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Vascular applications of quantitative optical coherence tomography
van der Meer, F.J.
Publication date
2005
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Citation for published version (APA):
van der Meer, F. J. (2005). Vascular applications of quantitative optical coherence
tomography.
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Q U A N T I T A T I V E M E A S U R E M E N T O F
A T T E N U A T I O N C O E F F I C I E N T S O F WEAKLY
SCATTERING MEDIA USING OPTICAL
C O H E R E N C E T O M O G R A P H Y
Dirk J. Faber, Freek J. van der Meer,
Maurice C.G. Aalders, and Ton G. van Leeuwen
MODELING THE O C T SIGNAL
F
r o m c a l i b r a t e d , w e a k l y s c a t t e r i n g t i s s u e p h a n t o m s (2-6 m m " ' ) , w e e x t r a c t t h e a t t e n u a t i o n c o e f f i c i e n t w i t h a n a c c u r a c y o f Ü.8 m m ' f r o m O C T d a t a in t h e clinically relevant 'fixed f o c u s ' g e o m e t r y . T h e data are analyzed u s i n g a single scattering m o d e l and a recently d e v e l o p e d d e s c r i p t i o n o f t h e confocal p o i n t spread f u n c t i o n ( P S F ) . We verify t h e validity o f t h e single s c a t t e r i n g m o d e l by a q u a n t i t a t i v e c o m p a r i s o n w i t h a multiple scattering m o d e l , and validate the use o f t h e P S F o n the calibrated s a m p l e s . I m p l e m e n t a t i o n o f this m o d e l for existing O C T s y s t e m s will b e straightforward. L o c a l i z e d quantitative m e a s u r e m e n t o f the a t t e n u a t i o n coefficient o f different tissues can significantly i m p r o v e t h e clinical value o f O C T .1. I N T R O D U C T I O N
T h e clinical value o f O p t i c a l C o h e r e n c e T o m o g r a p h y ( O C T )1 d e p e n d s o n high i m a g i n g
s p e e d t o p r o v i d e real t i m e in vivo i m a g i n g ,2 high spatial r e s o l u t i o n t o r e s o l v e small t i s s u e
s t r u c t u r e s ,3 a n d sufficient c o n t r a s t t o d i s c r i m i n a t e b e t w e e n t h o s e s t r u c t u r e s . C o n t r a s t in
O C T images originates from differences in reflectivity o f different tissues, which are c a u s e d by their variation in (complex) refractive i n d e x n. U n f o r t u n a t e l y , c o n t r a s t is limited b e c a u s e for m o s t tissues n only ranges from 1.3 t o 1.4. L o c a l i z e d m e a s u r e m e n t o f t h e a t t e n u a t i o n coefficient// can p r o v i d e additional i n f o r m a t i o n , and may increase the clinical p o t e n t i a l o f O C T by a l l o w i n g q u a n t i t a t i v e d i s c r i m i n a t i o n b e t w e e n different tissue t y p e s .
T h e a t t e n u a t i o n coefficient c a n b e m e a s u r e d f r o m t h e O C T signal by fitting a m o d e l r e l a t i o n to this signal from a r e g i o n o f i n t e r e s t in a n O C T i m a g e . C u r r e n t l y , t w o m o d e l s are available. W i d e l y used4-''-6 is t h e s o - c a l l e d single s c a t t e r i n g m o d e l , w h i c h a s s u m e s t h a t
only light that has b e e n b a c k s c a t t e r e d o n c e c o n t r i b u t e s t o t h e O C T signal. A m o d e l t a k i n g i n t o a c c o u n t multiple scattering was i n t r o d u c e d by T h r a n e eta/, and has recently b e e n u s e d
CHAPTER 2
to extract optical properties of atherosclerotic lesions'1 and human skin'.
Focusing optics in the sample arm suppress the detection of light scattered from outside the focal volume, similar to confocal microscopy. In clinically used probes and catheters, the optical components of the sample arm are fixed. Therefore, for quantitative extraction of//, the confocal properties of the O C T system have to be taken into account,
i.e. the change of the ()CT signal with increasing distance between the probed location in
the tissue and location of the focus."'" We have recently derived a general expression for the confocal axial point spread function (PSF) for single mode fiber (SMF) based O C T systems.12 The major advantage of this PSF is that it is described by one parameter only, the Rayleigh length, which can easily be determined experimentally.
In this paper we investigate the steps necessary to extract fx from O C T images of weakly scattering non-absorbing samples. This method provides a template that can be applied to other ranges of /x . In section 2, we discuss the general principles of non-linear least squares fitting and introduce test statistics to judge the significance of the best fit values. In section 3 we establish criteria for choosing an appropriate model for the O C T signal, and proceed to choose a model for weakly scattering media using calibrated samples. Section 4 investigates the range of validity of our PSF in scattering media. Section 5 combines these results to extract the attenuation coefficient from calibrated samples, in the clinically more relevant situation of a fixed focus. Section 6 discusses implications and limitations of this study.
2. CURVE FITTING
Discrimination of different tissues based on differences in their attenuation coefficient //_ requires its accurate measurement from O C T data. This is done by defining a functional relationship between the O C T signal as a function of depth and a , and then fitting this model to a region of interest in an O C T image. The curve fitting algorithms and statistics used throughout this paper can be found in textbooks.1'1 4 Suppose we fit a model / w i t h
Madjustable parameters a.to /Vdata points (.v.,v. ± Ay). The maximum likelihood estimate
of the model parameters a is found by minimizing the quantity X2 given by:
A
^=1
yi-f{*üa\...a
M)
2CT: (2-1)
i.e. by minimizing the sum of squared, weighted, residuals. It is appropriate to use the measurement errors for weighting, i.e. O = Ay, for experimental data. For non-linear models the ^-minimization is an iterative process, implemented by the Levenberg-Marquardt method. The number of degrees of freedom {dof) of the fit is defined as N—M. To judge the significance of the best fit values of the parameters a., uncertainty estimates
MODELING Ti ii-OCT SIGNAL
o f t h e s e values p l u s s o m e g o o d n e s s - o f - f i t statistics h a v e t o b e calculated. N o t e t h a t o f t e n u n i f o r m w e i g h t i n g is u s e d , i.e. each data p o i n t is a s s i g n e d e q u a l w e i g h t in the c u r v e fitting (G = 1 in e q u a t i o n 2-1).
