Frequency Domain Photoacoustic Imaging in Biomedicine Natalie Baddour Department of Mechanical Engineering, University of Ottawa, nbaddour@uottawa.ca
Abstract Photoacoustic imaging is a relatively new modality that has the capability to image organs, such as the breast and brain, with high contrast and high spatial resolution. In this article, the use of this modality for tomography is investigated by considering the obtainable information in the Fourier domain. 1. Introduction Photoacoustic signal generation is a new technique which has demonstrated great potential for non-invasive medical tomography. With this technique, a shortpulsed laser source is used to irradiate the sample. The energy absorbed produces a small temperature rise which induces a pressure inside the sample through thermal expansion. This pressure acts as an acoustic source and generates further acoustic waves which can be detected by ultrasound transducers positioned outside the sample. Since there is a large difference in optical absorption between blood and surrounding tissue, the laser irradiation induces an ultrasound wave on the inhomogeneities within the investigated volume. Hence, the acquired photoacoustic signals carry information about the optical absorption property of the tissue and in particular about the inhomogeneities in the sample volume. This approach is thus suitable for the imaging of the micro-vascular system or for tissue characterization [1]. Futhermore, this imaging technique has contrast similar to that of pure optical imaging and spatial resolution similar to that of pure ultrasonic imaging. It therefore combines the advantages of two imaging modalities in a single modality. 2. Equations in the Frequency Domain The physical principle behind this imaging modality is the photoacoustic effect. This entails the generation of an acoustic wave as a result of the absorption of light pulse. While optical energy can be converted to mechanical energy through various pathways, it is often the case that thermal expansion is the dominant mechanism. In pulsed photoacoustic tomography, the pulse duration is so short that the thermal conduction time is greater than the thermo-acoustic transit time and the effect of thermal conduction can be ignored [2].
The equation describing the thermoacoustic wave propagation with a thermal expansion source term is given in the temporal frequency domain by [3]
∇ 2 p (r, ω ) + k s2 p (r, ω ) = −
β s iω Cp
H (r, ω ) . (1)
Here, p is the pressure of the acoustic wave, Cp is the specific heat, H is the heating function defined as the thermal energy deposited by the energy source per unit time and volume, βs is the coefficient of thermal volume expansion and k s = ω 2
2
cs2 , where cs is the
speed of sound. The heating function can be written as the product of a spatial absorption function and a temporal illumination function of the source as
H ( r , ω ) = I o A ( r )η (ω ) ,
(2)
where Io is a scaling factor proportional to the incident radiation intensity and A describes the absorption properties of the medium -- essentially the inhomogeneity whose image is sought. The function η(ω) is the temporal Fourier transform of η(t), which describes the shape of the irradiating pulse and is a nonnegative function whose integration over time equals the pulse energy. 3. Geometry As for standard diffraction tomography theory [4], we assume a background medium infinite in extent and an inhomogeneity structure of finite extent. The previously-given equations for the pressure field are the most general form of the forward problem, valid for all points outside the inhomogeneity and for arbitrary source-detector configurations. Although the assumption of an infinite domain may not be the most physically realistic assumption, it is the simplest case for physical insight and can later be modified for different geometries. As it is also the assumption made for standard acoustic diffraction tomography, as well as for diffuse photonic wave tomography, this assumption allows for straightforward comparisons. We further specialize our formulation to the case where the acoustic wave is measured by a plane of detectors. We are thus interested in the Fourier transforms of the wave measured in the z = zd plane.
4. Solution For a plane detector far enough away from the inhomogeneity for the evanescent acoustic waves to have died, it can be shown that βsωIoη (ω ) iγ si z e F3D { A(r )} ω =γ z > zs z si C 2 γ (3) si p P (ωx , ωy , z, ω ) = βsωIoη (ω ) e−iγ si z F A(r ) } ωz =−γ si z < zs 3D { 2γ C si p In the above equation, zs is the location of the inhomogeneity, F3D denotes the 3D spatial Fourier transform, where ω x , ω y , ω z are the spatial Fourier frequencies and
γ si = k s2 − ω x2 − ω y2
assumed that ω x + ω y ≤ k s . 2
frequency such that
2
2
where it is
For values of spatial
ω x2 + ω y2 > k s2 ,
the evanescent
wave is heavily attenuated. Hence, the information received at the detection plane is a spatial low-pass version of the heterogeneity function. 4. Interpretation Equation (3) makes the statement that for z > z s , the 2D transform of the pressure detected on the z plane is proportional to the full 3D Fourier transform of the inhomogeneity evaluated on ω z = γ si . Similarly, for
z < zs , the spatial 2D Fourier transform of the pressure detected on the z plane is proportional to the full 3D Fourier transform of the inhomogeneity evaluated on ω z = −γ si . It should be noticed that
ω z = ±γ si
are the top and bottom hemispheres of the
full sphere in 3 dimensional Fourier Space, namely the surface that satisfies
ω z2 = γ si2 = k s2 − ω x2 − ω y2 → ω z2 + ω x2 + ω y2 = k s2 (4) This is a sphere in 3D spatial Fourier space that is
imply the rotation of the 3D spherical slice in Fourier space. Since this spherical slice is not centred at the origin, rotating it serves to obtain further coverage of the Fourier space. For ultrasonic tomography, if projections from all angles are given, their Fourier transforms will completely cover a low-pass version of the Fourier transform of the object function, with the low-pass filter cut-off frequency determined by the frequency of the incident ultrasound radiation. Thus, the object function can be determined by assembling the Fourier transforms at various views and obtaining enough coverage in spatial Fourier space to correctly back-transform to the space domain. For photoacoustic tomography, little (if any) new information is gained from different views and thus information must be obtained at different frequencies. The temporal frequencies, ω, that appear are those that are present in the input pulse. It should be obvious that the ‘optimal’ choice of input pulse is a true delta function pulse since its temporal Fourier transform is 1 for all frequencies. It can be seen that
indicates that all frequencies are equally present and the spherical slice in 3D spatial frequency space will eventually assume all possible radii. Thus, information about the entire 3D Fourier transform of the inhomogeneity function will be obtained, enabling Fourier backtransformation into the spatial domain to obtain an image of the inhomogeneity. 5. Conclusions From this interpretation of photoacoustic imaging in the frequency domain, a methodology for reconstructing the heterogeneity function emerges. Once sufficient data about the heterogeneity has been assembled in the spatial Fourier domain, it can be inverse Fourier transformed to yield an image of the heterogeneity in physical space variables. References [1]
centered at the origin and has radius k s . In other words, detection of the pressure on a plane is akin to extracting a spherical ‘slice’ of the 3D Fourier transform of the inhomogeneity. Since this sphere is centred at the origin, we note that rotating the sphere about the origin obtains no new information about the nature of the 3D transform of the object function away from that sphere. This is in contrast with standard ultrasonic diffraction tomography in which case the sphere in spatial Fourier space has centre at (0,0,ks) – so that measurements obtained at the same frequency but different views
η (ω ) =1
[2]
[3]
[4]
M. Xu and L. V. Wang, "Photoacoustic imaging in biomedicine," Review of Scientific Instruments, vol. 77, 2006. W. E. Gusev and A. A. A.A. Karabutov, Laser Optoacoustics. New York: American Institute of Physics, 1993. H. M. Lai and K. Young, "Theory of the pulsed optoacoustic technique," Journal of the Acoustical Society of America, vol. 72, pp. 2000-2007, 1982. M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging. Philadelphia: SIAM, 1988.