Inverse problems of ultrasound tomography in models with attenuation

Various physical problems can be described in terms of hyperbolic scalar wave equations. They include the inverse problems of ultrasound tomography in medicine, electromagnetic sounding, seismology, earthquake engineering, diagnostics of industrial products via acoustic radiation, etc. All these problems involve attenuation as a physical process inherent to wave propagation through media. We focus on solving the inverse problems of the diagnostics of objects via wave sounding. We consider the inverse problem as a coefficient inverse problem for a hyperbolic equation with attenuation effects. We analyze various models associated primarily with the propagation of ultrasound through inhomogeneous attenuating media.

Currently, ultrasound diagnosis is used for regular medical examinations, which cannot provide high resolution because of the limited range of observing angles. Unlike the mathematical models for x-ray tomographs, those of ultrasound tomography should take into account diffraction, refraction, multiple reflection of waves, etc. Attenuation of ultrasound waves in the diagnosed object is one of the main physical processes to address. Ultrasound pulses with frequencies greater than 20 MHz are used for diagnosing subdermis layers exclusively, because they are completely absorbed at a depth of the order of several millimeters. At frequencies between 0.3 and 1.5 MHz attenuation of the source signal in soft human tissues may amount to several factors of one for a 10–15 cm diagnosed region (Hendee and Ritenour 2002). This very frequency interval, 0.3–1.5 MHz, is used in the design of ultrasound tomographs.

One of the major medical problems is to reduce breast cancer mortality rate. Advanced cancer is diagnosed in more than 40% patients at the first examination. Common ultrasound instruments usually fail to reveal neoplasia with sizes smaller than 3 mm and the idea of using ultrasound tomographs is very appealing to this end. The possibility of examining the object from different sides gives hope for differential diagnostics of the disease at early stages using examinations involving no ionizing radiation. The aim of ultrasound tomography is to search for inhomogeneities in the diagnosed region. Ideally, one can try to reveal inhomogeneities both in wave velocity and attenuation. One of the aims of this paper is to assess the possibility of the simultaneous determination of the speed cross section and attenuation inhomogeneities by solving the inverse problem for the wave equation with attenuation.

The most promising research direction involves the development of efficient methods for solving inverse problems treating them as coefficient inverse problems. Most of the studies in this field are based on models without attenuation. Natterer and Wubbeling (1995) use the propagation–backpropagation method for various formulations of coefficient inverse problems without attenuation. Natterer (1997) solves the inverse problem in the 3D formulation in the Helmholtz equation approximation for the case of sounding by planar waves from different sides. The results of the coarse-grid computations reported by the above author do not allow reconstruction of high-resolution images. Natterer (2010) considered the formulation and solution methods for the tomography problem based on incomplete angular data.

Beilina and Klibanov (2008, 2012) developed solution methods for inverse problems of wave tomography using the hybrid globally convergent/adaptive inverse algorithm. The problem is considered as a coefficient inverse problem without attenuation. Kuzhuget et al (2012) analyzed the results of reconstruction based on the experimental data of electromagnetic wave sounding.

Attempts have been made to linearize the nonlinear inverse problem. Such approaches include the Born and Rytov approximations and other linearized models (Stotzka et al 2005, Jirik et al 2012, Huang and Quan 2007, van Dongen and Wright 2006, Backushinsky et al 1994, Goncharskii and Romanov 2000). Unfortunately, linearized models have a very limited potential. They can be used as initial approximations for iterative procedures of solving nonlinear inverse problems.

Only a few attempts have been undertaken to solve inverse problems in terms of models with attenuation. This is due to the fact that the development of solution methods for coefficient inverse problems, where unknowns include both functions characterizing velocity and functions characterizing attenuation, is a very challenging task. The authors of studies usually propose approximate methods and simplified models, where the degree of simplification depends on the capabilities of the algorithms and the available computing facilities. Thus Wiskin et al (2012) and Andre et al (2012) proposed methods for solving inverse problems of ultrasound tomography in terms of a model that includes both diffraction and attenuation effects. The above authors use, as the approximate model for the wave equation, the so-called parabolic model, which performs rather well for small diffraction and refraction angles and applies well to transmission ultrasonic tomography schemes. Only the transmission tomography scheme can be used in the framework of the parabolic model, and therefore two different problems have to be solved successively: one for reflection and one for transmission. In the model considered attenuation is frequency dependent, and models with different frequency dependences of attenuation can be used. The results were tested not only on simulated problems, but also on an ultrasound tomograph model.

Duric et al (2005, 2007), Pratt et al (2007), Schmidt et al (2011) have developed their ultrasound tomograph design solutions to the stage of prototypes. Various authors have addressed inverse problems in the ray formulation and in terms of wave models. Ray-model solutions are analyzed in Duric et al (2005, 2007). The authors of the above papers reconstruct the speed cross section based on the times of arrival, and this technique allows this task to be decoupled from that of reconstructing the attenuation properties. Ray models constitute a rather traditional direction in tomography. Such models are applicable at sufficiently high frequencies. Unlike what we have in x-ray tomography, inverse problems in ultrasound tomography are nonlinear irrespective of whether they are addressed in terms of the wave formulation or ray models, with all the subsequent consequences.

Pratt et al (2007) solve the inverse problem of ultrasound tomography as a coefficient inverse problem for the Helmholtz equation. They actually consider the coefficient of the Helmholtz equation as a complex-valued function, allowing both diffraction phenomena and frequency-independent attenuation to be described. The speed cross section is reconstructed in terms of a transmission tomography scheme. In one of the approximations used in the method the inverse problem of the reconstruction of the speed cross section is transferred to the domain of frequencies, which is subdivided into several ranges. The residual functional is minimized iteratively in each of the frequency ranges. The initial approximation is set equal to the point of the minimum of the functional found in the previous frequency range. This approximation actually plays an important part in the methods used.

Natterer (2008) and Goncharsky and Romanov (2013) used the formulation of the coefficient inverse problem for the wave equation incorporating frequency-independent attenuation. Unlike the authors of the above studies, they derived exact theoretical formulas for the gradient of the residual functional, allowing iterative algorithms to be constructed for the approximate solution of the nonlinear coefficient inverse problem with an allowance for frequency-independent attenuation.

The development of methods for solving inverse coefficient problems in attenuated models is also of interest for seismology tasks. Thus Tonn (1991) and Quan and Harris (1997) solve the seismic exploration problem in terms of a mathematical model that is close to the ray model with a certain attenuation law along the rays. In this case, the inverse problem of the reconstruction of the speed cross section and that of the reconstruction of the attenuating properties of the medium are artificially decoupled into two independent problems, and this approach from the formal mathematical viewpoint can be considered to be an approximation. In one of the basic models attenuation depends linearly on frequency. The development of methods for interpreting seismic data in terms of attenuated wave models is undoubtedly a promising direction in seismic exploration, where signal attenuation is an important factor.

Wave models are more consistent with reality. Until recently, ray models or linearized wave models have been the dominant techniques in ultrasound tomography. This is understandable given that the development of algorithms for solving nonlinear inverse problems is a challenging mathematical task. If high resolution has to be achieved, such algorithms are impossible to implement on a PC. Currently, the situation is changing in the development of both the algorithms and the computing facilities for the implementation of such algorithms.

We also point out the studies (Maia et al 1998, Gaul 1999, Bochud and Rus 2012, Szabo 1994, Chen and Holm 2003, 2004, Waters et al 2005), which involve theoretical analyses and model computations performed in terms of the direct problem for various models of acoustic-wave attenuation. In acoustic sounding problems attenuation is usually frequency dependent. Note that the higher the frequency, the stronger is attenuation. Various attenuation models have been considered, some of which even include fractional power frequency dependence. Wave equations in the formulations considered involve fractional time derivatives. Some of the models considered do not allow reasonable physical interpretation. Despite the large number of studies dealing with attenuation mechanisms and their physical models, many issues remain unclear in the mathematical modeling of wave attenuation in inhomogeneous media.

Thus scalar wave models are sufficiently good at describing the effects of diffraction, refraction and multiple reflection. Wave models have been well tested for describing effects in the optical, IR, centimeter-wave and radio wave domains of the spectrum. Unfortunately, physical models of attenuation are far from perfect, and therefore inverse problems have to be studied in the situation where wave models are somewhat uncertain as far as the nature of attenuation is concerned. In this paper we analyze several models with both frequency-independent attenuation and attenuation depending nonlinearly on frequency. We address the inverse problem in time domain tomography in terms of attenuated wave models.

