## 4 Discussion
This situation is analogous to a \"broad\" (i.e., the two dimensions of the illumination spot are much larger than \(\mu_{\mathrm{eff}}{}^{-1}\)) uniform illumination of a \"flat\" (i.e., the smaller radius of curvature of the tissue surface is much larger than \(\mu_{\mathrm{eff}}{}^{-1}\)) air-tissue interface. \[\phi(\mathbf{z})=\mathrm{Ekexp}(-\mu_{\mathrm{eff}}z), \tag{3}\]  
where E is the irradiance (mW/cm\({}^{2}\)), z (mm) is the distance from the air-tissue interface and k is a term that describes how backscattered light increases the irradiance delivered to the surface, yielding an elevated fluence rate near the surface. Its value, which is typically in the range of 3-6 for soft tissues in the red or NIR, depends on the tissue optical properties as well as on the refractive index matching conditions [41]. Its accurate value can be determined using Monte-Carlo simulations of the propagation of light [34; 42; 43]. In this case, however, the angular distribution of the radiance at the surface of the light distributor must be known, a situation illustrating the interest of performing directional radiometric measurements, as reported in this paper. Using optical coefficients compiled by [PERSON] [44] for the aorta at 630 nm, the value of k is about 5 if the refractive indexes are mismatched [41; 43]. It should be noted that expressions (2) and (3) are solutions of the diffusion approximation. Therefore, these solutions are only valid if the hypotheses that are at the origin of this approximation are fulfilled, in particular if the light flux in the tissues is isotropic. This condition is clearly satisfied at a much closer distance (less than \(\upmu_{\text{s}}\)\({}^{\prime-1}\)) from the light distributor if its emission is Lambertian rather than collimated, a fact that also illustrates the interest of multi-directional radiometric measurements. This concept is illustrated in Figure 4, which presents a comparison between the values of the normalized fluence rate determined according to Equation (3) or by Monte-Carlo simulations in the cases of broad illuminations of an air-tissue interface. This figure presents the evolution of the fluence with depth when the semi-infinite tissue, presenting the optical properties reported by [PERSON] [44] at 630 nm for the aorta, is subject to a collimated or a Lambertian illumination. The fluence rate is normalized in the sense that its absolute value is divided by the irradiance applied at the air-tissue interface. The Monte-Carlo simulation was based on commercially available software (Tracepro; Lambda Research Corporation, Littletown, MA, USA) and hardware described in details by [PERSON] et al. [45]. As illustrated in Figure 4, the shape of the fluence rate is significantly different, for an identical irradiance, at the tissue \"surface\" (where atherosclerotic plaques potentially sit) if the illumination is collimated or Lambertian. This difference is significant down to about 0.4 mm, i.e., \(\upmu_{\text{s}}\)\({}^{\prime-1}\), as expected (see the red squares). Importantly, the pre-exponential factor \"k\", used to fit the spatial evolution of the fluence rate with a solution of the diffusion approximation (Equation (3)), decreases from about 5.15 to 3.99 (more than 25% difference in our conditions) when a Lambertian illumination is considered instead of a collimated one. As a consequence, the nature of the tissue illumination (collimated or Lambertian) affects the fluence rate significantly, even in deep-seated tissues. If, depending on the light delivery design, the balloon or the diffuser radius is comparable to \(\upmu_{\text{eff}}\)\({}^{-1}\), as is the case when red light is used to treat plaque in the carotid, neither expression (2) nor (3) is valid. It should be noted, however, that the fluence rate can still be estimated analytically since it is confined within an interval defined by the values calculated with these two expressions, as discussed by [PERSON] et al. [40].