## Abstract

We present a novel procedure for localizing fluorescing-tagged objects embedded in turbid slab media from fluorescent intensity profiles acquired along a surface of interest. Using a numerical model based on a finite element code, we firstly develop a method devoted to lateral detection by varying the laser source position along one face of the tissue slab. Next, we mainly demonstrate the possibility to accurately assess the depth location by alternately changing the position of the source and the detector at the both sides of the slab. The dimensionless depth indicator derived from this procedure remains independent, over a wide range, on both the optical properties of the host tissue and the probe concentration. The overall findings validate the method in situations involving moderate size object-like tumors tagged with a new smart contrast agent (Cy 5.5) that offers high tumor-to-background contrast and great interest in early cancer diagnostic.

© 2006 Optical Society of America

## 1. Introduction

The use of optical imaging has attracted growing interest over the past few years. The major goal of the optical diagnostic strategies is to detect abnormalities such as cancerous and precancerous cells in biological tissue, while allowing to replace destructive biopsies by means of the development of non-invasive and faster optoelectronic devices. However, techniques that depend on only the intrinsic optical contrast between normal and diseased tissue are not sensitive enough to resolve an image deeper lesions having low volumes [1]. In the cases of cancer, when early detection is required to reduce morbidity, a novel element that can enhance the potential applications of optical imaging is the use of exogenous contrast agents, specifically in the near-infrared range where tissue exhibits low absorption, allowing deep probing of light into the tissue [2–4]. A number of optical imaging approaches have recently been described, some of which rely on fluorescence as a source of contrast (molecular beacon), while the volumetric images are reconstructed from comparison between forward diffuse optical data and measurements on the boundary [5]. However, except for studies that involve simplified diffusion model or refer to other modalities [6], diffuse optical imaging may require large computing time to reconstruct a map of optical properties through the investigated tissue. By another way, the inverse problem is always ill-posed [5, 7] and may lead to divergent solutions. These various difficulties have encouraged few research groups to study and report different methods devoted to the localization of fluorescent inclusions embedded in turbid media [8–22]. Kang et al [8] mainly observed the change in parameters (magnitude ratio, phase shift, and time constant) when a system has a localized spherical fluorescent absorber in a scattering tissue model considered as an infinite medium. Their data were compared to those acquired using a regular absorber, showing the interest to use fluorescent agent to enhance the capability of optical apparatus in detecting biological heterogeneities. Hull et al [9] demonstrated that the depth of a small fluorescent sphere embedded within a thick turbid medium can be accurately determined from the fitting of measurements on the surface by a theoretical model originally developed for steady-state diffuse reflectance spectroscopy [10]. Intes et al [11] performed investigations based on the use of two-interfering sources in order to enhance the lateral detection of an absorbing-fluorescent object centred inside a turbid slab medium. Foster et al [12] explored the concept of forming a ratio of acquired data from two different excitation beam diameters to assess the depth location of a fluorescent object.

In these first approaches, the authors only considered the fluorescent object to be localized as a enhanced absorber, thus ignoring fluorophore quantum yield, fluorescence lifetime, and optical properties at both excitation and emission wavelengths. Further, more complex investigations have been published in a variety of studies implying real animal tissue, tissue-simulating phantoms, or computational simulations. Wu et al [13] developed a time-resolved set-up for assessing the position of a fluorescent target in turbid media by evaluating early arriving photons. While this concept has the advantage to render the measurement practically independent on the fluorescence lifetime, the depth localization remains strongly dependent on the background optical properties. Gannot et al [14] described and experienced an analytical random walk model available for a semi-infinite medium, in a continuous wave mode. Next, by performing a fit between measured data and theory, they showed [15] that their approach was successful for the 3-D subsurface depth localization of a targeted fluorophore in the tongue of live mouse, but the semi-infinite geometry limits the extension of the analytical results to tissue slab having a finite thickness. Pfister and Scholz [16] used a multiple-signal classification algorithm called as MUSIC, to localize one or two fluorescent spots under tissue like scatter in a computational study. Milstein et al [17] presented a statistical approach for detecting and localizing a fluorescent tumor obscured underneath several millimetres of a turbid medium, from fluorescence measurements acquired above the surface. They validated the procedure in an experimental study involving an excised mouse tumor tagged with a new folate-indocyanine dye embedded in a tissue-simulating lipid suspension. D’Andrea et al [18] developed a set-up based on a CCD camera to localize fluorescent inclusions in diffusing media, in order to acquire a huge dataset along to directions. They also implemented a reconstruction algorithm to recover the position of one or two point-like fluorescent inclusions and to estimate their relative concentrations. The investigations of the usefulness of a depth-resolving technique based on spectral information have been the subject of two papers as reported by Swartling et al [19] and Svensson et al [20], respectively. These results suggest, however, that it is necessary to know the optical properties of the host tissue to assess the depth. More recently, Yuan and Zhu [21] showed the feasibility to reconstruct into two steps the structural and the functional information of a spherical fluorescing target embedded in a semi-infinite medium. Although the method provides useful quantification, the extracted data still seem required the knowledge of the optical properties of the tissues under interrogation.

