Abstract

The dynamic response behavior of red blood cells holds the key to understanding red blood cell related diseases. In this regard, an understanding of the physiological functions of erythrocytes is significant before focusing on red blood cell aggregation in the microcirculatory system. In this work, we present a theoretical model for a photoacoustic signal that results when deformed red blood cells pass through a microfluidic channel. Using a Green’s function approach, the photoacoustic pressure wave is obtained analytically by solving a combined Navier-Stokes and photoacoustic equation system. The photoacoustic wave expression includes determinant parameters for the cell deformability such as plasma viscosity, density, and red blood cell aggregation, as well as involving laser parameters such as beamwidth, pulse duration, and repetition rate. The effects of aggregation on blood rheology are also investigated. The results presented by this study show good agreements with the experimental ones in the literature. The comprehensive analytical solution of the extended photoacoustic transport model including a modified Morse type potential function sheds light on the dynamics of aggregate formation and demonstrates that the profile of a photoacoustic pressure wave has the potential for detecting and characterizing red blood cell aggregation.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (1)

2017 (4)

X. Bai, Y. Liang, H. Sun, L. Jin, J. Ma, B.-O. Guan, and L. Wang, “Sensitivity characteristics of broadband fiber-laser-based ultrasound sensors for photoacoustic microscopy,” Opt. express 25, 17616–17626 (2017).
[Crossref] [PubMed]

R. K. Saha, S. Karmakar, A. Adhikari, and M. C. Kolios, “Photoacoustic field calculation for nonspherical axisymmetric fluid particles,” Biomed. Phys. Eng. Express 3, 015017 (2017).
[Crossref]

D. Biswas, S. Vasudevan, G. C. Chen, P. Bhagat, N. Sharma, and S. Phatak, “Time–frequency based photoacoustic spectral response technique for differentiating human breast masses,” Biomed. Phys. Eng. Express 3, 035002 (2017).
[Crossref]

A. Yazdani, H. Li, J. D. Humphrey, and G. E. Karniadakis, “A general shear-dependent model for thrombus formation,” PLOS Comput. Biol. 13, e1005291 (2017).
[Crossref] [PubMed]

2016 (6)

T. Wang, U. Rongin, and Z. Xing, “A micro-scale simulation of red blood cell passage through symmetric and asymmetric bifurcated vessels,” Sci. Rep. 6, 20262 (2016).

L. Xiao, Y. Liu, S. Chen, and B. Fu, “Simulation of deformation and aggregation of two red blood cells in a stenosed microvessel by dissipative particle dynamics,” Cell Biochem. Biophys. 74, 513–525 (2016).
[Crossref] [PubMed]

E. M. Strohm, M. J. Moore, and M. C. Kolios, “Single cell photoacoustic microscopy: a review,” IEEE J. Sel. Top. Quantum Electron. 22, 137–151 (2016).
[Crossref]

T. Feng, Q. Li, C. Zhang, G. Xu, L. J. Guo, J. Yuan, and X. Wang, “Characterizing cellular morphology by photoacoustic spectrum analysis with an ultra-broadband optical ultrasonic detector,” Opt. Express 24, 19853–19862 (2016).
[Crossref] [PubMed]

H. S. Salehi, H. Li, A. Merkulov, P. D. Kumavor, H. Vavadi, M. Sanders, A. Kueck, M. A. Brewer, and Q. Zhu, “Coregistered photoacoustic and ultrasound imaging and classification of ovarian cancer: ex vivo and in vivo studies,” J. Biomed. Opt. 21, 046006 (2016).
[Crossref]

E. Aytac-Kipergil, A. Demirkiran, N. Uluc, S. Yavas, T. Kayikcioglu, S. Salman, S. G. Karamuk, F. O. Ilday, and M. B. Unlu, “Development of a fiber laser with independently adjustable properties for optical resolution photoacoustic microscopy,” Sci. reports 6, 38674 (2016).
[Crossref]

2015 (2)

E. M. Strohm and M. C. Kolios, “Classification of blood cells and tumor cells using label-free ultrasound and photoacoustics,” Cytom. Part A 87, 741–749 (2015).
[Crossref]

G. He, B. Li, and S. Yang, “In vivo imaging of a single erythrocyte with high-resolution photoacoustic microscopy,” Front. Optoelectron. 8, 122–127 (2015).
[Crossref]

2014 (9)

T. Wang, S. Nandy, H. S. Salehi, P. D. Kumavor, and Q. Zhu, “A low-cost photoacoustic microscopy system with a laser diode excitation,” Biomed. Opt. Express 5, 3053–3058 (2014).
[Crossref] [PubMed]

D. Kilinc and G. U. Lee, “Advances in magnetic tweezers for single molecule and cell biophysics,” Integr. Biol. 6, 27–34 (2014).
[Crossref]

W. Song, W. Zheng, R. Liu, R. Lin, H. Huang, X. Gong, S. Yang, R. Zhang, and L. Song, “Reflection-mode in vivo photoacoustic microscopy with subwavelength lateral resolution,” Biomed. Opt. Exp. 5, 4235–4241 (2014).
[Crossref]

E. M. Strohm, I. Gorelikov, N. Matsuura, and M. C. Kolios, “Modeling photoacoustic spectral features of micron-sized particles,” Phys. Med. Biol. 59, 5795 (2014).
[Crossref] [PubMed]

Y. Li, H. Fang, C. Min, and X. Yuan, “Analytic theory of photoacoustic wave generation from a spheroidal droplet,” Opt. Exp. 22, 19953–19969 (2014).
[Crossref]

T. Ye, N. Phan-Thien, B. C. Khoo, and C. T. Lim, “Dissipative particle dynamics simulations of deformation and aggregation of healthy and diseased red blood cells in a tube flow,” Phys. Fluids 26, 111902 (2014).
[Crossref]

H. Erkol, E. Aytac-Kipergil, and M. B. Unlu, “Photoacoustic radiation force on a microbubble,” Phys. Rev. E 90, 023001 (2014).
[Crossref]

M. Brust, O. Aouane, M. Thiébaud, D. Flormann, C. Verdier, L. Kaestner, M. Laschke, H. Selmi, A. Benyoussef, T. Podgorski, and et al., “The plasma protein fibrinogen stabilizes clusters of red blood cells in microcapillary flows,” Sci. reports 4, 4348 (2014).
[Crossref]

