Diffusion approximation of UPI in tissues

Photons that migrate in biological tissues can be described using numerical simulation by the Monte Carlo method or the analytical equation of radiation transfer (UPI). However, UPI is difficult to solve without applying simplifications (approximations). The standard method for simplifying the UPI is the diffusion approximation. The general solution of the diffusion equation for photons is obtained faster, but less accurately than the Monte Carlo method ^[1] .

Homogeneous case ^[2]

Absorbing Inhomogeneity ^[2]

Definitions

Dissipative Inhomogeneity ^[2]

UPI can mathematically model the transfer of photon energy within the tissue. The flow of radiation energy through a small area of an element in the radiation field can be characterized by the energy brightness :

 $L({\vec {r}},{\hat {s}},t)({\frac {W}{m^{2}sr}})$           $L$       $($             $r$       $\to$             $,$             $s$       $^$             $,$       $t$       $)$       $($           $W$           $m$         $2$           $s$       $r$             $)$           ${\ displaystyle L ({\ vec {r}}, {\ hat {s}}, t) ({\ frac {W} {m ^ {2} sr}})}$     $L({\vec {r}},{\hat {s}},t)({\frac {W}{m^{2}sr}})$   .

It is defined as the energy flux per unit area, per unit spatial angle, per unit time. Where

{\vec {r}}

{\ displaystyle {\ vec {r}}}

{\vec {r}}

denotes a radius vector,

{\hat {s}}

{\ displaystyle {\ hat {s}}}

\hat{s}

denotes the unit direction vector and

t

{\ displaystyle t}

t

indicates time. Other important physical quantities are based on determining the energy brightness of radiation. ^[1] Luminous flux density or intensity :

 $\Phi ({\vec {r}},t)=\int _{4\pi }L({\vec {r}},{\hat {s}},t)d\Omega ({\frac {W}{m^{2}}})$           $Φ$       $($             $r$       $\to$             $,$       $t$       $)$       $=$         $\int$         $four$       $π$           $L$       $($             $r$       $\to$             $,$             $s$       $^$             $,$       $t$       $)$       $d$       $Ω$       $($           $W$         $m$         $2$               $)$           ${\ displaystyle \ Phi ({\ vec {r}}, t) = \ int _ {4 \ pi} L ({\ vec {r}}, {\ hat {s}}, t) d \ Omega ({ \ frac {W} {m ^ {2}}}}$     $\Phi ({\vec {r}},t)=\int _{{4\pi }}L({\vec {r}},{\hat {s}},t)d\Omega ({\frac {W}{m^{2}}})$

Luminous flux ( fluence ):

 $F({\vec {r}})=\int _{-\infty }^{+\infty }\Phi ({\vec {r}},t)dt({\frac {J}{m^{2}}})$           $F$       $($             $r$       $\to$             $)$       $=$         $\int$         $-$       $\infty$           $+$       $\infty$           $Φ$       $($             $r$       $\to$             $,$       $t$       $)$       $d$       $t$       $($           $J$         $m$         $2$               $)$           ${\ displaystyle F ({\ vec {r}}) = \ int _ {- \ infty} ^ {+ \ infty} \ Phi ({\ vec {r}}, t) dt ({\ frac {J} { m ^ {2}}})}$

Energy flow :

{\vec {J}}({\vec {r}},t)=\int _{4\pi }{\hat {s}}L({\vec {r}},{\hat {s}},t)d\Omega ({\frac {W}{m^{2}}})

{\ displaystyle {\ vec {J}} ({\ vec {r}}, t) = \ int _ {4 \ pi} {\ hat {s}} L ({\ vec {r}}, {\ hat {s}}, t) d \ Omega ({\ frac {W} {m ^ {2}}})}

.

This is a vector that, similarly to the speed of light flux, indicates the direction of the energy flux.

