2.1.

Conceptual Basis of the Proposed Approach

When using CW light, two different situations are possible to explore a slab containing a turbid medium with inclusions.

Fig. 1

Sketch of the experimental setup (top view). A CW laser diode, operating at $830\mathrm{nm}$ , illuminates one face of the tank containing the milky diffusive solution and the inclusion. The collimated laser beam defines the optical axis of the system, which is coincident with the optical axis of the camera lens. A positioning device (not shown) allows placing the inclusion in the transverse direction $(x,y)$ and at different depths along the optical axis $\left(z\right)$ .

The proposed criterion is the result of fusing together these two approaches: a single picture is acquired that is simple and fast, and after that it is normalized to the background transmittance without inclusion, as happens in transilluminance.

As a proof of principles, we show in Figs. 2, 2, 2 the results obtained when using the theoretical model of a cylinder immersed in a slab of turbid media^{22, 23} for the two cases mentioned before. For this example we present in Fig. 2 the transilluminance and transmittance profiles, with no normalization, for a $1.4\text{-}\mathrm{cm}$ -diam absorbing cylinder [ ${\mu }_a^{\left(\mathrm{cyl}\right)}=0.28{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime \left(\mathrm{cyl}\right)}=10{\mathrm{cm}}^{-1}$ ] immersed in a $3\text{-}\mathrm{cm}$ turbid slab with optical properties ${\mu }_a=0.08{\mathrm{cm}}^{-1}$ and ${\mu }_s^{\prime }=10{\mathrm{cm}}^{-1}$ . The cylinder extends vertically along the $y$ direction and is located on the optical axis $(x=0)$ with its longitudinal axis at $z=1.5\mathrm{cm}$ depth (see Fig. 1 for the geometry). Note that the relative higher absorption of the cylinder is not obvious in the transmittance curve. Figure 2 shows the transmittance profile for the slab with the cylindrical inclusion of Fig. 2, together with the transmittance of the slab without inclusion (background transmittance). Figure 2 compares the quotient of the two curves of Fig. 2, which is the ansatz of this work, with the transilluminance profile of Fig. 2.

Fig. 2

Conceptual basis of this proposal. (a) Theoretical transmittance and transilluminance profiles when a cylindrical inclusion is present inside a slab. See Sec. 2.1 for the corresponding optical parameters used. (b) Theoretical transmittances with inclusion $\left[\mathbf{J}\left(\mathbf{r}\right)\right]$ and without inclusion $\left[{\mathbf{J}}_0\left(\mathbf{r}\right)\right]$ ; note that $\mathbf{J}\left(\mathbf{r}\right)$ alone does not gives a clear indication about the optical properties of the inhomogeneity. (c) Normalized fluxes: transilluminance normalized to the baseline value and transmittance normalized as proposed in this work [see Eq. 3].

As can be seen from Fig. 2, both curves tend to the same value far from the optical axis; therefore, the background transmittance becomes a natural “normalizer” for the transmittance profile. In fact, this quotient clearly shows the presence of the more absorbing inclusion, and this ratio has the same qualitative behavior as the transilluminance profile. For comparison, this last curve has been normalized to its own background (far from the optical axis). Thus, both constructions show the presence of the absorbing inclusion. Note that the normalized transilluminance profile has a better lateral resolution, while the modulation is similar to that of the normalized transmittance. However, as was already said, the construction of a transilluminance profile is much more time consuming, since it requires scanning the laser-detector pair.

From a formal point of view, because of the introduction of the inclusion, the photon flux at the detector position, which using conventional notation is given by $\mathbf{J}\left(\mathbf{r}\right)=-4\pi D\nabla \varphi \left(\mathbf{r}\right)$ ,⁶ will change with respect to the one when no inclusion is present ${\mathbf{J}}_0\left(\mathbf{r}\right)$ , according to

2.2.

Real Situations

2.2.2.