A. G. Tsai, B. Y. Salazar Vázquez, P. Cabrales, E. B. Kistler, D. M. Tartakovsky, S. Subramaniam, S. A. Acharya, and M. Intaglietta, “Replacing the transfusion of 1–2 units of blood with plasma expanders that increase oxygen delivery capacity: Evidence from experimental studies,” J. functional biomaterials 5, 232–245 (2014).
[Crossref]

2013 (6)

C. L. Bayer, S. Y. Nam, Y.-S. Chen, and S. Y. Emelianov, “Photoacoustic signal amplification through plasmonic nanoparticle aggregation,” J. Biomed. Opt. 18, 016001 (2013).
[Crossref]

M. J. Simmonds, H. J. Meiselman, and O. K. Baskurt, “Blood rheology and aging,” JGC 10, 291 (2013).
[PubMed]

P. Steffen, C. Verdier, and C. Wagner, “Quantification of depletion-induced adhesion of red blood cells,” Phys. review letters 110, 018102 (2013).
[Crossref]

R. K. Saha, S. Karmakar, and M. Roy, “Photoacoustic response of suspended and hemolyzed red blood cells,” Appl. Phys. Lett. 103, 044101 (2013).
[Crossref]

E. M. Strohm, E. S. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics. 1, 49–53 (2013).
[Crossref]

E. M. Strohm, E. S. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105, 59–67 (2013).
[Crossref] [PubMed]

2012 (7)

E. Hysi, R. K. Saha, and M. C. Kolios, “On the use of photoacoustics to detect red blood cell aggregation,” Biomed. Opt. Express 3, 2326–2338 (2012).
[Crossref] [PubMed]

M. Diez-Silva, Y. Park, S. Huang, H. Bow, O. Mercereau-Puijalon, G. Deplaine, C. Lavazec, S. Perrot, S. Bonnefoy, M. S. Feld, and et al., “Pf155 resa protein influences the dynamic microcirculatory behavior of ring-stage plasmodium falciparum infected red blood cells,” Sci. Rep. 2, 614 (2012).
[Crossref]

R. K. Saha, S. Karmakar, and M. Roy, “Computational investigation on the photoacoustics of malaria infected red blood cells,” PLoS One 7, e51774 (2012).
[Crossref] [PubMed]

R. P. Solano, F. I. Ramirez-Perez, J. A. Castorena-Gonzalez, E. A. Anell, G. Gutiérrez-Juárez, and L. Polo-Parada, “An experimental and theoretical approach to the study of the photoacoustic signal produced by cancer cells,” AIP Adv. 2, 011102 (2012).
[Crossref]

E. Hysi, R. K. Saha, and M. C. Kolios, “Photoacoustic ultrasound spectroscopy for assessing red blood cell aggregation and oxygenation,” J. Biomed. Opt. 17, 125006 (2012).
[Crossref] [PubMed]

A. Y. Maklygin, A. V. Priezzhev, A. Karmenian, S. Y. Nikitin, I. Obolenskii, A. E. Lugovtsov, and K. Li, “Measurement of interaction forces between red blood cells in aggregates by optical tweezers,” Quantum Electron. 42, 500 (2012).
[Crossref]

A. Ohlinger, A. Deak, A. A. Lutich, and J. Feldmann, “Optically trapped gold nanoparticle enables listening at the microscale,” Phys. review letters 108, 018101 (2012).
[Crossref]

2011 (5)

R. K. Saha and M. C. Kolios, “A simulation study on photoacoustic signals from red blood cells,” J. Acoust. Soc. Am. 129, 2935–2943 (2011).
[Crossref] [PubMed]

R. K. Saha and M. C. Kolios, “Effects of erythrocyte oxygenation on optoacoustic signals,” J. Biomed. Opt. 16, 115003 (2011).
[Crossref] [PubMed]

H. Bow, I. V. Pivkin, M. Diez-Silva, S. J. Goldfless, M. Dao, J. C. Niles, S. Suresh, and J. Han, “A microfabricated deformability based flow cytometer with application to malaria,” Lab Chip. 11, 1065–1073 (2011).
[Crossref] [PubMed]

Y. Z. Yoon, J. Kotar, A. T. Brown, and P. Cicuta, “Red blood cell dynamics: from spontaneous fluctuations to non-linear response,” Soft Matter 7, 2042–2051 (2011).
[Crossref]

J. L. Maciaszek, B. Andemariam, and G. Lykotrafitis, “Microelasticity of red blood cells in sickle cell disease,” J. Strain Anal. Eng. Des. 46, 368–379 (2011).
[Crossref]

2010 (4)

D. A. Fedosov, B. Caswell, and G. E. Karniadakis, “A multiscale red blood cell model with accurate mechanics, rheology, and dynamics,” Biophys. J. 98, 2215–2225 (2010).
[Crossref] [PubMed]

G. A. Barabino, M. O. Platt, and D. K. Kaul, “Sickle cell biomechanics,” Annu. Rev. Biomed. Eng. 12, 345–367 (2010).
[Crossref] [PubMed]

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, and G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. 107, 6731–6736 (2010).
[Crossref] [PubMed]

H. C. Kwaan, “Role of plasma proteins in whole blood viscosity: a brief clinical review,” Clin. Hemorheol. Microcirc. 44, 167–176 (2010).
[PubMed]

2009 (4)

M. Musielak, “Red blood cell-deformability measurement: review of techniques,” Clin. Hemorheol. Microcirc. 42, 47–64 (2009).
[PubMed]

G. Diebold, “Photoacoustic monopole radiation: waves from objects with symmetry in one, two and three dimensions,” Photoacoust. imaging spectroscopy 144, 3–17 (2009).
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J. Zhang, P. C. Johnson, and A. S. Popel, “Effects of erythrocyte deformability and aggregation on the cell free layer and apparent viscosity of microscopic blood flows,” Microvasc. Res. 77, 265–272 (2009).
[Crossref] [PubMed]

M. Fenech, D. Garcia, H. J. Meiselman, and G. Cloutier, “A particle dynamic model of red blood cell aggregation kinetics,” Ann. Biomed. Eng 37, 2299–2309 (2009).
[Crossref] [PubMed]

2008 (5)

G. Késmárky, P. Kenyeres, M. Rábai, and K. Tóth, “Plasma viscosity: a forgotten variable,” Clin. Hemorheol. Microcirc. 39, 243–246 (2008).
[PubMed]