Radiation Transfer Equation

UPI is a differential equation describing the brightness of radiation $L({\vec {r}},{\hat {s}},t)$ ${\ displaystyle L ({\ vec {r}}, {\ hat {s}}, t)}$ . This equation can be obtained using the law of conservation of energy . The UPI claims that a ray of light loses energy through divergence and attenuation (including absorption and scattering of a ray of light) and receives energy from light sources in the medium and scattering towards the ray. Coherence , polarization, and nonlinearity can be neglected. Optical properties such as refractive index $n$ ${\ displaystyle n}$ , absorption coefficient µ _a , scattering coefficient µ µ _s , scattering anisotropy parameter $g$ ${\ displaystyle g}$ are considered independent of time, but can vary in space. It is assumed that the scattering must be elastic. UPI ( Boltzmann equation ) is written in the form:

 ${\frac {\partial L({\vec {r}},{\hat {s}},t)/c}{\partial t}}=-{\hat {s}}\cdot \nabla L({\vec {r}},{\hat {s}},t)-\mu _{t}L({\vec {r}},{\hat {s}},t)+\mu _{s}\int _{4\pi }L({\vec {r}},{\hat {s}}',t)P({\hat {s}}'\cdot {\hat {s}})d\Omega '+S({\vec {r}},{\hat {s}},t)$                 $\partial$       $L$       $($             $r$       $\to$             $,$             $s$       $^$             $,$       $t$       $)$         $/$         $c$           $\partial$       $t$             $=$       $-$             $s$       $^$             $\cdot$       $\nabla$       $L$       $($             $r$       $\to$             $,$             $s$       $^$             $,$       $t$       $)$       $-$         $μ$         $t$           $L$       $($             $r$       $\to$             $,$             $s$       $^$             $,$       $t$       $)$       $+$         $μ$         $s$             $\int$         $four$       $π$           $L$       $($             $r$       $\to$             $,$               $s$       $^$             $'$         $,$       $t$       $)$       $P$       $($               $s$       $^$             $'$         $\cdot$             $s$       $^$             $)$       $d$         $Ω$       $'$         $+$       $S$       $($             $r$       $\to$             $,$             $s$       $^$             $,$       $t$       $)$           ${\ displaystyle {\ frac {\ partial L ({\ vec {r}}, {\ hat {s}}, t) / c} {\ partial t}} = - {\ hat {s}} \ cdot \ nabla L ({\ vec {r}}, {\ hat {s}}, t) - \ mu _ {t} L ({\ vec {r}}, {\ hat {s}}, t) + \ mu _ {s} \ int _ {4 \ pi} L ({\ vec {r}}, {\ hat {s}} ', t) P ({\ hat {s}}' \ cdot {\ hat { s}}) d \ Omega '+ S ({\ vec {r}}, {\ hat {s}}, t)}$

$c$ ${\ displaystyle c}$ is the speed of light in the tissue, determined taking into account the refractive index.
μ _t $=$ ${\ displaystyle =}$ μ _a + μ _{s is the} extinction coefficient.
$P({\hat {s}}',{\hat {s}})$ ${\ displaystyle P ({\ hat {s}} ', {\ hat {s}})}$ - phase function representing the probability of light scattering in the direction ${\hat {s}}'$ ${\ displaystyle {\ hat {s}} '}$ in spatial angle $d\Omega$ ${\ displaystyle d \ Omega}$ around ${\hat {s}}$ ${\ displaystyle {\ hat {s}}}$ . In most cases, the phase function depends only on the angle between the scattering direction ${\hat {s}}'$ ${\ displaystyle {\ hat {s}} '}$ and directions of incidence of light ${\hat {s}}$ ${\ displaystyle {\ hat {s}}}$ , i.e $P({\hat {s}}',{\hat {s}})=P({\hat {s}}'\cdot {\hat {s}})$ ${\ displaystyle P ({\ hat {s}} ', {\ hat {s}}) = P ({\ hat {s}}' \ cdot {\ hat {s}})}$ .
The scattering anisotropy parameter can be expressed as $g=\int _{4\pi }({\hat {s}}'\cdot {\hat {s}})P({\hat {s}}'\cdot {\hat {s}})d\Omega$ ${\ displaystyle g = \ int _ {4 \ pi} ({\ hat {s}} '\ cdot {\ hat {s}}) P ({\ hat {s}}' \ cdot {\ hat {s} }) d \ Omega}$ .
$S({\vec {r}},{\hat {s}},t)$ ${\ displaystyle S ({\ vec {r}}, {\ hat {s}}, t)}$ - describes the intensity of the light source.

