Applying radiobiological plan ranking methodology to VMAT prostate SBRT.

The impact of a rectal spacer and an increased near maximum target dose in VMAT prostate SBRT is studied. For a group of 11 patients (35Gy-in-five-fractions VMAT prostate SBRT) a set of 4 plans were generated, namely two VMAT plans, with D2%⩽37.5Gy (Hom) and with D2%⩽40.2Gy (Het), were created for each of two CT scans taken before (NoSpc) and after (Spc) transperineal spacer insertion. Consequently the methodology for parameter invariant TCP (tumor control probability) plan ranking was applied for comparison of the plans in terms of tumor control. NTCPs (normal tissue complication probabilities) were calculated for rectum and bladder using Lyman's model. For all 11 patients the TCP plan ranking has shown that the Het plans would perform considerably better in TCP terms than the Hom ones. The plans without rectal spacer were ranked worse compared to those with rectal spacer except for one set of Hom plans. The calculated NTCPs for rectum produced by the Het plans were quite similar to the NTCPs of the Hom ones. The rectal NTCPs of the Hom Spc plans were always lower than the NTCPs of the Hom NoSpc plans. The NTCP values for bladder were extremely low in all cases. The use of rectal spacer leads in general to lower risk of rectal complications, as expected, and even to better tumor control. Plans with increased near maximum target dose (D2%⩽40.2Gy) are expected to perform much better in terms of tumor control than those with D2%⩽37.5Gy.

Confidence limits of the probability of success in animal experiments and clinical studies: a Bayesian approach.

PURPOSE: To determine from the number of trials, n, and the number of observed successes, k the most probable value, the variance and the confidence limits of the probability of success, p, in animal experiments and clinical studies subject to binomial statistics. METHOD: In such experiments the probability of success is an unknown parameter. The Bayesian approach to the problem is advocated, based on constructed distribution of the probability of success. RESULTS: A simple Matlab code for the calculation of the confidence limits according to the proposed method is provided. The most probable, the mean, the variance and the confidence limits are calculated applying the usual definitions of these characteristics. CONCLUSION: The proposed method works for any number of trials--large and small and all possible values of the number of successes, including k=0 and k=n, providing exact formulae for the calculation of the confidence limits in all cases.

Limitations of a TCP model incorporating population heterogeneity.

The variation between individuals in their dose-response characteristics complicates attempts to extract estimates of radiobiological parameters (e.g. alpha, beta, etc) from fits to clinical dose-response data. The use of 'population' dose-response models that explicitly account for this variability is necessary to avoid obtaining skewed parameter estimates. In this work, we evaluated an example of a 'population' tumour control probability (TCP) model in terms of its ability to provide reliable parameter estimates. This was accomplished by performing fits of this population model to 'pseudo' data sets, which were generated with Monte Carlo techniques and based on preset values for the various radiobiological parameters. The fitting exercises illustrated considerable correlations between the model parameters. Especially significant was the large correlation observed between the parameter mu=alpha/sigmaalpha used to characterize the level of population heterogeneity in radiosensitivity and the alpha/beta parameter typically used to describe the response to fractionation. The results imply that fits to clinical data may not be able to distinguish between tumours exhibiting a high degree of heterogeneity and a strong beta-mechanism and those containing little heterogeneity and having a weak beta-mechanism. One implication is that basing the design of optimal fractionation regimes on such fitting results may be error-prone. If in vitro assays are to be used to independently determine biologically reasonable ranges for parameter values, an accurate knowledge of the relationship between in vitro and in vivo dose-response characteristics is required.

Radiation damage, repopulation and cell recovery analysis of in vitro tumour cell megacolony culture data using a non-Poissonian cell repopulation TCP model.

The effects of radiation damage, tumour repopulation and cell sublethal damage repair and the possibility of extracting information about the model parameters describing them are investigated in this work. Previously published data on two different cultured cell lines were analysed with the help of a tumour control probability (TCP) model that describes tumour cell dynamics properly. Different versions of a TCP model representing the cases of full or partial cell recovery between fractions of radiation, accompanied by repopulation or no repopulation were used to fit the data and were ranked according to statistical criteria. The data analysis shows the importance of the linear-quadratic mechanism of cell damage for the description of the in vitro cell dynamics. In a previous work where in vivo data were analysed, the employment of the single hit model of cell kill and cell repopulation produced the best fit, while ignoring the quadratic term of cell damage in the current analysis leads to poor fits. It is also concluded that more experiments using different fractionation regimes producing diverse data are needed to help model analysis and better ranking of the models.

Critical volume model analysis of lung complication data from different strains of mice.

