New Potential of the. Positron-Emission Tomography

Transcription

1 International Journal of Modern Physis and Appliation 6; 3(: ISSN: New Potential of the Positron-Emission Tomography Andrey N. olobuev, Eugene S. Petrov, Peter I. Romanhu, Paul K. Kuznetsov 3 Department of Physis, Samara State Medial University, Samara, Russia Department of Operative Surgery, Samara State Medial University, Samara, Russia 3 Department of Eletri-Drive, Samara State Tehnial University, Samara, Russia address volobuev7@yandex.ru (A. N. olobuev Keywords Annihilation, Eletron, Positron, Photon, Doppler s Effet, Angular Photon Distribution, Positron-Emission Tomography (PET Reeived: February, 6 Aepted: Marh, 6 Published: Marh, 6 Citation Andrey N. olobuev, Eugene S. Petrov, Peter I. Romanhu, Paul K. Kuznetsov. New Potential of the Positron-Emission Tomography. International Journal of Modern Physis and Appliation. ol. 3, No., 6, pp Abstrat On the basis of the reason analysis of the photon angular distribution formed at the eletron and positrons annihilation the new potential of the positron-emission tomography for detetion of tissue biophysial mehanial parameters is investigated. It is shown that photon angular distribution at eletron-positron annihilation is onsequene of Doppler s effet in the referene frame of the eletron and positron mass enter. In the referene frame bound with eletron the photon angular distribution is absent. But it is replaed by the Doppler s shift of photon frequenies. The estimation of the offered method sensitivity for a finding of tissue parameters is arried out.. Introdution Positron-emission tomograph (PET is the advaned diagnosti devie used for searh tumors and its metastases at the earliest stages of their ourrene. However in our opinion of a positron-emission tomography potential are not exhausted yet. The purpose of present paper is to show as with help PET it is possible to find density of a tissue. The analysis of angular distribution of flying out photons by energy ħ ω (ħ - Plan's onstant, ω - photon frequeny at annihilation of a positron e and eletron e has great importane for high-grade use of the PET. Unfortunately the mehanism of annihilative proess e e ħ ω of the eletron and positron is unnown. P. Dira has been offered model of this proess. Aording to Dira [, ] the annihilation it is possible to present as transformation of the eletron from a state with positive energy to the state with negative energy. Aording to the Dira s theory vauum holes the positron it is the hole in the field of vauum. Interation of the eletron and positron i.e. them annihilation is a filling vauum hole by the eletron. Thus energy as two quantums of eletromagneti radiation is alloated.. Angular and Power Distributions of Annihilative Radiations Quantum-eletrodynamial alulations of the annihilative proess have been arried out enough for a long time. They were repeatedly heed and reheed, inluding authors of the artile. As a result of these alulations two formulas for the differential effetive setion of eletromagneti radiation quantums sattering in a solid angle dω have been found. The first formula on time has been found by Heitler [3]. This formula loos lie:

