Comparison of modelling methods and why Monte Carlo was selected

The Kubelka M unk 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 l / |i / .

Although M onte 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

57

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 p./ from a measurement of the light distribution. This is known as an inverse problem, as opposed to a

forw ard 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 forw ard 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 forw ard 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 al55 starts from an analytical expression for the diffuse reflectance Rs at 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:

R(r)=j{r,\leff,\lt ) 1 5 8

The authors developed a method, that whilst not exactly solving the problem that needs to be 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 p eff and p., and hence fia and p./. 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, estimating p a and p / 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 modem PC is considerably

58 m o r e p o w e r f u l t h a n a n y o f t h e M i c r o V A X ’s , s o t o d a y it w i l l p r o b a b l y b e u n n e c e s s a r y t o u s e a n e u r a l n e t w o r k p u r e l y f o r s p e e d o f c o m p u t a t i o n , a l t h o u g h t h e i n c r e a s e o f a c c u r a c y m a y w a r r a n t it. H o w e v e r , a l t h o u g h F a r r e l l e t a l d o s t a t e t h a t a l t h o u g h t h e R M S e r r o r is s m a l l e r w i t h t h e n e u r a l n e t w o r k , it is n o t s i g n i f i c a n t l y s m a l l e r , s o e v e n t h i s b e n e f i t is d e b a t a b l e . W i t h t h e c o m p l e x g e o m e t r i e s i n v o l v e d in t h i s p r o j e c t , it is u n l i k e l y t h a t a n a n a l y t i c a l e x p r e s s i o n o f t h e f o r m in e q u a t i o n 1 . 5 8 w i l l e v e r b e f o u n d , b u t it w o u l d b e e a s y t o g e n e r a t e s e v e r a l h u n d r e d d a t a s e t s f o r t r a i n i n g a n e u r a l n e t w o r k , u s i n g t h e M o n t e C a r l o c o m p u t e r p r o g r a m . O n c e t h i s i s d o n e , t h e o p t i c a l p r o p e r t i e s o f t i s s u e c o u l d b e e s t i m a t e d w i t h t h e a i d o f t h e n e u r a l n e t w o r k . It s h o u l d b e n o t e d t h a t t h e m e t h o d u s e d b y F a r r e l l t o d e r i v e t h e o p t i c a l p r o p e r t i e s is a s i m p l e m e a s u r e m e n t o f c o n t i n u o u s r e f l e c t a n c e v e r s u s d i s t a n c e , n o t t i m e - d e p e n d e n t d a t a a s w o u l d b e g e n e r a t e d w i t h t h i s i n s t r u m e n t . H o w e v e r , t h e r e i s e v e r y r e a s o n t o b e l i e v e t h e s a m e p r i n c i p l e s c o u l d b e a p p l i e d . I n s t e a d o f s u p p l y i n g R s(r ) a s i n p u t s t o t h e n e u r a l n e t w o r k , w e w o u l d s u p p l y Y|/(r,t). A s s u c h , it is b e l i e v e d t h a t a s s u m i n g t h e t i m e d e p e n d e n t d a t a c a n b e m e a s u r e d , e s t i m a t i n g t h e o p t i c a l p r o p e r t i e s o f t h e t i s s u e f r o m t h e d a t a s h o u l d n o t b e t o o d i f f i c u l t . H o w e v e r , f i t t i n g t h e f u l l T P S F c a n b e e x p e c t e d t o t a k e l o n g e r t h a n f i t t i n g j u s t t h e r e f l e c t a n c e . M a r q u e t e t a l 56 a n d B e v i l a c q u a e t a l 57 h a v e b o t h d e s c r i b e d a m e t h o d o f e s t i m a t i n g p., F W H M L a s e r d i o d e Figure 1.10 S y s t e m d e v e l o p e d f o r e s t i m a t i n g p., a n d p / b y m e a s u r i n g t w o p a r a m e t e r s M a n d F W H M . C C D a rr a y I T y p i c a l o u t p u t i n t e n s i t y x ( m m ) a n d p / f r o m m e a s u r e m e n t s m a d e w i t h t h e h a r d w a r e s h o w n in f i g u r e 1 . 1 0 . T h i s u s e s a s a m p l e o f t i s s u e i n a c u v e t t e w i t h a l a s e r d i o d e ( A , = 6 7 0 n m ) i l l u m i n a t i n g t h e t i s s u e . T h e t r a n s m i t t e d l i g h t i n t e n s i t y w a s d e t e c t e d o n a C C D c a m e r a a n d t h e m a x i m u m i n t e n s i t y at t h e p e a k M a n d t h e f u l l w i d t h at h a l f - m a x i m u m o f t h e i n t e n s i t y ( i n m m ) w a s r e c o r d e d w i t h a C C D c a m e r a . T h e e x p e c t e d m a x i m u m i n t e n s i t y a n d f u l l w i d t h a t h a l f m a x i m u m w e r e t h e n e s t i m a t e d u s i n g a M o n t e C a r l o

59

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 both p a and p s' directly. This was said to have an accuracy of 5% for p a and 20% p s', 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.

1.6 Hardware for finding jLia and jlls without measuring the TPSF.

Before discussing how the TPSF can be measured then analysed to obtain p a and p s, it should be realised there are other ways to obtain the tissue optical parameters. One method, that of a CW based CCD system, was described in the last section, but others will now be considered. It is especially important to consider these alternatives, as time-domain methods are probably the more complex, and therefore may not be justified.