Velocity field based method for data processing in radionuclide studies

In this paper, we consider a method of construction of a velocity field based on an optimization approach. A general formulation of the problem is given. It is shown how, under the appropriate assumption, it reduces to known particular case. Two-dimensional case of the velocity field construction...

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Дата:	2018
Автори:	Kotina, E.D., Ovsyannikov, D.A.
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Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2018
Назва видання:	Вопросы атомной науки и техники
Теми:	Детекторы и детектирование ядерных излучений
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/147303
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Цитувати:	Velocity field based method for data processing in radionuclide studies / E.D. Kotina, D.A. Ovsyannikov // Вопросы атомной науки и техники. — 2018. — № 3. — С. 128-131. — Бібліогр.: 14 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1473032019-02-15T01:24:56Z Velocity field based method for data processing in radionuclide studies Kotina, E.D. Ovsyannikov, D.A. Детекторы и детектирование ядерных излучений In this paper, we consider a method of construction of a velocity field based on an optimization approach. A general formulation of the problem is given. It is shown how, under the appropriate assumption, it reduces to known particular case. Two-dimensional case of the velocity field construction is considered in detail under the gradient constancy assumption of the density of the radiopharmaceutical distribution. The problem is reduced to solving a sparse system of large dimension, and the convergence of the iterative algorithm to the solution is considered. This method can be used in the radionuclide data processing. Радіонуклідні методи є одними з сучасних методів функціональної діагностики різних органів і систем організму людини, які вимагають використання математичних методів обробки та аналізу даних, отриманих у ході дослідження. Тому розвиток сучасних методів обробки радіонуклідних зображень є актуальним завданням. У статті розглядається метод для обробки радіонуклідних зображень, заснований на побудові поля швидкостей. Даний метод може застосовуватися для корекції руху, побудови контурів, аналізу радіонуклідних зображень, також він може використовуватися для аналізу та формування динаміки заряджених частинок. Радионуклидные методы являются одними из современных методов функциональной диагностики различных органов и систем организма человека, которые требуют использования математических методов обработки и анализа данных, полученных в ходе исследования. Поэтому развитие современных методов обработки радионуклидных изображений является актуальной задачей. В статье рассматривается метод для обработки радионуклидных изображений, основанный на построении поля скоростей. Данный метод может применяться для коррекции движения, построения контуров, анализа радионуклидных изображений, также он может использоваться для анализа и формирования динамики заряженных частиц. 2018 Article Velocity field based method for data processing in radionuclide studies / E.D. Kotina, D.A. Ovsyannikov // Вопросы атомной науки и техники. — 2018. — № 3. — С. 128-131. — Бібліогр.: 14 назв. — англ. 1562-6016 PACS: 87.15.A http://dspace.nbuv.gov.ua/handle/123456789/147303 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Детекторы и детектирование ядерных излучений Детекторы и детектирование ядерных излучений
spellingShingle	Детекторы и детектирование ядерных излучений Детекторы и детектирование ядерных излучений Kotina, E.D. Ovsyannikov, D.A. Velocity field based method for data processing in radionuclide studies Вопросы атомной науки и техники
description	In this paper, we consider a method of construction of a velocity field based on an optimization approach. A general formulation of the problem is given. It is shown how, under the appropriate assumption, it reduces to known particular case. Two-dimensional case of the velocity field construction is considered in detail under the gradient constancy assumption of the density of the radiopharmaceutical distribution. The problem is reduced to solving a sparse system of large dimension, and the convergence of the iterative algorithm to the solution is considered. This method can be used in the radionuclide data processing.
format	Article
author	Kotina, E.D. Ovsyannikov, D.A.
author_facet	Kotina, E.D. Ovsyannikov, D.A.
author_sort	Kotina, E.D.
