Abstract

Perturbation Monte Carlo (pMC) has been previously proposed to rapidly recompute optical measurements when small perturbations of optical properties are considered, but it was largely restricted to changes associated with prior tissue segments or regions-of-interest. In this work, we expand pMC to compute spatially and temporally resolved sensitivity profiles, i.e. the Jacobians, for diffuse optical tomography (DOT) applications. By recording the pseudo random number generator (PRNG) seeds of each detected photon, we are able to “replay” all detected photons to directly create the 3D sensitivity profiles for both absorption and scattering coefficients. We validate the replay-based Jacobians against the traditional adjoint Monte Carlo (aMC) method, and demonstrate the feasibility of using this approach for efficient 3D image reconstructions using in vitro hyperspectral wide-field DOT measurements. The strengths and limitations of the replay approach regarding its computational efficiency and accuracy are discussed, in comparison with aMC, for point-detector systems as well as wide-field pattern-based and hyperspectral imaging systems. The replay approach has been implemented in both of our open-source MC simulators - MCX and MMC (http://mcx.space)

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Diffuse optical tomography (DOT) and fluorescence molecular tomography (FMT) with near-infrared light have been active research areas for both clinical and pre-clinical applications for over three decades. The appeal of these tomographic techniques resides in their high sensitivity, relatively low cost and unique utilization of intrinsic and exogenous functional biomarkers in charactering optically-thick tissue samples noninvasively [1]. These methods aim to recover the 3D distributions of either fluorophores or intrinsic absorption and scattering contrasts inside turbid media by employing measurement data collected on the surface of tissues and match them with an accurate light propagation forward model through solving an inverse problem [2]. Over the past years, DOT and FMT have been successfully applied to imaging human breast cancers [3], prostate cancers [4], muscle functions [5], joint conditions [6], peripheral vascular diseases [7], brain functions [8], small animals [9], and have also found applications in imaging ex vivo biological samples [10] as well as engineered tissues [11]. Regardless of the application, building an accurate and efficient forward model remains crucial to the quantitative imaging performance of inverse model-based techniques.

It has been widely agreed that the radiative transfer equation (RTE) provides an accurate model for light propagation in complex bio-tissues but it is known to be difficult to solve. Therefore, the diffusion equation (DE), a low-order expansion to the RTE, is usually adopted as an alternative to generate approximated solutions [12]. With the introduction of more sophisticated imaging strategies over the past decade, such as mesoscopic imaging [13], time-domain imaging with early photons [14] and whole-body imaging with structured light [15,16], DE becomes increasingly limited in both accuracy and flexibility. In these cases, the Monte Carlo (MC) method, a stochastic solver to the RTE, has gained increasing popularity.

Because MC solves for the RTE directly, in many studies, it was considered as the “gold standard” for modeling photon migration in biological tissues [17,18]. Nevertheless, MC was not deemed suitable for providing the forward solutions in DOT/FMT due primarily to two limitations. First, only limited types of domain configurations, such as layered and voxelated space, are supported by public MC packages, restricting their use in modeling complex tissues. This limitation was addressed by the recent development of mesh-based MC methods [19,20]. Second, MC has been associated with high computational expenses when simulating large number of photons especially in 3D applications. Thanks to the wide availability of modern accelerators such as general purpose graphics processing unit (GPGPU), the simulation time of MC has been reduced by orders of magnitude over the past years [21]. Additionally, efficient MC formulations such as the perturbation Monte Carlo (pMC) have been proposed to speed up the forward computation of MC [22,23]. Hence, the combination of efficient MC formulations with increased computation power has led to a rejuvenated interest in using MC to compute the forward model for DOT/FMT [24,25].

A Jacobian matrix, also known as the sensitivity matrix, is a central element in model-based DOT/FMT image reconstruction. It represents a map associating the changes of localized optical properties with the changes of measurements from selected source-detector pairs. Traditionally, the adjoint Monte Carlo (aMC) is adopted as an efficient method to calculate the Jacobians, especially when optodes are point-sources and point-detectors [25, 26]. The adjoint method works by multiplying the light fields of a forward simulation, computed by propagating photons from the source position, and an adjoint simulation, computed by propagating photons from an imaginary source at the detector position [27]. However, with the rise of more advanced DOT imaging architectures, such as those leveraging structured light illumination or detection [28], the aMC method becomes cumbersome with the rapid increase in data dimensionality via spatial, temporal and spectral multiplexing. The investigations to these expanded data dimensions have been driving the development of next generation DOT/FMT systems and resulted in significantly improved 3D imaging performances [29, 30]. In many of these studies, pMC was found to be well suited as it benefits greatly from rescaling strategies that do not require relaunching MC forward simulations. However, studies utilizing pMC have been primarily focused on analyzing domains with known segmentations; the perturbations being estimated are restricted to known tissue segments or regions-of-interest, and are not spatially distributed. Extending pMC to support spatially resolved image reconstructions can greatly benefit DOT in exploring additional measurement dimensions.

In this paper, we propose an approach to efficiently build the Jacobians via perturbation Monte Carlo with photon numerical “replay”, where the trajectory information of detected photons is first recorded from a “baseline” simulation and then utilized to build sensitivity profiles. By storing the pseudo random number generator (PRNG) seeds and recreating the photon trajectories in an MC simulation [31], we can avoid the prohibitive memory cost of storing photon “biographies” and efficiently generate spatially and temporally resolved Jacobians with only a small overhead. Additional optimization strategies are investigated for hyperspectral imaging and single-pixel detection to further accelerate the Jacobian calculation.

The remaining sections of the paper are organized as follows. In Section 2, we will briefly review perturbation Monte Carlo principles and derive spatial/temporal Jacobians for both absorption or scattering perturbations from pMC formulations. We will also provide a brief review on adjoint Monte Carlo. In Section 3, we will detail the “photon replay” algorithm for direct Jacobian constructions. We will particularly discuss two scenarios - camera-based wide-field detection and hyperspectral imaging. In Section 4, we validate the Jacobians from replay with those from the adjoint method - first provide a simple in silico DOT example as a demonstration, and then another example in solving an in vitro hyperspectral wide-field DOT problem. Detailed discussions and comparisons between the replay and adjoint methods are provided, followed by conclusions and future improvements in Section 5.

2. Perturbation Monte Carlo and adjoint Monte Carlo for DOT

2.1. Forward and inverse perturbation Monte Carlo

When a photon is launched from a source in a Monte Carlo simulation, it undergoes multiple scattering events before reaching a detector (or exits without being detected), and its weight decreases exponentially along the trajectory following the Beer-Lambert law [32]. Sometimes, the optical properties, i.e. absorption coefficient (μa), scattering coefficient (μs), and anisotropy factor (g), can be perturbed, (μ̂a, μ̂s, ĝ), from their background values (μa, μs, g) in a small region, such as in a tumor. In such cases, instead of running a new MC simulation with perturbed optical properties, we can predict the measurements by processing the recorded photon “biographies” from the background simulation, known as the perturbation Monte Carlo method [22,23]. For simplicity, here we assume g is not perturbed. In such case, the detected photon weight can be expressed as:

w^=w(μ^sμs)pexp((μ^sμs)L)exp((μ^aμa)L),
where w and ŵ are the photon weights before and after the perturbation and μt = μa + μs is the transport coefficient; L and p denote the total trajectory lengths and the number of scattering events of the photon inside the perturbed region, respectively. Note that we have implemented a continuous absorption weighting scheme [19,21], referred to as the “microscopic Beer-Lambert law” approach (mBLL) in [33]. In this method, a photon’s trajectory is solely determined by μs and the photon weight loss is solely determined by μa. This is different from the “albedo-weight” approach [33] commonly used in pMC literature [23,24], but was shown to be equivalent to the latter [33]. For a complex domain, we rewrite Eq. (1) into a spatially resolved form [24,34] as:
w^=wj=1M(μ^s(Ωj)μs(Ωj))p(Ωj)exp(δμs(Ωj)L(Ωj))exp(δμa(Ωj)L(Ωj)),
where Ωj (j = 1, 2, · · · , M) denote discretized spatial regions (such as segmentations, voxels or tetrahedra); δµs = μ̂sμs and δµa = μ̂aμa. Next, we can apply the rescaling relationship to all detected photons within a time gate, leading to the accumulated time-resolved form as:
W^(T)=k=1NTwkj=1M(μ^s(Ωj)μs(Ωj))pk(Ωj,T)exp(δμs(Ωj)Lk(Ωj,T))exp(δμa(Ωj)Lk(Ωj,T)),
where k = 1, 2, · · · , NT is the index of photons detected during the selected time gate T (out of Ntot photons). Note here that T is not the accumulative time-of-flight for the kth photon in region Ωj, instead, it is the total time-of-flight when the kth photon is captured by the detector.

