A simple and fast time-domain method for localizing inclusions, fluorescent optical probes or absorbers, is presented. The method offers new possibilities for situations where complete tomographic measurements are not permitted by the examined object, for example in endoscopic examination of the human prostate or the oesophagus. Feasibility has been envisioned with a phantom study conducted on a point-like fluorochrome embedded in a diffusing medium mimicking the optical properties of biological tissues.
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Over the last decade, interest in optical techniques for biological tissue screening has developed in that these techniques represent a non invasive and non ionizing method for diagnosis or imaging . Diffuse Optical Tomography (DOT) systems fall into three categories, according to the illumination light: i) continuous wave; ii) intensity modulated; iii) pulsed illumination. Time-resolved techniques are those conveying most information on the optical properties of the crossed tissue, and seem to be more suitable for deep tissue screening (several centimeters for brain or breast examination) [2,3]. However, they also involve the most sophisticated reconstruction techniques and may prove intractable for real time resolution of the inverse problem. Classical photon migration measurements consist of illuminating a diffusing medium (biological tissues) with a point-like source and collecting the reemitted optical signal at a given distance from the injection point. Distribution of the diffused photons depends on the optical properties (absorption and diffusion) of the screened medium. At each point, measurement is a time-dependent function known as the Temporal Point Spread Function (TPSF). The problem consists of extracting simple signatures from the TPSF, for which the theoretical expression is known (thanks to a forward model connecting the unknown parameters and these temporal signatures). Resolution of the inverse problem then leads to determination of the unknown parameters, which can be three-dimensional (3D) distribution of the optical parameters (absorption and diffusion coefficients) in DOT or/and the properties of an optical probe (concentration, lifetime), such as in Fluorescence DOT (FDOT). Nowadays, the trend is to increase the number of measurements with large numbers of source and detector positions, or multiple wavelength measurements, for better resolution [4–9]. However this results in very complex systems, with long acquisition times, and in memory consuming inverse problems, leading to computation times ranging from several minutes to hours, that are not compatible with real time information retrieval [6,8]. That could however be often convenient, for instance during surgery.
The problem we address in this paper is precisely the real time retrieval of information for assistance to diagnosis during surgery. By real time, we intend fast data acquisition, which supposes a limited number of source-detector pairs, together with a quasi-simultaneous result for the inverse problem. The addressed problem is also that, in some cases, the object or body to be examined quite simply prohibits the use of many sources and detectors, for example in endoscopy for the examination of the prostate or the oesophagus [4–9]. In these cases the space available for a measurement is reduced to few centimeters, and the acquisition can only be performed in reflectance mode, sources and detectors belonging to the same half-space.
Other groups have already proposed original simple approaches allowing to collect depth information of fluorescent optical probes , together with optical probes concentration or lifetime [11,12]. Hall et al.  proposed a method based on a single measurement of the maximum of the TPSF tmax and demonstrated experimentally, on calibrated measurements, knowing the optical properties of the host medium, that this temporal signature tmax i) does not depend on concentration of optical probes and ii) has a linear behavior as a function of the depth of the position a fluorescent point-like inclusion. For in vivo measurements, they suggest to fit the measured TPSF for a determination of the depth. The exploitation of the integral of the TPSF allows then the determination of the concentration. In this approach, the lateral relative positions of the source, detector and inclusion must be known, leaving only the depth as the unknown in the problem. Laidevant et al.  provided an analytical solution of the depth of a fluorescent inclusion as a function of the mean time-of-flight <t>, with a linear behavior if the pathlength source-fluorophore-detector is approximately twice the depth of the inclusion (that is distance source-detector small). Once again, the lateral relative positions of source, detector and inclusion have to be known, otherwise, as suggested by the authors, multiple measurements are required by scanning for example the source-detector pair according to a squared grid over a defined region of interest.
In the present work, we propose to generalize the approach described in  to the global determination of the three-dimensional position of an inclusion, with a limited and controlled number of measurements (three or four measurements are sufficient). The principle is inspired by the method of localization and imaging of cracks in concrete structures by ultrasounds .
