[An aggregation method for determination of parameters of mathematical models].

Two mathematical models for calculation of the probability of post-radiation complications in normal tissues are considered. The first model provides calculation of the probability of post-radiation complications within the entire dose and tissue volume range. The second model is approximative. It can be used for calculation of the probability of post-radiation complications that does not exceed 50%. Methods for determination of parameters of the mathematical models are described. These methods are based on solution of extremal problems. An original method for determination of three parameters of the mathematical models is suggested. It is based on solution of a complex automatization problem using aggregated clinical information. The method includes straightforward enumeration of values of the aggregation parameter p and solution of an extremal problem for determination of the other two parameters. This study was supported by the Russian Foundation for Basic Research, project No. 05-01-00326.

[Methods for geometric and nonlinear optimization of the structure of interstitial implants for interstitial x-ray therapy of malignant tumors].

In 1990 G. K. Edmundson suggested a geometric optimization method for determination of the effective structure of interstitial implants. In 1997 Y. Anacak, M. Essassolak, A. Aydin, et al. used this method for two-level interstitial implantation. It was shown in their work that the geometric optimization method provided higher efficiency of the dose distribution than conventionally used dose distributions produced by interstitial implants with equal exposure times. In 1983 the problem of optimization of the interstitial implant structure was formulated in our works. Its mathematical interpretation was also suggested. The goal of this work was to show that the geometric optimization method does not always provide high efficiency of the dose distribution, even in comparison with conventionally used dose distributions produced by interstitial implants with equal exposure times. The interstitial implant structure can be optimized by solving special mathematical programming problems.

[Synthesis of radiological models and radiological constants. Part 4: Synthesis of population-phenomenological models and Lyman model].

The goal of this work was to synthesize Lyman mathematical models (MM), which describe the probability of post-radiation complications (PRC) in tissue subjected to radiation therapy with given scheme of dose fractionating (DF), and population-phenomenological (PP) MMs PP3 and PP4, which describe equivalent DF schemes for a fixed PRC value. Construction of synthesized MMs (SMMs) becomes possible only on the basis of several assumptions requiring further clinical validation. Synthesized MMs can be used for determination of the optimal dynamic conditions of irradiation of malignant tumors. These conditions include the optimal physical plan of irradiation and the optimal time scheme of its implementation. Synthesis of MMs leads to determination of radiological invariants (constants), which can become a basis for a new branch of medical science, quantitative radiology.

[Synthesis of radiological models and radiological invariants (constants). (Part 3: synthesis of population-phenomenological models and Klepper model)].

The goal of this work was to synthesize Klepper mathematical models (MM), which describes the probability of post-radiation complications (PRC) in tissue subjected to radiation therapy with given scheme of dose fractionating (DF), and population-phenomenological (PP) MMs PP3 and PP4, which describe equivalent DF schemes for a fixed PRC value. Construction of synthesized MMs (SMMs) becomes possible only on the basis of several assumptions requiring further clinical validation. Synthesized MMs can be used for determination of the optimal dynamic conditions of irradiation of malignant tumors. These conditions include the optimal physical plan of irradiation and the optimal time scheme of its implementation. Synthesis of MMs leads to determination of radiological invariants (constants), which can become a basis for a new branch of medical science, quantitative radiology.

[Synthesis of radiological models and radiological invariants (constants). Part 2].

The goal of this work was to synthesize a mathematical model (MM) on the basis of Klepper and Lyman MMs, which describe the probability of post-radiation complications (PRC) in tissue subjected to radiation therapy with given scheme of dose fractionating (DF), and the LQ-model, which describes equivalent DF schemes for a fixed PRC value. Construction of synthesized MMs (SMMs) becomes possible only on the basis of several assumptions requiring further clinical validation. Synthesized MMs can be used for determination of the optimal dynamic conditions of irradiation of malignant tumors. These conditions include the optimal physical plan of irradiation and the optimal time scheme of its implementation. Synthesis of MMs leads to determination of radiological invariants (constants), which can become a basis for a new branch of medical science, quantitative radiology.

[An interactive method for optimization of irradiation plans in radiation therapy of malignant tumors].

Formation of a rational dose filed in contact radiation therapy of malignant tumors is, in the majority of cases, a multicriterion and multiextremal problem. A compromise plan of irradiation can be obtained by interactive matching of alternative criteria of radiation therapy. In this work, it is suggested to solve the problem of formation of effective therapeutic dose fields using synthesized methods, including modified trial-and-error method (visual optimization method), and mathematical programming methods. This combined approach provides better opportunities for formation of effective therapeutic dose fields in radiation therapy of malignant tumors.

[Synthesis of radiological models and radiological invariants (constants). Part 1].

Work is devoted to synthesis of mathematical models (MM) which describe probability of appearance of radiation complication (ARC) in a tissue at the Klepper's fixed scheme of dose fractionation (FD) and Ellis's MM, which describes equivalent schemes FD for fixed value ARC. Creation of synthesized mathematical models(S) appears possible at some assumptions which require a check on the basis of the clinical information are necessary for definition of optimum dynamic conditions of an radiation of cancerous growths which include the optimum physical radiation plan and the optimum scheme its realization in time. The synthesis of MM leads to natural appearance of radiological invariants (constants) which can become a basis for construction of a new scientific direction of quantitative radiology.