A distinguishing feature of the ultrasound tomography problems considered is that ultrasound properties of mamma differ little for healthy and tumor affected tissues. The speed of sound in water is c₀ = 1500 m s⁻¹, whereas in various breast tissues it differs from c₀ by no more than 10% (the approximate speed of sound in fat and muscular tissue is equal to 1450 and 1570 m s⁻¹, respectively). Different authors report different sound speed estimates in carcinomas. Published estimates are 1585–1630 m s⁻¹ (Wiskin et al 2012), 1530.8 m s⁻¹ (Chang et al 2007). Thus the speed of sound in a carcinoma differs from c₀ = 1500 m s⁻¹ by no more than 10%–15%. An anomalous region in the speed cross section is believed to be evidence if accompanied by an anomaly in the attenuation distribution. As a rule, sound attenuation in carcinoma is higher than in surrounding tissues (Duric et al 2007). Determining the shape of a new growth is a problem of potentially great importance as an unsmooth shape may indicate the presence of cancer.

In this paper we try to answer the following questions.

• (i)  
  As is well known, in x-ray tomography the standard set of tomographic data allows only one function to be reconstructed. The ray model of ultrasound tomography can be used to try to reconstruct both the speed cross section and the distribution of attenuating properties of the medium. Can ultrasound tomography made in terms of wave time domain model allow one to reconstruct not only the speed cross section, but also the function characterizing attenuation based on standard tomographic data acquired using detectors with different positions of the sources and receivers?
• (ii)  
  If the answer to the first question is positive, is the quality of reconstruction better for the speed cross section or for the function that characterizes attenuation?
• (iii)  
  To what extent do the answers to questions (i) and (ii) depend on the adopted attenuation model?
• (iv)  
  Does it make sense to try to reconstruct the velocity cross section in models without attenuation if attenuation is really present and significant?

Among the recent publications in the field of ultrasound tomography we point out the work of Hesse et al (2013), who analyze the possibility of reconstructing from tomographic data both the speed cross section and the mass density distribution inside the volume considered. They plan to continue this work by formulating and analyzing the inverse problem of reconstructing the speed cross section and the function that characterizes attenuation. It is this problem that we solve in this paper.

We solve direct and inverse problems of ultrasound tomography treating them as the problems of the reconstruction of the speed cross section and attenuation inhomogeneities in terms of the following mathematical models.

2.1. Wave models without attenuation

Acoustic field u(r, t) in a medium without attenuation is most often described in terms of model 1

$$\begin{eqnarray} &&{model\; {1}}\qquad c(r)u_{tt} (r,t)-\Delta u(r,t)=\delta (r-r_{0} ) f(t). \end{eqnarray} \tag{ 1 }$$

Here c^−0.5(r) = v(r) is the wave speed in the medium; r ∈ R³, the position of the point in space, and Δ, the Laplacian operator with respect to r . The pulse generated by the source is described by function f(t). Wave equation (1) efficiently describes such wave phenomena as diffraction, refraction and multiple reflection of waves in media without attenuation.

In this paper we use various attenuation models and approximately reconstruct the behavior of the frequency dependence of the attenuation coefficient.

2.2. Wave models with frequency-independent attenuation

The simplest model describing attenuation is model 2

$$\begin{eqnarray} &&{model\; 2}\qquad c(r)u_{tt} (r,t)+a(r)u_{t} (r,t)-\Delta u(r,t)=\delta (r-q) f(t) \end{eqnarray} \tag{ 2 }$$

where a(r) describes attenuation in the medium. We show below that in this case signal attenuation is independent of frequency. As a result, the pulse decreases in amplitude as it propagates, but its shape remains practically unchanged. In the one-dimensional case this equation corresponds to telegrapher's equation and models of electric field in a conducting medium or transverse electric waves in a homogeneous isotropic plasma (Szabo 1994). It is also used as an approximate model in ultrasound tomography of soft human tissues (Pratt et al 2007, Natterer 2008), and was also studied by Chen et al (2012) and Cox and Overton (1996).

Let us now determine how attenuation depends on frequency in the case of a planar wave for equation

$$\begin{eqnarray} &&cu_{tt} (r,t)+au_{t} (r,t)-u_{xx} (r,t)=0, \end{eqnarray} \tag{ 3 }$$

where c = const > 0, a = const ≠ 0. We seek the solution in the form

$$\begin{eqnarray} &&u(x,t)=\exp \left(-\mathrm{i}\omega \left(\frac{x}{v} -t\right)-bx\right), \end{eqnarray} \tag{ 4 }$$

where b ≠ 0 determines attenuation. We substitute equation (4) into equation (3) to obtain

$$\begin{equation*} c(\mathrm{i}\omega )^{2} -\left(-\frac{\mathrm{i}\omega }{v} -b\right)^{2} +a\mathrm{i}\omega =0. \end{equation*}$$

We separate the real and imaginary parts to find that in the first approximation in $\frac{a}{\omega }$ parameter b, which determines attenuation, is equal to

$$\begin{eqnarray} &&b= \frac{|a|}{2\sqrt{c}}, \end{eqnarray} \tag{ 5 }$$

i.e., is independent of frequency.

In problems of the ultrasound tomography of soft tissues attenuation actually depends on frequency. Various data (Szabo 1994, Holm and Sinkus 2010, Treeby and Cox 2010) suggest that the power-law exponent of the frequency dependence for longitudinal waves ranges from 1 to 2. Hence from a physical viewpoint model 2 does not describe the behavior of attenuation very well. This model is nevertheless used in ultrasound tomography of soft tissues (Pratt et al 2007, Natterer 2008).

2.3. Wave models with attenuation that depends quadratically on frequency

Consider now another wave model, where, unlike model 2, attenuation depends on frequency. We show below that in this case the inverse problem can also be reduced to a coefficient inverse problem for a partial differential equation.

We analyze the so-called Stokes attenuation in viscous media (Royer and Dieulesaint 1996, Shutilov 1988). Consider the propagation of ultrasound waves in a medium with loss of energy due to its partial transformation into heat. Dissipation of acoustic energy may be due to the thermal conductivity of the medium and various molecular processes. In most of the real media viscosity is the principal cause of ultrasound attenuation. We assume, in accordance with Newton's law for internal friction force, that viscous stresses are to a first approximation proportional to the rate of strain. In this case for an ideal isotropic medium without shear elasticity we add the viscous stress term $a{\rm \; }\frac{\partial }{\partial t} {\rm div}\xi$ to the formula for stress σ

$$\begin{eqnarray} &&\sigma =K{\rm div}\xi +a{\rm \; }\frac{\partial }{\partial t} {\rm div}\xi , \end{eqnarray} \tag{ 6 }$$

where Kdivξ describes elastic stress (Hooke's law for uniform compression); divξ is, in view of the continuity equation, the bulk dilatation; ξ, the vector of translation; K, the modulus of volume elasticity, and a, the viscosity coefficient. In our case the stress tensor has the form σ_ik = −pδ_ik, where p is the pressure in the medium (σ = −p). In accordance with Newton's second law, the equation of motion then has the form

$$\begin{eqnarray} &&-\nabla p=\rho _{0} {\rm \; }\frac{\partial ^{2} \xi }{\partial t^{2} } =\rho _{0} {\rm \; }\frac{\partial v}{\partial t} \end{eqnarray} \tag{ 7 }$$

where ρ₀ is density. In the case of vortex-free motion we introduce the velocity potential (Tikhonov and Samarskii 1990). It follows from equation (7) that

$$\begin{equation*} v(t)=v(0)- \frac{1}{\rho _{0}} \nabla \int _{0}^{t}p\,\mathrm{d}t , \end{equation*}$$

where v(0) is the initial velocity distribution. We now introduce the potential at t = 0 v(0) = -∇f. We then have

$$\begin{equation*} v(t)=-\nabla \left(f-\frac{1}{\rho _{0}} \int _{0}^{t}p\,\mathrm{d}t\right) =-\nabla F(t), \end{equation*}$$

where we introduce the velocity potential, which is equal to the expression in parentheses, and $\frac{\partial }{\partial t} F(t)= p/\rho _{0}$. We now differentiate both parts in formula (6) with respect to time to obtain, in view of σ = −p,

$$\begin{eqnarray} &&-p_{t} =-\rho _{0} F_{tt} =K{\rm div(}v{\rm )}+a{\rm \; }\frac{\partial }{\partial t} {\rm div(}v{\rm )}=-K{\rm div}\nabla F-a{\rm \; }\frac{\partial }{\partial t} {\rm div}\nabla F\nonumber \\ &&=-K\Delta F-a{\rm \; }\frac{\partial }{\partial t} \Delta F. \end{eqnarray} \tag{ 8 }$$

For slowly varying coefficients we derive the equation for velocity by taking the gradient, and the equation for pressure, by differentiating both parts of formula (8) with respect to time:

$$\begin{equation*} \rho _{0} v_{tt} =K\Delta v+a(\Delta v)_{t} , \end{equation*}$$

$$\begin{eqnarray} &&\rho _{0} p_{tt} =K\Delta p+a(\Delta p)_{t}. \end{eqnarray} \tag{ 9 }$$

