When a small single absorbing object is placed inside a diffusive medium (homogeneous or heterogeneous), it is possible to formulate an equivalence relation so that a small and completely black object can give the same result for the perturbation as that for a larger object with only moderate absorption change. By small object we mean an absorbing inclusion of volume placed in a position inside the diffusive medium so that the unperturbed Green function is nearly constant over . The time-dependent fluence rate can be expressed as39Display Formula
(1)where is the perturbed fluence, is the unperturbed fluence, is the unperturbed Green function, is the absorption variation of the object with respect to the background, is the field point, and denotes any point inside . This integral equation can be written in any geometry (slab, semi-infinite medium, cylinder, etc.) and for any distribution of the optical properties of the background medium. The simplest and most approximated evidence of the equivalence relation can be noticed within the Born approximation, i.e., assuming , so that Eq. (1) becomes Display Formula
(2)Equation (2) directly yields the Green function of the perturbed medium once the Green function of the unperturbed medium is known. Equation (2) can be written, assuming the unperturbed fluence and the Green’s function constant over , in the following form: Display Formula
(3)where Display Formula
(5) measures the strength of the object, and when is constant over , it is . The time-dependent function depends on the center of the absorbing object, , on the reduced scattering coefficient of the background medium, , and on the geometry of the surrounding medium, while for homogeneous media in term of absorption, it is independent of the absorption coefficient of the background since the Beer-Lambert attenuation factor is identical both for and for . By using the functions and , the time-dependent contrast can be expressed as Display Formula
(6)i.e., as the factorization of the function describing the temporal shape of the perturbation and the term accounting for the intensity of the perturbation. Also, the time-dependent contrast is independent of . Equation (3) expresses the equivalence relation since equivalent perturbations in terms of time-dependent contrast may be produced by different absorbing inclusions. A similar procedure can be repeated for a CW source, leading to a similar expression for the CW contrast Display Formula
(7)with the function describing the spatial shape of the perturbation. We stress that different from the time-dependent case, the dependence on in Eq. (7) cannot be factorized [see Eq. (4.22) of 39, that is, the expression of for the slab] and both and depend on the background absorption coefficient .