N e x t t o t h e b e s t e s t i m a t e s , the s t a n d a r d e r r o r o f e a c h fitted p a r a m e t e r is c a l c u l a t e d . If t h e s t a n d a r d e r r o r is small a n d t h e p a r a m e t e r is c h a n g e d a little, t h e c u r v e will fit m u c h w o r s e (i.e. higher %2). T h e m a g n i t u d e o f t h e w e i g h t s <X therefore influences the ' e l b o w r o o m '
of t h e p a r a m e t e r . C o n s e q u e n t l y , w h e n t h e s t a n d a r d e r r o r is t o be u s e d as a reliable e s t i m a t e o f t h e u n c e r t a i n t y of the fit p a r a m e t e r , w e i g h t i n g with the m e a s u r e m e n t e r r o r s is essential. F r o m t h e s t a n d a r d e r r o r s , 9 5 % c o n f i d e n c e i n t e r v a l s (c.i.) a r e calculated w h i c h a r e m o r e insightful as u n c e r t a i n t y e s t i m a t e s : if t h e fitting is r e p e a t e d o n a n o t h e r data set f r o m t h e s a m e s a m p l e , t h e b e s t fit value o f t h e p a r a m e t e r is e x p e c t e d t o fall w i t h i n this c.i. 9 5 o u t o f a 100 times. So-called p a r a m e t e r d e p e n d e n c i e s ( b e t w e e n 0 and 1) are also calculated: a value (very) close t o 1 indicates t h a t t h e fit d o e s n o t d e p e n d heavily o n the p a r a m e t e r , a n d m a y p o i n t t o o v e r - p a r a m e t e r i z a t i o n (i.e. a c h a n g e in t h e p a r a m e t e r c a n be c o m p e n s a t e d for b y c h a n g i n g the o t h e r p a r a m e t e r s ) .
N o t e t h a t a small c.i. c a n also b e f o u n d w h e n a b e s t fit d o e s n o t follow t h e d a t a v e r y well, for e x a m p l e d u e to an i n a p p r o p r i a t e m o d e l . T h e c o r r e l a t i o n coefficient R2 ( b e t w e e n 0
a n d 1) is calculated for e a c h fit. A R2 close to 1 indicates t h e b e s t fit c o m e s close t o t h e d a t a
p o i n t s . Arbitrarily, a R2> 0 . 8 is a s s u m e d as r e a s o n a b l e . H o w e v e r , a fit u s i n g physically
u n r e a l i s t i c p a r a m e t e r values can also h a v e a h i g h R2. T h e %2- m i n i m i z a t i o n ' a s s u m e s ' t h a t
the w e i g h t e d residuals have a G a u s s i a n d i s t r i b u t i o n and h a v e the s a m e s t a n d a r d d e v i a t i o n a l o n g t h e b e s t fit c u r v e ; a n d t h a t t h e d e v i a t i o n o f a p o i n t from t h e c u r v e is n o t c o r r e l a t e d t o t h e d e v i a t i o n from t h e n e x t o r p r e v i o u s p o i n t . T h e s e a s s u m p t i o n s a r e t e s t e d u s i n g a S h a p i r o - F r a n c i a t e s t and a r u n s - t e s t . T h e S h a p i r o - F r a n c i a t e s t c o m p u t e s t h e c o r r e l a t i o n coefficient W b e t w e e n a n o r m a l d i s t r i b u t i o n a n d t h e d i s t r i b u t i o n o f t h e w e i g h t e d r e s i d u -als, a n d it is i n t e r p r e t e d in t h e s a m e way as R2. T h e r u n s - t e s t c o m p a r e s the o b s e r v e d
n u m b e r o f r u n s (a series o f c o n s e c u t i v e data p o i n t s either a b o v e o r b e l o w t h e c u r v e ) to the e x p e c t e d n u m b e r o f r u n s . A g a i n arbitrary, W > 0.8 a n d p - v a l u e o f t h e r u n s t e s t pm n s > 0.05 are c o n s i d e r e d a c c e p t a b l e . T h e a s s u m p t i o n o f i n d e p e n d e n t s c a t t e r o f t h e w e i g h t e d residuals is a priori v i o l a t e d d u e t o l o w pass filtering (either in h a r d w a r e o r software) in the O C T data a c q u i s i t i o n , o r p o s s i b l e speckle a v e r a g i n g6 p r i o r to fitting. W h e n necessary, this
is d e a l t w i t h bv u s i n g e a c h ;/'th p o i n t o f t h e data set in t h e fitting, w h e r e // c o r r e s p o n d s to the n u m b e r o f p o i n t s in t h e d u r a t i o n o f t h e a v e r a g i n g t i m e o f t h e filters. If all p o i n t s are u s e d in the fit, p is d i s r e g a r d e d . All c u r v e fitting a l g o r i t h m s a n d statistical analysis a l g o r i t h m s a r e i m p l e m e n t e d in L a b V I E W 7. T h e a l g o r i t h m s w e r e v e r i f i e d against commercially available c u r v e fitting software (Microcal Origin 6.0, Microcal Software, Inc.).
3. C H O O S I N G A M O D F L FOR T H E O C T SIGNAL.
T h e m a i n q u e s t i o n in c h o o s i n g a m o d e l for t h e O C T signal is: c a n m u l t i p l e scattering effects b e i g n o r e d ? Since i m a g i n g d e p t h s generally d o n o t exceed = 1 m m , this may well be
CH \ITI R2
justified for weakly scattering media ("T < 6 mm ' for the samples used in this paper). We first compare the single scattering model4-'1,6 to a multiple scattering model using the same calibrated scattering samples with m ranging from 2 mm to 6 mm"1 described in ref. 15 and used in a similar analysis in ref. 8. The experiments are performed under the condition of dynamic focusing (i.e. the focal plane coincides with the probing location) such that the influence of the confocal properties during the depth scan is constant.
In OCT, interference between the light returning from the sample arm and the reference arm takes place only when the path length difference between both arms is matched within the coherence length of the light source (coherence gating). In the following, the O C T signal i{z) refers to the amplitude of the interference signal, and we define z = 0 at the sample interface. Ideally, z is the probing depth in the sample but the term 'location of the coherence gate in the sample' is more accurate. In the single scattering model, only light that has been backscattered once contributes to the O C T signal and the OCT signal is given by Beer's law:
/(z)cc Jexp(-2/j,z) (2-2)
T h e factor 2 accounts for round trip attenuation, the square root appears because the detector current is proportional to the sample field rather than intensity.
