2.1.

Absorption imaging

The total absorption coefficient can be written as a sum of a homogeneous background component μ_a0 and a spatially varying perturbation δμ_a0(r), thus, μ_a(r) = μ_a0 + δμ_a(r). As discussed above, we use the first Born approximation to the RTE and express the perturbed radiance [TeX:] $\delta I(r_{\rm d}, \hat \Omega _{\rm d})$ $\delta I(r_{\mathrm{d}},{\widehat{\Omega }}_{\mathrm{d}})$ , the intensity flowing in direction [TeX:] $\hat \Omega _{\rm d}$ ${\widehat{\Omega }}_{\mathrm{d}}$ and arriving at position r _d on the detector plane, as follows:

2.1.1.

Spatial resolution

To calculate the spatial resolution, we consider a single array of point absorbers evenly distributed throughout the cortical depth and with an absorption coefficient of μ_a0 + Δμ_a0, where Δμ_a0 is the extra absorption due to the functional activation; thus,

Eq. 4

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \delta \mu _{\rm a} (r) = \sum\limits_{0 \le z{}_i \le 1\,{\rm mm}} {\Delta \mu _{{\rm a}0} \delta (x,y,z - z_i)}, \end{equation}\end{document} $$\delta {\mu }_{\mathrm{a}}\left(r\right)=\sum _{0\le z{}_i\le 1\mathrm{mm}}\Delta {\mu }_{\mathrm{a}0}\delta (x,y,z-z_i),$$

where δ(x, y, z–z _i) is the Dirac δ function representing a point absorber at (0, 0, z _i) inside the tissue. In our calculation, each voxel is 10 μm in size, and we consider depths from 0 to 1 mm below the surface because the light intensity reaching the detector from tissues deeper than this is negligible for the wavelength range (in the visible up to ~670 nm) typically used in both absorption and fluorescence imaging. Therefore, there are 100 identical point absorbers [Fig. 2a]. Note that [TeX:] $I_0 (r,\hat \Omega)$ $I_0(r,\widehat{\Omega })$ in Eq. 1 only depends on depth z due to the collimated illumination and the homogeneous background optical properties; thus, [TeX:] $I_0 (r,\hat \Omega) = I_0 (z,\hat \Omega)$ $I_0(r,\widehat{\Omega })=I_0(z,\widehat{\Omega })$ . Combining Eqs. 4, 3, we obtain

Eq. 5

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \delta I(r_{\rm d}) = - \Delta \mu _{{\rm a0}} \sum\limits_{0 \le z \le 1\,\,{\mathop{\rm mm}\nolimits} } {\int {I_0 (z,\hat \Omega)G(r_{\rm d} |0,0,z,\hat \Omega)d\hat \Omega } } . \end{equation}\end{document} $$\delta I\left(r_{\mathrm{d}}\right)=-\Delta {\mu }_{\mathrm{a}0}\sum _{0\le z\le 1\mathrm{mm}}\int I_0(z,\widehat{\Omega })G\left(r_{\mathrm{d}}\right|0,0,z,\widehat{\Omega })d\widehat{\Omega }.$$

Here, δI(r _d) is the difference image caused by the single array of point absorbers. [TeX:] $G(r_{\rm d} |0,0,z,\hat \Omega)$ $G\left(r_{\mathrm{d}}\right|0,0,z,\widehat{\Omega })$ is the Green function solution for the transport equation for light emitting from a point (0, 0, z) inside the homogeneous tissue (where a point absorber is located) in the direction of [TeX:] $\hat \Omega$ $\widehat{\Omega }$ to a point r _d on the detector. Therefore, we can interpret δI(r _d) as follows [Fig. 5a]: collimated light illuminates the homogeneous tissue and leads to a radiance of [TeX:] $I_0 (z,\hat \Omega)$ $I_0(z,\widehat{\Omega })$ arriving in the direction of [TeX:] $\hat \Omega$ $\widehat{\Omega }$ at the point absorber located at (0, 0, z). The point absorber then acts as a “negative light source” with strength proportional to [TeX:] $ - \Delta \mu _{{\rm a}0} I_0 (z,\hat \Omega)$ $-\Delta {\mu }_{\mathrm{a}0}{I}_0(z,\widehat{\Omega })$ that propagates back toward the detector. We then obtain the intensity variation at the detector due to this point absorber alone by integrating the radiance over 4π [i.e., [TeX:] $ - \Delta \mu _{{\rm a}0} \int {I_0 (z,\hat \Omega)G(r_{\rm d} |0,0,z,\hat \Omega)d\hat \Omega }$ $-\Delta {\mu }_{\mathrm{a}0}\int I_0(z,\widehat{\Omega })G\left(r_{\mathrm{d}}\right|0,0,z,\widehat{\Omega })d\widehat{\Omega }$ ]. Hence, the total intensity variation at the detector caused by all the point absorbers is simply the summation of each individual contribution, as described by Eq. 5.