Location of the inclusion

Before MC simulations are carried out to retrieve the optical properties of the inclusion, its location (in 3-D) must be inferred. As an example of how this works in our proposal, we present in Fig. 3 an experiment for a $S=3\text{-}\mathrm{cm}$ -thick tank filled with a milky solution (see Sec. 3), the optical properties of which are given in Table 1 . A cylinder, $16\mathrm{mm}$ high and $16\mathrm{mm}$ in diameter, is placed inside the tank as an absorbing inclusion. Figure 3 is the complete transmittance image, with the slab containing the inclusion. A background transmittance image is also acquired without the inclusion. Please refer to Sec. 3 for the details about image acquisition. Figure 3 is the result of normalizing the transmittance image of Fig. 3 to the actual experimental transmittance background. Note that the location of the inhomogeneity (unknown a priori) relative to the optical axis results in Fig. 3. The horizontal and vertical profiles shown help to better visualize the location of the inclusion. In Fig. 3 the experimental background (which in real situations is not available) is replaced by one generated by MC using average optical properties of the slab. This is applied to all experiments presented in Sec. 3.2. Despite the fact that the modulations in the profiles of Fig. 3 are not the same as those in Fig. 3, the primary information concerning the location of the inclusion and its “more absorbing” nature are identical in both figures.

Fig. 3

2-D images, as seen by the camera, and the corresponding horizontal and vertical profiles to show how our proposal works. (a) A complete transmittance image of the tank containing an absorbing inclusion with unknown location; profiles are taken at the optical axis ( $x=0$ and $y=0$ ). (b) The result of normalizing the image shown in (a) to the actual experimental background (without inclusion); this was done in this controlled experiment to compare with the result shown in (c). (c) Same as (b) but normalized to a background generated by MC simulations using the optical properties given in Table 1. Please note that in (b) and (c), the location of the center of the inclusion is practically the same, found to be at $x\simeq 18\mathrm{mm}$ and $y\simeq 2\mathrm{mm}$ , relative to the optical axis (0,0). Additionally, a clue about the lateral extension of the inclusion can be inferred from the FWHM of these profiles.

Table 1

Measured optical properties of the bulk and inclusions (within 10%) corresponding to the different experiments. All measurements were made following the procedure described in Sec. 3.1.

	Experiment 1		Experiment 2		Experiment 3
	Tank	Cylinder	Tank	Cylinder	Tank	Cylinder
${\mu }_a\left[{\mathrm{cm}}^{-1}\right]$	0.07	$⪢0.07$	0.07	0.28	0.07	0.02
${\mu }_s^{\prime }\left[{\mathrm{cm}}^{-1}\right]$	9.04	—	9.40	9.40	9.40	9.40

Please note that rotation of the optical axis (or the slab) by $90\mathrm{deg}$ around the vertical axis produces a complementary view of the slab which, proceeding as described, gives the location of the inhomogeneity in the $z$ direction. For simplicity, in this example we have located the inhomogeneity at $z=S∕2$ , and thus only one view of the tank is needed to have all three coordinates of the center of the inclusion.

3.1.

Measurement of Optical Properties

In the experiments presented in this work, measurement of optical properties is mandatory. To this end, we used a time-resolved approach as described in the following. The advantage of this time-resolved method is that it is possible to retrieve both the absorption coefficient ${\mu }_a$ and the reduced scattering coefficient ${\mu }_s^{\prime }$ simultaneously.⁵ In this technique, a pulsed laser working at $50\text{-}\mathrm{MHz}$ repetition rate injects light pulses of very short duration (typically about $70\mathrm{ps}$ ) in an optically turbid medium via an optical fiber. Photons are diffused in the medium, collected at the exit by another optical fiber, and sent to a high sensitivity detector [photomultiplier tube (PMT)]. Due to dispersion inside the medium, photon trajectories are different from each other, and thus they last different times to reach the collecting optical fiber; this produces a temporal distribution of photons depending on the optical properties of the medium, which is called distribution of times of flight (DTOF). It is built up and displayed on a monitor by means of specific electronic equipment, which is the basis of the technique known as time correlated single photon counting (TCSPC), in which photons are assumed to arrive one by one at the detector. A detailed description of this technique can be found in the monograph by Becker.²⁴ Fitting the resulting DTOF with an adequate theoretical model allows us to obtain the desired optical properties; since we are dealing with slabs of turbid media, we use the model described by Contini, Martelli, and Zaccanti.⁶

It must be mentioned that due to the transit time of photons along the optical fibers and inside the detector, even if no turbid medium is present the laser pulses are temporally spread, giving rise to a certain instrument response function (IRF), which is also measured experimentally by placing the laser fiber face to face with that going to the PMT. The measured DTOF is then the convolution of this IRF with the actual desired temporal distribution; this must be taken into account when the fitting procedure is carried out. In symbols:

3.2.