2 Andrey N. olobuev et al.: New Potential of the Positron-Emission Tomography e p p sin θ p sin θ ( π p ( p os θ p os θ (. ( The formula is written in designations [] where there is its detailed dedution. The so-alled rational system of units whih speed of light and Plan's onstant are equal to unit ħ is used. In this system of the units the energy, impulse and mass have the idential dimension. In the formula ( e there is eletron harge (or positron with an opposite sign, - the photon energy, p - the eletron impulse, θ - the angle between impulses of the eletron and one of the radiated photons. The formula ( is found under ondition of summation on all diretions of photons polarization. At the dedution ( the referene frame onneted to the enter of mass interating the eletron and positron is used whih the impulses of eletron and positron are equal on the module and are opposite on the diretion p p p. Impulses of photons also are equal on the module and opposite on the diretion [3, ]. We shall note that in this referene frame the ondition of both photons supervision are idential. Figure. Angular distribution of annihilative radiations. In fig. the distribution f ( θ plotted aording to dω the formula ( is shown. This distribution is plotted for the p.. Fator before braets in ( is aepted ratio equal to unit. The graph in fig. has illustrative harater p α 5 ~ sine atually as it will be shown below where α there is fine-struture onstant [5]. In this 37 ase the distribution there is pratially spherial dω symmetri. The seond formula whih represents frequeny or power distribution of the radiated quantums has been offered a little later by Feynman [6]: e ω ω ω ( ee dω ( 6π m p ( E m ω ω The formula ( is written down in designations [6]. As well as in the previous variant ( the rational system of units is used. In the formula ( e and e there are unit vetors of photons polarization radiated at annihilation, ω and ω - the frequenies of the radiated photons, m - the mass of eletron (or positron, p - the module of the positron impulse, E - its energy. The formula ( is similar to Klein Nishina s formula for Compton s effet [6, 7]. The main differene there are before the third and fourth addends in the square braets the signs hanged on opposite. The major distintive ondition of the formula ( dedution is use of other referene frame in omparison with the formula ( dedution. The formula ( was dedued in the referene frame in whih eletron is at rest, and positron moves. This referene frame as a whole is equivalent to the referene frame oupled with PET. Therefore we shall name this referene frame as laboratory. The eletrons in the substane researhed in PET basially are in the bound state. Positrons there are result a β - positron radioative deay of the shortly-lived radiopharmaeutial isotopes, for example С, O, N, F, P Therefore eletrons in laboratory referene frame it is possible to assume motionless (if to exlude haoti thermal movement of moleules. Both formulas ( and ( were dedued with the help of Feynman standard diagram tehnique and diagrams of seond order of the perturbations theory. However results of the dedutions essentially differ. First, the formula ( assumes rather omplex angular distribution of intensity I of annihilative radiation, sine ~ dp ~ IdΩ where dp there is energy flux of radiation dp through the area ds, intensity I. And this distribution ds is onneted only to the eletron (positron impulse. The angle θ is present only at the omplex with impulse p. In the formula ( the distint form photons angular distribution in the obvious ind is absent. Seond, the formula ( assumes the opportunity of photons various energy at annihilation that is forbidden by the formula ( dedution owing to. Therefore, first of all, there is a question what nature of angular distribution of the annihilative radiation intensity in (? Whether this distribution with annihilative proess i.e. transformation substane energy is onneted or that is defined by other effets? Whether the given angular distribution of photons will be ept at transition to other referene frame oupled, for example, to the PET?

3 International Journal of Modern Physis and Appliation 6; 3(: The Reasons of Angular and Power Distribution of the Annihilative Radiations For researh of the angular dependene reason of differential effetive setion ( we shall onsider intermediate expression of the dedution whih is not summarized yet on diretions of the photons polarization []: ( ( pe pe e ( p ( p ( p ( p, (3 8π p ( ee ( pe ( pe ( ee ( p ( p where and there are impulses of photons. ariables in square braets: an impulse of eletron, impulses of photons, unit vetors of photons polarization are written down as - vetors. The formula (3 is simple for transforming to the ind: ( pe ( pe e ( ee. ( 8π p ( p ( p ( p ( p Let's transit in ( to spatial vetors using a rule ab a b ab where a and b there are three-dimensional vetors, vetors. a and b - in our ase power omponents of - Transiting to three-dimensional vetors, and also taing into aount absene power omponents at polarizing - vetors e the expression ( it is possible to present as: ( pe ( pe e ( ee 8π p ( p ( p e ( pe ( pe 8π p p p ( e e. (5 At dedution (5 the ondition of photons flying in strit opposite diretions also is used. Taing into aount, and also aording to the energy onservation law с m (for learly evident it is entered inside braets the speed of light с in the p formula (5 we shall replae speed of eletron. In result we shall reeive:, where - e ( pe ( pe 8π p Let's transit in (6 to the laboratory referene frame bound with eletron. In this ase p, and it is possible to examine as speed of a positron movement. The same there onerns and to value p in fator before braets (p positron impulse. In the given referene frame the formula (6 beomes simpler: ( e e. (6 e 8π p We researh an auxiliary tas. ( e e. (7 Figure. Supervision of the photons whih was emitted by the moving partile. The observer who are taing plae in "motionless" (onneted with the Earth referene frame, fig., examines some partile moving with a speed whih in ertain moment of time radiated two quantums opposite direted. At

