Research Papers: Imaging

Influence of absorption and scattering on the quantification of fluorescence diffuse optical tomography using normalized data

[+] Author Affiliations
Juan Felipe Perez-Juste Abascal, Juan Aguirre, Judit Chamorro-Servent, Juan J. Vaquero

Universidad Carlos III de Madrid, Departamento de Bioingeniería e Ingeniería Aeroespacial, 28911 Madrid, Spain

Martin Schweiger, Simon Arridge

University College London, Department of Computer Science, Gower Street, London WC1E 6BT, United Kingdom

Jorge Ripoll

Universidad Carlos III de Madrid, Departamento de Bioingeniería e Ingeniería Aeroespacial, 28911 Madrid, Spain

Institute of Electronic Structure and Laser Foundation for Research and Technology-Hellas, P.O. Box 1527, 71110 Heraklio, Greece

Manuel Desco

Universidad Carlos III de Madrid, Departamento de Bioingeniería e Ingeniería Aeroespacial, 28911 Madrid, Spain

Hospital General Universitario Gregorio Marañón, Unidad de Medicina y Cirugía Experimental, CIBER de Salud Mental (CIBERSAM), 28007 Madrid, Spain

J. Biomed. Opt. 17(3), 036013 (Apr 05, 2012). doi:10.1117/1.JBO.17.3.036013
History: Received October 25, 2011; Revised January 17, 2012; Accepted January 17, 2012.
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Open Access Open Access

* Authors contributed equally to the paper.

Abstract.  Reconstruction algorithms for imaging fluorescence in near infrared ranges usually normalize fluorescence light with respect to excitation light. Using this approach, we investigated the influence of absorption and scattering heterogeneities on quantification accuracy when assuming a homogeneous model and explored possible reconstruction improvements by using a heterogeneous model. To do so, we created several computer-simulated phantoms: a homogeneous slab phantom (P1), slab phantoms including a region with a two- to six-fold increase in scattering (P2) and in absorption (P3), and an atlas-based mouse phantom that modeled different liver and lung scattering (P4). For P1, reconstruction with the wrong optical properties yielded quantification errors that increased almost linearly with the scattering coefficient while they were mostly negligible regarding the absorption coefficient. This observation agreed with the theoretical results. Taking the quantification of a homogeneous phantom as a reference, relative quantification errors obtained when wrongly assuming homogeneous media were in the range +41 to +94% (P2), 0.1 to 7% (P3), and 39 to +44% (P4). Using a heterogeneous model, the overall error ranged from 7 to 7%. In conclusion, this work demonstrates that assuming homogeneous media leads to noticeable quantification errors that can be improved by adopting heterogeneous models.

Figures in this Article

Diffuse optical tomography in fluorescent mode (fDOT), also called fluorescent molecular tomography when fluorescent probes are used for molecular imaging, is becoming an important pre-clinical non-invasive method for small animal imaging.14 fDOT reconstruction algorithms model the propagation of light through the subject under study. The model is usually described by the diffusion approximation, since biological tissues present high scatter for wavelengths in the visible and near infrared spectrum ranges. Under this approximation, the optical properties of tissues are described statistically by the scattering coefficient μs and the absorption coefficient μa. While analytical solutions exist for media with homogeneous optical properties and simple geometries, more complex numerical solutions are required for media with heterogeneous optical properties or more general spatial domains.5,6

The existence of different scattering and absorption coefficients for different tissue types is a realistic assumption when facing the problem of modeling light transport in living subjects. One solution is to estimate these values using DOT7,8 and then perform fDOT.6,9,10 However, one cannot simultaneously recover absorption and scattering coefficients in constant-wave mode.11 Additionally, prior knowledge of the optical parameters in the frequency domain is necessary for accurate fDOT quantification.9 Some authors neglect the different values of the scattering parameter and only estimate the absorption coefficient with DOT. This coefficient is then used to reconstruct the fluorophore distribution.12,13 Another approach involves combining previously published values for tissue-specific optical parameters with high-resolution anatomical information provided by CT or MRI; however, there is no consensus in the literature on the actual values of tissue-specific optical parameters, and considerable inter-subject variability is reported.14

An alternative approach15 simplifies the problem by assuming homogeneous optical properties and normalizing fluorescence photon density to excitation photon density following the so-called normalized Born approximation. This approach is an adaptation of previous DOT algorithms16 and facilitates the use of fDOT for small animal imaging, thanks to its low sensitivity to inhomogeneities and straightforward implementation.2,17

However, only a few studies have assessed the quantification accuracy of the method in the presence of different levels of absorption18,19 and, to our knowledge, no studies using either simulations or experiments have assessed the effect of scattering heterogeneities on quantification accuracy. In addition, the effects of tissue-specific optical properties reported in previous DOT-fDOT studies without data normalization cannot be extrapolated to normalized data.6,9,10

