## 2 Physical Concepts  
Photon propagation in a highly diffusive medium (e.g., biological tissue) is primarily determined by two parameters, \(\mu_{s}\)' and \(\mu_{a}\)[49]. The former alters the paths of photons in the medium generating a collection of possible photon trajectories (random walks). The latter reduces progressively the photon survival rate with a typical exponentially decreasing probability given by the expression _exp_(\(-\mu_{a}\)-s) for homogeneous media, where \(s\) is the travelled pathlength (Lambert Beer law) [50; 51]. A typical DO measurement consists of injecting light at a given point on the surface of the diffusive medium (e.g., on the skin) and collecting the re-emitted photons at a different location on the surface. Depending on the measurement geometry, both transmittance (injection and collection on the opposite side of a diffusive slab, as for the compressed breast) or reflectance (injection and collection on the same side) arrangements are feasible [8; 52]. The signal conveyed by detected photons carries information from the regions visited by photons along their random path, and more specifically on the optical properties of the explored tissues (\(\mu_{a}\) and \(\mu_{s}\)'). More specifically, \(\mu_{a}\) is related to the tissue composition and functional status (e.g., water, lipid, collagen and blood content, oxygenation). Conversely, \(\mu_{s}\)' is related to discontinuities of the dielectric constant at the cellular and sub-cellular level, which can then be ascribed to the microstructure and to the presence of organelles (e.g., mitochondria). The adoption of a TD scheme for detection of diffused photons is possibly the most natural way to study diffusive media since it directly discriminates photons on the basis of their arrival time, and therefore of the length of the path travelled into the medium. In particular, the TD approach offers a series of advantages as summarized in the following:  
* For a homogeneous medium, \(\mu_{a}\) and \(\mu_{s}\)' affect differently the DTOF, and therefore can be more easily disentangled [52; 53; 54];
* A change in amplitude in the detected signal does not affect the shape of the DTOF [55; 56]. Therefore, a TD measurement relying only on the DTOF profile will be largely independent of the optical contact, the moving artefacts, the skin pigmentation, and also the laser power fluctuations;
* Space is encoded by time, meaning that the higher the photon travelling time, the farther the average distance from the injection point explored by photons. As a direct consequence, for reflectance measurements longer lived (late) photons propagate deeper into the medium [27; 57; 58; 59]. In the context of TD-DO, time-gating is particularly interesting for the possibility to extract few meaningful photons out of the multitude of collected photons. With reference to Figure 1, a proper time-gating permits for instance to select late photons arising from deeper structures. A standard continuous wave (CW) measurement would mix late photons with the overwhelming majority of other more \"superficial\" photons, thus diluting the weak information content. This feature opens the possibility to perform time-resolved reflectance measurements at null source-detector distances (\(\rho\)) [38]. In general, upon increasing \(\rho\), the diffusely reflected signal decreases at all photon arrival times, as shown in Figure 2. Thus, at least in principle, acquisition at \(\rho=0\) is always granting the maximum harvesting efficiency. The key problem is that the increase in signal at \(\rho=0\) is extreme for early photons so that for a classical detector the contribution of late photons would be overwhelming. The adoption of a hardware time-gating mechanism can solve this issue by gating-off early photons and enabling the detector only for late photons. The feasibility of TD diffuse reflectance at \(\rho=0\) enabled by hardware time-gating offers additional benefits.