2.1.

Creation of Mouse Phantom

We have generated the mouse phantom from MRI data using the following procedures.

2.1.1.

Boundary extraction

We extract the mouse boundary from MRI data through the application of image processing algorithms, as shown in Fig. 2 . The MRI data is first smoothed using a one-dimensional (1-D) Gaussian filter. A threshold is applied next on the filtered image volume. The crispness of the extracted boundary is particularly sensitive to the value of the threshold. Lower threshold values result in a boundary cluttered with undesirable clusters of points, whereas the higher threshold value results in the extraction of internal organs of the mouse along with the final boundary. In most real-world problems, foreground and background regions do not follow perfect bimodal intensity distributions, resulting in some overlap of intensities. We choose a threshold that eliminates as much of the background as possible without affecting the foreground object. Noise tends to persist despite thresholding and is subsequently removed through a process of erosion, image subtraction, and connected component analysis.

Fig. 2

Illustration of mouse boundary generation.

2.1.2.

Geometry generation

The original MRI data set of the mouse has 26 slices, each with dimensions $256\times 256$ . Resolution within each 2-D slice is $0.78\times 0.78\mathrm{mm}$ . The adjacent slices are separated by $2.0\mathrm{mm}$ . We do not consider some of the frames toward the two edges of the mouse for geometry creation because the entire boundary is not retrieved in these slices. The geometrical representation of the mouse is created from the extracted boundary through a process of selecting frames, contours, and edge segments and finally lofting the frames to generate a 3-D structure. For every two-dimensional (2-D) frame, we first generate a contour followed by a 2-D curve and a 2-D solid mesh. Last, we loft each 2-D solid mesh with appropriate spacing to create the geometry, as shown in Fig. 3 .

Fig. 3

Illustration of mouse geometry creation.

2.2.

Phantom Size and Mesh Resolution

We consider a range of optical properties in biological tissue and compute the derived optical properties $\mathit{mfp}$ and $\delta$ . The range of optical properties in biological tissue² is presented in Table 1 .

Table 1

Range of optical properties in biological tissue.

We first construct an equivalent cylindrical phantom with the same length and similar mesh resolution as that of the desired mouse phantom. In accordance with the rules of thumb, a single cylindrical phantom capable of simulating light propagation across the range of optical properties would need to have the size specified by the highest penetration depth and the mesh resolution specified by the lowest mean free path. This phantom should have at least a diameter of $40.4\mathrm{mm}$ and a length of $80.8\mathrm{mm}$ , and its FE model should have an average node-to-node distance of $<0.05\mathrm{mm}$ . Our current hardware ( $64\text{-}\text{bit}$ HP UNIX workstation) can provide a maximum mesh resolution of $0.98\mathrm{mm}$ for a cylinder of this size. Based on the capability of the hardware, we have designed a cylindrical phantom of $42\text{-}\mathrm{mm}$ diam and $82\text{-}\mathrm{mm}$ height represented by 199,414 nodes, 1,146,319 elements, and 30,690 boundary nodes.

The digital mouse phantom shown in Fig. 4 is constructed by simply shortening the mouse geometry by an equal distance from the axial extremities for the mouse phantom to have the same length and mesh resolution as that of the cylindrical phantom.

Fig. 4

Slices used in the comparative analysis in the mouse phantom. The source is located in the central slice 3.

2.3.

Optical Properties

Based on our earlier studies comparing light fluence computed by Monte Carlo simulation and the diffusion equation, we expect the attributes of the “design” phantom to be appropriate for prediction of photon transport for ${\mu }_s^{\prime }=0.5{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.01{\mathrm{mm}}^{-1}$ , and $n_r=1.38$ . In order to verify and assess the applicability of the phantom for light simulation at varying axial and angular distances from a point source located on the boundary, we measure simulated fluence on the boundary of five slices axially located at $\left[0\text{to}0.5\right]\mathrm{mm}$ and shifted by $\pm \left[3.0\text{to}3.5\right]\mathrm{mm}$ , $\pm \left[7\text{to}7.5\right]\mathrm{mm}$ on either side of a point source positioned approximately near the midaxis.

We use the comparative performance⁴⁰ of the FE diffusion equation and our MATLAB implementation of a 3-D analytical diffusion model²⁷ on a cylindrical geometry to verify the size and mesh resolution of the mouse phantom for the “design” optical parameters. We evaluate excitation fluence from the FE model (Fe) and compare it with the analytical solution (Fa) at all boundary points $(z,\alpha )$ in each $z$ slice for the optical properties corresponding to the phantom by computing $\mathrm{Fe}∕\mathrm{Fa}$ where $z$ is the axial coordinate and $\alpha$ is the source-detector angle in the $xy$ -plane. We then perform a sensitivity analysis of the “design” phantom by increasing the scattering and absorption coefficient by 5, 10, and 20% to study the variation in $\mathrm{Fe}∕\mathrm{Fa}$ . The results of the sensitivity analysis are presented in the next section.

2.4.

