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					      Monte Carlo Simulations of the Growth and Decay of Quasi-Ballistic
            Photon Fractions with Depth in an Isotropic Medium
                                  Nick Pfeiffer and Glenn H. Chapman
               Simon Fraser University, School of Engineering Science, 8888 University Ave,
                                     Burnaby, B.C., Canada V5A 1S6


                                                         ABSTRACT

  Quasi-ballistic or “snake” photons carry useable information on the internal structure of scattering mediums such as
  tissues. By defining quasi-ballistic photons to be those photons that have been scattered but have not exceeded a
  specified radial distance threshold from their initial trajectory (equivalent to the resolving limit of the quasi-ballistic
  photons) and by using the Henyey-Greenstein phase function, Monte Carlo modeling has shown that the number of
  quasi-ballistic photons increases with depth in an isotropic scattering medium until a maximum is reached and then the
  quantity decreases. The quantity of quasi-ballistic photons at a specified depth can be shown to be governed by two
  competing processes: the decay of ballistic photons into quasi-ballistic photons and the decay of quasi-ballistic photons
  into scattered photons. These well-defined behaviors allow one to write a rate equation governing the growth and decay
  in the quantity of quasi-ballistic photons with depth. It is found that as the anisotropy factor increases with forward
  scattering and as the resolution limit is widened, the quantity of quasi-ballistic photons begins to exceed the quantity of
  ballistic photons at a specified depth and the rate of decay of quasi-ballistic photon quantity decreases. The
  development of a rate equation for the formation of quasi-ballistic photons allows one to analyze how efficient various
  detection methods are in extracting these quasi-ballistic photons, and it can be seen that there is a compromise between
  desired resolution and the effective scattering ratio at a detector.

  Keywords: light scattering, optical tomography, tissue optics, collimating optics, quasi-ballistic, Monte Carlo

                                                     1. INTRODUCTION

  Optical methods have been extensively pursued as a replacement for x-rays to image objects within tissues1. Tissues are
  highly scattering media for light, and the radiative transport equation is often used to analyze the propagation of light
  inside such media. In such analysis, photons are assumed to transport energy through the media. Aggregates of these
  individual photons carry information on the internal structure of the media.

  In the case of transmissive illumination with a collimated beam normal to the surface of a plane sample, it is seen that a
  few photons pass through the media without scattering. These ballistic photons carry information about the internal
  objects within their path. Many more photons may scatter and rescatter at small angles, remaining close to their initial
  trajectory (forward scattering). These quasi-ballistic or “snake” photons also carry useable information about the
  internal objects within their path2. The majority of the photons scatter at large angles, or are absorbed, and provide no
  useable information.

  Existing optical techniques for imaging tissue (for example, optical coherence tomography3) rely upon the fact that both
  quasi-ballistic photon and ballistic photons can provide information. However, what quality of information is provided
  by the quasi-ballistic photons?

  To quantify the information contribution provided by quasi-ballistic photons in the above case of transmissive
  illumination, this paper adopts the following definition: a photon is considered quasi-ballistic if the maximum radial
  distance of its path through a media does not exceed a specified threshold distance r from its initial (ballistic) trajectory.
  The threshold distance r is equivalent to the resolving limit of the photon and is shown in Figure 1.




136                                                        Optical Interactions with Tissue and Cells XVI, edited by Steven L. Jacques,
                                                             William P. Roach, Proc. of SPIE Vol. 5695 (SPIE, Bellingham, WA, 2005)
                                                                                            1605-7422/05/$15 · doi: 10.1117/12.590512
                                    Figure 1 – Photon Paths through a Scattering Medium


With increasing depth z in a scattering media, the fraction of incident photons that lie within distance r of their initial
trajectory decreases as ballistic photons become quasi-ballistic and quasi-ballistic photons become scattered.

The main goal of the paper is to derive a simple approximation to determine the fraction of photons that are quasi-
ballistic (within a defined distance r) at a specified depth within an isotropic scattering medium.

                                          2. MONTE CARLO MODELING

The Monte Carlo method of modeling photon propagation in tissue is commonly employed to determine the photon
distribution at a detector for a specified geometry4. All simulations done for this paper were performed using the
Photon Transport Simulator, a general-purpose Monte Carlo ray-trace engine developed at Simon Fraser University to
explore the use of photon trajectory as a means of filtering quasi-ballistic photons from scattered photons5,6,. Primary
features of this software include arbitrary 3D geometry (sources, lenses, mirrors, media), Fresnel reflection/refraction,
diffraction, and flexible recording of specified photon information. Figure 2 shows the primary geometry used for the
Monte Carlo simulations.