Diffusion Theory

Assumptions

In UPI, six different independent variables are determined by the energy brightness of any space-time point ( $x$ ${\ displaystyle x}$ , $y$ ${\ displaystyle y}$ and $z$ ${\ displaystyle z}$ from ${\vec {r}}$ ${\ displaystyle {\ vec {r}}}$ polar angle $\theta$ ${\ displaystyle \ theta}$ and transition angle $\phi$ ${\ displaystyle \ phi}$ from ${\hat {s}}$ ${\ displaystyle {\ hat {s}}}$ and $t$ ${\ displaystyle t}$ ) Having put forward the corresponding hypotheses about the behavior of photons in a scattering medium, we can conclude that the number of independent variables can decrease. These assumptions lead to a diffusion theory (and to a diffusion equation) for photon migration. Two hypotheses allow the use of diffusion theory in UPI:

Regarding scattering events, there are very few absorption events. In addition, after numerous cases of scattering, several cases of absorption will occur, and the energy brightness will become almost isotropic. This hypothesis is sometimes called directional expansion.
Initially, the time of a significant change in current density is much longer than the transit time of the transport mean free path. Thus, the relative change in current density on the transport mean free path is much less than unity. This property is sometimes called temporary broadening.

It should be noted that both hypotheses are true only with a large surface reflection coefficient (mainly scattering) in the medium.

UPI in the diffuse approximation

Energy brightness can be expanded into a series of linearly independent spherical harmonics $Y$ ${\ displaystyle Y}$ _{n, m} . In diffusion theory, the energy brightness is most often taken as isotropic, therefore, only the isotropic component and the anisotropic component of the first order are used $L({\vec {r}},{\hat {s}},t)\approx \ \sum _{n=0}^{1}\sum _{m=-n}^{n}L_{n,m}({\vec {r}},t)Y_{n,m}({\hat {s}})$ ${\ displaystyle L ({\ vec {r}}, {\ hat {s}}, t) \ approx \ \ sum _ {n = 0} ^ {1} \ sum _ {m = -n} ^ {n } L_ {n, m} ({\ vec {r}}, t) Y_ {n, m} ({\ hat {s}})}$ Where $L$ ${\ displaystyle L}$ _{n, m is} the decomposition coefficient. The energy brightness is represented by 4 components, one for n = 0 (isotropic component) and 3 components for n = 1 (anisotropic component). Using the properties of spherical harmonics, determination of light flux density $\Phi ({\vec {r}},t)$ ${\ displaystyle \ Phi ({\ vec {r}}, t)}$ and determination of current density ${\vec {J}}({\vec {r}},t)$ ${\ displaystyle {\ vec {J}} ({\ vec {r}}, t)}$ isotropic and anisotropic components could be expressed as follows:

$L_{0,0}({\vec {r}},t)Y_{0,0}({\hat {s}})={\frac {\Phi ({\vec {r}},t)}{4\pi }}$ ${\ displaystyle L_ {0,0} ({\ vec {r}}, t) Y_ {0,0} ({\ hat {s}}) = {\ frac {\ Phi ({\ vec {r}} , t)} {4 \ pi}}}$
$\sum _{m=-1}^{1}L_{1,m}({\vec {r}},t)Y_{1,m}({\hat {s}})={\frac {3}{4\pi }}{\vec {J}}({\vec {r}},t)\cdot {\hat {s}}$ ${\ displaystyle \ sum _ {m = -1} ^ {1} L_ {1, m} ({\ vec {r}}, t) Y_ {1, m} ({\ hat {s}}) = { \ frac {3} {4 \ pi}} {\ vec {J}} ({\ vec {r}}, t) \ cdot {\ hat {s}}}$

Therefore, the energy brightness can be approximated as follows: ^[1]

L({\vec {r}},{\hat {s}},t)={\frac {1}{4\pi }}\Phi ({\vec {r}},t)+{\frac {3}{4\pi }}{\vec {J}}({\vec {r}},t)\cdot {\hat {s}}

{\ displaystyle L ({\ vec {r}}, {\ hat {s}}, t) = {\ frac {1} {4 \ pi}} \ Phi ({\ vec {r}}, t) + {\ frac {3} {4 \ pi}} {\ vec {J}} ({\ vec {r}}, t) \ cdot {\ hat {s}}}

Using this expression for energy brightness, the UPI can be written in scalar and vector form as follows (In the UPI, the integration of the term that describes the scattering is carried out in full spatial angle $4\pi$ ${\ displaystyle 4 \ pi}$ solid angle. UPI in vector form is multiplied by direction ${\hat {s}}$ ${\ displaystyle {\ hat {s}}}$ before evaluating it.)