J. Zhang, P. C. Johnson, and A. S. Popel, “Red blood cell aggregation and dissociation in shear flows simulated by lattice boltzmann method,” J. Biomech. 41, 47–55 (2008).
[Crossref]

L. V. Wang, “Tutorial on photoacoustic microscopy and computed tomography,” IEEE J. Sel. Top. Quantum Electron. 14, 171–179 (2008).
[Crossref]

A. B. Karpiouk, S. R. Aglyamov, S. Mallidi, J. Shah, W. G. Scott, J. M. Rubin, and S. Y. Emelianov, “Combined ultrasound and photoacoustic imaging to detect and stage deep vein thrombosis: phantom and ex vivo studies,” J. Biomed. Opt. 13, 054061 (2008).
[Crossref] [PubMed]

A. B. Goins, H. Sanabria, and M. N. Waxham, “Macromolecular crowding and size effects on probe microviscosity,” Biophys. journal 95, 5362–5373 (2008).
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2007 (4)

R. J. Talbert, S. H. Holan, and J. A. Viator, “Photoacoustic discrimination of viable and thermally coagulated blood using a two-wavelength method for burn injury monitoring,” Phys. Med. Biol. 52, 1815 (2007).
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J. Li, G. Lykotrafitis, M. Dao, and S. Suresh, “Cytoskeletal dynamics of human erythrocyte,” Proc. Natl. Acad. Sci. 104, 4937–4942 (2007).
[Crossref] [PubMed]

A. R. Fisher, A. J. Schissler, and J. C. Schotland, “Photoacoustic effect for multiply scattered light,” Phys. Rev. E 76, 036604 (2007).
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J. Zhang, P. C. Johnson, and A. S. Popel, “An immersed boundary lattice boltzmann approach to simulate deformable liquid capsules and its application to microscopic blood flows,” Phys. Biol. 4, 285 (2007).
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2006 (5)

K.-i. Tsubuto, S. Wada, and T. Yamaguchi, “Simulation study on effects of hematocrit on blood flow properties using particle method,” J. Biomech. Sci. Eng. 1, 159–170 (2006).
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Y. Liu and W. K. Liu, “Rheology of red blood cell aggregation by computer simulation,” J. Comput. Phys. 220, 139–154 (2006).
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C. Dubus and J.-B. Fournier, “A gaussian model for the membrane of red blood cells with cytoskeletal defects,” Eur. Lett. 75, 181 (2006).
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I. Dulińska, M. Targosz, W. Strojny, M. Lekka, P. Czuba, W. Balwierz, and M. Szymoński, “Stiffness of normal and pathological erythrocytes studied by means of atomic force microscopy,” J. Biochem. Biophys. Meth. 66, 1–11 (2006).
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M. Fornal, M. Lekka, G. Pyka-Fościak, K. Lebed, T. Grodzicki, B. Wizner, and J. Styczeń, “Erythrocyte stiffness in diabetes mellitus studied with atomic force microscope,” Clin. Hemorheol. Microcirc. 35, 273–276 (2006).
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2005 (2)

F. Rico, P. Roca-Cusachs, N. Gavara, R. Farré, M. Rotger, and D. Navajas, “Probing mechanical properties of living cells by atomic force microscopy with blunted pyramidal cantilever tips,” Phys. Rev. E 72, 021914 (2005).
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H. H. Lipowsky, “Microvascular rheology and hemodynamics,” Microcirculation. 12, 5–15 (2005).
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2004 (2)

Y. Liu, L. Zhang, X. Wang, and W. K. Liu, “Coupling of navier–stokes equations with protein molecular dynamics and its application to hemodynamics,” Int. J. Numer. Methods Fluids 46, 1237–1252 (2004).
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A. Kamruzzahan, F. Kienberger, C. M. Stroh, J. Berg, R. Huss, A. Ebner, R. Zhu, C. Rankl, H. J. Gruber, and P. Hinterdorfer, “Imaging morphological details and pathological differences of red blood cells using tapping-mode afm,” Biol. Chem. 385, 955–960 (2004).
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2003 (3)

J. P. Shelby, J. White, K. Ganesan, P. K. Rathod, and D. T. Chiu, “A microfluidic model for single-cell capillary obstruction by plasmodium falciparum-infected erythrocytes,” Proc. Natl. Acad. Sci. 100, 14618–14622 (2003).
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M. Dao, C. T. Lim, and S. Suresh, “Mechanics of the human red blood cell deformed by optical tweezers,” J. Mech. Phys. Solids 51, 2259–2280 (2003).
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O. K. Baskurt and H. J. Meiselman, “Blood rheology and hemodynamics,” “Proc Thromb. Hemost.”  29, 435–450 (2003).
[Crossref]

2002 (1)

B. Neu and H. J. Meiselman, “Depletion-mediated red blood cell aggregation in polymer solutions,” Biophys. journal 83, 2482–2490 (2002).
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2001 (1)

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767–784 (2001).
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2000 (1)

R. M. Hochmuth, “Micropipette aspiration of living cells,” J. Biomech. 33, 15–22 (2000).
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1999 (2)

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, “A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers,” Biophys. J. 76, 1145–1151 (1999).
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A. R. Bausch, W. Möller, and E. Sackmann, “Measurement of local viscoelasticity and forces in living cells by magnetic tweezers,” Biophys. J. 76, 573–579 (1999).
[Crossref] [PubMed]

1998 (1)

D. E. Discher, D. H. Boal, and S. K. Boey, “Simulations of the erythrocyte cytoskeleton at large deformation. ii. micropipette aspiration,” Biophys. J. 75, 1584–1597 (1998).
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1995 (1)

R. A. Kruger, P. Liu, Y. Fang, C. R. Appledorn, and et al., “Photoacoustic ultrasound (paus)–reconstruction tomography,” Med. physics 22, 1605–1609 (1995).
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1994 (1)

E. Evans and A. Yeung, “Hidden dynamics in rapid changes of bilayer shape,” Chem. Phys. Lipids 73, 39–56 (1994).
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1993 (1)

T. Somer and H. J. Meiselman, “Disorders of blood viscosity,” Ann. Med. 25, 31–39 (1993).
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1991 (2)