4 Andrey N. olobuev et al.: New Potential of the Positron-Emission Tomography the quantum frequeny is ω. The angle between speed of the partile and diretion of one quantum propagation is equal θ. In the observer diretion the partile has a omponent of speed. Due to Doppler s effet the quantum moving in the observer diretion will have the inreased frequeny [8]: ω ω. (8 For the quantum moving in an opposite diretion so-alled the red displaement of frequeny will be observed: ω ω. (9 Using (8 and (9 we shall find size of the omplex ω ω whih is inluded into the formula (: ω ω ( ω ω ω ω ω ω ω ω. ( Let's note that distintion in frequenies of quantums in the examined tas is determined by distintion in onditions of these quantums supervision: one quantum moves to the observer another leaves from him. In the formula (7 the onsidered auxiliary tas is atually realized. Thus the moving partile is meant as a positron, and the observer is on "motionless" eletron. Therefore substituting ( in (7 we shall find: e ( ee 8π p ω ω e ω ω ω ω 8π p ω ω e e. ( Let's note that at use of the formula ( we have atually refused the ondition. If in fator before braets in the formula ( to use E m ω the formulas ( and ( beome idential. We shall note one important point arising at transition from the referene frame of the eletron and positron mass enter to the referene frame bound with eletron or laboratory referene frame. If to divide the formula (9 on the formula (8 the result whih differs from the result reeived in monographies, for example, [9, ] turns out. At division (9 on (8 and aepting с we reeive: ω. ( ω In [9, ] the following ratio is offered: E p. (3 m Taing into aount E m and p m we find:. ( The formula ( differs from the formula ( a little. It is onneted by that the formula ( is reeived within the framewor of the first approximation of the perturbation theory. Therefore it is essentially inexat. The formula ( follows from exat formulas of Doppler s effet. Thus remaining only within the framewor of the first approximation of the perturbation theory it is impossible to establish equivalene of formulas ( and (. In summary we shall summarize the formula ( on photons polarization. Coming ba to polarizing -vetors with the aount e also using ( ee [], we shall find: e, e e ω ω Ω. (5 e ( ω ω dω 8π p ωω ( ee d 8π p ω ω The module used owing to standard use of the module of a ompound matrix element at the finding of the proess differential effetive setion [3].. Appliation of the Annihilative Radiations in a Positron-Emission Tomograph Taing into aount that in a positron-emission tomography the positrons speed is insignifiant, and also taing into aount (8 and (9 it is possible to write down: ωω ω ω. (6 os θ Substituting (6 in (5, we shall find: ( ω e 8π p ω, (7 Figure 3. The basi sheme of photons registration in the positron-emission tomography.