The critical volume (CV) normal tissue complication probability (NTCP) model was used to fit experimental data on radiation pneumonitis in mice to test the model and determine the values of the model parameters characterizing the lung structure: relative critical volume and cell radiosensitivity. The entire lungs of mice from ten different strains were irradiated acutely and homogeneously to different doses. The experimental animals from the different strains expressed different radiation sensitivities, forming ten well-defined dose-response curves. The most widely accepted biological NTCP model (the individual CV NTCP) readily applicable to cases of acute uniform irradiation was used to fit all the individual dose-response curves simultaneously. To account for the apparent difference in the response of the different strains, it was assumed that the strains differed in their (cell) radiosensitivity. The maximum likelihood method of fitting was used. The obtained fit was statistically highly acceptable. The best-fit value of the relative critical volume, mu, was 78%, which is extremely close to the histologically observed value of around 72%. The values of radiosensitivity, alpha, ranged between 0.26 and 0.37 Gy(-1) for the different strains. The best-fit numbers of functional subunits (FSU) constituting the lung, N, and the number of cells in an FSU, N(o), were implausibly low: N = 9 and N(o) = 23, respectively. The best-fit value of N(o)N was a very small number that was unlikely to correspond to the total number of cells comprising the lung, suggesting that a different interpretation of N and N(o) was required. The individual CV model provided a simultaneous description of the individual responses of different mouse strains through assumed interindividual variability in alpha only. A new interpretation is given to the entities corresponding to N(o) and N. N(o)N is interpreted as the number of certain elementary structures. These structures are considered to be equivalent to the classical functional subunit, which is much larger than a cell and plays a fundamental role in determining the radiation response of the organ. N is identified as the number of the few large subdivisions of the lungs, M = microN is the number that have to be damaged for the lung to fail. N(o) is interpreted as the mean number of elementary structures (FSU) per large subdivision. This imposes a picture of damage to large, contiguous subdivisions containing many FSU, which is consistent with the histological appearance of the lungs of mice in respiratory distress. This picture is in marked contrast to the random distribution of small areas of damage expected for the small size of an FSU. This random distribution is characteristic of earlier stages of the development of radiation pneumonitis, suggesting that some additional process spreads injury from damaged FSU to adjacent, undamaged FSU during the terminal phase.

Investigating the effect of clonogen resensitization on the tumor response to fractionated external radiotherapy.

In this work we further develop the modeling of tumor dynamics by proposing a mechanism of tumor resensitization that is based on the process of reoxygenation. Reoxygenation is modeled using the concept of nonstationary diffusion of oxygen. This leads to the derivation of an explicit expression for the radiosensitivity parameter that predicts a radiosensitivity that increases with time. To account for the resensitization mechanism, the time-dependent expression for the radiosensitivity is then incorporated within a tumor control probability (TCP) model that already includes tumor cell repopulation and repair. We fit a set of experimental animal TCP curves corresponding to several different fractionation regimes using both the modified (with resensitization) and unmodified (without resensitization) versions of the TCP model. In comparison to the unmodified model, the modified model produces statistically superior fits, and is able to describe an "inverse" dose-fractionation behavior present in the data.

Inverse treatment planning by physically constrained minimization of a biological objective function.

In the current state-of-the art of clinical inverse planning, the design of clinically acceptable IMRT plans is predominantly based on the optimization of physical rather than biological objective functions. A major impetus for this trend is the unproven predictive power of radiobiological models, which is largely due to the scarcity of data sets for an accurate evaluation of the model parameters. On the other hand, these models do capture the currently known dose-volume effects in tissue dose-response, which should be accounted for in the process of optimization. In order to incorporate radiobiological information in clinical treatment planning optimization, we propose a hybrid physico-biological approach to inverse treatment planning based on the application of a continuous penalty function method to the constrained minimization of a biological objective. The objective is defined as the weighted sum of normal tissue complication probabilities evaluated with the Lyman normal-tissue complication probability model. Physical constraints specify the admissible minimum and maximum target dose. The continuous penalty function method is then used to find an approximate solution of the resulting large-scale constrained minimization problem. Plans generated by our approach are compared to ones produced by a commercial planning system incorporating physical optimization. The comparisons show clinically negligible differences, with the advantage that the hybrid technique does not require specifications of any dose-volume constraints to the normal tissues. This indicates that the proposed hybrid physico-biological method can be used for the generation of clinically acceptable plans.

Investigating the effect of cell repopulation on the tumor response to fractionated external radiotherapy.