The paper is organized as follows: the second section presents the derivation of the proposed method; follows then, in section three, a description of the experimental implementation; section four presents the results commented and these results are then discussed in the fifth section. And finally we conclude.
2. Derivation of the method
The method described here is rewritten within the framework of FDOT, for a 3D localization of a fluorescent zone embedded in a diffusive volume, but can be transcribed for localizing an absorbing inclusion after a linearization step of the problem, using a perturbation approach. The equations for the fluorescence process are usually derived by considering a two step approach according to the Jablonski diagram: i) a photon is absorbed at the excitation wavelength ; ii) the photon is reemitted, after a delay known as the fluorescence lifetime τ, at a higher wavelength . With the same phenomenological description, absorption can be seen as a simplified fluorescence process: once absorbed, a photon is immediately reemitted (), at the same wavelength .Eqs. (1) are subject to Robin boundary conditions, to be satisfied at the system boundaries. Under these conditions, the solution to the set of Eqs. (1) can be written as a product of convolution of three functions:
V is the volume of the probed medium, stands for the temporal convolution product, andare the solutions to the time-domain diffusion equation:
In the remainder of this paper, equations will be written considering an infinite homogeneous diffusing medium. Under this assumption, is written as:16]. We consider the normalized time moments, , of the TPSF , defined as :
(respectively ) represents the mean time-of-flight of the photons traveling at the excitation (resp. emission) wavelength from the source point located at to the fluorescence inclusion at r (resp. from the fluorescence inclusion at r to the detector at ). thus represents the apparent speed of light within the diffusing medium at the excitation/emission wavelength.
In an unusual case, considering the same optical properties at both the excitation and emission wavelengths, which is a possible assumption when the excitation and emission wavelengths are close (few tens of nm), the following is obtained :
The above equation describes the 3D surface of an ellipsoid with focal points located at and . Its solution is the set of points r such that source-fluorochrome distance plus fluorochrome-detector distance is equal to a constant, noted in the present case d. This constant corresponds to the apparent speed of light in the diffusing medium ν multiplied by the mean time-of-flight corrected by the delay introduced by the lifetime τ. In other words, the location of a fluorochrome providing the measurement value belongs to this 3D surface. By increasing the number of measurements (at least three in 3D) and searching for the intersection between the set of the resulting 3D surfaces, the set of the possible positions of the fluorochrome is obtained.
Higher order moments can be considered. For instance, with the second order moment, it is possible to build the time variance expressed as , as well as observe that the resulting equation is also the 3D surface of an ellipsoid. Utilization of higher order moments should provide additional information  and in some way improve the method. Note that if the problem were reduced to location of an absorbing object, the 3D surface defining the set of the solutions would be obtained by setting in Eqs. (6) or (7).
In the general case, the optical properties at the excitation and emission wavelengths are different. Equation (6) still defines a 3D surface, and the idea is still to increase the number of measurements to find the intersection between the different 3D surfaces generated by each measurement. Note that location of a fluorochrome depends on lifetime, which is a parameter known to be sensitive to the biological environment. To get round this problem, differential measurements as detailed in Ref . can be considered: a reference measurement is subtracted from the others, discarding the dependency on the variable τ in Eq. (6). In this case, at least four measurements have to be considered, one of which will be dedicated to the reference measurement.
3. Experimental implementation
The experimental setup used (Fig. 1 ) is described in detail elsewhere . A picosecond laser diode operating at wavelength 635 nm with a repetition rate of 50 MHz (BHL-600, Becker&Hickl, Germany) is used for the study. The laser beam is injected in an excitation optical fiber to illuminate the sample; mean power is approximately 100 μW. The scattered reflected optical signal from the sample is collected by the detection optical fiber coupled to a fast photomultiplier tube (R3809U-50, Hamamatsu Photonics, Japan). A Time-Correlated Single Photon Counting (TCSPC) system (SPC-630, Becker&Hickl, Germany) is used to obtain time-dispersion curves. The TCSPC computer card is synchronized by a reference signal generated inside the laser diode box. The sample consists of a plastic cylindrical tank filled with a water solution of intralipid (Fresenius, France) as the scattering medium. The optical parameters are determined according to the method extensively described in , by using the time-dependent diffusion equation for an infinite homogeneous medium (proportional to Eq. (3)). The optical properties of the sample turn out to be = 8.7 cm−1 (D = 3.83x10−2 cm) and µa = 0.01 cm−1 at the excitation wavelength. These values fall within the wide range of optical parameters measurements for human prostate at approximately 635 nm [20,21], and are assumed to be the same at the emission wavelength. The fluorescent inclusion consists of two microliters of Cy5 fluorochrome (concentration 10 µM, Molecular Probes, Invitrogen, USA) placed at the tip of a thin glass capillary tube (3 cm long, 1 mm thick). Fluorescence lifetime is τ = 1.1 ns. The tube was inserted into the tank through a hole drilled in the bottom (see inset Fig. 1).
