PHYSICAL REVIEW E 84, 051915 (2011)
Angularly selective mesoscopic tomography Vadim Y. Soloviev,1,* Andrea Bassi,2,3 Luca Fieramonti,2 Gianluca Valentini,2,3 Cosimo D’Andrea,2,3 and Simon R. Arridge1 1
Departments of Computer Science, University College London, Gower Street, London WC1E 6BT, United Kingdom 2 Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy 3 Istituto di Fotonica e Nanotecnologie (IFN-CNR), Piazza Leonardo da Vinci 32, I-20133 Milano, Italy (Received 19 August 2011; revised manuscript received 12 October 2011; published 22 November 2011) We report three-dimensional tomographic reconstruction of optical parameters for the mesoscopic lightscattering regime from experimentally obtained datasets by employing angularly selective data acquisition. The approach is based on the assumption that the transport coefficient of a scattering medium differs by an order of magnitude for weakly and highly scattering regions. Datasets were obtained by imaging a weakly scattering phantom, which embeds a highly scattering cylinder of two to three photons’ mean path length in diameter containing light-absorbing inclusions. Reconstruction results are presented and discussed. DOI: 10.1103/PhysRevE.84.051915
PACS number(s): 87.57.−s, 42.30.Wb, 87.80.−y
I. INTRODUCTION
Light transport in a live object usually involves light scattering, the influence of which depends strongly on the scale at which imaging is being considered. At very small scales scattering may be negligible [1,2], whereas at large scales scattering is so dominant that light propagation can be considered as a diffusion process [3,4]. In between these extremes, for objects varying in size from several hundred micrometers to several millimeters, multiple scattering is present but not fully diffuse; this regime is known as the mesoscopic scale [4,5]. Two general techniques for modeling light propagation in the presence of multiple scattering are the stochastic Monte Carlo method [6] and the deterministic radiative transfer equation (RTE) [7,8]. Both methodologies are computationally very expensive, which precludes their routine application. For the mesoscopic scale, some approximations to the RTE were suggested recently. One of them is the Fokker-Planck equation [5,9–13], which assumes sharply peaked forward light scattering. Another approach takes into account only singly scattered photons with angularly selective intensity measurements, which allows application of the broken ray radon transform [14]. To provide a suitable model of the physical system used in our experimental study, we introduce an approximation to the RTE applicable to imaging at the mesoscopic scale when the scattering is not sharply peaked forward nor are all the photons singly scattered. This approximation results from some assumptions. We assume that scattering media consist of weakly and highly scattering regions, whose transport coefficients differ by an order of magnitude [15]. Such a situation is quite common for imaging in vivo. Thus, a live embryo or fetus is mostly transparent, with internal organs, bones, and brain being highly scattering. We also make an assumption that the phase function, which is included in the RTE collision term, can be approximated by three terms in its expansion over Legendre polynomials. Therefore, it contains the Rayleigh phase function as a special case. For the present, we neglect refractive index variation in the medium. In this paper, we consider a light-scattering phantom of a size comparable to the mean path length of a photon. In weakly scattering regions, ballistic and singly scattered
*
[email protected]
1539-3755/2011/84(5)/051915(9)
