1. 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

2. 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).

3. (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 cylinder Simulation results

4. Numerical simulation of 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

5. Boundary condition of the third kind (the Robin condition): where, Quantum Electronics (2014) sbm.

6. Software for numerical TPSF simulation The above model was implemented as dedicated software in LabVIEW Quantum Electronics (2014) p.174

7. Inhomogeneity localization and size Quantum Electronics (2014) p.174

8. Comparison of theoretical and experimental TPSF Quantum Electronics (2014) p.174

9. 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.

10. 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

11. Conformal mapping to cylindrical coordinates system To visualize the late arriving photons from all TPSF remove the leading area : where, Cylindrical representation is made using:

12. Cylindrical coordinate system Three-dimensional conformal mapping of TPFS for homogeneous (a) and inhomogeneous (b) cases (a) (b) Quantum Electronics (2014) p.174

13. Normalized function is modified: Then standard function is created: where K can be any real number except zero.

14. The resulting function is the conformal mapping into cylindrical coordinate system: where, Quantum Electronics (2014) sbm.

15. 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

16. (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 work Photon density in a sphere