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Biomedical Optics Express

Biomedical Optics Express

  • Editor: Joseph A. Izatt
  • Vol. 4, Iss. 10 — Oct. 1, 2013
  • pp: 2224–2230
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A least-squares fixed-point iterative algorithm for multiple illumination photoacoustic tomography

Tyler Harrison, Peng Shao, and Roger J. Zemp  »View Author Affiliations


Biomedical Optics Express, Vol. 4, Issue 10, pp. 2224-2230 (2013)
http://dx.doi.org/10.1364/BOE.4.002224


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Abstract

The optical absorption of tissues provides important information for clinical and pre-clinical studies. The challenge in recovering optical absorption from photoacoustic images is that the measured pressure depends on absorption and local fluence. One reconstruction approach uses a fixed-point iterative technique based on minimizing the mean-squared error combined with modeling of the light source to determine optical absorption. With this technique, convergence is not guaranteed even with an accurate measure of optical scattering. In this work we demonstrate using simulations that a new multiple illumination least squares fixed-point iteration algorithm improves convergence - even with poor estimates of optical scattering.

© 2013 OSA

1. Introduction

Several attempts have been made at non-iterative solutions [6

6. J. Ripoll and V. Ntziachristos, “Quantitative point source photoacoustic inversion formulas for scattering and absorbing media,” Phys. Rev. E 71, 031912 (2005). [CrossRef]

8

8. B. Banerjee, S. Bagchi, R. M. Vasu, and D. Roy, “Quantitative photoacoustic tomography from boundary pressure measurements: noniterative recovery of optical absorption coefficient from the reconstructed absorbed energy map,” J. Opt. Soc. Am. A 25, 2347–2356 (2008). [CrossRef]

], but due to the interrelation between absorption and local fluence, iterative approaches have also shown some success [9

9. 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. Opt. 45, 1866–1875 (2006). [CrossRef] [PubMed]

, 10

10. L. Yin, Q. Wang, Q. Zhang, and H. Jiang, “Tomographic imaging of absolute optical absorption coefficient in turbid media using combined photoacoustic and diffusing light measurements,” Opt. Lett. 32, 2556–2558 (2007). [CrossRef] [PubMed]

]. However, experimental work by Jetzfellner et al. [11

11. T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, K. H. Englmeier, and V. Ntziachristos, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett. 95, 013703 (2009). [CrossRef]

] concluded that errors in optical scattering estimates can result in non-convergence. Multiple wavelengths have been used by Cox et al. [12

12. B. T. Cox, S. R. Arridge, and P. C. Beard, “Estimating chromophore distributions from multiwavelength photoacoustic images,” J. Opt. Soc. Am. A 26, 443–455 (2009). [CrossRef]

] to recover absorption and scattering, and Bal and Ren [13

13. G. Bal and K. Ren, “On multi-spectral quantitative photoacoustic tomography in diffusive regime,” Inverse Probl. 28, 025010 (2012). [CrossRef]

] have proposed a method for also reconstructing the Grüneisen parameter.

2. Theory

2.1. Single illumination

Assuming that an initial pressure distribution p0 has already been reconstructed using any of the various techniques available [1

1. M. Xu and L. V. Wang, “Analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic or photoacoustic reconstruction,” Phys. Rev. E 67, 056605 (2003). [CrossRef]

3

3. Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15, 021311 (2010). [CrossRef] [PubMed]

], the problem for a single illumination is to reconstruct the optical absorption μa by modeling the fluence distribution Φ. The initial pressure distribution is actually a combination of μa, Φ and the Grüneisen parameter Γ which for simplicity we assume is uniform and constant. p0 takes the form in equation 1.
p0=ΓΦμa
(1)

Reconstruction is complicated by Φ, which should properly be written as Φ(r, μa, μ′s): a function varying over spatial position r, absorption μa(r), and the reduced scattering coefficient μ′s(r). For the simple iterative technique we are interested in [11

11. T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, K. H. Englmeier, and V. Ntziachristos, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett. 95, 013703 (2009). [CrossRef]

], μ′s is taken to be uniform. As in that work, we assume the diffusion equation is applicable, and use the finite element method solver available in the TOAST toolkit [20

20. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41 (1999). [CrossRef]

] to calculate Φ.