photons are present only. In highly scattering regions, light transport is considered to be a diffusion process. However, the diffusion approximation (DA) or the telegraph equation approximation (TE) is not a good approximation for scattering objects from which photons have a finite probability to leave without any scattering or absorption events. Realistic light transport models for small scattering objects should take into account both photon diffusion and light propagation in accordance with geometrical optics. Here we extend our previously reported approach [15] to the case of a more realistic phase function by keeping three terms in the phase-function expansion over Legendre polynomials and by verifying our methodology experimentally. The image reconstruction approach considered in this paper employs angularly selective intensity measurements [16,17]. Thus, optical properties are reconstructed by using optical projection datasets and scattered light outgoing from the medium at some angles with respect to the direct radiation. Datasets are obtained by imaging a weakly scattering phantom, which embeds a highly scattering cylinder of two to three photons’ mean path lengths in diameter and seven to eight mean path lengths in height containing light-absorbing inclusions. In the absence of scattering, inverse problems in biomedical optics can be formulated as the inversion of the radon transform or its analogs, such as the attenuated radon transform. For instance, optical projection tomography (OPT) was developed as an optical analog of x-ray computerized tomography (CT) and utilizes the inverse radon transform [18–23]. In the presence of scattering, inverse problems are more complex and mainly considered to be inverse-scattering problems. Inverse scattering is the problem of determining the characteristics of an object from surface measurement data of radiation scattered from the object. Except in special cases, reconstruction algorithms typically call for the solution of an optimization problem of an appropriately constructed cost functional. In this paper, the image reconstruction algorithm is based on the variational framework and involves repeated numerical solution of firstand second-order partial differential equations. This paper is organized as follows. In the next section, we discuss the direct problem and derive an approximation to the RTE, which is valid under our assumption. The direct solver is illustrated by computing camera images of a numerical phantom. Next, we briefly outline the reconstruction
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algorithm. Section III is devoted to experimental details, where the experimental setup and the light-scattering phantom are described. We present reconstruction results and discussions in Sec. IV. II. METHODOLOGY A. Direct problem
We start this section with the direct problem of light transport in turbid media. The most general description of light transport in scattering media is provided by the radiative transfer equation (RTE), which is the integrodifferential equation describing a balance of radiation along a given direction s (|s| = 1) [7,8]. The RTE in the Fourier domain (ω) can be written in the form of the first-order partial differential equation, s · ∇I + μI = λμB, where the function B in the source term reads B(r,s) = p(s · s )I (r,s )d 2 s + p(s · s0 )I0 (r,s0 ).
(1)
In Eqs. (1) and (2), (i) I denotes the intensity of light; (ii) I0 is the intensity of the direct radiation entering the domain along the direction s0 ; (iii) μ = μ + iω/c is the complex extinction coefficient; (iv) c is the speed of light in the medium; (v) μ is the transport coefficient, the quantity reciprocal to the mean path length of a photon in the medium between two successive scattering events; and (vi) λ denotes the albedo of a single scattering event. The albedo has a physical meaning of the probability for a photon to survive a scattering event and, therefore, λ ∈ [0,1]. The phase function p(s · s ) in Eq. (2) has a meaning of a probability distribution function. That is, the quantity p(s,s )d 2 s expresses a probability for a photon incident in the direction s to be scattered into the direction s. Because it is a probability distribution function, it integrates over the unit sphere to unity. In general, the phase function is anisotropic. In many physically meaningful cases, the phase function possesses azimuthal symmetry and depends only on a scalar