If we perturb the absorption or scattering coefficient one region (Ωj) at a time, we can obtain the corresponding change δW due to the perturbation as:

δWμ^a(Ωj)=W^μ^a(Ωj)W=k=1NTwk[exp(δμa(Ωj)Lk(Ωj,T))1],
δWμ^s(Ωj)=W^μ^s(Ωj)W=k=1NTwk(μs(Ωj)+δμs(Ωj)μs(Ωj))pk(Ωj,T)exp(δμs(Ωj)Lk(Ωj,T)),
respectively. We take the Taylor’s expansion for each right-hand-side term in Eqs. (45) and keep only the first order terms. Then we can compute the Jacobians in the form of limits as:
Jμa(Ωj,T)=limδμa(Ωj)0δWμ^a(Ωj)δμa(Ωj)=k=1NTwkLk(Ωj,T),
Jμs(Ωj,T)=limδμs(Ωj)0δWμ^s(Ωj)δμs(Ωj)=k=1NTwk(pk(Ωj,T)μs(Ωj)Lk(Ωj,T)).
Finally, if we define Jacobians as the sensitivity of the relative change of measurements, i.e. ΔW (T) = (Ŵ(T) − W(T))/W, to localized optical property changes, where W=k=1Ntotwk is the accumulated photon weights from all time gates, Eqs. (67) can be further simplified to
Jμa(Ωj,T)=L¯(Ωj,T),
Jμs(Ωj,T)=p¯(Ωj,T)μs(Ωj)L¯(Ωj,T),
where L¯(Ωj,T)=k=1NTwkLk(Ωj,T)k=1Ntotwk and p¯(Ωj,T)=k=1NTwkpk(Ωj,T)k=1Ntotwk are the average path lengths and scattering counts of all photons traversing Ωj in time gate T, respectively.

2.2. The Adjoint Monte Carlo

Derived from the diffusion equation [35,36] under the Rytov approximation, the continuous-wave (CW) Jacobians for absorption and scattering perturbations are expressed as

Jμa(Ωj)=ΩjG(rj,rs)Φ(rd,rj)Φ(rd,rs)dV,
Jμs(Ωj)=ΩjG(rj,rs)Φ(rd,rj)3μs2(1g)Φ(rd,rs)dV,
where G(rj, rs) is the Green’s function defined at any discretized region in the medium rj ∈ Ωj due to a source located at rs, obtained by solving the forward model; Φ(rd, rj) denotes the quantity measured at the detector located at rd as a result of the scattered light originated from the medium at rj. In previous literature, the measurement Φ is often defined as the fluence on the domain boundary, immediately inside the diffusive media. Note that the formulation to perform FMT is very similar to Eq. (10) when using the Born normalized approach. For simplicity, we will focus on DOT hereinafter but interested readers are referred to [34].

In many realistic DOT systems, the quantity being measured is not the fluence, but the total diffuse reflectance or transmittance (R) at a detector immediately outside the domain [37], i.e.

Φ(rd,r)=R(rd,r)=detectedwAdallw0,
where allw0 is the total simulated photon weights, detectedw is the total detected photon weights, and Ad is the area of the phantom surface that’s covered by the detector. In some other DOT systems, the measurement quantity is defined as the flux (J⃗) along the normal direction (n⃗) of the detector aperture. In these cases, we have
Φ(rd,r)=Jn(rd,r)=J(rd,r)n=detectedw(vn)Adallw0,
where v⃗ is the direction vector of a detected photon. Note that fluence, diffuse reflectance and flux all have the same unit of mm−2; one must use caution and choose the proper form for Φ.

If the measurement is the fluence inside the diffusive media, i.e. Φ(rd, rj) = G(rd, rj), it can then be simply replaced by the adjoint forward solution of the fluence G(rj, rd) as a result of reciprocity of the Green’s function, i.e. Φ(rd, rj) = G(rd, rj) = G(rj, rd). However, when the measured quantity is diffuse reflectance or normal flux on a refractive index mismatched boundary, reciprocity is not valid any more. In such cases, the relationship between R(rd, r), or Jn, and the adjoint fluence Green’s function G(rj, rd) is more complex, and is related to 1) the boundary condition, 2) the exiting photon angular distribution [38], 3) the asymmetry between the emitting profile of the adjoint source and the receiving profile of the detector, and 4) surface specular reflection. Although deriving an analytical relationship is possible in such cases for a planar boundary within the diffusion regime [12,39], the measurement can typically be expressed as a scaled version of the adjoint fluence solution, i.e. Φ(rd, rj) = αG(rj, rd). The scaling factor α can be numerically calibrated by two sets of MC simulations. Also note that ∇Φ (rd, rj) = −αG(rj, rd) because the two gradients have opposite directions. Thus, Eqs. (1011) can be ultimately re-written as

Jμa(Ωj)=αΩjG(rj,rs)G(rj,rd)Φ(rd,rs)dV,
Jμs(Ωj)=αΩjG(rj,rs)G(rj,rd)3μs2(1g)Φ(rd,rs)dV.
For a typical tissue-like domain (n = 1.37) on a flat surface, α is found to have a typical value of 8.47 when R is measured, and 12.21 when Jn is measured.

2.3. Solving the DOT inverse problem

After computing the sensitivity matrices from either the adjoint or perturbation MC, the DOT inverse problem typically requires solving the below equation

Δμ=J+ΔΦ,
where J+ is the pseudo-inverse of the sensitivity matrix Jμa and/or Jμs, Δμ is the vector of the perturbed absorption Δμa and/or scattering Δμs coefficients in all regions. The measurement perturbation ΔΦ is formulated with both unperturbed, Φ0, and perturbed measurements, Φ, under either the Born (ΔΦ = ΦΦ0) or Rytov (ΔΦ = (ΦΦ0)/Φ0) approximation. The Born approximation is valid when the perturbation is small compared to the background properties, whereas the Rytov approximation assumes that the perturbed field varies slowly. The latter often provides more accurate results in modeling biological tissues [35].

Due to the highly scattering nature of near-infrared light in the tissue, Eq. (16) is known to be ill-posed [2]. As a result, one must apply regularization to stabilize the solution. The standard Tikhonov regularization is often used. When there are more measurements than unknowns, i.e. over-determined, solving the below equation is more efficient [40]:

Δμ=(JTJ+λLTL)1JTΔΦ,
and that for the underdetermined cases (more unknowns than measurements) is
Δμ=(LTL)1JT[J(LTL)1JT+λI]1ΔΦ,
where λ is the regularization parameter, L is the regularization matrix, I is an identity matrix.

3. Building Jacobian matrices via photon replay

3.1. Photon replay algorithm

Based on Eqs. (89), the main challenge for building Jacobians with pMC is to find an efficient way to calculate the weighted average scattering event counts and photon trajectory lengths , both in time and space. A simple approach is to allocate a memory buffer with twice the numbers of voxels or elements, B, multiplied by the number of time gates, T, for each simulated photon, to store j, T) and j, T) during the propagation. However, the drawbacks of this method are apparent. First, the memory for storing 2B×T values needs to be pre-allocated for each detector. When the number of detectors is large, which is typical for DOT applications, or multiple photons are simulated simultaneously, such as GPU-based MC, the memory usage becomes unrealistic, especially for GPUs where memory utility is highly restrictive. Secondly, we do not know in advance if a photon will arrive at the detector. In many cases, only a small portion of the photons are captured, resulting in significant redundant calculations.

