  2] along with the end points of corresponding normal tissues / organs. In case of 2 point data, the tolerance doses, TD[5]/[5] and TD[50]/[5], are given only for single volume. Hence these parameters can not be computed, because for the purpose more than 2 point data are required. Due to unavailability of adequate data no attempt can be made to set correlation between NTCP and volume. To solve this problem, for simplicity, it is assumed that the organs which have only 2 point data do not show volume dependency with NTCP. So the value of k, for these organs, is set equal to zero. Using the computed values of aG, k and N0 parameters, the values of the tolerance doses for partial volumes of the organs are computed and are listed in [Table  3] along with the tolerance doses compiled by Emami et al.[8] Since the parameter aG is a factor of the coefficients a and b (or a/b), so to determine the values of these coefficients, an accurate value of a/b for a tissue/organ must be known. Hence, the published values of a/b, for different organs, are extracted from the literature, [54],[55],[56],[57],[58],[59],[60],[61],[62],[63],[64],[65],[66],[67],[68],[69],[70],[71],[72],[73],[74],[75],[76],[77],[78],[79],[80],[81] and are used to calculated the values of a and b. The extracted values of a/b of different tissues/organs, along with their reference(s) of the publication, and calculated values of a and b are listed in [Table  5]. In the calculation of the values of a and b from the factor aG, it is assumed that the dose per fraction is 2Gy for the conventional treatment schedule.
Survey of the literature reveals that there is a wide scattering in the normal tissue tolerance doses and no consensus on the issue among the radiation oncology community. In this study suitable tolerance dose data, for the organs, have been extracted from the literature and combined together with Emami et al's [8] data to compute the values of above said parameters. [Table  1] enlists the values of these parameters, i.e. aG, k and N0, for the listed organs, for the combined tolerance doses along with the source of references. With use of the values of aG, k and N0 parameters, from [Table  1], the values of the tolerance doses for 1/3, 2/3 & 3/3 partial volumes of all listed organs are computed and are listed in [Table  4]. In the brackets of the [Table  4] along with computed values of the tolerance doses, the 95 % confidence interval (CI) limits for published data are given. The limits of 95% CI are calculated using computed tolerance doses(TD[5]/[5] or TD[50]/[5]) and standard errors (s) of the published tolerance doses. The parameter aG is used to compute the values of a and b, for combined data set of the tolerance doses for each organ, the published values of a/b for different organs, as used for Emami et al's [8] data, have taken into account. The extracted values of a/b of different tissues/organs, along with their reference (s) of the publication, and calculated values of a and b are listed in [Table  5].
Using the values of aG, k and N0, from [Table  1][Table  2], 2 set of curves have been plotted between dose and computed NTCP for bladder, brain, heart and rectum for partial and whole volume and are shown in [Figure  1][Figure  2][Figure  3][Figure 4]. The solid lines of the curves are for the Emami et al's [8] data and broken lines are for combined data. In the curve fitting, a method of least square fit has been used. To plot the curve for Emami et al's [8] data with 2 points tolerance doses the parameter k is set to zero, because there is no conclusion could be made on volume dependency of the organ, and rest of the parameters are calculated from these data.
