This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).
Seven different radiobiological doseresponse models have been compared with regard to their ability to describe experimental data. The first four models, namely the critical volume, the relative seriality, the inverse tumor and the critical element models are mainly based on cell survival biology. The other three models: the Lyman (Gaussian distribution), the parallel architecture and the Weibull distribution models are semiempirical and rather based on statistical distributions. The maximum likelihood estimation was used to fit the models to experimental data and the χ^{2}distribution, AIC criterion and Ftest were applied to compare the goodnessoffit of the models. The comparison was performed using experimental data for rat spinal cord injury. Both the shape of the doseresponse curve and the ability of handling the volume dependence were separately compared for each model. All the models were found to be acceptable in describing the present experimental dataset (p > 0.05). For the white matter necrosis dataset, the Weibull and Lyman models were clearly superior to the other models, whereas for the vascular damage case, the Relative Seriality model seems to have the best performance although the Critical volume, Inverse tumor, Critical element and Parallel architecture models gave similar results. Although the differences between many of the investigated models are rather small, they still may be of importance in indicating the advantages and limitations of each particular model. It appears that most of the models have favorable properties for describing doseresponse data, which indicates that they may be suitable to be used in biologically optimized intensity modulated radiation therapy planning, provided a proper estimation of their radiobiological parameters had been performed for every tissue and clinical endpoint.
The study of the doseresponse relations in radiation therapy is important for improving quantification and knowledge about the mechanisms influencing the response of organs and tissues to radiation therapy. It is important to know the expected response level in normal tissues when irradiating a patient, since the aim of radiation therapy is to eradicate the tumor while sparing healthy normal tissues as far as possible. This is particularly important when using radiobiologically optimized radiation therapy where both the therapeutic effect and adverse normal tissue damage need to be accurately quantified in order to maximize the treatment outcome. Most often, the dose to the tumor is limited by the tolerance of the surrounding normal tissues. It is essential to understand the underlying biological processes for selecting the proper model, which can more accurately describe the normal tissue response and determine tissue tolerance in different situations. In this way, it will be possible to estimate the quality of life after the treatment by calculating the probability of tumor cure or local control and the associated risk of treatment related morbidity.
There exist several types of volume effects, defined by the decrease in tissue function or increase in the probability of having a specific endpoint with increasing irradiated volume. The response of a tissue to radiation depends on the organization of its sensitive functional subunits, the volume of the irradiated tissue and possible irradiation of associated organs, and finally on the ability of the different cell types to maintain the tissue or organ function. The latter is often dependent on the way different types of cells are organized into functional subunits (FSUs) [
Spinal cord is a critical normal tissue that almost at all cost should be spared during radiation therapy. It is an example of an organ with high serial arrangement of its functional subunits. It is built of nerve cells—Neurons, the axons of which are arranged in bundles along the organ. The characteristic Hshaped pattern on the spinal column cross section is a result of the arrangement of the nerve cell bodies and axons within the cord. The inner part, creating the Hletter shape consists of gray matter, while the white matter creates a more lipid rich, pale surrounding.
It is well known that the material building the gray matter is mainly nerve cell bodies, while axons and the associated myelin cells are the ones constituting the white matter. In mammalian nerve tissue, the axons (e.g., motor neurons or sensory neurons) are equipped with a special layer of insulation, namely the myelin sheath. The myelin sheath is created by Schwann cells surrounding the axons of a neuron, increasing the integrity, speed and information content of the transmitted signal.
There are often very serious consequences for exceeding the tolerance dose of normal tissues. As far as the latency period is concerned, radiation response can be divided into an early and a late occurring damage. There is a close correlation between the time of appearance of radiationinduced damage and the normal proliferative activity for a given tissue. The higher the rate of normal cell turnover, the faster the onset of the damage. In slowly proliferating tissues, such as spinal cord, the induction of radiation damage is considerably delayed in time. The late types of radiationinduced damage, in case of spinal cord myelopathy and paralysis, consist of two main endpoints: White matter necrosis and demyelination, occurring usually between six to eighteen months after irradiation, followed by vascular damage with an onset that ranges between one to four years.