title	Velocity field based method for data processing in radionuclide studies
title_short	Velocity field based method for data processing in radionuclide studies
title_full	Velocity field based method for data processing in radionuclide studies
title_fullStr	Velocity field based method for data processing in radionuclide studies
title_full_unstemmed	Velocity field based method for data processing in radionuclide studies
title_sort	velocity field based method for data processing in radionuclide studies
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2018
topic_facet	Детекторы и детектирование ядерных излучений
url	http://dspace.nbuv.gov.ua/handle/123456789/147303
citation_txt	Velocity field based method for data processing in radionuclide studies / E.D. Kotina, D.A. Ovsyannikov // Вопросы атомной науки и техники. — 2018. — № 3. — С. 128-131. — Бібліогр.: 14 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT kotinaed velocityfieldbasedmethodfordataprocessinginradionuclidestudies AT ovsyannikovda velocityfieldbasedmethodfordataprocessinginradionuclidestudies
first_indexed	2025-07-11T01:49:18Z
last_indexed	2025-07-11T01:49:18Z
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fulltext	ISSN 1562-6016. ВАНТ. 2018. №3(115) 128 VELOCITY FIELD BASED METHOD FOR DATA PROCESSING IN RADIONUCLIDE STUDIES E.D. Kotina, D.A. Ovsyannikov Saint-Petersburg State University, Saint-Petersburg, Russia E-mail: e.kotina@spbu.ru In this paper, we consider a method of construction of a velocity field based on an optimization approach. A general formulation of the problem is given. It is shown how, under the appropriate assumption, it reduces to known particular case. Two-dimensional case of the velocity field construction is considered in detail under the gradient constancy assumption of the density of the radiopharmaceutical distribution. The problem is reduced to solving a sparse system of large dimension, and the convergence of the iterative algorithm to the solution is considered. This method can be used in the radionuclide data processing. PACS: 87.15.A INTRODUCTION BASIC CONCEPTS The solution of various inverse problems has always been of great practical interest. In [1], the problem of determining velocity field from a given density of the distribution of charged particles was considered. In this paper, we propose to use this approach for the radionu- clide data processing, and to search for the velocity field from the known radiopharmaceutical distribution densi- ty [2]. Radionuclide studies are performed using gamma cameras and gamma tomographs [3]. Radionuclide methods are one of the most modern methods of func- tional diagnostics of diseases of the cardiovascular sys- tem, nephrology system, hepatobiliary systems etc. [2, 4, 5]. They require using of mathematical methods for data processing and analysis. РROBLEM STATEMENT Let us consider following system of differential equations fx  , (1) .0 3 1        divff xt i i i   (2) In accordance with the statement of the problem in article [1], we assume that the transport of the radio- pharmaceutical is described by equation (1) and the dis- tribution density of the radiopharmaceutical satisfies the Liouville’s equation (2). Here t is time,  Txxxx ),,( 321 the spatial coordinate’s vector,  ),,,(),( 321 xxxtxt  radiopharmaceutical dis- tribution density, Tfffxtff ),,(),( 321 is the ve- locity field and the superscript T denotes transposition of vector. We suppose that given function ),( xt satisfies the equation (2). The problem is to restore the velocity field of the system (1), i.e. to find function ),( xtf . In com- mon case, this is ill-posed problem [6]. So in [1] the regularization method is used and the corresponding variational problem is investigated. Further we will denote zxyxxx  321 ,, , .,, 321 wfvfuf  Following suggested approach [1] let us fix some moment t and formulate the problem of determination function ),( xtf as a minimization problem. We intro- duce the functional   ,),,( 222 dxdydzwvuJ M    (3) where ,))(( 22 zyxzyxt wvuwvu   ,2222222222 zyxzyxzyx wwwvvvuuu  α2 is a regularization parameter, M is a nonzero measure region in R 3 , , ,t x y   are notations for