> Discussion   
A number of models have been proposed to predict the NTCP of normal tissues/critical organs by many authors [52],[82],[53]. All the models predict an increase in NTCP with increasing absorbed dose and irradiated volume. The model, presented in this study, is the Kallman's[52] Poisson cell kill model, modified by Zaider and Amols,[53] which had a radiobiological base, because it is based on the linear quadratic model. Normal tissue tolerance doses of Emami et al's[8] and other authors (in combination of Emami et al's data[8]) have been used to fit into the transformed expression of the equation (1) to determine the values the parameters aG, k and N0. The values of these parameters were used to calculate the values of the tolerance doses for partial volumes of the organs and were named as the theoretically calculated tolerance doses, and are listed in [Table  3][Table  4] for both set of data. The theoretically calculated tolerance doses, for Emami et al's[8] data, are very close to the compiled tolerance doses[8] [Table  3]. The theoretical tolerance doses are also calculated for 1/3, 2/3 & 3/3 partial volumes of the organs using the values of aG, k and N0 from [Table  1] for the combined set of data. Values of k [Table  1][Table  2] indicate that the organs which has higher value of k have high volume dependency than that of the lower value of k. i.e. the volume dependency of the organs is directly proportional to the value of the k. No volume dependency could be estimated for the organs where only 2 point data are given. Such organs are femoral head and neck, rib cage, skin (telangiectasia), optic nerve, optic chiasma, cauda equina, eye lens, retina, ear (middle/external), parotid, larynx (edema), rectum and thyroid. The value of parameter, k, for these organs, is adjusted to zero. For the combined set of data, the correlation between tolerance dose and volume is similar to that for Emami et al's [8] data, except for 2 organs such as spinal cord and larynx (edema), where the value of k is negative which show that the tolerance dose increases with increasing the irradiated volume of these organs which is contradictory to the available data and our own experience.
The accuracy of the computed values of the parameters of the model depends on the accuracy of the complied tolerance doses and their end points, which are used to compute the parameters. The organs for which all 6 point tolerance doses are provided the calculated values of the parameters have better confidence. On the other hand, the values of the parameters became less accurate for the tolerance doses, where the tolerance doses are not provided for one or more partial volumes either at 5% or at 50% or at both NTCP levels. For these organs, the dependency of the parameters is more skewed towards data provided for the partial volumes and NTCP. For example, in case of Emami et al's [8] data of skin (necrosis) and brain stem, the tolerance doses at NTCP level of 5% are provided for all 3 partial volumes, while at NTCP level of 50% the data are provided only for whole organ. Hence the parameters, aG, k and N0, have more dependency on the tolerance doses provided for NTCP level of 5%. Similarly the dependency of the parameters can be seen for other data set. In the cases for which the tolerance doses are provided only for one partial volume for NTCPs at 5% and 50%, the volume dependent parameter, k, could not be computed, and hence there will be much less confidence in the results. For the cases for which only 2 point data are provided, the computation of the parameters, aG and N0, is done by adjusting k = 0 for the simplification. The values of the parameters, aG and N0, for 2 point data have less confidence. When other author's data were combined with the Emami et al's [8] data and the parameters, aG, k and N0, were computed, then it is seen that the values of these parameters become highly inaccurate. Because most of the data are for single volume of the organ and have a wide variation in their values, and even some of the data do not have their same endpoints, or may have different endpoint definition.
To get more accurate values of the parameters, aG, k and N0, it is necessary to have accurate and some more additional tolerance doses for all the organs. The best use of these parameters can be obtained if radiation oncologist compares the NTCP with his own experience. If the values the parameters match with his own values, then this suggests that the computed values of the parameters are reasonable and can be used to estimate the NTCP of critical organs. But if the computed values of the parameters consistently differ from that of the radiation oncologist, then new values of the parameters could be used to reflect the local experience.
The proposed model is connected with three variables viz. NTCP, delivered dose (D) and partial volume (v) of the irradiated organ. In 2 D graphical representation, a curve can be plotted between any two quantities while keeping the third one constant. To demonstrate the applicability of the model, a set of curves have been plotted between dose and NTCP for 1/3, 2/3 and 3/3 partial volumes for bladder, brain, heart and rectum and are shown in [Figure  1][Figure  2][Figure  3][Figure 4] respectively . It is clear from these Figures that the organs demonstrate threshold type behavior. In other words, the NTCP of the organ does not appreciate until a certain amount of radiation dose is delivered. The dose beyond that the NTCP is the function of dose is known as the 'threshold dose'. The pattern of the NTCP variation with dose depends on the behavior of the organ. The plot of the NTCP Vs dose for these organs have sigmoid shape. There is only difference in the threshold doses and increment in the NTCP with dose (after crossing the threshold dose) and can be seen between the curves of the organs. The 2 point tolerance data for rectum are reported only for one partial volume, hence the curve between NTCP and dose is a single line and does not show volume dependency [Figure 4].
