2014, 8(3): 811-829. doi: 10.3934/ipi.2014.8.811

Approximate marginalization of unknown scattering in quantitative photoacoustic tomography

1. 

Department of Applied Physics, University of Eastern Finland, P.O.Box 1627, 70211 Kuopio, Finland, Finland

2. 

University of Eastern Finland, Department of Applied Physics, P.O.Box 1627, 70211 Kuopio

3. 

Department of Mathematics, University of Auckland, Private Bag 92019, Auckland Mail Centre, Auckland 1142

4. 

Department of Medical Physics and Bioengineering, University College London, Gower Street, London WC1E 6BT, United Kingdom

5. 

Department of Computer Science, University College London, Gower Street, London WC1E 6BT, United Kingdom

Received  May 2013 Revised  June 2014 Published  August 2014

Quantitative photoacoustic tomography is a hybrid imaging method, combining near-infrared optical and ultrasonic imaging. One of the interests of the method is the reconstruction of the optical absorption coefficient within the target. The measurement depends also on the uninteresting but often unknown optical scattering coefficient. In this work, we apply the approximation error method for handling uncertainty related to the unknown scattering and reconstruct the absorption only. This way the number of unknown parameters can be reduced in the inverse problem in comparison to the case of estimating all the unknown parameters. The approximation error approach is evaluated with data simulated using the diffusion approximation and Monte Carlo method. Estimates are inspected in four two-dimensional cases with biologically relevant parameter values. Estimates obtained with the approximation error approach are compared to estimates where both the absorption and scattering coefficient are reconstructed, as well to estimates where the absorption is reconstructed, but the scattering is assumed (incorrect) fixed value. The approximation error approach is found to give better estimates for absorption in comparison to estimates with the conventional measurement error model using fixed scattering. When the true scattering contains stronger variations, improvement of the approximation error method over fixed scattering assumption is more significant.
Citation: Aki Pulkkinen, Ville Kolehmainen, Jari P. Kaipio, Benjamin T. Cox, Simon R. Arridge, Tanja Tarvainen. Approximate marginalization of unknown scattering in quantitative photoacoustic tomography. Inverse Problems & Imaging, 2014, 8 (3) : 811-829. doi: 10.3934/ipi.2014.8.811
References:
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[2]

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S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen and M. Vauhkonen, Approximation errors and model reduction with an application in optical diffusion tomography,, Inverse Probl., 22 (2006), 175. doi: 10.1088/0266-5611/22/1/010.

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G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics,, Inv. Probl., 26 (2010). doi: 10.1088/0266-5611/26/8/085010.

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[32]

J. Jose, R. G. H. Willemink, W. Steenbergen, C. H. Slump, T. G. van Leeuwen and S. Mahonar, Speed-of-sound compensated photoacoustic tomography for accurate imaging,, Med. Phys., 39 (2012), 7262. doi: 10.1118/1.4764911.

[33]

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J. Kaipio and V. Kolehmainen, Approximate marginalization over modeling errors and uncertainties in inverse problems,, in Bayesian Theory and Applications (eds. P. Damien, (2013), 544.

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V. Kolehmainen, M. Schweiger, I. Nissil, T. Tarvainen, S. R. Arridge and J. P. Kaipio, Approximation errors and model reduction in three-dimensional diffuse optical tomography,, J. Opts. Soc. Am. A, 26 (2009), 2257. doi: 10.1364/JOSAA.26.002257.

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V. Kolehmainen, T. Tarvainen, S. R. Arridge and J. P. Kaipio, Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,, Int. J. Uncertain. Quantif., 1 (2011), 1. doi: 10.1615/Int.J.UncertaintyQuantification.v1.i1.10.

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X. Li and H. Jiang, Impact of inhomogeneous optical scattering coefficient distribution on recovery of optical absorption coefficient maps using tomographic photoacoustic data,, Phys. Med. Biol., 58 (2013), 999. doi: 10.1088/0031-9155/58/4/999.

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S. Norton and M. Linzer, Ultrasonic Reflectivity Imaging in Three Dimensions: Reconstruction with Spherical Transducer Arrays,, Ultrasonic Imaging, 1 (1979), 210. doi: 10.1016/0161-7346(79)90017-8.

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show all references

References:
[1]

M. Agranovsky and P. Kuchment, Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed,, Inv. Probl., 23 (2007), 2089. doi: 10.1088/0266-5611/23/5/016.

[2]

M. A. Anastasio, J. Zhang, D. Modgil and P. J. La Rivière, Application of inverse source concepts to photoacoustic tomography,, Inv. Probl., 23 (2007). doi: 10.1088/0266-5611/23/6/S03.

[3]

S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen and M. Vauhkonen, Approximation errors and model reduction with an application in optical diffusion tomography,, Inverse Probl., 22 (2006), 175. doi: 10.1088/0266-5611/22/1/010.

[4]

G. Bal, A. Jollivet and V. Jugnon, Inverse transport theory of photoacoustics,, Inv. Probl., 26 (2010). doi: 10.1088/0266-5611/26/2/025011.