Boundary Modeling

As shown in Fig. 1, there are several image processing steps, each of which is a potential source of variation in the generation of a mouse phantom. For the current study, we focus on the number of edge segments that are used to describe the contours selected as the boundary for each frame. Local geometry and mesh resolution close to the source and detectors are known to have a strong influence on simulated fluence.^{11, 38} Sources and detectors are located on the boundary. Therefore, it is pertinent to assess the sensitivity of the simulated results to variations in the boundary modeled through a change in the number of edge segments. We consider the number of edge segments in the basic mouse design and generate mouse meshes with edge segments such that the number of boundary nodes change by approximately $\pm 5\%$ , $\pm 10\%$ , and $\pm 20\%$ from the number of boundary nodes in the designed case. The negative variation is generated by down-sampling the edge segments from the base design, and positive variation by up-sampling the edge variation from the base. The base design has 60 edge segments. The change in the number of segments leads to a change in the number of nodes in the FE model. The number of edge segments, nodes, and segment lengths in each design and their percent difference from the base design is shown in Table 2 .

Table 2

Attributes of design variants of the mouse model.

We compare the excitation light fluence for the six variants in boundary models with the base design in five slices axially located at $\left[0\text{to}0.5\right]\mathrm{mm}$ and shifted by $\pm \left[3.0\text{to}3.5\right]\mathrm{mm}$ , $\pm [7–7.5]\mathrm{mm}$ on either side of the point source positioned approximately near the midaxis, similar to that in the cylindrical phantom. As shown in Fig. 4, the source is located in the central slice 3. Each slice has a thickness of $0.5\mathrm{mm}$ . The results of the comparison are presented in the next section.

3.1.

Verification of Design Parameters

We compute the variation in excitation fluence⁴⁰ with source-detector angle from our 3-D FE implementation of the diffusion model (Fe) and the analytical solution of the 3-D diffusion equation (Fa) on the cylindrical geometry in all five slices over a $0\text{to}360\text{-}\mathrm{deg}$ range in the source-detector angles for the “design” optical properties ( ${\mu }_s^{\prime }=0.5{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.01{\mathrm{mm}}^{-1}$ , and $n_r=1.38$ ), ensure that $\mathrm{Fe}∕\mathrm{Fa}\sim 1$ at distances that are a few $\mathit{mfp}$ away from the source,¹¹ and observe that as the distance from the source $⪢\mathit{mfp}$ , $\mathrm{Fe}∕\mathrm{Fa}\to 1$ .

3.2.

Sensitivity to Variation in Optical Properties

For our analysis, we eliminate all nodes in the slice on the boundary that are within 2 $\mathit{mfp}$ of the source and compute mean $\mathrm{Fe}∕\mathrm{Fa}$ ratios for sixteen representative cases over the five slices with mean free paths corresponding to 0%, 5%, 10%, and 20% increases in ${\mu }_a$ and ${\mu }_s^{\prime }$ from their “design” values of $0.01{\mathrm{mm}}^{-1}$ and $0.5{\mathrm{mm}}^{-1}$ , respectively. The mean ratio is found to vary linearly with $\mathit{mfp}$ over the design space. The mean ratios increase with increase in scattering coefficient and decrease with increase in absorption within each slice.

The differences in the FE and the analytical diffusion model could arise from both meshing and the differences in the manner in which the boundary condition is incorporated in them.^{3, 27} We would expect $\mathrm{Fe}∕\mathrm{Fa}\to 1$ to represent the average pattern of comparison within an error margin. The error is defined as:

The error rises to a maximum of 5.7% for up to a 20% increase in the optical absorption and reduced scattering coefficients for source-detector distances varying from $1\mathrm{mm}$ to about $7\mathrm{cm}$ . However, as ${\mu }_s^{\prime }$ increases to $20{\mathrm{mm}}^{-1}$ ( $\sim 4000\%$ increase), our computations (not presented here) show that at a source-detector angle of $30\mathrm{deg}$ , FE fluence reduces seven orders of magnitude faster than analytical fluence at $z\sim 0$ , four orders of magnitude faster at $z\sim 3\mathrm{mm}$ , and two magnitude faster at $z\sim 7\mathrm{mm}$ . It could be expected that at large distances from the source, the FE fluence would converge to analytical fluence predictions and this trend would prevail as long as the photon transport is diffusive, i.e., ${\mu }_s^{\prime }⪢{\mu }_a$ . In measurements of fluorescence during frequency domain photon migration, Thomson and Sevick-Muraca have reported a mean accuracy of 5.4% in modulation depth and $0.3\mathrm{deg}$ in phase up to $1\text{-}\mathrm{cm}$ distance from a point source. Variability in measurements increases to 17% in modulation depth and $1.9\mathrm{deg}$ in phase for longer distances from the source.⁴¹ Our simulated variations in excitation fluence are of the order or less than typical measurement noise and permit the use of the cylinder size and mesh resolution to design the mouse phantom for optical properties within 20% of the design value.

3.3.