The “photon replay” method was inspired by the “reseeded” pMC method proposed by Sassaroli [31]. By taking advantage of the deterministic pseudo-random number generator (RNG) sequence followed by a seed, one only needs to store the photon’s initial RNG state at launch-time when it is captured by a detector. By reinitializing the photon with the saved RNG state, the photon will repeat its trajectory as in the baseline simulation and is guaranteed to be detected again at the same position. We therefore propose a photon “replay” algorithm combining the reseeded MC with the Jacobian formulations in Eqs. (89), with a schematic shown in Fig. 1.

 

Fig. 1 Schematic of the baseline MC (left) and replay MC (right) in a photon replay algorithm

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The photon replay algorithm is performed in two steps. First, a baseline Monte Carlo simulation is performed where a large number of photons with random seeds are launched. If a photon is captured by one of the specified detectors, its arrival time t, final weight w, RNG seed value s at launch, and the detector index are stored. After the number of photons reach the target, a second Monte Carlo simulation, i.e. the photon replay, is performed for each detector. This time, we initialize each new photon with the stored seed value sk, from the kth detected photon in the baseline simulation, so that they are guaranteed to be captured again. During its re-propagation, instead of depositing photon weight losses as in the baseline MC, we utilize the stored detected photon weight wk and deposit either weighted trajectory lengths wk Lj (“wl” mode) or weighted scattering numbers wk pj (“wp” mode) at any given discretized region (Ωj) and time gate (binned by photon’s total time-of-flight tk), as described in Eqs. (67). After all detected photons are replayed, the weighted average values (rj, t) and (rj, t) are generated and used to compute the temporal and spatial resolved Jacobians according to Eqs. (89).

3.2. Improved replay efficiency in wide-field and hyperspectral imaging

With the development of spatial light modulators (SLMs), structured light illumination such as spatial frequency domain imaging (SFDI) and camera based single-pixel detection strategies have gained increasing popularity [28]. This novel system design has shown great potential to improve data acquisition speed as well as signal-to-noise ratio (SNR) for DOT/FMT, making full-body small-animal imaging more efficient. Because the SNR of the Jacobians from replay largely depends on the number of detected photons, a single-pixel detection over a large field-of-view (FOV) can potentially benefit more from replay compared to point detectors.

Typically, dozens of detection patterns belong to the same class are used, such as 36 patterns in the quantized low-frequency base and 64 patterns in the Hadamard base [41]. Since they share the same detection area, we can run only one replay MC to generate Jacobians for all detection patterns with any given source configuration. To achieve this, the only additional data that need to be stored is a photon’s exiting bilinear position (xk, yk) in the detector aperture. Then, when depositing wk Lj and wk pj according to Eqs. (67), we simply loop over all detection patterns and modify wk to wkwdi, where wdi is the corresponding “detector weight”, defined as the ith detection pattern intensity (0 to 1) at (xk, yk). This approach is well suited for emerging methodologies in DOT/FMT such as compressive-sensing based or adaptive DOT where “optimal” patterns can be obtained either theoretically [42] or experimentally [43].

Another scenario where we can leverage the benefits of photon replay is multispectral or hyperspectral imaging, which provides enriched dataset at different wavelengths for unmixing various types of chromophores. The optical properties, either μa or μs, are usually slightly different as the wavelength changes, so we have to perform separate MC simulations at each wavelength if adjoint MC were to be used to build Jacobians. With the proposed replay method, we only need to run one baseline simulation and one replay for all wavelength channels, with the assumption that μs is a constant over the measured wavelength range. Both of our published MC simulators, MCX [21,44] and MMC [19,45], utilize the mBLL method. Therefore, as long as μs remains constant, we can exactly repeat the photon trajectory with the stored seeds and recalculate the detected photon weight wkμa for a new μa value using Eq. (1), without needing to rerun the baseline simulation [32]. Applying the rescaled wk in the replay will obtain the Jacobians of different wavelengths as proposed in [24].

4. Results

4.1. Validation of replay-derived Jacobians

We first compare the Jacobians calculated by the adjoint method (aMC) and photon replay for a single source-detector pair in a homogeneous domain. All simulations are performed with our GPU-accelerated MC simulator “MCX” [21] using an NVIDIA GTX TITAN V GPU. The domain has a volume of 60×60×60 mm3 and a voxel size of 1×1×1 mm3, with the optical properties μa= 0.005 mm−1, μs = 1 mm−1, g = 0 and n = 1.37. A pencil beam light source towards +z direction is placed at [29.5, 29.5, 0] mm and a detector with a radius 2.5 mm is located at [29.5, 39.5, 0] mm. To match the configurations of the two simulations, the light source for the adjoint MC simulation is set to a uniform disk source with 2.5 mm radius at the detector position, where the photons are launched uniformly within the disk towards −z direction. The constant α in Eqs. (1415) is determined to be 8.47 through calibration.

For aMC, 108 photons were launched for both the forward and adjoint simulations, each with a runtime of ∼ 1.90 s. For the replay approach, the baseline simulation with 109 photons took 22.6 s, in which around 7.30×106 photons were detected. The replay steps to obtain the “weighted trajectory length” (wl) and “weighted scattering number” (wp) took 0.39 s and 0.30 s, respectively. All runtimes reported are the average value of 5 repeated tests. To reduce stochastic noise, we apply Anscombe transform on the raw outputs, i.e. G(rj, rs), G(rj, rd) in aMC and j), j) in replay, before filtering them with a 3-D Gaussian kernel with σ = 1, followed by an exact inverse Anscombe transform [46,47].

In Fig. 2, we show the absorption and scattering Jacobians from the adjoint approach in (a–c), those from the replay approach are in (d–f), along with contour plot comparisons between the two methods in (h–i). Distributions of weighted average pathlengths j) (same as the replay adjoint Jacobian by definition) and weighted average scattering numbers j)/μsj) in (d, g), respectively. All plots are in log-scale with base 10 extracted at plane x = 29.5 mm.

 

Fig. 2 Comparison between the adjoint and replay Jacobians: (a–c) absorption Jacobian, positive and negative components of scattering Jacobian from the adjoint approach; (d–f) corresponding Jacobians from the replay approach; (d) is also the weighted average pathlength; (g) weighted average scattering; (h–i) comparisons between the adjoint and replay Jacobians.

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Several remarks can be made from Fig. 2. First, the distributions of (rj) in Fig. 2(d) and j) / μsj) in Fig. 2(g) are very similar to each other. This can be explained by the following relationship: the average distance traveled by a single photon between two scattering events can be calculated as l = L/p, and the photon “mean free path” happens to be the inverse of scattering coefficient 1μs (assuming g = 0) in an mBLL MC photon simulation. Therefore, the accumulated j) and j)/μsj) values for a collection of photons are also very close numerically at the same spatial location. Secondly, the overall magnitude of the scattering Jacobian is about 1–2 orders of magnitude lower than the absorption Jacobian. This is expected because the scattering Jacobian is calculated by subtracting images in (g) by (d), according to Eq. (9). For the same reason, the scattering Jacobian is much nosier than the absorption Jacobian, as the stochastic noise is relatively amplified with the smaller mean value.

Finally, in the scattering Jacobian, only a small region near the source-detector is negative while in majority of the space, it has a positive value. This is also expected. Mathematically, since ∇Φ(rj, rs) and ∇Φ (rj, rd) vectors are along the rjrs and rjrd directions, from Eq. (15) the sign of Jμsj) is determined by the inner product of these two vectors, so we can see the points with negative scattering sensitivity values are inside the circle with the diameter of |rsrd|. Physically, for photons launched from the source, a higher scattering coefficient along or close to the source-detector path tends to raise the possibility of deflecting photons away from the detector, thus reducing the total detected photon weight; on the other hand, a higher scattering coefficient far from the source-detector path tends to reflect photons back to the detector, thus increasing the total detected photon weight. Note here that, although g = 0 is used in this example, aMC and replay Jacobians are also expected to match when g is non-zero.

