Nonlinear approaches

Nonlinear approaches comprise a broad range of techniques used to recover optical and physiological parameters from PAT images. The majority of approaches involve casting the inversion as a least-squares problem. However, there are others that assume a known scattering coefficient to exploit the fact that the absorption coefficient can be expressed as a function of the PA image and unknown fluence, as performed in a fixed-point iteration.

The fact that the PA image appears as a term in the diffusion approximation can also be exploited by re-casting the DA assuming a known diffusion coefficient.

2.4.2.1 Fixed-point iteration

It is possible to write an expression for the absorption coefficient in terms of the measured image and the fluence, which is unknown:

µa= p^meas₀

ΓΦ . (2.32)

Assuming that Γ = 1 or known allows us to cast this as a fixed-point iteration as the current estimate of the absorption coefficient µ⁽ⁿ⁺¹⁾a will depend on the previous estimate of the fluence, Φ⁽ⁿ⁾:

µ⁽ⁿ⁺¹⁾_a = p^meas₀ Φ⁽ⁿ⁾(µ⁽ⁿ⁾a , µ_s) + %

, (2.33)

where % is a small number that ensures the denominator is never zero, without which the method could only be applied to regions of the image wher there is sufficient SNR. The absorption coefficient is estimated by iterating over this expression using a fluence model and iteration stops when the error between the measured image and the image estimate, µ⁽ⁿ⁺¹⁾a Φ⁽ⁿ⁺¹⁾, falls below some threshold. This method suffers from the significant draw-back that a priori knowledge of the scattering coefficient, which is generally unknown, is required. Accurate estimation of the absorption coefficient using this approach relies on the sensitivity of the PA image to the estimate of the scattering coefficient [97]; it has however been shown that estimates of the absorption coefficient made from noisy images and a correct guess of the background scattering coefficient is possible [98].

2.4.2.2 Re-arranging of the forward model

Banerjee et al. [99] noticed that the image divided by the Grüneisen parameter appears in the diffusion equation,

−∇ · D∇Φ + µ_aΦ = Q0, (2.34)

so making the substitution p^meas₀ /Γ = µaΦ,

−∇ · D∇Φ = Q₀− p^meas₀ /Γ, (2.35)

in turn allows this equation to be solved for Φ. The absorption coefficient can then be calculated via µa = ^pmeas0_ΓΦ . This however requires D to be approximated via ˆD =

1

3µs(1−g) = _3µ¹0

s, where the value of µ⁰_s is known, and of course requires Γ to be known.

In the situation where µ⁰_s is unknown, the accuracy of absorption estimates is then vulnerable to inaccuracies in the estimate of the reduced scattering coefficient. For this reason, schemes which can perform simultaneous estimation of the absorption and scattering coefficients are more attractive.

2.4.2.3 Linearisation of the forward model

Linearisation-type approaches can estimate the absorption and scattering coefficients simultaneously. This class of inversion expands the photoacoustic forward problem as a Taylor series about an estimate of the absorption and scattering coefficients, χ⁽⁰⁾ = [µ⁰_a µ⁰_s], which in practice can be their true value or some guess of their background value. If the photoacoustic image is formed by a product of the Grüneisen parameter and a forward operator F applied to χ such that

p₀(µ_a, µ_s; g) = ΓF (χ; g)[χ]. (2.36)

Then, the expansion is given by

p0(µa, µs; g) ≈ Γ h

F (χ; g)[χ⁽⁰⁾] + F⁰(χ)[χ − χ⁽⁰⁾] + F⁰⁰(χ)[χ − χ⁽⁰⁾]²+ . . . i

, (2.37) where F⁰ and F⁰⁰ are the first- and second- order Fréchet derivatives of the forward model. Therefore, despite the forward model being nonlinear, it can be linearised in the first Fréchet derivative by considering that p₀has been perturbed by some small changes

in µ_aand µ_s: ∆χ = [µ_a− µ⁽⁰⁾_a , µ_s− µ⁽⁰⁾_s ]

∆p0= ΓF⁰(χ)[∆χ], (2.38)

where ∆p0 = p^meas₀ − p₀. The update to χ is then given by

∆χ = 1

ΓF^0†[∆p0], (2.39)

where † is used to indicate the pseudoinverse (in practice this matrix inverse can be regularised via a number of methods to reduce noise or enforce smoothness in ∆χ) [100]. Perturbation-type methods such as those in Eq. (2.39) are only valid for small perturbations in the model parameters µaand µsand require the background absorption and scattering coefficients to be accurately known which limits the applicability of these techniques in general [101]. The formulation of the inversion in Eq. (2.39) can also be used within a minimisation scheme as achieved by Yao et al. [102], but the significant memory and algorithmic demands of computing F⁰restricts this approach to inversions of limited scale (e.g. few unknowns, low resolution of χ).