In this paper, we propose a novel procedure that uses a continuous wave laser excitation to improve the localization of a fluorescent object hidden in different scattering slab media. The demonstration refers to numerical simulations based on a finite element code previously described [22] and adapted here for the steady-state conditions.

The outline of the work is as follows. In Section 2, we develop the theoretical framework to account for a diffusive slab of finite thickness that contains a depth varying fluorescent cylindrical-shaped inclusion. The model based on the set of two steady-state coupled diffusion equations was derived for a refractive index mismatch between the slab and the surrounding medium by using extrapolated boundary conditions. A finite element approach allows us to compute the scanning change in fluorescence intensity resulting from the presence of the object, in case of continuous wave laser excitation. In Section 3, we report and discuss the computational results that show the possibility to infer the lateral position of the object and underline the interest of using the transmittance fluorescence mode. Ultimately, we plan to explore the accuracy and the application range of a dimensionless depth indicator based on a two-way transmittance determination from finite element calculations. Finally, a summary is provided in Section 4.

## 2. Model

#### 2.1 Light diffusion under fluorescence conditions

Consider a fluorescence measurement arrangement depicted Fig. 1, where a single continuous wave laser source acts at different positions y0 along the upper side (1) of a tissue slab to probe for a fluorescing embedded object at the opposite side (1’). The fluorophores, dispersed throughout the object to be localized, are excited with light at wavelength *λ*_{x}
and reemit light at a longer wavelength *λ*_{m}
. The problem linked to the fluorescence measurement can be well described by a set of two steady-state coupled diffusion equations [21, 22, 29] referring to a *y*- *z* coordinate system:

where Φ
_{x}
and Φ
_{m}
are the D.C. components of the diffuse photon density for excitation (subscript *x*) and emission (subscript *m*) light, *q*_{x}
and *q*_{m}
the excitation and emission source terms, *D*_{x}
and *D*_{m}
the optical diffusion coefficients for the excited and emitted lights, and *k*_{x}
and *k*_{m}
two coefficients that account for the total absorption in the medium at the wavelengths of the excited and emitted lights, such that *D*_{x,m}
=1/3(*µ*_{ax,m}
+*µ′*_{sx,m}
) and *k*_{x,m}
=(*µ*_{ax,m}
+*µ*_{fx,m}
).

Here, *µ*_{ax}
and *µ*_{am}
are the absorption coefficients of light in the medium at the excitation and emission wavelengths, while *µ′*_{sx}
and *µ′*_{sm}
are the reduced scattering coefficients of the medium at *λ*
_{x} and *λ*
_{m}. The fluorophores were also characterized by the values of the absorption coefficient at the excitation, *µ*_{fx}
, and emission, *µ*_{fm}
, wavelengths. Equations (1) and (2) are coupled through the fluorescent source term

where *ϕ* is the quantum efficiency for emission at *λ*_{m}
. The continuous wave laser source is modelled as a planar source [22, 23]

where *µ*_{tx}
=*µ*_{ax}
+*µ*_{sx}
, *µ*_{trx}
=*µ*_{ax}
+*µ′*_{sx}
are respectively the total attenuation and the transport coefficients at excitation wavelength and *g* the anisotropy coefficient. The best description of the air-tissue boundary related to the fluorescent signals in Fig. 1, is derived with an index-mismatched type III condition, in which the fluence rate at the edge of the tissue exits and does not return [24]

where *ξ* is a point along the segments 1,1′, 2 and 2′, and *A* a parameter whose quantity predicts the amount of light reflected or transmitted and the degree of anisotropy at the considered boundary

In the Eq. (6), the effective reflection coefficient *R*_{eff}
can exactly be calculated by numerical integration of functions involving the Fresnel reflection coefficient [24, 25].