                                        Figure 2 – Monte Carlo Simulation Geometry


In the simulation it is assumed that there are no interactions between photons and that scattering obeys the Henyey-
Greenstein phase function7. A point source is used to launch photons normal to a scattering sample having an index of
refraction fixed at 1.5. Approximately, 4% of the photons undergo specular reflection at the air/sample interface, with
the remaining photons entering the sample. The photons inside the sample undergo scattering and absorption according
to random walk theory based upon the anisotropy factor g, the scattering coefficient s, and the absorption coefficient
                                                                                          µ


                                                                                               Proc. of SPIE Vol. 5695   137
      µ a. In the simulated geometry, a series of detector planes (binned arrays) are placed at arbitrary locations, either within
      the sample or on either end, to record desired photon parameters such as number of scattering events, photon position,
      photon trajectory, photon path length, maximum photon radial distance from initial trajectory, etc. The results for each
      photon are binned for each detector plane and are available for analysis at the end of the simulation run. In this manner,
      it is possible to construct a wide variety of simulations and analyze the results throughout the medium.

                                    3. QUASI BALLISTIC PHOTON GROWTH AND DECAY

      From the underlying random walk theory of the Monte Carlo simulation, it is known that a photon has an exponentially
      increasing probability of becoming absorbed or scattered with increasing depth z. Using the previous definition of a
      quasi-ballistic photon, it can be shown that ballistic photons become quasi-ballistic photons for some finite distance
      following the first scattering event, provided that r>0. These quasi-ballistic photons may then undergo subsequent
      scattering that causes them to exceed the radial threshold distance r, at which point they cease to be considered quasi-
      ballistic and become scattered photons. Thus, there are at all times three pools of photons: ballistic, quasi-ballistic, and
      scattered. Photons successively pass through each pool and, once scattered, may never again become quasi-ballistic.

      As a first approximation, it is tempting to write a rate equation relating the rate of change in the quantity of quasi-
      ballistic photons Qqb with depth z to the quantity of ballistic photons Qb using growth coefficient A and decay
      coefficient B.

                                                            dQqb
                                                                       = AQb − BQqb                                            (1)
                                                             dz

      The Beer-Lambert Law states that ballistic photons decay exponentially with depth from the initial ballistic quantity
      Qb0. Using photon fractions qb and qqb in place of photon quantities and substituting the known exponential decay rate
      of ballistic photons for Qb, results in a revised rate equation.

                                                     dqqb
                                                             = Aqb 0 e −(u a +u s ) z − Bqqb                                   (2)
                                                      dz

      This equation can be solved by assuming that the fraction of quasi-ballistic photons grows and decays exponentially.
      Using the boundary condition that qqb=0 at z=0 yields the quasi-ballistic fraction qb as a function of a multiplying
      constant k, the decay rate of ballistic photons, and a quasi-ballistic decay parameter B.

                                                       q qb = k (e −( µ a + µ s ) z − e − Bz )                                 (3)

      Using the specified boundary condition, the multiplying factor k in the above equation is related to the growth
      coefficient A and the decay coefficient B by

                                                                           Aqb 0     
                                                                                     
                                                                          µa + µ s   
                                                            k=                       
                                                                                                                               (4)
                                                                          B
                                                                                −1
                                                                                         
                                                                                         
                                                                   
                                                                      µa + µ s           
                                                                                          


      The growth coefficient A is exactly equal to the scattering coefficient s and represents the change from a ballistic to a
                                                                                              µ
      quasi-ballistic photon. It should be noted that this fact arises from the specific definition of quasi-ballistic photon used,
      where each photon becomes quasi-ballistic at the first scattering event.

      It can be seen from the form of equation (3) that the quasi-ballistic photon fraction is zero at z=0 (as expected, since all
      photons are still ballistic) and that the fraction of quasi-ballistic photons grows to a maximum at some depth z and then



138       Proc. of SPIE Vol. 5695
decreases with increasing depth. It should be noted that in the case of no decay of the quasi-ballistic photon fractions
(B=0), equation (3) reduces to the scattered photon fraction predicted by the Beer-Lambert Law.