For the model equation with attenuation considered here model 3(a) can be used for any of the scalar components v and p

$$\begin{eqnarray*} &&{model\; 3({\rm a})}\qquad c(r)u_{tt} (r,t)+a(r)(\Delta u)_{t} -\Delta u(r,t)=\delta (r-q) f(t).\nonumber \end{eqnarray*}$$

We now find how attenuation depends on frequency in the model of plane wave for equation

$$\begin{eqnarray} &&cu_{tt} (r,t)+au_{xxt} (r,t)-u_{xx} (r,t)=0, \end{eqnarray} \tag{ 10 }$$

where c = const > 0, a = const ≠ 0. We seek for a solution in the form

$$\begin{eqnarray} &&u(x,t)=\exp (-\mathrm{i}\omega (x/v -t)-bx), \end{eqnarray} \tag{ 11 }$$

where b ≠ 0 determines attenuation. We substitute it into equation (10) to obtain

$$\begin{equation*} c(\mathrm{i}\omega )^{2} -\left(-\frac{\mathrm{i}\omega }{v} -b\right)^{2} +a\left(-\frac{\mathrm{i}\omega }{v} -b\right)^{2} \mathrm{i}\omega =0. \end{equation*}$$

We now isolate the real and imaginary parts to obtain, after simple algebraic transformations, the following exact formulas for b and v as functions of ω:

$$\begin{eqnarray} &&v=-2\frac{\omega ^{2} a^{2} +1}{c\omega ^{2} a} b,\qquad b^{2} =-\frac{c\omega ^{2} }{2(\omega ^{2} a^{2} +1)} \pm \frac{c\omega ^{2} }{2(\omega ^{2} a^{2} +1)} \sqrt{\omega ^{2} a^{2} +1}. \end{eqnarray} \tag{ 12 }$$

We now expand the root into a power series for small ωa $\sqrt{\omega ^{2} a^{2} +1} \approx 1+\frac{\omega ^{2} a^{2} }{2}$ to obtain for small ωa:

$$\begin{eqnarray} &&b_{1{\rm ,}2} =\pm (\sqrt{c} |a|\omega ^{2}) (2\sqrt{\omega ^{2} a^{2} +1} )^{-1}, \qquad v_{{\rm 1,2}} =\mp (|a|\sqrt{\omega ^{2} a^{2} +1}) (a\sqrt{c})^{-1}. \end{eqnarray} \tag{ 13 }$$

Thus in model 10 the coefficient b in 11 that is responsible for attenuation depends quadratically on frequency.

The next model can also be shown to have a similar quadratic dependence of attenuation on frequency

$$\begin{eqnarray} &&{model\; 3({\rm b})}\qquad c(r)u_{tt} (r,t)+a(r)u_{ttt} -\Delta u(r,t)=\delta (r-q) f(t). \end{eqnarray} \tag{ 14 }$$

Similarly to the case of model 3(a), the attenuation coefficient for equation (14) can be shown to have the form b = 0.5ω²|a|c^−0.5. Model 3(b) is also used to model the attenuation of ultrasound and acoustic waves (Holm and Sinkus 2010).

Thus the attenuation coefficient for models 3(a) and 3(b) depends quadratically on frequency ω.

2.4. Remark

3.1. Wave models without attenuation

Consider the wave equation that describes some acoustic field u(r, t) in the domain Ω⊂R^N, N = 2, 3 bounded by the surface S during time (0,T) with a point source located at point r₀. In the absence of attenuation the direct problem can be described by equations (15), (16)

$$\begin{eqnarray} &&{model\; 1}\qquad c(r)u_{tt} (r,t)-\Delta u(r,t)=\delta (r-r_{0} ) f(t) \end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray} &&u(r,t=0)=u_{t} (r,t=0)=0,\qquad \partial _{n} u|_{ST} =p(r,t). \end{eqnarray} \tag{ 16 }$$

Here ∂_nu|_ST is the derivative along the normal to the surface S in the domain S × (0, T), and p(r, t), a known function. We assume that inhomogeneities of the medium are due to velocity variations exclusively, whereas the velocity outside the inhomogeneity domain is equal to c^−0.5(r) = v(r) ≡ v₀ = const, where v₀ is known. The direct problem consists of computing the wave field in domain Ω in the case where radiation is produced by a point source. Formulas (16) constitute the boundary and initial conditions.

The inverse problem consists of finding the function c(r) that describes the inhomogeneity given the experimental data of the measurements of wave U(s, t) at the domain boundary S during time (0, T) with different source positions r₀.

Let us introduce the residual functional

$$\begin{eqnarray} &&\Phi (u(c))=\frac{1}{2} \Vert u|_{ST} -U\Vert ^{2} =\frac{1}{2} \int _{0}^{T} \int _{S}(u(s,t)-U(s,t))^{2} \,\mathrm{d}s\,\mathrm{d}t. \end{eqnarray} \tag{ 17 }$$

Here || · ||² is the squared norm in the L₂(S × (0, T)) space and U(s, t) are the experimental data at the domain boundary S during time (0, T). According to Natterer and Wubbeling (1995), Beilina and Klibanov (2012), Goncharskii and Romanov (2012), the formula for the gradient of functional (17) has the form

$$\begin{eqnarray} &&\Phi ^{\prime }_{C} (u(c)=\int _{0}^{T}w_{t} (r,t)u_{t} (r,t)\,\mathrm{d}t . \end{eqnarray} \tag{ 18 }$$

Here u(r, t) is the solution of the main problem (15), (16), and w(r, t), that of the following 'conjugate' problem for given c:

$$\begin{eqnarray} &&c(r)w_{tt} (r,t)-\Delta w(r,t)=0, \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} &&w(r,t=T)=w_{t} (r,t=T)=0,\qquad \partial _{n} w|_{ST} =u|_{ST} -U. \end{eqnarray} \tag{ 20 }$$

3.2. Wave models with frequency-independent attenuation

In this paper we propose efficient algorithms for solving the inverse problem of ultrasound tomography in media with different attenuation models. The direct (forward-time) problem in model 2 has the following form:

$$\begin{eqnarray} &&{model\; 2}\qquad c(r)u_{tt} (r,t)+a(r)u_{t} -\Delta u(r,t)=\delta (r-q) f(t) \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} &&u(r,t=0)=u_{t} (r,t=0)=0, \qquad \partial _{n} u|_{ST} =p(r,t). \end{eqnarray} \tag{ 22 }$$

Here a(r) describes attenuation in the medium. The inverse problem consists of finding the functions c(r) and a(r) that describe the inhomogeneity given in experimental data of the measurements of wave U(s, t) at the domain boundary S during time (0, T) with different source positions r₀.

Let us formulate the inverse problem as the problem of the minimization of the following quadratic functional

$$\begin{eqnarray} &&\Phi (u(c,a))=\frac{1}{2} \Vert u|_{ST} -U\Vert ^{2} =\frac{1}{2} \int _{0}^{T} \int _{S}(u(s,t)-U(s,t))^{2} \mathrm{d}s\,\mathrm{d}t \end{eqnarray} \tag{ 23 }$$

in functions c(r) and a(r). According to Goncharsky and Romanov (2013), the formula for functional (23) has the form

$$\begin{eqnarray} &&\Phi _{c}^{^{\prime }} (u(c))=\int _{0}^{T}w_{t} (r,t)u_{t} (r,t)\,\mathrm{d}t,\qquad \Phi _{a}^{^{\prime }} (u(a))=\int _{0}^{T}w_{t} (r,t)u(r,t)\,\mathrm{d}t. \end{eqnarray} \tag{ 24 }$$

Here u(r, t) is the solution of the main problem (21), (22), and w(r, t), that of the 'conjugate' problem (25), (26) for given c(r) and a(r).

$$\begin{eqnarray} &&Bw=c(r)w_{tt} (r,t)-a(r)w_{t} (r,t)-\Delta w(r,t)=0, \end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray} &&w(r,t=T)=w_{t} (r,t=T)=0, \qquad \partial _{n} w|_{ST} =u|_{ST} -U. \end{eqnarray} \tag{ 26 }$$

A different formulation of the coefficient inverse problem for hyperbolic equations with attenuation was used by Natterer (2008).

3.3. Wave models with attenuation that depends quadratically on frequency

The forward-time problem in model 3(a) has the form

$$\begin{eqnarray} &&{model\ 3({\rm a})}\qquad c(r)u_{tt} (r,t)+a(r)(\Delta u)_{t} -\Delta u(r,t)=\delta (r-q) f(t) \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray} &&u(r,t=0)=u_{t} (r,t=0)=0, \qquad \partial _{n} u|_{ST} =p(r,t). \end{eqnarray} \tag{ 28 }$$

We further assume that the speed outside the inhomogeneity domain is c^−0.5(r) = v(r) ≡ v₀ = const (where v₀ is known), and

$$\begin{eqnarray} &&a(r)|_{S} =0 \end{eqnarray} \tag{ 29 }$$

in the vicinity of the bounding surface S. If the object studied is located in water the condition a(r) ≡ 0 can be considered to be fulfilled near the boundary, because ultrasound attenuation in water is much weaker than in tissues. The inverse problem consists of finding the functions c(r) and a(r) that describe inhomogeneity given the experimental data of the measurements of wave U(s, t) at the domain boundary S during time (0, T) with different source positions r₀.