Assuming an initial guess of μ̂a = 0, the iterative method of [9

9. 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. Opt. 45, 1866–1875 (2006). [CrossRef] [PubMed]

] proceeds as follows over iterations i: use the forward model to estimate Φi; calculate an error parameter Δp0=p0μa(i)Φ(i); update μ^a(i+1)=p0Φ(i)+σ (where σ is a regularization parameter); and repeat with i = i + 1 until the error is within acceptable bounds, or the solution for μ̂a has converged.

2.2. Extension to multiple illuminations

A similar derivation can be used for multiple illuminations. Consider that a number of detectors are used to reconstruct the initial pressure distributions p^0k(r) into an N × N image due to illuminations k = 1,...,S. The reconstructed initial pressures can be modeled as p^0k(r)=ΓΦ^k(r)μ^a(r). Here, Φ̂k(r) is the estimated fluence due to illumination k, and μ̂a(r) is the estimated optical absorption coefficient distribution. For simplicity, Γ is considered to be spatially constant. From the M = N2 pixels of each of the S reconstructed images, one can form a vector of the observation data p^0[p^01(r1),,p^01(rM)p^0S(r1),,p^0S(rM)]T.

A vector equation may be formed to model computed initial pressure vectors: 0 = Âμ̂a, where μ̂a = [μ̂a(r1),..., μ̂a(rM)]T is an M × 1 column vector of estimated optical absorption coefficients, and where  = [Â1,...,ÂS]T with Âk = Γ × diag(Φ̂k), where Φ̂k = [Φ̂k(r1),..., Φ̂k(rM)]T.

The objective is to find μ̂a such that the error between the observations and the computed images ε(μ̂a) = ||0Âμ̂a||2 is a minimum. The least squares solution to this problem comes from solving (ÂTÂ)μ̂a =ÂT0 for the vector μ̂a. Here A^TA^=kA^kTA^k=Γ2×diag(kΦ^k2(r1),,kΦ^k2(rM)), and A^Tp^0=kA^kTp^0k with p^0=[p^0k(r1),,p^0k(rM). So, given the following form of the least-squares estimate: μ̂a = (ÂTÂ)−1ÂT0, we have μ^a=1Γdiag(1kΦ^k2(r1),,1kΦ^k2(rM))B with B=[kΦ^k(r1)p^0k(r1),,kΦ^k(rM)p^0k(rM)]T and thus we obtain Eq. (2).
μ^a(r)=1ΓkΦ^k(r)p^0k(r)kΦk2(r)
(2)

We can incorporate this into an iterative algorithm as follows: from initial fluence estimates Φ̂k(0)(r, μ^a(0)) computed using a zeroth iteration approximation of optical absorption distribution μ^a(0), we can compute a new estimate of optical absorption coefficients μ^a(1) using the least-squares estimate above. This new estimate can be used to form a new fluence estimate using the diffusion equation or radiative transfer equation, and the process can be iterated until the error is sufficiently small. In order to avoid numerical instability, we introduce a regularization parameter β. β is intended only to ensure long-term convergence of the algorithm which may otherwise diverge due to noise effects, or in the case of experimental work, non-idealities. This results in Eq. (3) for iteration i + 1. For this equation, the absorption coefficients are guaranteed to be non-negative given that reconstructed initial pressures are themselves non-negative.
μ^a(i+1)(r)=1ΓkΦ^k(i)(r)p^0k(r)k[Φk(i)(r)]2+β2
(3)

3. Simulations

Simulations were run in MATLAB using the TOAST [20

20. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41 (1999). [CrossRef]

] finite element-based forward solver for light propagation. In order to verify the simulation code, a similar absorption profile to that used in the Jetzfellner work [11

11. T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, K. H. Englmeier, and V. Ntziachristos, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett. 95, 013703 (2009). [CrossRef]

] was simulated. In order to mimic the circular illumination used, 512 point sources located at one transport mean free path within the absorbing body were used. For the multiple illumination simulations, these sources were partitioned to give the appropriate number of images.