{\frac {\partial \Phi ({\vec {r}},t)}{c\partial t}}+\mu _{a}\Phi ({\vec {r}},t)+\nabla \cdot {\vec {J}}({\vec {r}},t)=S({\vec {r}},t)

{\ displaystyle {\ frac {\ partial \ Phi ({\ vec {r}}, t)} {c \ partial t}} + \ mu _ {a} \ Phi ({\ vec {r}}, t) + \ nabla \ cdot {\ vec {J}} ({\ vec {r}}, t) = S ({\ vec {r}}, t)}

{\frac {\partial {\vec {J}}({\vec {r}},t)}{c\partial t}}+(\mu _{a}+\mu _{s}'){\vec {J}}({\vec {r}},t)+{\frac {1}{3}}\nabla \Phi ({\vec {r}},t)=0

{\ displaystyle {\ frac {\ partial {\ vec {J}} ({\ vec {r}}, t)} {c \ partial t}} + (\ mu _ {a} + \ mu _ {s} ') {\ vec {J}} ({\ vec {r}}, t) + {\ frac {1} {3}} \ nabla \ Phi ({\ vec {r}}, t) = 0}

In the UPI, the integration of the term that describes the scattering is carried out in the full solid angle 4π. UPI in vector form is multiplied by the direction until it is evaluated. The diffusion approximation can be applied only in cases where the reduced scattering coefficient is much larger than the absorption coefficient; as well as in cases where the size of the minimum layer thickness is comparable to several transport mean free paths.

Diffusion Equation

Note that, according to the second hypothesis of diffusion theory, the relative contribution of changes in current density ${\vec {J}}({\vec {r}},t)$ ${\ displaystyle {\ vec {J}} ({\ vec {r}}, t)}$ along one transport mean free path is negligible. In the vector representation of the diffusion theory, UPI reduces to Fick's law ${\vec {J}}({\vec {r}},t)={\frac {-\nabla \Phi ({\vec {r}},t)}{3(\mu _{a}+\mu _{s}')}}$ ${\ displaystyle {\ vec {J}} ({\ vec {r}}, t) = {\ frac {- \ nabla \ Phi ({\ vec {r}}, t)} {3 (\ mu _ { a} + \ mu _ {s} ')}}}$ , it determines the flux density in terms of the gradient of the particle transfer rate. Substituting Fick's law into the UPI scalar equation gives the diffusion equation: ^[1]

{\frac {\partial \Phi ({\vec {r}},t)}{c\partial t}}+\mu _{a}\Phi ({\vec {r}},t)-\nabla \cdot [D\nabla \Phi ({\vec {r}},t)]=S({\vec {r}},t)

{\ displaystyle {\ frac {\ partial \ Phi ({\ vec {r}}, t)} {c \ partial t}} + \ mu _ {a} \ Phi ({\ vec {r}}, t) - \ nabla \ cdot [D \ nabla \ Phi ({\ vec {r}}, t)] = S ({\ vec {r}}, t)}

$D={\frac {1}{3(\mu _{a}+\mu _{s}')}}$ ${\ displaystyle D = {\ frac {1} {3 (\ mu _ {a} + \ mu _ {s} ')}}}$ - diffusion coefficient ; μ ' _s $=(1-g)$ ${\ displaystyle = (1-g)}$ μ _s is the reduced coefficient. Note that in the diffusion equation there is no explicit dependence on the scattering coefficient. Instead, only the reduced scattering coefficient appears in the expression for $D$ ${\ displaystyle D}$ . It follows from this that diffusion does not depend on the anisotropy parameter, g, of the scattering medium, if the reduced scattering coefficient, μ's, remains constant. ^[one]

Solution of the diffusion equation

For various boundaries (for example, tissue layers) and locations of light sources, the diffusion equation can be solved by applying the appropriate boundary conditions and determining the characteristics of the source $S({\vec {r}},t)$ ${\ displaystyle S ({\ vec {r}}, t)}$ .

