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Mathematics of Photoacoustic and Thermoacoustic Tomography

Peter Kuchment, Leonid Kunyansky

arXiv:0912.2022v1math.APcs.NAmath-phmath.MPmath.NA

TL;DR

TAT/PAT combine electromagnetic and acoustic measurements to pair contrast with resolution, while this chapter surveys their mathematical models, problems, and algorithms. It presents uniqueness results under specified acquisition and sound-speed conditions, while noting limits on some theoretical results.

  • Problem

    The chapter examines how TAT data can support reconstruction and what mathematical conditions govern uniqueness, stability, incomplete data, and the range of possible measurements.

    Source: §2.3 · p. 7

  • Method

    The chapter surveys TAT/PAT mathematical models, reconstruction problems, and algorithms, including wave-equation and spherical-mean formulations.

    Source: §1 · p. 3, §2.3 · p. 7

  • Results

    The chapter reports uniqueness results for finite observation times and non-trapping sound speeds, as well as a harmonic-polynomial criterion for uniqueness in acoustically homogeneous media.

    Source: Uniqueness results for a finite… · p. 13, Uniqueness result for non-trapping… · p. 15, Uniqueness for acoustically… · p. 18

  • Takeaways & Limitations

    Combining electromagnetic absorption with ultrasound measurements pairs high contrast with sub-millimeter resolution in TAT.

    Source: §1 · p. 3

    For one uniqueness result, compact support is crucial to the known proof, and no proven analog is given for non-closed acquisition surfaces except specified analytic cases.

    Source: §3.1.1 · p. 13

Abstract

from arXiv · show

This is the manuscript of the chapter for a planned Handbook of Mathematical Methods in Imaging that surveys the mathematical models, problems, and algorithms of the Thermoacoustic (TAT) and Photoacoustic (PAT) Tomography. TAT and PAT represent probably the most developed of the several novel ``hybrid'' methods of medical imaging. These new modalities combine different physical types of waves (electromagnetic and acoustic in case of TAT and PAT) in such a way that the resolution and contrast of the resulting method are much higher than those achievable using only acoustic or electromagnetic measurements.

1 Introduction

TAT/PAT use electromagnetic irradiation to generate acoustic pressure waves, whose measurements support reconstruction of initial pressure. This hybrid approach combines electromagnetic contrast with ultrasound resolution.

1 Introduction

  • Electromagnetic absorption creates initial pressure differences between cancerous cells and healthy tissue, and acoustic measurements of the resulting wave support reconstruction of that pressure.The reconstructed initial pressure is described as carrying diagnostic information.Source: §1 · p. 3
  • TAT combines the higher contrast associated with electromagnetic absorption and the sub-millimeter resolution of ultrasound measurements.The chapter attributes these advantages to the complementary absorption and wavelength properties of the two wave types.Source: §1 · p. 3
  • The TAT/PAT physical principle was discovered in 1880, and imaging applications were suggested about a century later.Development as a viable medical imaging technique began in the mid-1990s.Source: §1 · p. 3
  • TAT/PAT is presented as a leading example of hybrid imaging, which combines radiation types to yield image quality unavailable from single-radiation modalities.The chapter situates its mathematical foundations in approximation theory, integral geometry, and sonar and radar.Source: §1 · pp. 3–4

2 Mathematical models of TAT

The chapter models TAT/PAT reconstruction as recovering initial pressure from measured acoustic data, using a wave-equation framework and, for constant sound speed, a spherical-mean formulation. It also outlines core mathematical questions and alternative detector geometries.