As above, we formulate the inverse problem as the problem of the minimization of quadratic functional (23). To minimize this functional, we use iterative gradient methods. In the appendix we derive a formula for the gradient of functional (23). The final formula for the gradient of functional Φ(u(ξ)) is

$$\begin{eqnarray} &&\Phi ^{\prime }_{c} (u(c))=\int _{0}^{T}w_{t} (r,t)u_{t} (r,t){\,\rm d}t ,\qquad \Phi ^{\prime }_{a} (u(a))=\int _{0}^{T}w_{t} (r,t)\Delta u(r,t){\,\rm d}t. \end{eqnarray} \tag{ 30 }$$

Here u(r, t) is the solution of the main problem (27), (28), and w(r, t), that of the 'conjugate' problem (A.4), (A.5) for given c(r) and a(r). Therefore, to compute the gradient of the functional, we have to solve the main and the 'conjugate' problem.

The forward-time problem in model 3(b) has the form

$$\begin{eqnarray*} &&{model\; 3({\rm b})}\qquad c(r)u_{tt} (r,t)+a(r)u_{ttt} -\Delta u(r,t)=\delta (r-q) f(t) \\ &&u(r,t=0)=u_{t} (r,t=0)=u_{tt} (r,t=0)=0,\qquad \partial _{n} u|_{ST} =p(r,t). \end{eqnarray*}$$

The 'conjugate' problem has the form

$$\begin{eqnarray*} &&Bw=c(r)w_{tt} (r,t)-a(r)w_{ttt} (r,t)-\Delta w(r,t)=0, \\ &&w(r,t=T)=w_{t} (r,t=T)=w_{tt} (r,t=T)=0 ,\qquad \partial _{n} w|_{ST} =u|_{ST} -U. \end{eqnarray*}$$

The formula for the gradient of the residual functional (23) for solving the inverse problem in model 3(b) can be easily shown to be

$$\begin{eqnarray*} &&\Phi _{c}^{ ^{\prime }} (u(c))=\int _{0}^{T}w_{t} (r,t)u_{t} (r,t){\,\rm d}t ,\qquad \Phi _{a}^{ ^{\prime }} (u(a))=\int _{0}^{T}w_{t} (r,t)u_{tt} (r,t){\,\rm d}t . \end{eqnarray*}$$

We thus have written out explicit formulas for the gradient of the residual functional for each of the models 1–3. With the gradient known, various iterative methods can be proposed for minimizing the residual functional. Our base method for solving inverse problems in models 1–3 was the modified version of the steepest descent method, which we have described in detail below.

A detailed description of numerical methods for solving the inverse problem in terms of model 1 without attenuation can be found in Goncharsky and Romanov (2013), Goncharsky et al (2013).

We solve the inverse problem in terms of model 2 in each layer using the finite-difference method. We introduce the discrete uniform mesh v_ijk = {(x_i, y_j, t_k): x_i = ih,   0 ⩽ i < n;   y_j = jh,   0 ⩽ j < n;    t_k = kτ,   0 ⩽ k < m} in the domain of arguments, where h is the mesh size in the horizontal coordinates and τ is the mesh size in time. The parameters h and τ are related by the CFL stability condition $c^{-0.5} \tau < h/ \sqrt{2}$. The parameters n, m specify the number of mesh points along the horizontal coordinates and time.

We use the following difference approximation of equation (21), which is second-order accurate in the source-free region

$$\begin{eqnarray*} c_{ij} \frac{u_{ij}^{k+1} -2u_{ij}^{k} +u_{ij}^{k-1} }{\tau ^{2} } &+a_{ij} \frac{u_{ij}^{k+1} -u_{ij}^{k-1} }{\tau } -\frac{u_{i+1j}^{k} -2u_{ij}^{k} +u_{i-1j}^{k} }{h^{2} } -\frac{u_{ij+1}^{k} -2u_{ij}^{k} +u_{ij-1}^{k} }{h^{2} } =0, \end{eqnarray*}$$

where the first, second and third terms approximate c(r)u_tt(r, t), a(r)u_t(r, t) and −Δu(r, t), respectively. Here $u_{ij}^{k}$ are the values of function u(r, t) at point (x_i, y_j) at time t_k; c_ij and a_ij are the values of functions c(r) and a(r) at point (x_i, y_j), respectively. We isolate the term $u_{ij}^{k+1}$ to derive the explicit difference scheme in time layers to compute the propagation of acoustic wave ('forward-time' computation) for equation (21)

$$\begin{eqnarray} &&u_{ij}^{k+1} =\bigg\lbrace 2\bigg(\frac{c_{ij}^{} }{\tau ^{2} } -\frac{2}{h^{2} } \bigg)u_{ij}^{k} +\frac{u_{i+1j}^{k} +u_{i-1j}^{k} +u_{ij+1}^{k} +u_{ij-1}^{k} }{h^{2} } \nonumber \\ &&+\bigg(\frac{a_{ij} }{2\tau } -\frac{c_{ij} }{\tau ^{2} } \bigg)u_{ij}^{k-1} \bigg\rbrace \bigg(\frac{a_{ij} }{2\tau } +\frac{c_{ij} }{\tau ^{2} } \bigg)^{-1} , \end{eqnarray} \tag{ 31 }$$

where the initial conditions are set in the form $u_{ij}^{0} =u_{ij}^{1} =0$. In this paper we adopt the boundary conditions for model computations in the form of the nonreflection condition at the boundary (Engquist and Majda 1977) ∂_nu|_ST = −c^0.5∂_tu|_ST, which in the difference approximation acquires the form

$$\begin{eqnarray*} &&\frac{u_{i+1j}^{k} -u_{i-1j}^{k} }{2h} =\pm \frac{u_{ij}^{k+1} -u_{ij}^{k-1} }{2\tau \sqrt{c_{ij}} } \qquad {for} \qquad i = 1 { or } (n - 2) \end{eqnarray*}$$

and similarly for subscript j for j = 1 or (n − 2).

In model computations the region studied is surrounded by a homogeneous medium (figure 1), where the process of the sounding pulse propagation is well known, allowing u(r, t) and u_t(r, t) to be computed for small t. The sounding pulse is set at a certain time instant t₁ > 0 to have the form of a wave radially propagating through medium L and defined by the following formula

$$\begin{eqnarray} &&\left\lbrace \begin{array}{@{}c@{}}u(r,t_{1} ) =K\exp (\alpha (r-R_{0} )){\rm sin}\left(\frac{4\pi (r-R_{0} )}{c_{0} T} \right)\qquad { for }\qquad (R_{0} -c_{0} T)<r<R_{0} \\ {u(r)=0\qquad \qquad \qquad \qquad \qquad \qquad { in other cases}} \end{array}\right. \end{eqnarray} \tag{ 32 }$$

where r is the distance from the current point to the pulse source; T,the pulse duration; c₀, the wave speed in medium L; R₀, the distance from the leading edge of the wave to the pulse source; K, the pulse amplitude, and α > 0 determines the steepness of the pulse envelope. We chose the parameters that define the pulse in our model computations so as to make the central frequency to be approximately equal to 300 kHz: α = 0.45, R₀ = 10 mm, c₀ = 1500 000 mm s⁻¹ and T = 0.000 006 66 s.

We similarly (with the adjustment for the reversal of the sign in front of a) write out the difference scheme for w(r,t) in backward time. In the conjugate problem the wave is also attenuated. The solution is found explicitly by time layers. We compute gradient (24) of functional (23) by the following difference formula

$$\begin{eqnarray} &&{\rm grad}C_{ij} =\sum _{k=1}^{m-2}\frac{u_{ij}^{k+1} -u_{ij}^{k} }{\tau } \frac{w_{ij}^{k+1} -w_{ij}^{k} }{\tau } \tau , \qquad {\rm grad}A_{ij} =\sum _{k=1}^{m-2}u_{ij}^{k} \frac{w_{ij}^{k+1} -w_{ij}^{k-1} }{\tau } \tau , \end{eqnarray} \tag{ 33 }$$

where gradC_ij is the gradient with respect to c at point (x_i, y_j), and gradA_ij, the gradient with respect to a at point (x_i, y_j). The residual is computed by the formula

$$\begin{eqnarray} &&F=\frac{1}{2} \sum _{k=0}^{m}\sum _{(i,j)\in S}\big(u_{ij}^{k} -U_{ij}^{k}\big)^{2} h\tau . \end{eqnarray} \tag{ 34 }$$

Here S is the boundary; $U_{ij}^{k}$, the values of U(r, t) at points (x_i, y_j) located at boundary S at time t_k.