Sensitivity to Variation in the Boundary

Since there is no point-to-point correspondence of boundary nodes among the six variant mouse phantom designs that we described in the preceding section, we subdivide each slice into four quadrants for the purposes of comparison. We use the mean of log of fluence in each quadrant in each slice as a representative estimate in each design and compare those estimates with the same measure in the basic design. Thus, the basis of our comparison is a ratio of the mean of log. Mean of log as an estimate makes sense since the fluence varies exponentially with source-detector angle.

In order to understand the nature of the variation, if any, between each design variant, we study the similarities and/or differences across the slices of each design. We ensure⁴² that each of our data sets follow a normal distribution while rejecting outliers that lie within 2 $\mathit{mfp}$ of the source. We use analysis of variance (ANOVA) tests⁴² to assess whether there are significant differences of ratios of mean of logs over all slices and all variant designs, over each slice across all designs, and over each design across all slices. The null hypothesis for the two-way ANOVA test is that the ratios of mean of logs over all mouse designs in all regions do not differ significantly. At a 95% confidence interval, we find no statistically significant difference between our six design variants. Change in light intensity as we move away from the source in each design is a more dominant source of variation $(\mathrm{p}-\text{value}=0.364)$ compared to the change in the FE model of the mouse $(\mathrm{p}-\text{value}=0.912)$ . This is also evinced by our results (not shown here) of a one-way unstacked ANOVA test for comparing the similarity of distribution of ratios within designs and within slices.

3.4.

Application

One of the important design goals for exogenous fluorescent contrast agents is to achieve an adequate specific binding to the disease target to enable detection through an optical imaging system.⁴³ Target-to-background ratio (TBR) is a measure of specific binding. TBR is a function of the sensitivity of the imaging system, optical properties in the interior of the tissue at the wavelengths of excitation and emission of the fluorescence, time after injection of contrast agent, and depth and size of the anomalous tissue and could be simulated using tissue mimicking phantoms.

To illustrate this, we embed a cylindrical fluorescent target of $2\text{-}\mathrm{mm}$ height and $3\text{-}\mathrm{mm}$ diam, $2\mathrm{mm}$ beneath the boundary of both the cylindrical and the mouse phantoms and illuminate it with an excitation point source placed on the boundary above the target. The absorption coefficient of the target is tenfold higher than the background and is assumed to be the same at the excitation and the emission wavelengths. We use our 3-D FE diffusion model to calculate both excitation and emission fluence in the reflectance geometry on the boundary close to the source for the two different target positions. The local geometry near the source for the two positions is shown in Fig. 5a . The heights are averaged over the nodes that are within an angular distance of $3\text{-}\mathrm{mm}$ on the boundary over every $1\text{-}\mathrm{mm}$ axial distance for $8\mathrm{mm}$ on either side of the source (at $0\mathrm{mm}$ for purposes of comparison). A similar averaging is used to plot the excitation and emission fluence profiles, shown in Figs. 5b and 5c.

Fig. 5

Detectability of a $2\text{-}\mathrm{mm}$ -deep fluorescent target located below a point source in the cylindrical phantom position 1 (---), position 2 (—), and the mouse phantom position 1 (--∗--), position 2 (−+−) designed for ${\mu }_s^{\prime }=0.5{\mathrm{mm}}^{-1}$ , ${\mu }_a=0.01{\mathrm{mm}}^{-1}$ (in the background), ${\mu }_a=0.1{\mathrm{mm}}^{-1}$ (in the target) for two source positions. (a) Height map of the boundary on either side of the source (located at $0\mathrm{mm}$ ). Note the uniform height of the cylinder surface and the nonuniform cross section of the mouse boundary. (b) Excitation fluence on the boundary near the source. Note that the peaks are marginally ( $\sim 15$ to 20%) higher for the mouse. (c) Emission fluence on the boundary near the source. Detectability is tenfold higher when the target is under a local hump compared to under a depression in the mouse phantom.

Table 3

Percent error for each test case in the sensitivity analysis study.

As shown in Fig. 5a, while the height of the boundary surface is uniform in the cylindrical phantom, the mouse boundary is definitely nonuniform. In Fig. 5b, we see that excitation fluence peaks over the mouse target are $\sim 15$ to 20% higher than the targets in the cylinder, due to the smaller size of the mouse cross section. In Fig. 5c, we see the local geometry near the source strongly influencing the profile of the emission fluence from the fluorescent target located just below the source. The detectability from a target located below a hump is about ten times higher than a target located under a concave depression on the mouse boundary. The profile of fluence is similar for the two positions on the cylindrical surface. While an appropriately modeled cylindrical surface to represent a small animal may be useful to determine a conservative measure of specific binding of a fluorescent target, the realistic boundary model of a small animal enables the prediction of the much wider range of detectability of a target based on its anatomical location. Thus simulations on a mouse phantom can aid in the imageability of fluorescent probes to help define bounds on “design” parameters and enhance the pace of development of targeted contrast by restricting the number of actual experiments and reducing the number of animals sacrificed.