The goal of our work was to assess the effect of absorption and scattering coefficients on quantification accuracy. In order to provide a theoretical framework, we obtained analytical expressions to explain the dependence of normalized data on changes in background absorption and scattering for homogeneous media using Green’s function for an infinite space and an infinite slab (Sec. 2.2). In principle, if normalized data are not affected by inaccuracies in optical property modeling, they should be invariant to changes in the scattering or absorption of the medium. We then studied the quantification error using four computer-simulated phantoms based on finite element methods. The first phantom was a homogeneous slab with different background absorption and scattering coefficients (Sec. 2.6.1) that was used to validate the previously obtained theoretical result. The second was a heterogeneous slab with a fluorescent cylindrical region in which absorption and scattering coefficients were defined as two, four and six times the background value (Sec. 2.6.2) to cover the wide range of absorption and scattering coefficients reported in the literature.14,20 The third was a computer-simulated mouse model that included different scattering coefficients for lung and liver and was used to assess the relationship between quantification accuracy and the tissue-specific values of the scattering coefficient (Sec. 2.6.3). Image reconstructions were performed with the algebraic reconstruction technique based on two different models: 1) assuming homogeneous media and 2) realistically modeling the heterogeneous optical properties using a Jacobian matrix for which heterogeneous optical properties can be explicitly modeled by the finite element method. The resulting reconstructed images were compared quantitatively.

We hypothesized that the assumption of homogeneous optical properties for modeling diffuse media with a highly heterogeneous scattering coefficient could lead to noticeable quantification errors despite data normalization. Accordingly, we also assessed whether a reconstruction method controlling for heterogeneous scattering coefficients could lead to an improvement in quantification accuracy.

Forward Problem
Diffusion approximation

Let μa(r,λ) be the absorption, μs(r,λ) the reduced scattering, and κ(r,λ)={3[μa(r,λ)+μs(r,λ)]}1 the diffusion coefficient for a wavelength λ at position r in a domain Ω. In a diffuse range, which is valid for highly scattering (μsμa) and weakly anisotropic media, photon density is approximated by the diffusion equation5. Hence, in fDOT, the excitation photon density Φex(r,λex) at the excitation wavelength λex and emission photon density Φem(r,λem) at the emission wavelength λem are given by the solution of a pair of coupled diffusion equations.6,21 The excitation photon density is emitted by an external source q0(rs) at a location rsΩ, and the emission photon intensity comes from a fluorescent region characterized by a fluorescent yield F(rfl), which accounts for its quantum efficiency, absorption parameter and the concentration of fluorophore. Assuming a time-stationary problem (frequency w=0) and that the absorption parameter is not affected by the presence of the fluorophore, excitation and emission photon densities are given by Display Formula

·κ(r,λex)Φex(r,λex)+μa(r,λex)Φex(r,λex)=q0(rs)(1)
Display Formula
·κ(r,λem)Φem(r,λem)+μa(r,λem)Φem(r,λem)=F(rfl)Φex(r,λex).(2)

In Eqs. (1) and (2), optical coefficients are different at excitation and emission wavelengths, and should be modeled differently in practice. However, since our main goal was to assess the impact of scattering, we believe that including additional problems such as optical properties mismatch at emission and excitation wavelengths would introduce additional variations. Hence, we opted for this simplified expression Display Formula

·κ(r)Φex(r)+μa(r)Φex(r)=q0(rs)(3)
Display Formula
·κ(r)Φem(r)+μa(r)Φem(r)=F(rfl)Φex(r).(4)
The diffusion equations [Eqs. (3) and (4)] can be solved analytically using Green’s function for homogeneous media and canonical geometries22 and by applying numerical methods, such as the finite element method, for heterogeneous media and/or general geometries.5,23

From a general perspective, let us define a Green function G(r,r) that solves the heterogeneous problem Display Formula

[·κ(r)+μa(r)]G(r,r)=δ(rr).(5)
The photon densities that solve Eqs. (3) and (4) are therefore given by Display Formula
Φex(r)=drG(r,r)q0(r)(6)
Display Formula
Φem(r)=drG(r,r)F(r)Φex(r).(7)
In this work, the solutions to Eqs. (3) and (4) are given for homogeneous canonical geometries by Green’s function and for both homogeneous and heterogeneous media with general geometries by a Galerkin finite element approach applied using TOAST.24,25 In the finite element implementation, photon density and concentration of fluorophore were approximated by piecewise-linear nodal basis functions. Sources [RHS on Eq. (3)] were modeled as isotropic point sources (located at a depth 1/μs below the surface), which resemble a collimated laser, as described in 25. Measurements were modeled by a Gaussian kernel centered at the detector location and computed as a linear operator M acting on the photon density at the boundary of the domain. Thus, measured excitation and emission photon densities at the detector position rd are given by Display Formula
Φexmeas(rd)=MΦex(r)andΦemmeas(rd)=MΦem(r).(8)

Normalized data: general case

Normalized data for fDOT, that is, the incorrectly termed Born ratio, are given15,16,18 by Display Formula

Ub(rd)=Φemmeas(rd)Φexmeas(rd)=drG(rd;r)F(r)Φex(r)drG(rd;r)q(r)=drG(rd;r)F(r)drG(r;r)q(r)drG(rd;r)q(r),(9)
where the M×1-vector Ub accounts for all source-detector measurement pairs.

Normalized Data: Homogeneous Media

In this section, we used Green’s function for homogeneous media to derive analytical expressions that describe the dependence on normalized data with background absorption and scattering coefficients. We theoretically solved two cases of interest: infinite space and infinite slab.