Fig. 5

(a) A point absorber at (0, 0, z) in the absorption imaging acts as a negative light source that has an anistropic radiance – [TeX:] $I_0 (z,\hat \Omega)$ $I_0(z,\widehat{\Omega })$ in the direction of [TeX:] $\hat \Omega$ $\widehat{\Omega }$ . (b) A fluorescent target at (0, 0, z) in the fluorescence imaging acts as an isotropic light source with an intensity of I ₀(z).

We employ Monte Carlo simulation to calculate both [TeX:] $I_0 (z,\hat \Omega)$ $I_0(z,\widehat{\Omega })$ and [TeX:] $G(r_{\rm d} |0,0,z,\hat \Omega)$ $G\left(r_{\mathrm{d}}\right|0,0,z,\widehat{\Omega })$ using the homogenous optical properties μ_s0, μ_a0, n₀, and g ₀. We need one Monte Carlo simulation for [TeX:] $I_0 (z,\hat \Omega)$ $I_0(z,\widehat{\Omega })$ , however, 100 different simulations for [TeX:] $G(r_d |0,0,z,\hat \Omega)$ $G\left(r_d\right|0,0,z,\widehat{\Omega })$ , one for each point absorber located at a different depth z. Because Monte Carlo simulation is time consuming, it would be advantageous to reduce the number of calculations. To this end, we take advantage of the translational invariance of the tissue, the principle of reciprocity, and the 4f configuration to further simplify Eq. 5 into

Eq. 6

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} \hspace*{-2pc}\delta I(x_{\rm d}, y_{\rm d}, z_{\rm d}) &=& - \Delta \mu _{a0} \sum\limits_{0 \le z \le 1\,{\rm mm}}\nonumber \\ && \hspace*{-.0pc}\times {\int\!\! {I_0 (z,\hat \Omega)G(x_{\rm d}, y_{\rm d}, z, - \hat \Omega |0,0,z_{\rm d})d\hat \Omega } } \end{eqnarray}\end{document} $$\begin{array}{ccc}\hfill \delta I(x_{\mathrm{d}},y_{\mathrm{d}},z_{\mathrm{d}})& =& -\Delta {\mu }_{a0}\sum _{0\le z\le 1\mathrm{mm}}\hfill \\ & & \times \int I_0(z,\widehat{\Omega })G(x_{\mathrm{d}},y_{\mathrm{d}},z,-\widehat{\Omega }|0,0,z_{\mathrm{d}})d\widehat{\Omega }\hfill \end{array}$$

[TeX:] $G(x,y,z, - \hat \Omega |0,0,z_d)$ $G(x,y,z,-\widehat{\Omega }|0,0,z_d)$ represents the radiance inside the tissue arriving at point (x, y, z) and traveling in a direction [TeX:] $ - \hat \Omega$ $-\widehat{\Omega }$ . It results from a point source located at the center of the detector plane (0, 0, z _d) and thus can be obtained from only one Monte Carlo simulation. Equation 6 shows that the difference image can be obtained from only two Monte Carlo simulations: one for [TeX:] $I_0 (z,\hat \Omega)$ $I_0(z,\widehat{\Omega })$ and the other for [TeX:] $G(x,y,z, - \hat \Omega |0,0,z_{\rm d})$ $G(x,y,z,-\widehat{\Omega }|0,0,z_{\mathrm{d}})$ .

2.2.