Monte Carlo simulations using 107 photons were performed to validate the above model for the growth and decay of
quasi-ballistic photons. Figures 3 and 4 show results for a 1 cm thick plane sample with a scattering coefficient of
              -1                                                 -1
µs=1.38 mm , absorption coefficients a of 0 and 1.38 mm , and anisotropy factors g of 0 and 0.9 with an initial
                                                                         µ
ballistic photon fraction qb0 of 0.96 (representing 4% loss due to specular reflection). It should be noted that the
scattering coefficient s =1.38 mm-1 was chosen as it represents a scattering ratio (ratio of scattered photons to ballistic
                         µ
photons at the exit of the sample, z=10 mm) of 106, and higher scattering ratios would have required larger numbers of
photons for the Monte Carlo simulations to have statistical validity. The fraction of ballistic photons in the non-
absorbing medium ( s =1.38 mm-1, a =0 mm-1) is also shown for comparison, and it can be seen that the fraction of
                     µ                                               µ
quasi-ballistic photons exceed the fraction of ballistic photons as the radial distance threshold r is increased.



                                                                   Quasi-Ballistic Photon Fractions vs. Depth
                                                                                     g= 0.0
                                                        0.05                                     r=100 um, us=1.38/mm, ua=0/mm
                         Quasi-Ballistic Fraction




                                                                                                 r=100 um, us=1.38/mm, ua=1.38/mm
                                                        0.04                                     r=10 um, us=1.38/mm, ua=0/mm
                                                                                                 r=10 um, us=1.38/mm, ua=1.38/mm
                                                                                                 Ballistic
                                                        0.03

                                                        0.02

                                                        0.01

                                                          0
                                                               0             0.5   1       1.5         2          2.5          3
                                                                                       Depth z (mm)

                              Figure 3 – Growth and decay of quasi-ballistic photon fractions with depth
                         for g=0.0, symbols are Monte Carlo results, thin lines are best fits using equation (3)


                                                                   Quasi-Ballistic Photon Fraction vs. Depth
                                                                                     g = 0.9
                                                         0.4                                     r=100 um, us=1.38/mm, ua=0/mm
                                                                                                 r=100 um, us=1.38/mm, ua=1.38/mm
                             Quasi-ballistic Fraction




                                                        0.35
                                                                                                 r=10 um, us=1.38/mm, ua=0/mm
                                                         0.3                                     r=10 um, us=1.38/mm, ua=1.38/mm
                                                        0.25                                     Ballistic

                                                         0.2
                                                        0.15
                                                         0.1
                                                        0.05
                                                           0
                                                               0             0.5   1       1.5         2          2.5          3
                                                                                       Depth z (mm)

                              Figure 4 – Growth and decay of quasi-ballistic photon fractions with depth
                         for g=0.9, symbols are Monte Carlo results, thin lines are best fits using equation (3)




                                                                                                                            Proc. of SPIE Vol. 5695   139
      The least-squares best fits of the B parameter using the error fraction of equation (3) to the simulation data are also
      shown in Figures 3 and 4. It can be seen that there are significant errors, with correlation coefficients ranging down to
      0.94. It is clear that the rate equation with constant coefficients A and B does not properly model the underlying
      mechanism of photon transport. Thus, it is proposed that coefficient B is not a constant due to the fact that the decay
      rate of a single photon from quasi-ballistic to scattered with increasing distance from the first scattering event varies
      significantly and can not be approximated as constant.

      To develop a better equation, we start in the following section to explore the modeling of the decay of a single photon
      from quasi-ballistic to scattered.

                                                    4. SINGLE PHOTON SCATTERING

      Figure 5 shows the possibilities for a single photon after the first scattering event. Prior to the first scattering event, the
      photon is assumed to be ballistic with a trajectory in the z direction. At each scattering event, the photon has a
      probability p(e ⋅ e ' ) of having the trajectory changed from e to new trajectory e ' .
                     r r                                                      r                        r




                                     Figure 5 – Possible Paths for a Single Photon After Initial Scattering


      For a photon following the Henyey-Greenstein phase function, the probability p(e ⋅ e ' ) is defined as7
                                                                                                           r r



                                                                           1          1− g2
                                               p(e ⋅ e ' ) = p(cos θ ) =
                                                    r r                                                    
                                                                                                                                 (5)
                                                                           2 (1 + g 2 − 2 g cos θ ) 3 / 2
                                                                                                           
                                                                              
                                                                                                           