F. Neumann, H. Katus, E. Hoberg, P. Roebruck, M. Braun, H. Haupt, H. Tillmanns, and W. Kübler, “Increased plasma viscosity and erythrocyte aggregation: indicators of an unfavourable clinical outcome in patients with unstable angina pectoris,” Heart. 66, 425–430 (1991).
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G. Nash, “Blood rheology and ischaemia,” Eye 5, 151 (1991).
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1990 (1)

T. Shiga, N. Maeda, and K. Kon, “Erythrocyte rheology,” Crit. Rev. Oncol. Hematol. 10, 9–48 (1990).
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1987 (2)

G. Lowe, “1 blood rheology in vitro and in vivo,” Baillière’s Clin. Haematol. 1, 597–636 (1987).
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T. Somer, “4 rheology of paraproteinaemias and the plasma hyperviscosity syndrome,” Baillière’s Clin. Haematol. 1, 695–723 (1987).
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1984 (1)

H. A. Cranston, C. W. Boylan, G. Carroll, S. P. Sutera, J. Williamson, I. Y. Gluzman, and D. J. Krogstad, “Plasmodium falciparum maturation abolishes physiologic red cell deformability,” Science 223, 400–404 (1984).
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1983 (1)

D. K. Kaul, M. Fabry, P. Windisch, S. Baez, and R. Nagel, “Erythrocytes in sickle cell anemia are heterogeneous in their rheological and hemodynamic characteristics,” J. Clin. Invest. 72, 22 (1983).
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1980 (1)

H. Schmid-Schönbein, “Blood rheology and physiology of microcirculation,” Ric. Clin. Lab. 11, 13–33 (1980).

1977 (1)

C. S. Peskin, “Numerical analysis of blood flow in the heart,” J. Comput. Phys. 25, 220–252 (1977).
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1975 (1)

E. A. Evans and P. L. La Celle, “Intrinsic material properties of the erythrocyte membrane indicated by mechanical analysis of deformation,” Blood. 45, 29–43 (1975).
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1972 (1)

C. S. Peskin, “Flow patterns around heart valves: a numerical method,” J. Comput. Phys. 10, 252–271 (1972).
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1963 (1)

L. Dintenfass, “Blood rheology in cardio-vascular diseases,” Nature. 199, 813–815 (1963).
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1957 (1)

V. M. Ingram, “Gene mutations in human haemoglobin: the chemical difference between normal and sickle cell haemoglobin,” Nature. 180, 326–328 (1957).
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A. G. Tsai, B. Y. Salazar Vázquez, P. Cabrales, E. B. Kistler, D. M. Tartakovsky, S. Subramaniam, S. A. Acharya, and M. Intaglietta, “Replacing the transfusion of 1–2 units of blood with plasma expanders that increase oxygen delivery capacity: Evidence from experimental studies,” J. functional biomaterials 5, 232–245 (2014).
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Adhikari, A.

R. K. Saha, S. Karmakar, A. Adhikari, and M. C. Kolios, “Photoacoustic field calculation for nonspherical axisymmetric fluid particles,” Biomed. Phys. Eng. Express 3, 015017 (2017).
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Aglyamov, S. R.

A. B. Karpiouk, S. R. Aglyamov, S. Mallidi, J. Shah, W. G. Scott, J. M. Rubin, and S. Y. Emelianov, “Combined ultrasound and photoacoustic imaging to detect and stage deep vein thrombosis: phantom and ex vivo studies,” J. Biomed. Opt. 13, 054061 (2008).
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Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, “The optical stretcher: a novel laser tool to micromanipulate cells,” Biophys. J. 81, 767–784 (2001).
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Andemariam, B.

J. L. Maciaszek, B. Andemariam, and G. Lykotrafitis, “Microelasticity of red blood cells in sickle cell disease,” J. Strain Anal. Eng. Des. 46, 368–379 (2011).
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Anell, E. A.

R. P. Solano, F. I. Ramirez-Perez, J. A. Castorena-Gonzalez, E. A. Anell, G. Gutiérrez-Juárez, and L. Polo-Parada, “An experimental and theoretical approach to the study of the photoacoustic signal produced by cancer cells,” AIP Adv. 2, 011102 (2012).
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Aouane, O.

M. Brust, O. Aouane, M. Thiébaud, D. Flormann, C. Verdier, L. Kaestner, M. Laschke, H. Selmi, A. Benyoussef, T. Podgorski, and et al., “The plasma protein fibrinogen stabilizes clusters of red blood cells in microcapillary flows,” Sci. reports 4, 4348 (2014).
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Appledorn, C. R.

R. A. Kruger, P. Liu, Y. Fang, C. R. Appledorn, and et al., “Photoacoustic ultrasound (paus)–reconstruction tomography,” Med. physics 22, 1605–1609 (1995).
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H. Erkol, E. Aytac-Kipergil, and M. B. Unlu, “Photoacoustic radiation force on a microbubble,” Phys. Rev. E 90, 023001 (2014).
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Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, and G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. 107, 6731–6736 (2010).
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D. K. Kaul, M. Fabry, P. Windisch, S. Baez, and R. Nagel, “Erythrocytes in sickle cell anemia are heterogeneous in their rheological and hemodynamic characteristics,” J. Clin. Invest. 72, 22 (1983).
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Bai, X.

Balwierz, W.

I. Dulińska, M. Targosz, W. Strojny, M. Lekka, P. Czuba, W. Balwierz, and M. Szymoński, “Stiffness of normal and pathological erythrocytes studied by means of atomic force microscopy,” J. Biochem. Biophys. Meth. 66, 1–11 (2006).
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Barabino, G. A.

G. A. Barabino, M. O. Platt, and D. K. Kaul, “Sickle cell biomechanics,” Annu. Rev. Biomed. Eng. 12, 345–367 (2010).
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Baskurt, O. K.

M. J. Simmonds, H. J. Meiselman, and O. K. Baskurt, “Blood rheology and aging,” JGC 10, 291 (2013).
[PubMed]

O. K. Baskurt and H. J. Meiselman, “Blood rheology and hemodynamics,” “Proc Thromb. Hemost.”  29, 435–450 (2003).
[Crossref]

Bausch, A. R.

A. R. Bausch, W. Möller, and E. Sackmann, “Measurement of local viscoelasticity and forces in living cells by magnetic tweezers,” Biophys. J. 76, 573–579 (1999).
[Crossref] [PubMed]

Bayer, C. L.

