### Theoretical Background

The intensity distribution at the surface of a turbid medium is described by a spatial radiance distribution (SRD). This SRD was estimated by solving the diffusion equation in an infinite homogeneous medium with solid spherical sources of radius *R* with its center located at depth *z*.

For deriving the theoretical SRD, an optical imaging system with the geometry depicted in Fig. 1 was chosen. It consisted of a slab of thickness d characterized by an absorption coefficient *μ*
_{
a
}and a reduced scattering coefficient , both with values corresponding to biological tissue.

A spherical light source is placed at the origin of a cylindrical coordinate system ρ, ϕ z. The SRD in a highly scattering medium is obtained from solving the diffusion equation for photon propagation [9, 10]:

where Φ(**r**) is the photon fluence (photons/cm^{2}) at location **r** with the radius given by for *z* ≤ *d*, Ω describes the region of interest, *S*(**r**) an isotropic source term, while the diffusion coefficient *D*(**r**) = 3(*μ*
_{
a
}(**r**) + *μ*
_{
s
}'(**r**)), accounts for diffusive photon propagation. This equation is derived from the adiative transfer equation [11] by applying the diffusion approximation and is valid for highly scattering media of a thickness greater than the photon mean-free path length.

Recently, the steady-state solution of the diffusion equation with a solid spherical source of radius *R* embedded in an infinite homogenous medium has been formulated as [12]:

with

where *P* is the source power denotes the effective attenuation coefficient, while *α*(*R*) stands for the exponential terms given in brackets.

Φ(**r**, *R*) at the interface between two media with different refractive indices can be approximated by an index-mismatched Robin-type boundary condition. This condition for Φ(**r**, *R*) on ∂Ω is described in the following formula [13, 14]:

The term **n** is the unit vector normal to a surface element ∂Ω and *A* is a parameter governing the internal reflection at the boundary. The value of *A* = (1+R)/(1-R) depends on the relative refractive index mismatch between tissue and air and can be derived from Fresnel's law, with R as [14]:

and *n* = *n*
_{
a
}/*n*
_{
t
}, *n*
_{
a
} and *n*
_{
t
} being the refractive indices for air and tissue, respectively. This equation yields *A* = 2.51 for the air-phantom interface with typical value of *n* = 1.33 [15]. Based on the boundary condition the measured fluence rate on ∂Ω is [13, 16].

Φ(**r**, *R*) can be converted to the radiance for a perfectly isotropic light distribution:

Because the radiance in biological tissue is slightly anisotropic and angle dependent due to the large number of scattering events, an angular dependent radiance *L*(**r**, *R*, ) is introduced which accounts for anisotropic optical properties of the tissue at **r** in direction of , i.e the expression for the radiance (Eq. (6)) is expanded according to [17]:

with the second term denoting the net energy flux **F**(**r**) = -*D*(**r**)▽Φ(**r**, *R*) at a distance r from the sphere in direction [16]. Combining the various equations one obtains:

which for **r** ∈ *S* corresponds to the radiance distribution at the surface *S* of the sample that is generated by a spherical light source of radius *R*, i.e to the SRD.

### Experimental Evaluation

For experimental evaluation, a commercial in-vivo fluorescence imaging system (Maestro, Cri Woburn, USA) has been used with a geometry depicted in Fig. 2. The system consists of a fiber-delivered Xenon excitation source (*λ* = 500-950 nm, 30 W), a standard excitation and emission filter set and in addition a liquid crystal tunable filter (LCTF). The LCTF transmits a narrow bandwidth (10 nm) of the emitted fluorescence light to a 1.3 megapixel CCD camera. A 615-665 nm excitation filter and a 700 nm long-pass emission filter were used for the evaluation.

The phantom was made from 6 g agarose (BioGene, Kimbolton, UK), 24 ml Intralipid 20% (Fresenius SE, Bad Homburg, Germany) and 18 μl Indian ink (Pelikan Holding, Schindellegi, Switzerland) [18] dissolved in 600 ml water. The optical properties of the phantom (with radius 10 cm) were measured with a frequency-domain tissue oximeter (oxiplexTS, ISS, Champaign, USA). The reduced scattering and absorption coefficients, at 692 nm, were then found to be = 8 cm^{-1} and μ_{a} = 0.11 cm^{-1} respectively. The same mixture was poured into four plexiglass molds of 6 × 6 cm and thickness ranging from 2 mm to 4 mm. The width of the mould was more than 10 penetration depths (0.6 cm) which fulfilled the requirement for a semi-infinite boundary medium [19]. After solidification, the agarose layers were removed from the molds and positioned by a fixed holder on the top of solid spherical source. The sources with radius 2-6 mm contained 5 mg/l near-infrared quantum dot (Qdot 705 ITK, Invitrogen, Basel, Switzerland) with emission wavelength at 705 nm.

The optical images were recorded for varying thickness *d* and denoised by a bandpass filter [20]. The bandpass filter consists of a Gaussian low-pass filter and a boxcar kernel which was used as a high-pass filter. First, a low pass filtered image was generated by convolving the diffuse image with the Gaussian filter, a high-pass filtered image was then obtained by convolving the diffuse image with the boxcar function. The difference between low-pass filtered image and high-pass filtered image allowed extraction of the final image for quantitative analysis.