Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Mathematical modeling of fluorescence diffuse optical imaging of cell membrane potential changes

Authors: Habib Ammari, Josselin Garnier and Laure Giovangigli
Journal: Quart. Appl. Math. 72 (2014), 137-176
MSC (2010): Primary 35R30, 35B30
DOI: https://doi.org/10.1090/S0033-569X-2013-01334-X
Published electronically: November 20, 2013
MathSciNet review: 3185136
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Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this paper is to provide a mathematical model for spatial distribution of membrane electrical potential changes by fluorescence diffuse optical tomography. We derive the resolving power of the imaging method in the presence of measurement noise. The proposed mathematical model can be used for cell membrane tracking with the resolution of the optical microscope.

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  • [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series, Vol. 55, 1964.
  • [2] Habib Ammari, Elena Beretta, Elisa Francini, Hyeonbae Kang, and Mikyoung Lim, Reconstruction of small interface changes of an inclusion from modal measurements II: the elastic case, J. Math. Pures Appl. (9) 94 (2010), no. 3, 322-339 (English, with English and French summaries). MR 2679030 (2011m:35412), https://doi.org/10.1016/j.matpur.2010.02.001
  • [3] Mathematical and statistical methods for imaging, Contemporary Mathematics, vol. 548, American Mathematical Society, Providence, RI, 2011. Papers from the NIMS Thematic Workshop held at Inha University, Incheon, August 10-13, 2010; Edited by Habib Ammari, Josselin Garnier, Hyeonbae Kang and Knut Sølna. MR 2868483 (2012h:65006)
  • [4] Habib Ammari, Pierre Garapon, François Jouve, Hyeonbae Kang, Mikyoung Lim, and Sanghyeon Yu, A New Optimal Control Approach for the Reconstruction of Extended Inclusions, SIAM J. Control Optim. 51 (2013), no. 2, 1372-1394. MR 3038017, https://doi.org/10.1137/100808952
  • [5] Habib Ammari, Josselin Garnier, Hyeonbae Kang, Mikyoung Lim, and Knut Sølna, Multistatic imaging of extended targets, SIAM J. Imaging Sci. 5 (2012), no. 2, 564-600. MR 2971173, https://doi.org/10.1137/10080631X
  • [6] H. Ammari, J. Garnier, and K. Sølna, Limited view resolving power of conductivity imaging from boundary measurements, SIAM J. Math. Anal. 45 (2013), 1704-1722.
  • [7] Habib Ammari and Hyeonbae Kang, Reconstruction of small inhomogeneities from boundary measurements, Lecture Notes in Mathematics, vol. 1846, Springer-Verlag, Berlin, 2004. MR 2168949 (2006k:35295)
  • [8] Habib Ammari and Hyeonbae Kang, Polarization and moment tensors, with applications to inverse problems and effective medium theory, Applied Mathematical Sciences, vol. 162, Springer, New York, 2007. MR 2327884 (2009f:35339)
  • [9] Habib Ammari, Hyeonbae Kang, Mikyoung Lim, and Habib Zribi, Conductivity interface problems. I. Small perturbations of an interface, Trans. Amer. Math. Soc. 362 (2010), no. 5, 2435-2449. MR 2584606 (2012f:35077), https://doi.org/10.1090/S0002-9947-09-04842-9
  • [10] Elena Beretta and Elisa Francini, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of thin inhomogeneities, Inverse problems: theory and applications (Cortona/Pisa, 2002) Contemp. Math., vol. 333, Amer. Math. Soc., Providence, RI, 2003, pp. 49-62. MR 2032006 (2005a:35263), https://doi.org/10.1090/conm/333/05953
  • [11] Elena Beretta, Elisa Francini, and Michael S. Vogelius, Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. A rigorous error analysis, J. Math. Pures Appl. (9) 82 (2003), no. 10, 1277-1301 (English, with English and French summaries). MR 2020923 (2004i:35021), https://doi.org/10.1016/S0021-7824(03)00081-3
  • [12] David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998. MR 1635980 (99c:35181)
  • [13] A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans, Optics Express, 15 (2007), 6696-6716.
  • [14] Marc Duruflé, Victor Péron, and Clair Poignard, Time-harmonic Maxwell equations in biological cells--the differential form formalism to treat the thin layer, Confluentes Math. 3 (2011), no. 2, 325-357. MR 2807112 (2012f:35520), https://doi.org/10.1142/S1793744211000345
  • [15] H. Egger, M. Freiberger, and M. Schlottbom, Analysis of forward and inverse models in fluorescence optical tomography, Aachen Institute for Advanced Study in Computational Engineering Science, November 2009.
  • [16] M. J. Eppstein, A.Godavarty, D. J. Hawrysz, R. Roy, and E. M. Sevick-Muraca, Influence of the refractive index-mismatch at the boundaries measured in fluorescence-enhanced frequency-domain photon migration imaging, Optics Express, 10 (2002), 653-662.
  • [17] R. Gowrishankar and J. C. Weaver, An approach to electrical modeling of single and multiple cells, Proc. Nat. Acad. Sci., 100 (2003), 3203-3208.