In addition, we also show the time-resolved absorption Jacobians Jμaj, T), built with the replay method for a transmission measurement setup, in Fig. 3. The size of the phantom is 60×60×30 mm3 with the same optical properties as above. The source remains at [29.5, 29.5, 0] mm and the detector locates at [29.5, 39.5, 30] mm. We recorded over 5 ns with 0.1 ns time step after launching 109 photons in the MC simulation, where the baseline and replay MC with 7.46×105 took 29.4 s and 0.11 s, respectively. We see that Jμa in an early time-gate (Fig. 3(a)) has a significantly narrower span compared to a late time gate with the same accumulated photon weights W(T), shown in Fig. 3(b), due to the ballistic photons with fewer scatterings [14]. Moreover, in Fig. 3(c), we compare the change of temporal point-spread functions (TPSFs) predicted by pMC and replay Jacobians, for a 20×20×10 mm3 inclusion with δµa = 10−4 mm−1 at the center of the phantom (20 ≤ x, y ≤ 40 mm, 10 ≤ z ≤ 20 mm). The TPSFs derived from pMC (as Eq. (3)) and the replay Jacobian, calculated as Jμa · Δμa, match almost exactly. This is not surprising as both were derived from the same set of detected photons.

 

Fig. 3 Validation of time-resolved replay Jacobian from (a) an early (0.25-0.35 ns) and (b) a late (1.15–1.25 ns) gate with the same accumulated photon weights, and (c) comparison of the change of TPSFs between pMC and replay with an inclusion of absorption perturbation. An animation of Jacobians from all time gates can be found in Visualization 1.

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4.2. Sample reconstructions using the replay Jacobians

In this section, we test the reconstruction performance of both absorption and scattering Jacobians obtained from photon replay with simulated data. The tested slab domain contains 40×40×20 isotropic 1×1×1 mm3 voxels. The optical properties for the background medium are μa = 0.01 mm−1, μs = 1 mm−1, g = 0 and n = 1.37. Although g is close to 0.9 in many biological tissues, in the diffusion regime, we used the equivalent settings, i.e. g = 0 and μs as the reduced scattering coefficient (μ′s), to speed up MC computation. Two 8 mm diameter tubes are placed side-by-side with their axes 10 mm below the surface and 16 mm apart. We varied the optical properties of the two cylindrical inclusions and ran multiple simulations. The light source is a collimated Gaussian beam with a waist radius of 2 mm, and 8×8 raster scanning positions are applied to the phantom bottom plane (z = 0 mm) within a rectangle region bounded by [6, 6, 0] mm and [34, 34, 0] mm. A total of 36 disk-shaped detectors of 3 mm radius are placed on the top plane (z = 20 mm) with centers located between 5 to 35 mm, with 6 mm spacing, in both x and y axes.

To generate the absorption and scattering Jacobians, we launched 109 photons from every source position in the baseline simulation and performed photon replays for each detector. On an NVIDIA TITAN V GPU, the average run-time for each baseline simulation was 9.63 s, where ∼ 4.1×107 photons on average were detected; the runtimes for replaying 36 detectors were 10.55 s and 7.20 s for the “wl” and “wp” steps, respectively, resulting in a total run-time of ∼ 29.2 minutes. The raw outputs were denoised as shown in Section 4.1 before constructing the full Jacobian of 2,304 (source-detector pairs) and 32,000 columns (voxel count). Meanwhile, we also generated 3 sets of simulated CW measurements (Φ0, Φ1 and Φ2), where Φ0 : the medium is homogeneous with background optical properties; Φ1 : the left tube has 3 times the background absorption μ̂a = 0.03 mm−1 and Φ2 : the right tube has twice the background scattering μ̂s = 2 mm−1. For each measurement set, 108 photons were used for simulations at each source position, taking an average of 1.05 s per simulation.

The inverse problems for absorption and scattering perturbations were solved based on Eq. (18) and optimal regularization parameters were chosen based on an L-curve approach. In Fig. 4, we show the reconstructed μa cross-sections with Born approximation in (a–c) and those with Rytov approximation in (d–f). The absorption and scattering reconstructions (positive component) on z = 10 mm and y = 20 mm planes are displayed in the first and second columns separately, and the comparison between the ground truth and reconstructed mean values along y dimension on z = 10 mm plane are plotted in (c,f). Note that only one of μa or μs was perturbed in each sample reconstruction; the case where both μa and μs are perturbed is more complicated, normally requiring time-resolved measurements, matrix rescaling or other techniques, and is thus beyond the scope of this paper.

 

Fig. 4 Reconstructions with Born (top row) and Rytov approximations (bottom): the recovered (a, d) absorption and (b, e) scattering profiles are shown on the z = 10 mm and y = 20 mm planes, and (c, f) compare reconstructed and ground truth values along y-axis on the z = 10 mm plane. The locations of 64 sources (green triangles) and 36 detectors (white crosses) are overlaid in (a).

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From Fig. 4, with either approximation, we have successfully reconstructed the true locations of the inclusions. However, the results from the Rytov approach appears to be more accurate, as evident when comparing Fig. 4(f) with (c). We also notice that the reconstructed scattering perturbation is noisier than absorption, as shown both on the surface and interior. This is expected because the scattering Jacobian is noisy as shown in Section 4.1. To suppress artefacts and obtain more accurate images, one can apply regularizations, adopt more advanced inverse solvers or utilize a priori structural information, which again are not the focus of this paper. We have demonstrated in silico that building Jacobians and solving inverse problem via photon replay are feasible for a typical DOT problem with many source-detector pairs.

4.3. Replay for hyperspectral wide-field DOT

Recently, we reported a novel hyperspectral time-resolved wide-field imaging system, capable of collecting in vivo small-animal whole-body data within 14 minutes [30]. One digital mirror device (DMD) is fiber-coupled with a pulsed laser to generate structured illumination patterns and a second DMD is coupled with a spectrophotometer-based photomultiplier tube detector (PMT) to enable single-pixel detection. In this section, we show that one can use the photon replay approach to efficiently generate Jacobians for this type of hyperspectral DOT system.

A liquid phantom of size 80×50×22.5 mm3 was prepared as the imaging target. The background medium contains 0.8% Intralipid, 0.008‰ Epolight 2735 (Ep), and 0.024‰ India Ink (Ink). Two tubular inclusions with 5 mm radius were suspended 10 mm below the surface, as shown with the black dash lines in Figs. 5(a–b). Both were filled with the same Intralipid concentration as the background, while the left one is 2.25× higher in Ep and the right one is 1.83× higher in Ink concentration. Therefore, both tubes have the same μ′s as the background but different μa values, and their ratio with respect to the unit concentrations is 1.23.

 

Fig. 5 The (a) top and (b) front view of reconstruction results with 5 wavelengths, compared to the ground truth (black dashed lines); (c) wavelength dependent absorption coefficients for background medium, 0.008‰ Epolight 2735 and 0.024‰ India Ink; (d) reconstructed crosstalk and concentration ratios of two absorbers with different wavelength channels.

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The experiment was performed under a transmission configuration, where the field-of-view (FOV) was set to 40×30 mm2 for both illumination (bottom) and detection (top). A total of 36 quantized low-frequency (QLF) illumination patterns and detection patterns were used in this experiment. These patterns was chosen for their high SNR at low photon counts [48].

CW data from 5 spectral channels, with the center wavelength ranging from 750 to 770 nm, were selected for unmixing two absorbers. Wavelength dependent optical properties of the background and two absorbing materials were calibrated for every channel. The μ′s value remained approximately constant at 0.84 mm−1, while μa of background medium increased from 0.015 mm−1 to 0.016 mm−1, μa of 0.008‰ Ep increased from 7.2×10−3 mm−1 to 8.6×10−3 mm−1, and μa of 0.024‰ Ink decreased from 9.5×10−3 mm−1 to 9.1×10−3 mm−1. We assume that all μa values change linearly with the wavelength, as shown in Fig. 5(c).