C. L. Bayer, S. Y. Nam, Y.-S. Chen, and S. Y. Emelianov, “Photoacoustic signal amplification through plasmonic nanoparticle aggregation,” J. Biomed. Opt. 18, 016001 (2013).
[Crossref]

Benyoussef, A.

M. Brust, O. Aouane, M. Thiébaud, D. Flormann, C. Verdier, L. Kaestner, M. Laschke, H. Selmi, A. Benyoussef, T. Podgorski, and et al., “The plasma protein fibrinogen stabilizes clusters of red blood cells in microcapillary flows,” Sci. reports 4, 4348 (2014).
[Crossref]

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A. Kamruzzahan, F. Kienberger, C. M. Stroh, J. Berg, R. Huss, A. Ebner, R. Zhu, C. Rankl, H. J. Gruber, and P. Hinterdorfer, “Imaging morphological details and pathological differences of red blood cells using tapping-mode afm,” Biol. Chem. 385, 955–960 (2004).
[Crossref] [PubMed]

Berndl, E. S.

E. M. Strohm, E. S. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics. 1, 49–53 (2013).
[Crossref]

E. M. Strohm, E. S. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105, 59–67 (2013).
[Crossref] [PubMed]

Best, C. A.

Y. Park, C. A. Best, K. Badizadegan, R. R. Dasari, M. S. Feld, T. Kuriabova, M. L. Henle, A. J. Levine, and G. Popescu, “Measurement of red blood cell mechanics during morphological changes,” Proc. Natl. Acad. Sci. 107, 6731–6736 (2010).
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D. Biswas, S. Vasudevan, G. C. Chen, P. Bhagat, N. Sharma, and S. Phatak, “Time–frequency based photoacoustic spectral response technique for differentiating human breast masses,” Biomed. Phys. Eng. Express 3, 035002 (2017).
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D. Biswas, S. Vasudevan, G. C. Chen, P. Bhagat, N. Sharma, and S. Phatak, “Time–frequency based photoacoustic spectral response technique for differentiating human breast masses,” Biomed. Phys. Eng. Express 3, 035002 (2017).
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Boal, D. H.

D. E. Discher, D. H. Boal, and S. K. Boey, “Simulations of the erythrocyte cytoskeleton at large deformation. ii. micropipette aspiration,” Biophys. J. 75, 1584–1597 (1998).
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Boey, S. K.

D. E. Discher, D. H. Boal, and S. K. Boey, “Simulations of the erythrocyte cytoskeleton at large deformation. ii. micropipette aspiration,” Biophys. J. 75, 1584–1597 (1998).
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M. Diez-Silva, Y. Park, S. Huang, H. Bow, O. Mercereau-Puijalon, G. Deplaine, C. Lavazec, S. Perrot, S. Bonnefoy, M. S. Feld, and et al., “Pf155 resa protein influences the dynamic microcirculatory behavior of ring-stage plasmodium falciparum infected red blood cells,” Sci. Rep. 2, 614 (2012).
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M. Rui, W. Bost, E. Weiss, R. Lemor, and M. C. Kolios, “Photoacoustic microscopy and spectroscopy of individual red blood cells,” in Biomedical Optics and 3-D Imaging, OSA Technical Digest (CD) (Optical Society of America, 2010), paper BSuD93.
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Figures (15)