Fluorescence imaging

In fluorescence imaging, a photon is first absorbed by a fluorescent target and subsequently emitted at a longer wavelength with certain fluorescent quantum efficiency. The functional activation can cause a local perturbation of the absorption coefficient, the fluorescence efficiency, or a combination of the two. The net result is a change of the local fluorescent intensity. For simplicity, we treat the tissue as a homogeneous scattering medium with optical properties of μ_s0, μ_a0, n₀, and g ₀ with its fluorescent quantum efficiency varying between η₀ and η₀ + δη(r) corresponding to without and with the functional activation, respectively. This local variation of the fluorescent quantum efficiency will lead to a difference fluorescent image δI. Similar to that described in the case of absorption imaging, we can find this difference image as follows [Fig. 5b]: collimated light propagates inside the tissue and arrives at position r, where a fluorescent target is located with an intensity of I ₀(r). The fluorescent target then emits incremental fluorescence due to the functional activation with an intensity δη(r)I ₀(r). This extra fluorescence propagates back toward the detector as described by the Green function G(r _d|r), resulting in an incremental fluorescent intensity at position r _d of the detector, δη(r)I ₀(r)G(r _d|r). The total incremental fluorescent intensity at position r _d, δI(r _d), is the summation of the individual contributions from all the fluorescent targets inside the tissue

Comparing Eqs. 3, 10, we can see that the difference image of the absorption imaging is sensitive to the angular distributions of both the radiance [TeX:] $I_0 (r,\hat \Omega)$ $I_0(r,\widehat{\Omega })$ and the Green function [TeX:] $G(r_{\rm d}, \hat \Omega _{\rm d} |r,\hat \Omega)$ $G(r_{\mathrm{d}},{\widehat{\Omega }}_{\mathrm{d}}|r,\widehat{\Omega })$ , whereas that for fluorescence imaging only depends on the total intensity. The negative sign in Eq. 3 means that the increase in tissue absorption reduces the intensity of the image on the detector plane, leading to a negative difference image, while the positive sign in Eq. 10 means that an increase in the fluorescence quantum efficiency increases the intensity of the image on the detector plane, leading to a positive difference image.

3.1.

Spatial profile of the difference Image Widens and the Spatial Rresolution Worsens with the Increase of NA and Focal Plane Depth

Figure 6 depicts the spatial profiles of the difference images due to the presence of a single array of point absorbers (or fluorescent targets) for both absorption and fluorescence imaging and under different optical configurations. In particular, Fig. 6a shows the spatial profiles corresponding to NA = 0.2 and focal plane depths at 100 (blue color), 300 (red color), and 500 μm (black color), respectively, for absorption imaging. As the imaging system focuses deeper into the tissue, the spatial profile broadens. Figure 6b shows the spatial profiles corresponding to three different NAs [0.1 (blue), 0.2 (red), and 0.4 (black)] and focal plane depth = 300 μm for absorption imaging. As the imaging system collects more light that exits the tissue with large angles, the spatial profile broadens. Figures 6c and 6(d) depict the corresponding profiles for fluorescence imaging, which exhibit the same trend. We explain the results of Fig. 6 as follows. Assume we have calculated the difference image on the tissue surface generated by the array of point absorbers of Fig. 2a. We would expect it to have features similar to that of Fig. 3b, thus a similar profile along the radial direction depicted in Fig. 3c. Now, let us examine how this profile would be imaged onto the detector plane by the 4f system. If the focal plane depth is 0, then a point on the tissue surface will be imaged onto a point on the detector plane (assuming there is no aberration). Therefore, the detector plane would have a similar profile. However, if the focal plane depth is d, then an area with a radius ~d tan[sin^–1(NA/n ₀)] will contribute to a point on the detector; thus, the profile on the detector would be “blurred.” The larger the focal plane depth d and NA are, the more blurring it will induce, thus leading to a wider profile and a lower spatial resolution (as can be seen in Fig. 7).

Fig. 6

Spatial profiles of the difference images due to the presence of a single array of point absorbers (or fluorescent targets) for both the absorption and fluorescence imaging. (a) The spatial profiles corresponding to NA = 0.2 and focal plane depths at 100 (blue), 300 (red), and 500 μm (black), respectively, for the absorption imaging. (b) The spatial profiles corresponding to three different NA at 0.1 (blue), 0.2 (red), and 0.4 (black), respectively, and focal plane depth = 300 μm for the absorption imaging. (c) The spatial profiles corresponding to NA = 0.2 and focal plane depths at 100 (blue), 300 (red), and 500 (black) μm, respectively, for the fluorescence imaging. (d) The spatial profiles corresponding to three different NA at 0.1 (blue), 0.2 (red), and 0.4 (black), respectively, and focal plane depth = 300 μm for the fluorescence imaging. In (c, d), straight and dashed lines represent the profiles corresponding to uniform and nonuniform distributions of the fluorescent targets, respectively.

