Figure 2
Undergraduate Category: Physical and Life Sciences Degree Level: Bachelor of Science Abstract ID #669 Bachelorβs Candidate Annie Dai Advisor Ting Zhou
Quantitative Thermo-acoustics Tomography Abstract
Goal
Result
Thermo-acoustic tomography is a medical imaging technique that combines the high contrast feature of electromagnetic (EM) wave imaging with the good resolution properties of ultrasound imaging. As shown in Figure 1, there are two steps, first is to inverting acoustic wave propagation to obtain the absorbed radiation by the tissue, second is inverting EM wave propagation to reconstruct the conductivity (absorption parameter) from the absorbed radiation. Quantitative thermo-acoustics tomography focuses on the second step, which involves an inverse problem to reconstruct the conductivity from the absorbed radiation.
Our goal for this research is to simulate the reconstruction algorithm of conductivity numerically on MATLAB, and analyze the conditions that the algorithm may or may not perform well in. In the end we want to improve the performance of the reconstruction algorithm.
In our examples, we set up a conductivity profile as following: 0.03, π₯ β π ππ’πππ π π₯ = 0.05, π₯ β πππ π 0.01, π₯ β ππ‘βπππ€ππ π
Figure 1: Process of thermo-acoustic tomography.
Method The absorbed radiation π» π₯ is modeled by π» π₯ =π π₯ π’2 π₯ where π π₯ is the conductivity, and π’ π₯ is the EM wave of the solution to the following differential equation, given a boundary condition π on the boundary ππ, and k the wavenumber: β³ π’ + π 2 π’ + πππ π₯ π’ = 0, π π’ = π, ππ After we get the radiation absorption from the previous step, we can now start the inverse problem to get the conductivity back. Let π β π = βπ 2 , then the reconstruction algorithm of π π₯ is the following iteration: β π+π π₯ lim ππ π₯ = π π» π₯ β β ππβ1 π₯ , πβ₯1 mββ
where π0 = 0, β π π₯ = π π₯ ππ + ππ + ππ ππ πβ π₯ π π’ β 1.
Background
and ππ =
Photo-Acoustic Effect The combination of optical and ultrasound imaging is based on By theory, if the boundary condition π is well chosen, this what we called photo-acoustic effect (Figure 2), which is a β π π₯ will form a contraction such that formation of a sound wave due to the absorption of light radiation. π π β π ββ π π β€πΆ πβπ π π where π = π 2 + πππ, π is a Sobolev space. In particular, taking π large enough,πΆ
Figure 2: Thermo-acoustic tomography using photo-acoustic effect. Image source: Wikipedia.
π= π 2 β π 2 + ππ , where π = 4. The wavenumber π is set to 1, and the boundary condition is set to π πβ π₯ . We first obtained the absorbed radiation and then reconstructed it back to the conductivity. The reconstruction algorithm converged in about 5 iterations and the result is as shown in Figure 3. The maximum error between the true conductivity and the reconstructed conductivity is about 2e-11.
π π π
< 1.
In order for the boundary condition π to be well chosen, it should be close enough to the so called CGO (Complex Geometrical Optics) solution. This CGO solution is unknown since the conductivity Ο should be unknown during the reconstruction. Furthermore, π cannot be too large since the oscillation of π πβ π₯ will has a higher frequency and amplitude, which causes further difficulty in numerical realization, especially when the boundary condition may not be well chosen.
Figure 3: On the left is the original conductivity profile we set up, on the right is the reconstructed conductivity using the reconstruction algorithm.
Conclusion Overall, the reconstruction algorithm performs well when π is less πβ π₯ than 5, otherwise π will generate overflow and the result of conductivity will not converged. However, the wavenumber π 2 cannot be larger than π because we want π β π = βπ . We set π to 1 in our example, but in reality it should be β 2π. Future study might focuses on how to solve this numerical challenges without affecting the performance of the algorithm. ACKNOWLEDGMENTS. This research was done as a project for Math 4020 Fall 2014 and MATH 4970 Spring 2015 at Northeastern University, with the support of Prof. Ting Zhou. I thank her for helping me though out the researching. The program codes used in this paper are modified from those used in [1], and I thank K. Ren for generously sharing the codes with us.