2.4.2.4 Noniterative methods

Bal et al. [103] proposed a noniterative method to uniquely recover the absorption and diffusion coefficients using the diffusion equation as the forward model. Here, the DA for two illumination positions (and thus two images H1 and H2) is re-arranged to yield

∇ · √

D µa

B

!

= 0, (2.40)

where

B = H1∇H₂− H₂∇H₁ = µ²_aΦ²₁∇ Φ₁ Φ₂



, (2.41)

and where Φ1 and Φ2 are the fluences corresponding to the two illuminations. This method allows stable reconstruction of D and µ_a when the vector field B connects every position in the domain to a point on the boundary, as demonstrated in a minimisa-tion based framework [104]. Nevertheless, the reliance of this method on the DA and sensitivity to noise due to the ratio of images would limit its application in practice.

2.4.2.5 Least-squares approaches

Minimisation-based approaches derive from the Bayesian framework (discussed in more detail later) and rely on the minimisation of an error function (χ) with respect to the parameter set, χ,

χ = arg min

χ

(χ) = arg min

χ

||y − F [χ]||. (2.42)

Note that in practice the problem in Eq. (2.42) is often subject to some regularisation, but the choice of regulariser is application-specific and is discussed in more detail where relevant.

Two classes of iterative least-squares minimisation approach are trust-region methods and linesearch methods [100]; trust-region methods use a local model of the error function at the current parameter estimate and identify a direction and step length within this region where the objective will undergo a sufficient change when the parameter set is updated. Linesearch methods, on the other hand, choose a search direction using gradient information and perform a 1D minimisation along the optimal trajectory.

Linesearch methods require a gradient to be computed to minimise Eq. (2.42). Two classes of linesearch method are: Newton and quasi-Newton methods. Newton methods

rely on calculation of the Hessian, the matrix of second-order partial derivatives of the error functional with respect to the model parameters. This approach suffers from the fact that computation of this matrix is not only time-consuming in an algorithmic sense, but memory demands scale quadratically with the number of unknowns meaning the Hessian is prohibitively large to store for large PA images which can contain millions of unknowns. In Newton methods, the descent direction is given by

p⁽ⁱ⁾ = −(∇²(χ)⁽ⁱ⁾)⁻¹∇(χ)⁽ⁱ⁾, (2.43)

where ∇²(χ)⁽ⁱ⁾ (the Hessian) and ∇(χ)⁽ⁱ⁾ (the gradient) are the second- and first-order derivatives of the error functional with respect to the model parameters at the i^th iteration. This approach has the advantage that the step-length, α⁽ⁱ⁾, is naturally 1 and has a quadratic rate of convergence. The computation of the derivatives of the error function are usually calculated using some numerical scheme, such as a finite difference algorithm (methods for gradient calculation are discussed more detail in Chapter 5) which can be computationally very costly to store and compute, particularly for the Hessian matrix.

Quasi-Newton methods only require calculation of the gradient and are therefore also termed ‘gradient-based’ methods. Under this scheme, the gradient is used to approximate the Hessian rather than explicitly computing it. A number of different algorithms exist for forming approximations to the Hessian matrix (e.g. Broyden-Fletcher-Goldfarb-Shanno (BFGS)) and the corresponding parameter updates. Convergence of quasi-Newton meth-ods is slower than the Newton approach, but is well-suited to large scale inverse problems where memory demands limit the range of algorithms that can be applied. Gradient-based approaches simply require a search direction p⁽ⁱ⁾ at the i^th iteration. The most intuitive of which is the gradient descent (or steepest decent) method whereby the de-scent direction is simply given by

p⁽ⁱ⁾(χ) = −∇⁽ⁱ⁾(χ). (2.44)

This scheme, unlike the Newton method, is not pre-conditioned and an appropriate step length, αⁱ, must be selected; αⁱcan either be pre-defined, obtained using a line search in the p direction, or can be found using some pre-conditioning scheme [105]. The step length must be chosen in a similar way when performing a nonlinear conjugate gradient (nCG) optimisation, which descends the error hypersurface using a series of steps along orthogonal directions with well-chosen step sizes [105] and typically requires fewer iterations to converge to the global minimum compared with the gradient descent method.

