Comparison of modelling methods and why Monte Carlo was selected

I I₀exp N i 1 µ(i) ali

regions are l₁, l₂... l_N. Therefore, it is not necessary to repeat calculations if only the absorption coefficient is changed, as long as the path lengths in the regions are known. However, any change of scattering coefficient or geometry requires a separate simulation.

The temporal profile of the exiting light is simple to calculate, since the time a photon would take to arrive at a position is simply given by

1.57 t 1 c N i 1 n_il_i

Since research often leads one to ask questions that were originally not considered, it is generally best to save as much information about the photon paths as possible, even if no immediate need for such data is known, since it requires very little extra time to record the information and disk space is relatively cheap nowadays. Once the information is collected, selecting only relevant data from all that is collected is a simple task. However, if one vital piece of information, that is later found to be necessary, was not recorded, it becomes necessary to repeat the simulations which may take months or even years of CPU time.

1.5.5 Comparison of modelling methods and why Monte Carlo was selected.

The Kubelka Munk theory has been shown to be fundamentally flawed and so while it is simple, it can not be expected to be very accurate.

The diffusion equation is an excellent predictor of light propagation in tissue, but analytical solutions, except for the simplest of geometries, are impossible to obtain. However, when coupled to a FEM model to obtain numerical values, it is very powerful. However, it can not be expected to be valid until the light has become diffuse, which is at a distance of approximately 1/µs′.

Although Monte Carlo modelling is a very slow technique, it really is the only acceptable method when data is required close to the source and detector, where other models just break down. In addition, the Monte Carlo method can deal with arbitrarily complex measurement geometries and can even take account of the detector geometry and spatial sensitivity characteristics. Hence for this project it was decided a Monte Carlo model would be

developed to accurately model the expected TPSF. By fitting the measured TPSF’s against the predicted TPSF using the accurate Monte Carlo model, it should be possible to determine the optical properties of the material measured.

For this project to work it is obviously essential, that it is possible to estimate µs′from a measurement of the light distribution. This is known as an inverse problem, as opposed to a forward problem of calculating the light distribution, given knowledge of the tissue optical properties. Bohren and Huffman compare the inverse problem to one of estimating the size and shape of a dragon, from its footsteps, which is difficult. The forward problem, of estimating the size and shape of a dragon’s footprints, given that you know what the dragon looks like, is easier. There are numerous papers which solve the forward problem of calculating light distributions, given the optical properties. A few authors have attempted the inverse problem of the sort described here and two specific examples are worth further discussion here.

The first by Farrell et al55starts from an analytical expression for the diffuse reflectance Rsat a radius r from a source incident on a semi infinite medium (see figure 1.7), which was a complicated non-linear equation of the form:

The authors developed a method, that whilst not exactly solving the problem that needs to be

1.58 R(r) f(r,µ_eff,µt)

tackled here, gives a hint of how the problem may be solved. Equation 1.58, like equation 1.39 is derived from the diffusion equation, but whereas equation 1.39 assumes a short pulse of light, equation 1.58, assumes a constant level of illumination, and as such has no time-dependence. However, instead of using intensity as a function of time, 800 sets of data of reflected intensity Rs(r) as a function of radius r for various tissue optical parameters typical of mammalian tissue were generated using equation 1.58. Random noise was then added, to simulate typical in vivo measurements. The noisy data was then fitted to equation 1.58, using a non-linear least squares fitting algorithm to estimate the tissue optical parameters µeff andµt and henceµa andµs′. The RMS error on the estimate of the accuracy of the optical properties was 10%, with the algorithm taking 4 s per calculation. A neural network algorithm was then written, to perform the same task of estimating the tissue parameters. The neural network was trained with 800 sets of noisy data, which took two hours. After training, the neural network was some 400 times faster than the non-linear least squares fit, estimatingµaandµs′in 10 ms, with an RMS error of 7% on the optical properties. It should be noted that the computer used by the authors was said to be a MicroVAX 1170, although no such computer exists! However, a modern PC is considerably

more powerful than any of the MicroVAX’s, so today it will probably be unnecessary to use a neural network purely for speed of computation, although the increase of accuracy may warrant it. However, although Farrell et al do state that although the RMS error is smaller with the neural network, it is not significantly smaller, so even this benefit is debatable.

With the complex geometries involved in this project, it is unlikely that an analytical expression of the form in equation 1.58 will ever be found, but it would be easy to generate several hundred data sets for training a neural network, using the Monte Carlo computer program. Once this is done, the optical properties of tissue could be estimated with the aid of the neural network. It should be noted that the method used by Farrell to derive the optical properties is a simple measurement of continuous reflectance versus distance, not time-dependent data as would be generated with this instrument. However, there is every reason to believe the same principles could be applied. Instead of supplying Rs(r) as inputs to the neural network, we would supply ψ(r,t). As such, it is believed that assuming the time dependent data can be measured, estimating the optical properties of the tissue from the data should not be too difficult. However, fitting the full TPSF can be expected to take longer than fitting just the reflectance. Marquet et al56 _{and Bevilacqua et al}57 _{have both described a method of estimating} µ

a

Laser diode

Sample

CCD array x

Received power / Source power

x (mm) Typical output intensity

(M)

FWHM

Figure 1.10 System developed for estimating µa and µs′ by measuring two parameters M and

FWHM.

andµ_s′ from measurements made with the hardware shown in figure 1.10. This uses a sample of tissue in a cuvette with a laser diode (λ=670 nm) illuminating the tissue. The transmitted light intensity was detected on a CCD camera and the maximum intensity at the peak M and the full width at half-maximum of the intensity (in mm) was recorded with a CCD camera. The expected maximum intensity and full width at half maximum were then estimated using a Monte Carlo

model for various tissue optical properties, assuming a geometry identical to the measurement system. By comparing the measured data to the Monte Carlo data, they could estimate bothµ_a andµs′directly. This was said to have an accuracy of 5% forµaand 20%µs′, although for some tissue types, especially brain white matter, the technique fails to work satisfactorily.

The basic method of comparing observed data to Monte Carlo simulations could in principle be extended to this work, although the form of the data (peak signal and full width at half maximum) will of course be different. Perhaps the full width at half max, mean time, or skew of the TPSF could be compared.