Absorbing and scattering inhomogeneity detection

using TPSF conformal mapping

Potlov A.Yu., Abdulkareem S.N., Frolov S.V.,

Proskurin S.G.

Biomedical engineering, TSTU, Russia

http://bmt.tstu.ru/ spros@tamb.ru

Saratov Fall Meeting 2014

Objectives

Quantum Electronics (2011) p.402

A time-resolved method of direct optical inhomogeneity detection in turbid media before the image reconstruction is described.

The key feature of the method is time point spread functions (TPSF) acquisition followed by their conformal mapping into surfaces in the cylindrical coordinate system.

Inhomogeneities’ express detection can be used in optical mammography and premature babies’ brain diagnostics.

The described technique can be implemented using the same hardware as for standard time-resolved diffuse optical tomography (TR-DOT).

(a) (b)

  Distribution of the photons in homogeneous (a) and inhomogeneous (b) cylindrical objects through 0.75 ns after light pulse injection (blue arrow)

Photon density in a cylinderSimulation results

.Ω∈z,y,x∀),t,z,y,x(S=

=)t,z,y,x()z,y,x(+)t,z,y,x(∇)z,y,x(D-t∂

)t,z,y,x(∂c1

a2 φμφ

φ

Numerical simulationof TPSF

TPSFs were obtained numerically using the model of a drop, i.e., the radiation

pulse containing a fixed initial number of photons that appears in the object near

its surface and diffuses within the object, decaying exponentially and moving

mainly towards its centre.

According to the diffusion equation, photon density is described as follows:

Photonics and Lasers in Medicine (2013 ) p.139

,qz,y,x,z,y,x,0)z,y,x(n

)t,z,y,x(F)z,y,x(D2)t,z,y,x(

( )[ ]{ } 1-sa )z,y,x()z,y,x(g-1+)z,y,x(3=)z,y,x(D μμ

2

c

3

c0

)Qcos(-1

)Qcos(+1-R-12

=F

2

2

0

1+

1-

=R

medium

object

medium

object

ν

ν

ν

ν

object

mediumc arcsinQ

Boundary condition of the third kind (the Robin condition):

where,

Quantum Electronics (2014) sbm.

Software for numerical TPSF simulation

The above model was implemented as dedicated software in LabVIEW

Quantum Electronics (2014) p.174

Inhomogeneity localization and size

Quantum Electronics (2014) p.174

Comparison of theoretical and experimental TPSF

Quantum Electronics (2014) p.174

Inhomogeneity detection

Cartesian coordinate system

We obtain information from the tail of the time point spread function

(TPSF) corresponding to the late arriving photons (LAP) and visualize

it in three-dimensional surface.

in Cartesian frame

homogenous case TPSF converge to a plane in the

inhomogeneous case, the curves also form a plane, but with a

crevasse near position of the inhomogeneity.

Cartesian coordinate system

Three-dimensional representation of TPSF for homogeneous (a) and inhomogeneous (b) cases

Slide 12

(a) (b)

Photonics & Lasers in Medicine (2013) p.139

Conformal mapping to cylindrical coordinates system

To visualize the late arriving photons from all TPSF remove the leading

area:

isot2 T,,n2,nt T,,nT,Tt isotisot3

23p t,R\t,Rt,R where,

Cylindrical representation is made using:

)t,N360(R

)t,(R)t,(R

3p

3p3n

Cylindrical coordinate system

Three-dimensional conformal mapping of TPFS for homogeneous (a) and inhomogeneous (b) cases

(a) (b)

Quantum Electronics (2014) p.174

Normalized function is modified:

( ) )t,(Rt,R 3n3st N°360

=α

1)t,(R,K1)-)t,((R1

1)t,(R,1)t,(R

3n3n

3n3k

Then standard function is created:

where K can be any real number except zero.

,qR)t,(R *st3st

,qR)t,(R *k3k

The resulting function

is the conformal mapping into cylindrical coordinate system:

3

23

2

tarctg

tq

where,

Quantum Electronics (2014) sbm.

Using cylindrical system of coordinates in the homogenous case,

TPSF are represented by a right cylindrical surface.

In the inhomogeneous case, conformal mapping shows a

convexity

near the angle where absorbing heterogeneity is located.

Such 3D mapping provides quick real time detection of

inhomogeneities prior to inverse problem solution.

Conclusion

(a) (b)

  Distribution of the photons in a homogeneous (a) and inhomogeneous (b) spherical objects through 0.75 ns after light pulse injection (blue arrow)

Future workPhoton density in a sphere