Gradient-based optimisation has been applied in conjunction with forward and adjoint models of diffusion [106] and radiative transport models [107]. As discussed previously, the diffusion model does not accurately account for the fluence in a region of interest for PAT and functional gradients _∂µ^∂

a and _∂D^∂ are not consistent with those obtained using forward and adjoint radiative transfer equations, meaning the reconstruction is limited in accuracy due to modelling error. Reconstructions using the RTE are accurate to within a few % of the true absorption and scattering distributions in the presence of 1% Gaussian noise [107]. However, this was achieved using four illumination positions and the reconstructed PA images were free from artefacts. In practice, it is rarely possible to illuminate the target from all sides and artefacts from acoustic reconstruction, such as limited view artefacts, will result in errors in the optical inversion. Possible means of remedying this issue are discussed later in this chapter.

2.4.2.6 Single-step methods

The optimisation-based approaches above aim to obtain the optical parameters (Γ, µa, µs) by inverting the absorbed energy density, H, which is first reconstructed from multiwave-length acoustic time-series p_d(x_d, t, λ)or time-series acquired by illuminating the target from different positions p_d(x_d, t, q(x_s, ˆs))where q(x_s)are sources at x_s positions. This one-step reconstruction can be achieved by combining the forward optical and acous-tic operators, which depend on (Γ, µ_a, µ_s)and c, respectively, into one single mapping, Λ(Γ, µa, µs; c). Ding et al. [108] employ this scheme using a FE model of diffusion for the light field and a finite difference model of acoustic propagation to reconstruct µ_a

and c, assuming Γ and the diffusion coefficient are known. The inversion is achieved using an optimisation scheme that computes gradients of Λ, _∂µ^∂Λ

a and ^∂Λ_∂c, using adjoint diffusion and wave equations and 8 distinct illuminations. A key benefit of this approach is that optical and acoustic parameters can be reconstructed simultaneously, however this relies on access to more data through the use of multiple illuminations in order to overcome any ill-posedness in this inversion. It was also found that run-time for single-step reconstruction of µa can be slower than the equivalent two-step (acoustic followed by optical) inversion [109].

2.4.2.7 Bayesian approach

Bayes relation states that given the data, y_m, the probability of parameters χ_nis propor-tional to the product of the prior distributions and the probability of the measurement given the parameters, or likelihood function, P (ym|χ_n)[110]:

P (χn|y_m) ∝ P (χn)P (ym|χ_n). (2.45)

P (χn) is an expression of the prior probability of the parameter set, while P (ym|χ_n) represents the likelihood of a measurement y_m being obtained given the parameters χn. Pior knowledge is included in the inversion through P (χn). In the presence of additive Gaussian-distributed noise, e = y_m− F (χ_n), the likelihood function is given by P_noise(y − F (χ_n)), where F is the forward model. It follows that P_noise is given by

P_noise(ym|χ_n) ∝ exp



−1

2[ym− F (χ_n)]^T Γ⁻¹_e [ym− F (χ_n)]



. (2.46)

which is normally distributed with covariance Γe. Written in this way Eq. (2.46) illus-trates the flexibility of the Bayesian approach in solving inverse problems corrupted by additive noise (it is applicable to problems affected by other noise models but this is not discussed here) because uncertainty on the priors and the likelihood function can incorporated. The Bayesian formulation of the inversion also allows derivation of the least-squares approach (Section 2.4.2.5), which can be achieved using the assumption

that Γe is diagonal and by taking the negative natural log of the expression for P_noise in Eq. (2.46) to reveal the error functional in Eq. (2.42).

In the estimation of the optimum parameter set χn, the maximum a posteriori (MAP) is often sought, i.e. the parameter set which maximises P (y_m|χ_n):

χ_{M AP} = arg max

χn

P (ym|χ_n). (2.47)

This approach has been demonstrated to remove artefacts in parameter estimates intro-duced by the acoustic inversion [111]. However, the memory and algorithmic demands of computing Γecan be significant for large datasets [112], meaning statistical inversion techniques are not always attractive for application in PAT.

In summary, there are two broad classes of inversion: those that neglect the impact of the fluence or assume it is constant with wavelength, and those that account for the fluence using some analytic or numerical model. Chapter 3 investigates the validity of the first class of inversion, while Chapters 5 and 6 consider novel ways of tackling model-based inversion and examine which optimisation scheme performs best for gradient estimates made using a MC model of light transport. Both types of inversion approach are not without their challenges, discussed in Section 2.4.3, and knowledge of these obstacles is important in maximising the accuracy of these inversion schemes.