T h e contribution of multiple scattering to the O C T signal has been described by Thrane et al. hollowing their terminology, the O C T signal for dynamic focusing is expressed as the root mean square heterodyne signal current:
1
/ -v x 2exp(-// -)[l-exp(-// z)]
r,
v,->wf
i(z) oc exp(-2/Az) +
1V / s / L,
I V^
/ J+[ 1 - exp(-7i,z)]--4
w~ Wi
T h e last two terms under the square root describe the contribution due to multiple scattering. I lere, ws I wj = ll + [2WQ/po{z)]~ ) where ir, and ir are the 1 / e intensity radii of the p r o b e beam with and without scattering, respectively, w.is the 1/e intensity radius at the focusing lens andpnis the lateral coherence length, given by: PQ(Z)- yj3///5z(AQ/x0nm \nf'jz), where Xlt is the center wavelength of the light source, ƒ is the focal length of the objective lens and 0 s is the root-mean-square scattering angle. Note that /u^ in equation 2-3 is the
MODELING THE O C T SIGNAL
s c a t t e r i n g coefficient. F o r n o n a b s o r b i n g s a m p l e s as u s e d in this p a p e r ,"s—/^t- F o r t h e
e p o x v s a m p l e s A l - E l , t h e a t t e n u a t i o n coefficient /J.I i n c r e a s e s f r o m ~2 mm"' to ~7 m m ';
n = l .55. T h e s c a t t e r i n g a n i s o t r o p y g = 0.75 w h i c h m e a n s that e q u a t i o n 3 is slightly o u t s i d e o f the range o f validity o f small-angle forward scattering. H o w e v e r , in ref. 8 g o o d a g r e e m e n t b e t w e e n e q u a t i o n 2 - 3 a n d e x p e r i m e n t w a s f o u n d u s i n g t h e e x a c t s a m e s a m p l e s , at 1300 n m . 0 ^ in e q u a t i o n 2-2 is a p p r o x i m a t e l y g i v e n by "v (2(1 -g)) = 0 . 7 1 .
T h e S M F ( F i b e r c o r e , S M 7 5 0 , m o d e field d i a m e t e r 5.3 m m . ) b a s e d O C T s e t u p u s e d i n t h e e x p e r i m e n t s i n c l u d e s a T i r S a p p h i r e laser ( F e m t o l a s e r s , A.n = 8 0 0 n m , AX = 125 n m
F W H M ) . R e f e r e n c e m i r r o r a n d the f o c u s i n g lens in t h e s a m p l e a r m are m o u n t e d o n t w o v o i c e c o i l t r a n s l a t o r s ( Q u i c k S c a n V102.2L, Physik I n s t r u m e n t e ) . Scan s p e e d w a s 1 A -s c a n / -s . D y n a m i c r a n g e wa-s 111 d B . T h e d e t e c t o r c u r r e n t i-s d e m o d u l a t e d u -s i n g a l o c k - i n amplifier and l o w - p a s s filtered in software p r i o r t o s t o r a g e . All data a c q u i s i t i o n software is w r i t t e n in L a b V I E W 6. T h e c o l l i m a t i n g lens a n d f o c u s i n g lens in t h e s a m p l e a r m are b o t h E d m u n d O p t i c s A c h r o m a t s P 4 5 - 7 9 3 , f = 2 5 m m , N A = 0 . 0 8 . C h r o m a t i c a b e r r a t i o n e x p r e s s e d as m a x . - m i n . effective focal l e n g t h is 10 /xva in t h e b a n d w i d t h o f o u r light s o u r c e . D e p t h o f focus in air is 1 2 6 ± 6 / a n ( c o r r e s p o n d i n g t o 2 x Rayleigh l e n g t h m e a s u r e d in air). T h e lateral resolution ( d e t e r m i n e d by t h e s p o t size o f the focused s a m p l e b e a m ) is a p p r o x i m a t e l y 7 fxm. We r e c o r d e d O C T i m a g e s o f e p o x y s a m p l e s A l - E l . O C T i m a g e s c o n t a i n e d 500 A-s c a n A-s o f 4 0 9 6 p o i n t A-s ( 0 . 2 4 , « m axial, 20 u r n lateral i n c r e m e n t ) . After i m a g i n g , a 6 x 6 pixel m o v i n g average filter ( a p p r o x i m a t e l y c o r r e s p o n d i n g to 1 c o h e r e n c e length) was applied t o the data t o r e d u c e s p e c k l e . N o t e t h a t this a v e r a g i n g r e d u c e d t h e s t a n d a r d d e v i a t i o n o f t h e data by a f a c t o r v6. A larger a v e r a g i n g k e r n e l w o u l d lead t o f u r t h e r r e d u c t i o n b u t also to u n d e s i r a b l e loss o f r e s o l u t i o n . T h e a v e r a g e a n d s t a n d a r d d e v i a t i o n o f 50 A - s c a n s w a s calculated for use in t h e c u r v e fitting. A l t h o u g h a v e r a g i n g m o r e A - s c a n s w o u l d yield a s m o o t h e r d a t a set for t h e fitting, for clinical O C T i m a g e s w e e x p e c t t o b e able t o use only t h e limited n u m b e r o f 5 0 - 1 0 0 A - s c a n s for a specific tissue r e g i o n .