For the diffuse light, we adopt the same boundary conditions as given by the Eq. (5) (Φ_{x} is now substituted to Φ_{m}), except for the impact laser source area, around Y_{0}, where the following equation holds [23]

The output fluorescence photon flux *J*_{m}
(*y,z*), reflected
^{(R)}
or transmitted
^{(T)}
, is then computed from the gradient of the fluence, ∇→Φ
_{m}
(*y, z*), at either the surface of interest of the slab (z=0) or (z=d). This yields, from the Fick’s law

We note that, by combining Eqs. (8) and (9) with Eq. (5), the computed flux is simply proportional to the fluence at the surface.

#### 2.2 Finite element implementation

A finite element code was used to compute solution of the governing Eq. (1) and Eq. (2) together with the boundary conditions (5) and (7). For this, the rectangular bounded domain Ω was divided into non overlapping elements of simple shape, such as triangles joined at N nodes.

We note that the solution to Eq. (2) priorly depends on the solution to Eq. (1), through the coupling term expressed by the Eq. (3). Consequently, to predict fluorescence emission fluence Φ_{m} (or *J*
_{m}) at both sides of interest, one first solves the excitation Eq. (1) with the boundary conditions Eq. (5) and Eq. (7), to compute the excitation fluence Φ_{x} at all the nodes of the meshed domain, in presence to the planar exciting source [Eq. (4)]. The predicted excitation Φ_{x} is subsequently used in the fluorescence source term [Eq. (3)] for solving the Eq. (2), subject to boundary conditions Eq. (5), for the fluorescence emission Φ_{m}.

A solution of this procedure was obtained by using the so-called Galerkin approach, which yields the weak formulation of the problem (see Refs 26, 27, and 23). Both functions Φ_{x} and Φ_{m} are then approximated by a linear combination of known functions and the solution consists of determining the parameters of these combinations.

Setting Φ
_{x}
≈ ${\mathrm{\Sigma}}_{j=1}^{N}$
*α*_{j}*φ*_{xj}
, *α*_{j}
∊ ℜ and Φ
_{m}
≈ ${\mathrm{\Sigma}}_{j=1}^{N}$
*β*_{j}*φ*_{mj}
, *β*_{j}
∊ ℜ, the left hand side of the set (1) and (2) turns into matrix-vector equations where the vector contains the unknown coefficients *α*_{j}
and *β*_{j}
, while the matrix contains the scalar products only depending on the test functions. Further, by an appropriate choice of these functions, the matrix can be inverted quickly, resulting in a solution for *α*_{j}
and *β*_{j}
. The system was implemented on the FLEX PDE software package [28].

## 3. Results and discussion

#### 3.1 Basic requirements

The numerical results reported in this section refer to a scattering slab of length 100mm, with a thickness fixed at 40mm and 60mm, respectively. Whereas the considered geometry and the optical properties of the domain are of practical interest for slightly breast tissues investigated in the near-infrared range, the absorption and the reduced scattering coefficients, at excitation and emission wavelengths, were altered to evaluate the sensitivity of our method to such variations. In that way, the set of optical properties which serves as input of the different simulations reported below will be noticed in the text. However, under the assumption that the Stokes shift is small, the equality of the coefficients for both wavelengths can be accepted, i.e. *Dx*=*D*_{m}
, *µ*_{ax}
=*µ*_{am}
and *µ′*_{sx}
=*µ′*_{sm}
. The radius of the fluorescent cylindrical-shaped inclusion located midway between the both ends of the slab, but at varying depths, ranges from r_{t}=1mm to r_{t}=10mm. The concentration C (expressed in µM) of the uniformly dispersed fluorophores inside the object, was calculated according to the data given by Sadoqi et al [29], that is *µ*_{fx}
=2.3C*ε*_{fx}
and *µ*_{fm}
=2.3C*ε*_{fm}
, where *ε*_{fx}
(M^{-1} mm^{-1}) and *ε*_{fm}
(M^{-1} mm^{-1}) are respectively the molar extinction coefficients of the fluorochrome at excitation and emission wavelengths. Consideration based on irregular shape of the target that contains an inhomogeneous fluorophore concentration is outside the scope of this paper. However, we note that the information carried by a cylindrical size with a variable radius remain statistically acceptable.

