applied sciences Article Ultrasonic Tomographic Technique and Its Applications Takashi Takiguchi Department of Mathematics, National Defense Academy of Japan, Yokosuka 239-8686, Japan; [email protected] Received: 30 December 2018; Accepted: 22 February 2019; Published: 11 March 2019 Abstract: X-ray tomography and magnetic resonance imaging (MRI) are excellent techniques for non-destructive or non-invasive inspections, however, they have shotcomings including the expensive cost in both the devices themselves and their protection facilities, the harmful side effects of the X-rays to human bodies and to the environment. In view of this argument, it is necessary to develop new, inexpensive, safe and reliable tomographic techniques, especially in medical imaging and non-destructive inspections. There are new tomographic techniques under development such as optical tomography, photo-acoustic tomography, ultrasonic tomography and so on, from which we take ultrasonic tomography as the topic in this paper. We introduce a review of the known ultrasonic tomographic techniques and discuss their future development. Keywords: ultrasonic tomography; medical imaging; non-destructive inspection 1. Introduction In this paper, we review the known techniques in the development of ultrasonic tomography and discuss their future development. As for reliable and minute tomographic techniques, the X-ray computerized tomography (X-ray CT) and the magnetic resonance imaging (MRI) are well known and frequently applied, especially for non-invasive inspections in medical imaging. They are superior techniques in non-invasive and non-destructive inspections, however, they have deficiencies such as the expense of both the devices themselves and their protective facilities, the harmful side effects of the X-rays to human bodies, the unsuitability of an MRI for a non-destructive inspection of those containing steel inside their bodies and so on. In order to avoid such shortcomings, tomographic techniques other than X-ray CT and MRI are under investigation; for example, electrical impedance tomography, acoustic tomography, optical tomography, photo-acoustic tomography, ultrasonic tomography and so on. In the next section, we briefly review tomographic techniques without X-ray nor MRI, among which includes ultrasonic tomography which we shall be interested in. In the third section, we first study why G. N. Hounsfield was successful in practicalizing CT. We briefly review his idea and give a theoretical proof why his idea worked well, based on which, we propose a new algorithms for ultrasonic CT for its further development. In the last section, we summarize the conclusion in this paper. 2. New Tomographic Techniques As mentioned in the previous section, new tomographic techniques distinct from X-ray and MRI are under investigation from the viewpoint of both theory and practice. In this section, we review such new tomographic techniques, among which, we shall be interested in ultrasonic tomography (or ultrasonic computerized tomography (USCT)). We also introduce why we shall be interested in ultrasonic tomography at the end of this section. Appl. Sci. 2019, 9, 1005; doi:10.3390/app9051005 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1005 2 of 14 2.1. Acoustic Tomography The history of acoustic tomography is old. It is based on inverse problems of the wave equation as Equation (1); 2 (¶t − C(x)D)u(x, t) = 0, (1) where C(x) is the velocity of the wave at the point x 2 W with some domain W ⊂ R3, and t > 0. In this problem, by controlling and observing each of the two boundary data as Equation (2) u (2) ¶W or as Equation (3) ¶u g , (3) ¶n ¶W we try to reconstruct the function C(x). Mathematical research in acoustic tomography became the origin of the study of what are called “Neumann-to-Dirichlet” and “Dirichlet-to-Neumann” maps. Acoustic tomography is an inverse problem to reconstruct the wave velocity at each space point from the control and observation of the boundary data. Consult the second section in [1] for the review of the acoustic tomography. This idea is applied for geophysical analysis [2], ocean acoustic tomography [3] and others. 