Geodesic
On inverse reconstruction problems of erythrocyte size distribution in laser diffractometry
Matematičeskoe modelirovanie, Tome 29 (2017) no. 3, pp. 51-62.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we have analyzed inverse problems of erythrocyte size distribution reconstruction when laser diffractometry data is given. We consider two geometrical models of erythrocyte — flat disk and biconcave disk. We have found that Tikhonov regularization technique allows one to reconstruct correct erythrocyte size distributions in cases of normal blood, microcytoses and macrocytoses when using a priori information about smoothness, finiteness and positivity of the solution. We also considered the case when the inverse problem is solved assuming the shape of the cells to be flat disks and the diffraction pattern is calculated according to biconcave disk model. In this case, first three statistical moments of the erythrocyte size distributions are obtained with error which is directly proportional to value of flexure in biconcave disk model. In addition, inversion of the corresponding integral equation leads to a solution, which has generally the same shape as the true distribution but is shifted along the horizontal axis. This allows one to calculate true solution by using a priori information about average value of erythrocyte size distribution.
Keywords: inverse problem, laser diffractometry, erythrocyte, 1st kind integral equation, Tikhonov regularization method. 
@article{MM_2017_29_3_a4,
     author = {V. D. Ustinov},
     title = {On inverse reconstruction problems of erythrocyte size distribution in laser diffractometry},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {51--62},
     publisher = {mathdoc},
     volume = {29},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2017_29_3_a4/}
}
TY  - JOUR
AU  - V. D. Ustinov
TI  - On inverse reconstruction problems of erythrocyte size distribution in laser diffractometry
JO  - Matematičeskoe modelirovanie
PY  - 2017
SP  - 51
EP  - 62
VL  - 29
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2017_29_3_a4/
LA  - ru
ID  - MM_2017_29_3_a4
ER  - 
%0 Journal Article
%A V. D. Ustinov
%T On inverse reconstruction problems of erythrocyte size distribution in laser diffractometry
%J Matematičeskoe modelirovanie
%D 2017
%P 51-62
%V 29
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2017_29_3_a4/
%G ru
%F MM_2017_29_3_a4
V. D. Ustinov. On inverse reconstruction problems of erythrocyte size distribution in laser diffractometry. Matematičeskoe modelirovanie, Tome 29 (2017) no. 3, pp. 51-62. http://geodesic.mathdoc.fr/item/MM_2017_29_3_a4/

[1] J. D. Bessman, R. K. Johnson, “Erythrocyte volume distribution in normal and abnormal subjects”, Blood, 46:3 (1975), 369–379

[2] Ye Yang, Z. Zhang, X. Yang, J. H. Yeo, L. J. Jiang, D. Jiang, “Blood cell counting and classification by nonflowing laser light scattering method”, Journal of Biomedical Optics, 9:5 (2004), 995–1001 | DOI

[3] S. Yu. Nikitin, A. E. Lugovtsov, A. V. Priezzhev, V. D. Ustinov, “Relation between the diffraction pattern visibility and dispersion of particle sizes in an ektacytometer”, Quantum Electronics, 41:9 (2011), 843–846 | DOI

[4] S. Yu. Nikitin, A. V. Priezzhev, A. E. Lugovtsov, V. D. Ustinov, A. V. Razgulin, “Laser ektacytometry and evaluation of statistical characteristics of inhomogeneous ensembles of red blood cells”, Journal of Quantitave Spectroscopy and Radiative Transfer, 146 (2014), 365–375 | DOI

[5] S. Yu. Nikitin, A. V. Priezzhev, A. E. Lugovtsov, V. D. Ustinov, “Measuring skewness of red blood cell deformability distribution by laser ektacytometry”, Quantum Electr., 44:8 (2014), 774–778 | DOI

[6] J. S. Geert, A. G. Hoekstra, R. M. Heethaar, “Anomalous diffraction by arbitrarily oriented ellipsoids: applications in ektacytometry”, Applied Optics, 33:31 (1994), 7288–7296 | DOI

[7] K. S. Shifrin, Vvedenie v optiku okeana, Gidrometeoizdat, L., 1983, 280 pp.

[8] A. Doicu, T. Wriedt, Y. A. Eremin, Light Scattering by Systems of Particles. Null-Field Method with Discrete Sources: Theory and Programs, Springer, 2006, 333 pp. | Zbl

[9] M. A. Yurkin, A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations”, Journal of Quantitative Spectroscopy and Radiative Transfer, 112:13 (2011), 2234 | DOI

[10] J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing”, Applied Optics, 30:33 (1991), 4800–4817 | DOI

[11] T. Wriedt, J. Hellmers, E. Eremina, R. Schuh, “Light scattering by single erythrocyte: Comparison of different methods”, J. of Quantitative Spectroscopy Radiative Transfer, 100 (2006), 444–456 | DOI

[12] V. I. Dmitriev, E. V. Zakharov, Metod integralnykh uravnenii v vychislitelnoi elektrodinamike, Maks Press, M., 2008, 316 pp.

[13] S. Yu. Nikitin, A. V. Priezzhev, A. E. Lugovtsov, Advanced Optical flow Cytometry: Methods and Disease Diagnoses, ed. V. V. Tuchin, Wiley-VCH, Weinheim, 2011, 133 | DOI

[14] V. A. Trenogin, Funktsionalnyi analiz, Nauka, M., 1980, 496 pp.

[15] M. Bertero, E. R. Pike, “Particle Size Distributions from Fraunhofer Diffraction”, Optica Acta, 30:8 (1983), 1043–1049 | DOI | MR

[16] J. G. McWhirter, E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind”, J. Phys. A: Math. Gen., 11:9 (1978), 1729–1745 | DOI | MR | Zbl

[17] A. N. Tikhonov, V. V. Stepanov, A. G. Yagola, Numerical methods for solving incorrect problems, Kluwer Academic Publishers, Dordrecht, 1995, 253 pp.

[18] K. Miller, “Least Squares Methods for Ill-Posed Problems with a Prescribed Bound”, SIAM Journal on Mathematical Analysis, 1 (1970), 52–74 | DOI | MR | Zbl

[19] S. Iu. Nikitin, Iu. S. Iurchuk, V. D. Ustinov, “Lasernaia difraktometriia mazka krovi i izmerenie raspredeleniia eritrocitov po razmeram”, Physiologiia krovoobrashcheniia, VI Vserossiiskaia s mezhdunarodnym uchastiem shkola-konferenciia, Tezicy dokladov (Moskva, 2–5 fevralia 2016), MAKS Press, M., 2016, 117–118