## Reconstruction of optical absorption coefficient maps of heterogeneous media by photoacoustic tomography coupled with diffusion equation based regularized Newton method

Optics Express, Vol. 15, Issue 26, pp. 18076-18081 (2007)

http://dx.doi.org/10.1364/OE.15.018076

Acrobat PDF (294 KB)

### Abstract

We describe a novel reconstruction method that allows for quantitative recovery of optical absorption coefficient maps of heterogeneous media using tomographic photoacoustic measurements. Images of optical absorption coefficient are obtained from a diffusion equation based regularized Newton method where the absorbed energy density distribution from conventional photoacoustic tomography serves as the measured field data. We experimentally demonstrate this new method using tissue-mimicking phantom measurements and simulations. The reconstruction results show that the optical absorption coefficient images obtained are quantitative in terms of the shape, size, location and optical property values of the heterogeneities examined.

© 2007 Optical Society of America

## 1. Introduction

^{1–17}While conventional PAT can image tissues with high spatial resolution, it provides only the distribution of absorbed light energy density that is the product of both the

*intrinsic*optical absorption coefficient and

*extrinsic*optical fluence distribution. Thus the imaging parameter of conventional PAT is clearly not an intrinsic property of tissue. It is well known, however, that it is the tissue absorption coefficient that directly correlates with tissue physiological/functional information. These physiological parameters including hemoglobin concentration, blood oxygenation and water content are critical for accurate diagnostic decision-making.

^{11–14}However, there are several limitations associated with these methods. First, in these methods, one has to know the exact boundary reflection coefficients as well as the exact strength and distribution of incident light source which require careful experimental calibration procedures. It is often difficult to obtain these initial parameters accurately. Second, the recovered results strongly depend on the accuracy of the distribution of absolute absorbed energy density from conventional PAT. As such, to overcome the limitations mentioned above, in this paper we propose a novel reconstruction approach that combines conventional PAT with diffusion equation based regularized Newton method for accurate recovery of optical properties. This work represents the first application of the diffusion equation based iterative nonlinear algorithms that couple the conventional Tikhonov regularization with a priori spatial information-based regularization schemes for reconstruction of absorption coefficient using tomographic photoacoustic measurements. We demonstrate this method using a series of phantom experiments.

## 2. Methods and materials

^{10,17}By incorporating the recovered absorbed energy density distribution into the photon diffusion equation, the absorption coefficient map is then extracted using a diffusion equation based regularized Newton method. The core procedure of our PAT algorithm can be described by the following two equations

*p*is the pressure wave;

*k*

_{0}=

*ω/c*

_{0}is the wave number described by the angular frequency,

*ω*and the speed of acoustic wave in the medium,

*c*is the thermal expansion coefficient;

_{0}; β*C*is the specific heat; Φ is absorbed light energy density that is the product of optical absorption coefficient,

_{p}*µ*and optical fluence or photon density, Ψ (i.e., Φ=

_{a}*µ*Ψ);

_{a}**p**

^{o}=(

*p*

^{0}

_{1},

*p*

^{0}

_{2},…,

*p*

^{0}

*)*

_{M}*,*

^{T}**p**

*=(*

^{c}*p*

_{c}_{1},

*p*

^{c}

_{2},…,

*p*)

^{c}_{M}*, where*

^{T}*p*and

^{o}_{i}*p*are observed and computed complex acoustic field data for i=1, 2…,

^{c}_{i}*M*boundary locations; Δ

*χ*is the update vector for the absorbed optical energy density; ℑ is the Jacobian matrix formed by ∂p/∂Φ at the boundary measurement sites;

*λ*is a Levenberg-Marquardt regularization parameter

^{15}and

**I**is the identity matrix. Thus here the image formation task is to update absorbed energy density distribution via iterative solution of Eqs. (1) and (2) so that a weighted sum of the squared difference between computed and measured acoustic data can be minimized.

*µ*from the absorbed energy density, Φ, the photon diffusion equation as well as the Robin boundary conditions (BCs) can be written in consideration of Φ=

_{a}(r)*µ*Ψ,

_{a}^{16}

**L**is the regularization matrix or filter matrix,

*β*the regularization parameter and

**E**

_{0}the initial guess of the inverse of optical absorption coefficient.

