An overview of radiotherapy (RT) induced normal tissue complication probability (NTCP) models is presented. NTCP models based on empirical and mechanistic approaches that describe a specific radiation induced late effect proposed over time for conventional RT are reviewed with particular emphasis on their basic assumptions and related mathematical translation and their weak and strong points.

Modern radiotherapy techniques allow unprecedented levels of accuracy, precision, and conformity in target localization, patient setup, and dose delivery thanks to the aid of many different imaging modalities.

Contemporary treatment strategies almost always involve delivering higher doses to the targeted tissue with the aim of improving tumor control, but before such approaches can be safely implemented an accurate and reliable knowledge on toxic effects on surrounding tissues has to be secured.

With the aim of normal tissue preservation many models have been proposed to describe radiation induced complications mostly focusing on late complications which, being irreversible, are considered to have the highest impact on the patient quality of life.

In this, as in most overviews [

This overview focuses on the description of tissue organization without any initial assumption on the subunit response to radiation. Our approach stresses the mathematical translation of the features of the presented models to better expose their versatility and the opportunities for further developments. Indeed, radiobiological modeling needs a quite complex mathematical toolbox, mirroring the complexity of the biological systems. Each organ is not just an agglomerate of cells but it has an underlying architecture/organization that is the very basis of its functional role, enabling many different strategies (renewal/replacement of damaged cells, intricate microscopic repair pathways, etc.) to successfully deal with radiation damage [

Many of these radiobiological models have been integrated into Treatment Planning Systems [

The most widely used NTCP model in clinical radiobiology is the Lyman model [

The core of the model is a fit of frequency data collected for chosen clinical toxicity endpoints and for a particular class of dose irradiation patterns, namely, those patterns where a given portion of an organ or tissue volume absorbs a spatially uniform dose and the rest of its volume absorbs (ideally) no dose at all [

In more technical terms, if we define a random indicator

Having described the radiobiological consequences of the partial uniform dose distributions, the model needs to be extended to arbitrary distributions; that is, we want to know the probability

The range of

The key idea of the Kutcher-Burman interpolation is to consider each couple

Note also that in the above reasoning the values

When the Lyman model was proposed, patient irradiation had a simple ballistician that delivered dose distributions with sharp penumbras. As a consequence the basic assumption that dose delivery was performed on a dichotomic basis, with irradiated tissue absorbing all the dose at the intended fractionation and the rest of the tissues absorbing no dose at all, was fairly accurate. In a few years, however, technological advances allowed irradiations strategies that, in spite of a much better performance in concentrating the high dose region around the tumour tissue, could induce highly nonuniform low dose deposit in surrounding healthy tissues. This motivated, at least partially, the introduction of the Kutcher-Burman reduction scheme, which then needed to accommodate also the fact that different portions of the irradiated volume receive the accumulated dose in different fraction sizes according to their position in the inhomogeneous dose map. The most used approach to solve this issue is a nonlinear rescaling of the dose axis according to the BED formula of the LQ model [

The Critical Volume Model [

It is also assumed that a definite collection of

It is only from this viewpoint that the FSUs are considered as “atomic” in the sense that all the information about their interaction with ionizing radiation is summarized by the inactivation probability

If we define

In the standard formulation of the model, the FSUs are assumed independent, and the behavior of their macroscopic aggregate, that is, the tissue, is described by the variable

From the assumed independence of the FSUs it follows that if

Given

From expression (

A different expression is often used for (

From (

The probability

The customary interpretation of the described probability distributions and the related nomenclature is such that when

The number of FSUs is usually very large and approximate expressions for (

It can be shown [

If we introduce the random variable

Even if no specification other than monotonic increase is made about the functional dependence of the FSU inactivation probability on radiation dose, it is clear that, upon choice of a given tissue, that is, a value of

To explain the gentler slope of

In formulas, if we rewrite (

Having shown the assumptions that regulate the organizational response of tissue in the Critical Volume Model and discussed their consequences we must now specify the dose dependence of the FSU inactivation probabilities

The simplest approach, which however somehow spoils the mechanistic attitude of the CV model, is to assume a generic sigmoid shaped dose response curve whose parameters may or may not depend on the position of the FSU within the organ. In the former case we have some means to introduce a spatial feature within the organ, so that the position and shape of the irradiated volume become relevant.

A more mechanistically oriented assumption is the replication of the Critical Volume approach at a smaller scale: the FSU is considered as an aggregate of

The idea of nesting one Poisson-binomial probability distribution into another allows modeling increasing complexity of tissue organization.

At the level of complexity immediately next to the Critical Volume Model, an organ is viewed as a bundle of

Single FSU inactivation can be described, for example, by expression (

In formulas for the

The main aim of the LKB model is to accommodate the available clinical data into a reasonably manageable function, in order to help the clinician in assessing the odds of safe and successful treatment. As long as the clinical needs are involved, any understanding of the underlying biological phenomena is pursued only as long as it is instrumental to the above purpose. From the viewpoint of detailed radiobiological modeling, however, this model can be taken as a summary of the available clinical knowledge.

Quite general features of a normal tissue complication probability model which are already present in the LKB model are the following.

The observation of endpoint

The only parameter of the Bernoulli distribution, that is, the probability of occurrence of the event, is given by a functional defined on

At this stage, then, a rather featureless sample space is enough to describe experimental data. The TPB models add some details to the sample space

The mechanistic models, here exemplified by the TPB models, have an intrinsic ability to include microscopic biological features of the radiation induced damage; however the parameters needed to describe such details have not been determined for most treatment sites and thus cannot be used in daily clinical practice. Accurate determination of fine structure parameters in a clinical setting is quite a formidable task, even when foreseeable biasing is taken into account in the model framework. In addition, even the QUANTEC parameter database for the widely used LKB model suffers from indeterminacies due to nonuniformity of volume definitions (e.g., hollow organs) and heterogeneity in the quality of data and in the radiobiological assumptions

All these models and the proposed parameters summarizing the features of a specific tissue/organ have originated in the years of two-dimensional and three-dimensional (conformal) radiotherapy. Nevertheless they currently are the only available approaches to predict the expected toxicities deriving from the modern intensity modulated delivery techniques such as IMRT and VMAT.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Marco D’Andrea, Marcello Benassi, and Lidia Strigari contributed equally to this work.