We use the following iterative process in our model computations.

Our chosen initial approximation is c⁽⁰⁾ = c₀ = const, which corresponds to the 1.5 km s⁻¹ speed of sound in pure water, and

$$\begin{equation*} a^{(0)} =\left\lbrace \begin{array}{@{}lll@{}} a_{0} ={\rm const}&\quad {\rm for }\ (i, j)\in {\rm G} \\ 0 &\quad {\rm for }\ (i, j)\notin {\rm G}, \end{array}\right. \end{equation*}$$

where a₀ is the mean attenuation in region G, whose boundary is known.

The following operations are performed at each iteration (p).

• (i)  
  Computation of the initial pulse of the source using formula (32).
• (ii)  
  Solution of the direct problems (21), (22) for the current iterative approximation (c^(p), a^(p)). The propagation of ultrasound wave u^(p)(r, t) is computed by formula (31) and the u(r, t) values are computed at each detector.
• (iii)  
  Computation of residual F^(p) = F(u^(p)(r, t)) by formula (34).
• (iv)  
  Solution of the conjugate problems (25), (26) for w^(p)(r, t).
• (v)  
  Computation of the gradient (gradC^(p), gradA^(p)) by formula (33) for all sources.
• (vi)  
  Updating the current approximation c^{(p + 1)} = c^(p) + γ^(p)gradC^(p), a^{(p + 1)} = a^(p) + γ^(p)gradA^(p). The process returns to item (ii).

The iterative process stops either when the residual becomes smaller than some preset threshold, which corresponds to the a priori known error of input data, or when the rate of decrease of the residual reaches the given minimum threshold.

The problem considered here is ill posed. In principle, e.g., in model 1, the Green's function technique can be used to reduce the task to solving the two nonlinear operator equations that constitute an ill-posed problem (Goncharskii et al 2010). The theory of ill-posed problems is one of the most outstanding results of 20th century mathematics. In this theory not only methods for the approximate solution have been developed, but also a number of purely mathematical problems have been solved that involve the approximation of functions whose arguments are known with errors (Tikhonov and Arsenin 1979). Exhaustive results have been obtained concerning in which cases a solution can or cannot be constructed. The problem of constructing regularizing algorithms that are optimal in a certain sense has been solved (Bakushinsky and Goncharsky 1994). The most natural technique to solve nonlinear problems is by using iterative procedures, like, e.g., the method of successive iteration, gradient methods, regularized Newton's method, etc. In this study we use the steepest descent method to construct the approximate solution. The use of iterative algorithms for solving ill-posed problems has specificities of its own, which have been analyzed in detail by many authors (Bakushinsky and Goncharsky 1994, Bakushinsky et al 2011). One can use procedures based on Tikhonov regularization techniques. Our regularizing parameter has the form of the iteration stopping rule, where the number of iterations is determined by the level of input data errors.

The initial step γ⁽⁰⁾of gradient descent is chosen based on a priori considerations. A more accurate determination of the step of the steepest descent requires additional iterations, which would increase the computing time by a factor of two or even more. If residual F^(p)at the next iteration is greater than F^(p−1), the step γ^(p) is decreased by a factor of 1.5. After each iteration we set a^(p) = 0  for  (i, j)∉G, which corresponds to a priori information about the absence of attenuation outside region G.

We then write out the difference scheme for equation (27) in model 3(a). We use a scheme with four time layers, which produce a stable solution for wave equation (27) with attenuation. For the terms c(r)u_tt(r, t) − Δu(r, t) we use the same difference approximation as above and propose the following scheme for the a(r)(Δu)_t term:

$$\begin{eqnarray*} &&\frac{a_{ij} }{\tau h^{2} } \big\lbrace \big(u_{i+1j}^{k} -u_{ij}^{k+1} -u_{ij}^{k-1} +u_{i-1j}^{k} +u_{ij+1}^{k} -u_{ij}^{k+1} -u_{ij}^{k-1} +u_{ij-1}^{k} \big) \\ &&-\big(u_{i+1j}^{k-1} -u_{ij}^{k} -u_{ij}^{k-2} +u_{i-1j}^{k-1} +u_{ij+1}^{k-1} -u_{ij}^{k} -u_{ij}^{k-2} +u_{ij-1}^{k-1} \big)\big\rbrace . \end{eqnarray*}$$

We isolate the term $u_{ij}^{k+1}$ to obtain an explicit difference scheme in time layers for computing the propagation of acoustic wave ('forward-time' computation) for equation (27)

$$\begin{eqnarray*} &&u_{ij}^{k+1} =\bigg(\frac{c_{ij} }{\tau ^{2} } -\frac{2a_{ij} }{\tau h^{2} } \bigg)^{-1} \bigg[\frac{c_{ij} }{\tau ^{2} } \big(2u_{ij}^{k} -u_{ij}^{k-1} \big)-\frac{a_{ij} }{\tau h^{2} } \big\lbrace \big(u_{i+1j}^{k} -2u_{ij}^{k-1} +u_{i-1j}^{k} +u_{ij+1}^{k} +u_{ij-1}^{k} \big) \\ &&-\big(u_{i+1j}^{k-1} -2u_{ij}^{k} -2u_{ij}^{k-2} +u_{i-1j}^{k-1} +u_{ij+1}^{k-1} +u_{ij-1}^{k-1} \big)\big\rbrace \\ &&+\frac{1}{h^{2} } \big(u_{i+1j}^{k} +u_{i-1j}^{k} +u_{ij+1}^{k} +u_{ij-1}^{k} -4u_{ij}^{k} \big)\bigg]. \end{eqnarray*}$$

The conjugate problem is described by equation (A.4). We propose the following scheme to approximate the term Δ{a(r)w_t(r, t)}:

$$\begin{eqnarray*} &&\frac{1}{\tau h^{2} } \big[a_{i+1j} \big(w_{i+1j}^{k+1} -w_{i+1j}^{k} \big)-a_{ij} \big\lbrace \big(w_{ij}^{k+2} +w_{ij}^{k} \big)-\big(w_{ij}^{k-1} +w_{ij}^{k+1} \big)\big\rbrace \\ &&+\,a_{i-1j} \big(w_{i-1j}^{k+1} -w_{i-1j}^{k} \big) + a_{ij+1} \big(w_{ij+1}^{k+1} -w_{ij+1}^{k} \big) \\ &&-\,a_{ij} \big\lbrace \big(w_{ij}^{k+2} +w_{ij}^{k} \big)-\big(w_{ij}^{k-1} +w_{ij}^{k+1} \big)\big\rbrace +a_{ij-1} \big(w_{ij-1}^{k+1} -w_{ij-1}^{k} \big)\big]. \end{eqnarray*}$$

The difference scheme for wave w(r, t) in the conjugate problem can be written out in the following explicit form by time layers in backward time:

$$\begin{eqnarray*} &&w_{ij}^{k-1} =\bigg(\frac{c_{ij} }{\tau ^{2} } -\frac{2a_{ij} }{\tau h^{2} } \bigg)^{-1} \bigg[\frac{c_{ij} }{\tau ^{2} } \big(2w_{ij}^{k} -w_{ij}^{k+1} \big)+\frac{1}{\tau h^{2} } \big\lbrace a_{i+1j} \big(w_{i+1j}^{k+1} -w_{i+1j}^{k} \big) \\ &&+\,a_{i-1j} \big(w_{i-1j}^{k+1} -w_{i-1j}^{k} \big) + a_{ij+1} \big(w_{ij+1}^{k+1} -w_{ij+1}^{k} \big)-2a_{ij} \big(w_{ij}^{k+2} +w_{ij}^{k} -w_{ij}^{k+1} \big) \\ &&+\,a_{ij-1} \big(w_{ij-1}^{k+1} -w_{ij-1}^{k} \big)\big\rbrace +\frac{1}{h^{2} } \big(w_{i+1j}^{k} +w_{i-1j}^{k} +w_{ij+1}^{k} +w_{ij-1}^{k} -4w_{ij}^{k} \big) \bigg]. \end{eqnarray*}$$

We compute gradient (A.10) of the residual functional by the following difference formula:

$$\begin{eqnarray*} &&{\rm grad}C_{ij} =\sum _{k=1}^{m-2}\frac{u_{ij}^{k+1} -u_{ij}^{k} }{\tau } \frac{w_{ij}^{k+1} -w_{ij}^{k} }{\tau } \tau , \\ &&{\rm grad}A_{ij} =\sum _{k=1}^{m-2}\frac{u_{i+1j}^{k} +u_{i-1j}^{k} +u_{ij+1}^{k} +u_{ij-1}^{k} -4u_{ij}^{k} }{h^{2} } \frac{w_{ij}^{k+1} -w_{ij}^{k-1} }{\tau } \tau , \end{eqnarray*}$$

where gradC_ij is the gradient with respect to c at point (x_i, y_j), and gradA_ij, the gradient with respect to a at point (x_i, y_j). All other computations and the iterative process are performed in the same way as described above for model 2.