Modeling of normal tissue response to radiation has become an important domain of modern radiation therapy. Numerous models have been developed during the years to help in determining the optimal treatment. The process of creating such models usually involves many simplifying assumptions. The damage induction is considered stochastic, whereas the survival of cells follows either binomial or poisson statistics. The organ response is assumed to depend on either the response of individual cells and/or the response of the FSU. All the cells as well as all the FSUs are assumed to respond identically. The isoeffect relationships do not depend on the level of response and equal dose fractions are assumed to cause equal effects, provided the time separation is sufficient. Two connected levels of radiation response are generally modeled, namely survival of cells and response of an organ. Many models originate from an expression that describes cell survival and they incorporate this expression in the formula that describes the relation between dose and organ function. However, other models are purely phenomenological, which means that there is not any explicit formula for cell survival included. A radiobiological model, to be considered reliable, has to fulfill certain requirements. It should appropriately predict the shape of the doseresponse curve as well as it should duly handle the volume and fractionation effects.
Numerical quantitative comparisons of existing doseresponse models have been done by many authors [
Doseresponse models can be categorized into several groups based on the statistical distribution they use for describing the sigmoid shape of the doseresponse curve (
Assuming
The Poisson distribution is the limiting case of the binomial distribution when
Using the normal distribution function (the probit function) for the response results in the following expression:
The logit distribution is an analytical sigmoidal shaped curve commonly used in biology defined in the following way:
In this model the mathematical expression for NTCP,
The model intercomparison was performed both with and without considering the volume effect. This was made in order to separately evaluate and compare: (a) the accuracy by which the different models fit the shape of the doseresponse curves from uniform dose irradiation; and (b) the ability of the different models to account for the volume effect. To be able to judge each of the above phenomena individually, we tried to separate them by removing the volume effect. This was achieved through a separate fit of the models for each spinal cord length, which was done by making a fit to each of the irradiated length of spinal cord separately without taking the volume dependence of the model into consideration, assuming that each partially irradiated length is a separate unit. Each of the models had a total of six free parameters to be estimated due to the cancelling of the ones describing the volume dependence.
The Lyman (Gaussian) model is the most widely used model in the literature. For this reason the authors chose to use this model as a reference in order to be able to project the findings of the present study to other clinical studies where the Lyman model has been used for analyzing the treatment outcome data.
To perform the fitting of the models to experimental data, the maximum likelihood method was used [
After fitting the models to the clinical data, the goodness of fit of the models and their parameters was evaluated by the χ^{2}test, which was applied as suggested by Baltas and Grassman [
An intercomparison of the fitting of the models to the experimental data was done using the Ftest method. The main principle of this method is to perform a comparison between a given model and the Lyman model with volume effect, which is the reference model, by comparing their fitted results to the experimental data. The Ftest value is a probability distribution calculated for the χ^{2} value of the reference model divided by the χ^{2} value for the model under investigation. The smaller the Ftest value, the better the compared model is in comparison to the reference model. The value
This analysis was based on a goodnessoffit evaluation. Under the assumption of Gaussian errors around the true function describing survival, the model behavior was studied at different dose ranges for each clinical endpoint. The chisquare values were calculated and the corresponding
Also, the Akaike's information criterion [
The models were fitted to experimental data for paralysis after irradiation of spinal cord of rats [
The best estimates as well as the 68% confidence intervals of the parameters of all the models are given in
These values were determined using the maximum likelihood method to fit the complete set of experimental data (with volume effect) and the different spinal cord segment lengths, separately (without volume effect) in each case (white matter necrosis, vascular damage). For white matter necrosis, the degreesoffreedom (DF) is 14 −