product s · s = cos ϑ. In such cases, the phase function can be expanded over Legendre polynomials. Practically, series over Legendre polynomials are truncated. Retaining only the first three terms, we write the phase function in the form 3 1 32 2 p(s · s ) + 1 cos ϑ + cos ϑ , (3) 4π 3 + 2 3 + 2 where the coefficient 3/ (3 + 2 ) is included for normalization. Parameters 1 and 2 vary as 1 ∈ [−1,1] and 2 ∈ [0,∞). The case 1 = 1 and 2 = 0 gives the simplest form of the anisotropic phase function, which was considered earlier [15]. The case 1 = 0 and 2 = 1 results in the Rayleigh phase function [7,8,24]. Retaining more terms in the phase-function expansion over the Legendre polynomials makes the approach more general and flexible. The formal solution of the RTE, Eq. (1), is well known and given in the form of integration along rays: lmax I (r,s) = I0 |s·s0 =1 + λ (r − sl) μ (r − sl) 0
0
I u + 3s · q +
(2)
(4π)
l μ(r − sl )dl dl, × B(r − sl,s) exp −
where the integration is performed along a light ray from the observation point r in the reverse direction, −s, and lmax denotes the maximum distance of a ray’s path contributing to the intensity. Substitution of the function B, Eq. (2), into Eq. (4) results in the Fredholm integral equation of the second kind for the intensity I . To avoid solving the integral equation, an approximation to the function B can be found self-consistently with the RTE for some special cases. Let us assume that the medium consists of weakly and highly scattering regions, whose transport coefficients differ by an order of magnitude. We further assume that recorded photons coming from weakly scattering regions are scattered only once, i.e., 1/μ is a length scale on the order of physical dimensions of the scattering domain. For such a case, the method of successive approximations provides the function B in the form of the singly scattered direct radiation p(s · s0 )I0 [7]. On the other hand, in highly scattering regions the intensity in Eq. (2) is approximated by three terms,
(4)
15 ◦ ss: g, 2
(5)
where (i) s = (sin θ cos ϕ, sin θ sin ϕ, cos θ )T is the unit vector in the spherical system of coordinates; (ii) θ and ϕ are the ◦ polar and azimuthal angles, respectively; and (iii) ss denotes the nondivergent dyadic tensor, ⎛ 2 ⎞ sin θ cos2 ϕ − 1/3 12 sin2 θ sin 2ϕ 12 sin 2θ cos ϕ ◦ ⎜ ⎟ ss =⎝ 12 sin2 θ sin 2ϕ sin2 θ sin2 ϕ − 1/3 12 sin 2θ sin ϕ⎠. 1 2
sin 2θ cos ϕ
1 2
sin 2θ sin ϕ
cos2 θ − 1/3 (6)
The symbol “:” denotes
the double product of two tensors a and b such as a : b = ij aij bj i . The average intensity, u, the flux q, and the analog of the stress tensor g in Eq. (5) are defined according to 1 u= I (s) d 2 s, (7) 4π (4π) 1 q= sI (s) d 2 s, (8) 4π (4π) 1 ◦ g= ssI (s) d 2 s. (9) 4π (4π) ◦
The nondivergent dyadic tensor ss contains only five linearly independent entries. Therefore, the expansion of I over the ◦ orthogonal basis (1,s,ss) in Eq. (5) is completely analogous to the expansion of the intensity, I , over the spherical harmonics Ylm (θ,ϕ), where l and m are integers and 0 l 2 and −l m l, due to the linear dependence of functions forming ◦ the basis (1,s,ss) and Ylm (θ,ϕ). Then, the substitution of the approximate intensity, Eq. (5), into the function B, Eq. (2), gives 32 1 si sj − δij gij . B = p (s · s0 ) I0 + u + 1 si qi + 3 + 2 3 (10) Here and in the rest of this section, a summation over repeated indices is assumed. In Eq. (10), si are the components of the unit vector s, and the tensor gij is nondivergent.
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Next, we find equations satisfied by the average intensity u, the flux q, and the analog of the stress tensor g by taking moments of the RTE, Eq. (1). That is, multiplying the RTE conse◦ quently by 1, s, and ss and integrating over the whole solid angle, we obtain the system of first-order partial differential equations, ∂qi + ( μ − λμ) u = λμu0 , ∂xi ∂gij ∂u − 3κ + 1 λμ i , qi = −κ ∂xi ∂xj ∂qj σ ∂qi λμ2 σ ∂qk gij = − + + ij , + δij 2 ∂xj ∂xi 3 ∂xk 3 + 2
∂u ∂l 2 σ 32 σ ∂ ∂u κ − ∇ · κ∇u. + 3 + 2 ∂l ∂l 3 + 2