Assuming a point source q(r)=δ(rrs) and a fluorescent point F(r)=δ(rrf), after substitution in Eq. (9), leads to a simplified form, as follows: Display Formula

u(rd;rs,rf)=drG(rd;r)δ(r-rf)drG(r;r)δ(rrs)drG(rd;r)δ(rrs)=G(rd;rf)G(rf;rs)G(rd;rs).(10)

Infinite space

The Green function G(r,r) for an infinite space22 is Display Formula

G(r,r)=1(2π)322κexp(3μa(μa+μs)|rr|)1|rr|,(11)
and the normalized data are Display Formula
u(rd;rs,rf)=1(2π)322κexp[3μa(μa+μs)(|rfrs|+|rdrf||rdrs|)]|rdrs||rsrf||rdrf|,(12)
which in diffusive media, assuming that μsμa, can be simplified as Display Formula
u(rd;rs,rf)=3μs2(2π)32exp[3μaμs(|rfrs|+|rdrf||rdrs|)]|rdrs||rsrf||rdrf|.(13)

Thus, normalized data are invariant with absorption but almost linearly related to scattering, as the exponential term in Eq. (12) is less dominant (see results, Sec. 3.1).

Infinite slab

The Green function G(r,r) for an infinite slab22 is Display Formula

Gslab(r,r)=1(2π)322κn=[exp(3μa(μa+μs)ρ12+n)ρ12+n-exp(-3μa(μa+μs)ρ12-n)ρ12-n](14)
Display Formula
=1(2π)322κn=h~(μa,μs,r,r)=1(2π)322κh(μa,μs,r,r),(15)
where h~ and h are used for abbreviation, Display Formula
ρ12+n2=(x2x1)2+(y2y1)2+(2nd+z2+z1)2(16)
Display Formula
ρ12n2=(x2x1)2+(y2y1)2+(2nd+z2z1)2(17)
and d is the slab height.

Thus, the normalized data are given by Display Formula

u(rd;rs,rf)=1(2π)322κh(μa,μs,rs,rf)h(μa,μs,rf,rd)h(μa,μs,rs,rd)=1(2π)322κf(μa,μs,rs,rf,rd),(18)
which for μsμa is simplified as Display Formula
u(rd;rs,rf)=3μs2(2π)32f(μa,μs,rs,rf,rd).(19)

Hence, we obtain a relationship similar to that described in the case of infinite space.

Jacobian Matrix

The emission photon density Φem is linearly dependent on the fluorophore concentration F under certain assumptions. This can be seen by taking its Fréchet derivative with respect to the concentration of fluorophore and by assuming that a variation of fluorophore δF induces a variation of emission photon density δΦem while the excitation photon density Φex (and implicitly the optical parameters) remains invariant and there is no re-emission by the fluorophore, Display Formula

τ[Φem(F+τδF)Φex]|τ=0=UbδF.(20)
By differentiating both sides of Eq. (7) and discretizing the integral as the sum of all finite elements Ωj, the variation of emission photon density is given by Display Formula
δΦem(rd)=jΩjdrjG(rd,rj)Φex(rj,rs)δFj=jΩjdrjΦ~(rj,rd)Φex(rj,rs)δFj,(21)
where Φex(rj,rs) is the excitation photon density at rj due to a source at rs, reciprocity G(rj,rd)=G(rd,rj) is used,24 and Φ~(rj,rd) is the adjoint field at rj due to a source q~0 located at the detector position rd, which solves an adjoint equation Display Formula
[·κ(r)+μa(r)]Φ~(r)=q~0(rd).(22)
Comparing Eqs. (20) and (21), the Jacobian matrix entry Jij that relates each measurement (Ub)i (where the subindex i accounts for each source-detector pair) to the concentration of fluorophore at the element Ωj is Display Formula
Jij=(Ub)iFj=1Φexmeas(rd,rs)ΩjdrjΦ~(rj,rd)Φex(rj,rs).(23)
Let J be an M×N-matrix after it has been mapped to a uniform grid and F an N×1-vector of fluorophore coefficients, where N is the number of voxels. Then, the fDOT linear system can be expressed as6,21,26Display Formula
Ub=JF.(24)

The Jacobian matrix in Eq. (23) was computed using the finite element method, which enables modeling of heterogeneous optical properties.

Inverse Problem and Image Analysis

Reconstruction was performed using a randomized algebraic reconstruction technique which has been previously applied in fDOT.7,18,27,28 A relaxation parameter of 0.1 and 20 iterations was applied to all the reconstructions.

Image quantification was performed by defining a Region of Interest (RoI) that corresponded to the known target distribution and by taking the average of the voxels within this RoI. The final concentration of fluorophore is expressed as a percentage of the corresponding value in the homogeneous phantom, which was used as a reference. Thus, concentrations become expressed in a relative scale as Display Formula

FFhomo,(25)
where F and Fhomo are the recovered concentrations of fluorophore corresponding to the heterogeneous and homogeneous model, respectively. The measurement error is also expressed as a percentage, given by 100(FFhomo)/Fhomo.

In addition, profiles were drawn along the reconstructed images as a test of image quality and positional accuracy.

Five percent additive white noise was incorporated into normalized data before reconstruction.

Phantoms and Acquisition Protocol
Slab phantom

We defined a heterogeneous slab-geometry phantom with two regions, a cylindrical fluorescent region in which optical parameters could be varied and a background region. The slab was 10 mm thick and the cylinder diameter was 5 mm (Fig. 1). A homogeneous phantom was used as a reference.