partial deriva- tives of first order in yxt ,, respectively. Thus, the problem of finding velocity field is con- sidered as a functional minimization problem (3) [1, 7]. If we put in the equation (2) 0divf , we get a case of the so-called optical flow, when the density of the indicator remains constant along the trajectories of the system (1) [8, 9]. Further we consider two-dimensional case ,, 21 yxxx  Tvuf ),( . The Euler-Lagrange equations for the integral func- tional (3) were written in [1] for the general three- dimensional case. For the two-dimensional case, taking into account 0divf , we obtain the well-known equa- tions for finding the velocity field [8]: 2 2 2 2 , , x x y t x y x y t y u u v v v u                         (4) here  Laplace operator. As a result of this approach, the problem for deter- mining functions vu, reduces to solving the system (4) under the appropriate boundary conditions. In this paper we will consider in more detail the case of the gradient constancy assumption, i.e. when the gra- dient of the distribution density along the trajectories of the system (1) remains constant: , , ytyyyx xtxyxx vu vu     (5) here , , , ,xx xt xy yy yt     are notations for partial deriva- tives of the second order. In this case, the integral functional (3) will be con- sidered for the two-dimensional case and we put .)()( 222 vuvu yyyxytxyxxxt   ISSN 1562-6016. ВАНТ. 2018. №3(115) 129 The Euler-Lagrange equations will have following form . )()( , )()( 222 222 ytyyxtxy yyxxxyyyxy ytyxxtxx yyxxxyyxxx uvv vuu         (6) To find the velocity field of the system (1), it is nec- essary to solve the system (6) under the appropriate boundary conditions. DATA PROCESSING CONSTRUCTION OF THE SPARSE SYSTEM OF SPECIAL FORM We consider the system (6), where Myx ),( and functions u, v are defined at the boundary of the region M. The dynamic data acquisition mode allows observ- ing radiopharmaceutical density distribution in the stud- ied system as a function of time [2]. As a result we ob- tain radiopharmaceutical density distribution as a func- tion of time and spatial coordinates ),,( yxt  ,  Tt ,0 , Myx ),( . Taking into account the discreti- zation with respect to time and spatial coordinates we get the sequence of matrices. We denote the density distribution of the radiophar- maceutical at the point located at the intersection of i-th line, j-th column, and k-th moment of time as ),,( ji .1,,0,  Nji  The solution of system (6) can be con- sidered at the nodes of a square grid with a step equal to the one pixel change in the distance along any axis. In the grid point ),( ji the approximation to the solution of system (6) can be written as ),(),,( jivjiu . Laplacians in (6) then can be changed with finite differences and partial derivatives of the second order can be calculated using the density values in the neighboring grid points according to the chosen scheme. So we obtain linear system of equations                        ).,(),(),(),( ),(),()),(),(( ),()),(),(4())1,( )1,(),1(),1(( ),,(),(),(),( ),(),()),(),(( ),()),(),(4())1,( )1,(),1(),1(( 222 2 222 2 jijijiji jiujijiji jivjijijiv jivjivjiv jijijiji jivjijiji jiujijijiu jiujiujiu ytyyxtxy xyyyxx yyxy ytyxxtxx xyyyxx yxxx         (7) .,,1, Nji  According to our assumption that the functions vu, are given on the boundary of the region, only 22N at the interior points of the grid are unknown in equations (7). Thus, we obtain a linear system of 22N equations, the solution of which gives an approximation to the so- lution of the system (6) at grid points. Further we will denote ijij vjivujiu  ),(,),( . As a result of the discretization of the system of par- tial differential equations (6), a linear system of differ- ence equations (7) was obtained, and then we consider its solution. Let introduce the following notation  Tnzzzzz ,...,,, 321 , where ,       s s s v u z ,),...,,...,,...,,,...,( ),...,,( 1221111 21 T NNNNN T n uuuuuu uuu  .),...,,...,,...,,,...,( ),...,,( 1221111 21 T NNNNN T n vvvvvv vvv  The system (7) can be written in the following form qHz  . (8) The matrix H and right-hand side and vector of unknowns are portioned as follows                                             