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Photo-acoustic tomography in a rotating measurement setting

Photo-acoustic tomography (PAT) aims to leverage the photo-acoustic coupling between optical absorption of light sources and ultrasound (US) emission to obtain high contrast reconstructions of optical parameters with the high resolution of sonic waves. Quantitative PAT often involves a two-step procedure: first the map of sonic emission is reconstructed from US boundary measurements; and second optical properties of biological tissues are evaluated. We consider here a practical measurement setting in which such a separation does not apply. We assume that the optical source and an array of ultrasonic transducers are mounted on a rotating frame (in two or three dimensions) so that the light source rotates at the same time as the US measurements are acquired. As a consequence, we no longer have the option to reconstruct a map of sonic emission corresponding to a given optical illumination. We propose here a framework where the two steps are combined into one and an absorption map is directly reconstructed from the available ultrasound measurements.


Introduction
Photo-acoustic tomography (PAT) is a novel medical imaging modality that aims to image the optical properties of biological tissues with high resolution. It combines the high contrast of optical (mostly absorption) parameters with the high resolution of ultrasound. As radiation propagates into tissues, a small fraction is transformed into sonic waves by the photo-acoustic effect. These sonic waves propagate through the domain and are measured by an array of transducers at the boundary of the domain of interest Ω ⊂ R n , where n is spatial dimension.
Mathematically, sound propagation is modeled by the following scalar wave equation: Here the function c(x) is assumed to be a smooth function and H(x) is the amount of acoustic signal generated by the absorbed radiation of a short pulse of light propagating throughout the domain. Its expression is given by where for each x ∈ Ω, u(x) is the density of radiation reaching point x, σ(x) is the absorption coefficient, and λ(x) is the Grüneisen coefficient, which characterizes the amount of ultrasound generated by each absorbed photon. A reasonable model for the propagation of radiation is given by the following secondorder elliptic equation where Ω is a smooth bounded domain in R n (n ≥ 2), γ(x) is a (scalar) diffusion coefficient and f describes how light enters the domain Ω. Photoacoustic Tomography aims to reconstruct (γ(x), σ(x)) as well as possibly λ(x) from the measurements v(t, x)| ∂Ω of the acoustic pressure leaving the domain Ω at each point x ∈ ∂Ω and each positive time t > 0; as well as for each available probing illumination f . Such reconstructions are typically done in two steps. In a first step, the sonic source H is reconstructed from the measurements v(t, x)| ∂Ω . This is a well-posed problem, at least in the presence of complete data and when sound speed c(x) is known; see, e.g., [11,12,14,15,19,20] and their references for experimental and theoretical works on the first, qualitative, step of PAT.
Once H(x) has been reconstructed for one or more illuminations f on ∂Ω, the second, quantitative, step of PAT allows us to obtain explicit reconstructions of γ, σ, and λ in some cases. We refer to [2,3,5,6,7,22] as well as their references for several works on the problem. Note that high resolution reconstructions can typically not be achieved by purely optical measurements, which are modeled by a problem that has similar stability properties to the standard electrical impedance tomography problem; see [1].
In this paper, we consider an experimental setting [4,21] in which such a separation into first and second steps is not feasible. The reason is that the source of radiation f and the small array of transducers performing the acoustic measurements are mounted on a rotating frame. In other words, as the pressure v(t, x) is measured by a rotating array of transducers on ∂Ω, it corresponds to a radiation source f that also rotates. We thus no longer acquire a pressure v(t, x) that corresponds to a single illumination f and cannot even define a meaningful initial condition H(x). The objective of this paper is to present a mathematical framework for this experimental setting, whose main advantage is that it allows for a clear spatial separation between the light source and the array of detectors.
As we mentioned above, the qualitative and quantitative steps of PAT need to be merged into one reconstruction. In order to simplify the presentation, we assume that the Grüneisen coefficient is known and set to 1. We also assume that the diffusion coefficient γ is known and is normalized to 1. Under these assumptions, we present a theory for the reconstruction of the absorption coefficient σ from knowledge of pressure measurements at the domain's boundary in the aforementioned rotating setting.
The rest of the paper is structured as follows. The measurement setting and our main results are presented in Section 2. The proof of our main result is split into a proof of a linearized version in Section 3 and a proof of the full nonlinear inverse problem in Section 4.