References: [1] G. Bal, K.Ren, G. Uhlmann, T. Zhou, Quantitative Thermo-acoustics and related problems. Inverse Problems, vol. 27 (2011), 055007.
Quantitative Thermo-acoustics Tomography Abstract
Goal
Result
Thermo-acoustic tomography is a medical imaging technique that combines the high contrast feature of electromagnetic (EM) wave imaging with the good resolution properties of ultrasound imaging. As shown in Figure 1, there are two steps, first is to inverting acoustic wave propagation to obtain the absorbed radiation by the tissue, second is inverting EM wave propagation to reconstruct the conductivity (absorption parameter) from the absorbed radiation. Quantitative thermo-acoustics tomography focuses on the second step, which involves an inverse problem to reconstruct the conductivity from the absorbed radiation.
Our goal for this research is to simulate the reconstruction algorithm of conductivity numerically on MATLAB, and analyze the conditions that the algorithm may or may not perform well in. In the end we want to improve the performance of the reconstruction algorithm.
In our examples, we set up a conductivity profile as following: 0.03, π₯ β π ππ’πππ π π₯ = 0.05, π₯ β πππ π 0.01, π₯ β ππ‘βπππ€ππ π
Figure 1: Process of thermo-acoustic tomography.
Method The absorbed radiation π» π₯ is modeled by π» π₯ =π π₯ π’2 π₯ where π π₯ is the conductivity, and π’ π₯ is the EM wave of the solution to the following differential equation, given a boundary condition π on the boundary ππ, and k the wavenumber: β³ π’ + π 2 π’ + πππ π₯ π’ = 0, π π’ = π, ππ After we get the radiation absorption from the previous step, we can now start the inverse problem to get the conductivity back. Let π β π = βπ 2 , then the reconstruction algorithm of π π₯ is the following iteration: β π+π π₯ lim ππ π₯ = π π» π₯ β β ππβ1 π₯ , πβ₯1 mββ
where π0 = 0, β π π₯ = π π₯ ππ + ππ + ππ ππ πβ π₯ π π’ β 1.
Background
and ππ =
Photo-Acoustic Effect The combination of optical and ultrasound imaging is based on By theory, if the boundary condition π is well chosen, this what we called photo-acoustic effect (Figure 2), which is a β π π₯ will form a contraction such that formation of a sound wave due to the absorption of light radiation. π π β π ββ π π β€πΆ πβπ π π where π = π 2 + πππ, π is a Sobolev space. In particular, taking π large enough,πΆ
Figure 2: Thermo-acoustic tomography using photo-acoustic effect. Image source: Wikipedia.
π= π 2 β π 2 + ππ , where π = 4. The wavenumber π is set to 1, and the boundary condition is set to π πβ π₯ . We first obtained the absorbed radiation and then reconstructed it back to the conductivity. The reconstruction algorithm converged in about 5 iterations and the result is as shown in Figure 3. The maximum error between the true conductivity and the reconstructed conductivity is about 2e-11.
π π π
< 1.
In order for the boundary condition π to be well chosen, it should be close enough to the so called CGO (Complex Geometrical Optics) solution. This CGO solution is unknown since the conductivity Ο should be unknown during the reconstruction. Furthermore, π cannot be too large since the oscillation of π πβ π₯ will has a higher frequency and amplitude, which causes further difficulty in numerical realization, especially when the boundary condition may not be well chosen.
Figure 3: On the left is the original conductivity profile we set up, on the right is the reconstructed conductivity using the reconstruction algorithm.
Conclusion Overall, the reconstruction algorithm performs well when π is less πβ π₯ than 5, otherwise π will generate overflow and the result of conductivity will not converged. However, the wavenumber π 2 cannot be larger than π because we want π β π = βπ . We set π to 1 in our example, but in reality it should be β 2π. Future study might focuses on how to solve this numerical challenges without affecting the performance of the algorithm. ACKNOWLEDGMENTS. This research was done as a project for Math 4020 Fall 2014 and MATH 4970 Spring 2015 at Northeastern University, with the support of Prof. Ting Zhou. I thank her for helping me though out the researching. The program codes used in this paper are modified from those used in [1], and I thank K. Ren for generously sharing the codes with us.
References: [1] G. Bal, K.Ren, G. Uhlmann, T. Zhou, Quantitative Thermo-acoustics and related problems. Inverse Problems, vol. 27 (2011), 055007.