2.1 Point detectors and the wave equation model

  • The point-detector model seeks the initial pressure f(x) from measured pressure data g(y,t), with f naturally supported inside the domain bounded by a closed acquisition surface.The model assumes point-like omnidirectional transducers and uses the measured data as a boundary condition for the wave equation.Source: §2.1 · p. 4

2.2 Acoustically homogeneous media and spherical means

  • For constant sound speed, the wave-equation data are equivalent to spherical means of f on the acquisition surface; this relation breaks down when sound speed varies.The chapter treats both the wave-equation and integral-geometric forward operators.Source: §2.2 · pp. 5–6

2.3 Main mathematical problems arising in TAT

  • The central mathematical questions include uniqueness, stability, incomplete data, forward-operator range, and recovery of sound speed.The range of the forward operator can inform inversion, data completion, and error correction, and is connected to sound-speed determination.Source: §2.3 · pp. 6–7

2.4 Variations on the theme: planar, linear, and circular integrating detectors

  • Planar detectors measure integrals of initial pressure over planes, which can be recovered from measurements and used as planar Radon data.The described planar model assumes constant sound speed and a detector sufficiently large to be treated as infinite.Source: §2.4 · pp. 7–8
  • Linear detectors measure line integrals; repeating measurements across directions yields data sufficient to reconstruct f through inversion of the X-ray transform.The approach solves a two-dimensional wave problem for each direction before recovering the line integrals.Source: §2.4 · p. 9
  • Circular integrating detectors have a closed-form inverse solution, but the approach has not yet produced numerical examples or real-data reconstructions.Alternative thin, long, or wide detectors are motivated by finite-size transducer limits on image resolution.Source: §2.4 · pp. 7, 9

3 Mathematical analysis of the problem

This section addresses most of the mathematical issues introduced in Section 2.3, while deferring reconstruction algorithms to Section 4.

3 Mathematical analysis of the problem

  • The section addresses most of the previously listed TAT problems, except reconstruction algorithms, which are discussed in Section 4.The algorithms are deferred rather than included in this section.Source: §3 · p. 9

3.1 Uniqueness of reconstruction

The section characterizes when TAT measurements uniquely determine the initial pressure, considering acquisition-surface geometry, sound speed, and observation time. It establishes broad uniqueness results while identifying geometric and temporal limits.

3.1.1 Acoustically homogeneous media

  • Theorem 5 gives uniqueness on closed S even when f is not supported inside it; every non-uniqueness set must admit a vanishing nonzero harmonic polynomial.Sets that are uniqueness sets for harmonic polynomials are sufficient for unique TAT reconstruction, though this does not guarantee practical reconstruction.Source: §3.1.1 · p. 11
  • A set is a uniqueness set if it is nonalgebraic or contains an open part of a closed analytic surface; a plane is non-unique, but recovery is unique when f is supported entirely on one side.In R2, a set is a non-uniqueness set for compactly supported f if and only if it belongs to the union of a Coxeter cross and a finite set of points.Source: §3.1.1 · pp. 12–13

Uniqueness results for a finite observation time

  • For constant speed, data collected through 0.5D uniquely determines f on a smooth closed S [42]; shorter observation on a sphere leaves a central neighborhood unrecoverable.The finite-time result follows because each singularity emits two rays, at least one of which reaches S within D/2.Source: Uniqueness results for a finite… · pp. 13–14

Trapping and non-trapping

  • For variable sound speed, non-trapping means every ray with nonzero frequency tends to infinity; radial parabolic sound speeds provide a trapping example.Source: Trapping and non-trapping · p. 14

Local energy decay estimates

  • With compactly supported f and non-trapping c, local energy decays polynomially in even dimensions and exponentially in odd dimensions after T0 exceeds the domain diameter.Source: Local energy decay estimates · p. 15

Uniqueness result for non-trapping speeds

  • Smooth non-trapping c and closed S give unique recovery even when f is not fully enclosed by S [3].The reconstruction argument starts the wave problem at t = ∞ with zero conditions and measured data as boundary conditions.Source: Uniqueness result for non-trapping… · p. 15

Uniqueness results for finite observation times

3.2 Stability

TAT reconstruction stability depends on the sound-speed geometry and, for constant speed, whether the object is supported inside the acquisition surface. Non-trapping speeds permit sufficient stability, whereas trapping and invisible singularities lead to severe instability.