[Estimated equivalent radiation conditions (EERC), more useful clinical data and estimation of potential resorption of lesion nodule with respect to its volume and total focal dose]. / Ispol'zovanie metoda rascheta ékvivalentnykh uslovii oblucheniia dla uvelicheniia ob''ema poleznoi klinicheskoi informatsii i dlia rascheta veroiatnosti rezorbtsii ochaga porazheniia v zavisimosti ot ego ob''ema i summarnoi ochagovoi dozy.

Assumptions on properties of tumor cells are defined. On the basis on them the survival rate of tumor cells is described by the LQ function and the dependence of resorption probability (RP) in the lesion nodule on the number survived tumor cells (tumor volume) and on the total focal dose (TFD) is described by the Poisson function. An analysis of the above approach to the determination of lesion-nodule RP resulted in designing a calculation method for EERC that can be used to estimate the lesion-nodule RP as a function of its volume and TFD. An analytical or graphical description of the RP/TFD dependence for a fixed volume of lesion nodule is needed for the implementation of EERC. There is no need to determine the cell structure of tissue or the radiobiological properties of its cells. The possibility to construct the RP section function and to extend, thus, the volume of useful clinical data belongs to the method's advantages.

[Mathematical modeling of optimal dose fields in radiotherapy of malignant tumors. Part 2 (Contact methods of radiotherapy)]. / Matematicheskoe modelirovanie optimal'nykh dozovykh polei v luchevoi terapii zlokachestvennykh opukholei. Chast' 2 (Kontaktnye metody luchevoi terapii).

The principles of mathematical modeling of optimal dose fields in contact radiotherapy (RT) of malignant tumors are investigated. The point dose additivity provides for presetting the permissible dose field in an irradiated organism as a system of linear limitations to doses in the control points (CP) distributed in the lesion focus and in healthy organs and tissues. It was shown as impossible to shape a dose field by linear limitations to doses in CP in using the RT contact methods with the irradiation sources being implanted into lesion focus. A mathematic interpretation was suggested for the task (with its solution by an iterative algorithm) of forming an optimal dose field in the lesion focus with implanted irradiation sources, which is based on maximizing the factor of dose-field homogeneity. It was further demonstrated that linear limitations, if added to the dose in healthy organs and tissues, make the task even more complicated if not insoluble. Finally, it is suggested to use the method of shaping an effective dose field by the iterative method with interactive visual optimization of the dose field.

[The mathematical modeling of the optimal dose fields in radiation therapy of malignant tumors. Part 1 (Distance radiotherapy)]. / Matematicheskoe modelirovanie optimal'nykh dozovykh polei v luchevoi terapii zlokachestvennykh opukholei. Chast' 1 (distantsionnaia luchevaia terapiia).

The specificity of mathematical modeling of optimal dose fields in radiation therapy of malignant tumors is under consideration. The permissible dose field is set now for irradiated body as a system of linear limitations to the doses at control spots (CS) distributed in the lesion focus and in the healthy organs and tissues. It is for the first time that an issue related with choosing an adequate number and method of CS distribution, based on uniform continuity of the dose field, is addressed in the paper. An iterative procedure of building the local CS networks is suggested. The method of linear programming (LP) can be used to select an optimal irradiation plan. A method of serial input of limitations, which cuts both the LP task scope and its computer-aided solution, is described.

[Interactive input and correction of data about the structure of irradiated organism in the planning of radiotherapy for malignant tumor]. / Interaktivnyi vvod i korrektsiia informatsii o stroenii obluchaemogo organizma pri planirovanii luchevoi terapii zlokachestvennykh opukholei.

A method is suggested for the interactive input, conversion and presentation of data about the structure of irradiated organism preset as Data Reference Planes (DRP). The possibility to build a library of DRP and the simplicity of their adjustment to each disease case witness to that the method of input, storage, conversion and presentation of data about the irradiated organism is promising in tumor radiotherapy.

[A method of the interactive visual optimization of the therapeutic dose field in contact radiation therapy of malignant tumors (theoretical aspects of the problem)]. / Metod interaktivnoi vizual'noi optimizatsii terapevticheskogo doznogo polia v kontaktnoi luchevoi terapii zlokachestvennykh opukholei (teoreticheskie aspekty problemy).

The mathematical and interpretation tasks of a directed shaping of dose fields in the contrast radiation therapy of malignant tumors are defined on the basis of the dose-field homogeneity parameter. A schematic iterative algorithm of how to solve the tasks is described. A method for the visual optimization of such field is elaborated; it is based on preset limits to the dose field in the lesion focus and in the healthy organs and tissues. The dose field is shaped by an applicator with multiple terminal fixed positions of irradiation sources--the effect is achieved due to variability of their exposure duration.