nnnnnnn n n q q q z z z HHH HHH HHH      2 1 2 1 1,1 22221 11211 , (9) the partitioning of q and z into subvectors iq and iz of size 2 are identical and compatible with the partition- ing of H . Here H block matrix of size ,nn ,2Nn  with square blocks of second order        ssss ssss ss cb ba H , ),(),(4 222 jijia yxxxss   , )),,(),()(,( jijijib yyxxxyss   ),(),(4 222 jijiс yxyyss   , Nji ,1,  ,        sr sr sr c a H 0 0 , rs  , ,srsr ca  and nonzero ele- ments are equal to 2 ;        s s s e d q , 2,1, Nrs  , ,),...,,...,,...,,,...,( ),...,,( 1221111 21 T NNNNN T n dddddd ddd  here ),,(),(),(),( jijijijid ytyxxtxxji   Nji ,1,  . ,),...,,...,,...,,,...,( ),...,,( 1221111 21 T NNNNN T n eeeeee eee  here ),,(),(),(),( jijijijie ytyyxtxyji   Nji ,1,  . Fig. 1 shows the matrix H scheme for the mesh 66.                                                                                                     Fig. 1. Pattern of matrix associated with the 66 mesh ISSN 1562-6016. ВАНТ. 2018. №3(115) 130 System (8) is large sparse linear system, and we will solve it by iterative block Gauss-Seidel method [10]. BLOCK GAUSS-SEIDEL METHOD. THE CONVERGENCE OF METHOD Let us represent matrix H in following way FEDH  . (10) Matrix D block diagonal, E and F lower tri- angular and upper triangular block matrices, respective- ly.                nnH H H D 00 0 00 00 22 11     ,                    0 0 0 1,1 221 112 nnn n n HH HH HH FE     , note, that zeros here are for zero blocks of second order. Block Gauss-Seidel method for solving system (8), corresponding to introduced partition (9), has the fol- lowing form ,...,1,0,,1 ,11       kni qzHzHzH i n ij k jij k j n ij ij k iii (11) or taking into account (10) and using matrix notations qzFzED kk  1)( , ,...,2,1,0k . (12) For the convergence of the method it is sufficient that the conditions obtained in [11] be satisfied: .")" , 22 ,1 , 22 ) ,,1 ,0,0,0) 2 2 1 1 2 2 2 conditioneirreduciblblockc b ca a ca ssomeforandns b ca a ca b ns cabcaa ss ssss n ir r sr ssss n sr r ss ssss sr ssss ssssssssss                           (13) It is easy to verify that conditions (13) are satisfied for system (8), and thus the Gauss-Seidel method con- verges to a unique solution of system (8) for any initial approximation. METHOD APPLICATIONS FOR DATA PROCESSING IN RADIONUCLIDE STUDIES This method can be used to process radionuclide images. Important stages of processing radionuclide studies are motion correction [12 - 14], contour con- struction and analysis of the regions of interest (ROI). This method can be used to solve these problems. Figs. 2 - 5 give examples of the results of constructing the velocity field for radionuclide images. In Fig. 2, we see two images obtained in the study of the human hepatobiliary system in which there is a shift of the studied organ. In the region indicated by the dotted line, the velocity field has the form shown in Fig. 3. Fig. 2. Two radionuclide images of hepatobiliary system 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Fig. 3. Velocity field In Fig. 4, we see the images of region of interest at the moment of time k and 1k , these are fragments of images of cardiac radionuclide research. In Fig. 5 the velocity field constructed in this region is represented. 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Fig. 4. Images of region of interest (ROI) 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Fig. 5. Velocity field The data illustrate the possibility of using this meth- od for the motion detecting and its subsequent correc- tion, and also for constructing the contours of the inves- tigated objects on radionuclide images. CONCLUSIONS The method of constructing the velocity field pro- posed in the article can be used in solving the problems of data processing of radionuclide studies. The variants of application given in the article can be used for the analysis of radionuclide images, on their basis it is pos- sible to build algorithms of construction of contours of the investigated areas of interest, and also to carry out ISSN 1562-6016. ВАНТ. 2018. №3(115) 131 correction of motion for dynamic radionuclide research- es. This method can also be used in other problems, for example, in the study of dynamics of charged particles. REFERENCES 1. D.A. Ovsyannikov, E.D. Kotina. Determination of velocity field by given density distribution of charged particles // Problems of Atomic Science and Technology. Series “Nuclear Physics Investiga- tions”. 