T h r e e different m o d e l s w e r e fit independently to t h e a v e r a g e A - s c a n s u s i n g t h e s t a n d a r d d e v i a t i o n as w e i g h t s : (1) t h e single s c a t t e r i n g m o d e l of e q u a t i o n 2-2 w i t h an a d d e d offset / and m u l t i p l i e r ^ . F o r each fit, / was fixed to the average noise level of t h e data set; A and
(x w e r e t h e free r u n n i n g p a r a m e t e r s . M o d e l (II) is b a s e d o n e q u a t i o n 2 - 3 , with a d d e d offset
/' a n d m u l t i p l i e r ^ . 0 s w a s fixed t o its t h e o r e t i c a l v a l u e a n d /'0was fixed to t h e average
n o i s e level o f the d a t a set, b u t w a s a l l o w e d t o vary if this i m p r o v e d the fit. A a n d ^r were
the free r u n n i n g p a r a m e t e r s . M o d e l (III) is the s a m e as I I , w i t h all p a r a m e t e r s /'0, A,/nt and
0 allowed to vary. F o r all m o d e l s , c o n v e r g e n c e o f t h e a l g o r i t h m is c h e c k e d by using different initial g u e s s v a l u e s for t h e m o d e l p a r a m e t e r s . I n all fits, e v e r y 2 0 ' t h data point w a s u s e d , c o r r e s p o n d i n g t o t h e r e s p o n s e t i m e o f t h e s o f t w a r e l o w - p a s s filter (see section 2).
T h e r e s u l t s o f t h e b e s t fit v a l u e s o f /<c, u s i n g m o d e l s I, I I a n d I I I are s u m m a r i z e d in
Fig. 2 - 1 . T h e e r r o r b a r s r e p r e s e n t t h e 9 5 % c o n f i d e n c e i n t e r v a l s o f t h e fitted /u^. T h e first c r i t e r i o n in c h o o s i n g a m o d e l is w h e t h e r o r n o t t h e b e s t fit v a l u e s o t t h e p a r a m e t e r s , and
CHAPTER 2
their c.i. and dependency, are physically reasonable. Both model 1 and II give physically acceptable values for the attenuation coefficient and corresponding confidence intervals. In the fits of model II to samples C l - E l the offset Z(| was allowed to vary, because it reduced
X2 compared to fitting with fixed /(|. However, this causes larger confidence intervals of ,i/. compared to model I because the fit is now not as 'tight': within the limits given bv the measurement errors, a change in ju can be compensated for to a larger extent bv a change in
A and /'0. The fits of model 111 did not yield phvsicallv possible values for 9 (which tended to unrealistically large values, effectively annihilating the multiple scattering contributions). Both the offset /. and the multiple scattering contributions in equation 2-3 result in the O C T signal approachinga constant value, with increasing depth. Therefore, variation of iQ and 0.n has the same effect, fixing iQ to the noise level of the data set did not resolve this problem. We conclude that model III does not model the O C T signal for this range of scattering coefficients, relatively low scattering anisotropv, and our measurement setup very well.
Both model I and II appear appropriate for describing the O C T signal. The goodness-of-fit of both models is judged by their Revalue which was 0.8 for sample Al and larger than 0.95 for samples Bl-F.1, for both models. Violations of the assumptions of non-linear least squares fitting are checked by judging W and the p-value of the runs-test. There is no evidence these models arc inappropriate since W' > 0.86 and p > 0.05 for all
E, 7 6 5 4 3 2 1 -0 //'J integtating sphere I B model I (eq. 2)
I I model II (eq. 3; 8,„.s fixed)
I I model III (eq. 3)
È
A1 B1 C1
Sample
D1 E1
Figure 2-1: comparison of attenuation coefficients extracted from epoxv samples Al-El using integrating sphere measurements (from ref [15]), and curve fitting using models I,II and III. The error bars represent 95% confidence intervals of the extracted fit parameter.
M( (DELING THE O C T SIGNAL
samples. Consequently, the model yielding the smallest value of fj should be chosen. By definition, variations in yf < 1 are not significant.13 A fit using a model with more parame-ters (less dof likely has smaller %2 and an F-test can assess whether this reduction in y} is worth the cost of having less dof. A comparison per sample shows that %2 is not reduced between model I and II; whereas model II has an additional fit parameter for the larger scattering coefficients. Therefore the simpler model I is chosen.
We conclude that there is no evidence against using the single scattering model I for describing the O C T signal with depth, for/^ < 6 mm"1 and our measurement setup, and we will use this model in the remainder of this paper, f o r other ranges o(/u the analysis outlined in this section should be repeated to establish the appropriate model.
4. MODELING OF THE CONFOCAL PSF
In clinically used probes and catheters, dynamic focusing is in general not possible, and the influence of the confocal point spread function on the O C T signal has to be accounted for to quantitatively extract attenuation coefficients. We have recently introduced a general expression for the PSF of single mode fiber based O C T systems. In this section, we investigate the range of validity of this expression in scattering media.
The axial confocal PSF for these OCT systems h(z) is given by:
z-zcf
{
ZR )
2 > + 1
J
Here, Zcf is the position of the confocal gate and zR is the 'apparent' Rayleigh length used to characterize the PSF. The Rayleigh length zn of a Gaussian beam is given by z =
nnof/XQ with CO the beam waist at the focus and XQ the center wavelength of the light source. The apparent Rayleigh length is related to z0 through zR = az() where a is used to distinguish specular reflection (0C=1) from diffuse reflection (a=2).1 2 This distinction is based on theoretical grounds assuming single backscattering (or more generally, assuming the beam is not distorted prior to and after backscattering). The Rayleigh length of our system measured on a mirror in air is Z= 63 ± 3 /mi.
The PSF can be measured by moving a reflector through the focus of the sample beam and recording the detector output. F.quivalently, the reflector can be held at a fixed position
z while moving the focusing lens. In OCT, we can use the coherence gate to select a
'reflector' inside a sample. To systematically evaluate the PSF for specular and diffuse reflections inside scattering media, O C T images of the samples described below were recorded for 100 different positions of the confocal gate z as illustrated in Fig. 2-2(A). From each image the average A-scan was calculated (Fig. 2-2(15); red curves represent fits as discussed in section 5 below) and the average A-scans were combined to a data set shown
CHAPTER 2
as a g r a y scale i m a g e in 2 - 2 ( C ) , w h e r e the h o r i z o n t a l axis c o r r e s p o n d s t o t h e p o s i t i o n o f t h e c o n f o c a l gate z a n d the vertical axis to t h e position of the c o h e r e n c e gate (or reflector') z. A similar d a t a set c o n t a i n i n g t h e s t a n d a r d d e v i a t i o n o f t h e O C T signal w a s also c o n s t r u c t e d for use as w e i g h t s in t h e curve fitting. E a c h r o w of this data set ( c o n s t a n t z) is t h u s p r o p o r t i o n a l t o t h e P S F at fixed p o s i t i o n o f a ' r e f l e c t o r ' inside t h e s a m p l e . F i g u r e 2 - 3 ( A ) s h o w s t h e d a t a set for e p o x v sample A l a n d t h e d o t t e d line c o r r e s p o n d s t o a 'diffuse reflector' at d e p t h z = 0.3 m m . The a p p a r e n t Rayleigh length zR is then extracted at
d i f f e r e n t d e p t h s bv fitting e q . 2 - 4 t o rows o f the data set (see fig 2-3B).