Due to the overlap of absorption and emission spectra of the considered markers (especially for the ICG and the Cy5.5), a notable fraction of fluorescence photons is reabsorbed by the dye itself, therefore, *µ*_{fm}
=*µ*_{fx}
/2. The anisotropy factor g was fixed at 0.8, whereas the mismatch in the refractive index has been set to 1.4. This last value corresponds to the mean of typical measured values (1.37–1.45) for various biological tissues [30], and agrees well with recent data reported for diseased breast tissues [31]. Haskell et al [24] have calculated the effective Fresnel coefficient *R*_{eff}
to be 0.4935 when the air-tissue interfaces n=1.4. This leads finally *A*=2.948.

The optimization of the grid size used for the finite element code received a full attention. The accuracy of the numerical data were validated by ensuring that the results are independent of the mesh size. A final arrangement based on 4611 nodes with 2224 cells was selected for all reported computations.

#### 3.2 Lateral localization of the object

Figures 2, 3(a), and 3(b) show computational results relating to fluorophore dispersed inside a cylindrical inclusion (r_{t}=3mm) of varying depth (Z_{t}), for different positions of the source (Y_{0}). The optical properties of the domain (d=40mm) were chosen to model situations often reported by the literature, where the dye is contained in a transparent tube [2, 16, 18]. This has the advantage to cancel the residual fluorescence background effect, but also to calibrate the method while providing the main features. Background optical properties were also modelled as homogeneous, considering the case of fluorescence tomography based firstly on the use of fluorophore Indocyanine Green, with excitation λ_{x}=785nm and emission λ_{m}=830nm. At the basic fluorophore concentration *C*=1µM, the fluorescence optical properties of the object refer to *µ*_{fx}
=0.023mm^{-1} and *µ*_{fm}
=0.0115mm^{-1}, according to *ε*_{fx}
=10^{4} M^{-1} mm^{-1} and *ε*_{fx}
=0.5 10^{4} M^{-1} mm^{-1}, with a quantum efficiency *ϕ*=0.016. The scans depicted Fig. 2 were computed using *µ*_{ax,m}
=0.015mm^{-1}, *µ′*_{sx,m}
=0.8mm^{-1}, and *µ*_{fx}
=*µ*_{fm}
=0 inside the host tissue. It is shown that the transverse location of the object (r_{t}=3mm, C=1µM, Z_{t}=20mm) is easily inferred, independently of the source position, while the higher peak profile value occurs when the source is aligned with the object axis. Previous works using analytical solutions available for a point-like fluorescent inclusion have also shown the possibility to infer the transverse position of embedded targets, in either reflectance [14] or transmittance [18] geometry. The main advantage of the numerical scheme based on the use of the finite element method is to compute the fluorescence intensity profile without restrictions on the radius of the object and to account for perfect and imperfect uptakes (see Section 3/*3.3*).

The maximum fluorescence intensity depends on the medium properties through which the fluorescent light passed. Typically, the magnitude of the signal decreases with the increase of the reduced scattering coefficients of the host tissue. Indeed, the higher the background absorption, the higher will be the attenuation of the laser source and hence the lower the magnitude of the fluorescence signal. Moreover, the whole curve of the fluorescence intensity profile is also strongly influenced by the probe concentration. The effect of the variation of the dye concentration on the reflected and transmitted fluorescent signals is depicted in Figs.3(a) and 3(b), for three different depth positions of the cylindrical object (Z_{t}=5mm, 20mm, and 35mm). The objective of these plots is to show the concentration at which both fluorescence signals reach their maximum values and to highlight the advantage of the transmittance mode over the reflectance mode.