                                                                                                            


      At a distance z from the initial point of first scattering, there is a probability Pp1 that the photon will pass within radial
      distance r of the z-axis (neglecting rescattering and absorption). For an aggregate of photons having first scattering at
      point z=0, the probability Pp1 can be replaced by photon fraction qp1. For photons with scattering trajectory probability
      described by equation (5), the fraction qp1, representing those photons that pass within the threshold distance r at z, can
      be determined by

                                                                                                                        
                                                                                                                        
                                                   1− g2           1                           1
                                    q p1 ( z ) =                              −
                                                                                                                        
                                                                                                                               (6)
                                                    2g         1 + g 2 − 2g                                          r
                                                                                                            −1
                                                                                  1 + g − 2 g cos tan
                                                                                      2                                
                                                                                                                         
                                                                                                                          
                                                           
                                                           
                                                                                                   
                                                                                                                   z    


      At a distance z from initial scattering, there is a probability Pb that the photon will be ballistic. There is also a
      probability Ps that the photon will have been scattered (at least once).




140     Proc. of SPIE Vol. 5695
                                                            Pb ( z ) = e −( µa + µ s ) z                                                    (7)


                                                  Ps ( z ) =
                                                                  µs
                                                               µa + µ s
                                                                            (1 − e   −( µa + µs ) z
                                                                                                      )                                     (8)


When taking absorption and scattering into account, the fraction of photons reaching z from the initial scattering is
reduced from qp1 by the probability that the photon is still ballistic. For z >> r (small angle approximation), the
probability that the scattered photon is ballistic at z is equal to Pb(z).

Some fraction Ps(z) of the photons will have rescattered before reaching z. The photon distribution of each of these
scattering points may be integrated to determine the fraction of rescattered photons that reach z with the radial
threshold8. Some of the rescattered photons will again scatter before reaching z and these scattering points may again
be integrated. The final fraction of photons reaching z within the radial threshold r is the sum of the fraction reaching z
from the initial scattering plus an infinite series with each subsequent term representing the contribution from each
further scattering event (successive order multiple scattering9).

A lumped parameter model for the total photon fraction qp is proposed that condenses the contribution from scattering
events other than the initial scattering event into a single effective scattering event occurring on the z-axis at distance z’.

                                     q p ( z ) = q p1 ( z ) Pb ( z ) + Ps ( z )q p1 ( z − z ' ) Pb ( z − z ' )                              (9)

It can be seen that the photon fraction reaching z within threshold r is comprised of the initial fraction, modified by the
amount rescattered or absorbed, plus an additional fraction representing the amount of all photons rescattered before
reaching z. Replacing z’ by (1-c)z, allows the lumped parameter model to be expressed as:

                                         q p ( z ) = q p1 ( z ) Pb ( z ) + Ps ( z )q p1 (cz) Pb (cz)                                       (10)

The parameter c represents the strength of the contribution from rescattered photons and is a function of scattering
coefficient s, radial distance threshold r, and anisotropy factor g. Smaller values of c place z’ closer to z and result in
            µ
stronger contributions by the rescattered photons, whereas larger values of c result is smaller contributions by the
rescattered photons. While an analytic function for c has not yet been determined, it can be seen that c decreases as r
and g increase. Monte Carlo simulations with modest numbers of photons can be used to determine c.

Monte Carlo simulations using 106 and 107 photons were performed to validate the above model of single photon
scattering. In these simulations, each photon was scattered at z=0 and its subsequent position tracked. Figures 6 and 7
show results for a 1 cm thick plane sample with a scattering coefficient of s=1.38 mm-1, absorption coefficients a of 0
                                                                                                      µ                               µ
and 1.38 mm-1, and anisotropy factors g of 0 and 0.9.

It can be seen there is excellent correspondence between a least-squares best fit of parameter c of the error fraction of
equation (10) to the simulation data. The fitted parameter c varies between 0.76 and 2.1 for the specific simulations
performed, indicating that a virtual source of rescattered photons located at a distance close to the original scattering
point approximately describes the decay of quasi-photons originating from the single point z=0. The worst-case
correlation coefficient for the above least-squares fits is 0.99989. Thus, the single parameter c enables good fits to the
Monte Carlo data using equation (10), for the range of cases tested.