T o m e a s u r e z for s p e c u l a r reflection, a m i r r o r was placed at z — 3 /um i n s i d e d i l u t e d I n t r a l i p i d s o l u t i o n s w i t h varying,:/ a n d the P S F at t h e m i r r o r w a s e x t r a c t e d from t h e data s e t s . T h e s a m p l e s w e r e p r e p a r e d t r o m a s t o c k s o l u t i o n (n = 1.35, c h a n g e s in n d u e t o d i l u t i o n are n e g l e c t e d ) . T h e a t t e n u a t i o n coefficient of t h e s t o c k s o l u t i o n w a s d e t e r m i n e d a t / / = 4.6 ± 0.2 m m ' bv d y n a m i c focusing O C T a n d a fit u s i n g m o d e l I. I m a g e s w e r e r e c o r d e d for 100 d i f f e r e n t , fixed p o s i t i o n s of t h e c o n f o c a l g a t e (-1.0 t o 1.0 m m with r e s p e c t t o sample b o u n d a r y , m e a s u r e d in air) and 20 A-scans o f 2048 p o i n t s (axial increment 0 . 7 3 ,um, lateral i n c r e m e n t 1 0 / / m ) . Signal f r o m the m i r r o r is i n t e g r a t e d by a 50x1 m o v i n g a v e r a g e filter. E q u a t i o n 2-4 is e x p a n d e d with an offset z' and multiplier A. In t h e fitting, / w a s fixed at the average noise level; A, z and z were free r u n n i n g p a r a m e t e r s . T h e calculated s.d. o f t h e O C T d a t a is u s e d for weighting. Figure 2-3(C), b l u e d o t s , s h o w s t h e best tit v a l u e s for the a p p a r e n t Rayleigh length zR and 9 5 % c.i. as a f u n c t i o n o f t h e e x p e c t e d n u m b e r o f m e a n free p a t h s {mfp) = u^-z. T h e v a l u e o f zR at {mfp) = 0 c o r r e s p o n d s t o a
focus position zcf j ... k ... 1
depth
F i g u r e 2-2: Schematic illustration of the O C T measurement method. A: O C T images of the scattering samples are taken for different positions z of the focus in the sample (indexed i,/,k). B: From each image, the average A-scan is calculated. C: All average A-scans are combined into a data set, shown as a gray scale image, where the horizontal axis corresponds the focus position in the sample (i.e. ro the location of the confocal gate), and the vertical axis corresponds to depth (i.e. to the location of the coherence gate o r position of a 'reflector').
MODELING THE OCT SIGNAL
m e a s u r e m e n t in water. We expect zR for specular reflection in Intralipid to be
n~xzn = 85 ± 4,um which corresponds well to the experimental data. In all fits, R2 > 0.93 and W > 0.87. However, values of pn m s were all < 0.05. This is due to small asymmetry of the PSF.
To measure zR for diffuse reflection, data sets were collected for the calibrated samples A l - E l . O C T images were recorded for 100 different positions of the confocal gate (-0.33 to 1 mm; with respect to sample boundary) and contained 100 A-scans of 4096 samples (axial increment 0.73,um, lateral increment 40,urn). Equation 2-4 was fitted to rows of the data sets, i.e. at different depths inside the samples, with an offset iQ and multiplier^. All parameters are free running in the fitting. To prevent loss of spatial resolution, no speckle averaging was performed. Consequently, the s.d. of the OCT data was too large to serve as weighting factors and uniform weighting was used. As a result, the calculated 9 5 % c.i. are unreliable estimates of the accuracy of the fitting parameter (see section 2). Fits with unrealistic best fit values for the fit parameters, low R2 or W' were discarded. Consequently,
zR could be measured up to 3.5 (mfp). Data from A l - E l were combined, and the average zR and its s.d. for different (mfp) intervals was calculated. The results are shown in Fig. 2-3(C), red s q u a r e s . In all fits, R2 > 0.80 and \X" > 0.80. Most values of p w e r e
* " r u n s < 0.05 due to small asymmetry of the PSF. Measured values of z are larger than expected
based on theory (»x2Xz0 = 195 ± 9,urn). O n average, CX= 2.6 ± 0.8. About 10% of the difference can be accounted for by the samples being placed under an angle with respect to the probe beam, to avoid high excess noise levels. Oblique incidence of the beam leads to a broader waist in the sample,16 and consequently a larger zR. Furthermore, in the derivation of equation 2-4 for diffuse reflection, it is assumed that the beam is not distorted by the sample prior to, and after being reflected. Most likely this assumption is not true. T h e total signal can be split into a contribution due to single (back)scattering and multiple scattering (as is done in ref. 7). Theoretically, the first contribution can then be described using equation 2-4 and (X=2. This description also suffices if the beam is not distorted due to scattering, i.e. when multiple purely forward scattering occurs. To model any other multi-ple scattering contribution, an expression for the multimulti-ple scattered beam and its waist position should be derived, and the analysis given in appendix A of ref. 12 should be repeated. Such an expression will inherently include the phase function of the scattering particles. From the data of the previous section it seems that /';/ the focus the single scattering assumption is valid. Outside the focus the contribution of multiple scattered light to the O C T signal may be larger, which would lead to broadening of the PSF, i.e. to our larger observed apparent Rayleigh length.
For further analysis, we extracted intensity vs. depth profiles from the datasets corresponding to a fixed distance between the coherence and confocal gate (Fig. 2-4(A)), and subsequently fitted the single scattering model I to the data. In all fits, R2 > 0.8. Prior to fitting a 4x4 moving average filter was used to reduce speckle. The offset in the fitting was fixed to the average noise level. Figure 2-4(B) shows the fitted attenuation coefficients vs. offset from the focus (all 9 5 % confidence intervals < 0.1 mm"1; not plotted for clarity).