The explicit formulas for the gradient of the residual functional for models 1–3 can be used to build efficient iterative procedures for reconstructing c(r) and a(r). The formulas derived for the gradient of the residual allow solving the inverse problem both in the 2D and 3D formulations. In this paper we analyze two-dimensional problems of ultrasound tomography and our computations are based on mathematical models with and without attenuation described by equations (15), (21) and (27) in terms of models 1–3.

Because of the large amount of computations involved, the most efficient way to solve inverse problems is to run them on supercomputers. Supercomputers allow problems to be solved on grids with up to several thousand points along each coordinate in the reconstructed layer, and in this study we performed our modeling on a uniform 500 × 500 grid. Test computations were run on the 'Lomonosov' supercomputer of the Supercomputing Center of Lomonosov Moscow State University (Voevodin et al 2012). Let us list some of its parameters. 'Lomonosov' is a general-purpose supercomputer with 52000 general-purpose CPU cores and a peak performance of 1.7 PFlops, which ranks 27th in the TOP500 list. Our programs are based on MPI technology. For our simulations we used a small number (256) of cores, which allowed us to reduce the computing time by a factor of ∼100 compared to running the program on a single core. It took about 3.5 h to compute 500 iterations in the steepest descent algorithm. A problem with 64 sources on a 500 × 500 grid is efficiently parallelizable on even more cores. Thus parallelizing it over 2000 cores would result in a factor of ∼1000 speed gain. The computing time for real-life problems when run on GPU-based supercomputers may be ∼0.5 h.

Figure 1 shows the experimental design with the sources and receivers denoted by '1' and '2', respectively. Sources are located at the sides of the computation domain at equal intervals, and detectors are also placed along the perimeter of the computation domain at intervals ∼λ/3, where λ is the wavelength of pulse central frequency. This is a typical transducer size. A distinguishing feature using such transducers is their wide power-beam pattern. The domain G studied, which contains inhomogeneities, is located at the center of the square-shaped computation domain and is surrounded by nonabsorbing medium L with a known speed v₀ = 1500 m s⁻¹.

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Experimental studies were performed on a computer-synthesized 2D object with model inhomogeneities. We used the same cross sections of the ultrasound wave propagation speed v(x, y) in the object studied for models 1–3. The minimum size of an inhomogeneity is 3 mm. Variation of the speed c(x, y) did not exceed 20%. Attenuation is absent in model 1 whereas in models 2 and 3 it is characterized by function a(x, y). The variation of attenuation coefficient a(x, y) did not exceed 50% within the object. The wave amplitude decreases by a factor of about 3 when computed with the allowance for attenuation. The adopted ranges of parameter variation correspond to those observed in soft human tissues (Hendee and Ritenour 2002).

In our model computations we solved the direct problem of the propagation of ultrasound waves using difference-approximation numerical methods. Figure 2(a) shows the form of the sounding pulse as a function of time. The pulse is localized and has a broad spectrum with a central frequency of 300 kHz. Figure 2(b) shows the signals recorded at one of the transducers as functions of time computed in terms of models 1–3 after passage through the object. The solid line corresponds to the wave that has crossed the medium without attenuation in model 1, the dots show the wave computed allowing for attenuation in terms of model 2, and the dashed line, the wave computed with allowance for attenuation in terms of model 3(a). It is evident that the wave amplitude decreases by a factor of about 3 when computed with allowance for attenuation. In model 3(a) the frequency spectrum of the wave contains mostly low frequencies, whereas the frequency spectrum in model 2 did not change compared to that of the initial wave because attenuation is frequency independent.

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In this paper we consider the inverse problem in terms of time domain tomography. In this case the sounding signal has the form of a pulse and the signal recorded by the detectors is a function of time. In frequency terms this implies a wide frequency spectrum. The smaller the width of the frequency spectrum, the poorer are the results of the solutions of inverse problems. In the worst case the usage of signal is on only one frequency. In this limiting case the attenuation dependence from frequency loses its sense.

We solved the inverse problem for the data obtained, without introducing extra noise. The parameters of the computational model are: radiation wavelength 5.0 mm; signals recorded at 2.0 mm intervals in space and the horizontal size of the ultrasound sounding region is 200 × 200 mm².

To solve the inverse problem, we used the iterative process starting with the initial approximation c(r) = const = c₀, where c₀ is known outside the inhomogeneity region. In the presence of attenuation we chose its initial approximation in the form a(r) = const = a₀ in the diagnosed region, where we set a₀ equal to the average attenuation coefficient for soft tissues. We set a(r) equal to zero everywhere outside the inhomogeneity region, where attenuation is absent. We performed our computations for 64 radiation sources.

Figure 3 shows the simulated cross sections of ultrasound wave propagation speed v(x, y) in the object studied (left) and ultrasound wave attenuation function a(x, y) (right) as functions of x, y for model 2.

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Figures 4 and 5 show the results of the reconstruction of function v(x, y) in terms of model 2 after 500 and 3500 iterations.

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Figure 6 shows for model 3 the simulated cross sections of ultrasound wave propagation speed v(x, y) in the object studied (left) and ultrasound wave attenuation function a(x, y) (right) as functions of x, y.

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Figures 7 and 8 show the results of the reconstruction of function v(x, y) in terms of model 3(a) after 500 and 3500 iterations. It is evident that both models with attenuation allow simultaneous reconstruction of the two unknown functions—velocity and attenuation. Although models 2 and 3 are difficult to compare because of different attenuation behavior, we could not see from our solutions whether there is any difference between the reconstruction of the function that describes attenuation in models 2 and 3. It is, however, fair to say that for the given parameters the speed function can be reconstructed with fewer iterations and with higher quality than the attenuation function.

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The fact that speed function is reconstructed with higher quality than the attenuation function is also evident from figure 9, which shows the plots of the exact and reconstructed cross sections along the AA line in figure 6 for speed (a) and attenuation (b).

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Figure 10 shows the results of the reconstruction after 500 iterations in model 3 with superimposed normally distributed random noise with a standard deviation of 0.02, which corresponds to 10% of the amplitude of the transmitted wave. After 500 iterations the standard deviation decreased to 0.000 396, which agrees well with the dispersion of superimposed noise, which is equal to 0.0004. One may get the impression that 500 iterations is sufficiently large. However, this is not quite the case. The fastest algorithm in this domain is the conjugate-gradient method, which minimizes the quadratic functional in n iterations, where n is the number of unknowns. In our problem the number of unknowns is equal to 500 000. In addition, the problem considered is nonlinear. As mentioned above, supercomputers allow the computing time to be reduced by a factor of several hundred.

As is evident from figure 10, the distribution of speed again can be reconstructed better than that of attenuation. The quality of velocity reconstruction is high even in the case of sufficiently high noise levels.

It follows from figure 11 that the reduction of the number of sources to four (compared to 64) also has a stronger effect on attenuation, the quality of which becomes rather poor.

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We now try to answer the question of the importance of using a model with attenuation. We found that if the model is not the right one, attenuation cannot be reconstructed at all. However, speed is reconstructed rather well, as is apparent from figure 12, where we show the reconstruction of v(r) and a(r) in terms of model 2 using the data obtained from model 3. Speed in this case is reconstructed fairly well, whereas attenuation is not reconstructed at all. Moreover, speed is reconstructed rather well even if attenuation is completely ignored, as it follows from figure 13, where we show the reconstruction of speed in terms of model 1 (without attenuation) using the data obtained, as above, from model 3. The results shown in figure 13 were obtained in the case where attenuation decreases the amplitude of the signal recorded by detectors by a factor of up to 3!

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An important task to accomplish in solving ill-posed problems is to develop algorithms that allow an approximate solution to be obtained for various levels of input data errors. Here we present only one computation of a large number of model problems with randomly generated data errors at the detectors (figure 10). In reality, we have not only detector input data errors to deal with, but the model itself can be to a certain degree inadequate for the actual physical situation. Mathematically, this can be viewed as an error in the operator if the inverse problem is considered as an operator equation of the first kind. Which of the errors has greater effect on the accuracy of reconstruction? From a mathematical viewpoint the theory of solving ill-posed problems allows an approximate solution to be obtained only if both these errors tend to zero (Bakushinsky and Goncharsky 1994). The problem of greatest interest for us was to find out how stable the results are against the uncertainties of the model. It turned out that the speed cross section can be reconstructed better than the functions that characterize attenuation. Furthermore, reconstruction of the speed cross section proved to be rather stable against the uncertainty of the attenuation model.