In order to measure the accuracy and convergence of MIPAT, we use a normalized root-mean-squared error (NRMSE) of the reconstructed absorption profile. NRMSE is calculated as: NRMSE=|μ^aμ^a(i)|2|μ^a|2 where the summation is over the entire image, based on the ideally reconstructed absorption, μ^a=kp^0kkΦk (using the known Φk from the forward simulation), and μ̂a(i), the current quantitative image. This is the same measure as the second quality measure from the work of Jetzfellner et al. [11

11. T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, K. H. Englmeier, and V. Ntziachristos, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett. 95, 013703 (2009). [CrossRef]

]. With this metric, we are interested in investigating three things: does increasing the number of sources improve convergence; what effect does the regularization parameter have; and how robust is the technique to noise. Convergence is not guaranteed to be a global minimum, only that the iteration will reach a stopping condition (i.e. further iteration no longer significantly changes the resulting reconstruction).

Fig. 1 (a) μa phantom; Single uniform illumination: (b) PA image (top) and true Φ (bottom), (c) first iteration, and (d) 30th iteration. Four illumination MIPAT: (e) PA images (top) and true Φ (bottom), (f) first iteration, and (g) 30th iteration. μa in cm−1.

Figure 2 shows more detailed results of the MIPAT simulation, comparing normalized root-mean-squared error (NRMSE) to the number of iterations for different numbers of illuminations. From this figure, we see that the previously non-convergent cases can be made to converge by increasing the number of illuminations using MIPAT techniques. Additionally, the speed of convergence improves with additional illuminations if not the absolute error.

Fig. 2 Simulated results of MIPAT with (a) 4, (b) 16, and (c) 512 illuminations with β = 0.032.

Finally, we investigated the effect of noise on MIPAT reconstruction. As it turns out, β plays an important role in ensuring convergence in this case, so we explored the problem space in two dimensions: both over signal-to-noise ratio (SNR) and β. To make a fair comparison between different MIPAT illumination patterns, for each of S illuminations the total optical power delivered was 1S the power used in the uniform illumination case (Fig. 1(b)) while noise levels per image remained the same. More precisely, the standard deviation of the noise added is based on the uniform illumination case at a level to produce the SNR defined as twenty times the base-10 logarithm of a ratio of signal (maximum signal in this case) to the standard deviation of the noise. This results in an SNR that is lower for each individual image in the MIPAT technique than the uniform illumination image. This should be somewhat equivalent to an experimental setup where portions of the delivered light are blocked to achieve each unique illumination. In cases with convergent solutions, 30 iterations was typically more than enough to demonstrate that behavior, so the root-mean-square error was measured at the 30th iteration for varying values of β and σ. Figure 3 shows our results with 4 and 16 illuminations.

Fig. 3 Simulated results of MIPAT after 30 iterations over different image signal-to-noise-ratios (SNR) and β values with (a) 4 and (b) 16 illuminations. The high SNR portion of (b) is presented in (c). Note that here, the standard deviation of the noise added to individual images is equal to that of the uniform illumination case.

4. Discussion and conclusions

Figure 1(a–b) illustrates the problem that has been previously observed in experimental circumstances: even with a good estimate of μ′s, a simple iterative technique does not necessarily result in a convergent solution. Our simulated results are in very good agreement with experiments from Jetzfellner et al. [11

11. T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, K. H. Englmeier, and V. Ntziachristos, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett. 95, 013703 (2009). [CrossRef]

]. Non-convergence with good knowledge of background optical properties is highly problematic, and combined with the non-uniqueness problem that has previously been explored may prevent this technique from being practical.

MIPAT has already been demonstrated as a practical solution to the non-uniqueness problem [16

16. P. Shao, B. Cox, and R. J. Zemp, “Estimating optical absorption, scattering, and grueneisen distributions with multiple-illumination photoacoustic tomography,” Appl. Opt. 50, 3145–3154 (2011). [CrossRef] [PubMed]

], and our simulated results in Fig. 1 and Fig. 2 show how introducing multiple images from different optical excitations improves convergence - even where μ′s is not well-known. While the speed of convergence is similar for the cases presented here, this figure shows that in all cases, the root-mean-squared error is minimized with the correct estimate for μ′s as one would expect. Figure 2 illustrates an interesting behavior of the numerical method: for all values of μ′s used in the reconstruction, a local minimum appears to be reached after two or three iterations before the error measure increases to eventually either diverge or converge to a different value. Indeed, fixed-point iteration numerical methods are not guaranteed to converge at a global minimum, and in fact are not guaranteed to converge at all.