In this work we study the descriptive power of the main tumor control probability (TCP) models based on the linear quadratic (LQ) mechanism of cell damage with cell recovery. The Poisson, binomial, and a dynamic TCP model, developed recently by Zaider and Minerbo are considered. The Zaider-Minerbo model takes cell repopulation into account. It is shown that the Poisson approximation incorporating cell repopulation is conceptually incorrect. Based on the Zaider-Minerbo model, an expression for the TCP for fractionated treatments with varying intervals between two consecutive fractions and with cell survival probability that changes from fraction to fraction is derived. The models are fitted to an experimental data set consisting of dose response curves that correspond to different fractionation regimes. The binomial TCP model based on the LQ mechanism of cell damage solely was unable to fit the fractionated response data. It was found that the Zaider-Minerbo model, which takes tumor cell repopulation into account, best fits the data.

Derivation of the expressions for gamma50 and D50 for different individual TCP and NTCP models.

This paper presents a complete set of formulae for the position (D50) and the normalized slope (gamma50) of the dose-response relationship based on the most commonly used radiobiological models for tumours as well as for normal tissues. The functional subunit response models (critical element and critical volume) are used in the derivation of the formulae for the normal tissue. Binomial statistics are used to describe the tumour control probability, the functional subunit response as well as the normal tissue complication probability. The formulae are derived for the single hit and linear quadratic models of cell kill in terms of the number of fractions and dose per fraction. It is shown that the functional subunit models predict very steep, almost step-like, normal tissue individual dose-response relationships. Furthermore, the formulae for the normalized gradient depend on the cellular parameters alpha and beta when written in terms of number of fractions, but not when written in terms of dose per fraction.

Generalization of a model of tissue response to radiation based on the idea of functional subunits and binomial statistics.

This work investigates the existing biological models describing the response of tumours and normal tissues to radiation, with the purpose of developing a general biological model of the response of tissue to radiation. Two different types of normal tissue behaviour have been postulated with respect to its response to radiation, namely critical element and critical volume behaviour. Based on the idea that an organ is composed of functional subunits, models have been developed describing these behaviours. However, these models describe the response of an individual, a particular patient or experimental animal, while the clinically or experimentally observed quantity is the population response. There is a need to extend the models to address the population response, based on the ideas we have about the individual response. We have attempted here to summarize and unify the existing individual models. Finally, the population models are investigated by fitting to pseudoexperimental sets of data and comparing them with each other in terms of goodness-of-fit and in terms of their power to recover the values of the population parameters.

Modelling the dose-volume response of the spinal cord, based on the idea of damage to contiguous functional subunits.

PURPOSE: To investigate the response of the spinal cord of experimental animals to homogeneous irradiation, the main purpose being to propose a new version of the Critical Volume Normal Tissue Complication Probability (NTCP) model, incorporating spatial correlation between damaged functional subunits (FSU). METHOD: The standard Critical Volume NTCP model and its modified version, the Contiguous Damage model promoted here, are described in mathematical terms. Also, a fiber-like structure of the spinal cord is considered, which is a more complex structure than the standard Critical Volume NTCP model assumes. It is demonstrated that the Contiguous Damage model predicts different responses to two-segment irradiation and to single-segment irradiation to the same combined length as observed in experiments on rats, a result that cannot be described by the standard Critical Volume NTCP model. RESULTS AND CONCLUSIONS: Both the Critical Volume model and the Contiguous Damage model, are fitted to two sets of canine spinal cord radiation data corresponding to two different fractionation regimes of irradiation. Whole-organ irradiation as well as partial irradiation to different lengths are considered, allowing the investigation of dose-volume effects. Formal goodness-of-fit investigation shows that both models fit the canine spinal cord data equally well.

A study of objective functions for organs with parallel and serial architecture.

An objective function analysis when target volumes are deliberately enlarged to account for tumour mobility and consecutive uncertainty in the tumour position in external beam radiotherapy has been carried out. The dose distribution inside the tumour is assumed to have logarithmic dependence on the tumour cell density which assures an iso-local tumour control probability. The normal tissue immediately surrounding the tumour is irradiated homogeneously at a dose level equal to the dose D(R) delivered at the edge of the tumour. The normal tissue in the high dose field is modelled as being organized in identical functional subunits (FSUs) composed of a relatively large number of cells. Two types of organs--having serial and parallel architecture are considered. Implicit averaging over intrapatient normal tissue radiosensitivity variations is done. A function describing the normal tissue survival probability S0 is constructed. The objective function is given as a product of the total tumour control probability (TCP) and the normal tissue survival probability S0. The values of the dose D(R) which result in a maximum of the objective function are obtained for different combinations of tumour and normal tissue parameters, such as tumour and normal tissue radiosensitivities, number of cells constituting a normal tissue functional unit, total number of normal cells under high dose (D(R)) exposure and functional reserve for organs having parallel architecture. The corresponding TCP and S0 values are computed and discussed.

A new method for optimum dose distribution determination taking tumour mobility into account.