                                                                                                                 Proc. of SPIE Vol. 5695    141
                                                                                                  Single Photon Quasi-ballistic Fraction vs.
                                                                                                     Distance from First Scattering (g=0)
                                                                                             1

                                                                                            0.1
                                                                                                                                            r=100 um, us=1.38/mm, ua=0/mm


                                  Quasi-ballistic Fraction
                                                                                           0.01                                             r=100 um, us=1.38/mm, ua=1.38/mm
                                                                                                                                            r=10, us=1.38/mm, ua=0/mm
                                                                                          0.001                                             r=10 um, us=1.38/mm, ua=1.38/mm

                                                                                         0.0001

                                                                                        0.00001

                                                                          0.000001

                                                             0.0000001
                                                                                                  0        1             2              3           4           5             6
                                                                                                                   Distance z from First Scattering (mm)


                                   Figure 6 –Decay of single photon quasi-ballistic fractions with distance
                             for g=0.0, symbols are Monte Carlo results, thin lines are best fits using equation (10)


                                                                                                  Single Photon Quasi-ballistic Fraction vs.
                                                                                                    Distance from First Scattering (g=0.9)
                                                                                                  1
                                                                                                                                             r=100 um, us=1.38/mm, ua=0/mm
                                                                                                                                             r=100, us=1.38/mm, ua=1.38/mm
                                                                                              0.1
                                                             Quasi-ballistic Fraction




                                                                                                                                             r=10 um, us=1.38/mm, ua=0/mm
                                                                                             0.01                                            r=10 um, us=1.38/mm, ua=1.38/mm


                                                                                            0.001

                                                                                           0.0001

                                                                                          0.00001

                                                                                         0.000001
                                                                                                      0        1             2          3           4           5             6

                                                                                                                   Distance z from First Scattering (mm)

                                   Figure 7 –Decay of single photon quasi-ballistic fractions with distance
                             for g=0.9, symbols are Monte Carlo results, thin lines are best fits using equation (10)


                                                                                            5. A QUASI-BALLISTIC SCATTERING MODEL

      The success of the lumped parameter model in predicting single photon scattering allows one to expect that the concept
      can be extended to predictions of macroscopic quasi-ballistic scattering. A quasi-ballistic scattering model can be
      developed by multiplying the single photon fraction qp by the rate of change of scattering from ballistic to quasi-ballistic
      fractions with distance and integrating over the depth z.

                                                                                                                     z
                                                                                                                                        dq s
                                                                                                          qqb ( z ) = q p ( z − z 0 )
                                                                                                                                            
                                                                                                                                            dz 0
                                                                                                                                                                                 (11)
                                                                                                                     ∫
                                                                                                                                        dz  
                                                                                                                     0


      From equation (2), the rate of change of scattering from ballistic to quasi-ballistic with distance is


142     Proc. of SPIE Vol. 5695
                                                                                   dq qb
                                                                                               = u s q b 0 e − (u a + u s ) z                                                  (12)
                                                                                    dz

Therefore, the final equation to describe the macroscopic growth and decay of quasi-ballistic photon fractions is


                                                                                                                   (                )
                                                                                           z
                                                                     qqb ( z ) = u s qb 0 q p ( z − z 0 ) e −( ua +us ) z0 dz 0
                                                                                           ∫                                                                                   (13)
                                                                                           0


This integration of qqb may be performed analytically or numerically. The quasi-ballistic photon fractions predicted by
equation (13), using numerical integration and the c values determined by single photon simulations, were compared to
the Monte Carlo simulations of Figures 3 and 4.


                                                                 Quasi-Ballistic Photon Fractions vs. Depth
                                                                                   g= 0.0
                                                      0.05                                                                r=100 um, us=1.38/mm, ua=0/mm
                       Quasi-Ballistic Fraction




                                                                                                                          r=100 um, us=1.38/mm, ua=1.38/mm
                                                      0.04                                                                r=10 um, us=1.38/mm, ua=0/mm
                                                                                                                          r=10 um, us=1.38/mm, ua=1.38/mm
                                                                                                                          Ballistic
                                                      0.03

                                                      0.02

                                                      0.01

                                                        0
                                                             0          0.5              1                    1.5               2          2.5          3
                                                                                                    Depth z (mm)

                           Figure 8 – Growth and decay of quasi-ballistic photon fractions with depth
                      for g=0.0, symbols are Monte Carlo results, thin lines are best fits using equation (13)