[Shaping-up of optimal dose fields in a stretch by means of dot-type and linear sources of irradiation (theoretical aspects)]. / Formiroanie optimal'nykh doznykh polei na otrezke s pomoshch'iu tochechnykh i lineinykh istochnikov izlucheniia (teoreticheskie aspekty problemy).

Shaping-up of a dose field in a stretch by means of dot-type and linear irradiation sources is mathematically interpreted. A criterion of an optimal dose field is formulated for the contact radiotherapy applicable to malignant tumors, which is based on homogeneity of such therapy. The task of forming an optimal dose field in a stretch by the dot-type and linear irradiation sources is insoluble on the basis of analytical methods. The task properties were investigated and an iterative method designed to solve it was elaborated on the basis of such properties.

[Formation of optimum dose fields in contact radiation therapy of malignant tumors]. / Formirovanie optimal'nykh dozovykh polei v kontaktnoi luchevoi terapii zlokachestvennykh opukholei.

The definition of the homogeneity of a dose field in the contact radiation therapy for malignant tumors is introduced. The mathematical interpretation of problems in the formation of optimum dose fields, to which the maximum homogeneity of a dose field at the site of lesion corresponds, is presented. It is shown that the problems in the formation of optimum dose fields may be divided into two subsets in relation to whether the sources of radiation are located at the site of lesion or adjacent to the latter (application techniques of radiation). An analytical method for solving a problem in the formation of an optimal dose field in the ring circle by means of one ring source of radiation (the first type of problems). The investigation was conducted with the support of the Russian Fund of Fundamental Investigations (RFFI 01-01-00137).

[A summarized method for calculating the possibility of resorption of a focal lesion as a function of its volume and irradiation dose]. / Obobshchennyi metod rascheta veroiatnosti rezorbtsii ochaga porazheniia kak funktsii ot ego ob"ema i dozy oblucheniia.

The mathematical model of calculation of equivalent conditions of irradiation (CECI) can be used in those cases, when it is impossible to divide clearly the tumor cells into radiation sensitive (RS) and radiation resistant (RR) and when the spectrum of radio-biological properties of tumor cells can be regarded as quasi-continuous.

[Approximate methods for calculating the probability of radiation complications. The PKLQ method]. / Priblizhennye metody rascheta veroiatnosti luchevogo oslozhneniia. Metod PKLK.

A method of approximate calculation of the probability of resorption of a lesion focus by means of three mathematical models: the Poisson model, the Klepper model, and the LQ-model (the PKLQ method) is described. The method is based on a procedure for reducing SOD to the preset scope of a lesion focus. It is suggested that radio-sensitive (RS) cells predominate in the focus of lesion; radio-resistant cells are available in small quantities or their radiobiological properties differ from RS cells.

[Approximate methods for calculation of the likelihood of radiation complications. The generalized PKLQ method (GPKLQM)]. / Priblizhennye metody rascheta veroiatnosti luchevogo oslozhneniia. Obobshchennyi metod PKLK (OMPKLK).

Search for rational schedules of radiation therapy for malignant tumors is a topical problem of modern radiology. It cannot be solved without prognostic estimates of radiation affecting tumor and normal organs and tissues. The aim of the study was to develop mathematical models to be used for approximate evaluation of radiation affecting a tumor focus and normal organs and tissues. This paper provides a theoretical rationale for the generalized PKLQ (GPKLQ) method for a random count of tumor tissue radio-resistant cells, which is based on the view of the effective count of radio-sensitive cells (the effective volume of a damage focus).

[Probability of tissue cell death, integral cellularity and likelihood of radiation-induced tissue complications]. / Veroiatnosti gibeli kletok tkani, integral'naia kletochnost' i veroiatnost' vozniknoveniia luchevogo oslozhneniia v tkani.

The paper describes a mathematical model for determining the likelihood of radiation-induced complications in the tissue as a function of the number of surviving cells. It is suggested the irradiated tissue cannot be regarded as a structureless point set of cells and that there are repairable and nonrepairable spatial configurations formed by surviving tissue cells. The probability of none tissue radiation complications may be considered is that of formation of a repairable structure from the surviving tissue cells. The paper shows why the radiation effect of radiation may be coupled (graded) in units of integral cellularity. Recurrent equations have been derived for calculating the number of repairable structures in relation to the conditions of tissue radiation. The values of a number of repairable structures as a function of an area and single dose of radiation are calculated as an example.

[Method of calculating the equivalent tumor dose as a function as to irradiated tumor tissue volume]. / Metod rascheta ékvivalentnoi opukholevoi dozy kak funktsii ot ob"ema obluchennoi opukholevoi tkani.

Based on the assumption that tumor tissue consists of normal and radiation-resistant, that the survival of both types of tumor cells may be described by LQ functions and that the count of radiation-resistant cells is in proportion to that of tumor cells, the author has developed a method for calculating the equivalent tumor dose as a function as to irradiated tumor tissue volume for the fixed value of a single dose in the session of radiation. The developed formalism may be used to test the hypothesis that the count of clonogenic and radiation-resistant cells is in proportion to the baseline number of the cells in the tumor tissue.