Graphic Jump LocationF1 :

(a) Finite element heterogeneous model corresponding to the physical slab-geometry phantom simulating a cylindrical heterogeneous region filled with fluorochrome. (b) Source positions (detector positions on the upper plane) on the finite element mesh used for simulating data.

The acquisition protocol used a configuration of 9×9sources located on the lower plane covering a surface of 12 by 12mm2 (Fig. 1), together with an equivalent configuration of detectors located at the corresponding positions on the upper plane

Computer-simulated data for each experimental phantom were generated using a fine tetrahedral finite element mesh (115 350 nodes and 540 000 tetrahedra). A coarser mesh (55 061 nodes and 253 125 tetrahedra) was used for calculation of the Jacobian matrix, which was resampled into a uniform mesh of 20×20×20 for the reconstruction.

Mouse model phantom

A heterogeneous model of a mouse that simulates differences in lung and liver scattering was designed based on the Digimouse atlas.29 Resembling the experimental set-up proposed in 1, we simulated a parallel-plate imaging chamber, where the mouse is gently compressed by moving plates filled with intralipid. The model comprised four tissue types: lung, liver, surrounding mouse tissue, and intralipid. It was created by selecting lung and liver regions on the Digimouse model and mapping them into a slab-geometry finite element mesh measuring 12.5 mm in thickness (Fig. 2). The heterogeneous model was created by assigning different μs for these four regions: 2.12  mm1 for the lung and 0.65mm1 for the liver as reported in 14 (for a wavelength of 700 nm); 1.2mm1 for the surrounding tissue, which was also used in 19; and 1.5mm1 for the intralipid region selected as a middle value. The absorption parameter was set to μa=0.01mm1 and kept constant for all models.

Graphic Jump LocationF2 :

(a) Digimouse mesh showing lungs, liver and the surrounding tissue; and (b) its mapping into the slab-geometry finite element mesh used for simulation.

The simulated acquisition configuration consisted of 10×10 sources located at the inferior plane covering a surface of 15×15mm2, and the same configuration of detectors was used at the corresponding positions on the superior plane.

Finally, two spherical fluorophore spots measuring 2 mm in diameter were simulated and placed inside the right lung and the liver at the same z-level (z=6.5mm) (Fig. 2).

Quantification Analysis
Homogeneous slab phantom

For the homogeneous slab phantom described in Sec. 2.5.1, adopting the parameters μa=0.01mm1 and μs=0.8mm1 as a reference, quantification errors were studied as a function of an n-fold change in the background absorption coefficient and an n-fold change in the scattering coefficient (n=0.5, 1.6, 2.7, 3.8, 4.9, 6). All data sets corresponded to identical spherical fluorophore distributions measuring 2.5 mm in diameter and were reconstructed assuming the reference coefficients.

Heterogeneous slab phantom

Heterogeneous slab-geometry phantoms (Fig. 1) were simulated by increasing the scattering coefficient (2μs, 4μs and 6μs) or the absorption coefficient (2μa, 4μa and 6μa) in the cylindrical region with respect to the background optical parameters (μa=0.01mm1 and μs=0.8mm1).

All data sets were reconstructed 1) by assuming a homogeneous medium and 2) by modeling the differences in scattering and absorption coefficients. The reconstructed images were quantitatively compared with the reconstruction corresponding to the homogeneous phantom as detailed above.

Mouse model phantom

Four reconstructions of this phantom were generated. The first one (reference phantom) made use of the homogeneous model with μs=1mm1, for both the simulation and the reconstruction. The remaining three were as follows: a) a heterogeneous model reconstructed using the heterogeneous model, b) a homogeneous model with optical parameters chosen as in 19 (μs=1mm1), and c) a homogeneous model with scattering computed as the volume-weighted average of the heterogeneous tissue values (μs=1.27mm1). The reconstructed images were quantitatively compared with the reconstruction corresponding to the homogeneous phantom, as detailed above.

Normalized Data: Homogeneous Media

The dependence of the normalized data on background absorption and scattering was studied for the homogeneous infinite space solution [Eq. (12)]. Normalized data were generated for a point source located at (0,0,0), a point fluorophore located at (0,0,5) and a moving position of the detector along the x-axis (located at (x,0,10), x varying from 0 to 6 mm). We found that the exponential term in Eq. (12) is equal to or less than one and decreases slightly when n increases, being equal to one for x=0. Hence, the change in normalized data is almost linear with a change in scattering and is almost unaffected by a change in absorption (Fig. 3).

Graphic Jump LocationF3 :

Normalized data [Eq. (12)] for a homogeneous infinite space solution corresponding to a point source located at (0,0,0), a point fluorophore located at (0,0,5), and a position of the detector moving along the x-axis (located at (x,0,10), x varying from 0 to 6 mm), for an n-fold change in the background (a) scattering coefficient and (b) absorption coefficient.

Quantification Analysis
Homogeneous slab phantom

Quantification errors increased almost linearly with a change in background scattering and increased slightly with a change in background absorption (Fig. 4). For the homogeneous case, the ratio of these two should be linear, as it can be extrapolated from Eq. (18), where the exponential term and the spatial dependence vanished.