While this improved convergence is an important result, the simulations in Fig. 1 and Fig. 2 do not include any noise or manipulation of the β regularization parameter. Figure 3 demonstrates the effects of SNR and β for 4 and 16 illuminations. In general, the trend that can be seen from these simulations is that increasing β can improve convergence in noisy situations, but that comes at the expense of reconstruction quality. The results seen in the figure are fairly intuitive, showing that low SNR hurts reconstruction quality, as does increasing β. However, there seems to be an ideal β for this set of simulations around 0.01 that provides reasonably accurate convergent solutions in low SNR conditions, while not significantly impacting reconstruction in high SNR conditions. While Fig. 3 demonstrates a range of SNR values, low SNR will be more typical of practical systems. In this case, more illuminations and larger regularization parameters are shown to be helpful. The choice of the regularization level using experimental data should be a topic of future work. A value of β that is too high will tend to cause the algorithm to converge in as little as a single iteration while providing inaccurate reconstruction. The choice of these regularization parameters (β for MIPAT and σ for the single illumination case) is certainly non-trivial, and in this work we were guided by the experimental work by Jetzfellner et al. [11

11. T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, K. H. Englmeier, and V. Ntziachristos, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett. 95, 013703 (2009). [CrossRef]

], and we do find that the value of β that produced stable, yet accurate results is on the same order of magnitude as the σ = 0.001 used in that work if we use σ = β2.

In comparing MIPAT with uniform illumination PAT, we chose to effectively divide the incident optical power into S sources each having 1S of the available laser power. Even better performance might be expected if all energy of the incident laser were directed to each illumination location, however ANSI standards might limit fluence at each location in practice.

Our proposed algorithm makes no assumptions of simplified models of light transport or weak optical property perturbations. However, for simplicity, the light-propagation simulations used in this manuscript presently use the diffusion approximation. Like the work of Cox et al. [21

21. B. Cox, T. Tarvainen, and S. Arridge, “Multiple illumination quantitative photoacoustic tomography using transport and diffusion models,” in “Tomography and Inverse Transport Theory,” G. Bal, D. Finch, J. Schotland, P. Kuchment, and P. Stefanov, eds. (American Mathematical Society, Providence, RI, USA, 2012), pp. 1–12.

], future work should assess robustness to cases where optical absorption coefficients are large or comparable to scattering coefficients and cases close to the point of entry, where traditional diffusion-regime approaches fail.

In the present form of the algorithm, reconstructed initial pressure distributions are taken as relatively faithful reconstructions of true initial pressures and it is assumed that there are known calibration factors relating reconstructed signals to true initial pressures. Additionally, no attempt was presently made to account for transducer spatio-temporal impulse-response or to otherwise account for non-ideal initial pressure reconstruction, and is a topic that should be given careful consideration in future work. Despite these limitations, faithful reconstructions were obtained on data which was extremely similar to the experimental data of Jetzfellner et al. [11

11. T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, K. H. Englmeier, and V. Ntziachristos, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett. 95, 013703 (2009). [CrossRef]

], offering significant promise.

Acknowledgments

We gratefully acknowledge funding from NSERC ( 355544-2008, 375340-2009, STPGP 396444, student scholarships), Terry-Fox Foundation and the Canadian Cancer Society ( TFF 019237, TFF 019240, CCS 2011-700718), Alberta Advanced Education & Technology (student scholarships), and China Scholarship Council scholarships for graduate student Peng Shao.

References and links

1.

M. Xu and L. V. Wang, “Analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic or photoacoustic reconstruction,” Phys. Rev. E 67, 056605 (2003). [CrossRef]

2.

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

3.

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15, 021311 (2010). [CrossRef] [PubMed]

4.

B. Cox, J. G. Laufer, S. R. Arridge, and P. C. Beard, “Quantitative spectroscopic photoacoustic imaging: a review,” J. Biomed. Opt. 17, 061202 (2012). [CrossRef] [PubMed]

5.

G. Bal and K. Ren, “Multi-source quantitative photoacoustic tomography in a diffusive regime,” Inverse Probl. 27, 075003 (2011). [CrossRef]

6.

J. Ripoll and V. Ntziachristos, “Quantitative point source photoacoustic inversion formulas for scattering and absorbing media,” Phys. Rev. E 71, 031912 (2005). [CrossRef]

7.

