finite-element-based photoacoustic tomography: phantom and chicken bone experiments

Finite-element-based photoacoustic tomography:phantom and chicken bone experiments

Zhen Yuan, Hongzhi Zhao, Changfeng Wu, Qizhi Zhang, and Huabei Jiang

We describe a photoacoustic image reconstruction algorithm that is based on the finite-element solutionto the photoacoustic wave equation in the frequency domain. Our reconstruction approach is an iterativeNewton method coupled with combined Marquardt and Tikhonov regularizations that can extract thespatial distribution of optical-absorption property in heterogeneous media. We demonstrate this algo-rithm by using phantom and chicken bone measurements from a circular scanning photoacoustic tomog-raphy system. The results obtained show that millimeter-sized phantom objects and chicken bonesand�or joints can be clearly detected using our finite-element-based photoacoustic tomographymethod. © 2006 Optical Society of America

OCIS codes: 170.0170, 170.3010, 170.5120, 170.6960, 110.5120.

1. Introduction

Photoacoustic tomography (PAT) promises theunique capability of imaging biological tissues withhigh optical contrast and high ultrasound resolutionin a single modality. PAT has shown the potential forimaging soft tissues including the breast, skin, andbrain.1–4 Here we show that PAT can also be used forimaging hard tissues such as bones and associatedsoft tissues. Diseases related to bones and joints suchas osteoporosis and arthritis are major causes of mor-bidity in the population over 50 years old, affectingmillions of people over the world. Conventional x-rayradiography has a very limited ability for detectingbone and joint related diseases from mild to severestatus.

PAT is actually concerned with an inverse problemwhere a single pulsed light beam illuminates an ob-ject and the light-induced acoustic fields in multiplelocations around the object are measured. The geom-etry of the object and spatial distribution of the opti-cal and�or acoustic property can be obtained from the

measured scattered fields by using a reconstructionalgorithm. The core of the reconstruction algorithm isa model describing light-induced acoustic wave prop-agation in tissue, which provides a tractable basis forimage reconstruction by using multiply scatteredfields. The Helmholtz-like photoacoustic wave equa-tion has been commonly used as an accurate modelfor PAT reconstructions. Thus far several algorithmshave been implemented,5–10 most of which have beensuccessfully tested using phantom and in vivo data.Almost all of these algorithms, however, rely on an-alytical solutions to the photoacoustic wave equationin a regularly shaped imaging domain without ap-propriate boundary conditions applied. In addition tothe possible impact on imaging accuracy, one limita-tion is that the analytic solution-based algorithmsmay not allow for the development of more sophisti-cated tissue excitation and data collection strategies.Furthermore, these methods generally require theuse of a reference medium with known optical prop-erties. Finally, the analytical methods have to as-sume that the heterogeneities constitute only a smallperturbation in a homogeneous background medium.

To overcome the potential limitations of the exist-ing reconstruction methods, in this paper, we developa finite-element-based reconstruction algorithm thatis able to provide a more accurate solution to thephotoacoustic wave equation subject to the well-known radiation or absorbing boundary conditions(BCs) in an arbitrary problem geometry. Togetherwith an iterative Newton method, our algorithm isable to offer a stable inverse solution for photoacous-tic tomography. In Section 2, we first describe the

The authors are with the Department of Biomedical Engineer-ing, University of Florida, Gainesville, Florida 32611-6131. Z.Yuan’s e-mail address is [email protected], H. Zhao’s e-mail ad-dress is [email protected], C. Wu’s e-mail address [email protected], and Q. Zhang’s e-mail address is [email protected]. H. Jiang ([email protected]) is the author to whomcorrespondence should be addressed.

Received 20 April 2005; revised 17 August 2005; accepted 21September 2005.

0003-6935/06/133177-07$15.00/0© 2006 Optical Society of America

1 May 2006 � Vol. 45, No. 13 � APPLIED OPTICS 3177

reconstruction algorithm and then evaluate the algo-rithm using phantom and chicken bone experimentsin Sections 3 and 4.