**Φ**

^{o}=(

**Φ**

^{o}

_{1},

**Φ**

^{o}

_{2},…,

**Φ**

^{o}

_{N})

^{T}and

**Φ**

^{c}=(

**Φ**

^{c}

_{1},

**Φ**

^{c}

_{2},…,

**Φ**

^{c}

_{N})

^{T}, where

**Φ**

^{0}

_{i}is the normalized absorbed energy density obtained from PAT, and

**Φ**

^{c}

_{i}is the absorbed energy density computed from Eqs. (3) and (4) for

*i*=1, 2…, N locations within the entire PAT reconstruction domain. It should be noted that the reconstruction of the inverse of optical absorption coefficient using Eqs. (3) and (4) will make the inverse computation easier. The initial estimate of the inverse of absorption coefficient can be updated based on iterative Newton method as follows,

**J**is the Jacobian matrix formed by ∂Φ/∂E inside the whole reconstruction domain including the boundary zone. The practical update equation resulting from Eq. (6) is utilized with

*β*=1,

**L**shown in Eq. (7). In this study, Laplacian-type filter matrix is employed and constructed according to the region or tissue type it is associated based on derived priors. This filter matrix is able to relax the smoothness constraints at the interface between different regions or tissues, in directions normal to their common boundary so that the co-variance of nodes within a region is basically realized. As such, the elements of matrix

**L**,

*L*, is specified as follows

_{ij}^{16}:

*NN*is the total node number within one region or tissue. It should be noted the last term in Eq. (6) is not routinely used in the reconstruction and including the term would reduce the sharpness of known edges given a homogeneous initial guess. Thus the absorption coefficient distribution is reconstructed through the iterative procedures described by Eqs. (3) and (7).

*µ*=0.04 mm

_{a}^{-1}and

*µ′*=1.0 mm

_{s}^{-1}while optical properties for the background were

*µ*=0.01 mm

_{a}^{-1}and

*µ′*=1.0 mm

_{s}^{-1}. In the simulation, a homogeneous distributed area source is utilized to illuminate the whole imaging domain from its top surface, which is the same as in our experiments (see below). A total of 120 ultrasound receivers were equally distributed along the boundary of background region. While PAT signals carry a wide range of acoustic frequencies, only 50 frequencies (frequency range: 50~540 kHz) were used for our PAT reconstruction.

^{12}pulsed light from a Nd: YAG laser (wavelength: 532nm, pulse duration: 3–6ns) were coupled into the phantom via an optical subsystem and generated acoustic signals. The transducer (1MHz central frequency) and phantom were immersed in a water tank. A rotary stage rotated the receiver relative to the center of the tank. The incident optical fluence was controlled below 10mJ/cm

^{2}and the incident laser beam diameter was 5.0cm. For the first two experiments, we embedded two objects with a size ranging from 2.0–5.5 mm in diameter in a 50.8 or 40.0 mm-diameter solid cylindrical phantom. We then immersed the object-bearing solid phantom into a 110.6 mm-diameter water background. The phantom materials used consisted of Intralipid as scatterer and India ink as absorber with Agar powder (1–2%) for solidifying the Intralipid and India ink solution. The background phantom had

*µ*=0.01 mm

_{a}^{-1}and µ

*′*=1.0mm

_{s}^{-1}while the two targets had

*µ*=0.03 mm

_{a}^{-1}and µ

*′*=2.0 mm

_{s}^{-1}for test 1, and

*µ*=0.07 mm

_{a}^{-1}and µ

*′*=3.0mm

_{s}^{-1}for test 2. In the next two experiments, we placed a single-target-containing phantom into the water, aiming to test the capability of resolving target having different optical contrasts relative to the background phantom. The target size was 1.0 and 2.0mm in diameter for tests 3 and 4, respectively. The target had

*µ*=0.03 mm

_{a}^{-1}and µ

*′*=2.0 mm

_{s}^{-1}for test 3, and

*µ*=0.015 mm

_{a}^{-1}and µ

*′*=2 mm

_{s}^{-1}for test 4. In the image reconstructions for the four experiments, we assumed scattering coefficient known as constant (1.0mm

^{-1}). The initial guesses of optical absorption coefficient for the target(s) and background medium were 0.02mm

^{-1}and 0.01mm

^{-1}, respectively. Although a single transducer is used, the transducer has a bandwidth that allows us to use multiple frequencies by simply Fourier transforming the detected time domain acoustic signals. In this work, 50 frequencies (frequency range: 50~540 kHz) were used for our PAT reconstruction. It required about 30 minutes to finish the two-step reconstruction computation.