5 International Journal of Modern Physis and Appliation 6; 3(: 39-3 Let's find the frequenies differene of radiated photons, i.e. size ω ω ω, using formulas (8 and (9: ω ω ω ω. (8 On fig. 3 the basi sheme of photons registration in the positron-emission tomography [] is shown. If the angle θ, i.e. a positron moves on the line onneting detetors γ - radiation D and D, the differene of the photons frequenies ω will be greatest and the formula (8 will be transformed to the ind: ω ω max Taing into aount <<, we shall find: ω max ω. (9. ( The size ω an be found from the approahed equality ħ ω m. In this ase: ω max, ( Ż ħ 3 where Ż 3,8659 m there is Compton s length m of an eletron wave []. Let's estimate the size ωmax. The positron at disintegration used in PET a radiopharmaeutial element taes off with very big veloity lose by light speed. But twophoton annihilation it is possible only at small veloity of partiles by means of positronium formation [5]. At the big energy of the positron there originate mesons or hadrons. Therefore, for ourrene two-photon annihilation the positron should be slowed down very muh. The positronium energy is equal E m α. Believing the positron energy p Ep in eletron eletri field to equal half of m positronium energy we find the positron impulse from the α p expression m. As positron impulse is small 8 m p m α the positron veloity is equal α. Using сα ( we shall find ωmax, 57 s. The found Ż differene of frequenies is equivalent 3.75e. If to use energy of an annihilative photon 5e the reeived differene of frequenies maes.73 %. The given size ompletely is aeptable to measurement by modern detetors of γ -radiation. Earlier at the estimation of the real form of annihilative radiation distribution, fig., the ratio p α 5 ~ has been used. Really from the formula m p p p α. α follows 8 m m с The researhed objet is plaed in the ring of detetors. At the annihilation of a positron and eletron, taing plae in a point a, the two quantums energies ħ ω and ħ ω in opposite diretions radiated (Plan's onstant ħ used for learing. If the quantums flying on line A-A, are registered by detetors D and D simultaneously the point of quantums emission is in the middle between detetors D and D. Detetors in a ring from the point of Doppler s effet view in the referene frame bound with eletrons play a role of motionless observers. By number of the quantums whih are radiated in different diretions proess is spherial symmetri. Therefore the density of detetors in a ring should be uniform. However the quantum frequenies and onsequently also their energy depending on a diretion on the detetor (observer due to the Doppler s effet an be differ on size ω ω ω. Measuring the frequenies or energies differene of the quantums whih radiated opposite diretions also using the maximal value of this differene during measurement ωmax it is possible to find the speeds of positrons movement under the formula (. The positron veloity depends on mehanial parameters of a tissue through whih it moves: density, visosity, et. Therefore estimating the positron veloity it is possible to reeive the information on mehanial parameters of the tissue in tumor. This additional information an be reeived during diagnostis of an organism with the help of the positron-emission tomograph. 5. Conlusion By results of the arried out analysis we an draw the following onlusions. Formulas Heitler ( and Feynman ( it is adequate in different referene frames desribe sattering photons at annihilation of eletron and positron. In the laboratory referene frame bound with eletron the angular distribution of number photons is spherial symmetri however due to distintion in onditions of quantums supervision owing to Doppler s effet there is a distintion in frequenies of the radiated quantums. At transition in the referene frame bound to the enter of mass of eletron and positron the distintion in frequenies of the radiated quantums is redued in angular distribution of annihilative photons intensity whih also is onsequene of Doppler s effet. Investigating the angular distribution of eletromagneti radiation intensity at annihilation of a positron and eletron in the referene frame of their mass enter we researh not annihilation, and other physial phenomenon Doppler s effet whih aompanies with annihilative radiation. Hene

6 Andrey N. olobuev et al.: New Potential of the Positron-Emission Tomography the first not disappearing amendment of the perturbation theory reeived on the basis of a holes Dira s hypothesis does not result in onfirmation or denying of this hypothesis even if experiments onfirm angular distribution of the annihilative radiation intensity. Measuring the frequenies or energies differene of the quantums whih have flung out opposite diretions it is possible to find the speeds of positrons movement, see formula (. Taing into aount the size of positron veloity in the pathologial tissue through whih it moves it is possible to reeive the information on mehanial parameters of this tissue. Referenes [] Dira, P. A. M. Diretion in Physis, Edited by H. Hora and J. R. Shepansi, John Wiley & Sons, New Yor ( p. [] Thomson M. Modern Partile Physis, University Printing House, Cambridge (3, p [3] Heitler, W. Quantum Theory of Radiation, In. Lit., Mosow, (956 pp [] Bogolubov, N. N. & Shirov, D.. Introdution in the Theory of Quantum Fields, Siene, Mosow (976, pp [5] Berestetsij. B., Lifshits E. M., L. Pitaevsij L. P. Quantum Eletrodynamis, Siene, Mosow (989, p.. [6] Feynman, R. P. Quantum Eletrodynamis. A Leture Note, Boo House LIBROKOM, Mosow (9, pp [7] olobuev, A. N. & Tolstonogov, A. P. Malus Law for X-ray Radiation, Mosow (3, Journal of Surfae Investigation, X-ray, Synhrotron and Neutron Tehniques, ol. 7, No., pp [8] Landau, L. D. & Lifshits, E. M. Theory of Field, Siene, Mosow (967, p. 56. [9] Itzuson, C. & Zuber, J-B. Quantum Field Theory, World, Mosow (98, p. 8. [] Bjoren, J. D. & Drell, S. D. Relativisti Quantum Theory, Siene, Mosow (978, p. 38. [] olobuev, A. N. Bases of Medial and Biologial Physis, Textboo, Samara Publishing House, Samara (, p [] Javorsy, B. M. & Detlavs, A. A. Handboo on Physis, Siene, Mosow (99, p. 576.