CHAPTER 2 (A, (C) E E 0.5 -0.51 0.00 0.50 1.00 1.55 z_(mm) 500-, 400' 300 % 200 N 100-1 0 . ' • 1.00-. 0.75 0.50' V
r
jy (B) -0.5 0.0 0.5 1.0 1.5 z . (mm) 2 3 4 5 6 <nfp>Figure 2-3: A: Gray scale image of recorded data set of sample Al; B: confocal PSF (dots) along dashed line in A, and best fit (solid); C, blue circles: zR for specular reflection and 95% c.i; red squares: average zR and s.d. for diffuse reflection in different (wfp) intervals. Dashed lines: expected zR for both cases.
A negative offset from the focus indicates that the coherence gate is located between the lens and the location of the confocal gate. From the samples with higher /u , e.g. CI - E l , we see the fitted attenuation coefficient is indeed less for the out-of-focus situation, indicating a larger contribution of multiply scattered light. For the other samples this effect is less distinct. The asymmetry of the curves of Fig. 2-4(13) around the zero-offset point explains the slight asymmetry found in the PSF mentioned above. These measurements indicate an increased contribution of multiple scattering in the tails of the PSF and a resulting increase of the apparent Ravleigh length. In any case, the PSF is distinctly broadened for diffuse reflection compared to specular reflection. From the curve fits we find no evidence that the PSF of equation 2-4 should be inappropriate to describe the PSF in scattering media even though a is larger then theoretically predicted. Furthermore, the off-focus fitted attenuation coefficients are within 1 mm"1 from the corresponding dynamic focusing value for all samples, which suggest that the single scattering model may still yield results of reasonable accuracy
We conclude that the combination of confocal and coherence gating to a large extent suppresses the multiple scattered part of the specularly reflected beam at up to 7 (tnfp) and of the diffuse reflected beam, e.g. at tissue interfaces, at least up to 3.5 {mfp) and expect that the combination of our 'single scattering' point spread function and the single scattering model for the O C T signal can describe the O C T signal recorded in fixed-focus geometry with reasonable accuracy.
M( )OI-l.l\C; THF. O C T SIGNAL
5. EXTRACTION OF JU_ FROM FIXED-FOCUS OCT DATA
From the results of sections 3 and 4, we expect that a model based on the PSF as given by equation 2-4, using (X=2, in combination with the single scattering model of equation 2-2 will allow extraction of attenuation coefficients of weakly scattering samples in the fixed-focus geometry with sufficient accuracy. The OCT signal as a function of depth i(z) is then proportional to the product of the PSF and single scattering model:
i(z) cc ,-2/V
(z-z^
1 ZR
• f l (2-5)
To verify equation 2-5, it was fitted to columns of the recorded data sets of the epoxy samples A l - E l , described in section 4, which are the average A-scans at a specific position of the confocal gate z inside the sample (see Fig. 2-2). Prior to constructing the data set for the present analysis, speckle in each of the OCT images was reduced by a 4x4 moving average filter (corresponding to approximately 1 coherence length). In the curve fitting, equation 2-4 was expanded with an offset / and multiplier A; /' was fixed at the average noise level, z , was fixed at its pre-set position and zn was fixed at nX2Xz = 195 mm (0t=2). A and,u are free running parameters in the fit. Standard deviation of the O C T data was used for weighting. We expect this method to be close to what will be possible for clinical images. Figure 2-5 shows an average A-scan of sample Bl and the best fit to the data (red curve). From this fit, p. — 3.92 ± 0.15 mm ' which corresponds well to the attenuation coefficient measured using dynamic focusing (jn — 3.69 + 0.37 m m ' ) . T o
(A) (B) 6.0-1 5.5- 5.0- 4.5-_~ 4.0-1 3.5-- 3.03.5-- 3.0- 2.5- 2.0-1 5 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 offset from focus (mm)
F i g u r e 2-4, A: extraction of data with a fixed distance between coherence and confocal gates. The dashed lines represent zero (1) distance i.e. corresponding to dynamic focusing, and non-7.ero (negative) distance (2). B: fitted attenuation coefficient vs distance between confocal and coherence gate for all 5 epoxv samples.
CHAPTER2
f u r t h e r i l l u s t r a t e t h e d i s t i n c t i o n b e t w e e n s p e c u l a r a n d diffuse reflection in h(z), the b l u e l i n e s h o w s a fit u s i n g zR fixed at 9 8 m m (0C=1) w h i c h gives // = 3.09 ± 0.17 mm"1. A s
e x p e c t e d , Ct=2 p r o v i d e s t h e b e t t e r tit. O t h e r e x a m p l e s are s h o w n in figure2-2(B). T h e 9 5 % c o n f i d e n c e i n t e r v a l s o f /< a r e s m a l l e r t h a n t h e c.i. o b t a i n e d for d y n a m i c f o c u s i n g in s e c t i o n 3. In t h e p r e s e n t data, 5 0 a d d i t i o n a l A - s c a n s w e r e u s e d for a v e r a g i n g , w h i c h w o u l d m a x i m a l l y r e d u c e t h e s.d. of t h e O C T data by v50 ~ 7 for fully d e v e l o p e d s p e c k l e s . Recall t h a t the size o t the m e a s u r e m e n t e r r o r s u s e d for w e i g h t i n g d e t e r m i n e s t h e s e n s i t i v i t y o f yj t o small c h a n g e s in t h e fit p a r a m e t e r , w h i c h is e x p r e s s e d in the m a g n i t u d e o f t h e c o n f i d e n c e intervals. N e v e r t h e l e s s , t h e 9 5 % c.i. may n o t r e p r e s e n t a reliable estimate o t the accuracy with which the attenuation coefficient can b e d e t e r m i n e d , because o f possible d e p e n d e n c e o f t h e fitted // o n t h e focus p o s i t i o n .