In this example, we used MMC [45] for both the baseline and replay calculations using 36 CPU nodes in parallel on the Blue Gene/Q cluster, each consisting of a 16-core 1.6 GHz A2 processor at the Center for Computational Innovations (CCI) at Rensselaer Polytechnic Institute (RPI). The phantom mesh consisted of 17,073 nodes and 100,241 elements [49]. The baseline MC with 108 photons and μa = 0.015 mm−1 took 16.8 minutes on average on the CPU cluster, where 5.3×106 photons were collected by the detector. With the technique introduced in Section 3.2, a single replay MC session created Jacobians for all 36 detection patterns in 15.7 minutes. Furthermore, under a constant scattering coefficient, we ran five replay simulations for different wavelength channels. Therefore, the total simulation time is 16.8 + 15.7×5 = 95.3 min.

Rytov approximation was again used to compute the measurement vector, and an optode calibration method was adopted [50] to reduce artifacts caused by fluctuations of pattern intensities in the experiment. The inverse problem was solved with a MATLAB built-in solver CGS with the default tolerance [51]. The reconstructed Cep and Cink at 50% isovolume are shown in Figs. 5(a–b). Despite distortions along the z-axis - an expected result of the fixed projection angle - the two absorbers were successfully differentiated with accurate locations and low surface artefacts. We calculate the reconstructed absorber concentrations as the mean of elements larger than 50% of the maximum value, and define the crosstalk of Ep as max(C¯ink(Ωep))max(C¯ep(Ωep)), where ep and ink are normalized to their own maximum and Ωep is the ground truth volume of Ep (vice versa for Ink). From Fig. 5(d), we found that with measurements from more wavelength channels, the crosstalk between Epolight 2735 and India Ink kept decreasing (from 100% to 50.4% and 18.0%), whereas the concentration ratio between two absorbers kept approaching the ground truth (123%). More details about further improving resolution/ratio and reducing crosstalk can be found at [48].

5. Discussions and conclusions

In this section, we discuss the pros and cons of the replay and adjoint approaches. First, we discuss the accuracy of the Jacobians from the two approaches. The replay Jacobians was derived based on pMC, built upon the direct connection between the individual detected photon data and the actual measurement changes. By contrast, the adjoint method relies on the reciprocity of light [27]. Thus, when sufficient number of photons are detected, we consider the replay to be more accurate than the adjoint method. We recognize that several factors may contribute to the derivations between the two Jacobians. Firstly, the adjoint method requires assuming an imaginary source that is located at the detector position. Isotropic or pencil-beam sources have been widely used as the adjoint sources, but the light emission profiles from these source forms are obviously different from the light reception profile of common detectors, such as an optical-fiber. To accurately simulate an adjoint source, a spatially and angularly distributed light source must be used, which is not commonly available. Secondly, the formulation of the adjoint Jacobian depends on the actual type of the measurement, such as fluence, flux or diffuse reflectance. Choosing different measurement quantities can result in different scaling factors, α, as we discussed in Section 2.2. In reality, the scaling factor due to the chosen measurement is effectively removed as part of the data calibration [52].

Next, we want to compare the two methods in terms of computational efficiency. If we follow the procedures described above, if launching M photons at each of the Ns sources and Nd non-overlapping detectors, the run-time for the adjoint method satisfies tad ∝ [(Ns + Nd) M]; that for the replay method satisfies tre ∝ [(1 + f) NsM], where f is the fraction of the photons being detected in the baseline simulation. However, this comparison does not consider the differences of the two Jacobians in terms of SNRs due to their differences in the number of photons involved. Assuming a shot-noise model, we found that the SNR of the adjoint Jacobian is proportional to2M while that for the replay is proportional to fM/Nd. Therefore, to produce a Jacobian that matches the SNR of the adjoint computation, the replay method needs to launch 2Ndf times more photons in the baseline simulation. This leads to an equivalent replay time of tre ∝ [2NdNs (1 + f) M/f].

In the cases where the detected photons are shared among the detectors, such as in the single-pixel camera setup, if we assume that every photon is on average shared between 1 ≤ gdNd detectors (we call gd as the detection reusability gain), the average SNR of the replay becomes fgdM/Nd, and the resulting replay time is tre[2NdNs(1+f)Mfgd] with the assumption that the memory operation cost is neglectable compared to photon propagation. Similarly, in multi- /hyper-spectral imaging with only absorption perturbations, the detected photons can be replayed once for all Nλ spectral channels, leading to a new running time of 2NdNs(1+f)MfgdNλ. Lastly, in applications where iterative reconstruction is performed, the baseline simulation is only needed once at the first iteration if the replay method is used. The forward model outputs and Jacobians of the subsequent iterations can be computed rapidly using only the detected photons, yielding an overall tre2NdNs(1+Kf)MfgdNλ for K iterations. In comparison, the adjoint method requires repeated simulations at all sources and detectors in all iterations, giving tad ∝ (Ns + Nd) KM.

In Fig. 6, we show 3 sample configurations to demonstrate the pros and cons of the replay algorithm in terms of computational expense. The contour plots of the runtime ratios between the replay and the adjoint methods for building Jacobians, i.e. tre/tad, are presented, with a range of source and detector numbers 1 ≤ Ns, Nd ≤ 20. In Fig. 6(a), we show a case with single wavelength (Nλ = 1), non-overlapping point detectors (gd = 1) and f = 0.1. In Fig. 6(b), we show the runtime ratios for a multispectral single-pixel camera detection system with gd = Nd, Nλ = 10 and f = 0.1. In Fig. 6(c), we show an example of iterative reconstruction (K = 10) for a hyperspectral (Nλ = 10) point-detector (gd = 1) system with f = 0.3.

 

Fig. 6 Contour plots of the replay run-times divided by the adjoint run-times at various source-detector numbers: (a) a point-source/detector system, (b) a single-pixel camera hyperspectral system, and (c) a multispectral point-detection system with iterative reconstructions.

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From Fig. 6(a), it appears that if the Jacobian SNR is the major concern, the replay method performs poorly in terms of speed compared with the adjoint method in point-source/detector configurations: when Ns = Nd = 10, the replay takes 100–120 times longer than the adjoint to build a Jacobian with same SNR. In fact, the larger the source and detector numbers, the more efficient is the adjoint method. However, according to Fig. 6(b), when more detected photons are shared between detection channels, either in the form of spatial patterns or spectral channels, the replay method can be more efficient under the condition that NsNd<Nλ1+f(2Nλ). Even for point-detection systems, utilizing iterative reconstruction and multispectral data can also make the replay method more effective compared to the adjoint, as suggested by Fig. 6(c).

In summary, we have proposed a novel approach named “photon replay” to directly and conveniently build time-resolved Jacobians for both absorption and scattering perturbations for diffuse optical tomography. The replay Jacobians were validated with the traditional adjoint approach and show efficiency in several scenarios, such as hyperspectral and single-pixel detection systems. We also demonstrated that the absorption Jacobians from replay can be practically useful for solving real-world DOT reconstructions; however, the scattering Jacobian suffers from stochastic noise despite large photon counts and additional denoising. We are currently working on new strategies to improve the quality of the scattering Jacobian.

The “replay” feature has been implemented and tested in both our voxel- and mesh-based MC platforms. The updated software, MCX and MMC, are freely available at http://mcx.space/. We strongly believe that these tools could contribute to a wide spectrum of explorations in diffuse optical imaging.

Funding

National Institute of Health (NIH) (R01-GM114365, R01-CA204443 (QF), and R01-EB19443, R01 BRG CA207725 (XI)); National Science Foundation (NSF) Career Award CBET 1149407 (XI); NVIDIA GPU Grant Program.

Disclosures

The authors declare that there are no known conflicts of interest related to this article.

