August  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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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.  Google Scholar

[2]

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[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.  Google Scholar

[4]

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[5]

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[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.  Google Scholar

[7]

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[8]

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[9]

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[10]

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[11]

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[12]

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[13]

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[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.  Google Scholar

[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.  Google Scholar

[16]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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