Experimental data were recorded at different positions between the fibers and the fluorescent inclusion in space (XYZ) (see Fig. 2 ). For each case, the fibers were embedded 3.5 cm into the medium. This distance was found to be sufficient to ensure the infinite medium geometry required . The distance between the fiber plane and the fluorescent inclusion is set at 0.5 cm. The light is driven from a laser by an optical fiber, the beam is infinitely narrow. It is commonly admitted that the system is equivalent to an isotropic source placed at a distance of one transport mean free path from the actual source, that is, in the present case 0.11 cm from the tip of the fiber. Five different configurations of source (S) and detector (D), reported on Fig. 2a, have been considered for this experiment.
No particular arrangement for source and detector positions was chosen, except that i) of course, the fluorescent signal should be measured; and ii) at least one SD pair should be placed on a configuration perpendicular to the two others, allowing optimum intersection in 3D of the three ellipsoids required. Indeed, the intersection between two ellipsoids corresponding to two different positions of the optodes in 3D is a curve. A wrong positioning of the third optode would be to chose its position collinear to the two others, because under these conditions, the intersection between the three ellipsoids would still be a curve, not a single point. However, a systematic study should highlight optimum configurations.
The corresponding measured TPSFs are presented on Fig. 2b, on the right. Note that the measured curves have been preprocessed as recommended in Ref . The Instrument Response Function (IRF) was measured by aligning the two fibers separated by a known distance in air, which allowed us to compensate for the time lag. The IRF had a temporal width of ~80 ps (FWHM). The theoretical model corresponds then to the convolution of the IRF and the expression calculated with Eq. (2). A first fluorescence measurement was taken without the fluorescence target in order to measure the autofluorescence of the host medium. This measurement is then subtracted to the raw fluorescence data.
The measured TPSFs (Fig. 2b) correspond to the following calculated values of the distance d = (2.27; 2.01; 2.73; 2.39; 2.53) cm in Eq. (7). To solve the inverse problem, the medium is meshed into voxels. In the present case, the mesh chosen is defined as follows: X direction, 50 nodes from −0.5 to 2 cm; Y direction, 30 nodes from −0.5 to 1 cm; Z direction, 50 nodes from −6 to −3.5 cm (corresponding to cubic voxels of size 0.05 cm). Figure 3 presents an example of a 3D surface defined by the set of points solution of Eq. (3) for the measurement S1D1. In the present case, as the medium is assumed to be infinite and the optical properties are the same at the excitation and emission wavelengths, the 3D surface describes an ellipsoid.
We first considered three of the available measurements, and found the intersection of the sets of solutions of the three corresponding equations, i. e. the set of points r belonging to the three ellipsoids and satisfying Eq. (7), for each above-mentioned measured distance d. Figure 4 illustrates the three corresponding ellipsoids, and their intersections corresponding to the experimentally determined possible positions of the point-like fluorochrome. Computation time for the chosen mesh is less than 3 s (computation of distance d for the 3 measurements 2.6 s, computation of the three ellipsoids 0.12 s and determination of the intersection 0.04 s, with Matlab® software, and processor Intel Core2, 2.13GHz, 1Gb RAM).
The fluorochrome was found to be located at (, , ) cm while the actual position is (0,0,-4) cm. We then considered the whole set of five measurements. The results are reported on Fig. 5 . The fluorochrome was found to be located at (, , ) cm which is a slightly more refined solution, with 2.25% error in z position, 6% in x and 12% in y. All in all, we estimated the global accuracy in the 3D localization to be higher than 2 mm.