                                                                 Quasi-Ballistic Photon Fraction vs. Depth
                                                                                   g = 0.9
                                                       0.4                                                               r=100 um, us=1.38/mm, ua=0/mm
                           Quasi-ballistic Fraction




                                                                                                                         r=100 um, us=1.38/mm, ua=1.38/mm
                                                      0.35                                                               r=10 um, us=1.38/mm, ua=0/mm
                                                       0.3                                                               r=10 um, us=1.38/mm, ua=1.38/mm
                                                      0.25                                                               Ballistic

                                                       0.2
                                                      0.15
                                                       0.1
                                                      0.05
                                                         0
                                                             0          0.5              1                   1.5                2         2.5           3
                                                                                                   Depth z (mm)

                           Figure 9 – Growth and decay of quasi-ballistic photon fractions with depth
                      for g=0.9, symbols are Monte Carlo results, thin lines are best fits using equation (13)




                                                                                                                                                     Proc. of SPIE Vol. 5695    143
      Figures 8 and 9 show results for a 1 cm thick plane sample with a scattering coefficient of s=1.38 mm-1, absorption             µ
      coefficients a of 0 and 1.38 mm-1, and anisotropy factors g of 0 and 0.9 with an initial ballistic photon fraction qb0 of
                  µ
      0.96 (representing 4% loss due to specular reflection).

      It can be seen that there is good agreement between a least-squares best fit of equation (13) and the simulation data,
      with a worst-case correlation coefficient of 0.9998. It is important to note that the excellent agreement of the
      macroscopic growth and decay of quasi-ballistic photon fractions to the Monte Carlo results were obtained using
      equation (13) with the single parameter c that was obtained from the simpler, single photon simulations.

                                                                          6. QUASI-BALLISTIC RESOLUTION

      From the definition of a quasi-ballistic photon, it can be seen that the resolution of the image resulting from collection
      of quasi-ballistic photons is determined by the radial distance threshold used to categorize the photon as quasi-ballistic.
      One use for the quasi-ballistic scattering model of equation (13) is to determine the tradeoffs between desired image
      resolution and detector sensitivity (or effective scattering ratio which is defined as the ratio of scattered photons exiting
      the sample to the number of ballistic plus quasi-ballistic photons exiting the sample at the detector).

      Monte Carlo simulations so far shown in this paper have been limited to 107 to 108 photons due to computer resource
      and time constraints. This has limited scattering coefficients to relatively low values (1.3 mm-1) such that a statistically
      significant number of photons exit the sample at the detector. However, equation (13) allows one to estimate the
      parameter c and then model how more realistic scattering coefficients can affect the quasi-ballistic photon fraction qqb.

      To illustrate how the quasi-ballistic photon fractions predicted by equation (13) can be used, the following worked
      example of a more realistic tissue scattering coefficient3 of 10 mm-1 is presented.

      A single Monte Carlo simulation using 107 photons was performed to determine the parameter c for a scattering
      coefficient s of 10 mm-1 and an anisotropy factor g of 0.0. From the simulation data, a best fit of parameter c as a
                  µ
      function of r was determined and is shown in Figure 10.


                                                                  Single Photon Quasi-ballistic Fraction vs.
                                                                     Distance from First Scattering (g=0)
                                                                  1
                              Quasi-ballistic Fraction




                                                                                                      r=10 um, c=1.4     r=25 um, c=1.12
                                                                0.1
                                                                                                      r=50 um, c=1.05    r=75 um, c=0.89
                                                               0.01                                   r=100 um, c=0.74

                                                             0.001

                                                            0.0001

                                                           0.00001

                                                         0.000001
                                                                      0       0.25        0.5        0.75        1         1.25           1.5
                                                                                  Distance z from First Scattering (mm)

                                                         Figure 10 – Decay of single photon quasi-ballistic fractions with distance for
                                                          g=0.0, s=10 mm-1, a=0 mm-1, symbols are Monte Carlo results, thin lines
                                                                 µ            µ
                                                                               are best fits using equation (10)




144     Proc. of SPIE Vol. 5695
                                                               Quasi-Ballistic Photon Fraction vs. Depth
                                                                     g = 0.0, us=10/mm, ua=0/mm
                                                     0.2



                         Quasi-ballistic Fraction
                                                                                                    Ballistic            r=10 um

                                                    0.15                                            r=25 um              r=50 um

                                                                                                    r=75 um              r=100 um
                                                     0.1


                                                    0.05


                                                      0
                                                           0            0.2             0.4             0.6              0.8              1
                                                                                        Depth z (mm)

                     Figure 11 – Growth and decay of estimated quasi-ballistic photon fractions with depth
                         for g=0.0, s=10 mm-1, a=0 mm-1, symbols are Monte Carlo results, thin lines
                                                               µ            µ
                                are best fits using equation (13) based on the results of Figure 10


Figure 11 shows the calculated quasi-ballistic photon fractions for a range of radial threshold distances r. It can be seen
that as r increases, the fraction of quasi-ballistic photons increases and eventually exceeds the fraction of ballistic
photons qb for r=100 um at z=0.6 mm.