For the white matter necrosis, as expected, the maximum value of the Loglikelihood function is higher in the fittings without volume effect since more parameters are used to fit each spinal cord segment length separately compared to the fittings with volume effect in the models. However, among the different models, the higher values were received by the Parallel architecture (−34.3) and the Relative Seriality (−35.3) models with volume effect and the Weibull (−31.6) and the Relative Seriality (−31.9) models without volume effect. However, the maximum value of the Loglikelihood function is only an approximate descriptor of the goodnessoffit and it does not account for the mean and variance of the Loglikelihood function distribution around its maximum.
Similar results hold for the vascular damage. Among the different models, the higher values of the maximum of the loglikelihood function were received by the Relative Seriality (−26.1) and the Critical volume (−26.2) models with volume effect and the Weibull (−24.7) and the Parallel architecture (−24.7) models without volume effect.
For the white matter necrosis, based on the χ^{2} and
For the vascular damage, based on the χ^{2} and
The qualitative part of this information can be observed in
Currently, in most clinical practices, when evaluating the fitness of a plan, the mean and maximum or minimum doses, isodose distributions and DVH are typically examined. However, these data do not take into account the biological characteristics of the examined tissue. That is because different treatment plans may deliver different dose distributions to a given normal tissue getting the same response rate. Just as the dose volume histogram chart is a good illustration of the volumetric dose distribution delivered to the patient, so is the doseresponse plot as a measure of the expected clinical outcome.
It has been reported that radiobiological evaluation is more sensitive to small changes in dose distribution and the differences observed in the doseresponse diagrams comparing different treatment plans are not always reflected in the DVH plots. This is because the way a certain dose distribution affects an organ depends on its radiobiological characteristics. Using the doseresponse diagrams together with the dosimetric diagrams, a more complete picture of the effectiveness of a given treatment plan may be given. Consequently, there is an obvious need for radiobiological models that are able to describe quantitatively the normal tissue response and its dependence on the irradiated volume and dose level. In the present study, a comparison of available models has been made. The results of the fit of these models to the experimental data describing rat paralysis caused by white matter necrosis or vascular damage after irradiation gave a good indication that these models show a suitable behavior in describing relevant experimental data.
The results and conclusions of this study are strongly dependent on the accuracy of the radiobiological models and the parameters describing the doseresponse relation of the different tissues. However, it is known that all the existing models are based on certain assumptions or take into account certain only biological mechanisms. Furthermore, in clinical practice the determination of the model parameters expressing the effective radiosensitivity of the tissues is subject to uncertainties imposed by the inaccuracies in the patient setup during radiotherapy, lack of knowledge of the interpatient and intrapatient radiosensitivity and inconsistencies in treatment methodology. Consequently, the determined model parameters and the corresponding doseresponse curves are characterized by confidence intervals. So, the expected response of a tissue is known with some uncertainty, which has become clinically acceptable for many cancer sites.
Until now, in clinical practice the different tissues are generally assumed to have homogeneous radiosensitivity. DVHs are a good illustration of the volumetric dose distribution delivered to an organ but spatial dosimetric information gets lost. However, there is increasing evidence that the spatial information of the dose distribution is important in determining treatment complications. Therefore, although DVH simplifies a 3D dose distribution into a 2D plot, such a plot may not be representative of the efficacy of the given treatment technique and therefore may have not a close relation with treatment outcome. The presented models cannot account for any spatially distributed radiosensitivity variation in their present forms and they have to be further developed in order to incorporate such information.
In order to compare the fit of the models to the experimental data four methods were used (the maximum likelihood method, the χ^{2} distribution, the AIC method and the Ftest). The results for all the examined models based on the above statistical methods are shown in
From the presented results one can clearly see that the range of differences between the different models in fitting the experimental data is large in the case that volume effect is accounted for (Ftest: 0.43–0.90 for white matter necrosis and 0.08–0.53 for vascular damage) whereas it is small in the case that the different spinal cord segment lengths are fitted separately (Ftest: 0.84–0.99 for white matter necrosis and 0.88–0.99 for vascular damage). One should also observe that the confidence intervals of the determined parameters are rather large. Fine differences in the results may be caused by small errors within the experimental dataset. Based on the above described results the following general conclusions can be drawn regarding the weak and strong points of the different models.