2. Reconstruction Algorithm

The photoacoustic wave equation in the frequencydomain can be stated as follows11:

�2p(r, k) � k2p(r, k) � ik�c�(r)

Cp, (1)

where p is the acoustic pressure field, c is the soundspeed, � is the isobaric volume expansion coefficient,� is the product of the optical-absorption coefficientand excitation light distribution, Cp is the specificheat, k is the acoustic wavenumber, and k � ��c (� isthe angular frequency).

To solve Eq. (1) with the finite-element method, weneed to define p and � as the sum of coefficientsmultiplied by a set of basis functions:

p � �i�1

N

pi�i, ��i�1

N

�i�i, (2)

where N is the node number of the finite-elementmesh and �i is the basis function. Then the weightedweak form of Eq. (1) can be stated as

�V

�i(�2p � k2p)dV ��

V

�i(ik�c��Cp)dV � 0, (3)

where the integration is performed over the entireproblem domain. Substituting Eq. (2) into Eq. (3), wecan obtain the finite-element discretization of Eq. (1)and associated boundary conditions,

Ap � b�, (4)

where the elements of matrix A are expressed as

amn ��V

(��m · � �n � k2�m�n)dV

��

(��m��n · n̂)d; (5)

the elements of matrix b are specified as

bmn ��V

(�ik�c�m�n�Cp)dV, (6)

vectors � and p are expressed by their real and imag-inary components as

��r, 1, �i, 1, �r, 2, �i, 2, . . . , �r, N, �i, N�T,

p � �pr, 1, pi, 1, pr, 2, pi, 2, . . . , pr, N, pi, N�T, (7)

and the second-order absorption BCs have been ap-plied on the boundary (Ref. 12),

��n · n̂ � �n � ��2�n

�2 , (8)

where � �ik � 3�2� � i3�8k�2�1 � i�k�, � � �i�2k�2�1 � i�k�, and � is the angular coordinate at aradial position �.

If the acoustic field is Taylor expanded as a seriesof functions of absorption properties, we then get thefollowing equation:

po � pc ��pc

��r��r �

�pc

��i��i � · · · . (9)

Thus Eq. (9) can be truncated to obtain the followingmatrix equation:

� � � � po � pc, (10)

where po � �p1o, p2

o, . . . , pMo�T, pc � �p1

c, p2c,

. . . pMc�T, and pi

o and pic are observed and computed

complex acoustic field data for i � 1, 2, . . ., M bound-ary locations. �� r,1, ��r,2, . . . , ��r,nn,��i,1,��i,2, . . . , ��i,nn�T. � is the Jacobian matrix and spec-ified as

� �

�p1

��R,1· · ·

�p1

��R,n

�p1

��I,1· · ·

�p1

��I,n

�p2

��R,1· · ·

�p2

��R,n

�p2

��I,1· · ·

�p2

��I,n

É Ì É É Ì É

�pM

��R,1· · ·

�pM

��R,n

�pM

��I,1· · ·

�pM

��I,n

. (11)

The elements of the Jacobian matrix in Eq. (11) canbe computed by the following matrix equations:

�AR �AI

AI AR

�pR

��R

�pI

��I

� � 0 ��BI

��R

�BI

��R0 ��10, (12a)

�AR �AI

AI AR

�pR

��I

�pI

��I

� � 0 ��BI

��I

�BI

��I0 ��01. (12b)

To realize an invertible system of equations for ��,Eq. (10) is left multiplied by the transpose of � toproduce

(�T� � �I)�� T(po � pc), (13)

where regularization schemes are invoked to stabi-

3178 APPLIED OPTICS � Vol. 45, No. 13 � 1 May 2006

lize the decomposition of �T�: I is the identity matrixand � is the regularization parameter determined bycombined Marquardt and Tikhonov regularizationschemes. The regularization parameter used was� � 0.8, which was selected based on trial and error.We found that any value of � between 0 and 1 gavesimilar reconstruction results, suggesting that it isinsensitive to the chosen value of �. In PAT, the im-age formation task is to update absorption properties(the real and imaginary parts of �) through the iter-ative solutions of Eqs. (4) and (13) so that a weightedsum of the squared difference between computed andmeasured data can be minimized.