[5]

G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics,, Inv. Probl., 26 (2010). doi: 10.1088/0266-5611/26/8/085010.

[6]

G. Bal and K. Ren, Multi-source quantitative photoacoustic tomography in a diffusive regime,, Inv. Probl., 27 (2011). doi: 10.1088/0266-5611/27/7/075003.

[7]

G. Bal and K. Ren, On multi-spectral quantitative photoacoustic tomography in diffusive regime,, Inv. Probl., 28 (2012). doi: 10.1088/0266-5611/28/2/025010.

[8]

B. Banerjee, S. Bagchi, R. M. Vasu and D. Roy, Quantitative photoacoustic tomography from boundary pressure measurements: noninterative recovery of optical absorption coefficient from the reconstructed absorbed energy map,, J. Opt. Soc. Am. A, 25 (2008), 2347.

[9]

P. Beard, Biomedical photoacoustic imaging,, Interface Focus, 1 (2011), 602. doi: 10.1098/rsfs.2011.0028.

[10]

S. Bu, Z. Liu, T. Shiina, K. Kondo, M. Yamakawa, K. Fukutani, Y. Someda and Y. Asao, Model-based reconstruction integrated with fluence compensation for photoacoustic tomography,, IEEE. Trans. Biomed. Eng., 59 (2012), 1354.

[11]

A. Buehler, A. Rosenthal, T. Jetzfellner, A. Dima, D. Razansky and V. Ntziachristos, Model-based optoacoustic inversions with incomplete projection data,, Med. Phys., 38 (2011), 1694. doi: 10.1118/1.3556916.

[12]

P. Burgholzer, G. J. Matt, M. Haltmeier and G. Paltauf, Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface,, Phys. Rev. E, 75 (2007). doi: 10.1103/PhysRevE.75.046706.

[13]

K. M. Case and P. F. Zweifel, Linear Transport Theory,, Addison-Wesley, (1967).

[14]

W. F. Cheong, S. A. Prahl and A. J. Welch, A review of the optical properties of biological tissues,, IEEE J. Quantum Electron, 26 (1990), 2166. doi: 10.1109/3.64354.

[15]

B. T. Cox, S. R. Arridge, K. P. Köstli and P. C. Beard, Two-dimensional quantitative photoacoustic image reconstruction of absorption distributions in scattering media by use of a simple iterative method,, Appl. Optics, 45 (2006), 1866. doi: 10.1364/AO.45.001866.

[16]

B. T. Cox, S. R. Arridge and P. C. Beard, Photoacoustic tomography with a limited-aperture planar sensor and a reverberant cavity,, Inv. Probl., 23 (2007). doi: 10.1088/0266-5611/23/6/S08.

[17]

B. T. Cox, S. R. Arridge and P. C. Beard, Estimating chromophore distributions from multiwavelength photoacoustic images,, J. Opt. Soc. Am. A, 26 (2009), 443. doi: 10.1364/JOSAA.26.000443.

[18]

B. T. Cox and B. E. Treeby, Artifact trapping during time reversal photoacoustic imaging for acoustically heterogeneous media,, IEEE Trans. Med. Imaging, 29 (2010), 387. doi: 10.1109/TMI.2009.2032358.

[19]

B. Cox, T. Tarvainen and S. Arridge, Multiple illumination quantitative photoacoustic tomography using transport and diffusion models,, Contemp. Math., 559 (2011), 1. doi: 10.1090/conm/559/11067.

[20]

D. Finch, S. K. Patch and Rakesh, Determining a Function from Its Mean Values Over a Family of Spheres,, SIAM J. Math. Anal., 35 (2004), 1213. doi: 10.1137/S0036141002417814.

[21]

D. V. Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball,, Inv. Probl., 22 (2006), 923. doi: 10.1088/0266-5611/22/3/012.

[22]

D. V. Finch, M. Haltmeier and Rakesh, Inversion of Spherical Means and the Wave Equation in Even Dimensions,, SIAM J. Math. Anal., 68 (2007), 392. doi: 10.1137/070682137.

[23]

D. V. Finch and Rakesh, The spherical mean value operator with centers on a sphere,, Inv. Probl., 23 (2007). doi: 10.1088/0266-5611/23/6/S04.

[24]

H. Gao, H. Zhao and S. Osher, Bregman Methods in Quantitative Photoacoustic Tomography,, UCLA CAM Report 10-42, (2010), 10.

[25]

H. Gao, H. Zhao and S. Osher, Quantitative photoacoustic tomography,, Lecture Notes in Math., 2035 (2012), 131. doi: 10.1007/978-3-642-22990-9_5.

[26]

M. Haltmeier, T. Schuster and O. Scherzer, Filtered backprojection for thermoacoustic computed tomography in spherical geometry,, Math. Meth. Appl. Sci., 28 (2005), 1919. doi: 10.1002/mma.648.

[27]

J. Heino, E. Somersalo and J. Kaipio, Compensation for geometric mismodelling by anisotropies in optical tomography,, Opt. Express, 13 (2005), 296. doi: 10.1364/OPEX.13.000296.