The sigmoid shape of doseresponse for the Weibull distribution is the reverse of that of the Poisson model and may therefore be better suited for some normal tissue responses where mild damage may be more clinically relevant. On the other hand, the Lyman model is a symmetric sigmoid, which may be suboptimal for describing the response of normal tissues. The relative seriality model, having “finetuning” ability with the relative seriality parameter,
In the case of white matter necrosis most of the models showed low but acceptable fitting results. The Weibull distribution model was the only model giving better overall fit than the Lyman model, which was the reference model in the Ftest. Furthermore, the Lyman, Parallel architecture and Weibull distribution models were best in handling the volume effect, giving the best Ftest results. On the other hand, the fitting results of the inverse tumor and the critical element models were considerably worse (showing much lower Loglikelihood values) than the ones of other models, when the volume effect was taken into account, whereas when making a fit without the volume effect the Lyman model gave inferior results with the lowest Loglikelihood value.
Similarly, in the case of vascular damage, the models showed acceptable fitting results, which for many models were very good. More specifically, the Relative Seriality, Critical element, Critical volume and Parallel architecture models showed considerably better fitting results (higher Loglikelihood values) than the Inverse tumor, Lyman and Weibull distribution models when the volume dependence was taken into account. On the other hand, when the volume dependence was disregarded, apart from the Lyman and Critical volume models, the results of all the models were similar. The Critical volume, Relative Seriality, Critical element and Parallel architecture models gave the best overall fits handling volume effect in the best way, showing the lowest Ftest values among all the models. On the other hand, the Lyman, Parallel architecture and Relative Seriality models gave the best fitting results (lowest Ftest values) when the different spinal cord segment lengths were fitted separately.
In clinical radiotherapy there is an increasing need for accurate models capable of describing the normal tissue response as a function of the dose and the irradiated volume. The present study is an overview of the existing models that are most frequently used in scientific reports or clinical studies. However, sill more effort has to be given on radiobiological studies that could develop new improved models, which would be able to more accurately account for further biological mechanisms and will become especially suitable for biologically optimized radiotherapy.
By applying the different doseresponse models on the same experimental data, their inherent structural differences could be revealed. Furthermore, the accuracy by which the volume effect is accounted for in the different models is examined. These issues have not been investigated in such a clear and systematic way before. Also, the expression of the basic doseresponse parameters of the different models in terms of
In the literature there are several articles that deal with radiobiological studies which have evaluated and improved models based on biologically optimized clinical radiotherapy treatment plans. Although such studies may be more suitable for the extraction of clinical results in relation to a given radiobiological model, the nature of those studies is mainly to determine the values of the radiobiological model parameters regarding a given treatment technique and clinical endpoint. However, the structure of such clinical data is not suitable for performing the type of analysis that is performed in the present study in order to make a more clear intercomparison that will reveal the inherent characteristics and differences of the examined models.
Today, all organs are assumed to be totally uniform and amorphous, whereas we know that some organ regions are more sensitive and others more tolerant to radiation. In the future, such variations need to be considered, e.g., by splitting the hilus region from the rest of the organ in most organs of mixed serialparallel organization. Fortunately, this has not been a major problem in this study. However, if white and gray matter had been separately irradiated this would have been the case here, too.
Statistical distributions used in NTCP models to describe the shape of the doseresponse curve.
Volume and doseresponse curves for white matter necrosis of different lengths of rat cervical spinal cord. The solid lines give the combined best fitting. The dashed lines have been fitted to each of the irradiated spinal cord segment lengths separately,
Volume and doseresponse curves for vascular damage of different lengths of rat cervical spinal cord. The solid lines give the combined best fitting. The dashed lines have been fitted to each of the irradiated spinal cord segment lengths separately,
Overview of the examined doseresponse models together with a summary of their inherent parameters.
Critial volume  (1− 

