# American Institute of Mathematical Sciences

February  2013, 7(1): 253-265. doi: 10.3934/ipi.2013.7.253

## Quantitative photoacoustic tomography with variable index of refraction

 1 Department of Mathematics, University of Washington, Seattle, WA 98195-4350, United States

Received  March 2012 Revised  June 2012 Published  February 2013

Photoacoustic tomography is a rapidly developing medical imaging technique that combines optical and ultrasound imaging to exploit the high contrast and high resolution of the respective individual modalities. Mathematically, photoacoustic tomography is divided into two steps. In the first step, one solves an inverse problem for the wave equation to determine how tissue absorbs light as a result of a boundary illumination. The second step is generally modeled by either diffusion or transport equations, and involves recovering the optical properties of the region being imaged.
In this paper we, address the second step of photoacoustics, and in particular, we show that the absorption coefficient in the stationary transport equation can be recovered given certain internal information about the solution. We will consider the variable index of refraction case, which will correspond to an inverse transport problem on a Riemannian manifold with internal data and a known metric. We will prove a stability estimate for a functional of the absorption coefficient of the medium by finding a singular decomposition for the distribution kernel of the measurement operator. Finally, we will use this estimate to recover the desired absorption properties.
Citation: Lee Patrolia. Quantitative photoacoustic tomography with variable index of refraction. Inverse Problems and Imaging, 2013, 7 (1) : 253-265. doi: 10.3934/ipi.2013.7.253
##### References:
 [1] G. Bal, A. Jollivet and V. Jugnon, Inverse transport theory of photoacoustics, Inverse Problems, 26 (2010), 025011, 35 pp. doi: 10.1088/0266-5611/26/2/025011. [2] G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics, Inverse Problems, 26 (2010), 085010, 20 pp. doi: 10.1088/0266-5611/26/8/085010. [3] M. Choulli and P. Stefanov, An inverse boundary value problem for the stationary transport equation, Osaka J. Math., 36 (1999), 87-104. [4] D. Finch, S. K. Patch and Rakesh, Determining a function from its mean value over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240. doi: 10.1137/S0036141002417814. [5] Y. Hristova, P. Kuchment and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 24 (2008), 055006, 25 pp. doi: 10.1088/0266-5611/24/5/055006. [6] P. Kuckment and L. Kunyansky, Mathematics of thermoacoustic tomography, Euro. J. Appl. Math., 19 (2008), 191-224. doi: 10.1017/S0956792508007353. [7] S. McDowall, An inverse problem for the transport equation in the presence of a Riemannian metric, Pacific Journal of Math., 216 (2004), 303-326. doi: 10.2140/pjm.2004.216.303. [8] S. McDowall, P. Stefanov and A. Tamasan, Stability of the gauge equivalent classes in inverse stationary transport, Inverse Problems, 26 (2010), 025006, 19 pp. doi: 10.1088/0266-5611/26/2/025006. [9] S. McDowall, P. Stefanov and A. Tamasan, Stability of the gauge equivalent classes in inverse stationary transport in refractive media, in "Tomography and Inverse Transport Theory," Contemporary Mathematics, 559, Amer. Math. Soc., Providence, RI, (2011), 85-100. doi: 10.1090/conm/559/11074. [10] P. Stefanov and G. Uhlmann, Theromoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16 pp. doi: 10.1088/0266-5611/25/7/075011. [11] P. Stefanov and G. Uhlmann, Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27 (2011), 045004, 26 pp. doi: 10.1088/0266-5611/27/4/045004. [12] L. Wang, ed., "Photoacoustic Imaging and Spectroscopy," CRC Press, Boca Raton, FL, 2009.

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##### References:
 [1] G. Bal, A. Jollivet and V. Jugnon, Inverse transport theory of photoacoustics, Inverse Problems, 26 (2010), 025011, 35 pp. doi: 10.1088/0266-5611/26/2/025011. [2] G. Bal and G. Uhlmann, Inverse diffusion theory of photoacoustics, Inverse Problems, 26 (2010), 085010, 20 pp. doi: 10.1088/0266-5611/26/8/085010. [3] M. Choulli and P. Stefanov, An inverse boundary value problem for the stationary transport equation, Osaka J. Math., 36 (1999), 87-104. [4] D. Finch, S. K. Patch and Rakesh, Determining a function from its mean value over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240. doi: 10.1137/S0036141002417814. [5] Y. Hristova, P. Kuchment and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 24 (2008), 055006, 25 pp. doi: 10.1088/0266-5611/24/5/055006. [6] P. Kuckment and L. Kunyansky, Mathematics of thermoacoustic tomography, Euro. J. Appl. Math., 19 (2008), 191-224. doi: 10.1017/S0956792508007353. [7] S. McDowall, An inverse problem for the transport equation in the presence of a Riemannian metric, Pacific Journal of Math., 216 (2004), 303-326. doi: 10.2140/pjm.2004.216.303. [8] S. McDowall, P. Stefanov and A. Tamasan, Stability of the gauge equivalent classes in inverse stationary transport, Inverse Problems, 26 (2010), 025006, 19 pp. doi: 10.1088/0266-5611/26/2/025006. [9] S. McDowall, P. Stefanov and A. Tamasan, Stability of the gauge equivalent classes in inverse stationary transport in refractive media, in "Tomography and Inverse Transport Theory," Contemporary Mathematics, 559, Amer. Math. Soc., Providence, RI, (2011), 85-100. doi: 10.1090/conm/559/11074. [10] P. Stefanov and G. Uhlmann, Theromoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16 pp. doi: 10.1088/0266-5611/25/7/075011. [11] P. Stefanov and G. Uhlmann, Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27 (2011), 045004, 26 pp. doi: 10.1088/0266-5611/27/4/045004. [12] L. Wang, ed., "Photoacoustic Imaging and Spectroscopy," CRC Press, Boca Raton, FL, 2009.
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