A method for determining the optimum dose distribution in the planning target volume is proposed when target volumes are deliberately enlarged to account for tumour mobility in external beam radiotherapy. The optimum dose distribution is a dose distribution that will result in an acceptable level of tumour control probability (TCP) in most of the arising cases of tumour dislocation. An assumption is made that the possible shifts of the tumour are subject to a Gaussian distribution with mean zero and known variance. The idea of a reduced (mean in ensemble) tumour cell density is introduced. On this basis, the target volume and dose distribution in it are determined. The tumour control probability as a function of the shift of the tumour has been calculated. The Monte Carlo method has been used to simulate TCP distributions corresponding to tumour mobility characterized by different variances. The obtained TCP distributions are independent of the variance of the mobility because the dose distribution in the planning target volume is prescribed so that the mobility variance is taken into account. For simplicity a one-dimensional model is used but three-dimensional generalization can be done.

A variational approach to the problem of optimizing the radiation dose distribution in tumours.

This paper presents a precise mathematical formulation of a biological criterion by which the radiation dose distribution in tumours homogeneous or heterogeneous in cell density and radiosensitivity can be optimized. The criterion is formulated as search for a dose distribution that would minimize the mean dose delivered to the tumour under the constraint that the tumour control probability reaches a given desired value. Using a method from the calculus of variations it has been proven that a homogeneous dose distribution is the solution in case of tumours homogeneous in radiosensitivity independent of their cell spatial density status. Thus the usual requirement for homogeneous dose distribution in case of homogeneous tumours is proven if the leading clinical criterion is the described one. The formula for the dose distribution in case of tumours heterogeneous in cell radiosensitivity is given too.

A mathematical approach to optimizing the radiation dose distribution in heterogeneous tumours.

This paper offers a general mathematical approach to dose distribution optimization which allows tumours with different degrees of complexity to be considered. Two different biological criteria - A) keeping the control probability of the different parts of the tumour (local tumour control probability) uniform throughout the tumour and B) minimizing the mean dose delivered to the tumour are studied. For both criteria we impose the requirement that the whole tumour control probability be kept on a certain desired level. It is proved that the adoption of the first criterion requires a dose distribution logarithmic with the cell density and proportional to the inverse of the cell radiosensitivity while the adoption of the second criterion necessitates a homogeneous dose distribution when the cell radiosensitivity is constant. The corresponding formula for the dose distribution in case of heterogeneous cell radiosensitivity is also given. The two criteria are compared in terms of local tumour control probability and mean dose delivered to the tumour. It is concluded that maintaining constant local tumour control probability (criterion A) may be of greater clinical importance then minimizing the mean dose (criterion B).

Binding of ketoprofen to human serum albumin studied by circular dichroism.

The protein binding of ketoprofen has been studied using circular dichroism titration method as well as the new algorithm proposed by the authors for the treatment of data obtained. The quantitative parameters association constants (k) and number of binding sites (N) have been determined. It is proved that the protein binding of Ketoprofen is going through separate stages and the number of binding sites probably arises. It is acceptable that a high affinity binding takes place primarily (kI = 3.8 x 10(6) l.mol-1). Later, due to the conformational changes in the protein molecule the binding areas are modified and the number of binding sites considerably arises (NI = 3.5 and NII = 14), while the binding affinity reduces 100-fold (kII = 5.10(4) l.mol-1). The number of binding sites has been studied and an identification of the chromophore taking part in the drug-protein interaction has been performed on the base of UV- and CD spectra. A mechanism of the interaction is proposed which coincides with the stepwise binding model.

Binding of sulindac to human serum albumin studied by circular dichroism.

The protein binding of sulindac (CAS 38194-50-2) was studied using circular dichroism (CD). By the new algorithm for the analysis of proposed data the association constants (k) and number of binding sites (N) were determined. The binding was found to go through separate stages, where the binding affinity tends to become lower; the first step characterized by kI = 7.6 x 10(6) l.mol-1 and NI = 1.4; while for the second step kII = 1.7 x 10(6) l.mol-1 and NII = 6.6. On the basis of the CD-data and using UV-spectra the nature of binding sites was studied. It may be stated that the binding sites are situated in the region of asymmetrically perturbed chromophore of the drug, which made a positive contribution to the Cotton effect. The results obtained suggest a mechanism of interaction which is consistent with the stepwise binding model.

New algorithm for analysis of data obtained by means of circular dichroism titration method.

A new method for analysis of circular dichroism titration data used for drug-protein binding investigations is proposed. A square equation between molar ellipticity change and total drug concentration is obtained which is examined analytically. A new minimization algorithm for determining the binding sites and association constants of each type of drug-protein complexes is applied.