Experiments, Simulations, and Discussion

To illustrate how our proposal works, we present three controlled experiments for CW light transmittance in turbid media slabs, in which diffusive light intensity, experimentally measured or simulated, is mapped on a bidimensional array at the exit face of the slab. From these 2-D intensity arrays (images), 1-D profiles are taken at a constant vertical position for better visualization and simpler processing. Inclusions consist of cylindrical inhomogeneities placed at different depths inside a homogeneous slab. Optical properties of both, the bulk and the inhomogeneities for all three experiments, are summarized in Table 1. They were measured using the method described in the previous section. This constitutes the a-priori information to test the goodness of the approach.

Experiments were carried out using a $25\times 25\text{-}\mathrm{cm}$ and $3\text{-}\mathrm{cm}$ -thick cuvette filled with a solution of milk, distilled water, and 1:1000 diluted india ink in proportions 1:3:0.04. This mix gives a reduced scattering coefficient $\left({\mu }_s^{\prime }\right)$ and an absorption coefficient $\left({\mu }_a\right)$ , which are very close to those of biological tissues. They were measured previous to the transmittance experiments by fitting DTOF of photons obtained by TCSPC, as described before. The inclusions, when present, were held in place inside the milky solution by a rigid mount attached to a translation stage for proper positioning. Illumination was provided by the collimated beam of a CW, $10\text{-}\mathrm{mW}$ laser diode operating at $\lambda =830\mathrm{nm}$ , and the images were collected by a $14\text{-bit}$ , $768\times 512$ $(\text{horizontal}\times \text{vertical})$ pixel, digital charge coupled device (CCD) camera and stored in the computer. A zoom lens was used to achieve a magnification $m=0.1$ ; thus, the resulting images map an area $76.8\times 51.2{\mathrm{mm}}^2$ of the exit face of the tank. Figure 1 is a schema showing the experimental situation.

The initial MC simulations were carried out using ${10}^{10}$ photons. This is important in transmittance experiments; at the wings of the profiles, far from the optical axis, the number of photons is poor and decreases quickly. Simulating the paths of ${10}^{10}$ photons requires about $30\text{to}40\min$ using the GPU. This time must be compared to the one taken by the Core 2 Duo CPU for performing the same number of simulations, which is about $12\text{days}$ $(\sim 1.7\times {10}^4\min )$ , giving an acceleration factor close to 600. However, after our first results were evaluated, it could be concluded that it is enough to use ${10}^9$ photons in each simulation; thus, calculation time is reduced to a few minutes. So, it is possible to obtain a representative set of MC outcomes (say ten) in a reasonable lapse, which allows us to infer the optical parameters of the inclusion, as well as re-estimate its size and location. Accordingly to this, all MC results presented in Figs. 4, 5, 6, 7, 8, 9, 10 correspond to simulations with ${10}^9$ photons. Clearly the exact number of photons required varies accordingly with both the optical properties and the thickness of the slab and the type of inclusion.

Fig. 4

Highly absorbing inclusion (black painted wooden stick), ${\varphi }_{\mathrm{inc}}=0.7\mathrm{cm}$ , placed next to the light entrance face of the tank. (a) Normalized experimental intensity profile (circles) traced at vertical pixel 256 $(y=0)$ , shown together with the normalized outcome of the MC simulation (solid curve). (b) Experiment and MC profiles given in (a) divided by a common background generated by MC simulations, accordingly with the proposal explained in the text.

Fig. 5

Same as in Fig. 4 for the highly absorbing inclusion placed at the middle of the tank.

Fig. 6

Same as in Fig. 4 for the highly absorbing inclusion placed next to the light exit face of the cuvette.

Fig. 7

Absorbing agarose cylinder of ${\varphi }_{\mathrm{inc}}=1.4\mathrm{cm}$ placed next to the light entrance face of the tank. (a) Normalized experimental intensity profile (circles) traced at vertical pixel 256 $(y=0)$ , shown together with the normalized outcome of the MC simulation (solid curve). (b) Experiment and MC profiles given in (a) divided by a common background generated by MC simulations, accordingly with the proposal explained in the text.

Fig. 8

Same as in Fig. 7 for the agarose cylinder at the middle of the tank.

Fig. 9

Same as in Fig. 7 for the agarose cylinder next to the light exit face of the tank.

Fig. 10

Small diameter agarose cylinder less absorbing than the surrounding medium placed next to the light exit face of the tank. (a) Normalized experimental intensity profile (circles) traced at vertical pixel 256 $(y=0)$ , shown together with the normalized outcome of the MC simulation (solid curve). (b) Experiment and MC profiles given in (a) divided by a common background generated by MC simulations, accordingly with the proposal explained in the text.