Stability

  • The chapter treats stability through Sobolev-space error bounds and singular-value decay, with faster decay indicating greater instability.Lipschitz and Hölder stability are introduced; logarithmic stability is not addressed.Source: §3.2 · p. 16
  • Forward smoothing makes inversion less stable: the stronger the smoothing, the less stable the reconstruction, with super-algebraic or exponential singular-value decay marking extreme instability.The Radon transform example smooths by adding (n − 1)/2 derivatives and is Lipschitz stable between the corresponding Sobolev spaces.Source: §3.2 · p. 16
  • For constant sound speed, reconstruction is Lipschitz stable when f is supported inside the closed acquisition surface, but recovery outside it is not even Hölder stable [103].Source: §3.2 · p. 17
  • For variable non-trapping sound speed, time-reversal reconstruction is Lipschitz stable for observation times exceeding T(Ω).The passage describes solving the wave equation in reverse time, starting from t = ∞.Source: §3.2 · p. 17
  • Trapping sound speed makes reconstruction not even Hölder stable, with super-algebraically decaying singular values because some singularity information never reaches the acquisition surface [91].The passage states this applies regardless of how long data are collected.Source: §3.2 · p. 17

3.3 Incomplete data

Incomplete TAT data raises questions of uniqueness, stability, and reconstruction quality: an open acquisition surface may still uniquely determine the object, but invisible singularities degrade reconstruction and stability. The chapter surveys sufficient uniqueness conditions and relates visibility to whether rays reach the observation surface.

3.3 Incomplete data

  • Incomplete TAT data can result from an acquisition surface that does not fully surround the object or from trapping, and the central questions are uniqueness and its effects on stability and image quality.Source: §3.3 · p. 17

Uniqueness for acoustically homogeneous media

  • For a non-closed acquisition surface in a homogeneous medium, uniqueness is guaranteed under any of several conditions, including non-algebraicity, being a uniqueness set for harmonic polynomials, or specified analytic-surface or support geometries.The sufficient conditions also include an open piece of a closed analytic surface and a support restriction relative to a tangent plane.Source: Uniqueness for acoustically… · p. 18

Uniqueness for acoustically inhomogeneous media

  • In variable-speed media, an open part of a strictly convex domain’s boundary determines f uniquely for T > T(Ω), when the speed is smooth and equals 1 outside the domain and f’s support does not reach its boundary.A modification not requiring strict convexity is also available [124].Source: Uniqueness for acoustically… · p. 18

3.3.2 “Visible” (“audible”) singularities

  • A singularity is visible when its ray reaches the observation surface in finite time; a region is in the visibility zone when all singularities based there are visible.In the illustrated inhomogeneous example, rays escape through an opening in the observation surface, leaving the flat phantom’s wavefront mostly invisible.Source: §3.3.2 · p. 19
  • Partial-data reconstructions preserve visible fronts better than invisible ones: vertical fronts are reconstructed with endpoint blurring, while a horizontal phantom with mostly invisible wavefront fails to reconstruct well.For a square phantom, invisible horizontal-boundary singularities blur while vertical boundaries remain clear.Source: §3.3.2 · pp. 19–20, Figure 6 · p. 20, Figure 7 · p. 20

3.3.3 Stability of reconstruction for incomplete data problems

  • If invisible singularities exist, inversion is not Hölder stable in any Sobolev spaces and the forward operator’s singular values decay super-algebraically [91]; visible singularities can be reconstructed with Lipschitz stability.The chapter also states that parts of f outside a closed observation surface are unstable to reconstruct.Source: §3.3.3 · p. 21
  • Replacing missing data with zeros can recover visible singularities, with smoothing at data junctions to avoid artifact singularities, but invisible singularities preclude stable recovery of correct f values.Source: §3.3.3 · p. 21

3.4 Discussion of the visibility condition

Visibility depends on whether geometric paths from a point reach the acquisition surface: lines in constant-speed media and rays in variable-speed media. Trapped rays can leave interfaces blurred even when the surface surrounds the object.