Fig. 1
Fig. 1 The schematic of the model.
Fig. 2
Fig. 2 Aggregation force (F) corresponding to the various aggregation rates vs function of separation between cell surfaces (β(d−r0)) for (a) case 1 [65, 72]: D = 3 × 10−17 J/m2, β = 8 × 105 m−1, r0 = 3 μm; case 2 [65, 72]: D = 2.1 × 10−17 J/m2, β = 8 × 105 m−1, r0 = 2.5 μm; case 3 [65, 72]: D = 2.1 × 10−17 J/m2, β = 6 × 105 m−1, r0 = 2 μm, (b) Dextran 70 [74]: D = 5.75 × 10−17 J/m2, β = 6 × 105 m−1, r0 = 2 μm; Dextran 150 [74]: D = 56.4 × 10−17 J/m2, β = 8 × 105 m−1, r0 = 3 μm.
Fig. 3
Fig. 3 Normalized ∇·F (modified aggregation forces) vs function of separation between cell surfaces (β(d−r0)) for case 1 [65, 72]: D = 3 × 10−17 J/m2, β = 8 × 105 m−1, r0 = 3 μm; case 2 [65,72]: D = 2.1 × 10−17 J/m2, β = 8 × 105 m−1, r0 = 2.5 μm; case 3 [65,72]: D = 2.1 × 10−17 J/m2, β = 6 × 105 m−1, r0 = 2 μm.
Fig. 4
Fig. 4 Normalized ∇·F which represents the different red blood cell aggregation conditions with different intercellular strengths vs time scale of acoustic wave, t(s) for case 1 [65,72]: D = 3 × 10−17 J/m2, β = 8 × 105 m−1, r0 = 3 μm; case 2 [65,72]: D = 2.1 × 10−17 J/m2, β = 8 × 105 m−1, r0 = 2.5 μm; case 3 [65, 72]: D = 2.1 × 10−17 J/m2, β = 6 × 105 m−1, r0 = 2 μm.
Fig. 5
Fig. 5 Normalized PA wave p(r,t)/p0 generated by first source vs normalized time cst/R, an ultrasonic detector located at position r = 2R for the two different pulse durations (a) τ = 1 and (b) 5 ns with different beamwidths σ = 6 μm (dashed line) and 8 μm (solid line), where R = 8 μm.
Fig. 6
Fig. 6 Normalized power spectral density of the photoacoustic wave in Fig. 5, an ultrasonic detector located at the position r = 2R for the two different pulse durations (a) τ = 1 and (b) 5 ns with the different beamwidths σ = 6 μm (dashed line) and 8 μm (solid line), where R = 8 μm.
Fig. 7
Fig. 7 Normalized PA wave p(r,t)/p0 generated by first source vs time time (μs), an ultrasonic detector located at position r = 2 mm for the two different pulse durations (a) τ = 1 with different beamwidths σ = 8 μm (dashed line) and 6 μm (solid line), where R = 8 μm and (b) 5 ns with different beamwidths σ = 8 μm (dashed line) and 6 μm (solid line), respectively where R = 8 μm.
Fig. 8
Fig. 8 The effects of viscosity on normalized PA wave p(r,t)/p0 vs normalized time cst/R, an ultrasonic detector located at r = 2R for two different pulse durations (a)–(b) τ = 1, (c)–(d) 5 ns with different beamwidths σ = 6 (dashed line) and 8 μm (solid line), where R = 8 μm ((a)–(c) η=1.2 cP, ζ=6 cP, η′=7.6 cP and, (b)–(d) η=1.3 cP, ζ=6 cP, η′=7.7 cP) [64,85].
Fig. 9
Fig. 9 The effects of viscosity on normalized PA wave p(r,t)/p0 vs time (μs), an ultrasonic detector located at r = 500 μm for two different pulse durations (a)–(b) τ = 1 and (c)–(d) 3 ns with different beamwidths σ = 16 (dashed line) and 14 μm (solid line), where R = 16 μm ((a)–(c) η=1.2 cP, ζ=6 cP, η′=7.6 cP and, (b)–(d) η=1.3 cP, ζ=6 cP, η′=7.7 cP) [64,85].
Fig. 10
Fig. 10 The effects of viscosity on normalized PA wave p(r,t)/p0 generated by three sources for high level of aggregation force (case 1) (dashed line) and non-aggregation force (i.e., without force (wf)) (solid line) vs normalized time cst/R, an ultrasonic detector located at r = 2R, for the same pulse duration (a)–(b) τ = 5 ns with the same beamwidth σ = 8 μm for both cases, where R = 8 μm ((a) η=1.2 cP, ζ=6 mPa.s, η′=7.6 mPa.s and, (b) η=1.2 cP, ζ=20 mPa.s, η′=21.6 mPa.s) [64,85,87].
Fig. 11
Fig. 11 Normalized PA wave p(r,t)/p0 generated by three sources for case 2 (dashed line) and without force (wf) (solid line) vs normalized time cst/R, an ultrasonic detector located at r = 2R, for two different pulse durations (a) τ = 1 and (b) 5 ns with different beamwidths (i) and (ii) represent σ = 6 and 8 μm, respectively; where R = 8 μm (η=1.2 mPa.s, ζ=5 mPa.s, η′=6.6 mPa.s) [64,85].
Fig. 12
Fig. 12 Normalized PA wave p(r,t)/p0 generated by three sources p(r,t)/p0 for case 3 (dashed line) and without force (wf) (solid line) vs normalized time cst/R, an ultrasonic detector located at r = 2R, for two different pulse durations (a) τ = 1 and (b) 5 ns with different beamwidths (i) and (ii) represent σ = 6 and 8 μm, respectively; where R = 8 μm (η=1.2 mPa.s, ζ=4.5 mPa.s, η′=6.1 mPa.s) [64,85].
Fig. 13
Fig. 13 Normalized PA waves p(r,t)/p0 generated by three sources p(r,t)/p0 for without force (solid line), Dextran 150 (dashed line), and Dextran 70 (dashed line) regarding the chosen force at the values of function of between cell surfaces β(dr0) = (a) 0.4, (b) 0.6, (c) 0.8, (d) 1, (e) 1.2, (f) 1.4, (g) 1.6, (h) 1.8 vs normalized time cst/R. Here, an ultrasonic detector is located at r = 2R, or the pulse duration and the beamwidth are τ = 5 ns and σ = 8 μm, and the values of viscosity are η=6.3 mPa.s, ζ=10 mPa.s, η′=18.4 mPa.s for Dextran 70 [74,83,84], and η=6.3 mPa.s, ζ=22.9 mPa.s, η′=31.3 mPa.s for Dextran 150 [74,84], respectively.
Fig. 14
Fig. 14 Normalized PA wave p(r,t)/p0 generated by three sources p(r,t)/p0 for (a) Dextran 70 (14–23 pN) (dashed line), (b) Dextran 150 (43–169 pN) (solid line), (c) both Dextran 70 (dashed line), and Dextran 150 (solid line) vs the chosen force at the values of function of between cell surfaces, (d) aggregation force (F) corresponding to the various aggregation rates vs function of separation between cell surfaces (β(dr0)) for Dextran 70 [74,83,84] (dashed line) and Dextran 150 [74,84] (solid line). Here, an ultrasonic detector is located at r = 2R, or the pulse duration and the beamwidth are τ = 5 ns and σ = 6 μm, and the values of viscosity are η=6.3 mPa.s, ζ=10 mPa.s, η′=18.4 mPa.s for Dextran 70 [83], and η=6.3 mPa.s, ζ=22.9 mPa.s, η′=31.3 mPa.s for Dextran 150 [84], respectively.
Fig. 15
Fig. 15 Normalized PA wave p(r,t)/p0 generated by three sources p(r,t)/p0 for (a) case 1 (solid line), (b) case 3 (dashed line), (c) both case 1 (solid line) and case 3 (dashed line) vs the chosen force values, (d) aggregation force (F) corresponding to the various aggregation rates vs function of separation between cell surfaces (β(dr0)) for case 1 (solid line) and case 3 (dashed line) [65,72]. Here, an ultrasonic detector is located at r = 2R, for the pulse duration and the beamwidth are τ = 5 ns, σ = 8 μm, and the values of viscosity are η=1.2 mPa.s, ζ=6 mPa.s, η′=6.7 mPa.s for case 1, and η=1.2 mPa.s, ζ=4.5 mPa.s, η′=6.1 mPa.s for case 3, respectively [64,85].

Tables (4)

Tables Icon

Table 1 Model parameters [67,68,70].

Tables Icon

Table 2 Model parameters [65,72].

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Table 3 Constitutive parameters for the three various forces resulting from the RBC aggregation [65,72,73].

Tables Icon

Table 4 The model parameters for the measured dextran-induced aggregation forces of red blood cells [74].