  • [18] D. Gross, L. M. Loew, and W. W. Webb, Optical imaging of cell membrane potential changes induced by applied electric fields, Biophysical J., 50 (1986), 339-348.
  • [19] C. L. Hutchinson, J. R. Lakowicz, and E. M. Sevick-Muraca, Fluorescence life-time based sensing in tissues: a computational study, Biophys. J., 68 (1995), 1574-1582.
  • [20] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473 (34 #3324)
  • [21] Abdessatar Khelifi and Habib Zribi, Asymptotic expansions for the voltage potentials with two-dimensional and three-dimensional thin interfaces, Math. Methods Appl. Sci. 34 (2011), no. 18, 2274-2290. MR 2861742 (2012j:35029), https://doi.org/10.1002/mma.1529
  • [22] Rainer Kress, Linear integral equations, 2nd ed., Applied Mathematical Sciences, vol. 82, Springer-Verlag, New York, 1999. MR 1723850 (2000h:45001)
  • [23] Dionisios Margetis and Nikos Savva, Low-frequency currents induced in adjacent spherical cells, J. Math. Phys. 47 (2006), no. 4, 042902, 18. MR 2226331 (2006m:78018), https://doi.org/10.1063/1.2190333
  • [24] V. A. Markel and J. C. Schotland, Inverse problem in optical diffusion tomography. II. Role of boundary conditions, J. Opt. Soc. Amer. A, 19 (2002), 558-566.
  • [25] V. A. Markel and J. C. Schotland, Multiple projection optical diffusion tomography with plane wave illumination, Phys. Med. Biol., 50 (2005), 2351-2364.
  • [26] A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, Fluorescence optical diffusion tomography, Applied Optics, 42 (2003), 3081-3094.
  • [27] Jean-Claude Nédélec, Acoustic and electromagnetic equations, integral representations for harmonic problems, Applied Mathematical Sciences, vol. 144, Springer-Verlag, New York, 2001. MR 1822275 (2002c:35003)
  • [28] V. Ntziachristos, Fluorescence molecular imaging, Annu. Rev. Biomed. Eng., 8 (2006), 1-33.
  • [29] M. A. O'Leary, D. A. Boas, X. D. Li, B. Chance, and A. G. Yodh, Fluorescence lifetime imaging in turbid media, Opt. Lett., 21 (1996), 158-160.
  • [30] M. S. Patterson, B. Chance, and B. C. Wilson, Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties, Appl. Opt., 28 (1989), 2331-2336.
  • [31] M. S. Patterson and B. W. Pogue, Mathematical model for time resolved and frequency-domain fluorescence spectroscopy in biological tissues, Appl. Opt., 33 (1994), 1963-1974.
  • [32] Clair Poignard, Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer, Math. Methods Appl. Sci. 31 (2008), no. 4, 443-479. MR 2387417 (2009a:35049), https://doi.org/10.1002/mma.923
  • [33] Clair Poignard, About the transmembrane voltage potential of a biological cell in time-harmonic regime, Mathematical methods for imaging and inverse problems, ESAIM Proc., vol. 26, EDP Sci., Les Ulis, 2009, pp. 162-179 (English, with English and French summaries). MR 2498146 (2010j:92046), https://doi.org/10.1051/proc/2009012
  • [34] C. Poignard, P. Dular, R. Perrussel, L. Krähenbühl, L. Nicolas, and M. Schatzman, Approximate conditions replacing thin layers, IEEE Trans. Mag., 44 (2008), 1154-1157.
  • [35] M. C. W. van Rossum and Th. M. Nieuwenhuizen, Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion, Rev. Modern Phys., 71 (1999), 313-371.
  • [36] R. Roy and E. M. Sevick-Muraca, Truncated Newton's optimization schemes for absorption and fluorescence optical tomography: Part I, theory and formulation, Optics Express, 4 (1999), 353-371.
  • [37] E. M. Sevick and C. L. Burch, Origin of phosphorescence signals reemitted from tissues, Opt. Lett., 19 (1994), 1928-1930.
  • [38] John C. Schotland, Path integrals and optical tomography, Mathematical and statistical methods for imaging, Contemp. Math., vol. 548, Amer. Math. Soc., Providence, RI, 2011, pp. 77-84. MR 2868489 (2012j:65307), https://doi.org/10.1090/conm/548/10837
  • [39] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44 #7280)
  • [40] D. J. Stephens and V. J. Allan, Light microscopy techniques for live cell imaging, Science, 300 (2003), 82-86.

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Additional Information

Habib Ammari
Affiliation: Department of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005 Paris, France
Email: habib.ammari@ens.fr

Josselin Garnier
Affiliation: Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris VII, 75205 Paris Cedex 13, France
Email: garnier@math.jussieu.fr

Laure Giovangigli
Affiliation: Department of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005 Paris, France
Email: laure.giovangigli@ens.fr

DOI: https://doi.org/10.1090/S0033-569X-2013-01334-X
Keywords: Resolving power, stability and resolution analysis, fluorescence diffuse optical tomography, cell tomography, cell membrane, electric field, layer potential techniques
Received by editor(s): April 13, 2012
Published electronically: November 20, 2013
Additional Notes: This work was supported by ERC Advanced Grant Project MULTIMOD–267184
Article copyright: © Copyright 2013 Brown University

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