Using c(r) from Figure 10 in equation (13), the quasi-ballistic photon fraction at a hypothetical detector located at
z = 10 mm can be determined. Figure 12 shows the predicted fraction of quasi-ballistic and ballistic photons as a
function of radial threshold r at the detector.


                                                       Estimated Detector Sensitivity vs. Quasi-ballistic
                                                                 Radial Threshold Distance
                                                                 g=0.0, us=10/mm, ua=0/mm
                                                      1E-34
                      Ballistic + Quasi-
                      ballistic Photon




                                                      1E-36
                           Fraction




                                                      1E-38

                                                      1E-40

                                                      1E-42

                                                      1E-44
                                                                   0   10     20   30     40       50    60         70    80       90   100
                                                                                Radial Threshold Distance r (um)

                                                       Figure 12 – Estimated Ballistic + Quasi-ballistic Photon Fractions
                                                            Exiting a 1 cm sample, g=0.0, s=10 mm-1, a=0 mm-1
                                                                                               µ                µ


It can be seen that as the radial threshold distance r increases, the total ballistic plus quasi-ballistic photon fraction
increases and the effective scattering ratio decreases. Thus, for a given detector sensitivity, the maximum observed
resolution can be estimated.


                                                                                                                                        Proc. of SPIE Vol. 5695   145
      The worked example with g=0.0 shows that the fraction of photons captured by a detector can be increased by
      approximately 8 orders of magnitude as the radial threshold distance r→100 m. This is clearly a much worse case than
                                                                                    µ
      tissue, where the anisotropy factor g is much higher, which would have significantly higher fractions of quasi-ballistic
      photons.

                                                          7. DISCUSSION

      Light propagation through tissue has been well studied, but analytic modeling of quasi-ballistic photon growth and
      decay is not widely available. While Monte Carlo and other numerical methods can give exact results for arbitrary
      conditions, the computer resource and time requirements are often prohibitive.

      By adopting a rigorous definition for quasi-ballistic photons, the results of the present paper can be used to determine
      approximate photon fractions for applications in which a plane slab is transmissively illuminated by a collimated beam.
      These simplified photon fractions calculations, equally applicable to both coherent and incoherent illumination, can
      subsequently be utilized to examine detector sensitivity and resolution compromises.

      While a lumped parameter model inherently introduces approximations, it is believed that the quasi-ballistic photon
      fractions predicted by such a model are sufficiently accurate to be useful for the range of scattering coefficients,
      absorption coefficients, and anisotropy values found in tissues. While the lumped parameter approximation has only
      been validated for the Henyey-Greenstein phase function, it may well be appropriate for other phase functions.

                                                         8. CONCLUSIONS

      There are many practical biomedical applications in which a sample is transmissively illuminated by a collimated beam
      and the information collected by a CCD array or equivalent photo-detector. Within these applications, this paper has
      proposed an exact definition of a quasi-ballistic photon that can be directly compared to a resolution limit and a detector
      sensitivity.

      It was found that while quasi-ballistic photon fractions do grow and decay with depth inside an isotropic scattering
      medium, the decay does not follow a simple rate equation. It was found that the quasi-ballistic fraction decay of a
      single photon model can be approximated by a lumped parameter model consisting of the contribution due to the first
      scattering event plus a contribution due to all other scattering events from a virtual source. This single photon model
      allowed the development of a straightforward formula to predict the quasi-ballistic photon fraction at an arbitrary depth.

      Some uses for the simplified quasi-ballistic photon fraction model have been explored, including the examination of the
      trade-off between detector sensitivity and resolution limit. It should be noted that a sample sensitivity/resolution
      comparison was performed at a scattering level ( s=10 mm-1) that was not feasible to calculate within the existing
                                                          µ
      Monte Carlo simulation constraints.

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