B (0) p (s · s0 ) I0 + u − 1 κ (11) (12) (13)
where the complex diffusion coefficient, κ, and an analog of the viscosity coefficient, σ , are defined as κ = (3 μ − λμ1 )−1 , λμ2 −1 5 μ− . σ = 2 3 + 2
Equations (19) and (20) are substituted into the function B, Eq. (10). Performing summation, we take into account that the terms ∂/∂xj κ∂u/∂xi are counted twice and make use of the identity si ∂u/∂xi = ∂u/∂l, where l is the distance parameter along the ray’s path. This gives
Moreover, the last term −∇ · κ∇u in Eq. (21) is replaced with μu in accordance with Eqs. (11) and (19). The λμ(u0 + u) − average intensity, u, entering the function B (0) is found by solving the TE, which is derived from Eqs. (11) and (19) by eliminating the flux in Eq. (11). In the compact form, the TE is written as
(14) (15)
1 I0 , 4π
1 i = κs0,i I0 , 4π 1 1 s0,i s0,j − δij σ I0 , ij = 4π 3
(16) (17) (18)
where s0,i (i = 1,2,3) denote the components of the unit vector s0 . We notice that the source terms i and ij in Eqs. (12) and (13) can be neglected. Thus, bearing in mind a weak formulation of the direct problem, we integrate Eq. (11) over an infinitesimally small volume and apply Gauss’ theorem to the first term. This gives q · ndA, where the integration is performed over the surface of the infinitesimal volume A. Then, we make use of Eqs. (12) and (13) for computing q · ndA. Because the volume is infinitesimally small, the surface integrals λμκI0 (s0 ·n)dA and λμσ I ndA in source terms can be approximated by 0 Furthermore, it is seen λμκI0 (s0 ·n)dAand λμσ I0 ndA. that the integrals (s0 ·n)dA and ndA vanish. In general, it is worth solving the system (11)–(13) numerically. However, solving the system requires a sufficient amount of memory for allocation of a nine-dimensional solution vector at every grid point of the domain. Moreover, in the context of the inverse problem, solving the system repeatedly could be exceedingly expensive. Therefore, we seek an approximate solution of this system. Departing from the telegraph equation approximation (TE) [3], we define the zero-order approximation for the flux as qi(0)
∂u −κ , ∂xi
u = λμu0 ,
(22)
= −∇ · κ∇ + μ − λμ.
(23)
where
The source terms in Eqs. (11)–(13) are given by u0 =
Summarizing, we solve Eq. (22) for the average intensity u and compute the intensity I at the observation point r according to Eq. (4) by using Eq. (21) and a ray-tracing algorithm. Some details on implementation of the direct solver are briefly outlined below. B. Implementation details
High-resolution imaging imposes certain constraints on mesh density. A dense mesh requires high-performance algorithms for solving the direct and inverse problems. In many cases, good performance is achieved by using the simplest and most computationally inexpensive approaches. In order to spare computational resources, efficient dynamic memory allocation is employed. In this study, a Cartesian mesh has been chosen. The entire computational domain is split into computational cells (voxels), whose dimensions correspond to a pixel’s dimension of the CCD array. All functions are approximated by piecewise constant functions having constant values in each computational cell. The Helmholtz equation (22) is solved by employing the finite-volume method [25,26]. Computation of the intensity in the scattering medium involves integrations along the rays’ paths. A ray integration in Eq. (4) is performed by using Siddon’s algorithm [27], which is the ray-tracing algorithm designed for Cartesian grids. The choice of an anisotropic phase function in the form (3) requires numerical evaluation of the terms l ∂u exp − μ(r − sl )dl dl, ∂l 0 0 l lmax ∂ ∂u exp − κ λμσ μ(r − sl )dl dl, ∂l ∂l 0 0
lmax
λμκ
(19)
which results in the zero-order approximation for the analog of the stress tensor, σ ∂ ∂u ∂ σ ∂ ∂u ∂u − δij gij(0) κ + κ κ . (20) 2 ∂xj ∂xi ∂xi ∂xj 3 ∂xk ∂xk
(21)
(24) (25)
along a ray according to Eqs. (21) and (4). Straightforward computation of derivatives ∂u/∂l as s · ∇u is not only inefficient but also results in a significant computational error.