Point sources in infinite homogeneous media

This section presents the solution of the diffusion equation in the simple case of a pulsed point source for a homogeneous infinite medium. The characteristic of the radiation source in the diffusion equation is as follows: $S({\vec {r}},t,{\vec {r'}},t')=\delta ({\vec {r}}-{\vec {r'}})\delta (t-t')$ ${\ displaystyle S ({\ vec {r}}, t, {\ vec {r '}}, t') = \ delta ({\ vec {r}} - {\ vec {r '}}) \ delta (t-t ')}$ where ${\vec {r}}$ ${\ displaystyle {\ vec {r}}}$ , the coordinate of the point at which the light flux density is measured, ${\vec {r'}}$ ${\ displaystyle {\ vec {r '}}}$ source coordinate. Impulse peak is determined by time $t^{'}$ ${\ displaystyle t '}$ . To determine the light flux density, the diffusion equation is solved as follows:

$\Phi ({\vec {r}},t;{\vec {r'}},t)={\frac {c}{[4\pi Dc(t-t')]^{3/2}}}\exp \left[-{\frac {\mid {\vec {r}}-{\vec {r'}}\mid ^{2}}{4Dc(t-t')}}\right]\exp[-\mu _{a}c(t-t')]$ ${\ displaystyle \ Phi ({\ vec {r}}, t; {\ vec {r '}}, t) = {\ frac {c} {[4 \ pi Dc (t-t')] ^ {3 / 2}}} \ exp \ left [- {\ frac {\ mid {\ vec {r}} - {\ vec {r '}} \ mid ^ {2}} {4Dc (t-t')}} \ right] \ exp [- \ mu _ {a} c (t-t ')]}$

Factor $\exp \left[-\mu _{a}c(t-t')\right]$ ${\ displaystyle \ exp \ left [- \ mu _ {a} c (t-t ') \ right]}$ describes the exponential attenuation in the light flux density due to absorption in accordance with the Bouguer-Lambert-Beer law . The remaining factors represent source expansion due to scattering. Given the above solution, an arbitrary source can be characterized as a superposition of short pulsed point sources. By removing the time dependence from the diffusion equation, we obtain a time-independent solution for a point source

$S({\vec {r}})=\delta ({\vec {r}})$ ${\ displaystyle S ({\ vec {r}}) = \ delta ({\ vec {r}})}$ :

\Phi ({\vec {r}})={\frac {1}{4\pi Dr}}\exp(-\mu _{eff}r)

{\ displaystyle \ Phi ({\ vec {r}}) = {\ frac {1} {4 \ pi Dr}} \ exp (- \ mu _ {eff} r)}

$\mu _{\mathrm {eff} }={\sqrt {\frac {\mu _{a}}{D}}}$ ${\ displaystyle \ mu _ {\ mathrm {eff}} = {\ sqrt {\ frac {\ mu _ {a}} {D}}}}$ effective absorption coefficient , which shows the attenuation rate of the light flux density in space.

Boundary conditions

Luminous flux density at the border. The application of boundary conditions allows the use of diffusion equations to solve the problems of light propagation in media of limited size (where the boundary between the bioobject and the environment must be taken into account). In order to solve problems at the boundary, you need to understand what happens when photons in the medium reach its boundary (that is, the surface). The radiation directed inward to the medium and directionally integrated at the boundary is equal to the directionally integrated radiation at the boundary directed from the medium and multiplied by the reflection coefficient $R_{F}$ ${\ displaystyle R_ {F}}$ :

\int _{{\hat {s}}\cdot {\hat {n}}<0}L({\vec {r}},{\hat {s}},t){\hat {s}}\cdot {\hat {n}}d\Omega =\int _{{\hat {s}}\cdot {\hat {n}}>0}R_{F}({\hat {s}}\cdot {\hat {n}})L({\vec {r}},{\hat {s}},t){\hat {s}}\cdot {\hat {n}}d\Omega

{\ displaystyle \ int _ {{\ hat {s}} \ cdot {\ hat {n}} <0} L ({\ vec {r}}, {\ hat {s}}, t) {\ hat { s}} \ cdot {\ hat {n}} d \ Omega = \ int _ {{\ hat {s}} \ cdot {\ hat {n}}> 0} R_ {F} ({\ hat {s} } \ cdot {\ hat {n}}) L ({\ vec {r}}, {\ hat {s}}, t) {\ hat {s}} \ cdot {\ hat {n}} d \ Omega }

Where ${\hat {n}}$ ${\ displaystyle {\ hat {n}}}$ normal to the border, outward. The diffusion approximation gives an expression for radiation $L$ ${\ displaystyle L}$ in terms of light flux density $\Phi$ ${\ displaystyle \ Phi}$ and current density ${\vec {J}}$ ${\ displaystyle {\ vec {J}}}$ .