2.2. Electrical Impedance Tomography (EIT) In EIT, we study inverse problems of the equation as Equation (4) r · g(x, w)ru(x) = 0, (4) where u(x) is the electric potential at the point x 2 W in the body W ⊂ R3, g(x, w) = s(x, w) + iwe (x, w) is the admittivity, s(x, w) is the electric conductivity, e(x, w) is the electric permittivity and w is the angular frequency of the applied current. In Equation (4) if we control the input current density Equation (3) where ¶W is the boundary of W and n is the outward normal vector on ¶W, and we observe the voltage Equation (2) on the boundary then we obtain a Neumann-to-Dirichlet type problem. Conversely, if we control Equation (2) and observe Equation (3) then we obtain a Dirichlet-to-Neumann type problem. EIT is an inverse problem to reconstruct the admittivity at each space point and a frequency from Neumann-to-Dirichlet or Dirichlet-to-Neumann type data. Consult [4] for the review of EIT technique and [5], for example, for an application of EIT for medical imaging. 2.3. Optical Tomography In optical tomography, we project the infrared lazer beams to an object and observe their advection through and scattering by the object. From the observation on the boundary of the object, we try to reconstruct the interior information, which is formulated an an inverse problems in the transport equation as Equation (5) Z 0 0 0 ¶tu(x, q, t) + v · qru(x, q, t) − mau(x, q, t) = ms f(q , q)u(x, q , t)dq , (5) S2 where x, v 2 R3, t ≤ 0, q 2 S2, v is the velocity vector, u(x, q, t) is the density of photon at (x, t) in 2 the direction q 2 S , ma is the absorption constant and ms is the scattering constant. The right hand side of Equation (5) represents the effect of scattering, the last term in the right hand side represents the effect of absorption and the second term in the right hand side represents the advection effect. In general it being difficult to exactly know the scattering kernel f(q0, q), suitable a priori informations in accordance with the phenomena are known. In the study of optical tomography, reconstruction of Appl. Sci. 2019, 9, 1005 3 of 14 any of v, ma, ms, with some a priori assumption on the others, can be of interest. For the introductive and general theory of the optical tomography, consult [6]. If the advection and the absorption effects are small, then it is possible to approximate the Equation (5) by the diffusion equation as Equation (6) ¶tu(x, q, t) − r · D(x)ru(x, q, t) = 0 (6) and reconstruction of the diffusion coefficient D(x) gives a rough sketch of the interior information, for which consult [7]. 2.4. Photoacoustic Tomography (PAT) PAT is based on the photoacoustic effect. If non-ionized lazer beams are projected to a human body, they go through the skin and reach the interior organs and cells, where lazer energy is absorbed and changed into heat, which projects ultrasound waves. We can observe this ultrasound and change it into the image of the human body. In this imaging process, we solve an inverse problem for the photoacoustic wave equation as Equation (7) 2 (¶t − C(x)D)u(x, t) = gE(x, t), (7) where C(x) is the velocity of the wave at the point x 2 W of the human body W ⊂ R3, t ≥ 0, E(x, t) is the amount of absorbed light energy at (x, t) and g is what is called Grüneisen parameter which represents the rate where E(x, t) is changed into ultrasound. It looks that there are a number of unsolved, or even unestablished, mathematical problems for further development of PAT technology. Consult [8] for the survey of PAT. 2.5. Ultrasonic Computerized Tomography (USCT) Ultrasonic tomography is a kind of an acoustic tomography. In this technique, we apply ultrasound whose frequency differs in accordance with the object for inspection. If the object is concrete structures, we apply ultrasound of the frequency about 50 KHz, while ultrasound of the frequency about 2 MHz is applied for the human body. Though we are able to study the same inverse problems in acoustic tomography, in this paper, let us study other problems than acoustic tomography, which are very special to ultrasonic tomography. For the propagation of the ultrasound, the following property is known. Property 1. Let W ⊂ R3 be a domain where the object locates, and C(x) be the propagation speed of the ultrasound at the point x 2 W. For a, b 2 ¶W, we denote by ga,b a route from a to b contained in W. The primary wave of the ultrasound which travels from the point a 2 ¶W to the point b 2 ¶W takes the route where the travel time is given by Equation (8) Z min 1/C(x)dg, (8) g a,b ga,b This route is called ‘the fastest route’. For this property consult [9]. By virtue of this property, the mathematical problem for ultrasonic tomography is given as Problem 1 (Problem to develop USCT). Let W ⊂ R3 be a domain and C(x) be the propagation speed of the ultrasonic wave at the point x 2 W. For a, b 2 ¶W, we denote by ga,b a route from a to b contained in W. In this case, reconstruct C(x)(x 2 W) out of the data as Equation (9) Z min 1/C(x)dg, (9) g a,b ga,b Appl.