## 4 Discussion
Importantly, the deviations from ideal longitudinal profiles have a reduced effect if the distributor surface and the tissue are not in contact since the irradiance applied to the lesion results from the contribution of a large surface of the diffuser surface. Wavelengths in the red or NIR are usually preferred to treat atherosclerotic plaque by PDT because of their deeper penetration in most soft tissues compared to shorter wavelengths [37]. In addition, their homogenous distribution in these tissues makes these wavelengths optimal for endovascular applications. Since their absorption is much smaller than their scattering in the blood vessel wallor in the surrounding tissues, the fluence rate can be calculated using the diffusion approximation, as proposed by [PERSON] [38]. The steady-state expression to calculate the fluence rate in the tissue is:  
\[(\Delta-\mu_{\mathrm{eff}}{}^{2})\phi(\mathbf{r})=-q(\mathbf{r})/D, \tag{1}\]  
where \(\phi(\mathbf{r})\) is the fluence rate (mW/cm\({}^{2}\)), \(\mu_{\mathrm{eff}}=[3\mu_{\mathrm{a}}(\mu_{\mathrm{a}}+\mu_{\mathrm{s}}(1-g)] ^{1/2}\) is the effective attenuation coefficient (mm\({}^{-1}\)), D = \(\mu_{\mathrm{a}}/\mu_{\mathrm{eff}}{}^{2}\) is the optical diffusivity (mm), and q is the diffusive photon density (mW/cm\({}^{3}\)). The absorption and scattering coefficients, \(\mu_{\mathrm{a}}\) (mm\({}^{-1}\)) and \(\mu_{\mathrm{s}}\) (mm\({}^{-1}\)), correspond to the inverse of the photon's mean free path before they are absorbed or scattered, respectively. g is the scattering anisotropy factor, which relates the reduced scattering \(\mu_{\mathrm{s}}\)\({}^{\prime}\) (mm\({}^{-1}\)) and the scattering coefficients in the expression: \(\mu_{\mathrm{s}}\)\({}^{\prime}\) = \(\mu_{\mathrm{s}}(1-g)\). The solution of Equation (1) for the fluence rate \"far\" (i.e., at several times \(\mu_{\mathrm{s}}\)\({}^{\prime-1}\)) from the source and interfaces produced by an infinitely long light diffuser inserted in a tissue in a cylindrically symmetric configuration is given by [39]:  
\[\phi(\mathbf{r})=[(P/2\pi\mathrm{a})/(D\mu_{\mathrm{eff}}K_{1}(\mu_{\mathrm{ eff}}\,\mathrm{a}))]\,K_{0}(\mu_{\mathrm{eff}}\,\mathrm{r}), \tag{2}\]  
where P is the optical power per unit length (mW/mm) of the light diffuser with radius a (mm). K\({}_{0}\) and K\({}_{1}\) are the modified Bessel K functions of the zero and first order, respectively. The distance from the diffuser axis is expressed as r (mm). Since the length of light diffusers used to treat atherosclerotic plaque by PDT in coronary arteries ranges between 10 and 40 mm, and since their diameter is much smaller (between several hundreds of microns and one millimeter), expression (2) can be used if the diffuser is in contact with the vessel wall. This expression can also be used if the light diffuser is inserted at the center of a transparent balloon, provided that the radius of this balloon is much smaller than \(\mu_{\mathrm{eff}}{}^{-1}\) and the length of the illuminated section of the artery. In that rare situation, which prevails only when NIR light is emitted by a small (diameter of 1 mm or less) balloon, the parameter a (mm) corresponds to its radius. This means that expression (2) can be used to calculate the fluence rate in the wall and around coronary arteries assuming uniform optical coefficients. The picture is different if PDT is performed in larger vessels, as is the case for peripheral atherosclerotic disease. In that case, expression (2) is no longer valid since the radius of the vessel can be in the order of the diffuser's length. An estimate of the fluence rate in the tissues at depths \"z\" larger than 1 mm is given by expression (3) if the balloon radius and the diffuser's length are much larger than \(\mu_{\mathrm{eff}}{}^{-1}\)[40; 41].