F i g u r e 2-6, left p a n e l , s h o w s t h e best tit v a l u e s o f t h e a t t e n u a t i o n coefficient for t h e fixed f o c u s g e o m e t r y vs. t h e l o c a t i o n o f the f o c u s w i t h r e s p e c t t o the s a m p l e b o u n d a r y . I n all fits, 0C=2 is u s e d . F o r all s a m p l e s , the a t t e n u a t i o n c o e f f i c i e n t is u n d e r e s t i m a t e d w h e n the f o c u s is located n e a r t h e s a m p l e boundary, and o v e r e s t i m a t e d w h e n the focus is located inside t h e sample. T h i s effect is larger for l a r g e r / / . F o r each fit, R2 > 0.8 a n d W > 0 . 8 (except
for s a m p l e A l w h i c h h a d R2 > 0.7). Since all p o i n t s o f a n a v e r a g e A - s c a n w e r e u s e d in t h e
c u r v e fitting, t h e r u n s - t e s t did n o t yield valid results (section 2), therefore the 'reasonability' o f t h e b e s t fit v a l u e s , R2 a n d \X" w e r e used a s t h e o n l y g o o d n e s s - o f - f i t criteria.
T h e a c c u r a c y o f t h e , a . m e a s u r e m e n t s r e g a r d l e s s o f t h e f o c u s p o s i t i o n is calculated by a v e r a g i n g t h e fixed f o c u s / / o v e r z ; t h e result is s h o w n in t h e right p a n e l o f Fig. 2 - 6 . F o r all s a m p l e s , t h e s.d. i n / / w a s < 0.8 m m ' . T h e c o r r e s p o n d e n c e with t h e d y n a m i c f o c u s i n g // 's s h o w n in Fig. 2-6 is excellent. Moreover, t-tests s h o w e d that the m e a n s o f the m e a s u r e d d i s t r i b u t i o n s of// w e r e all significantly different from each o t h e r ( p < 0 . 0 0 2 ) . T h i s accuracy is e x p e c t e d t o b e sufficient for d i s c r i m i n a t i o n b e t w e e n tissue s t r u c t u r e s e.g. for calcified a n d lipid l e s i o n s o f a t h e r o s c l e r o t i c tissue.1 8 T h e s y s t e m a t i c u n d e r e s t i m a t i o n of// w h e n
t h e f o c u s is l o c a t e d n e a r t h e s a m p l e b o u n d a r y c a n partly b e c a u s e d by the i n f l u e n c e o f t h e 1.00 -| ro TO c O) to
\-O O 0.00 J 1 1 1 1 1 . 1 . 1 0.00 0.25 0.50 0.75 1.00 z (mm)
F i g u r e 2-5: Average A-scan of sample B l , with the focus fixed at z = 0 . 3 mm; best fits to the data with equation 2-5 using 0C=1 and CL—2.
0 . 7 5 0.50 0 . 2 5
-Ml IDHI.ING II IF-: O C T SIGNAL 8-, 7- 6- 5- 4-3 2-1 1-0 -0.5 0.0 0.5 z., (mm) 1.0
^
rt-'"
y B1 D1 E1 —i i — i — i — • — i — > — i — • — i — • — i — • — i 1.5 2 3 4 5 6 7 8 DF M, (mm')Figure 2-6: Left panel: 'Fixed focus' attenuation coefficient vs. position of the focus in the sample (with respect to sample boundary). Right panel: 'Fixed focus' attenuation coefficient, averaged over all focus positions vs. attenuation coefFicient determined by dynamic focusing (DF). The line y=x is drawn as a guide to the eye. In all Fits, 0C=2 is used.
sample boundary itself, or may point to under-parameterization of our model. However, from the fits no statistical evidence was found that shows equation 2-5 to be inappropriate for describing the O C T signal in the fixed focus geometry, for this range of scattering coefficients and our setup.
We conclude that we can accurately retrieve the attenuation coefficients of weakly scattering samples using the single scattering model in combination with the PSF of
eq. 2-5.
6. D I S C U S S I O N AND CONCLUSION
In this studv we investigate the steps necessary for extracting the attenuation coefficient from weakly scattering samples (u < 6 mm"1) by fitting a model describing the O C T signal to a region of interest in the O C T image. For this range of attenuation coefficients, we compare a single scattering model to a model taking into account multiple scattering using OCT measurements on calibrated scattering samples. The scattering anisotropy of this samples (g=0.75) is lower than that generally found in biological tissues. The multiple scattering model is valid only for small-angle forward scattering, which may explain why the full multiple scattering model III performed worse then e.g. model I taking into account only single scattering. O n the other hand, in ref. 8 exactly the same samples were used in a similar analysis at 1300 nm; here valid results were indeed obtained. N o statistical evidence is found opposing the use of a single scattering model for this /j.t range. For larger
CHAPTER2
IJ. t h e analysis s h o u l d b e r e p e a t e d , and this will b e t h e s u b j e c t o f f u r t h e r r e s e a r c h . It is
i m p o r t a n t t o n o t e t h a t c h o i c e s in the fitting p r o c e d u r e (e.g. size o f s p e c k l e r e d u c t i o n k e r n e l , n u m b e r of A - s c a n s for averaging) i n f l u e n c e the c o n f i d e n c e intervals of the b e s t fit v a l u e s . We c h o o s e t h e s e p a r a m e t e r s in a c c o r d a n c e w i t h w h a t we e x p e c t t o be r e l e v a n t for clinical i m a g e s ; i n c r e a s i n g k e r n e l size or n u m b e r o f A - s c a n s w o u l d lead t o u n a c c e p t a b l e l o s s o f r e s o l u t i o n o r i m a g e s p e e d .