In the algorithm, we have also adapted a dualmeshing method for efficient inverse computation.This dual meshing method, detailed elsewhere,13 ex-ploits two separate meshes—one fine mesh for accu-rate wave propagation and one coarse mesh forparameter recovery. The dual meshing scheme al-

lows a significant reduction of the problem size dur-ing the reconstruction, thus increasing overallcomputational efficiency. For example, to obtain thereconstruction results in this study, we just neededa fine mesh with 39 680 triangular elementsand 20 081 nodes for the forward computation, and acoarse mesh with 2480 elements and 1301 nodes forthe inverse solution. The characteristic sizes of thesemeshes are then approximately 144 and 16 nodeswithin a 1 cm2 domain for the fine mesh and coarsemesh, respectively. In this case it would take ninetimes more computational cost to complete a recon-struction if both forward and inverse computationsused the fine mesh. We determined the trade-off be-tween the mesh sizes (and hence computation time)and quality of the image reconstructions based onnumerical simulation studies that gave satisfactoryreconstruction results with similar mesh sizes. Im-plementation of the dual meshing scheme affects twocomponents of the reconstruction algorithm: (1) theforward solution at each iteration for the pressurefield where the optical property profiles are definedon the coarse mesh while the forward solution calcu-lation is based on the fine mesh, and (2) calculation ofthe Jacobian matrix, which is used to update theoptical property profile estimates during the inversesolution procedure. Thus, for the forward solution,the integral computation in Eq. (4) is performed overthe elements of the fine mesh, while �R and �I needto be expanded in the basis functions that are definedover the coarse mesh. For example, for an arbitrarynode i of the fine mesh, which is embedded in a coarsemesh element with nodes L1, L2, and L3, the values of

Fig. 1. Schematic of the photoacoustic tomography system. BS,beam splitter; PC, personal computer.

Fig. 2. Measured acoustic signal in the temporal domain for the two-target rubber test for a typical detector position. Y axis, acousticpressure in relative scale; X axis, time.


�R and �I at node i become

�R(xi, yi) � �n�1

3

�R, Ln�Ln

(x, y),

�I(xi, yi) � �n�1

3

�I, Ln�Ln

(x, y), (14)

where �Lnis the Lagrangian basis function over the

coarse mesh. The second impact of the dual meshingmethod appears during the construction of the Jaco-bian matrix, �, which is used to update the objectprofile values. The elements of � are composed of thepartial derivatives of the scattering field at the ob-servation sites with respect to the values of �R and �I

at each node within the coarse mesh.

Fig. 3. (a), (c), (e) Test geometry of the phantom experiments where dimensional units are in millimeters. The transducer path was alonga circle of 60 mm in diameter for the one-target cases, while the path was along a circle of 110.6 mm in diameter for the two-target cases.(b), (d), (f) Reconstructed photoacoustic images for the one- and two-target cases.


3. Experimental Materials and Methods

The experimental system used is based on a mechan-ical scanning mechanism, schematically shown inFig. 1. In this system, pulsed light from a Nd:YAGlaser (wavelength 532 nm, pulse energy 360 mJ,pulse duration 3–6 ns; Altos, Bozeman, Montana) iscoupled into the phantom via an optical subsystemand generates an acoustic pressure wave. While360 mJ is the maximum energy available from thelaser, we actually used only 120 mJ and a beam size

of 3.8 cm diameter for the experiments. So the actualenergy at the phantom surface was approximately10 mJ�cm2, which is far below the safety standard. A500 kHz transducer (Valpey Fisher, Hopkinton, Mas-sachusetts) is used to receive the acoustic signals.The transducer and the phantom and�or bones areimmersed in a water tank. A rotary stage rotates thereceiver relative to the center of the tank. One set ofdata is taken at 120 positions when the receiver isscanned circularly over 360°. The complex wave fieldsignal is first amplified by a preamplifier (gain: 17 dB,5 kHz–25 MHz, Onda, Corporation, Sunnyvale,California), and then amplified further by a pulser–receiver (GE Panametrics, Waltham, Massachu-setts). A data-acquisition board converts the signalinto a digital one, which is fed to a PC. The radius ofthe receiver motion path was 55.3 mm in the phan-tom experiments conducted. The entire data acquisi-tion is realized through C programming. In thecurrent version of the system, data collection for atotal of 120 measurements requires approximately 2min. The sound speed of the phantom is specified at1495 � 1.03 m⁄s. One representative measured acous-tic signal in the temporal domain for a two-targetrubber experiment is provided in Fig. 2, where thedetected temporal relative acoustic pressure (Y axis)for the first detector (0°) is plotted versus time (Xaxis, unit, 0.05 � 10�6 s).