Graphic Jump LocationF4 :

Homogeneous slab phantom: Quantification of the reconstructed concentration of fluorophore F for several values of the background optical properties (changing the absorption and scattering by a factor n) relative to the quantification for reference properties (μa=0.01mm1 and μs=0.8mm1).

Heterogeneous slab phantom

Figure 5 shows the relative fluorophore concentration values [Eq. (25)] for changes in absorption and in scattering in the cylindrical heterogeneous region. Reconstructed images are shown in Fig. 6. Quantification errors induced by assuming homogeneous media ranged from +41 to +94% and from +0.1 to 7% for the different values used for scattering coefficients (2μs, 4μs, 6μs) and absorption coefficients (2μa, 4μa, 6μa), respectively. Using the reconstruction model that accounted for the previous optical differences led to lower quantification errors in the range 4 to +4%, for changes in both scattering and absorption.

Graphic Jump LocationF5 :

Heterogeneous slab phantom: Image quantification of the relative fluorophore concentration for a slab-geometry phantom with a cylindrical heterogeneity in which the absorption coefficients (blue squares) and scatter coefficients (red circles) vary from twice to six times the background value. Images were reconstructed (a) by assuming a homogeneous model and (b) by modeling the differences in scattering.

Graphic Jump LocationF6 :

Heterogeneous slab phantom: Axial slice (at height z=5mm) of the reconstructed images for the following: (a) an n-fold change in scattering and (b) an n-fold change in absorption (n=1, 2, 4, and 6), using a homogeneous reconstruction model (HO) and a heterogeneous reconstruction model (HE). The actual concentration is the same in all cases. Same results as in Fig. 5.

The profiles drawn along the x-axis on the images in Fig. 6 are also shown (Fig. 7). It seems that positional accuracy is not affected by cylindrical heterogeneity. In contrast, quantification is highly affected by scattering heterogeneity.

Graphic Jump LocationF7 :

Heterogeneous slab phantom: Profiles are drawn along the x-axis (across the cylindrical target), at y=0mm and z=5mm, for the reconstructed images in Fig. 6. Plots correspond to the following: an n-fold change in scattering, reconstructed with a) a homogeneous reconstruction model and b) a heterogeneous reconstruction model; and to an n-fold change in absorption (n=1, 2, 4, and 6), reconstructed with c) a homogeneous reconstruction model and d) a heterogeneous reconstruction model.

Mouse model phantom

Quantification results of the relative concentration of fluorophore in the right lung and liver and its ratio are presented in Table 1. In this case, the reference concentration in Eq. (25) is the average between the recovered concentrations in the lung and liver for the homogeneous model. Quantification errors for the lung and liver analyzed separately were in the range 39 to +44% when assuming a homogeneous model. Accounting for the scattering differences improved quantification, with an error of 7 to +7%. The quantification error for the ratio between concentrations of fluorophore in the liver and in the lung was 46% for the homogeneous model, 13% for the heterogeneous model, and +6% for the reference model. In the reference approach, a homogeneous model was adopted for both data simulation and reconstruction.

Table Grahic Jump Location
Table 1Image quantification of the relative fluorophore concentration [Eq. (25)] in a computer-simulated mouse model with two fluorophore spots, one inside the lung (LU) and one inside the liver (LI). The first row corresponds to simulation and reconstruction using the homogeneous model (homogeneous, μs=1mm1), where the average between the recovered concentrations in LU and LI was used as a reference for the quantification. The second to fourth rows correspond to heterogeneous model simulations reconstructed with the heterogeneous model (second row), the reference homogeneous model (third row), and a homogeneous model with scattering value computed as a weighted average of the scattering values (homogeneous μs=1.27mm1) (fourth row).

Assuming a homogeneous model led to a decrease in image quality with a distortion from the theoretical spherical shape, as compared with the heterogeneous model (Fig. 8).

Graphic Jump LocationF8 :

Axial slice (at height z=7mm) of the reconstructed images for a computer-simulated mouse phantom with two spherical fluorophore spots located inside the right lung and the liver. Figure a) corresponds to simulation and reconstruction using the homogeneous model (homogeneous, μs=1mm1), where the average between the recovered concentrations in the lung and liver was used as a reference for the quantification. Figures b–d) correspond to heterogeneous model simulations reconstructed with b) the heterogeneous model, c) the reference homogeneous model and d) a homogeneous model with scattering value computed as a weighted average of the scattering values (homogeneous* μs=1.27mm1).

Profiles for the reconstructed images in Fig. 8 were drawn along the x-axis across the two spherical targets in the lung and liver. As the liver has a smaller scattering coefficient (μs=0.65mm1) than the background and the lung has a larger one (μs=2.12mm1), the fluorophore concentration can be both underestimated and overestimated (Fig. 9).

Graphic Jump LocationF9 :

Mouse model phantom: Profiles for the reconstructed images in Fig. 8 were drawn along the x-axis, across the two spherical targets in the lung and liver. HO-HO stands for simulation and reconstruction using the reference homogeneous model (μs=1mm1), HE-HO for simulation with the heterogeneous model and reconstruction with the reference homogeneous model, HE-HE for simulation and reconstruction using the heterogeneous model, and HE-HO* for simulation with the heterogeneous model and reconstruction with a different homogeneous model (μs=1.27mm1).

