Monte Carlo Simulation of Photon Propagation Through Diffusive Medium

In a semi-infinite homogeneous medium, when the source and detector are separated by a distance of ρ, the number of photons per unit area and time emerging from tissue after having reached at least depth z is given by [1]:

where, R(ρ, t) = k t ̶ ̶ 5/2 exp(-μa υt) exp( ̶ 𝜌

2

4 𝐷𝜐𝑡 ) S(D,s0,t)

R(ρ,t) is the time-resolved reflectance i.e., the total number of photons per unit area and time emerging from tissue at distance ρ, and p(z,t)is the time-resolved probability that photons penetrate at a certain depth before being detected at the tissue surface. k is a constant, S(D, S0, t) is a dipole term taking into account boundary conditions and thickness S0 of the probed medium, D = (3μs’)-1 is the diffusion coefficient, μs’ is the reduced scattering coefficient, μa is the absorption coefficient, and v is the speed of light in the medium. Since p(z, t) is independent of the source-detector separation ρ and of μa , we can write that the relative percentage ratio in photons, at a given time, due to the use of a finite source-detector separation ρ as compared to ρ = 0 is : G(ρ,t) = 𝑁 (𝜌,𝑡) 𝑁(0,𝑡) = 𝑝(𝑧,𝑡) 𝑅(𝜌,𝑡) 𝑝(𝑧,𝑡) 𝑅(0,𝑡) = 𝑅 (𝜌,𝑡) 𝑅 (0,𝑡) = exp[- (ρ/ρ0) 2_{] (3.2)}

where, ρ0 is √4Dvt. We observe a significant loss of photons when we perform Time-resolved reflectance measurements with values for ρ in the range 20 – 40 mm as compared to that in the case of a null source detector separation. Also depth information is not degraded when we reduce the source-detector distance because p(z, t) does not depend on ρ. We have tested the proposed approach (ρ= 0, Null Distance Source-Detector) against the classical one (ρ>>0) by means of Monte Carlo simuations. The latter deals with the transport of an infinitely narrow photon beam perpendicularly incident on a multi-layered tissue. The medium is defined as a cube with cube-shaped voxels, typically a 100x100x100 voxel cube. Each voxel is assigned an integer value which identifies a particular type of tissue with unique optical properties of μa, μs and g. We considered a simple adult head model consisting of four different homogenous media including a surface, CSF, Gray-matter and White-matter layers. The surface layer imitates the scalp and skull. The thickness of each layer is 10mm, 2mm, 4mm and semi-infinite respectively (Fig. 3.3). In the simulations the source and the detector were placed on the surface separated by a certain distance. A large amount of photons (106 to 107) were injected into the medium and a certain percentage of the injected photons are detected. The diffused photons probe a volume corresponding to a “banana shape”. The index value depicts the total number of photons that are coming from the depth (x=14mm) divided by the total number of detected photons. The depth is chosen in the gray mater. The index values are shown for different positions and depth both for the classical method (source-detector distance, ρ=40) and the proposed method (Null

source detector distance, ρ=0). Sensitivity profiles are shown in Fig. 3.4. Reducing source- detector separation results in a better localization of photons in the proposed approach as compared to the classical one and also provides better spatial resolution. As seen in Fig. 3.4, when the source-detector is separated by 40mm, the index value reduces to a great extent indicating a significant loss of photons as compared to the null source-detector separation. The most severe problem as observed in Fig. 3.4 is the presence of early photons at zero source- detector distance, that increases upon decreasing ρ at a much faster rate as compared to the late photons. One possible approach is to use a detection system operated with a time gate with rise time in the range of 100-200 ps and delayed appropiately with respect to the initial burst of photons. This enables the detection of only the late photons while rejecting the early photons that are coming from the surface of the brain and so do not contain any relevant information. To support the experimental feasibility of the proposed gating approach, for short source- detector separation, the detector is gated from 0.4ns to 1.5ns i.e., the detector is turned off for the first 0.4ns. So, we can observe the decrease in the number of detected early photons. As the time gate shifts from 400ps to 600ps, the number of detected photons (both early and late) further decreases as is inferred from the index values. The major advantage of gating is to avoid the detection of the early burst of photons coming from the surface area due to which the detector will be completely blinded and henceforth will not be able to detect the late photons. One more inference, which can be drawn after seeing the index values, is that the number of detected photons is independent on the source-detector distance.

Figure 3.3. Illustration of the sensitivity of different source-detector separations, modeled and simulated by Monte Carlo, using fluorescent light propagation in the defined multilayers diffusive medium

(a) ρ = 40mm ρ = 1mm

(b) ρ = 40mm ρ = 1mm

(c) ρ = 40mm ρ = 1mm

Figure 3.4. Comparison of the Sensitivity profiles for the classical approach (ρ = 40mm) and for the novel approach (ρ = 1mm) (a) no Time gating, (b) detecting photons after 400ps and (c) 600ps