A series of phantom and chicken bone experimentswere performed to evaluate our reconstruction algo-rithm by using the imaging system described above.In the phantom experiments, we embedded one ortwo rubber objects (3 mm in diameter) in a 10 or 25mm diameter solid cylindrical phantom (1% In-tralipid � India ink � distilled water � 2% Agarpowder) [see Figs. 3(a), 3(c), and 3(e) for the geometryof the phantom experiments]. We then immersed theobject-bearing solid phantom in a 110.6 mm diameterwater background. The recovered images and quan-titative results are presented in Figs. 3(b), 3(d), 3(f),and 4, respectively, for this rubber phantom test. Thechicken bone experiments conducted were similar tothe phantom experiments except that a pair ofchicken bones was embedded in the 10 mm diametersolid phantom instead of rubbers as objects. We per-formed the bone experiments by using two differentpairs of chicken bones with different shapes and sizes[see Figs. 5(a) and 5(c) for the geometry of the chickenbone experiments]. Based on the present mesh sizesand PC performance, it took approximately 30 min-utes to finish a set of reconstruction computationsfrom 120 measurements.

4. Results and Discussion

Figures 3(b) and 3(d) show the reconstructed imagesfor one small target embedded at two different posi-tions, while Fig. 3(f) presents the reconstructed im-age of two small targets (we chose to plot the absolutevalue of the recovered �, i.e., ��r

2��i2 in the images

shown in this paper). We can see that the targets areclearly detected in all three cases. These rubber tar-

Fig. 4. Recovered parameter profiles along transect AB [see Figs.3(a), 3(c), and 3(e)] for the images shown in Figs. 3(b), 3(d), and 3(f),respectively.


gets have a relatively high optical absorption, as canbe seen from the gray scale in the images. The recov-ered target position and shape for these cases areaccurate compared to the actual rubber geometry.The two small targets are clearly separated from Fig.3(f). The shadows noted around the target(s) shouldcorrespond to the Intralipid phantom, while we alsonote some artifacts especially in the water back-ground. In Figs. 4(a)–4(c) we present the recoveredparameter profiles plotted along line AB shown inFigs. 3(a), 3(c), and 3(e) for the images shown in Figs.3(b), 3(d), and 3(f). We estimated the size of targetsbased on the full width at half-maximum (FWHM)and found the recovered target size to be 3.4, 3.2, and4.3 mm for the images shown in Figs. 3(b), 3(d), and3(f), respectively. These recovered sizes are in goodagreement with the exact target sizes of 3.0, 3.0, and4 mm, respectively.

Figures 5(b) and 5(d) give the reconstructed photo-acoustic images for two different pairs of chickenbones. We see that the arbitrary shape and size of thebones are correctly imaged. We also note that thejoint between the two bones is detectable for bothcases. This suggests that PAT might have the poten-tial to diagnose bone and joint related disease be-

cause normal and diseased bones and�or jointsexhibit distinct optical properties.14,15

In this study we have assumed that the density andspeed of sound are uniform within the phantom. Ourprior simulation studies have shown that this as-sumption may cause distortion of the target shapeand introduce acoustic field patternlike artifacts inthe background. However, these effects can begreatly reduced when more frequencies are used.Since we have used a sufficient number of frequenciesin this work, we believe the impact of this assumptionis minimized.

In summary, we have developed a finite-elementreconstruction algorithm for photoacoustic tomogra-phy. Successful photoacoustic images have beenachieved from experimental data by using phantomsand chicken bones. We plan to in vivo test the recon-struction method described here in the near future.

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Fig. 5. (a), (c) Test geometry of the chicken bone experiments. The transducer path was along a circle of 110 mm in diameter for bothcases. (b), (d) Reconstructed photoacoustic images for two pairs of chicken bones.


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finite-element-based photoacoustic tomography: phantom and chicken bone experiments

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