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References

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  1. A. Gibson, J. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1 (2005).
    [Crossref] [PubMed]
  2. S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
    [Crossref]
  3. D. Grosenick, H. Rinneberg, R. Cubeddu, and P. Taroni, “Review of optical breast imaging and spectroscopy,” J. of Biomed. Optics 21, 091311 (2016).
    [Crossref]
  4. D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
    [Crossref]
  5. D. Contini, L. Zucchelli, L. Spinelli, M. Caffini, R. Re, A. Pifferi, R. Cubeddu, and A. Torricelli, “Brain and muscle near infrared spectroscopy/imaging techniques,” J. Near Infrared Spectrosc. 20, 15–27 (2012).
    [Crossref]
  6. A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
    [Crossref] [PubMed]
  7. M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
    [Crossref]
  8. A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
    [Crossref] [PubMed]
  9. A. H. Hielscher, “Optical tomographic imaging of small animals,” Current Opinion in Biotech. 16, 79–88 (2005).
    [Crossref]
  10. M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).
  11. M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
    [Crossref]
  12. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [Crossref]
  13. M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
    [Crossref]
  14. M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
    [Crossref] [PubMed]
  15. V. Venugopal, J. Chen, F. Lesage, and X. Intes, “Full-field time-resolved fluorescence tomography of small animals,” Opt. Lett. 35, 3189–3191 (2010).
    [Crossref] [PubMed]
  16. N. Ducros, C. D’andrea, G. Valentini, T. Rudge, S. Arridge, and A. Bassi, “Full-wavelet approach for fluorescence diffuse optical tomography with structured illumination,” Opt. Lett. 35, 3676–3678 (2010).
    [Crossref] [PubMed]
  17. L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [Crossref] [PubMed]
  18. C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. of Biomed. Optics 18, 050902 (2013).
    [Crossref]
  19. Q. Fang, “Mesh-based Monte Carlo method using fast ray-tracing in Plücker coordinates,” Biomed. Opt. Express 1, 165–175 (2010).
    [Crossref] [PubMed]
  20. H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. & Biol. 55, 947 (2010).
    [Crossref]
  21. Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17, 20178–20190 (2009).
    [Crossref] [PubMed]
  22. A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).
    [Crossref]
  23. C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
    [Crossref]
  24. J. Chen and X. Intes, “Time-gated perturbation Monte Carlo for whole body functional imaging in small animals,” Opt. Express 17, 19566–19579 (2009).
    [Crossref] [PubMed]
  25. J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography–computational efficiency,” Med. Phys. 38, 5788–5798 (2011).
    [Crossref] [PubMed]
  26. A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. of Biomed. Optics 19, 065003 (2014).
    [Crossref]
  27. Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
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2018 (1)

L. Yu, F. Nina-Paravecino, D. Kaeli, and Q. Fang, “Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms,” J. of Biomed. Optics 23, 010504 (2018).
[Crossref]

2017 (1)

Q. Pian, R. Yao, N. Sinsuebphon, and X. Intes, “Compressive hyperspectral time-resolved wide-field fluorescence lifetime imaging,” Nat. Photonics 11, 411 (2017).
[Crossref] [PubMed]

2016 (3)

D. Grosenick, H. Rinneberg, R. Cubeddu, and P. Taroni, “Review of optical breast imaging and spectroscopy,” J. of Biomed. Optics 21, 091311 (2016).
[Crossref]

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

R. Yao, X. Intes, and Q. Fang, “Generalized mesh-based Monte Carlo for wide-field illumination and detection via mesh retessellation,” Biomed. Opt. Express 7, 171–184 (2016).
[Crossref] [PubMed]

2015 (3)

2014 (2)

A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. of Biomed. Optics 19, 065003 (2014).
[Crossref]

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

2013 (3)

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
[Crossref]

C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. of Biomed. Optics 18, 050902 (2013).
[Crossref]

V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. of Biomed. Optics 18, 036006 (2013).
[Crossref]

2012 (3)

2011 (5)

J. Chen, V. Venugopal, and X. Intes, “Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt. Express 2, 871–886 (2011).
[Crossref] [PubMed]

A. Sassaroli, “Fast perturbation Monte Carlo method for photon migration in heterogeneous turbid media,” Opt. Lett. 36, 2095–2097 (2011).
[Crossref] [PubMed]

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography–computational efficiency,” Med. Phys. 38, 5788–5798 (2011).
[Crossref] [PubMed]

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. on Image Processing 20, 99–109 (2011).
[Crossref]

2010 (5)

2009 (3)

2008 (3)

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
[Crossref] [PubMed]

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. of Biomed. Optics 13, 041304 (2008).
[Crossref]

P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (2)

A. Gibson, J. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1 (2005).
[Crossref] [PubMed]

A. H. Hielscher, “Optical tomographic imaging of small animals,” Current Opinion in Biotech. 16, 79–88 (2005).
[Crossref]

2004 (1)

Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
[Crossref] [PubMed]

2002 (1)

2001 (2)

2000 (1)

1999 (1)

F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. & Biol. 44, 1257 (1999).
[Crossref]

1998 (1)

1995 (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[Crossref] [PubMed]

1994 (1)

1989 (1)

P. Sonneveld, “CGS, a fast Lanczos-type solver for nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 10, 36–52 (1989).
[Crossref]

Aikawa, E.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
[Crossref] [PubMed]

Alerstam, E.

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. of Biomed. Optics 13, 041304 (2008).
[Crossref]

Alianelli, L.

Andersson-Engels, S.

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. of Biomed. Optics 13, 041304 (2008).
[Crossref]

Angelo, J. P.

J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

Arridge, S.

Arridge, S. R.

Bartels, K. E.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

Bassi, A.

Beuthan, J.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

Bevilacqua, F.

Blaschke, S.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

Blumetti, C.

Boas, D. A.

Bunting, C. F.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

Caffini, M.

Chen, C.-W.

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

Chen, J.

M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography–computational efficiency,” Med. Phys. 38, 5788–5798 (2011).
[Crossref] [PubMed]

J. Chen, V. Venugopal, and X. Intes, “Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt. Express 2, 871–886 (2011).
[Crossref] [PubMed]

V. Venugopal, J. Chen, F. Lesage, and X. Intes, “Full-field time-resolved fluorescence tomography of small animals,” Opt. Lett. 35, 3189–3191 (2010).
[Crossref] [PubMed]

J. Chen and X. Intes, “Time-gated perturbation Monte Carlo for whole body functional imaging in small animals,” Opt. Express 17, 19566–19579 (2009).
[Crossref] [PubMed]

V. Venugopal, J. Chen, and X. Intes, “Robust imaging strategies in time-resolved optical tomography,” in “Optical Tomography and Spectroscopy of Tissue X,” (Int. Society for Optics and Photonics, 2013), p. 857827.

Chen, S.-J.

J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

Chen, Y.

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

Choe, R.

Contini, D.

Corlu, A.

Cubeddu, R.

Culver, J.

Culver, J. P.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

D’andrea, C.

Dai, G.

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
[Crossref]

Dayal, R.

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

de Kleine, R. H.

M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
[Crossref] [PubMed]

Dehghani, H.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
[Crossref] [PubMed]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Dehghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
[Crossref]

Ducros, N.

Dunn, A.

Dunn, A. K.

Durduran, T.

Eggebrecht, A. T.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

Fang, Q.

L. Yu, F. Nina-Paravecino, D. Kaeli, and Q. Fang, “Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms,” J. of Biomed. Optics 23, 010504 (2018).
[Crossref]

R. Yao, X. Intes, and Q. Fang, “Generalized mesh-based Monte Carlo for wide-field illumination and detection via mesh retessellation,” Biomed. Opt. Express 7, 171–184 (2016).
[Crossref] [PubMed]

Q. Fang, “Mesh-based Monte Carlo method using fast ray-tracing in Plücker coordinates,” Biomed. Opt. Express 1, 165–175 (2010).
[Crossref] [PubMed]

Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17, 20178–20190 (2009).
[Crossref] [PubMed]

Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
[Crossref] [PubMed]

Q. Fang and D. A. Boas, “Tetrahedral mesh generation from volumetric binary and grayscale images,” in “IEEE Int. Symp. on Biomed. Imaging: From Nano to Macro, 2009. ISBI’09,” (IEEE, 2009), pp. 1142–1145.

Feng, T.-C.

Ferradal, S. L.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

Fisher, J. P.

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

Foi, A.

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. on Image Processing 20, 99–109 (2011).
[Crossref]

Gardner, A. R.

A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. of Biomed. Optics 19, 065003 (2014).
[Crossref]

Gaudette, T.