Reconstruction based on normalized data has previously been found to be robust in the presence of absorption differences.18 To our knowledge, this is the first report presenting evidence of the effect of scattering heterogeneities and quantification accuracy using normalized data. Our results suggest that scattering heterogeneity has a noticeable negative effect on quantification accuracy.

For a homogeneous infinite space, normalized data present a dominant linear relationship with the scattering coefficient while remaining invariant with the absorption coefficient. Accordingly, for a homogeneous slab phantom, a reconstruction assuming incorrect optical parameters leads to quantification errors that increase almost linearly with the scattering coefficient while increasing slightly with the absorption coefficient.

For heterogeneous slab phantoms with different scattering coefficient in a cylindrical fluorescent region, we found that quantification errors induced by assuming a homogeneous medium ranged from +41 to +94%, taking quantification from a homogeneous phantom as a reference. Additionally, the presence of regions with absorption heterogeneities led to much smaller quantification errors, ranging from +0.1 to 7%, which agrees with the findings of previous reports.18 Quantification errors decreased significantly (range, 4 to +4%) when using a reconstruction model that accounted for the differences in scattering and absorption coefficients.

For the reconstruction of a computer-simulated mouse model that included different liver and lung scattering coefficient, we obtained quantification errors ranging from 39 to +44% when assuming a homogeneous medium and 7 to +7% when using a reconstruction model that accounted for the differences in scattering. Thus, neglecting the effect of scattering heterogeneities led to a systematic overestimation of the calculated concentration of fluorophore; however, the effect of absorption was remarkably smaller. In more realistic situations, the actual effect may be more complicated.

These results are consistent with those of a previous study that analyzed the effect of modeling errors simultaneously in both scattering and absorption parameters.19 The model used in this study was a computer-simulated reconstruction of a mouse torso comprising five regions (liver, heart, lung, bone and background tissues), and relative 2-norm errors only increased 0.74% when a heterogeneous reconstruction model was used. The averaged optical property values in this model were taken from the literature. However using a homogeneous model, errors increased 4.8%. These errors were defined as 2-norm errors that accounted for the entire image and were relative to the error associated with the exact model; consequently, they do not correspond exactly to our quantification errors. Our results correspond to a model that accounted for two organs with distinct scattering (reduced scattering in the range 0.65 to 2.34mm1 at 700 nm). Larger errors should be expected when the biological tissue diversity in real animal experiments is taken into account.

From the profiles drawn on the reconstructed images for the heterogeneous slab phantom and for the mouse model phantom, it seems that positional accuracy is not much affected by optical heterogeneity. In contrast, quantification is highly affected by scattering heterogeneity.

The heterogeneous phantoms used in the present study emulate quantification in two organs with different scattering coefficients. Hence, these results cannot be generalized to other arrangements, such as quantification in the presence of organs with a distinct scattering coefficient. Further studies, especially in vivo validations, would help to establish the limits of use for normalized data in preclinical studies.

In conclusion, the present study makes use of theoretical results for homogeneous media and computer simulations for heterogeneous models to demonstrate a) that inaccuracies in modeling the scattering parameter may lead to noticeable quantification errors when using a normalized data reconstruction approach and b) that more sophisticated models of the forward problem yield much better results.

This study was supported by Ministerio de Ciencia e Innovación (FPI program, TEC 2008-06715, and CENIT AMIT CEN-20101014), Comunidad de Madrid and European Regional Development Fund ARTEMIS S2009/DPI-1802, and EU-FP7 project FMTXCT-201792.

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Tan  Y., Jiang  H., “Diffuse optical tomography guided quantitative fluorescence molecular tomography,” Appl. Opt.. 47, (12 ), 2011 –2016 (2008). 0003-6935 CrossRef
Arridge  S. R., Lionheart  W. R. B., “Non-uniqueness in diffusion-based optical tomography,” Opt. Lett.. 23, (11 ), 882 –884 (1998). 0146-9592 CrossRef
Garofalakis  A. et al., “In vivo validation of free-space fluorescence tomography using nuclear imaging,” Opt. Lett.. 35, (18 ), 3024 –3026 (2010). 0146-9592 CrossRef
Herve  L. et al., “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt.. 46, (22 ), 4896 –4906 (2007). 0003-6935 CrossRef
Alexandrakis  G., Rannou  F. R., Chatziioannou  A. F., “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.. 50, (17 ), 4225 –4241 (2005). 0031-9155 CrossRef
Ntziachristos  V., Weissleder  R., “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized born approximation,” Opt. Lett.. 26, (12 ), 893 –895 (2001). 0146-9592 CrossRef
Ntziachristos  V. et al., “Diffuse optical tomography of highly heterogeneous media,” IEEE Trans. Med. Imag.. 20, (6 ), 470 –478 (2001). 0278-0062 CrossRef
Weissleder  R., Ntziachristos  V., “Shedding light onto live molecular targets,” Nat. Med.. 9, (1 ), 123 –128 (2003). 1078-8956 CrossRef
Soubret  A., Ripoll  J., Ntziachristos  V., “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized born ratio,” IEEE Trans. Med. Imag.. 24, (10 ), 1377 –1386 (2005). 0278-0062 CrossRef
Hyde  D. et al., “Performance dependence of hybrid x-ray computed tomography/fluorescence molecular tomography on the optical forward problem,” J. Opt. Soc. Am. A. 26, (4 ), 919 –923 (2009). 0740-3232 CrossRef
Cheong  W.-F., Prahl  S. A., Welch  A. J., “A review of the optical properties of biological tissues,” J. Quant. Elect.. 26, (12 ), 2166 –2185 (1990). 0018-9197 CrossRef
Milstein  A. B. et al., “Fluorescence optical diffusion tomography,” Appl. Opt.. 42, (16 ), 3081 –3094, (2003). 0003-6935 CrossRef
Arridge  S. R., “Photon-measurement density functions. part i: analytical forms,” Appl. Opt.. 34, (31 ), 7395 –7409 (1995). 0003-6935 CrossRef
Arridge  S. R., Schweiger  M., “Photon-measurement density functions. Part 2: finite-element-method calculations,” Appl. Opt.. 34, (34 ), 8026 –8037 (1995).  0003-6935 CrossRef
Schweiger  M., Application of the finite element method in infrared image reconstruction of scattering media, Ph.D. Thesis, University College London (1994).
Schweiger  M. et al., “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.. 22, (11 ), 1779 –1792 (1995). 0094-2405 CrossRef
Schulz  R. B. et al., “Hybrid system for simultaneous fluorescence and x-ray computed tomography,” IEEE Trans. Med. Imag.. 29, (2 ), 465 –473 (2010). 0278-0062 CrossRef
Intes  X. et al., “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol.. 47, (1 ), N1 –N10 (2002). 0031-9155 CrossRef
Strohmer  T., Vershynin  R., “A randomized kaczmarz algorithm with exponential convergence,” J. Fourier Anal. Appl.. 15, (2 ), 262 –278 (2009). 1069-5869 CrossRef
Dogdas  B. et al., “Digimouse: a 3-D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.. 52, (3 ), 577 –587 (2007). 0031-9155 CrossRef
© 2012 Society of Photo-Optical Instrumentation Engineers