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A better way to evaluate the path derivative is the following. We note that a piecewise constant function u satisfies ∂ u (r − sl) = (26) [u]li δ (l − li ) , ∂l i where [u]li = u− − u+ denotes a jump of u across a cell’s interface at l = li along the reverse direction, −s, where superscripts − (+) denote left (right) values of u at the intersected cell’s interface. Use of Eq. (26) in Eq. (24) gives ⎛ ⎞ imax i {λμκ}li [u]li exp ⎝− μj lj ⎠ . (27) i=0
j =0
Here, (i) the symbol {a}li denotes the cell’s interface value of a quantity a at l = li , which ischosen here as the interface average {a}li = (1/2) a + + a − ; (ii) the distance lj is the length of the ray’s path within a cell provided by Siddon’s algorithm, (iii) the index j enumerates cells on the ray path, and (iv) μj is the complex extinction coefficient of the j th cell. Furthermore, integration by parts by using ∂u/∂l|li ±li /2 = 0 and Eq. (26) in the second term, Eq. (25), results in ⎛ ⎞ imax i {κ}li [u]li ({λμ μσ }li − [λμσ ]li /li ) exp ⎝− μj lj⎠. i=0
j =0
(28)
To illustrate the effect of the phase function, we simulated camera images of a scattering phantom. The numerical phantom is a weakly scattering cylinder, whose transport coefficient is 0.1 mm−1 and the value of the albedo is 0.999. It contains a highly scattering cylinder with μ = 1.0 mm−1 and the same value of the albedo. The highly scattering cylinder is tilted and contains two light-absorbing inclusions. Inclusions are rods with λ = 0.25. One of the rods is half the length of the other. The direct radiation is modeled as parallel rays entering the domain in the direction s0 = (1,0,0)T . The camera rotates around the embedding weakly scattering cylinder. The viewing direction of the camera is defined by the outward normal to the CCD array n, which points toward the object. Images are shown for φ = {0,π/2,π,3π/2} in Fig. 1, where φ is the angle between s0 and n. The first row, Figs. 1(a)–1(d), displays scattered light from the numerical phantom in accordance with the Rayleigh phase function (1 = 0 and 2 = 1). The second row, Figs. 1(e)–1(h), shows recorded intensity scattered according to 1 = 1 and 2 = 0 in Eq. (3). As is seen, the choice of the phase function mainly affects the contrast in the intensity coming from weakly and highly scattering regions. The highest contrast gives the second case, 1 = 1 and 2 = 0, for φ = 0, Fig. 1(e), where the intensity coming from weakly scattering regions is 0. In both cases, two absorbing rods are visible on projection images, φ = π , in Figs. 1(c) and 1(g). On the other hand, the backscattering case, φ = 0, is useless for reconstruction of absorbing inclusions, as is illustrated in Figs. 1(a) and 1(e).
FIG. 1. (a)–(d) Scattered light from the numerical phantom in accordance with the Rayleigh phase function (1 = 0 and 2 = 1). (e)–(h) Scattered light when 1 = 1 and 2 = 0. Each row displays images for four angles φ = {0,π/2,π,3π/2}, where φ is the angle between the direction of the incident light, s0 , and the camera normal, n. 051915-4
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C. Inverse problem
In this subsection, we briefly outline the reconstruction algorithm, which was discussed in detail earlier [15]. The image reconstruction algorithm is based on the minimization of an appropriately constructed cost functional. The cost functional is built from (i) the error norm (or the objective function), which is the L2 norm of difference between measured and computed intensities, (ii) the Lagrangian terms, which are inner products of Lagrange multipliers with corresponding zero-valued functions, and (iii) regularization terms, which are required for correcting the ill-posedness of the inverse problem. Setting the first variation of the functional to zero results in the system of partial differential equations, whose solution is used for iterative reconstruction of optical parameters. We consider a simple experimental setup wherein positions of the source plane and the CCD camera are fixed, but the object under study is rotated. Computationally, it is much simpler to rotate the source and camera with fixed spatial orientation of the object. Then, the variational problem is formulated as a minimization problem of the cost functional: F = ς (ω) (E + L) dω + ϒ. (29) In Eq. (29), E is the error norm given by E = ξ (s) d 2 s χ (r) |IE − I |2 d 3 r,
(30)
V
where IE and I are experimentally recorded and computed excitation intensities in the direction s, respectively. The function ξ (s) represents sampling of the source camera’s positions, (31) δ (s − sn ) , ξ (s) = 0n