2012, № 3, p. 122-125. 2. E.D. Kotina. Data processing in radionuclide studies // Problems of Atomic Science and Technology. Se- ries “Nuclear Physics Investigations”. 2012, № 3, p. 195-198. 3. M.A. Arlychev, V.L. Novikov, A.V. Sidorov, A.M. Fialkovskii, E.D. Kotina, D.A. Ovsyannikov, V.A. Ploskikh. EFATOM Two-Detector One- Photon Emission Gamma Tomograph // Technical Physics. 2009, v. 54, № 10, p. 1539-1547. 4. E.D. Kotina, V.A Ploskikh. Data Processing and Quan- titation in Nuclear Medicine // Proceedings of Ru- PAC2012.http://accelconf.web.cern.ch/AccelConf/ru pac2012/, 2012, p. 526-528. 5. E.D. Kotina, V.A. Ploskikh, A.V Babin. Radionu- clide Methods Application in Cardiac Studies // Problems of Atomic Science and Technology. Series “Nuclear Physics Investigations”. 2013, № 6, p. 179-182. 6. A.N. Tikhonov, V.Y. Arsenin. Methods for Solving Ill-posed Problems. M: “Nauka”, 1979, p. 288. 7. D.A. Ovsyannikov, E.D. Kotina. Reconstruction of velocity field // Proceedings of ICAP2012, http://accelconf.web.cern.ch/AccelConf/ICAP2012, 2012, p. 256-258. 8. B.K.P. Horn, B.G. Schunck. Determining optical flow // Artificial intelligence. 1981, v. 17, p. 185-203. 9. E. Kotina, G. Pasechnaya. 3D velocity field for heart tomography // 2015 International Conference on “Stability and Control Processes” in Memory of V.I. Zubov, SCP 2015 – Proceedings 7342231, 2015, p. 646-647. 10. Y. Saad. Iterative methods for sparse linear systems. Philadelphia: Siam, 2003, 552 p. 11. E.D. Kotina. On convergence of block iterative methods // Izvestiya Irkutskogo gosudarstvennogo universiteta, 2012, v. 5, № 3, p. 41-55 (in Russian). 12. D.A. Ovsyannikov, E.D. Kotina, A.Yu. Shirokolobov. Mathematical Methods of Motion Correction in Ra- dionuclide Studies // Problems of Atomic Science and Technology. Series “Nuclear Physics Investiga- tions”. 2013, № 6, p. 137-140. 13. G. Germano, T. Chua, P. Kavanagh, et al. Detection and correction of patient motion in dynamic and static myocardial SPECT using a multi-detector camera // Journal of Nuclear Medicine. 1993, v. 34, p. 1394-1395. 14. R.D. Folks, D. Manatunga, E.V. Garcia, A.T. Taylor. Automated patient motion detection and correction in dynamic renal scintigraphy // J. Nucl Med Tech- nol. 2011, v. 39(2), p. 131-139. Article received 05.03.2018 МЕТОД ПОЛЯ СКОРОСТЕЙ ДЛЯ ОБРАБОТКИ ДАННЫХ РАДИОНУКЛИДНЫХ ИССЛЕДОВАНИЙ Е.Д. Котина, Д.А. Овсянников Радионуклидные методы являются одними из современных методов функциональной диагностики раз- личных органов и систем организма человека, которые требуют использования математических методов обработки и анализа данных, полученных в ходе исследования. Поэтому развитие современных методов обработки радионуклидных изображений является актуальной задачей. В статье рассматривается метод для обработки радионуклидных изображений, основанный на построении поля скоростей. Данный метод может применяться для коррекции движения, построения контуров, анализа радионуклидных изображений, также он может использоваться для анализа и формирования динамики заряженных частиц. МЕТОД ПОЛЯ ШВИДКОСТЕЙ ДЛЯ ОБРОБКИ ДАНИХ РАДІОНУКЛІДНИХ ДОСЛІДЖЕНЬ O.Д. Котіна, Д.А. Овсянников Радіонуклідні методи є одними з сучасних методів функціональної діагностики різних органів і систем організму людини, які вимагають використання математичних методів обробки та аналізу даних, отриманих у ході дослідження. Тому розвиток сучасних методів обробки радіонуклідних зображень є актуальним за- вданням. У статті розглядається метод для обробки радіонуклідних зображень, заснований на побудові поля швидкостей. Даний метод може застосовуватися для корекції руху, побудови контурів, аналізу радіонуклід- них зображень, також він може використовуватися для аналізу та формування динаміки заряджених части- нок.

Velocity field based method for data processing in radionuclide studies

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