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/cm2) 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 μa = 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.