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: Recovery of optical absorption coefficient maps of heterogeneous media,” Appl. Phys. Lett. 88, 231101 (2006). [CrossRef]

8.

B. Banerjee, S. Bagchi, R. M. Vasu, and D. Roy, “Quantitative photoacoustic tomography from boundary pressure measurements: noniterative recovery of optical absorption coefficient from the reconstructed absorbed energy map,” J. Opt. Soc. Am. A 25, 2347–2356 (2008). [CrossRef]

9.

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. Opt. 45, 1866–1875 (2006). [CrossRef] [PubMed]

10.

L. Yin, Q. Wang, Q. Zhang, and H. Jiang, “Tomographic imaging of absolute optical absorption coefficient in turbid media using combined photoacoustic and diffusing light measurements,” Opt. Lett. 32, 2556–2558 (2007). [CrossRef] [PubMed]

11.

T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, K. H. Englmeier, and V. Ntziachristos, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett. 95, 013703 (2009). [CrossRef]

12.

B. T. Cox, S. R. Arridge, and P. C. Beard, “Estimating chromophore distributions from multiwavelength photoacoustic images,” J. Opt. Soc. Am. A 26, 443–455 (2009). [CrossRef]

13.

G. Bal and K. Ren, “On multi-spectral quantitative photoacoustic tomography in diffusive regime,” Inverse Probl. 28, 025010 (2012). [CrossRef]

14.

G. Bal and G. Uhlmann, “Inverse diffusion theory of photoacoustics,” Inverse Probl. 26, 085010 (2010). [CrossRef]

15.

R. J. Zemp, “Quantitative photoacoustic tomography with multiple optical sources,” Appl. Opt. 49, 3566–3572 (2010). [CrossRef] [PubMed]

16.

P. Shao, B. Cox, and R. J. Zemp, “Estimating optical absorption, scattering, and grueneisen distributions with multiple-illumination photoacoustic tomography,” Appl. Opt. 50, 3145–3154 (2011). [CrossRef] [PubMed]

17.

P. Shao, T. Harrison, and R. J. Zemp, “Iterative algorithm for multiple illumination photoacoustic tomography (mipat) using ultrasound channel data,” Biomed. Opt. Express 3, 3240–3249 (2012). [CrossRef] [PubMed]

18.

H. Gao, S. Osher, and H. Zhao, “Quantitative photoacoustic tomography,” in “Mathematical Modeling in Biomedical Imaging II: Optical, Ultrasound, and Opto-Acoustic Tomographiess,” vol. 2035 of Lecture Notes in Mathematics: Mathematical Biosciences Subseries, H. Ammari, ed. (Springer-Verlag, Berlin, 2011), pp. 131–158.

19.

K. Ren, H. Gao, and H. Zhao, “A Hybrid Reconstruction Method for Quantitative PAT,” SIAM J. Imaging Sci. 6, 32–55 (2013). [CrossRef]

20.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41 (1999). [CrossRef]

21.

B. Cox, T. Tarvainen, and S. Arridge, “Multiple illumination quantitative photoacoustic tomography using transport and diffusion models,” in “Tomography and Inverse Transport Theory,” G. Bal, D. Finch, J. Schotland, P. Kuchment, and P. Stefanov, eds. (American Mathematical Society, Providence, RI, USA, 2012), pp. 1–12.

OCIS Codes
(100.0100) Image processing : Image processing
(100.3010) Image processing : Image reconstruction techniques
(110.5120) Imaging systems : Photoacoustic imaging
(110.6960) Imaging systems : Tomography

ToC Category:
Image Reconstruction and Inverse Problems

History
Original Manuscript: July 3, 2013
Revised Manuscript: September 5, 2013
Manuscript Accepted: September 11, 2013
Published: September 24, 2013

Citation
Tyler Harrison, Peng Shao, and Roger J. Zemp, "A least-squares fixed-point iterative algorithm for multiple illumination photoacoustic tomography," Biomed. Opt. Express 4, 2224-2230 (2013)
http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-4-10-2224