After substitution, we obtain an estimate for the above integrals:

{\frac {\Phi ({\vec {r}},t)}{4}}+{\vec {J}}({\vec {r}},t)\cdot {\frac {\hat {n}}{2}}=R_{\Phi }{\frac {\Phi ({\vec {r}},t)}{4}}-R_{J}{\vec {J}}({\vec {r}},t)\cdot {\frac {\hat {n}}{2}}

{\ displaystyle {\ frac {\ Phi ({\ vec {r}}, t)} {4}} + {\ vec {J}} ({\ vec {r}}, t) \ cdot {\ frac { \ hat {n}} {2}} = R _ {\ Phi} {\ frac {\ Phi ({\ vec {r}}, t)} {4}} - R_ {J} {\ vec {J}} ({\ vec {r}}, t) \ cdot {\ frac {\ hat {n}} {2}}}

$R_{\Phi }=\int _{0}^{\pi /2}2\sin \theta \cos \theta R_{F}(\cos \theta )d\theta$ ${\ displaystyle R _ {\ Phi} = \ int _ {0} ^ {\ pi / 2} 2 \ sin \ theta \ cos \ theta R_ {F} (\ cos \ theta) d \ theta}$
$R_{J}=\int _{0}^{\pi /2}3\sin \theta (\cos \theta )^{2}R_{F}(\cos \theta )d\theta$ ${\ displaystyle R_ {J} = \ int _ {0} ^ {\ pi / 2} 3 \ sin \ theta (\ cos \ theta) ^ {2} R_ {F} (\ cos \ theta) d \ theta}$

We do the substitution using Fick's law ( ${\vec {J}}({\vec {r}},t)=-D\nabla \Phi ({\vec {r}},t)$ ${\ displaystyle {\ vec {J}} ({\ vec {r}}, t) = - D \ nabla \ Phi ({\ vec {r}}, t)}$ ), at a distance from the border, z = 0, we obtain:

\Phi ({\vec {r}},t)=A_{z}{\frac {\partial \Phi ({\vec {r}},t)}{\partial z}}

{\ displaystyle \ Phi ({\ vec {r}}, t) = A_ {z} {\ frac {\ partial \ Phi ({\ vec {r}}, t)} {\ partial z}}}

$A_{z}=2D{\frac {1+R_{eff}}{1-R_{eff}}}$ ${\ displaystyle A_ {z} = 2D {\ frac {1 + R_ {eff}} {1-R_ {eff}}}}$
$R_{\mathrm {eff} }={\frac {R_{\Phi }+R_{J}}{2-R_{\Phi }+R_{J}}}$ ${\ displaystyle R _ {\ mathrm {eff}} = {\ frac {R _ {\ Phi} + R_ {J}} {2-R _ {\ Phi} + R_ {J}}}}$

Literature

LV Wang , HI Wu (2007). Biomedical Optics. Wiley. ISBN 978-0-471-74304-0 .

Notes

↑ ¹ ² ³ ⁴ ⁵ LV Wang and HI Wu. Biomedical Optics. - Wiley, 2007 .-- ISBN 978-0-471-74304-0 .
↑ ¹ ² ³ A.Yu. Potlov, SG Proskurin, SV Frolov ,. SFM'13 - Saratov Fall Meeting, 2013 (Neopr.) .

Links

A.Yu. Potlov, SG Proskurin, SV Frolov ,. 2014 Quantum Electron. 44,174 (neopr.) .

SG Proskurin ,. Quantum Electron. 41,402 (neopr.) . (2011)

A.Yu. Potlov, SG Proskurin, SV Frolov ,. SFM'13 - Saratov Fall Meeting, 2013 (Neopr.) .

[Wang_BioMedBooK-1] ¹ ² ³ ⁴ ⁵ LV Wang and HI Wu. Biomedical Optics. - Wiley, 2007 .-- ISBN 978-0-471-74304-0 .

[SFM13-2] ¹ ² ³ A.Yu. Potlov, SG Proskurin, SV Frolov ,. SFM'13 - Saratov Fall Meeting, 2013 (Neopr.) .