Citation

Juan Felipe Perez-Juste Abascal ; Juan Aguirre ; Judit Chamorro-Servent ; Martin Schweiger ; Simon Arridge, et al.
"Influence of absorption and scattering on the quantification of fluorescence diffuse optical tomography using normalized data", J. Biomed. Opt. 17(3), 036013 (Apr 05, 2012). ; http://dx.doi.org/10.1117/1.JBO.17.3.036013


Figures

Graphic Jump LocationF2 :

(a) Digimouse mesh showing lungs, liver and the surrounding tissue; and (b) its mapping into the slab-geometry finite element mesh used for simulation.

Graphic Jump LocationF3 :

Normalized data [Eq. (12)] for a homogeneous infinite space solution corresponding to a point source located at (0,0,0), a point fluorophore located at (0,0,5), and a position of the detector moving along the x-axis (located at (x,0,10), x varying from 0 to 6 mm), for an n-fold change in the background (a) scattering coefficient and (b) absorption coefficient.

Graphic Jump LocationF4 :

Homogeneous slab phantom: Quantification of the reconstructed concentration of fluorophore F for several values of the background optical properties (changing the absorption and scattering by a factor n) relative to the quantification for reference properties (μa=0.01mm1 and μs=0.8mm1).

Graphic Jump LocationF5 :

Heterogeneous slab phantom: Image quantification of the relative fluorophore concentration for a slab-geometry phantom with a cylindrical heterogeneity in which the absorption coefficients (blue squares) and scatter coefficients (red circles) vary from twice to six times the background value. Images were reconstructed (a) by assuming a homogeneous model and (b) by modeling the differences in scattering.

Graphic Jump LocationF6 :

Heterogeneous slab phantom: Axial slice (at height z=5mm) of the reconstructed images for the following: (a) an n-fold change in scattering and (b) an n-fold change in absorption (n=1, 2, 4, and 6), using a homogeneous reconstruction model (HO) and a heterogeneous reconstruction model (HE). The actual concentration is the same in all cases. Same results as in Fig. 5.

Graphic Jump LocationF7 :

Heterogeneous slab phantom: Profiles are drawn along the x-axis (across the cylindrical target), at y=0mm and z=5mm, for the reconstructed images in Fig. 6. Plots correspond to the following: an n-fold change in scattering, reconstructed with a) a homogeneous reconstruction model and b) a heterogeneous reconstruction model; and to an n-fold change in absorption (n=1, 2, 4, and 6), reconstructed with c) a homogeneous reconstruction model and d) a heterogeneous reconstruction model.

Graphic Jump LocationF8 :

Axial slice (at height z=7mm) of the reconstructed images for a computer-simulated mouse phantom with two spherical fluorophore spots located inside the right lung and the liver. Figure a) corresponds to simulation and reconstruction using the homogeneous model (homogeneous, μs=1mm1), where the average between the recovered concentrations in the lung and liver was used as a reference for the quantification. Figures b–d) correspond to heterogeneous model simulations reconstructed with b) the heterogeneous model, c) the reference homogeneous model and d) a homogeneous model with scattering value computed as a weighted average of the scattering values (homogeneous* μs=1.27mm1).

Graphic Jump LocationF9 :

Mouse model phantom: Profiles for the reconstructed images in Fig. 8 were drawn along the x-axis, across the two spherical targets in the lung and liver. HO-HO stands for simulation and reconstruction using the reference homogeneous model (μs=1mm1), HE-HO for simulation with the heterogeneous model and reconstruction with the reference homogeneous model, HE-HE for simulation and reconstruction using the heterogeneous model, and HE-HO* for simulation with the heterogeneous model and reconstruction with a different homogeneous model (μs=1.27mm1).