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References

  1. M. Xu and L. V. Wang, “Analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic or photoacoustic reconstruction,” Phys. Rev. E67, 056605 (2003). [CrossRef]
  2. L. Wang, “Tutorial on photoacoustic microscopy and computed tomography,” IEEE J. Sel. Top. Quant.14, 171–179 (2008). [CrossRef]
  3. Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt.15, 021311 (2010). [CrossRef] [PubMed]
  4. B. Cox, J. G. Laufer, S. R. Arridge, and P. C. Beard, “Quantitative spectroscopic photoacoustic imaging: a review,” J. Biomed. Opt.17, 061202 (2012). [CrossRef] [PubMed]
  5. G. Bal and K. Ren, “Multi-source quantitative photoacoustic tomography in a diffusive regime,” Inverse Probl.27, 075003 (2011). [CrossRef]
  6. J. Ripoll and V. Ntziachristos, “Quantitative point source photoacoustic inversion formulas for scattering and absorbing media,” Phys. Rev. E71, 031912 (2005). [CrossRef]
  7. Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: Recovery of optical absorption coefficient maps of heterogeneous media,” Appl. Phys. Lett.88, 231101 (2006). [CrossRef]
  8. B. Banerjee, S. Bagchi, R. M. Vasu, and D. Roy, “Quantitative photoacoustic tomography from boundary pressure measurements: noniterative recovery of optical absorption coefficient from the reconstructed absorbed energy map,” J. Opt. Soc. Am. A25, 2347–2356 (2008). [CrossRef]
  9. 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. Opt.45, 1866–1875 (2006). [CrossRef] [PubMed]
  10. L. Yin, Q. Wang, Q. Zhang, and H. Jiang, “Tomographic imaging of absolute optical absorption coefficient in turbid media using combined photoacoustic and diffusing light measurements,” Opt. Lett.32, 2556–2558 (2007). [CrossRef] [PubMed]
  11. T. Jetzfellner, D. Razansky, A. Rosenthal, R. Schulz, K. H. Englmeier, and V. Ntziachristos, “Performance of iterative optoacoustic tomography with experimental data,” Appl. Phys. Lett.95, 013703 (2009). [CrossRef]
  12. B. T. Cox, S. R. Arridge, and P. C. Beard, “Estimating chromophore distributions from multiwavelength photoacoustic images,” J. Opt. Soc. Am. A26, 443–455 (2009). [CrossRef]
  13. G. Bal and K. Ren, “On multi-spectral quantitative photoacoustic tomography in diffusive regime,” Inverse Probl.28, 025010 (2012). [CrossRef]
  14. G. Bal and G. Uhlmann, “Inverse diffusion theory of photoacoustics,” Inverse Probl.26, 085010 (2010). [CrossRef]
  15. R. J. Zemp, “Quantitative photoacoustic tomography with multiple optical sources,” Appl. Opt.49, 3566–3572 (2010). [CrossRef] [PubMed]
  16. P. Shao, B. Cox, and R. J. Zemp, “Estimating optical absorption, scattering, and grueneisen distributions with multiple-illumination photoacoustic tomography,” Appl. Opt.50, 3145–3154 (2011). [CrossRef] [PubMed]
  17. P. Shao, T. Harrison, and R. J. Zemp, “Iterative algorithm for multiple illumination photoacoustic tomography (mipat) using ultrasound channel data,” Biomed. Opt. Express3, 3240–3249 (2012). [CrossRef] [PubMed]
  18. H. Gao, S. Osher, and H. Zhao, “Quantitative photoacoustic tomography,” in “Mathematical Modeling in Biomedical Imaging II: Optical, Ultrasound, and Opto-Acoustic Tomographiess,” vol. 2035 of Lecture Notes in Mathematics: Mathematical Biosciences Subseries, H. Ammari, ed. (Springer-Verlag, Berlin, 2011), pp. 131–158.
  19. K. Ren, H. Gao, and H. Zhao, “A Hybrid Reconstruction Method for Quantitative PAT,” SIAM J. Imaging Sci.6, 32–55 (2013). [CrossRef]
  20. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl.15, R41 (1999). [CrossRef]
  21. B. Cox, T. Tarvainen, and S. Arridge, “Multiple illumination quantitative photoacoustic tomography using transport and diffusion models,” in “Tomography and Inverse Transport Theory,” G. Bal, D. Finch, J. Schotland, P. Kuchment, and P. Stefanov, eds. (American Mathematical Society, Providence, RI, USA, 2012), pp. 1–12.

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