Geimer, S. D.

Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
[Crossref] [PubMed]

Gibson, A.

A. Gibson, J. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1 (2005).
[Crossref] [PubMed]

Gioux, S.

J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

Grosenick, D.

D. Grosenick, H. Rinneberg, R. Cubeddu, and P. Taroni, “Review of optical breast imaging and spectroscopy,” J. of Biomed. Optics 21, 091311 (2016).
[Crossref]

Haskell, R. C.

Hassanpour, M. S.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

Hayakawa, C. K.

A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. of Biomed. Optics 19, 065003 (2014).
[Crossref]

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
[Crossref]

Hebden, J.

A. Gibson, J. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1 (2005).
[Crossref] [PubMed]

Hershey, T.

A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
[Crossref] [PubMed]

Hielscher, A.

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

Hielscher, A. H.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

A. H. Hielscher, “Optical tomographic imaging of small animals,” Current Opinion in Biotech. 16, 79–88 (2005).
[Crossref]

Hoi, J.

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

Holyoak, G. R.

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

Illing, G.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

Intes, X.

Q. Pian, R. Yao, N. Sinsuebphon, and X. Intes, “Compressive hyperspectral time-resolved wide-field fluorescence lifetime imaging,” Nat. Photonics 11, 411 (2017).
[Crossref] [PubMed]

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
[Crossref]

R. Yao, X. Intes, and Q. Fang, “Generalized mesh-based Monte Carlo for wide-field illumination and detection via mesh retessellation,” Biomed. Opt. Express 7, 171–184 (2016).
[Crossref] [PubMed]

R. Yao, Q. Pian, and X. Intes, “Wide-field fluorescence molecular tomography with compressive sensing based preconditioning,” Biomed. Opt. Express 6, 4887–4898 (2015).
[Crossref] [PubMed]

Q. Pian, R. Yao, L. Zhao, and X. Intes, “Hyperspectral time-resolved wide-field fluorescence molecular tomography based on structured light and single-pixel detection,” Opt. Lett. 40, 431–434 (2015).
[Crossref] [PubMed]

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
[Crossref]

V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. of Biomed. Optics 18, 036006 (2013).
[Crossref]

M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography–computational efficiency,” Med. Phys. 38, 5788–5798 (2011).
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J. Chen, V. Venugopal, and X. Intes, “Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt. Express 2, 871–886 (2011).
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V. Venugopal, J. Chen, F. Lesage, and X. Intes, “Full-field time-resolved fluorescence tomography of small animals,” Opt. Lett. 35, 3189–3191 (2010).
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J. Chen and X. Intes, “Time-gated perturbation Monte Carlo for whole body functional imaging in small animals,” Opt. Express 17, 19566–19579 (2009).
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J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

M. I. Ochoa, Q. Pian, and X. Intes, “Comparison of Compressive Basis for Quantitative Single-Pixel Fluorescence Lifetime Imaging,” in “Optical Tomography and Spectroscopy,” (Optical Society of America, 2018), pp. OTu4D–4.

Q. Pian, R. Yao, and X. Intes, “Hyperspectral Single-Pixel Wide-Field Time Domain Diffuse Optical Tomography,” in “Bio-Optics: Design and Application,” (Optical Society of America, 2015), pp. BM2A–6.

V. Venugopal, J. Chen, and X. Intes, “Robust imaging strategies in time-resolved optical tomography,” in “Optical Tomography and Spectroscopy of Tissue X,” (Int. Society for Optics and Photonics, 2013), p. 857827.

Ismaelli, A.

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
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M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
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D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
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L. Yu, F. Nina-Paravecino, D. Kaeli, and Q. Fang, “Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms,” J. of Biomed. Optics 23, 010504 (2018).
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M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
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A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
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M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
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M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
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P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
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M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
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M. J. Niedre, R. H. de Kleine, E. Aikawa, D. G. Kirsch, R. Weissleder, and V. Ntziachristos, “Early photon tomography allows fluorescence detection of lung carcinomas and disease progression in mice in vivo,” Proc Natl Acad Sci U S A 105,19126–19131 (2008).
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M. I. Ochoa, Q. Pian, and X. Intes, “Comparison of Compressive Basis for Quantitative Single-Pixel Fluorescence Lifetime Imaging,” in “Optical Tomography and Spectroscopy,” (Optical Society of America, 2018), pp. OTu4D–4.

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M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
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M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
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Q. Pian, R. Yao, N. Sinsuebphon, and X. Intes, “Compressive hyperspectral time-resolved wide-field fluorescence lifetime imaging,” Nat. Photonics 11, 411 (2017).
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Q. Pian, R. Yao, L. Zhao, and X. Intes, “Hyperspectral time-resolved wide-field fluorescence molecular tomography based on structured light and single-pixel detection,” Opt. Lett. 40, 431–434 (2015).
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Q. Pian, R. Yao, and X. Intes, “Hyperspectral Single-Pixel Wide-Field Time Domain Diffuse Optical Tomography,” in “Bio-Optics: Design and Application,” (Optical Society of America, 2015), pp. BM2A–6.

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D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
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M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).

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P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
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M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
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Q. Pian, R. Yao, N. Sinsuebphon, and X. Intes, “Compressive hyperspectral time-resolved wide-field fluorescence lifetime imaging,” Nat. Photonics 11, 411 (2017).
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D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
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A. T. Eggebrecht, S. L. Ferradal, A. Robichaux-Viehoever, M. S. Hassanpour, H. Dehghani, A. Z. Snyder, T. Hershey, and J. P. Culver, “Mapping distributed brain function and networks with diffuse optical tomography,” Nat. Photonics 8, 448 (2014).
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J. P. Angelo, S.-J. Chen, M. Ochoa, U. Sunar, S. Gioux, and X. Intes, “Review of Structured Light in Diffuse Optical Imaging,” J. of Biomed. Optics (in press).

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V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. of Biomed. Optics 18, 036006 (2013).
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M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).

J. Chen, V. Venugopal, and X. Intes, “Monte Carlo based method for fluorescence tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt. Express 2, 871–886 (2011).
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V. Venugopal, J. Chen, F. Lesage, and X. Intes, “Full-field time-resolved fluorescence tomography of small animals,” Opt. Lett. 35, 3189–3191 (2010).
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V. Venugopal, J. Chen, and X. Intes, “Robust imaging strategies in time-resolved optical tomography,” in “Optical Tomography and Spectroscopy of Tissue X,” (Int. Society for Optics and Photonics, 2013), p. 857827.

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P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
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M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
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Q. Pian, R. Yao, L. Zhao, and X. Intes, “Hyperspectral time-resolved wide-field fluorescence molecular tomography based on structured light and single-pixel detection,” Opt. Lett. 40, 431–434 (2015).
[Crossref] [PubMed]

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
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Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
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C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. of Biomed. Optics 18, 050902 (2013).
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Zucchelli, L.

Zwaka, P. A.