Equations (51)

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ρ 0 c p T ( r , t ) t κ 2 T ( r , t ) = H ( r , t )
ρ t + ( ρ v ) = ρ β T ( r , t ) t
ρ [ v t + v ( v ) ] = p + η 2 v + ( ζ + 1 / 3 η ) ( v )
ρ t + ρ v = β H ( r , t ) c p .
δ ρ t + ρ 0 v = β H ( r , t ) c p
ρ 0 v t = δ p + η 2 v + ( ζ + 1 3 η ) ( v ) .
ρ 0 v t = δ p + η 2 v + ν ( v ) + f ( r , t ) ,
f ( r , t ) = n = 1 N [ F n Δ ( r Y n ( t ) ) ]
1 c s 2 2 p ( r , t ) t 2 = 2 p ( r , t ) + η ρ 0 c s 2 [ 2 p ( r , t ) t Γ 2 H ( r , t ) ] f ( r , t ) + β c p H ( r , t ) t
2 p ( r , t ) + η ρ 0 c s 2 2 p ( r , t ) t 1 c s 2 2 p ( r , t ) t 2 = S ( r , t )
S ( r , t ) = β c p H ( r , t ) t + Γ η ρ c s 2 2 H ( r , t ) + f ( r , t ) .
2 p ˜ ( r , ω ) + ω 2 p ˜ ( r , ω ) c s 2 ( 1 i ω B ) = S ˜ ( r , ω ) .
G ˜ ( r , r ; ω ) = 1 4 π | r r | exp ( i k 1 | r r | )
H ( r , t ) = A ( r ) I ( t ) .
A ( r ) = p 0 ( r ) c p c s 2 β .
p 0 ( r ) = β T ( r ) c κ
I ( t ) = exp ( t 2 / 2 τ 2 ) 2 π τ 2
S 1 ( r , t ) = p 0 ( r ) c s 2 t [ exp ( t 2 / 2 τ 2 ) 2 π τ 2 ] = 1 2 π τ 3 p 0 ( r ) c s 2 t exp ( t 2 / 2 τ 2 ) .
S ˜ 1 ( r , ω ) = 1 2 π τ 3 p 0 ( r ) c s 2 + t exp ( t 2 / 2 τ 2 ) exp ( i ω t ) d t = p 0 ( r ) i ω c s 2 exp ( τ 2 ω 2 / 2 ) .
S 2 ( r , t ) = η T ρ 0 c s 2 [ exp ( t 2 / 2 τ 2 ) 2 π τ 2 ] 2 p 0 ( r ) .
S ˜ 2 ( r , ω ) = η Γ ρ c s 2 2 p 0 ˜ ( r ) + exp ( t 2 / 2 τ 2 ) 2 π τ 2 exp ( i ω t ) d t = η Γ ρ c s 2 2 p 0 ˜ ( r ) exp ( τ 2 ω 2 / 2 ) .
f ( r , t ) = n = 1 N [ F n Δ ( r Y n ( t ) ) ]
Δ ( X ) = ( 2 π h 2 ) 3 / 2 exp [ ( r r ) 2 2 h 2 ] exp ( t 2 / 2 τ 2 ) 2 π
f ( r , t ) = 2 β D ( e 2 β ( r 0 d ) e β ( r 0 d ) ) ( 2 π h 2 ) 3 / 2 exp [ ( r r ) 2 2 h 2 ] exp ( t 2 / 2 τ 2 ) 2 π
p 0 ( r ) = p 0 exp ( r 2 2 σ 2 ) θ ( r ) θ ( r + R )
p ˜ ( r , ω ) = G ˜ ( r , r ; ω ) S ˜ ( r , ω ) d 3 r
p 1 ˜ ( r , ω ) = i p 0 4 π ω c s 2 exp ( τ 2 ω 2 2 ) 0 2 π d ϕ 0 R ( r ) 2 d r 1 + 1 exp [ i k 1 | r r | ] | r r | d μ
1 + 1 exp [ ( i k 1 ) ( r 2 + r 2 2 r r μ ) ] ( r 2 + r 2 2 r r μ ) d μ = 1 i r r k 1 [ exp [ i k 1 ( r r ) ] exp [ i k 1 ( r + r ) ] ] .
p 1 ( r , t ) = p 0 2 2 π r c s 2 0 R d r r exp ( r 2 2 σ 2 ) × exp [ τ 2 ω 2 2 + i k 1 ( r ± r ) i ω t ] [ c s 1 i ω B ω ] ω d ω ,
1 i ω B = 1 i B ω 2 + 1 8 ( B ω ) 2 + O ( ( B ω ) 3 ) ,
I = exp [ τ 2 ω 2 2 + ( i ω ( r ± r ) c s 1 i ω B ) i ω t ] d ω i B 2 ω exp [ τ 2 ω 2 2 + ( i ω ( r ± r ) c s 1 i ω B ) i ω t ] d ω .
exp [ i ω ( r ± r ) c s 1 i ω B ] ~ exp [ i ω ( r ± r ) c s ] B ω [ ω ( r ± r ) e i ω ( r ± r ) c s ] 2 c s .
p 1 ( r , t ) = p 0 2 2 π 1 r c s 0 R d r r exp ( r 2 2 σ 2 ) × { exp [ τ 2 ω 2 2 + ( i ω ( r ± r ) c s ) i ω t ] d ω i B 2 ω exp [ τ 2 ω 2 2 + ( i ω ( r ± r ) c s ) i ω t ] d ω } .
p 1 ( r , t ) = [ p 0 2 2 π 1 r c s ] [ J 1 a + i B 2 J 1 b ]
J 1 a = 0 R d r r exp ( r 2 2 σ 2 ) [ 2 π exp ( ( c s t + r r ) 2 2 τ 2 c s 2 ) τ 2 π exp ( ( c s t + r r ) 2 2 τ 2 c s 2 ) τ ]
J 1 b = 0 R d r r exp ( r 2 2 σ 2 ) [ i 2 π ( c s t + r + r ) exp ( ( c s t + r + r ) 2 2 τ 2 c s 2 ) τ 3 c s + i 2 π ( c s t + r r ) exp ( ( c s t + r r ) 2 2 τ 2 c s 2 ) τ 3 c s ] .