Measurement setting and main results
We consider the stable reconstruction of the absorption coefficient σ(x) from multiple partial measurements under the assumption γ = λ = 1 inside a ball in R n (n ≥ 2). More precisely, let B ρ be the ball of radius ρ in R n and assume Ω ⋐ B ρ . Let σ ∈ W 1,∞ 0 (Ω), and define Θ : a finite number of rotations around the origin in R n . Now fix 0 where We shall assume that g is not identically zero and nonnegative. In practical applications, we may envision g to have a small support on ∂B ρ , and certainly to be supported away from the location of the ultrasound transducers we now consider.
Let v i be the solution of the wave equation where c ≥ c 0 > 0 is the sound speed, and c − 1 is assumed to be supported inB ρ . In general the time-dependent wave solution at the boundary of the ball B ρ is given by In practical setting that we consider here, we have access to v i on [0, T ]× Γ for some Γ ⋐ ∂B ρ and not on the whole domain (0, ∞) × ∂B ρ . Typically the support of the ultrasound transducers Γ is relatively small and away from the support of the optical source to avoid measurement interferences.
To model this restriction, fix Γ ⊂ ∂B ρ and define Here, we are interested to know if the absorption coefficient σ(x) can be stably determined from the finite number of rotating partial measurements where This is a reasonably faithful model for the experimental setups described in [4,21], to which we refer the reader for additional details.
We now introduce additional hypotheses and notation in order to state our main result; theorem 2.1 below. One of our main theoretical tools is a description of acoustic wave propagation in the domain B ρ mostly following the presentation in [15]. As in [15], and for a given h, we define v to be the unique solution of where ϕ is the harmonic extension of h(T, ·) in B ρ . We also define the wave solution at time t = 0, and set where s(y) is a continuous function on ∪ R i ∈Θ Γ i indicating how long measurements need to last at every measurement point. It is know from, e.g., the work in [15] that longer times are necessary for stability purposes than for injectivity purposes. As in ( [15]) and to guarantee injectivity of the measurement operator, we will assume that there exists j ∈ {1, 2, ..., m} such that ∀x ∈ Ω, ∃y ∈ Γ j with dist(x, y) < s(y), where dist(x, y) denotes the distance with respect to the metric c −2 g (g is the Euclidean metric in R n ). This assumption is partially technical: it imposes that the measurements for one of the rotation step j are taken for a sufficiently long duration that the measurement operator (mapping σ to the available measurements) is injective.
To guarantee stability of the reconstruction of σ, we need the following stronger assumption on s(y): ξ))) ∈ G for η = + or η = −, (12) where γ η (x, ξ) are the integral curves of the corresponding Hamilton vector field (see Chapter 6 in [8]) and We refer to [15] for additional details on this assumption, which here simply means that any singularity of σ at position x ∈ B ρ and in direction ξ propagates to a singularity in the measurement set G.
Note that since Ω is compact there exists an open set G ′ ⊂ G such that (12) still holds. Define where π x is the projection map. Now fix the cut-off functions The following is the main result of this paper.
Theorem 2.1 Let B ρ be the ball of radius ρ in R n , Ω ⋐ B ρ , and σ ∈ H 1 0 (Ω). Let m ∈ N and assume that (11) and (12) hold. If where C Ω is the best constant in the classical Poincaré inequality on H 1 0 (Ω). Then the following stability estimate holds where C > 0 is independent ofσ and σ.

Remark 2.2 Notice that
C Ω is small for a small region Ω (Ω ⊂ B r for some small r), and therefore the condition (13) is satisfies if the support of σ is small in B ρ . This is consistent with the experiments in [4] where the method is applied on small animals.