When we set about to prepare the paper and perform model computations, our goal was to demonstrate that in real problems with attenuation it is absolutely useless to try solving the inverse problem in terms of models without attenuation. The results of model computations presented in figures 12 and 13 show that the situation is not as bad as it seems. Even in the presence of significant attenuation the speed cross section reconstructed in terms of a model without the allowance for attenuation agrees rather well with the exact solution. Of course, the result obtained cannot be compared to the perfect picture obtained in terms of the corresponding model with attenuation. However, the fact that the reconstructed speed cross section does not depend very strongly on the attenuation model inspires hope.

At the physical level of rigor this result can possibly be explained by the fact that the dependence of the residual functional on speed variations is stronger than its dependence on the variations of the coefficients in the attenuation model. Imagine that sounding is performed, for the sake of simplicity, by very short pulses. Then even for small velocity variations the signal recorded by the detector and its delayed counterpart in the perturbed version should be orthogonal in L₂ if their carriers do not intersect. A change of the attenuation coefficient in the model would change only the amplitude of the signal arriving at the detector.

To simultaneously determine the inhomogeneities characterizing attenuation as well, mathematical models should be used that adequately describe the physical processes of attenuation.

• (1)  
  In this study we have developed efficient methods for solving inverse problems of ultrasound tomography in models with attenuation. We have addressed the inverse problem as a coefficient inverse problem for unknown coordinate-dependent functions characterizing both the speed cross section and the coefficients of the wave equation describing attenuation in the diagnosed region. The main difference between the models is in the frequency dependence of the attenuation of sounding radiation. In model 2, unlike model 3, the frequency composition of the sounding signal does not change as the signal propagates through the medium. The choice of the attenuation model is primarily determined by the physics of attenuation processes.
• (2)  
  We have derived exact formulas for the gradient of the residual functional with respect to both the speed and the coefficient of the wave equation as a function of coordinates, which characterizes attenuation in models 2 and 3. We have developed efficient algorithms for minimizing the residual functional based on the computation of the gradient by solving the conjugate problem. In the formulation considered there is no need to artificially subdivide the data into the transmission and reflection measurements and implement algorithms stage by stage. For each arrangement of the sources detector data are collected. We use these data with our developed algorithms to construct approximate solutions both for the speed cross section and for the function that characterizes attenuation.
• (3)  
  We have performed all computations on general-purpose processors of the 'Lomonosov' supercomputer with up to 52000 cores available for the task. The use of supercomputers of this type is a promising technique at the initial stage of the development of algorithms and the modeling of tomographs. The potential of using general-purpose supercomputers in real tomographs is limited if for no other reason than because their power consumption amounts to several MW. However, GPU-based supercomputers have become a promising alternative. Such supercomputers are efficient when used to run highly parallelizable tasks, and are much less expensive and power hungry for a comparable performance. A single rack can accommodate a GPU-based computing facility capable of solving problems of the type considered here within the time considered reasonable in medical practice.
• (4)  
  An important problem in constructing iterative methods is the choice of the initial approximation. We have solved all model problems starting with the initial approximation where all the required coefficients are set to be constant inside the diagnosed region.
• (5)  
  Our model computations showed that in the case of small data errors it is possible to reconstruct not only the speed cross section, but also the coefficients of the wave equation that characterize attenuation, as functions of coordinates. How does this result agree with the fact that only one function can be reconstructed in x-ray tomography based on standard tomographic data? Unlike what we have in x-ray tomography, in ultrasound tomography each detector measures the signal that is a one-dimensional function of time.
• (6)  
  In both models with attenuation the speed cross section is reconstructed better than the coefficients a(r) that characterize attenuation. With increasing data errors one can anticipate the situation where the solution reconstructs the speed cross section, but practically fails to reconstruct the coefficients of the wave equation as functions of coordinates.
• (7)  
  Our result that the reconstruction of the speed cross section does not depend very strongly on the adopted attenuation model allows us to look with optimism to the future of using ultrasound tomographs in medicine for the determination of inhomogeneities of the distribution of acoustic-wave propagation speed. To determine the inhomogeneities that characterize attenuation as well, mathematical models are needed that adequately describe the physical processes of attenuation.
• (8)  
  We have proposed efficient methods for solving coefficient inverse problems of seismology, earthquake engineering and electromagnetic diagnosis of subsurface Earth layers.

It is our pleasant duty to thank the Corresponding Member of the Russian Academy of Sciences Vl V Voevodin for numerous discussions of the work. We are also grateful to Academician V A Kubyshkin, Director of Vishnevsky Institute of Surgery of the Russian Academy of Medical Sciences, for advice concerning medical aspects of the diagnosis of oncological diseases. This work was supported by the Russian Foundation for Basic Research (project no 13-07-00824-a).

Let us write out the mathematical problem that will allow us to compute the gradient of functional (23). We denote by ξ the pair ξ = (c, a), u_ξdξ = u_cdc + u_ada. Let us now find the part of the increment of functional (23) that is linear in arbitrary variation dξ = (dc, da)

$$\begin{eqnarray} &&{\rm d}\Phi ^{\prime }(u(\xi ){\rm ,}\xi )=\int _{ST}(u-U)(u_{\xi } \mathrm{d}\xi ) {\,\rm d}s\,{\rm d}t, \end{eqnarray} \tag{ A.1 }$$

where u_ξ is the Frechet derivative.

The function u(r, t) is the solution of problem (27), (28) for some ξ(r) (i.e., u(r, t) is an implicit function of ξ(r)), and therefore total differentiation with respect to ξ(r) in equation (27) yields

$$\begin{eqnarray} &&c(r)(u_{c} {\rm d}c)_{tt} +a(r)\Delta (u_{c} {\rm d}c)_{t} -\Delta (u_{c} {\rm d}c)+u_{tt} (r,t){\,\rm d}c=0, \nonumber \\ &&c(r)(u_{a} {\rm d}a)_{tt} +a(r)\Delta (u_{a} {\rm d}a)_{t} -\Delta (u_{a} {\rm d}a)+\Delta u_{t} (r,t){\,\rm d}a=0. \end{eqnarray} \tag{ A.2 }$$

We then introduce operator P to restrict the function to the domain t = 0, and derive from equation (28) that u(r, t = 0) ≡ P(u(r, t)) = 0 = const. Differentiation with respect to ξ(r) yields $(P(u(r,t)))_{\xi }^{^{\prime }} {\rm d}\xi =P(u_{\xi } {\rm d}\xi )=(u_{\xi } {\rm d}\xi )(r,t=0)=0$. Similarly, we derive from equation (28) that

$$\begin{eqnarray} &&(u_{\xi } {\rm d}\xi )(r,t=0)=(u_{\xi } {\rm d}\xi )_{t} (r,t=0)=0,\qquad \partial _{n} (u_{\xi } {\rm d}\xi )|_{ST} =0. \end{eqnarray} \tag{ A.3 }$$

Hence function (u_ξdξ)(r, t) for arbitrary variation dξ = (dc, da) is a solution of problem (A.2), (A.3). We now introduce operator A: Au = c(r)u_tt(r, t) + a(r)Δu_t(r, t) − Δu(r, t).

Let us consider what we refer to as the 'conjugate' problem to the main problem (27), (28)

$$\begin{eqnarray} &&Bw=c(r)w_{tt} (r,t)-\Delta \lbrace a(r)w_{t} (r,t)\rbrace -\Delta w(r,t)=0, \end{eqnarray} \tag{ A.4 }$$

$$\begin{eqnarray} &&w(r,t=T)=w_{t} (r,t=T)=0, \qquad \partial _{n} w|_{ST} =u|_{ST} -U, \end{eqnarray} \tag{ A.5 }$$

where u is the solution of the main problem (27), (28). We denote $(u_{\xi } {\rm d}\xi )(r,t)=\tilde{u}(r,t)$ for some variation dξ and consider the scalar product $(Bw,\tilde{u})$. We use formulas (29), (A.3) and (A.5), as well as the relation Δw(r, t = T) = 0, to obtain