[28]

J. Heiskala, V. Kolehmainen, T. Tarvainen, J. P. Kaipio and S. R. Arridge, Approximation error method can reduce artifacts due to scalp blood flow in optical brain activation imaging,, J. Biomed. Opt., 17 (2012). doi: 10.1117/1.JBO.17.9.096012.

[29]

Y. Hristova, P. Kuchment and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media,, Inv. Probl., 24 (2008). doi: 10.1088/0266-5611/24/5/055006.

[30]

A. Ishimaru, Wave Propagation And Scattering In Random Media,, Academic Press, (1978).

[31]

X. Jin and L. V. Wang, Thermoacoustic tomography with correction for acoustic speed variations,, Phys. Med. Biol., 51 (2006), 6437. doi: 10.1088/0031-9155/51/24/010.

[32]

J. Jose, R. G. H. Willemink, W. Steenbergen, C. H. Slump, T. G. van Leeuwen and S. Mahonar, Speed-of-sound compensated photoacoustic tomography for accurate imaging,, Med. Phys., 39 (2012), 7262. doi: 10.1118/1.4764911.

[33]

J. Kaipio and E. Somersalo, Statistical And Computational Inverse Problems,, Springer, (2005).

[34]

J. Kaipio and E. Somersalo, Statistical inverse problems: Discretization, model reduction and inverse crimes,, J. Comput. Appl. Math, 198 (2007), 493. doi: 10.1016/j.cam.2005.09.027.

[35]

J. Kaipio and V. Kolehmainen, Approximate marginalization over modeling errors and uncertainties in inverse problems,, in Bayesian Theory and Applications (eds. P. Damien, (2013), 544.

[36]

V. Kolehmainen, M. Schweiger, I. Nissil, T. Tarvainen, S. R. Arridge and J. P. Kaipio, Approximation errors and model reduction in three-dimensional diffuse optical tomography,, J. Opts. Soc. Am. A, 26 (2009), 2257. doi: 10.1364/JOSAA.26.002257.

[37]

V. Kolehmainen, T. Tarvainen, S. R. Arridge and J. P. Kaipio, Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,, Int. J. Uncertain. Quantif., 1 (2011), 1. doi: 10.1615/Int.J.UncertaintyQuantification.v1.i1.10.

[38]

L. A. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform,, Inv. Probl., 23 (2007), 373. doi: 10.1088/0266-5611/23/1/021.

[39]

L. A. Kunyansky, A series solution and a fast algorithm for the inversion of the spherical mean Radon transform,, Inv. Probl., 23 (2007). doi: 10.1088/0266-5611/23/6/S02.

[40]

J. Laufer, B. Cox, E. Zhang and P. Beard, Quantitative determination of chromophore concentrations from 2D photoacoustic images using a nonlinear model-based inversion scheme,, Appl. Optics, 49 (2010), 1219. doi: 10.1364/AO.49.001219.

[41]

C. Li and L. V. Wang, Photoacoustic tomography and sensing in biomedicine,, Phys. Med. Biol., 54 (2009). doi: 10.1088/0031-9155/54/19/R01.

[42]

X. Li and H. Jiang, Impact of inhomogeneous optical scattering coefficient distribution on recovery of optical absorption coefficient maps using tomographic photoacoustic data,, Phys. Med. Biol., 58 (2013), 999. doi: 10.1088/0031-9155/58/4/999.

[43]

A. V. Mamonov and K. Ren, Quantitative photoacoustic imaging in radiative transport regime,, Communications in Mathematical Sciences, 12 (2014), 201. doi: 10.4310/CMS.2014.v12.n2.a1.

[44]

S. J. Matcher, M. Cope and D. T. Delpy, Use of the water absorption spectrum to quantify tissue chromophore concentration changes in near infrared spectroscopy,, Phys. Med. Biol., 39 (1994), 177. doi: 10.1088/0031-9155/39/1/011.

[45]

S. J. Matcher, C. E. Elwell, C. E. Cooper, M. Cope and D. T. Delpy, Performance comparison of several published tissue near-infrared spectroscopy algorithms,, Anal. Biochem., 227 (1995), 54. doi: 10.1006/abio.1995.1252.

[46]

F. Natterer, Photo-acoustic inversion in convex domains,, Inverse Probl. Imag., 6 (2012), 315. doi: 10.3934/ipi.2012.6.315.

[47]

S. Norton and M. Linzer, Ultrasonic Reflectivity Imaging in Three Dimensions: Reconstruction with Spherical Transducer Arrays,, Ultrasonic Imaging, 1 (1979), 210. doi: 10.1016/0161-7346(79)90017-8.

[48]

S. Norton, Reconstruction of a two-dimensional reflecting medium over a circular domain: exact solution,, J. Acoust. Soc. Am., 67 (1980), 1266. doi: 10.1121/1.384168.

[49]

S. J. Norton and M. Linzer, Ultrasonic Reflectivity Imaging in Three Dimensions: Exact Inverse Scattering Solutions for Plane, Cylindrical, and Spherical Apertures,, IEEE. Trans. Biomed. Eng., BME-28 (1981), 202. doi: 10.1109/TBME.1981.324791.

[50]

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