A good accuracy in the depth recovery of a fluorescent inclusion has been demonstrated with the proposed method: a submillimeter error in z has been found, and the accuracy can be improved by studying more systematically the relative positioning of the fibers and the inclusion. Compared to results obtained by other groups [11,12], with comparable localization methods, with however smaller interfiber distances (few millimetres) and larger fluorescent objects, the obtained results are satisfying. In the present study, a point-like inclusion (maximum diameter 1mm) has been preferred for the demonstration because for larger objects of finite size, as pointed out in Ref  and also experienced with the present method,the depth determined by the point model is close to the top surface of the actual inclusion.
The method is very versatile and shows that it is possible to adopt multiresolution schemes, according to the expected resolution. Note that if source and detector are very close, in an infinite medium, according to Eq. (3), the 3D surface described becomes a sphere. This configuration has been recently cited as a very advantageous situation for photon migration measurements , when deep tissue screening is envisioned, with better spatial resolution, higher signal intensity, all this by preserving the same contrast as if the interfibers distance was larger (2cm). The reason invoked is the fact that the density distribution of photons detected at null interfibers distance is more spatially confined . This is possible provided a time gating is applied on the TPSF measurement to reject early-arriving photons that do not meet the requirements for using the diffusion approximation.
Most of the computation time (few seconds) is spent in the resolution of the forward model and depends on the mesh chosen, corresponding in our case to 75000 nodes, and in the number of measurements considered, at least three here. The inversion in itself is achieved in only 0.04 s which is, to the best of our knowledge, much faster than reconstruction results obtained with conventional tomographic methods.
We have described a fast and accurate method for real time localization of compact absorbing or fluorescing zones embedded within turbid media. It is fast in the sense that only three TPSFs measurements are needed, and computation time is less than 3s for localization with accuracy > 2mm in the described experiment (submillimeter accuracy in depth), which, although not accurate as a complete tomographic process, falls within acceptable limits to be considered a tool to help improving diagnostics. Accuracy should nevertheless be increased by studying the limits of the method (optimum configuration of sources and detectors, optimum space discretization, etc). This method is otherwise very easy to introduce as an initialization step in a more sophisticated reconstruction algorithm. By using the proposed method, the surgeon can at least diagnose the presence of a tumor and have a relatively precise idea of its location, all this in less than one minute counting the measurement and calculation time. Having made these considerations, one can indeed qualified the method as a real time diagnostic method.
For simplicity, the demonstration, theoretical and experimental, has been conducted assuming a homogeneous diffusing and absorbing infinite medium as the host medium, with a point-like fluorescent inclusion inside. For a semi-infinite medium, which is a more practical situation for in vivo experiments, an analytical expression can still be easily derived for the mean time-of-flight as a function of the distances and . Nevertheless, the whole derivation relies on knowledge of the expression of the Fourier transform in expression (1), which can be computed numerically for a non homogeneous medium, with arbitrary shape, allowing a generalization of the method. The resulting 3D surfaces would be more complicated than simple ellipsoids.
Overall, our findings demonstrate the experimental feasibility of the approach for diagnosis, opening up new investigation directions in the field of photon migration based approaches of diagnostics and imaging. For instance it can be easily coupled with an ultrasound transrectal probe and used for guiding biopsies in prostate cancer detection or therapy  and could be an add-on to other other conventional endoscopic tomographic systems [4–9].
The method is, however, limited to the localization of a single object as the theoretical model is strictly only valid for a point inclusion at a given depth. Although this hypothesis might be considered as oversimple for complex biodistributions, it is still useful for many situations such as in vivo early tumor detection. Moreover, it does not provide, in itself, quantitative measurements. However, as proposed by other authors [11,12], once localized, the concentration and/or lifetime of biomarkers can be determined afterwards. The theoretical model is strictly only valid for a point inclusion at a given depth. Its robustness has to be studied, especially when the contrast between the background and the target is weak.
This work was supported by the French National Research Agency (ANR) through Carnot funding.
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