## 3. Results and discussion

_{Φ}). An optimization scheme was then applied to search for the boundary conditions coefficient,

*α*and the relative source strength as described previously.

^{15}As such, the reconstruction of optical properties with our algorithms does not depend on the absolute values of absorbed energy density and optical fluence as well as the boundary parameter. For example, even though the values/scales of the absorbed energy density for experiments 1 and 2 are very different as shown in Figs. 2(c) and 2(d), the algorithm is still able to recover the absorption coefficient images quantitatively in terms of the location, size, and absorption coefficient value of the objects. In addition, our method is able to resolve the issue of negative absorbed energy density values often seen in conventional PAT. For our previous methods

^{12}, the negative values must be specified as zero, which may affect the quantitative accuracy of the recovered absorption coefficient images. In this study we demonstrate experimental evidence that it is possible to obtain absolute optical absorption coefficient image using photoacoustic tomography coupled with diffusion equation based regularized Newton method. The methods described are able to quantitatively reconstruct absorbing objects with different sizes and optical contrast levels.

## Acknowledgments

## References and links

1. | R.G.M. Kollkman, W. Steenbergen, and T. G. van Leeuwen, “Reflection mode photoacoustic measurement of seed of sound,” Opt. Express |

2. | R. I. Siphanto, K. K. Thumma, R.G.M. Kolkman, T.G. van Leeuwen, F.F.M. de mul, J.W. van Neck, L.N.A. van Adrichem, and W. Steenbergen, “Serial noninvasive photoacoustic imaging of neovascularization in tumor angiogenesis,” Opt. Express |

3. | Z. Chen, Z. Tang, and W. Wan, “Photoacoustic tomography imaging based on a 4f acoustic lens imaging systems” Opt. Express |

4. | Z. Yuan, Q. Zhang, and H. Jiang, “Simultaneously reconstruction of acoustic and optical properties of heterogeneous medium by quantitative photoacoustic tomography,” Opt. Express |

5. | R.A. Kruger, D. Reinecke, and G. Kruger, “Thermoacoustic computed tomography-technical considerations”, Med. Phys. |

6. | G. Paltauf, J. Viator, S. Prahl, and S. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging”, J. Acoust. Soc. Am. |

7. | S.J. Norton and T. Vo-Dinh, “Optoacoustic diffraction tomography: analysis of algorihms”, J. Opt. Soc. Am. A |

8. | A.A. Karabutov, E. Savateeva, and A. Oraevsky, “Imaging of layered structures in biological tissues with opto-acoustic front surface transducer”, Proc. SPIE |

9. | C.G.A. Hoelen, F.F. de Mul, R. Pongers, and A. Dekker, “Three-dimensional photoacoustic imaging of blood vessls in tissue”, Opt. Lett. |

10. | H. Jiang, Z. Yuan, and X. Gu, “Spatially varying optical and acoustic property reconstruction using finite element-based photoacoustic tomography,” J. Opt. Soc. Am. A |

11. | J. Ripoll and V. Ntziachristos, “Quantitative point source photoacoustic inversion formulas for scattering and absorbing medium,” Phys. Rev. E |

12. | Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: recovery of optical absorption coefficient map of heterogeneous medium,” Appl. Phys. Lett. |

13. | B. Cox, S. Arridge, K. Kostli, and P. Beard, “2D quantitative photoacoustic image reconstruction of absorption distributions in scattering medium using a simple iterative method,” Appl. Opt. |

14. | L. Yin, Q. Wang, Q. Zhang, and H. Jiang, “Tomographic imaging of absolute optical absorption coefficient in turbid medium using combing photoacoustic and diffusing light measurements,” Opt. Lett. |

15. | N. Iftimia and H. Jiang, “Quantitative optical image reconstruction of turbid media by use of direct-current measurements,” Appl. Opt. |

16. | P. Yalavarthy, H. Dehghani, B. Pogue, and K.D. Paulsen, “Wight-matrix structured regularization provide optimal generalized least-square in diffuse optical tomography,” Med. Phys. |

17. | Z. Yuan and H. Jiang, “Three-dimensional finite element-based photoacoustic tomography: Reconstruction algorithm and simulations,” Med. Phys. |