J 1 a = τ σ 2 c s 2 ( σ 2 + τ 2 c s 2 ) 3 / 2 exp [ 2 R ( r c s t ) + 2 ( r c s t ) 2 + R 2 2 τ 2 c s 2 R 2 2 σ 2 ] ( 2 π σ ( r c s t ) { erf [ σ 2 ( r + R + c s t ) + R τ 2 c s 2 2 τ σ c s σ 2 + τ 2 c s 2 ] × exp [ ( r + R c s t ) 2 2 τ 2 c s 2 + σ 2 ( r c s t ) 2 2 τ 2 c s 2 ( σ 2 + τ 2 c s 2 ) + R 2 2 σ 2 ] + erf [ σ 2 ( r + R c s t ) + R τ 2 c s 2 2 τ σ c s σ 2 + τ 2 c s 2 ] exp [ ( σ 2 ( r + R c s t ) + R τ 2 c s 2 ) 2 σ 2 ( σ 2 + τ 2 c s 2 ) + ( r c s t ) 2 2 τ 2 c s 2 ] ] 2 τ c s σ 2 + τ 2 c s 2 exp [ ( r c s t ) 2 2 τ 2 c s 2 ] exp [ 2 R ( r c s t ) τ 2 c s 2 ] 1 } ) × θ ( r | R c s t | ) θ ( r + R + c s t )
p 2 ( r , t ) = η T i ρ c s 2 0 R r d r 2 p 0 ( r ) × exp [ τ 2 ω 2 2 + i k 1 ( r ± r ) i ω t ] [ c s 1 i ω B ω ] d ω .
p 2 ( r , t ) = η Γ i ρ c s 0 R r d r 2 p 0 ( r ) × { exp [ τ 2 ω 2 2 + i ω ( r ± r ) c s i ω t ] ω d ω i B 2 exp [ τ 2 ω 2 2 + i ω ( r ± r ) c s i ω t ] d ω } .
J = exp [ τ 2 z 2 2 + i z ( r ± R ) c s i z t ] z d z ,
J = { π i lim z 0 z exp [ τ 2 z 2 2 + i z ( ( r ± R ) c s t ) ] z if ( r ± R c s t ) > 0 π i lim z 0 z exp [ τ 2 z 2 2 + i z ( r ± R c s t ) ] z if ( r ± R c s t ) < 0 .
p 2 ( r , t ) = B η Γ p 0 4 2 π ρ r c s 0 R d r r [ r 2 exp ( r 2 2 σ 2 ) σ 4 exp ( r 2 2 σ 2 ) σ 2 ] × [ 2 π exp [ ( c s t + r + r ) 2 2 τ 2 c s 2 ] τ 2 π exp [ ( c s t + r r ) 2 2 τ 2 c s 2 ] τ ] .
p 2 ( r , t ) = [ B η Γ p 0 4 2 π ρ r c s ] ( J 2 a J 2 b )
J 2 a = 0 R d r r [ ( r ) 2 exp ( r 2 2 σ 2 ) σ 4 exp ( r 2 2 σ 2 ) σ 2 ] × [ 2 π exp [ ( c s t + r r ) 2 2 τ 2 c s 2 ] τ ]
J 2 b = 0 R d r r [ ( r ) 2 exp ( r 2 2 σ 2 ) σ 4 exp ( r 2 2 σ 2 ) σ 2 ] × [ 2 π exp [ ( c s t + r r ) 2 2 τ 2 c s 2 ] τ ] .
p 3 ( r , t ) = c s τ 4 i 2 π r 0 R r d r f ( r , ω ) × { exp [ τ 2 ω 2 2 + i ω ( r ± r ) c s i ω t ] ω d ω i B 2 exp [ τ 2 ω 2 2 + i ω ( r ± r ) c s i ω t ] d ω } .
p 3 ( r , t ) = c s τ B 4 2 π r ( J 3 a J 3 b )
J 3 a = β D [ e 2 β ( r 0 d ) e β ( r 0 d ) ] π h 5 0 R d r r ( r r ) exp [ ( r r ) 2 2 h 2 ] × [ 2 π exp [ ( c s t + r + r ) 2 2 τ 2 c s 2 ] τ ]
J 3 b = β D [ e 2 β ( r 0 d ) e β ( r 0 d ) ] π h 5 0 R d r r ( r r ) exp [ ( r r ) 2 2 h 2 ] × [ 2 π exp [ ( c s t + r r ) 2 2 τ 2 c s 2 ] τ ] .
J 3 a = β D ( e β ( r 0 d ) 1 ) π τ h 5 × { τ h 2 c s 2 ( τ 2 c s 2 + h 2 ) 5 / 2 exp [ r ( r t c s ) τ 2 c s 2 + h 2 r 2 c s 2 + t 2 2 τ 2 + β ( r 0 d ) r 2 2 h 2 ] × [ 2 τ h 2 c s τ 2 c s 2 + h 2 ( t c s 2 r ) exp [ τ 2 r 2 c s + h 2 r t τ 4 c s 3 + τ 2 h 2 c s ] + exp ( τ 4 r 2 c s 4 + 2 τ 2 h 2 r t c s 3 + h 4 r 2 + h 4 t 2 c s 2 2 τ 4 h 2 c s 4 + 2 τ 2 h 4 c s 2 ) 2 π h × ( h 2 ( c s 2 ( τ 2 + t 2 ) + 2 r 2 3 r t c s ) + τ 2 c s 2 ( τ 2 c s 2 2 r 2 + r t c s ) ) × erf [ τ 2 r c s 2 h 2 ( r t c s ) 2 τ h c s τ 2 c s 2 + h 2 ] ] τ h 2 c s 2 ( τ 2 c s 2 + h 2 ) 5 / 2 exp [ r ( r t c s ) τ 2 c s 2 + h 2 r 2 + R 2 + t 2 c s 2 2 b 2 c s 2 + β ( r 0 d ) r 2 + R 2 2 h 2 ] × ( 2 τ c s τ 2 c s 2 + h 2 ( τ 2 R c s 2 + h 2 ( 2 r + R + t c s ) ) × exp ( r ( r t c s ) τ 2 c s 2 + h 2 + t c s ( r + R ) r R τ 2 c s 2 + r R h 2 ) + exp [ τ 4 c s 4 ( r 2 + R 2 ) + 2 τ 2 h 2 c s 2 ( r t c s + R 2 ) + h 4 ( r 2 + R 2 + t 2 c s 2 ) 2 τ 2 h 2 c s 2 ( τ 2 c s 2 + h 2 ) ] × 2 π h ( h 2 ( c s 2 ( τ 2 + t 2 ) + 2 r 2 3 r t c s ) + τ 2 c s 2 ( τ 2 c s 2 2 r 2 + r t c s ) ) × erf [ τ 2 c s 2 ( r R ) h 2 ( r + R t c s ) 2 τ h c s τ 2 c s 2 + h 2 ] ) θ ( r | R c s t | ) θ ( r + R + c s t ) } .
E = D ( e 2 β ( r 0 d ) 2 e β ( r 0 d ) ) .

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