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
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Ann. Biomed. Eng. (1)

M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen, and X. Intes, “Mesoscopic fluorescence molecular tomography for evaluating engineered tissues,” Ann. Biomed. Eng. 44, 667–679 (2016).
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Appl. Opt. (1)

Biomed. Opt. Express (4)

Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML - Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[Crossref] [PubMed]

Current Opinion in Biotech. (1)

A. H. Hielscher, “Optical tomographic imaging of small animals,” Current Opinion in Biotech. 16, 79–88 (2005).
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Eur. J. Vasc. Endovasc. Surg. (1)

M. Khalil, H. Kim, J. Hoi, I. Kim, R. Dayal, G. Shrikhande, and A. Hielscher, “Detection of peripheral arterial disease within the foot using vascular optical tomographic imaging: a clinical pilot study,” Eur. J. Vasc. Endovasc. Surg. 49, 83–89 (2015).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

D. Piao, K. E. Bartels, Z. Jiang, G. R. Holyoak, J. W. Ritchey, G. Xu, C. F. Bunting, and G. Slobodov, “Alternative transrectal prostate imaging: a diffuse optical tomography method,” IEEE J. Sel. Top. Quantum Electron. 16, 715–729 (2010).
[Crossref]

IEEE Trans. Med. Imaging (1)

A. H. Hielscher, H. K. Kim, L. D. Montejo, S. Blaschke, U. J. Netz, P. A. Zwaka, G. Illing, G. A. Muller, and J. Beuthan, “Frequency-domain optical tomographic imaging of arthritic finger joints,” IEEE Trans. Med. Imaging 30, 1725–1736 (2011).
[Crossref] [PubMed]

IEEE Trans. on Image Processing (1)

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. on Image Processing 20, 99–109 (2011).
[Crossref]

IEEE Trans. on Medical Imaging (1)

Q. Fang, P. M. Meaney, S. D. Geimer, A. V. Streltsov, and K. D. Paulsen, “Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation,” IEEE Trans. on Medical Imaging 23, 475–484 (2004).
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Inverse Probl. (1)

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[Crossref]

J. Near Infrared Spectrosc. (1)

J. of Biomed. Imaging (1)

M. Pimpalkhare, J. Chen, V. Venugopal, and X. Intes, “Ex vivo fluorescence molecular tomography of the spine,” J. of Biomed. Imaging 2012, 942326 (2012).

J. of Biomed. Optics (7)

M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence molecular tomography of reporter genes in bioprinted thick tissue,” J. of Biomed. Optics 18, 100501 (2013).
[Crossref]

D. Grosenick, H. Rinneberg, R. Cubeddu, and P. Taroni, “Review of optical breast imaging and spectroscopy,” J. of Biomed. Optics 21, 091311 (2016).
[Crossref]

A. R. Gardner, C. K. Hayakawa, and V. Venugopalan, “Coupled forward-adjoint Monte Carlo simulation of spatial-angular light fields to determine optical sensitivity in turbid media,” J. of Biomed. Optics 19, 065003 (2014).
[Crossref]

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. of Biomed. Optics 13, 041304 (2008).
[Crossref]

C. Zhu and Q. Liu, “Review of Monte Carlo modeling of light transport in tissues,” J. of Biomed. Optics 18, 050902 (2013).
[Crossref]

V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. of Biomed. Optics 18, 036006 (2013).
[Crossref]

L. Yu, F. Nina-Paravecino, D. Kaeli, and Q. Fang, “Scalable and massively parallel Monte Carlo photon transport simulations for heterogeneous computing platforms,” J. of Biomed. Optics 23, 010504 (2018).
[Crossref]

J. Opt. Soc. Am. A (3)

Med. Phys. (2)

J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence molecular tomography–computational efficiency,” Med. Phys. 38, 5788–5798 (2011).
[Crossref] [PubMed]

P. K. Yalavarthy, D. R. Lynch, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Implementation of a computationally efficient least-squares algorithm for highly under-determined three-dimensional diffuse optical tomography problems,” Med. Phys. 35, 1682–1697 (2008).
[Crossref] [PubMed]

Nat. Photonics (2)

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Supplementary Material (1)

NameDescription
» Visualization 1       The complete profile time-gated Jacobians for all 50 time gates, associated with Figure 3.

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Figures (6)

Fig. 1
Fig. 1 Schematic of the baseline MC (left) and replay MC (right) in a photon replay algorithm
Fig. 2
Fig. 2 Comparison between the adjoint and replay Jacobians: (a–c) absorption Jacobian, positive and negative components of scattering Jacobian from the adjoint approach; (d–f) corresponding Jacobians from the replay approach; (d) is also the weighted average pathlength; (g) weighted average scattering; (h–i) comparisons between the adjoint and replay Jacobians.
Fig. 3
Fig. 3 Validation of time-resolved replay Jacobian from (a) an early (0.25-0.35 ns) and (b) a late (1.15–1.25 ns) gate with the same accumulated photon weights, and (c) comparison of the change of TPSFs between pMC and replay with an inclusion of absorption perturbation. An animation of Jacobians from all time gates can be found in Visualization 1.
Fig. 4
Fig. 4 Reconstructions with Born (top row) and Rytov approximations (bottom): the recovered (a, d) absorption and (b, e) scattering profiles are shown on the z = 10 mm and y = 20 mm planes, and (c, f) compare reconstructed and ground truth values along y-axis on the z = 10 mm plane. The locations of 64 sources (green triangles) and 36 detectors (white crosses) are overlaid in (a).
Fig. 5
Fig. 5 The (a) top and (b) front view of reconstruction results with 5 wavelengths, compared to the ground truth (black dashed lines); (c) wavelength dependent absorption coefficients for background medium, 0.008‰ Epolight 2735 and 0.024‰ India Ink; (d) reconstructed crosstalk and concentration ratios of two absorbers with different wavelength channels.
Fig. 6
Fig. 6 Contour plots of the replay run-times divided by the adjoint run-times at various source-detector numbers: (a) a point-source/detector system, (b) a single-pixel camera hyperspectral system, and (c) a multispectral point-detection system with iterative reconstructions.

Equations (18)

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w ^ = w ( μ ^ s μ s ) p exp ( ( μ ^ s μ s ) L ) exp ( ( μ ^ a μ a ) L ) ,
w ^ = w j = 1 M ( μ ^ s ( Ω j ) μ s ( Ω j ) ) p ( Ω j ) exp ( δ μ s ( Ω j ) L ( Ω j ) ) exp ( δ μ a ( Ω j ) L ( Ω j ) ) ,
W ^ ( T ) = k = 1 N T w k j = 1 M ( μ ^ s ( Ω j ) μ s ( Ω j ) ) p k ( Ω j , T ) exp ( δ μ s ( Ω j ) L k ( Ω j , T ) ) exp ( δ μ a ( Ω j ) L k ( Ω j , T ) ) ,
δ W μ ^ a ( Ω j ) = W ^ μ ^ a ( Ω j ) W = k = 1 N T w k [ exp ( δ μ a ( Ω j ) L k ( Ω j , T ) ) 1 ] ,
δ W μ ^ s ( Ω j ) = W ^ μ ^ s ( Ω j ) W = k = 1 N T w k ( μ s ( Ω j ) + δ μ s ( Ω j ) μ s ( Ω j ) ) p k ( Ω j , T ) exp ( δ μ s ( Ω j ) L k ( Ω j , T ) ) ,
J μ a ( Ω j , T ) = lim δ μ a ( Ω j ) 0 δ W μ ^ a ( Ω j ) δ μ a ( Ω j ) = k = 1 N T w k L k ( Ω j , T ) ,
J μ s ( Ω j , T ) = lim δ μ s ( Ω j ) 0 δ W μ ^ s ( Ω j ) δ μ s ( Ω j ) = k = 1 N T w k ( p k ( Ω j , T ) μ s ( Ω j ) L k ( Ω j , T ) ) .
J μ a ( Ω j , T ) = L ¯ ( Ω j , T ) ,
J μ s ( Ω j , T ) = p ¯ ( Ω j , T ) μ s ( Ω j ) L ¯ ( Ω j , T ) ,
J μ a ( Ω j ) = Ω j G ( r j , r s ) Φ ( r d , r j ) Φ ( r d , r s ) d V ,
J μ s ( Ω j ) = Ω j G ( r j , r s ) Φ ( r d , r j ) 3 μ s 2 ( 1 g ) Φ ( r d , r s ) d V ,
Φ ( r d , r ) = R ( r d , r ) = detected w A d all w 0 ,
Φ ( r d , r ) = J n ( r d , r ) = J ( r d , r ) n = detected w ( v n ) A d all w 0 ,
J μ a ( Ω j ) = α Ω j G ( r j , r s ) G ( r j , r d ) Φ ( r d , r s ) d V ,
J μ s ( Ω j ) = α Ω j G ( r j , r s ) G ( r j , r d ) 3 μ s 2 ( 1 g ) Φ ( r d , r s ) d V .
Δ μ = J + Δ Φ ,
Δ μ = ( J T J + λ L T L ) 1 J T Δ Φ ,
Δ μ = ( L T L ) 1 J T [ J ( L T L ) 1 J T + λ I ] 1 Δ Φ ,

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