**OCIS Codes**

(170.0110) Medical optics and biotechnology : Imaging systems

(170.5120) Medical optics and biotechnology : Photoacoustic imaging

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: September 10, 2007

Revised Manuscript: December 12, 2007

Manuscript Accepted: December 13, 2007

Published: December 18, 2007

**Virtual Issues**

Vol. 3, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Zhen Yuan, Qiang Wang, and Huabei Jiang, "Reconstruction of optical absorption coefficient
maps of heterogeneous media by photoacoustic
tomography coupled with diffusion equation
based regularized Newton method," Opt. Express **15**, 18076-18081 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-26-18076

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### References

- R. G. M. Kollkman, W. Steenbergen, and T. G. van Leeuwen, "Reflection mode photoacoustic measurement of seed of sound," Opt. Express 15, 3291-3300 (2007). [CrossRef]
- R. I. Siphanto, K. K. Thumma, R. G. M. Kolkman, T. G. van Leeuwen, F. F. M. de mul, J. W. van Neck, L. N. A. van Adrichem, and W. Steenbergen, "Serial noninvasive photoacoustic imaging of neovascularization in tumor angiogenesis," Opt. Express 13, 89-95 (2005). [CrossRef]
- Z. Chen, Z. Tang, and W. Wan, "Photoacoustic tomography imaging based on a 4f acoustic lens imaging systems," Opt. Express 15, 4966-4976 (2007). [CrossRef]
- Z. Yuan, Q. Zhang, and H. Jiang, "Simultaneously reconstruction of acoustic and optical properties of heterogeneous medium by quantitative photoacoustic tomography," Opt. Express 14, 6749-6753 (2006). [CrossRef]
- R. A. Kruger, D. Reinecke, and G. Kruger, "Thermoacoustic computed tomography-technical considerations," Med. Phys. 26, 1832-1837 (1999). [CrossRef]
- G. Paltauf, J. Viator, S. Prahl, and S. Jacques, "Iterative reconstruction algorithm for optoacoustic imaging," J. Acoust. Soc. Am. 112, 1536-1544 (2002). [CrossRef]
- S. J. Norton and T. Vo-Dinh, "Optoacoustic diffraction tomography: analysis of algorihms," J. Opt. Soc. Am. A 20, 1859-1866 (2003). [CrossRef]
- A. A. Karabutov, E. Savateeva, and A. Oraevsky, "Imaging of layered structures in biological tissues with opto-acoustic front surface transducer," Proc. SPIE 3601, 284-295(1999).
- C. G. A. Hoelen, F. F. de Mul, R. Pongers, and A. Dekker, "Three-dimensional photoacoustic imaging of blood vessls in tissue," Opt. Lett. 23, 648-650 (1998).
- H. Jiang, Z. Yuan, and X. Gu, "Spatially varying optical and acoustic property reconstruction using finite element-based photoacoustic tomography," J. Opt. Soc. Am. A 23, 878-888 (2006). [CrossRef]
- J. Ripoll and V. Ntziachristos, "Quantitative point source photoacoustic inversion formulas for scattering and absorbing medium," Phys. Rev. E 71, 031912 (2005). [CrossRef]
- Z. Yuan and H. Jiang, "Quantitative photoacoustic tomography: recovery of optical absorption coefficient map of heterogeneous medium," Appl. Phys. Lett. 88, 231101 (2006). [CrossRef]
- B. Cox, S. Arridge, K. Kostli and P. Beard, "2D quantitative photoacoustic image reconstruction of absorption distributions in scattering medium using a simple iterative method," Appl. Opt. 45, 1866-1875 (2006). [CrossRef]
- L. Yin, Q. Wang, Q. Zhang, H. Jiang, "Tomographic imaging of absolute optical absorption coefficient in turbid medium using combing photoacoustic and diffusing light measurements," Opt. Lett. 32, 2556-2558 (2007). [CrossRef]
- N. Iftimia and H. Jiang, "Quantitative optical image reconstruction of turbid media by use of direct-current measurements," Appl. Opt. 39, 5256-5261 (2000).
- P. Yalavarthy, H. Dehghani, B. Pogue and K. D. Paulsen, "Wight-matrix structured regularization provide optimal generalized least-square in diffuse optical tomography," Med. Phys. 34, 2085-2098 (2007). [CrossRef]
- Z. Yuan and H. Jiang, "Three-dimensional finite element-based photoacoustic tomography: Reconstruction algorithm and simulations," Med. Phys. 34, 538-546 (2007). [CrossRef]

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