Visibility for acoustically homogeneous media

  • A point is visible in constant-speed media exactly when every line through it intersects the acquisition surface.With detectors on only the left half-circle, interfaces whose normals do not cross the surface are blurred.Source: Visibility for acoustically… · p. 21, Figure 8 · p. 22

Visibility for acoustically inhomogeneous media

  • In variable-speed media, a point is visible exactly when at least one of the two rays from each initial direction reaches the acquisition surface.The ray-based criterion replaces the line-based test for constant speed.Source: Visibility for acoustically… · p. 22
  • Trapped rays can prevent visibility even when the acquisition surface completely surrounds the object, leading to instability and blurred interfaces.The centered square in the trapping-speed example reconstructs very well because none of the rays carrying its singularities is trapped.Source: Visibility for acoustically… · p. 22

3.5 Range conditions

Range conditions characterize which data can arise from the forward operators and are useful for reconstruction, error correction, and incomplete-data completion. Complete descriptions are established for the spherical mean operator and, under narrower conditions, for the forward operator.

3.5 Range conditions

  • Ideal data satisfy infinitely many mandatory identities, and range descriptions support reconstruction, error correction, and incomplete-data completion.Range descriptions are also related to recovery of the unknown sound speed.Source: §3.5 · p. 22

3.5.2 The range of the forward operator W.

  • The forward operator W is harder to characterize, and complete range descriptions are unknown in even dimensions and for variable sound speed.In even dimensions, non-local transformations make the data fail to vanish at large times and have led only to clumsy descriptions using other operators.Source: §3.5 · p. 23, §3.5.2 · p. 25

3.5.1 The range of the spherical mean operator M.

  • Moment conditions for the spherical mean operator are unnecessary because they follow from the other range conditions.Smoothness and support together with moment conditions are incomplete; orthogonality conditions provide the additional constraints.Source: §3.5.1 · p. 24
  • For the spherical mean operator on a unit sphere, smoothness, support, and orthogonality conditions are necessary and sufficient for data to be representable.The result also holds at finite smoothness with the corresponding Sobolev-space requirements.Source: §3.5.1 · p. 24
  • For non-spherical acquisition surfaces, the complete range of the spherical mean operator has not been established, even though moment and orthogonality conditions hold.Appropriate smoothness and support conditions can be formulated at least for convex surfaces, but their completeness is unproven.Source: §3.5.1 · pp. 24–25

3.5.2 The range of the forward operator W.

  • For odd dimensions with a unit-sphere acquisition surface, data belong to the range of W exactly when the auxiliary time-reversed solution satisfies qt(x, 0) = 0 throughout the unit ball.The auxiliary problem imposes the measured data on the boundary and zero terminal values at time 2.Source: §3.5.2 · p. 25

3.6 Speed of sound reconstruction

Estimating sound speed matters because using an average for a varying speed can distort reconstructed values and tissue-interface shapes. The chapter discusses simultaneous speed and image recovery, but reports that the problem still requires significant analysis.

3.6 Speed of sound reconstruction

  • Replacing a varying sound speed with its average can significantly distort reconstructed values and tissue-interface shapes.The chapter asks whether sound speed and the tomogram can be recovered simultaneously, noting that determining the speed alone would permit subsequent image inversion.Source: §3.6 · p. 26
  • Numerical algebraic iterative inversions indicate that recovering both sound speed and initial pressure might be possible.The chapter says substantial analysis is still required and that direct use of range conditions has so far yielded weak, impractical results.Source: §3.6 · pp. 26–27
  • If two speeds are pointwise ordered and produce the same nonzero TAT data for some initial functions, the speeds must be equal.The stated condition is c1(x) ≥ c2(x) throughout the domain.Source: §3.6 · p. 26
  • If the initial function has nonzero Laplacian and two speeds produce the same TAT data, the speeds are equal.Source: §3.6 · p. 26

4 Reconstruction formulas and procedures

Most reconstruction techniques require measurements on a closed surface surrounding the object; when that geometry is unavailable, approximate reconstruction techniques are used.

4 Reconstruction formulas and procedures

  • Most reconstruction methods require data collected on a closed surface surrounding the object, while incomplete-surface settings call for approximate reconstruction techniques.The chapter distinguishes constant-speed methods from methods for known variable sound speed.Source: §4 · p. 27

4.1 Full data (closed acquisition surfaces)

For constant sound speed, reconstruction from closed-surface measurements reduces to inversion of the spherical mean transform. The section reviews series, eigenfunction, and closed-form approaches, each with distinct geometry or data conditions.