In experiment 1 (Figs. 4, 5, 6), the inclusion was a highly absorbing black painted wooden stick $({\varphi }_{\mathrm{inc}}=0.7\mathrm{cm})$ placed at different depths $z$ . In all cases, part (a) refers to the normalized transmittance profiles with inclusion obtained by both experiment and MC simulation. Part (b) shows the results after applying the analysis of Sec. 2.2.2. As stated, profiles in (b) are obtained after division by a background generated by MC, since in general, the experimental background is not known.

The important features that simulations are expected to reproduce are both full width at half maximum (FWHM) and intensity modulation of the intensity profiles. That means that those profiles obtained from MC must match the corresponding experimental ones. When this condition is reached, the best set of optical parameters of the inclusion has been found. These retrieved values for the experiments presented next are summarized in Table 2 . Note that intensity modulation as well as FWHM of the transmittance profile are both expected to change for different depths of the same inclusion. This is because the most probable paths for the photons (the “banana”) have no constant section inside the tank, and thus the number of photons intersected by the inclusion changes for different depths of it.

Table 2

Retrieved optical properties of the inclusions corresponding to the different experiments according to the proposal of this work. The given uncertainties are not of statistical nature; they arise because for each experiment, three positions of the inclusion must be considered simultaneously.

In Fig. 4, the inclusion was placed inside the tank next to the face where light enters the slab. Both FWHM and intensity modulation retrieved from MC agree with the experimental data, as can be directly seen from Fig. 4. Notice that this figure is ambiguous about the optical properties of the inclusion with respect to the bulk. However, after processing [Fig. 4], it becomes evident that an inclusion, which is more absorbing than the bulk, is present.

Figures 5 and 6 are analogous to Fig. 4 but with the highly absorbing inclusion placed in the middle of the tank $(z=S∕2)$ and close to the wall facing the camera, respectively.

Please note the ambiguity of Figs. 5 and 6, which at first sight could lead to the conclusion that two inclusions that are less absorbing than the bulk are present inside the tank. This ambiguity is solved in both cases after normalization to the background, as shown in Figs. 5 and 6. Note also that, as expected, Figs. 4, 5, 6 show that the more distant the inclusion from the camera, the smaller the intensity modulation and the wider the profile.

As another test much closer to an actual situation, we present experiment 2, where the inclusion is a cylinder made of agarose mixed with a milky solution similar to that of the tank and india ink to achieve the desired absorption. The optical properties of the inclusion could not be measured directly from the inclusion itself because of its relatively small dimensions and its shape. Instead, we used the same mixture of agarose and milky solution to fabricate, simultaneously with then inclusion, a $3\text{-}\mathrm{cm}$ -thick slab from which we measured the optical properties by fitting the DTOF of photons, as described before. Since the process of solidification of agarose inside the small plastic cylindrical form used to fabricate the inclusion may differ from the corresponding one when generating the slab (much greater volume), slight departures of the actual optical properties of inclusion from those of the slab could be expected. Results for this case are presented in Figs. 7, 8, 9. For this series of experiments, we used the same tank filled with the same milky solution as experiment 1. The inclusion, a $25\text{-}\mathrm{cm}$ -long cylinder of diameter ${\varphi }_{\mathrm{inc}}=1.4\mathrm{cm}$ had an absorption coefficient four times higher than that of the solution filling the tank. Refractive index of both bulk and inclusion were assumed equal.

Again, the comparisons between experiment and simulations for transmittance data and processed profiles show very good agreement. Sensitivity with depth is also demonstrated from the series of Figs. 7, 8, 9, which show changes in the slenderness of the profiles.

As a final example, we present in experiment 3 (Fig. 10) the case of a slightly perturbing less absorbing inclusion, consisting of an agarose cylinder of ${\varphi }_{\mathrm{inc}}=0.5\mathrm{cm}$ immersed in the cuvette filled with the same milky solution used for the preceding experiments. It was placed next to the light exit face of the slab. Notice that the processed profile [Fig. 10] produces a peak that is higher than the background, clearly indicating the presence of a less absorbing inclusion.

Note that in Figs. 4, 5, 6, 7, 8, 9, 10, all transverse positions are measured with respect to the optical axis defined by the line joining the laser and the camera (Fig. 1).