− 
 
Relative seriality 


 
Inverse tumor 


 
Critical element 


 
Lyman 


 
Parallel architecture 




Weibull distribution 



 
Binomial  Critial volume  (1− 

D0, N0, M, N  − 

Relative seriality 


D0, N0, s 
 
Poisson  Inverse tumor 


D0, N0, k 
 
Critical element 


D0, N0 
 
Lyman 


D50, m, n 
 
Parallel architecture 


D50, k, v50, σ 

k/4  
Weibull distribution 


A1, b, A2 


Doseresponse data for developing white matter related spinal cord paralysis (white matter necrosis) within 30 weeks and paralysis or histological evidence of vascular lesions (vascular damage) after a latent interval of >30 weeks after single dose irradiation of the rat spinal cord [
16  20  0  6  
21  3  6  
22  3  6  
23  6  6  
 
8  22  0  6  
24  1  6  
28.5  2  6  
32.5  4  6  
40  5  5  
 
4  39.1  0  6  
42.7  1  6  
47.8  2  6  
54.5  4  6  
70  6  6  
 
16  18  0  6  
20  3  6  
21  3  3  
 
8  20.4  1  6  
22  4  6  
24  4  5  
28.5  4  4  
 
4  25.3  4  6  
30  4  6  
35.9  5  6  
39.1  6  6 
Model parameter values for white matter necrosis. The best estimates of the parameter values are given with their 68% confidence intervals. The values of the Loglikelihood function, χ^{2}, degreesoffreedom (DF) and probability of χ^{2} distribution (
Critical volume  20.7 (20.3–21.2)  3.20 (2.6–3.3)  With  −35.6  12.75  10  0.24  20.75  0.57  
Without  −32.0  4.57  8  0.80  16.57  0.92  
Relative seriality  21.3 (21.1–21.5)  4.0 (3.6–4.4)  With  −35.3  14.09  11  0.23  20.09  0.63  
Without  −31.9  4.84  8  0.77  16.84  0.93  
Inverse tumor  19.5 (17.5–21.5)  1.0 (0.8–1.2)  With  −40.2  16.89  11  0.11  22.89  0.73  
Without  −31.9  4.84  8  0.77  16.84  0.96  
Critical element  24.2 (23.1–26.0)  1.0 (0.7–1.3)  With  −47.3  25.76  12  0.01  29.76  0.90  
Without  −31.9  4.84  8  0.77  16.84  0.99  
Without  −33.4  5.56  8  0.70  17.56  0.84  
Parallel architecture  21.2 (18.0–30.4)  3.9 (3.2–5.1)  With  −34.3  11.52  10  0.32  19.52  0.51  
Without  −32.1  5.02  8  0.76  17.02  0.87  
Weibull distribution  22.8 (22.6–23.4)  3.2 (2.5–4.0)  With  −35.4  10.33  11  0.50  16.33  0.43  
Without  −31.6  4.26  8  0.83  16.26  0.89 
Model parameter values for vascular damage.
Critical volume  19.3 (19.0–19.7)  5.3 (4.1–6.2)  With  −26.2  4.11  7  0.77  12.11  0.08  
Without  −25.1  2.21  5  0.82  14.21  0.93  
Relative seriality  19.9 (19.4–20.4)  6.7 (5.5–8.0)  With  −26.1  5.03  8  0.75  11.03  0.11  
Without  −24.9  1.51  5  0.91  13.51  0.90  
Inverse tumor  20.2 (19.0–20.8)  2.0 (1.5–2.5)  With  −30.6  13.17  8  0.11  19.17  0.53  
Without  −24.9  1.51  5  0.91  13.51  0.99  
Critical element  20.2 (19.7–20.3)  4.9 (3.9–6.1)  With  −26.8  5.62  9  0.78  9.62  0.13  
Without  −24.9  1.51  5  0.91  13.51  0.92  
Without  −26.7  4.19  5  0.52  16.19  0.88  
Parallel architecture  19.9 (19.0–21.5)  7.9 (5.9–8.2)  With  −26.7  5.50  7  0.60  13.5  0.15  
Without  −24.7  1.80  5  0.88  13.8  0.88  
Weibull distribution  22.9 (21.9–23.5)  1.6 (1.2–2.0)  With  −31.1  12.20  8  0.14  18.2  0.49  
Without  −24.7  1.97  5  0.85  13.97  0.97 
This work has been mainly supported by grants from the Cancer Society in Stockholm, The King Gustaf V Jubilee Fund, Stockholm, within the Center of Excellence by The Swedish National Board for Industrial and Technical Development.