[Figure  1] shows that the calculated the NTCP, for Emami et al's [8] data, increases sharply with dose than that of the combined data. The threshold doses for 1/3, 2/3 and 3/3 partial volumes of combined data are in the range of 3540 Gy, which are quite lower than that predicted for Emami et al's [8] data. For Emami et al's [8] data, the threshold doses are 85 Gy, for 1/3 volume; 70 Gy, for 2/3 volume and 60 Gy for 3/3 volume and the window of variation of tolerance doses between the partial volumes, at all NTCP levels, is wider than that of the combined data set, which demonstrates that the NTCP in bladder is highly volume dependent. On the other hand, a narrow window for combined data set indicates that the NTCP in bladder is less volume dependent. At all dose levels there is a wide variation in the predicted NTCP for both the data sets, which is highly confusing to decide that which data set should be used in the practice. This is also a problem to consider whether the NTCP in bladder is a highly volume dependent or less volume dependent. Hence it is recommended that to predict NTCP in bladder, the radiation oncologist should use his own experience.
It is seen in [Figure  2] that the predicted the NTCP in brain for 2 sets of data in the therapeutic range is reasonably accurate. The threshold dose, for these sets of data, are almost at the same level and window of variation of tolerance doses is similar between partial volumes. The gap between the curves for the partial volumes reveals that the NTCP of the brain is the function of the volume, i.e. the brain NTCP is having volume dependency. From these curves, it can be suggested that any set of predictions can be used in the clinical practice, if the doses are in the therapeutic range. At higher doses, beyond the therapeutic range, the predicted NTCP, for Emami et al's [8] data, is higher than that of the combined set of the data, hence this portion of the curves left physician indecisive.
Curves, in [Figure  3], show that the predicted NTCP in heart, for 2 sets of data, is fairly accurate at all doses. The threshold doses are almost same for both sets of data and window of variation of tolerance doses is similar between partial volumes. Hence any set of prediction can be used in the practice. Here also the gap between the curves for the partial volumes of the heart indicates that the NTCP of is the function of the volume, i.e. the heart NTCP is volume dependent.
In case of rectum, Emami et al's [8] have provided 2 point tolerance doses from which no correlation could be made between the NTCP and volume. Hence for simplification, it is assumed that the rectum NTCP may not be volume dependent, so the value of the parameter k is adjusted equal to zero. While some other reports show that the NTCP increases with increasing the volume of the rectum.[49], [83], [50], [48], [18] Using combined set of tolerance data of Emami et al's [8] and other author's, the value of k was found equals to 0.2001, which shows that the NTCP is a function of irradiated volume of rectum. The values of all 3 parameters, aG, k and N[0], of 2 sets of data, are used to generate the curves between dose and NTCP [Figure 4]. In [Figure 4], the solid line is for Emami et al's [8] data, while broken lines are for combined set of data. [8],[9],[10],[11],[12],[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[25],[26],[27],[28],[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40],[41],[42],[43],[44],[45],[46],[47],[48],[49],[50],[51] It is clear from these curves that tolerance doses of Emami et al's [8] do not show volume dependency for rectum which is contrary to our own experience. While combined data set have shown volume dependency, but the window of tolerance doses between partial volumes is narrow, hence the NTCP in rectum could be considered to be volume independent. The Emami et al's [8] data predicts a sharp increase in NTCP and at higher doses and is more than that of the combined set of data [951]. While in therapeutic range of doses, both set of data predict NTCP reasonably accurate.
