# Optical Characterization of Homogeneous and Heterogeneous Intralipid-Based Samples
## 3 Materials and Methods
### Experimental Techniques and Data Analysis Procedures
#### 3.2.2 Time-Resolved Transmittance Experiments  
An experimental apparatus used for time-resolved measurements of Intralipid suspensions and inclusions is generally composed by an ultrashort laser source as a Ti: Sa laser that is able to give pulses with a time duration of hundreds of femtoseconds or pulsed diode laser with a pulse duration of tens of picosecond. In both cases, near-infrared radiation is used, i.e., with wavelengths around 800 nm. The detectors are streak cameras or time-correlated single-photon counting (TSPC) systems. Streak cameras offer a time resolution of a few picoseconds, while TSPC systems usually ensure a time resolution of tens of picoseconds. Detailed descriptions of similar experimental set-ups are reported in [19; 20; 43; 44]. For measurements on liquid suspensions, cuvettes of various thicknesses were used to satisfy all the assumptions of the diffusion approximation even for Intralipid low concentrations. For all the examined samples, the cuvette size prevented boundary-reflection problems. By means of the TRT technique, the optical properties (namely, the absorption coefficient, \(\mu_{a}\), and the reduced scattering coefficient, \(\mu_{s}\)') of the homogeneous Intralipid suspensions were obtained,by using the analytical expression for the transmitted light reported in Equation (14). The expression was rewritten to introduce a set of three fitting parameters: RA0, an amplitude factor; \(\mu_{a}\); and \(\mu_{s}\)', forming the parameter vector \(\alpha\) = (RA0, \(\mu_{a}\), \(\mu_{s}\)) [19; 44]. After subtraction of the background noise, the experimental data were fitted to the theoretical prediction by minimization of the \(\chi^{2}\) parameter. The minimization procedure was carried out by the use of a home-made code employing Migrad, and Nelder and Mead minimization procedures. If these methods failed to find a real minimum, a Monte Carlo minimization procedure was also employed. It is worth to note that TRT experiments allow the determination of \(\mu_{s}\)' and \(\mu_{a}\) that can be related to the photon diffusion constant D. It is interesting to note that the photon diffusion coefficient D that is present in Equation (14) that describes the photon propagation inside turbid media can be related to another interesting scattering property named the transport mean free path, \(\ell^{\star}\). The mean free path represents the distance travelled by photons between two subsequent scattering events [45]. In particular, the following relationship can be considered: D = (I/3) \(\
u_{\mathrm{e}}\ell^{\star}\), where \(\
u_{\mathrm{e}}\) is the transport velocity. For measurements on absorptive cylindrical inclusions placed inside the host medium, a rectangular cell with an internal thickness of 40 mm, a height of 145 mm, and a width of 130 mm was adopted. The cell was fabricated using black polyvinyl chloride with two small transparent apertures to reduce inner boundary reflections. Each cylindrical inclusion was placed in the center of the cell by means of a nylon wire. The photon source, the cylinder axis and the detector were kept aligned within 0.5 mm. The time-resolved transmittance data were accumulated at intervals of 5 ps. At the beginning of data processing, the center of each reference pulse was determined [19], and the temporal axis of each profile was arranged so that the time at which the beam hit the phantom surface was assumed equal to zero. The errors related to the identification of pulse centers can cause noise in the final data. Point measurements were carried on by translating the inclusion inside the host medium along the x-direction from x = \(-\)40 mm to x = 40 mm with a 1 mm step, transversally to the source-detector axis. The obtained profiles described the temporal distributions of the transmitted light through the phantom and were employed to determine the absorption coefficient of the inclusions. For each inclusion, the Gaussian perturbation model was used for performing the fit with the experimental profiles for each scanning position.