$$\begin{eqnarray*} &&0 =(Bw,\tilde{u})=\int _{\Omega T}(Bw)\tilde{u} {\,\rm d}r\,{\rm d}t \\ &&{\hphantom{0 }}=\int _{\Omega }\int _{0}^{T}c(r)w_{tt} (r,t)\tilde{u}(r,t) {\,\rm d}t\,{\rm d}r-\int _{\Omega }^{}\int _{0}^{T}\Delta \lbrace a(r)w_{t} (r,t)\rbrace \tilde{u}(r,t) {\,\rm d}t\,{\rm d}r \\ &&-\int _{0}^{T}\int _{\Omega }\Delta w(r,t)\tilde{u}(r,t) {\,\rm d}r\,{\rm d}t\\ &&{\hphantom{0 }} =-\int _{\Omega }\int _{0}^{T}c(r)w_{t} (r,t)\tilde{u}_{t} (r,t) {\,\rm d}t\,{\rm d}r \\ &&-\int _{0}^{T}\int _{S}\partial _{n} \lbrace a(s)w_{t} (s,t)\rbrace \tilde{u}(s,t) {\,\rm d}s\,{\rm d}t +\int _{0}^{T}\int _{\Omega }\nabla \lbrace a(r)w_{t} (r,t)\rbrace \nabla \tilde{u}(r,t){\rm d}r\,{\rm d}t \\ &&-\int _{0}^{T}\int _{S}\partial _{n} w(s,t)\tilde{u}(s,t) {\,\rm d}s\,{\rm d}t +\int _{0}^{T}\int _{\Omega }\nabla w(r,t)\nabla \tilde{u}(r,t) {\,\rm d}r\,{\rm d}t \\ &&{\hphantom{0 }}=-\int _{\Omega }\int _{0}^{T}c(r)w_{t} (r,t)\tilde{u}_{t} (r,t) {\,\rm d}t\,{\rm d}r+\int _{0}^{T}\int _{\Omega }\nabla \lbrace a(r)w_{t} (r,t)\rbrace \nabla \tilde{u}(r,t) {\,\rm d}r\,{\rm d}t \\ &&-\int _{ST}(u(s,t)-U(s,t))\tilde{u}(s,t) {\,\rm d}s\,{\rm d}t+\int _{0}^{T}\int _{\Omega }\nabla w(r,t)\nabla \tilde{u}(r,t) {\,\rm d}r\,{\rm d}t . \end{eqnarray*}$$

We thus have

$$\begin{eqnarray} &&\int _{ST}(u(s,t)-U(s,t))\tilde{u}(s,t) {\,\rm d}s\,{\rm d}t =-\int _{\Omega }\int _{0}^{T}c(r)w_{t} (r,t)\tilde{u}_{t} (r,t) {\,\rm d}t\,{\rm d}r \nonumber \\ &&+\int _{0}^{T}\int _{\Omega }\nabla \lbrace a(r)w_{t} (r,t)\rbrace \nabla \tilde{u}(r,t) {\,\rm d}r\,{\rm d}t +\int _{0}^{T}\int _{\Omega }\nabla w(r,t)\nabla \tilde{u}(r,t) {\,\rm d}r\,{\rm d}t . \end{eqnarray} \tag{ A.6 }$$

We then consider the scalar product $(w,A\tilde{u})$. We use the relations (28), (A.2), (A.5) and Δu(r, t = 0) = 0 to obtain

$$\begin{eqnarray} &&(w,A\tilde{u})=\int _{\Omega T}w(r,t)A(\tilde{u}(r,t)) {\,\rm d}r\,{\rm d}t \nonumber \\ &&{\hphantom{(w,A\tilde{u})}}=-\int _{\Omega }\int _{0}^{T}\lbrace w(r,t)u_{tt} (r,t){\rm d}c(r)+w(r,t)\Delta u_{t} (r,t){\,\rm d}a(r)\rbrace {\,\rm d}t\,{\rm d}r \nonumber \\ &&{\hphantom{(w,A\tilde{u})}}=\int _{\Omega }\int _{0}^{T}\lbrace w_{t} (r,t)u_{t} (r,t){\rm d}c(r)+w_{t} (r,t)\Delta u(r,t){\,\rm d}a(r)\rbrace {\,\rm d}t\,{\rm d}r. \end{eqnarray} \tag{ A.7 }$$

On the other hand, we use relations (29), (A.3) and (A.5) to obtain

$$\begin{eqnarray} &&(w,A\tilde{u})=\int _{\Omega }\int _{0}^{T}w(r,t)c(r)\tilde{u}_{tt} (r,t) {\,\rm d}t\,{\rm d}r \nonumber \\ &&+\int _{\Omega }^{}\int _{0}^{T}w(r,t)a(r)\Delta \tilde{u}_{t} (r,t) {\,\rm d}t\,{\rm d}r -\int _{0}^{T}\int _{\Omega }w(r,t)\Delta \tilde{u}(r,t) {\,\rm d}r\,{\rm d}t \nonumber \\ &&{\hphantom{(w,A\tilde{u})}}=-\int _{\Omega }\int _{0}^{T}c(r)w_{t} (r,t)\tilde{u}_{t} (r,t) {\,\rm d}t\,{\rm d}r-\int _{\Omega }^{}\int _{0}^{T}a(r)w_{t} (r,t)\Delta \tilde{u}(r,t) {\,\rm d}t\,{\rm d}r \nonumber \\ &&-\int _{0}^{T}\int _{S}w(s,t)\partial _{n} \tilde{u}(s,t) {\,\rm d}s\,{\rm d}t +\int _{0}^{T}\int _{\Omega }\nabla w(r,t)\nabla \tilde{u}(r,t) {\,\rm d}r\,{\rm d}t \nonumber \\ &&{\hphantom{(w,A\tilde{u})}}=-\int _{\Omega }\int _{0}^{T}c(r)w_{t} (r,t)\tilde{u}_{t} (r,t) {\,\rm d}t\,{\rm d}r-\int _{0}^{T}\int _{S}a(s)w_{t} (s,t)\partial _{n} \tilde{u}(s,t) {\,\rm d}s\,{\rm d}t \nonumber \\ &&+\int _{\Omega }^{}\int _{0}^{T}\nabla \lbrace a(r)w_{t} (r,t)\rbrace \nabla \tilde{u}(r,t) {\,\rm d}t\,{\rm d}r +\int _{0}^{T}\int _{\Omega }\nabla w(r,t)\nabla \tilde{u}(r,t) {\,\rm d}r\,{\rm d}t \nonumber \\ &&{\hphantom{(w,A\tilde{u})}}=-\int _{\Omega }\int _{0}^{T}c(r)w_{t} (r,t)\tilde{u}_{t} (r,t) {\,\rm d}t\,{\rm d}r+\int _{\Omega }^{}\int _{0}^{T}\nabla \lbrace a(r)w_{t} (r,t)\rbrace \nabla \tilde{u}(r,t) {\,\rm d}t\,{\rm d}r \nonumber \\ &&+\int _{0}^{T}\int _{\Omega }\nabla w(r,t)\nabla \tilde{u}(r,t) {\,\rm d}r\,{\rm d}t. \end{eqnarray} \tag{ A.8 }$$

We derive from formulas (A.7) and (A.8)

$$\begin{eqnarray} &&\int _{\Omega }^{}\int _{0}^{T}\lbrace w_{t} (r,t)u_{t} (r,t){\rm d}c(r)+w_{t} (r,t)\Delta u(r,t){\,\rm d}a(r)\rbrace {\,\rm d}t\,{\rm d}r \nonumber \\ &&=-\int _{\Omega }\int _{0}^{T}c(r)w_{t} (r,t)\tilde{u}_{t} (r,t) {\,\rm d}t\,{\rm d}r+\int _{\Omega }^{}\int _{0}^{T}\nabla \lbrace a(r)w_{t} (r,t)\rbrace \nabla \tilde{u}(r,t) {\,\rm d}t\,{\rm d}r \nonumber \\ &&+\int _{0}^{T}\int _{\Omega }\nabla w(r,t)\nabla \tilde{u}(r,t) {\,\rm d}r\,{\rm d}t. \end{eqnarray} \tag{ A.9 }$$

We then derive from formulas (A.1), (A.6) and (A.9)

$$\begin{eqnarray*} {\rm d}\Phi ^{\prime }(u(\xi )) & =\int _{ST}(u-U)(u_{\xi } \mathrm{d}\xi ) {\,\rm d}s\,{\rm d}t \\ & =\int _{\Omega }^{}\int _{0}^{T}\lbrace w_{t} (r,t)u_{t} (r,t){\rm d}c(r)+w_{t} (r,t)\Delta u(r,t){\rm d}a(r)\rbrace {\,\rm d}t\,{\rm d}r \\ & =\int _{\Omega }\left[\left\lbrace \int _{0}^{T}w_{t} (r,t)u_{t} (r,t){\rm d}t \right\rbrace {\,\rm d}c(r)+\left\lbrace \int _{0}^{T}w_{t} (r,t)\Delta u(r,t){\rm d}t \right\rbrace {\,\rm d}a(r)\right]{\rm d}r. \end{eqnarray*}$$

We now isolate the parts linear in variations dc and da to derive the final formula for the gradient of functional Φ(u(ξ))

$$\begin{eqnarray} &&\Phi ^{\prime }_{c} (u(c))=\int _{0}^{T}w_{t} (r,t)u_{t} (r,t){\,\rm d}t ,\qquad \Phi ^{\prime }_{a} (u(a))=\int _{0}^{T}w_{t} (r,t)\Delta u(r,t){\,\rm d}t. \end{eqnarray} \tag{ A.10 }$$

Here u(r, t) is the solution of the main problem (27), (28), and w(r, t), that of the 'conjugate' problem (A.4), (A.5) for given c(r) and a(r).