Fig. 7

Spatial resolution versus focal plane depth of the (a) absorption and (b) fluorescence imaging for NA of 0.1 (circle), 0.2 (star), and 0.4 (triange), respectively. In (b), straight and dashed lines represent the profiles corresponding to uniform and non-uniform distributions of the fluorescent targets, respectively.

The uniform [straight line, Figs. 6c and 6d] and nonuniform [dashed line, Figs. 6c and 6d] distributions of the fluorescent targets have similar profiles, with the exception that the profile of the nonuniform distribution has slightly longer tails than that of the uniform distribution under the same optical configuration. As discussed below, only fluorescent targets within the top ~500 μm of the tissue have a major effect on the spatial profile. Figures 2c and 2d illustrate that, within the top 500 μm, more fluorescent targets lie in the deeper layers in the nonuniform distribution, resulting in more light scattering and thus longer tails in this case.

The spatial profiles of the absorption imaging have longer tails than their fluorescence imaging counterparts under the same optical configuration. Equation 5 indicates that radiance [TeX:] $I_0 (z,\hat \Omega)$ $I_0(z,\widehat{\Omega })$ from the incidence light arriving at the point absorber at (0, 0, z) and flowing in the direction [TeX:] $\hat \Omega$ $\widehat{\Omega }$ . The point absorber absorbs it and subsequently acts as a negative light source with an emission that propagates toward the detector in the same direction [TeX:] $\hat \Omega$ $\widehat{\Omega }$ [Fig. 5a]. Equation 14 describes a similar process for fluorescence imaging except that after the radiance [TeX:] $I_0 (z,\hat \Omega)$ $I_0(z,\widehat{\Omega })$ is absorbed, the fluorescent target emits isotropic fluorescence [Fig. 5b]. Because we are interested in light propagating from near to within a few scattering lengths from the surface and the scattering is highly forward (anisotropic factor g is 0.9), the point absorbers act as anisotropic sources that emit radiation preferentially toward deeper tissue layers. Consequently, this light would travel deeper into the tissue and thus experience more scattering as compared to the emission from fluorescent targets that act as isotropic light sources. Therefore, the spatial profile in the former case has longer tails.

Figure 7 shows the spatial resolution of absorption (a) and fluorescence imaging (b) versus focal plane depth under different NAs. In both cases, the spatial resolution worsens with the increase of the NA and focal plane depth. Note that the spatial resolutions of absorption and fluorescence imaging are similar, with the exception that those of absorption imaging are slightly worse under the same optical configurations, which is consistent with their spatial profiles. In general, as we vary the NA from 0.1 to 0.4 and the focal plane depth from 20 to 300 μm, we change the spatial resolution from <20 to 200 μm, which is still sufficient to resolve many individual functional columns. For example, the horizontal size of many vertebrate columns ranges from 50 to 800 μm.²⁵ We can compare our result to the spatial resolution obtained by Polimeni using the following tissue parameters: μ_s0= 35.4 and 53.2 mm⁻¹, μ_a0= 0.27 and 0.22 mm^–1, n₀ = 1.4, and g ₀ = 0.94 and 0.82 for gray and white matter, respectively; and NA ~0.4 and focal plane depth = 300 μm.⁹ They used two lenses coupled directly instead of the 4f configuration. They obtained a spatial resolution of 240 μm for the summed contributions within a tissue depth of 200–500 μm.⁹ This is similar to the 200 μm, which we obtained for NA = 0.4, focal plane depth = 300 μm, and a tissue depth of 200–500 μm. The difference can be attributed to the slight variation in the tissue parameters and the difference between their lenses being directly coupled versus our 4f configuration.

The profiles shown in Fig. 6 have larger tails than that of a Gaussian with the same FWHM, which is consistent with that reported by Polimeni.⁹ Furthermore, Figs. 6c, 6d, 7b show that modifying the density distribution of the fluorescent dyes across the cortical depth from Figs. 2c to 2d can result in a spatial profile with longer tails and yet a slightly smaller FWHM width (i.e., the spatial resolution). This suggests that, in addition to spatial resolution, it is also important to consider a cross talk between signals generated in adjacent cortical columns, which is larger in a case of diffusely reflected light than in the case of Gaussian-like signal profiles.

3.2.