4.1.1 Constant speed of sound

  • Known constant sound speed reduces reconstruction to inverting the spherical mean transform from spherical integrals measured on a closed surface.The initial pressure is supported inside the region bounded by the acquisition surface.Source: §4.1.1 · p. 27

Series solutions for spherical geometry

  • Series inversion for circular and spherical acquisition surfaces uses harmonic decompositions, but Bessel-function zeros make noisy-data division require regularization.Replacing J0 with a Hankel function avoids division by real zeros; the method extends to 3D using spherical harmonics and spherical Bessel functions.Source: Series solutions for spherical geometry · pp. 27–28

Eigenfunction expansions for a general geometry

  • Eigenfunction expansion applies to closed surfaces with known Dirichlet-Laplacian eigenpairs, and is especially efficient for cubes with explicit sine-product eigenfunctions.The approach reconstructs f inside the acquisition surface exactly even when its support extends outside, unlike the closed-form FBP methods discussed here.Source: Eigenfunction expansions for a general… · pp. 28–30

Closed form inversion formulas

  • Closed-form inversions include odd-dimensional spherical formulas, a 3D universal backprojection formula valid for planes and cylinders, and formulas for arbitrary n ≥2.The formulas (26)–(29) are not equivalent on data outside the range of the spherical mean transform.Source: Closed form inversion formulas · pp. 30–31

Greens’ formula approach and some symmetry considerations

The section relates closed-form inversions to Green’s formula and symmetry, then reviews alternatives and their practical constraints. It also describes reconstruction in variable-speed media and the computational cost of an eigenfunction approach.

Greens’ formula approach and some symmetry considerations

  • Green’s theorem would reconstruct the initial pressure from boundary pressure and its normal derivative, but closed-form methods eliminate the unmeasured normal derivative.In 3D, observation through the wave’s escape time suffices; the approach motivates the closed-form inversions reviewed in the section.Source: Greens’ formula approach and some… · p. 32
  • Symmetry of boundary integrals yields the kernel symmetry used in inversion derivations and explains the universal backprojection formula’s validity for spheres, infinite cylinders, and planes.The authors consider other acquisition surfaces unlikely to admit closed-form formulas because the required symmetry is shared only by these three surfaces.Source: Greens’ formula approach and some… · pp. 32–33

Algebraic iterative algorithms

  • Algebraic iterative reconstruction discretizes the data equation and solves the resulting linear system; it can improve images but is notoriously slow.It has been used for partial-data reconstructions.Source: Algebraic iterative algorithms · p. 33

Parametrix approaches

  • Parametrix methods recover interface jumps and often their locations, but not reliably lower spatial frequencies, and their approximations do not converge as discretization is refined and noise removed.They can serve as approximate images or starting points for iterative algorithms; using a parametrix as a preconditioner can accelerate convergence.Source: Parametrix approaches · p. 34

Numerical implementation and computational examples

  • Discretized exact formulas use stable backprojection and mildly unstable differentiation, with noise sensitivity reported as similar to classical X-ray FBP.In 2D, the filtration in formulas (34)–(36) can be computed with two Hilbert transforms in the frequency domain.Source: Parametrix approaches · p. 34
  • For formula (34), smoothing noisy simulated data suppresses noise but also blurs edges.The example uses 257 equally spaced projections, each with 129 spherical integrals, and noise at 15% of signal L2 intensity.Source: Parametrix approaches · p. 35, Figure 10 · p. 35

4.1.2 Variable speed of sound

  • Variable-speed reconstruction is important when tissues such as soft tissue and bone cause significant acoustic inhomogeneity, which can distort values and interface locations if ignored.The reviewed methods assume known, smooth, positive, non-trapping sound speed; time reversal also works for variable speed and even dimensions, using a finite cutoff in practice despite exact reconstruction requiring infinite time.Source: §4.1.2 · pp. 35–36, Time reversal · p. 36

Time reversal

  • The variable-speed eigenfunction expansion generalizes the method of without requiring the whole-space Green’s function, but computing many eigenpairs and summing the series may be too time-consuming.The time-reversal formula is valid under the non-trapping condition, though its operator-function form may limit usefulness.Source: Time reversal · p. 37

4.2 Partial (incomplete) data

Partial acquisition surfaces make accurate reconstruction difficult, especially when interfaces are invisible. The section reviews approximations for constant sound speed and identifies variable-speed partial-data reconstruction as largely open.