Inspection of the curves referring to Z_{t}=20 and 35mm, in Fig. 3(a), shows that the magnitude of the fluorescent reflected signal increases rapidly as the fluorophore concentration increases until it reaches a maximum at about 3–4µM, and then begins to decrease gradually. However, when the object is embedded near the top surface of the slab (Z_{t}=5mm) the reflected signal increases slowly from 6µM to 10µM and reaches its maximum at about 9.6µM. At fixed concentration, the magnitude of the reflected signal is still greatly influenced by the depth location of the probe. Therefore, a free factor equal respectively to 10^{2} and 5.10^{3} was necessary to scale the data linked to Z_{t}=20mm and 35mm together with the plots corresponding to Z_{t}=5mm

As depicted Fig. 3(b), the evolution of the transmitted fluorescence response on the fluorophore concentration seems different. Although the peak of the signal varies for different depth positions, following a ratio that equals about 1.5, the emission maximum stays relatively close to C=2.5µM. Another interesting point is that a fluorescent object located close to the faces of the slab (Z_{t}=5mm and 35mm) has a much stronger effect on the transmitted signal than does one located in the middle plane. Additionally, there is a slight discrepancy between the curves computed with the object located at Z_{t}=5mm and 35mm. This discrepancy can be perhaps attributed to boundary conditions that are applied along both faces of the slab.

We note from these both graphs that the signal detected along the surfaces of interest of the slab, increases almost linearly with low concentrations and then begins to decrease more or less gradually for large concentrations. This kind of behaviour seems surprising at first glance, but it has been also observed in cases where the fluorophores are somewhat uniformly distributed inside a scattering medium [22, 29, 33, 34]. Few experimental results on fluorescence probe calibration in diffuse transmittance optical tomography are available in the literature for comparison purposes. Nonetheless, our findings are similar to the results of Patwardhan et al [32] that reported a linear measurement from 1nM to 1µM concentrations of ICG contained inside a 3mm diameter tube phantom located at the centre of a plane imaging volume (depth of 7.5mm from the detector window).

For large amount of fluorophores in the target (C >2.5µM), the absorption at the emission wavelength will increase, and the fluorophore will begin to absorb its own fluorescence by self-quenching mechanism. To deepen this question, we computed the reemitted fluorescence light contour plots, for a probe (r_{t}=3mm) located at Z_{t}=20mm, using three selected concentrations C=0.1µM, 1µM, and 10µM. The results are shown in Figs. 4(a), 4(b) and 4(c).

As the concentration of the object becomes larger, most of the laser light is absorbed in a region around the object due to the high density of the absorbing fluorophores. This limits effectively the emission of fluorescence from only a small curved region near the surface of the object facing the laser source [see Fig. 4(c)], and therefore induces the fall in the magnitude of the transmitted fluorescence signal. The light absorption that may occur in case of high fluorophore concentration leads to a well-known phenomenon called as “Inner-cell effect.” The text book of Guilbault [35] provides a description about this effect.

Above all, the calculations reported in this Section clearly show that the reflectance arrangement is more appropriate to perform the lateral detection of a fluorescent object compared with the reflectance geometry if the object is closer to the illuminated slab surface. However, the reflectance sensitivity decreases rapidly as the object reaches the middle plane of the slab. The interest of the transmittance geometry is based on the fact that the opposite plane to the source can be easily scanned without any obstacle, while the acquired fluorescence intensity profile remains practically not affected by the depth location of the object inside the investigated tissue. Next, we tentatively explore the possibility to assess the depth localization of the fluorescent object using the width of the transmitted fluorescence intensity profile.

#### 3.3 Depth localization

Figure 5(a) shows the simulated fluorescent transmission measurements (normalized to their peak intensity) due to an emitting object (*r*_{t}
=1mm) located at four different depths (*Z*_{t}
=2, 20, 30 and 36 mm). We mainly note that the FWHM of the scan profiles decreases as the distance between the centre of the object and the plane 1′ decreases. For small distances, the computed profiles seemingly tend to a narrow profile acquired along the middle plane (*Z*_{t}
=20 mm) and containing the same object. It is also shown in Fig. 5(b) that the scan intensity profile remains practically invariant to the size of the object, suggesting that the fluorescing object acts like a point source, especially for radii extending up to about 5–6mm. As shown Fig. 6(b), for larger sizes, this simple model fails, because the part of the fluorescent photons mainly located at the periphery of the object facing the laser source, contributes to mediate the apparent radius of the embedded inclusion and to shift the centroid of the fluorescent object towards the source. These simulations were performed with the same set of optical parameters as used in Fig. 2. Similar results as those presented in Figs. 5(a)–5(b) have already been reported with either phantom experiments or calculations. D’Andrea et al [18] have shown, by using a CCD technique, that the fluorescence intensity distribution in the output plane strongly depends on the depth of the inclusion, while it is less sensitive to its volume. The same result was reported in reflectance geometry for a spherical fluorescent object by Milstein at al [17]. This behaviour is also in agreement with the results found by Gannot et al [14, 15], who used another method to solve the fluorescence diffusion equation in a semi-infinite medium. In Figs. 7(a) and 7(b) we explore the possibility to determine the depth location of the fluorescent object by forming the ratio