Depth Sensitivity Depends on NA and Focal Plane Depth when the Functionally Activated Column is Small and Independent of NA and Focal Plane Depth when the Activated Column is Large

Here, we examine how the NA and focal plane depth affect the depth sensitivity for absorption (Figs. 8 and 9) and fluorescence imaging (Figs. 10 and 11). Figure 8 depicts the depth sensitivity for functionally activated columns of four different cross-sectional areas [Fig. 8a: 10 × 10, Fig. 8b: 50 × 50, Fig. 8c: 100 × 100, and Fig. 8d: 200 × 200 μm²]. In Figs. 8a, 8b, 8c, 8d, NA is fixed at 0.2 and focal plane depth varies among 100 (circle), 300 (star), and 500 μm (triangle). Here, we plot the percent contribution of each layer to the total signal at the detector center using Eqs. 8, 9. For example, for absorption imaging with NA of 0.2 and focal plane depth of 100 μm, the top 100 μm (0–100 μm) of the tissue contributes to ~85% of the total signal (contributions from 0–1000 μm). In Fig. 8a, where the activated column represents a single array of point absorbers [see Fig. 2a] (in our simulation, the horizontal size of one voxel is 10 μm), the depth sensitivity varies drastically with the focal plane depth. In particular, the increase of the focal plane depth from 100 to 300 μm results in a significant increase in the percent contribution from the deeper tissues; the increase of the focal plane depth from 300 to 500 μm, however, results in a smaller increase of the percent contribution from the deeper tissues. For example, the contributions of the tissue within the top 100 μm are 84, 46, and 37%, while those from 200 to 300 μm are 2, 10, and 16%, corresponding to focal plane depths of 100, 300, and 500 μm, respectively. This trend of increasing percent contribution from deeper layers with the focal plane depth reaches a plateau for focal plane depth of >500 μm (not plotted here).

Fig. 8

Depth sensitivity for functionally activated columns of four different cross-sectional areas: (a) 10 × 10, (b) 50 × 50, (c) 100 × 100, and (d) 200 × 200 μm², respectively, for absorption imaging. In (a)–(d) NA is fixed at 0.2 and the focal plane depth is varied among 100 (blue), 300 (circle), and 500 (triangle) μm.

Fig. 9

Depth sensitivity for functionally activated columns of four different cross-sectional areas: (a) 10 × 10, (b) 50 × 50, (c) 100 × 100, and (d) 200 × 200 μm², respectively, for absorption imaging. In (a)–(d), the focal plane depth NA is fixed at 300 μm and NA is varied among 0.1 (circle), 0.2 (star), and 0.4 (triangle).

Fig. 10

Depth sensitivity for functionally activated columns of four different cross-sectional areas: (a) 10 × 10, (b) 50 × 50, (c) 100 × 100, and (d) 200 × 200 μm², respectively, for fluorescence imaging with the nonuniform distribution of the fluorescent targets. In (a)–(d), NA is fixed at 0.2 and the focal plane depth is varied among 100 (circle), 300 (star), and 500 (triangle) μm.

Fig. 11

Depth sensitivity for functionally activated columns of four different cross-sectional areas: (a) 10 × 10, (b) 50 × 50, (c) 100 × 100, and (d) 200 × 200 μm², respectively, for fluorescence imaging with the nonuniform distribution of the fluorescent targets. In (a)–(d), the focal plane depth is fixed at 300 μm and NA is varied among 0.1 (circle), 0.2 (star), and 0.4 (triangle).

As the cross-sectional area of the activated column increases, the depth sensitivity still varies, but less drastically, with the focal plane depth [Figs. 8b and 8c]. When the area of the activated column reaches 200 × 200 μm² [Fig. 8d], the depth sensitivity barely depends on the focal plane depth. Note that the horizontal size 200 μm is larger than the spatial resolutions for the given NA and focal plane depths. As can be seen from all panels, >97% of the signal comes from the top 500 μm of tissue; thus, only the depth sensitivity from the top 500 μm is plotted in all panels and the Figs. 8, 9, 10, 11.

Figure 9 depicts the depth sensitivity for functionally activated columns with different cross sectional areas [Fig. 9a: 10 × 10, Fig. 9b: 50 × 50, Fig. 9c: 100 × 100, and Fig. 9d: 200 × 200 μm²), when the focal plane depth is fixed at 300 μm and the NA varies between 0.1 (circle), 0.2 (star), and 0.4 (triangle). The results are similar to those of Fig. 8. In Fig. 9a, where the activated column represents a single array of point absorbers, the depth sensitivity varies significantly with the NA. In particular, the increase of the NA from 0.1 to 0.2 results in a large increase of the percent contribution from the deeper tissues; the increase of the NA from 0.2 to 0.4, however, results in a smaller increase of the percent contribution from the deeper tissues. This trend of increasing percent contribution from deeper layers with the NA reaches a plateau for NA ≥ 0.4 (not plotted here). The depth sensitivity for the column with a cross-sectional area of 50 × 50 μm² [Fig. 9b] is similar to that of [Fig. 9a]. When the area of the activated column increases to 100 × 100 μm², the depth sensitivity still varies, but less drastically, with the NA. When the area of the activated column reaches 200 × 200 μm² [Fig. 9d], the depth sensitivity barely depends on the NA.