4.2 Partial (incomplete) data

  • Most exact inversion methods for closed surfaces do not extend to partial data, where accurate reconstruction requires visibility of the relevant interfaces.When visibility fails, invisible interfaces are smoothed; when it holds, reconstruction is mildly unstable [103].Source: §4.2 · p. 38, §4.2.1 · p. 38

4.2.1 Constant speed of sound

  • Partial-data parametrices zero-fill missing measurements and weight backprojections to compensate for duplicated directions; they recover jumps correctly but are not exact.Iterative refinement further improves images in numerical experiments; another method generates missing data from moment range conditions but does not recover values well.Source: §4.2.1 · pp. 38–39
  • For constant sound speed, the method of approximates plane waves using single-layer potentials and can yield very accurate reconstructions when every point in the region is visible.Its 2D examples use detectors on part of a circular arc and show accurate-data, noisy-data, and smoothed noisy-data reconstructions.Source: §4.2.1 · pp. 39–40, Figure 11 · p. 40
  • The [83] method’s costly potential densities need computing only once for a fixed acquisition surface, after which reconstruction is extremely fast.Recovered Fourier-transform values can be inverted numerically or through an FBP reduction.Source: §4.2.1 · pp. 39–40

4.2.2 Variable speed of sound

  • Variable-speed reconstruction from open surfaces remains largely open: some characteristics may miss the detectors, and generalizing may require prohibitively costly position-dependent Green’s functions.A time-reversal initialization with missing data filled by zeros or an extension, followed by fixed-point refinement, is proposed but its key questions remain open; a reverberant cavity is another approach.Source: §4.2.2 · pp. 40–41

5 Final remarks and open problems

The chapter closes by identifying open questions in uniqueness, inversion, and forward-operator range, alongside practical constraints in measurement and unresolved steps between initial pressure and tissue properties.

Open problems

  • Uniqueness acquisition sets are practically resolved, but proofs for rapidly decaying noncompact support and nonclosed surfaces remain open; higher-dimensional Conjecture 9 is also unresolved.The known theorem proof relies on compact support, while techniques in and might help address the open questions.Source: §5 · p. 41
  • Closed-form inversions are known only for spherical, cylindrical, and planar surfaces, and do not work correctly when the sought function extends outside the acquisition surface.Time-reversal and eigenfunction-expansion methods do not share this support-related deficiency.Source: §5 · pp. 41–42
  • Reliable reconstruction methods for partially surrounding detectors in acoustically inhomogeneous media remain unavailable apart from algebraic iterative approaches.This contrasts with the acoustically homogeneous case.Source: §5 · p. 42
  • Complete range descriptions remain unknown in even dimensions and are not established for non-spherical observation surfaces or variable sound speed.For constant sound speed and arbitrary closed surfaces, moment and orthogonality conditions are available but incomplete; inhomogeneous media also have incomplete orthogonality conditions.Source: §5 · p. 42
  • Unique determination of sound speed from TAT data remains largely open, and the chapter's analysis omits attenuation compensation, which is not fully resolved.Attenuation modeling and compensation are addressed in [74].Source: §5 · p. 42
  • Full Cauchy data would enable unique determination and simple reconstruction of f, but transducers do not measure the required normal derivative.Techniques relying on full Cauchy data therefore require further mathematical and experimental study.Source: §5 · p. 42
  • An added high-frequency filter in FBP-type formulas implements local tomography, stably emphasizing singularities such as tissue interfaces even with incomplete data.With incomplete data, invisible parts are lost.Source: §5 · p. 42
  • Recovering tissue optical parameters from the reconstructed initial pressure f is non-trivial because f is related to, but not identical to, those features.The problem is addressed, probably for the first time, in [20].Source: §5 · pp. 42–43
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