## 2 Physics of Time Domain Diffuse Optical Spectroscopy (TD-DOS)  
The photon propagation in a diffusive medium can be described by the radiative transfer equation (RTE), originally developed to describe the transport of non-interacting neutrons in nuclear science [54]. The assumption of non-interacting photons is still valid because, inside a diffusive medium, a multiple scattering regime is present, and consequently the light coherence vanishes after a few scattering events. Hereafter the RTE equation:  
\[\frac{1}{v}\frac{\partial L}{\partial t}=-\mathbf{s}\cdot\
abla L(\mathbf{r},\mathbf{s},t) -(\mu_{a}+\mu_{s})L(\mathbf{r},\mathbf{s},t)+\mu_{s}\int_{4\pi}p(\mathbf{s},\mathbf{s}^{\prime} )\;L(\mathbf{r},\mathbf{s},t)d\mathbf{s}^{\prime}+Q(\mathbf{r},\mathbf{s},t), \tag{1}\]  
where \(L(\mathbf{r},\mathbf{s},t)\) is the radiance at the position \(\mathbf{r}\), which is the power flowing in the direction of the unit vector \(\mathbf{s}\), per unit of solid angle and area normal to \(\mathbf{s}\). \(v\) is the photon velocity in the medium, \(\mu_{a}\) and \(\mu_{s}\) is are the absorption and scattering coefficients, \(p(\mathbf{s},\mathbf{s}^{\prime})\) is the scattering phase function, which gives the probability that a photon running in the direction \(\mathbf{s}\) is scattered to the direction \(\mathbf{s}\prime\), and \(Q(\mathbf{r},\mathbf{s},t)\) is the spatial, temporal, and angular distribution of the source [55].  
Figure 1: Timeline of important events to exemplify the evolution and potential of broadband time domain diffuse optical spectroscopy (TD-DOS): the top section shows the events in the advancement of instrumentation and the bottom three sections cover various applications of broadband TD-DOS.  
When a short pulse is injected in a diffusive medium and detected on the external surface, Equation (1), together with proper boundary conditions, rules the distribution of the photon time-of-flight (DTOF) after injection of a short pulse. The DTOF is related both to the volume probed by the light in the medium and to its optical properties (\(\mu_{a}\) and \(\mu_{s}\)). In fact, photons having travelled long paths give information on a larger volume than photon having travelled short paths (Figure 2). This fact is key for detecting information from deep regions of a tissue when the source and detector are placed on the surface on the same side, which is the most used configuration for non-invasive monitoring of biological tissues [56].  
Another interesting advantage of the time domain approach is the fact that the shape of the DTOF is strongly modified by the absorption and scattering properties of the probed portion of the medium. Photons running a long path in the medium carry information about the probability of absorption, thus affecting the tail of the DTOF (Figure 3a). Instead, photons running a short path are more sensitive to the scattering properties of the medium, thus modifying the temporal position of the barycenter of the DTOF (Figure 3b).  
These two effects open the possibility to estimate the spatial distribution of both optical properties (\(\mu_{a}\) and \(\mu_{s}\)) simultaneously from the shape of time-resolved DTOF using analytical, numerical, or stochastic models. Another advantage of such an approach relies on the fact that the estimation is unaffected by the amplitude of the DTOF, thus rejecting every source of error related to the optical coupling between tissue and delivery and/or detection optics.  
Figure 2: Sensitivity profiles (\(\log 10\)), which represent the relative change in the detected signal when a localized defect is placed at every position of the volume, in a homogeneous diffusive medium. At three time intervals (0.5, 1, 1.5 ns) for: (**a**) reflectance geometry (3 cm source-detector distance) and (**b**) collinear transmittance geometry. Each white line represents the contour edge of the contrast every decade.