In summary, Figs. 8 and 9 demonstrate that the smaller the cross-sectional area of the activated column is, the more sensitive the depth sensitivity is to the NA and focal plane depth. Interestingly, the depth sensitivity does not depend on the NA or focal plane depth when the horizontal size of an activated column is larger than its spatial resolution, which can be explained as follows: δI _{z 0}(0, 0, z _d), the individual contribution to the intensity at the detector center, is proportional to the summation of the Green function G(x, y, z|0, 0, z _d) over the cross-sectional area of the activated column. Because this Green function represents the light distribution inside the tissue from a point source at the center of the detector, its summation over a cross-sectional area with a horizontal size larger than spatial resolution is approximately the total power transmitted through the tissue, whose profile versus depth (that is, the depth sensitivity) does not change with either NA or focal plane depth.

Next, we present the results for fluorescence imaging. For fluorescence imaging with the uniform distribution of fluorescent targets, these are similar to the results shown in Figs. 8 and 9. Therefore, we only show the results for the nonuniform distribution of fluorescent targets [as depicted in Fig. 2d]. Here, we observe the same trend of the depth sensitivity change with the change of focal plane depth (Fig. 10) and NA (Fig. 11). In addition, there is a notable difference between fluorescence imaging of tissues with uniform and nonuniform staining profile. Figure 10d depicts a higher percent contribution (53%) to the signal from deeper tissues (200–500 μm) than that shown in Fig. 8d (26%), a direct consequence of the different spatial distributions of the fluorescent targets in these two cases, as shown in Figs. 2c and 2d. This illustrates the strong effect of the dye-concentration profile on the depth sensitivity. In many functional fluorescence imaging experiments, the dyes are externally administered. Hence, different dye concentration profiles may be obtained depending on the specific experimental protocols.^{20, 21, 22}

3.3.

Larger NA and Focal Plane Depth Suppress the Contribution from Surface Vessels in the Absorption Imaging

As we have seen from Figs. 8d and 10d, the depth sensitivity does not depend on the NA and focal plane depth when the activated column is large. This is only true, however, when the activated columns are devoid of surface vessels above them. In this case, we can assume a uniform distribution of absorption coefficient in the tissue. However, there are areas in the brain tissue that are covered with surface vessels, which can significantly affect the absorption coefficient of that area. Here, we consider only the situation in which the horizontal size of the activated column is larger than the spatial resolution: in particular, in which it is 400 μm. The vessel of 50 × 50 × 50 μm³ is embedded within the top 400 μm of the tissue, and a blood tissue volume ratio of ~1% is used. Hence, the extra point absorbers due to the presence of the vessel is

Figure 12a depicts the percent contribution from different depths of tissue and the surface vessel in the absorption imaging for NA of 0.2 and focal plane depths of 100 (circle), 300 (star), and 500 μm (triangle). As we focus deeper into the tissue, the contribution of the surface vessels decreases. For example, the contribution from the surface vessel is suppressed from ~36 to 14% as the focal plane depth changes from 100 to 500 μm. A similar trend is observed for focal plane depth = 300 μm and NA of 0.1 (circle), 0.2 (star), and 0.4 (triangle), as shown in Fig. 12b. Note that when the surface vessels are either very large (≥100 μm) or small (~10 μm), the contribution of the surface vessels to the signal will be either too large or too small; thus, the effect of changing NA and focal plane depth will be negligible.

Fig. 12

Percent contributions from different depths of the tissue and the surface vessel in the absorption imaging. (a) NA = 0.2 and focal plane depths of 100 (circle), 300 (star), and 500 (triangle) μm. (b) focal plane depth = 300 μm, NA = 0.1 (circle), 0.2 (star), and 0.4 (triangle). In (a, b), the horizontal size of the activated column is 400 μm. The vessel of 50 × 50 × 50 μm³ is embedded within the top tissue and a blood tissue volume of ~1% is used. SV: surface vessel.