Stability of the Linearized Problem
In this section we study the linearized problem associated with (4)- (7). Fix σ ∈ C ∞ 0 (Ω) and letσ ∈ C ∞ 0 (Ω). Assume u i ,ũ i be the corresponding solutions of (4). Then where δu i = u i −ũ i and δσ = σ −σ. Thus q(x, ξ) for all α, β ∈ N n . See Chapter 1 in [8] for more details about symbols and oscillatory integrals. Now let δp i (t, x) be the solution of the wave equation Notice that σu i −σũ i = ∆(δu i ).

Hence modulo smooth terms
where the phase function ϕ ± are homogeneous of order 1 and solve the eikonal equations a ± are amplitudes of order zero satisfying the corresponding transport equations (see equation V.1.50 in [17]). Now define F ± to be the Fourier integral operators and P i to be the pseudodifferential operator Then Eliminate the dependence on y in the symbol a(x, y, ξ) to get The composition of the Fourier integral operator F ± with the pseudodifferential operator P i is a Fourier integral operator with the same phase ϕ τ and amplitude b τ (t, x, y, ξ) = u i (y)a τ (t, x, ξ) mod S −1 (B ρ × R n ) (see [17]). Therefore The measurements are modeled by the operator The above pseudodifferential calculus in (17) indicates that H δσ := m i=1 χ i Λ(u i δσ) is the higher order term in Λ * (σ −σ), and L δσ := m i=1 χ i Λ(σδu i ) may be controlled by H δσ for small δσ (see Section 4). Hence we first study invertibility of the operator (18) and find an approximate inverse. Let A be the back-propagation operator defined in (9) and set Proposition 3.1 The operator κ is a zero order pseudo-differential operator in a neighborhood of Ω with principal symbol Consequently if (11) and (12) hold, then κ is an elliptic Fredholm operator on H 1 0 (Ω), and there exists C > 0 such that Proof. First note that κ(δσ) = Σ m i=1 A (χ i δp i (t, x)) . It follows from Theorem 3 in [15] that the principle symbol of κ is given by (20). It follows from maximum principle that u i ≥ β > 0 in Ω for all 1 ≤ i ≤ m. Since G satisfies (12), Thus the operator κ is elliptic, and therefore it follows from the mapping properties of the back-propagation operator A (see [10] and [15]) that Since (11) holds for some 1 ≤ j ≤ m, by Theorem 2 in [15], the measurement χ j Λ(σu j ) uniquely determines σu j in Ω. On the other hand, u j is the unique solution of −∆u j = −σu j u j | ∂Bρ = g j ≥ 0.
By strong maximum principle we have u j > 0 in Ω. Hence if (11) holds, then σ = σu j u j is uniquely determined from a single measurement χ j Λ(σu j ). Consequently the operator m i=1 χ i Λ(u i δσ) is also injective. Therefore it follows from Proposition V.3.1 in [18] that the estimate (21) holds for some constant C > 0, possibly different from the above constant.

Stability of the Nonlinear Problem
In this section we present the proof of Theorem 2.1. Let B ρ be the ball of radius ρ in R n and assume σ andσ are essentially bounded in B ρ and Ω := supp(σ −σ) ⋐ B ρ . For fixed 0 ≤ g ∈ H 1/2 (∂B) and for R i ∈ Θ define u i (x) to be the unique solution of Similarly letũ i be the unique solution of Then Proof of Theorem 2.1. Since the mapping Λ : [15]), it is enough to prove the theorem for σ,σ ∈ C ∞ 0 (Ω). The general result will follow from a standard density argument.
Multiply (22) by δu i and integrate by parts and use Hölder's inequality to get By Poincaré inequality, there exists C Ω such that where C ρ is dependent of δu i . Thus we have ∇δu i L 2 (Bρ) ≤ C ρ u i δσ L 2 (Bρ) . Therefore where C Ω is the best constant in the classical Poincaré inequality on H 1 0 (Ω). Thus we have σδu i H 1 0 (Ω) ≤ C ρ C Ω σ W 1,∞ (Ω) u i δσ H 1 0 (Ω) .
On the other hand, since the mapping Λ : where M g is the maximum of g on ∂Ω. Therefore there exists η > 0 such that if for some C * > 0 independent ofσ and σ.