I n clinical p r a c t i c e , d y n a m i c focusing will n o t be p o s s i b l e a n d t h e i n f l u e n c e o f t h e f o c u s i n g o p t i c s o n t h e O C T signal h a s to b e a c c o u n t e d for. We have recently i n t r o d u c e d a d e s c r i p t i o n o f t h e axial c o n f o c a l p o i n t spread f u n c t i o n o f single m o d e fiber b a s e d O C T s y s t e m s . T h e m a j o r a d v a n t a g e o f this e x p r e s s i o n is t h a t it is c h a r a c t e r i z e d by o n e single p a r a m e t e r , t h e Rayleigh l e n g t h , w h i c h for e x i s t i n g p r o b e s a n d c a t h e t e r s can b e easily m e a s u r e d . T h e r e f o r e , i m p l e m e n t a t i o n of t h i s P S F in d a t a - a n a l y s i s a l g o r i t h m s for clinical s y s t e m s will b e s t r a i g h t f o r w a r d . O u r model p r e d i c t s different a p p a r e n t Rayleigh l e n g t h s , c h a r a c t e r i z e d by t h e p a r a m e t e r a , for specular (Ot=l) a n d diffuse (0t=2) r e f l e c t i o n . T h i s d i f f e r e n c e is e x p e r i m e n t a l l y verified; however, the Rayleigh length for diffuse reflection was larger t h a n e x p e c t e d (OC=2.6±0.8). P a r t of t h e difference ( a b o u t 10%) is c a u s e d by the fact t h a t t h e s a m p l e is placed u n d e r an angle with r e s p e c t to t h e b e a m . Analysis o f o u t - o f - f o c u s d a t a i n d i c a t e s an i n c r e a s e d c o n t r i b u t i o n of m u l t i p l y s c a t t e r e d light t o the tails o f t h e PSF, w h i c h also leads t o b r o a d e n i n g . A m o r e c o m p r e h e n s i v e m o d e l o f the P S F would therefore h a v e t o t a k e i n t o a c c o u n t 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 s a m p l e , and m o r e specifically t h e p h a s e f u n c t i o n o f t h e s c a t t e r e r s . T h i s will b e subject o f f u r t h e r study. We n o t e h e r e t h a t t h e d e r i v a t i o n o f e q . 2-5 is b a s e d o n G a u s s i a n o p t i c s w h i c h is only valid in the paraxial a p p r o x i m a t i o n (low N A ) a n d o n t h e a s s u m p t i o n t h a t t h e p r o b e b e a m is n o t d i s t o r t e d by t h e t i s s u e . O t h e r d e s c r i p t i o n s for t h e confocal P S F t o r O C T s y s t e m s h a v e b e e n given in l i t e r a t u r e and w e r e d i s c u s s e d in ref. 12. In m o s t r e p o r t e d P S F ' s n o d i s t i n c t i o n is m a d e b e t w e e n specular a n d diffuse reflection. This d i s t i n c t i o n is m a d e in t h e O C T signal analysis o f T h r a n e et a/.7, h o w e v e r c o n t r a r y t o our findings, the P S F for diffuse reflection is n o t
b r o a d e n e d a c c o r d i n g t o t h i s t h e o r y ; i.e. in t e r m s o f o u r m o d e l , (X~ 1 for b o t h diffuse a n d s p e c u l a r reflection.
C o m b i n i n g t h e single s c a t t e r i n g model a n d this axial P S F allows e x t r a c t i o n o f t h e a t t e n u a t i o n c o e f f i c i e n t o f c a l i b r a t e d samples at d i f f e r e n t fixed p o s i t i o n s o f t h e focus i n s i d e t h e scattering m e d i u m . T h e accuracy with w h i c h the a t t e n u a t i o n coefficient could be d e t e r m i n e d , r e g a r d l e s s o f t h e f o c u s position, is a p p r o x i m a t e l y 0.8 m m '. T h e p r e c i s i o n o f t h e i n d i v i d u a l m e a s u r e m e n t s (i.e. d e f i n e d as 9 5 % c o n f i d e n c e i n t e r v a l s in this p a p e r ) is m u c h h i g h e r and in g e n e r a l less t h e n 1 0 % , c o m p a r a b l e to for e x a m p l e ref. 11 a n d ref. 17. T h e P S F and t h e O C T signal d e s c r i p t i o n are i n d e p e n d e n t in o u r m o d e l . T h i s allows e a s y d e t e r m i n a t i o n o f JX in multi-layered tissue; i.e. in small r e g i o n s o f i n t e r e s t in t h e O C T i m a g e . I n m o d e l s b a s e d o n e.g. e q u a t i o n 2-3 this e x t r a c t i o n m a y b e less s t r a i g h t f o r w a r d . In clinical i m a g e s , we e x p e c t t o h a v e a b o u t 50-100 A-scans ( c o r r e s p o n d i n g t o a certain tissue) available for averaging. T h e r e f o r e , in the experiments w e use t h e same n u m b e r for averaging.
M ( )DFJ.1N( i Tl IE O C T SIGNAL
Clinical implications
Preliminary studies indicate we can discriminate between lipid rich and calcified lesions in ex vivo images of human atherosclerotic tissue due to a 3-fold lower attenuation coefficient of the former. This identification was not possible based on gray levels and structural appearance in the O C T image alone. Figure 2-7 shows a typical example of the samples used in that study. The thick grey line represents the average of 100 A-scans corresponding to the highlighted area in the O C T image in the inset. The arrow marks the known position of the focus. The attenuation coefficient of regions of interest (identified as intima and a lipid rich lesion in histology) was fitted using equation 2-4 and the procedu-res outlined in this paper. The extracted attenuation coefficients for this sample are indicated in the figure, including 95% confidence intervals of the fitted/z.
In conclusion, we have shown that a single scattering model accurately retrieves attenuation coefficients for dynamic focusing OCT data up to 7 mm'1; we have verified our expression for the confocal PSF at specular and diffuse reflection inside scattering media; and have shown that for attenuation coefficients up to 6 mm"1, our PSF with a single scattering model can extract// with an accuracy of about 0.8 mm ' for the clinically relevant fixed focus geometry. This PSF can easily be implemented for existing clinical OCT systems which may allow quantitative discrimination between different tissues in vivo, and therefore increase the clinical potential of OCT.
oo Intima: H, = 5.6 + 0.1 mm-' Lipid-rich: u, = 2.7 ± 0.8 mm-1
"Si
X
v
0.4 0.6 0.1 depth (mm)Figure 2-7: Average OCT A-scan data (thin gray line) of the region depicted in the OCT image in the inset; and the fitted signal using equation 2-4 (thick black lines) and corresponding attenuation coefficients for the inimal region and a lipid rich region. The arrow shows the location of the focus in the sample.
CHAPTER 2
A c K N ( ) W L E D G E M H N T S
This work is part of the research programme of the 'Stichting voor Fundamenteel Onderzoek der Materie' (FOM), which is financially supported by the 'Nederlandse Orga-nisatie voor Wetenschappelijk Onderzoek' (NW'O). We acknowledge the use of the epoxy samples, part of the Furopean framework "Medphot" (QLG1-CT-2000-01464). Part of this research is sponsered by the Netherlands Heart Foundation (grant 99.199). We acknowledge the Interuniversity Cardiology Institute of the Netherlands ( I O N ) for financial support.
MODELING TUI; OCT SIGNAL
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