Here, we consider absorption imaging only because in many fluorescence imaging experiments the wavelength is chosen so that the absorption of the hemoglobin, and thus, of the surface vessels, is negligible.

4.1.

Method validation

The Monte Carlo code tMCimg itself has been validated against an accepted analytic solution for a semi-infinite medium, and cross-validated for a slab geometry with an absorbing inclusion.¹⁹

We evoke the first Born approximation in our calculation. In order for this approximation to be valid, the spatial perturbation to the absorption coefficient, δμ_a(r), needs to be much smaller than the background absorption coefficient μ_a0.¹⁶ As discussed above, δμ_a(r) is often one order of magnitude smaller than μ_a(r); thus, the first Born approximation is valid and δI in Eq. 1 can truly represent the difference image. When the perturbation is large, however, a full Monte Carlo simulation needs to be employed. In order to reduce the number of Monte Carlo simulations for calculating the spatial resolution, we take advantage of the 4f configuration so as to simplify Eqs. 5, 10 into Eqs. 6, 14. Here, we examine what will happen if the imaging system does not have a 4f configuration. In general, the two lenses L₁ and L₂ can have different focal lengths f ₁ and f ₂, respectively and are separated by a distance d. Let us examine two situations: (i) d = f ₁ + f ₂ and 2) d ≠ f ₁ + f ₂. When d = f ₁ + f ₂, the imaging system can be viewed as a general 4f configuration with magnification M = f ₂/f ₁. The spatial profiles and spatial resolutions shown in Figs. 6 and 7 are valid as long as we understand that they apply directly to the brain tissue. For example, if M = 2, then a spatial resolution of 200 μm means that we can resolve two cortical columns separated by 200 μm, though on the detector plane they will actually be separated by 400 μm (200 μm × 2). When d ≠ f ₁ + f ₂, the spatial resolution provides good approximation. For example, the spatial resolution calculated by Polimeni ⁹ using a tissue model that is similar to a particular configuration used in our simulation, though with a notable difference: in their model, f ₁ = f ₂, but d ≪ f ₁ + f ₂, which represents a case very different from the 4f configuration. However, their result of 240 μm is similar to our result of 200 μm. In addition, Orbach and Cohen¹⁰ reported an experimental result of 280 μm using a condition similar to that of Polimeni.⁹ Hence, our simulation improves the efficiency while maintaining results that agree with other calculations and experimental measurements. Note that we do not need to assume a 4f configuration in deriving the equations for calculating depth sensitivity; thus, the depth sensitivity for other optical configurations will be the same as shown in Figs. 8, 9, 10, 11, 12.

4.2.

Computational Efficiency

We have simulated 21 different optical configurations—three values for NA (0.1–0.4) and seven for focal plane depth (20–600 μm)—covering the typical range encountered in camera-based optical brain-imaging experiments. We have considered the optical disturbance within the top 1 mm of the brain tissue and have chosen a voxel size of 10 μm. Therefore, we must calculate the contributions from 100 locations in order to obtain the spatial resolution.

As discussed above, in order to obtain the difference images, we calculate the Green function originating from the center of the detector toward the tissue [TeX:] $G(x,y,z,\hat \Omega |0,0,z_{\rm d})$ $G(x,y,z,\widehat{\Omega }|0,0,z_{\mathrm{d}})$ instead of that originating from the point absorbers (or fluorescent targets) inside the tissue toward the detector [TeX:] $G(x_{\rm d}, y_{\rm d}, z_{\rm d} |0,0,z,\hat \Omega)$ $G(x_{\mathrm{d}},y_{\mathrm{d}},z_{\mathrm{d}}|0,0,z,\widehat{\Omega })$ . Here, we will compare the number of simulations needed for both approaches. In the previous case, we need one simulation for each combination of NA and focal plane depth; however, the contributions from different locations inside the tissue are automatically taken care of by [TeX:] $G(x,y,z,\hat \Omega |0,0,z_{\rm d})$ $G(x,y,z,\widehat{\Omega }|0,0,z_{\mathrm{d}})$ . Hence, we need a total of 21 simulations for estimating the spatial resolution and depth sensitivity of both the absorption and fluorescence imaging. In the latter case, we need 100 simulations. By using the former approach, we improve the efficiency by five times.

4.3.