obtained by using alternately the source and the detector positioned at the plane 1 or 1′, but aligned with the object axis. The curve depicted in Fig. 7(a) shows the variation of the dimensionless indicator *F*_{W}*(Z*_{t}*)* with the object depth *Z*_{t}
using different set of optical properties, whereas the fluorophore concentration is kept equal to 1µM. An almost linear behaviour of *F*_{W}*(Z*_{t}*)* can be seen, that implies the feasibility of assessing the depth location with such source-detector combination. In another way, the sign of *F*_{W}*(Z*_{t}*)* gives useful information on the object position. A positive value means that the object is closer to the laser source, a very low value reveals a position around the middle plane of the slab, whereas a negative value indicates that the object is deeply embedded nearness of the detector.

More precisely, inside a domain bounded at least from 15 mm to 25 mm, the slope of the curve *F*_{W}*(Z*_{t}*)*/|*Z*_{t}
-*d*/*2*| is constant and equals 0.031 mm^{-1}. This feature allows to accurately determine the depth location of deeply embedded objects having moderate sizes [Fig. 7(b)]. Outside of the above mentioned domain, the slope of the curve slightly deviates from a linear behaviour, mainly due to the influence of the surface boundaries. Nevertheless, with our data, the relative error does not exceed 15–20% in the case limits. Interestingly, these findings confirm that the depth indicator is independent on the optical properties of the host tissue with quite large ranges of scattering and absorption coefficients including those available for breast tissue. Notice that the slope change shown in Fig. 7(b) for an object of radius 5mm can be explained by the forward-shift of the target centroid location with respect to the object centre. Other markers used in optical imaging than the ICG molecules such as the Cy5.5 dye, the fluorescein dye or the GFP proteins, have fluorescence quantum yields with different efficiencies, depending both on the wavelength and on the conjugated material to the fluorophore (Gannot et al [38], Ntziachristos et al [39]). Moreover according to the improvement of the targeting of tumor cell membrane receptors, the fluorescence from the pathologic tissue can overcome the one that is excited by the background. Several biologic progress were accomplished with the receptor-specific near infrared molecular probes and they allow by different mechanisms to be accumulated inside or around the tumor (Achilefu [40]). Therefore, the influences of both the fluorescence molecule properties and the concentration of these contrast agents inside the cancerous lesions, describing so the statistically average number of the fluorescence molecular probes bound with the tumor cells receptors, must be studied.

In Figs. 8(a) and 8(b) we test the dependence of the ratio *F*_{W}*(β)* with the dimensionless depth *β*=*Z*_{t}
/*d* for two types of markers (ICG and Cy5.5) distributed with various concentrations C=0.1µM, 1µM and 3µM inside a probe of radius r_{t}=3mm. Two sets of optical parameters were selected as representing typical breast tissues at both ranges of wavelengths: 785–830nm (ICG) and 675–695nm (Cy5.5). Consequently, for the case corresponding to the use of the ICG the following parameters were designed for the slab background *µ*_{ax,m}
=0.003mm^{-1}, *µ′*_{sx,m}
=1mm^{-1} in agreement with the mean values as measured by Heusmann et al [37] at around 800nm. For the case related to the use of the Cy5.5, the optical parameters refer to *µ*_{ax,m}
=0.005mm^{-1} and *µ′*_{sx,m}
=1.1mm^{-1} as reported by Spinelli et al [36], with a quantum efficiency *ϕ*=0.23.

The concentrations of both markers were calculated according to the relationship given in the sub-section *3.1*, but the molar extinction coefficient, *ε*_{fx}
, linked to the Cy5.5, was fixed at 2.5 10^{4} M^{-1} mm^{-1} [2] at excitation wavelength of 675nm, with *ε*_{fm}
=*ε*_{fx}
/2 at emission wavelength of 695nm.