It can be seen from above said [Table  3][Table  4] that some of the organs show wider window of variation in the tolerance doses between partial volumes, and some have very narrow window, while others do not have any variation in the tolerance doses with the change in partial volume. The organs which have very narrow window of tolerance dose variation with the change in partial volume or no window of tolerance dose variation, show that even if a small volume of a organ is irradiated to a sufficiently high dose, a whole organ NTCP will occur, which is independent of the irradiation to the rest of the organ. On the other hand, the organs where window of tolerance doses is wider and vary with the change in partial volume, show that the NTCP is a function of dose and volume. In other words, the intensity of the NTCP depends on the amount of radiation dose and irradiated volume of the organ i.e. a smaller volume of the organ could tolerate a higher amount of radiation dose than does a large volume in order to cause same NTCP.
Burman et al [84] used Emami et al's [8] data to generate the NTCP curves for these organs. In their study, the Lyman's[82] NTCP model has been used to compute its parameters and to generate the NTCP curves. Since the Lyman's [82] model is based on the normal distribution of the tolerance data and do not have any correlation with radiobiological processes and findings, hence can not be accounted for varying tissue specific radiobiological parameters. In the present model, the factor aG has two tissue specific radiobiological coefficients, such as a and b, which account for acell kill (lethal damage) and acell kill (sublethal damage) of the LQ model. For a conventional treatment schedule where 2 Gy per fraction radiation dose is delivered to the organ, the value of the factor aG can directly be used from [Table  1][Table  2] to interpret the NTCP of the organ, for any amount of the radiation dose and partial volume of the organ, if the delivered dose is uniform throughout the irradiated volume of the organ. When an altered dose fractionation schedule is used to irradiate the organ, then radiobiological coefficients, a & b (a/b), play an important role in the prediction of the NTCP for a particular dose and volume of the organ. Burman et al [84] did not say any thing about altered fractionation schedules that by using Lyman's [82] model how one could predict NTCP.
To compute the values of a & b from the factor aG, the published values of a/b extracted from the literature, are used and listed in [Table  5] along with their source of reference. The main difficulty with the choice of a/b is that in the literature there is no definite value of a/b is reported. Always one can find a range of a/b values reported by different researchers, which made our work somewhat difficult during the search of the literature. We have taken the values of a/b from the published reports, but in the prediction of NTCP for altered fractionation schedules the radiation oncologist must use the value of a/b of his own choice with careful selection to match his own experience.
> Conclusion   
A radiobiological model of NTCP, presented in this study, was used to fit the normal tissue tolerance data compiled by Emami et al's [8] and combined data of Emami et al [8] and some other investigators. [9],[10],[11],[12],[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[25],[26],[27],[28],[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40],[41],[42],[43],[44],[45],[46],[47],[48],[49],[50],[51] These data sets have provided reasonable estimate of the values of the parameters (aG, k and N0) of the model for all the listed organs. In this model volume correction factor is represented by a powerlaw and the curves between dose and NTCP are presented. However, volume wise response of the tissue is a complicated process and is not well understood. There have been attempts other than the powerlaw to understand the volume dependent complication process.[85] It has been discussed that in some cases there are insufficient data to determine the values of the parameters (aG k and N0) more accurately. Hence the calculated values of the parameters represent a substantial extrapolation of the normal tissue tolerance data, like in case of rectum the tolerance data are given only for one volume which show no volume effect which is not true, because in some studies [49],[83],[50],[48],[18] it is seen that rectum has volume dependency. In case of spinal cord and larynx (edema) the value of k, for combined data set, is negative which shows that the tolerance dose, for these organs, increases with increasing the volume of the organ, which is contrary to our experience. This is because of wider variation in tolerance doses of these organs. Hence to find out appropriate reasonable values of the tolerance doses for the organs, more normal tissue tolerance data are required, and widely accepted values of the tolerance doses will be estimated. The model used in this study can be used to estimate the outcome of altered multifractionation schedules because it has a radiobiological basis. The generated curve can be used to estimate the NTCP for a fractional (partial) volume of the organ if it is being irradiated uniformly and match with local experience. The values of a and b along with two other parameters of the model could be used to compute the value of the NTCP for an altered fractionation schedules.
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[Figure  1], [Figure  2], [Figure  3]
[Table  1], [Table  2], [Table  3], [Table  4], [Table  5]
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