Graphic Jump LocationF1 :

(a) Finite element heterogeneous model corresponding to the physical slab-geometry phantom simulating a cylindrical heterogeneous region filled with fluorochrome. (b) Source positions (detector positions on the upper plane) on the finite element mesh used for simulating data.

Tables

Table Grahic Jump Location
Table 1Image quantification of the relative fluorophore concentration [Eq. (25)] in a computer-simulated mouse model with two fluorophore spots, one inside the lung (LU) and one inside the liver (LI). The first row corresponds to simulation and reconstruction using the homogeneous model (homogeneous, μs=1mm1), where the average between the recovered concentrations in LU and LI was used as a reference for the quantification. The second to fourth rows correspond to heterogeneous model simulations reconstructed with the heterogeneous model (second row), the reference homogeneous model (third row), and a homogeneous model with scattering value computed as a weighted average of the scattering values (homogeneous μs=1.27mm1) (fourth row).

References

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Lin  Y. et al., “Fluorescence diffuse optical tomography with functional and anatomical a priori information: feasibility study,” Phys. Med. Biol.. 52, (18 ), 5569 –5585 (2007). 0031-9155 CrossRef
Tan  Y., Jiang  H., “Diffuse optical tomography guided quantitative fluorescence molecular tomography,” Appl. Opt.. 47, (12 ), 2011 –2016 (2008). 0003-6935 CrossRef
Arridge  S. R., Lionheart  W. R. B., “Non-uniqueness in diffusion-based optical tomography,” Opt. Lett.. 23, (11 ), 882 –884 (1998). 0146-9592 CrossRef
Garofalakis  A. et al., “In vivo validation of free-space fluorescence tomography using nuclear imaging,” Opt. Lett.. 35, (18 ), 3024 –3026 (2010). 0146-9592 CrossRef
Herve  L. et al., “Noncontact fluorescence diffuse optical tomography of heterogeneous media,” Appl. Opt.. 46, (22 ), 4896 –4906 (2007). 0003-6935 CrossRef
Alexandrakis  G., Rannou  F. R., Chatziioannou  A. F., “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.. 50, (17 ), 4225 –4241 (2005). 0031-9155 CrossRef
Ntziachristos  V., Weissleder  R., “Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized born approximation,” Opt. Lett.. 26, (12 ), 893 –895 (2001). 0146-9592 CrossRef
Ntziachristos  V. et al., “Diffuse optical tomography of highly heterogeneous media,” IEEE Trans. Med. Imag.. 20, (6 ), 470 –478 (2001). 0278-0062 CrossRef
Weissleder  R., Ntziachristos  V., “Shedding light onto live molecular targets,” Nat. Med.. 9, (1 ), 123 –128 (2003). 1078-8956 CrossRef
Soubret  A., Ripoll  J., Ntziachristos  V., “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized born ratio,” IEEE Trans. Med. Imag.. 24, (10 ), 1377 –1386 (2005). 0278-0062 CrossRef
Hyde  D. et al., “Performance dependence of hybrid x-ray computed tomography/fluorescence molecular tomography on the optical forward problem,” J. Opt. Soc. Am. A. 26, (4 ), 919 –923 (2009). 0740-3232 CrossRef
Cheong  W.-F., Prahl  S. A., Welch  A. J., “A review of the optical properties of biological tissues,” J. Quant. Elect.. 26, (12 ), 2166 –2185 (1990). 0018-9197 CrossRef
Milstein  A. B. et al., “Fluorescence optical diffusion tomography,” Appl. Opt.. 42, (16 ), 3081 –3094, (2003). 0003-6935 CrossRef
Arridge  S. R., “Photon-measurement density functions. part i: analytical forms,” Appl. Opt.. 34, (31 ), 7395 –7409 (1995). 0003-6935 CrossRef
Arridge  S. R., Schweiger  M., “Photon-measurement density functions. Part 2: finite-element-method calculations,” Appl. Opt.. 34, (34 ), 8026 –8037 (1995).  0003-6935 CrossRef
Schweiger  M., Application of the finite element method in infrared image reconstruction of scattering media, Ph.D. Thesis, University College London (1994).
Schweiger  M. et al., “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.. 22, (11 ), 1779 –1792 (1995). 0094-2405 CrossRef
Schulz  R. B. et al., “Hybrid system for simultaneous fluorescence and x-ray computed tomography,” IEEE Trans. Med. Imag.. 29, (2 ), 465 –473 (2010). 0278-0062 CrossRef
Intes  X. et al., “Projection access order in algebraic reconstruction technique for diffuse optical tomography,” Phys. Med. Biol.. 47, (1 ), N1 –N10 (2002). 0031-9155 CrossRef
Strohmer  T., Vershynin  R., “A randomized kaczmarz algorithm with exponential convergence,” J. Fourier Anal. Appl.. 15, (2 ), 262 –278 (2009). 1069-5869 CrossRef
Dogdas  B. et al., “Digimouse: a 3-D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.. 52, (3 ), 577 –587 (2007). 0031-9155 CrossRef

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