One of the most significant features that appears in Fig. 8(a) is the independence of the ratio *F*_{W}*(β)* on the fluorophore concentration of the probe to be localized, indifferently to the nature of the marker. Figure 8(b) shows that a fluorescent probe tagged by a low concentration (0.1µM) of Cy5.5 can easily be localized from the ratio *F*_{W}*(β)* inside a slab medium having two different thicknesses often encountered in optical mammography [37].

According to a recent report that related fluorescent attenuation rates of breast tissue [2], we note that our procedure is feasible owing to the thickness of the slab and the size of the cylindrical objects simulating diseased tissue (tumor).

Practically, the presence of either dispersed or neighboring fluorophores in the surrounding tissues can affect the contrast and requires appropriate subtraction schemes mainly based on the normalized Born approximation [41].

In order to test the potentially of our method, we adopted three typical concentration contrasts 1:0, 1:0.01, and 1:0.005 that were reported in the literature [21, 42, 43]. The Ref [43] treats especially of a detailed review about this subject. The first mentioned ratio 1:0 results of the perfect uptake of fluorophore into the target, while the more imperfect uptake results from a target to background concentration ratio equal to 1:0.01. We performed different simulations based on the same sets of optical properties as those used in Figs. 8(a)–8(b) for the two markers (ICG and Cy5.5). The fluorophore concentrations inside the target of radius 3mm were set respectively as 1µM of ICG and 0.1µM of Cy5.5 [2], whereas the corresponding values for the background were recalculated according to the different mentioned uptakes. The method is based on the use of the raw ratios of the FWHM of the transmitted profiles as calculated from Eq. (10). The results of these investigations are depicted in Fig. 9. It is clearly shown that the procedure requiring the two-way transmittance determination can correct the effects due to the residual fluorescence background for moderate uptakes as is the case for the two different tested fluorochromes ICG (1:0.05) and Cy5.5 (1:0.01). The computed data (red-points and blue-points) are not practically distinct to those (green-diamonds) that results from a perfect uptake of fluorophores into the target.

The effects of the background concentration is more pronounced when the target to background concentration equals 1:0.01 for the ICG (yellow-points). This limitation can however be overcome by subtracting the background signal from the fluorescence intensity profiles before calculating the dimensionless depth indicator *F*_{W}*(β)*. It is worthy of note that the use of new “smart” contrast agents such as the Cy5.5 at uptakes comparable to bio - distribution studies [2], can provide accurate results for fluorescent objects (like tumors) detection in a tissue model.

## 4. Conclusion

In this work, an extensive analysis has been performed to investigate the interaction of the light with heterogeneous tissues with the goal to localize fluorescent tagged objects. A model based on a set of two time-independent photon diffusion equations, the transport of the continuous wave laser source and the transport of the induced fluorescent light excited by the source and originating from the object to be localized, was firstly described. This problem was efficiently solved by using a finite element code based on a two-dimensional meshed domain along which boundary conditions relative to air-tissue interfaces were applied.

In a second step, different computations were performed to explore the potentialities of a novel method that allows to determine both lateral and depth localizations of fluorescent probes embedded in turbid slab media. The procedure refers to the detection of the fluorescence peak intensity value by varying the laser source position along one side of the slab and relies on the dependence of the full-width-at-half-maximum (FWHM) of the transmitted fluorescence intensity profile as a function of the target depth. Next, a dimensionless depth indicator was derived from the data obtained by positioning alternately the source and the detector at both output planes of the slab, but aligned with the object axis. Such ratio help in providing depth information in situations where both the optical properties of the surrounding tissues and the fluorophore concentration inside the object to be localized are unknown. In addition, the effects due to the residual fluorescence background can be corrected for moderate uptakes. These possibilities, combined with new fluorescent contrast agents could improve the performance in optical diagnosis. An important future step would be to apply our analysis to recover the localization of inhomogeneous fluorescent objects with a well defined elongated shape, using a more accurate 3-D finite element scheme. Further studies are also required to experimentally validate this procedure in tissue arrangement where the relevant data are only collected along two plates as is the case in breast tissue investigations.

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