Strategies to Increase the Penetration Depth of Absorption and Fluorescence Imaging

As discussed in Sec. 3, changing NA and focal plane depth does not affect depth sensitivity when the functionally activated cortical areas are large and devoid of large surface vessels. However, as can be seen in Fig. 12, when imaging cortical areas covered with large surface vessels, focusing deeper into the tissue and/or enlarging the NA may suppress the contributions from these vessels and effectively enhance the penetration depth of the absorption imaging. In fact, this practice has been employed in functional imaging to reduce the effect of the surface vessels, as demonstrated in Ref. 15.

In many camera-based optical brain imaging experiments, large cortical areas are often excited during the peak response. For example, electrically stimulating the forepaw of a rat can lead to functional activation of a cortical area well beyond 1 mm². Hence, changing the NA and focal plane depth does not affect the penetration depth (Fig. 8). On the other hand, with the advent of new technologies, such as optogenetics in which optical methods are used to control genetically targeted neurons within intact neural circuits, it is likely that selective activation of a small cross-sectional area ≤ 100 × 100 μm² may be soon achieved.²⁶ As shown in Figs. 8, 9, 10, 11, focusing deeper into the tissue and/or enlarging the NA would enhance the penetration depth of both absorption and fluorescence imaging of such small cortical areas.

In fluorescence imaging, where dyes are applied externally, it may be possible to enhance the contributions from specific cortical layers by regulating the density distribution of the dyes across the cortical depth. For example, in voltage sensitive dye imaging, Ferezou ²⁰ reported a density distribution shown in Fig. 2d, where deeper layers (200–500 μm) had more dye than the shallow layers (0–200 μm). Therefore, the image would consist of more contribution from deeper layers [comparing the two profiles of the depth sensitivity, in Figs. 8d and 10d].²⁰

4.4.

Depth Sensitivity is Important in Data Interpretation, Cross Modality and Species Comparison of Functional Imaging Results

It is common to compare functional brain-imaging data from different imaging modalities (such as IOI and VSDI) or across species (such as mouse and rat). In general, the mammalian neo-cortex consists of six layers, each with a unique thickness depending on the species and cortical areas.²⁷ To compare fairly, data collected from different modalities or species should come from the same cortical layer(s). Hence, it is critical to know quantitatively the contribution from each layer with a specific imaging modality (e.g., IOI) on a particular animal model (e.g., mouse). Here, we discuss two examples and determine for each whether it is appropriate to compare data obtained from two different modalities or species.

We first compare the data from the primary somato-sensory cortex (SI) of a rat obtained from two different modalities: IOI and VSDI. For simplicity, we only consider the case when the activated column is larger than or equal to the spatial resolution and the staining profile for VSDI is nonuniform, as shown in Fig. 2d. As discussed earlier, 97% of the contribution to the signal comes from the top 500 μm. Therefore, we need to consider only layer I (0–150 μm) and II/III (150–650 μm) of the rat SI.²⁷ By using Eqs. 8, 9 for IOI and Eqs. 17, 18 for VSDI, we have obtained the percent contribution from layer I and II/III to the total signal to be 58 and 41% when using IOI, and 26 and 73% when using VSDI. Hence, although the signal is dominated by layer I in IOI, it is dominated by layer II/III in VSDI.

The second example examines the IOI results obtained from the same area of two different species: the SI of a mouse and a rat. For simplicity, we only consider the case when the activated column is larger than the spatial resolution. The quantitative analysis of the depth sensitivity shows that 38 and 56% of the signal are from layer I (0–100 μm) and II/III (100–400 μm) of the mouse SI, respectively.²⁷ Once again, although we see that the signal is dominated by layer I in the rat, it is dominated by layer II/III in the mouse.

Both examples reveal that caution has to be taken in comparing data across modalities and species. In general, we need to estimate the contribution from each cortical layer of an individual case in order to compare meaningfully data across different imaging modality and species.

4.5.

Knowledge of Spatial Resolution and Depth Sensitivity Helps Guide Optical Imaging Optimization

When choosing a camera-based optical imaging system, we need to consider important parameters such as NA, focal plane depth, and pixel size of the detector. Our results (Figs. 7, 8, 9, 10, 11, 12) provide both qualitative and quantitative guidance as to how to optimize these parameters: the increase of NA and focal plane depth results in a lower spatial resolution, more suppression of the contribution from the surface vessels, and more contribution from deeper tissues (when the activated column is sufficiently small). In general, larger NA also leads to a stronger signal. Here, we consider some typical examples and give quantitative estimation for each case, as follows: