## Quantitative measurement of attenuation coefficients of weakly scattering media using optical coherence tomography

Optics Express, Vol. 12, Issue 19, pp. 4353-4365 (2004)

http://dx.doi.org/10.1364/OPEX.12.004353

Acrobat PDF (1328 KB)

### Abstract

From calibrated, weakly scattering tissue phantoms (2–6 mm^{-1}), we extract the attenuation coefficient with an accuracy of 0.8 mm^{-1} from OCT data in the clinically relevant ‘fixed focus’ geometry. The data are analyzed using a single scattering model and a recently developed description of the confocal point spread function (PSF). We verify the validity of the single scattering model by a quantitative comparison with a multiple scattering model, and validate the use of the PSF on the calibrated samples. Implementation of this model for existing OCT systems will be straightforward. Localized quantitative measurement of the attenuation coefficient of different tissues can significantly improve the clinical value of OCT.

© 2004 Optical Society of America

## 1. Introduction

1. D. Huang, E. A. Swanson, C.P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C.A. Puliafito, and J.G. Fujimoto, “Optical coherence tomography,” Science **254**, 1178–1181 (1991). [CrossRef] [PubMed]

*in vivo*imaging [2

2. N. Nassif, B. Cense, B.H. Park, S.H. Yun, T.C. Chen, B.E. Bouma, G.J. Tearney, and J.F. de Boer, “In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography,”Opt. Lett , **29**, 480–482 (2004). [CrossRef] [PubMed]

3. A. Unterhuber, B. Povazay, B. Hermann, H. Sattmann, W. Drexler, V. Yakovlev, G. Tempea, C. Schubert, E. M. Anger, P. K. Ahnelt, M. Stur, J. E. Morgan, A. Cowey, G. Jung, T. Le, and A. Stingl, “Compact, low-cost TiAl 2 O 3 laser for in vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett, **28**, 905–907 (2003). [CrossRef]

*n*. Unfortunately, contrast is limited because for most tissues

*n*only ranges from 1.3 to 1.4. Localized measurement of the attenuation coefficient μ

_{t}can provide additional information, and may increase the clinical potential of OCT by allowing quantitative discrimination between different tissue types.

4. J. M. Schmitt, A. Knüttel, M. Yadlowsky, and M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. **39**, 1705–1720 (1994). [CrossRef] [PubMed]

5. R. O. Esenaliev, K. V. Larin, I. V. Larina, and M. Motamedi, “Noninvasive monitoring of glucose concentration with optical coherence tomography,” Opt. Lett. **26**, 992–994 (2001). [CrossRef]

6. A.I. Kholodnykh, I.Y. Petrova, K.V. Larin, M. Motamedi, and R.O. Esenaliev, “Precision of Measurement of Tissue Optical Properties with Optical Coherence Tomography”, Appl. Opt. **42**, 3027–3037 (2003). [CrossRef] [PubMed]

*et al*[7

7. L. Thrane, H. T. Yura, and P. E. Andersen, “Analysis of optical coherence tomography systems based on the extended HuygensFresnel principle,” J. Opt. Soc. Am. A **17**, 484–490 (2000). [CrossRef]

8. D. Levitz, L. Thrane, M.H. Frosz, P.E. Andersen, C.B. Andersen, S. Andersson-Engels, J. Valanciunaite, J. Swartling, and P.R. Hansen, “Determination of optical scattering properties of highly-scattering media in optical coherence tomography images,” Opt.Express **12**, 249–259 (2004). [CrossRef] [PubMed]

_{t}, the confocal properties of the OCT system have to be taken into account, i.e., the change of the OCT signal with increasing distance between the probed location in the tissue and location of the focus [10

10. J. A. Izatt, M.R. Hee, G.M. Owen, E.A. Swanson, and J.G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. **19**, 590–592 (1994). [CrossRef] [PubMed]

11. A.I. Kholodnykh, I.Y. Petrova, M. Motamedi, and R.O. Esenaliev, “Accurate measurement of total attenuation coefficient of thin tissue with optical coherence tomography”, IEEE J. Sel. Top. Quantum Electron. **9**, 210–221 (2003) [CrossRef]

12. T.G. van Leeuwen, D.J. Faber, and M.C. Aalders, IEEE J. Sel. Top. Quantum Electron. **9**, 227–233 (2003). [CrossRef]

_{t}from OCT images of weakly scattering non-absorbing samples. This method provides a template that can be applied to other ranges of μ

_{t}. 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 OCT 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

_{t}requires its accurate measurement from OCT data. This is done by defining a functional relationship between the OCT signal as a function of depth and μ

_{t}, and then fitting this model to a region of interest in an OCT image. The curve fitting algorithms and statistics used throughout this paper can be found in textbooks [13,14]. Suppose we fit a model

*f*with

*M*adjustable parameters

*a*

_{j}to

*N*data points (

*x*

_{i},

*y*

_{i}±Δy

_{i}). The maximum likelihood estimate of the model parameters

*a*

_{j}is found by minimizing the quantity

*χ*

^{2}given by:

_{i}= Δy

_{i}, for experimental data. For non-linear models the

*χ*

^{2}-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*

_{j}, uncertainty estimates of these values plus some goodness-of-fit statistics have to be calculated. Note that often uniform weighting is used, i.e., each data point is assigned equal weight in the curve fitting (σ

_{i}= 1 in equation 1).

*χ*

^{2}). The magnitude of the weights σ

_{i}therefore influences the ‘elbowroom’ of the parameter. Consequently, when the standard error is to be used as a reliable estimate of the uncertainty of the fit parameter, weighting with the measurement errors is essential. From the standard errors, 95% confidence intervals (c.i.) are calculated which are more insightful as uncertainty estimates: if the fitting is repeated on another data set from the same sample, the best fit value of the parameter is expected to fall within this c.i. 95 out of a 100 times. So-called parameter dependencies (between 0 and 1) are also calculated: a value (very) close to 1 indicates that the fit does not depend heavily on the parameter, and may point to over-parameterization (i.e. a change in the parameter can be compensated for by changing the other parameters).

## 3. Choosing a model for the OCT signal.

_{t}< 6 mm

^{-1}for the samples used in this paper). We first compare the single scattering model [4

4. J. M. Schmitt, A. Knüttel, M. Yadlowsky, and M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. **39**, 1705–1720 (1994). [CrossRef] [PubMed]

5. R. O. Esenaliev, K. V. Larin, I. V. Larina, and M. Motamedi, “Noninvasive monitoring of glucose concentration with optical coherence tomography,” Opt. Lett. **26**, 992–994 (2001). [CrossRef]

6. A.I. Kholodnykh, I.Y. Petrova, K.V. Larin, M. Motamedi, and R.O. Esenaliev, “Precision of Measurement of Tissue Optical Properties with Optical Coherence Tomography”, Appl. Opt. **42**, 3027–3037 (2003). [CrossRef] [PubMed]

7. L. Thrane, H. T. Yura, and P. E. Andersen, “Analysis of optical coherence tomography systems based on the extended HuygensFresnel principle,” J. Opt. Soc. Am. A **17**, 484–490 (2000). [CrossRef]

_{t}ranging from 2 mm

^{-1}to 6 mm

^{-1}described in [15

15. J. Swartling, J. S. Dam, and S. Andersson-Engels, “Comparison of Spatially and Temporally Resolved Diffuse-Reflectance Measurement Systems for Determination of Biomedical Optical Properties” Appl. Opt. **42**, 4612–4620 (2003). [CrossRef] [PubMed]

8. D. Levitz, L. Thrane, M.H. Frosz, P.E. Andersen, C.B. Andersen, S. Andersson-Engels, J. Valanciunaite, J. Swartling, and P.R. Hansen, “Determination of optical scattering properties of highly-scattering media in optical coherence tomography images,” Opt.Express **12**, 249–259 (2004). [CrossRef] [PubMed]

*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 OCT signal and the OCT signal is given by Beer’s law:

*et al*[7

7. L. Thrane, H. T. Yura, and P. E. Andersen, “Analysis of optical coherence tomography systems based on the extended HuygensFresnel principle,” J. Opt. Soc. Am. A **17**, 484–490 (2000). [CrossRef]

*w*

_{0}/

*ρ*

_{0}(

*z*)]

^{2}) where

*w*

_{h}and

*w*

_{s}are the 1/e intensity radii of the probe beam with and without scattering, respectively,

*w*

_{0}is the 1/e intensity radius at the focusing lens and

*ρ*

_{0}is the lateral coherence length, given by:

*λ*

_{0}is the center wavelength of the light source,

*f*is the focal length of the objective lens and θ

_{rms}is the root-mean-square scattering angle. Note that μ

_{s}in equation 3 is the scattering coefficient. For non absorbing samples as used in this paper μ

_{s}=μ

_{t}. For the epoxy samples A1-E1, the attenuation coefficient μ

_{t}increases from ≈2 mm

^{-1}to ≈7 mm

^{-1}; n=1.55. The scattering anisotropy g = 0.75 which means that Eq. (3) is slightly outside of the range of validity of small-angle forward scattering. However, in [8

8. D. Levitz, L. Thrane, M.H. Frosz, P.E. Andersen, C.B. Andersen, S. Andersson-Engels, J. Valanciunaite, J. Swartling, and P.R. Hansen, “Determination of optical scattering properties of highly-scattering media in optical coherence tomography images,” Opt.Express **12**, 249–259 (2004). [CrossRef] [PubMed]

_{rms}in Eq. (2) is approximately given by √(2(1-g)) = 0.71.

_{0}= 800 nm, Δλ= 125 nm FWHM). Reference mirror and the focusing lens in the sample arm are mounted on two voice-coil translators (QuickScan V-102.2L, Physik Instrumente). Scan speed was 1 A-scan/s. Dynamic range was 111 dB. The detector current is demodulated using a lock-in amplifier and low-pass filtered in software prior to storage. All data acquisition software is written in LabVIEW 6. The collimating lens and focusing lens in the sample arm are both Edmund Optics Achromats P45-793, f=25 mm, NA=0.08. Chromatic aberration expressed as max.-min. effective focal length is 10 μm in the bandwidth of our light source. Depth of focus in air is 126±6 μm (corresponding to 2 × Rayleigh length measured in air). The lateral resolution (determined by the spot size of the focused sample beam) is appropimately 7 μm. We recorded OCT images of epoxy samples A1-E1. OCT images contained 500 A-scans of 4096 points (0.24 μm axial, 20 μm lateral increment). After imaging, a 6x6 pixel moving average filter (approximately corresponding to 1 coherence length) was applied to the data to reduce speckle. Note that this averaging reduced the standard deviation of the data by a factor √6. A larger averaging kernel would lead to further reduction but also to undesirable loss of resolution. The average and standard deviation of 50 A-scans was calculated for use in the curve fitting. Although averaging more A-scans would yield a smoother data set for the fitting, for clinical OCT images we expect to be able to use only the limited number of 50–100 A-scans for a specific tissue region.

*independently*to the average A-scans using the standard deviation as weights: (I) the single scattering model of Eq. (2) with an added offset i

_{0}and multiplier A. For each fit, i

_{0}was fixed to the average noise level of the data set; A and μ

_{t}were the free running parameters. Model (II) is based on Eq. (3), with added offset i

_{0}and multiplier A. θ

_{rms}was fixed to its theoretical value and i

_{0}was fixed to the average noise level of the data set, but was allowed to vary if this improved the fit. A and μ

_{t}were the free running parameters. Model (III) is the same as II, with all parameters i

_{0}, A, μ

_{t}and θ

_{rms}allowed to vary. For all models, convergence of the algorithm is checked by using different initial guess values for the model parameters. In all fits, every 20’th data point was used, corresponding to the response time of the software low-pass filter (see Section 2).

_{t}. The first criterion in choosing a model is whether or not the best fit values of the parameters, and their c.i. and dependency, are physically reasonable. Both model I and II give physically acceptable values for the attenuation coefficient and corresponding confidence intervals. In the fits of model II to samples C1-E1 the offset i

_{0}was allowed to vary, because it reduced

*χ*

^{2}compared to fitting with fixed i

_{0}. However, this causes larger confidence intervals of μ

_{t}compared to model I because the fit is now not as ‘tight’: within the limits given by the measurement errors, a change in μ

_{t}can be compensated for to a larger extent by a change in A and i

_{0}. The fits of model III did not yield physically possible values for θ

_{rms}(which tended to unrealistically large values, effectively annihilating the multiple scattering contributions). Both the offset i

_{0}and the multiple scattering contributions in Eq. (3) result in the OCT signal approaching a constant value, with increasing depth. Therefore, variation of i

_{0}and θ

_{rms}has the same effect. Fixing i

_{0}to the noise level of the data set did not resolve this problem. We conclude that model III does not model the OCT signal for this range of scattering coefficients, relatively low scattering anisotropy, and our measurement setup very well.

^{2}value which was 0.8 for sample A1 and larger than 0.95 for samples B1-E1, 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 are inappropriate since W’ > 0.86 and p

_{runs}> 0.05 for all samples. Consequently, the model yielding the smallest value of

*χ*

^{2}should be chosen. By definition, variations in

*χ*

^{2}< 1 are not significant [13]. A fit using a model with more parameters (less

*dof*) likely has smaller

*χ*

^{2}and an F-test can assess whether this reduction in

*χ*

^{2}is worth the cost of having less

*dof*. A comparison per sample shows that

*χ*

^{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.

_{t}< 6 mm

^{-1}and our measurement setup, and we will use this model in the remainder of this paper. For other ranges of μ

_{t}the analysis outlined in this section should be repeated to establish the appropriate model.

## 4. Modeling of the confocal PSF

*h*(

*z*) is given by:

*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, OCT images of the samples described below were recorded for 100 different positions of the confocal gate

*z*

_{cf}as illustrated in Fig. 2(a). From each image the average A-scan was calculated (Fig. 2(b); red curves represent fits as discussed in Section 5 below) and the average A-scans were combined to a data set shown as a gray scale image in Fig. 2(c), where the horizontal axis corresponds to the position of the confocal gate

*z*

_{cf}and the vertical axis to the position of the coherence gate (or ‘reflector’)

*z*. A similar data set containing the standard deviation of the OCT signal was also constructed for use as weights in the curve fitting. Each row of this data set (constant

*z*) is thus proportional to the PSF at fixed position of a ‘reflector’ inside the sample. Figure 3(a) shows the data set for epoxy sample A1 and the dotted line corresponds to a ‘diffuse reflector’ at depth

*z*= 0.3 mm. The apparent Rayleigh length

*z*

_{R}is then extracted at different depths by fitting eq. 4 to rows of the data set (see Fig 3(b)).

*z*

_{R}for specular reflection, a mirror was placed at

*z*= 3 mm inside diluted Intralipid solutions with varying μ

_{t}and the PSF at the mirror was extracted from the data sets. The samples were prepared from a stock solution (n = 1.35, changes in n due to dilution are neglected). The attenuation coefficient of the stock solution was determined at μ

_{t}= 4.6 ± 0.2 mm

^{-1}by dynamic focusing OCT and a fit using model I. Images were recorded for 100 different, fixed positions of the confocal gate (-1.0 to 1.0 mm with respect to sample boundary, measured in air) and 20 A-scans of 2048 points (axial increment 0.73 μm, lateral increment 10 μm). Signal from the mirror is integrated by a 50×1 moving average filter. Equation (4) is expanded with an offset i0 and multiplier A. In the fitting, i

_{0}was fixed at the average noise level; A, z

_{R}and z

_{cf}were free running parameters. The calculated s.d. of the OCT data is used for weighting.

_{R}and 95% c.i. as a function of the expected number of mean free paths 〈

*mfp*〉 = μ

_{t}×

*z*. The value of z

_{R}at 〈

*mfp*〉 = 0 corresponds to a measurement in water. We expect z

_{R}for specular reflection in Intralipid to be

*n*×z

_{0}= 85 + 4 μm which corresponds well to the experimental data. In all fits, R

^{2}> 0.93 and W’ > 0.87. However, values of p

_{runs}were all < 0.05. This is due to small asymmetry of the PSF.

*z*

_{R}for diffuse reflection, data sets were collected for the calibrated samples A1-E1. OCT 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 μm, lateral increment 40 μm). Equation (4) was fitted to rows of the data sets, i.e. at different depths inside the samples, with an offset i

_{0}and multiplier A. 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 95% 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 R

^{2}or W’ were discarded. Consequently,

*z*

_{R}could be measured up to 3.5 〈

*mfp*〉. Data from A1-E1 were combined, and the average

*z*

_{R}and its s.d. for different 〈

*mfp*〉 intervals was calculated. The results are shown in Fig. 3(c), red squares. In all fits, R

^{2}> 0.80 and W’ > 0.80. Most values of p

_{runs}were < 0.05 due to small asymmetry of the PSF. Measured values of z

_{R}are larger than expected based on theory (

*n*×2×z

_{0}= 195 + 9 μm). On average, α= 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

16. G.A. Massey and A.E. Siegman, “Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,” Appl. Opt. **8**, 975–978 (1969) [CrossRef] [PubMed]

*z*

_{R}. Furthermore, in the derivation of Eq. (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. The total signal can be split into a contribution due to single (back)scattering and multiple scattering (as is done in [7

**17**, 484–490 (2000). [CrossRef]

12. T.G. van Leeuwen, D.J. Faber, and M.C. Aalders, IEEE J. Sel. Top. Quantum Electron. **9**, 227–233 (2003). [CrossRef]

*in*the focus the single scattering assumption is valid. Outside the focus the contribution of multiple scattered light to the OCT signal may be larger, which would lead to broadening of the PSF, i.e., to our larger observed apparent Rayleigh length.

^{2}> 0.8. Prior to fitting a 4×4 moving average filter was used to reduce speckle. The offset in the fitting was fixed to the average noise level. Figure 4(b) shows the fitted attenuation coefficients vs. offset from the focus (all 95% confidence intervals < 0.1 mm

^{-1}; not plotted for clarity). 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 μ

_{t}, e.g., C1 - E1, 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. 4(b) 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 Rayleigh 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 Eq. (4) should be inappropriate to describe the PSF in scattering media even though α 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.

*mfp*〉 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 OCT signal can describe the OCT signal recorded in fixed-focus geometry with reasonable accuracy.

## 5. Extraction of μ_{t} from fixed-focus OCT data

*i*(

*z*) is then proportional to the product of the PSF and single scattering model:

*z*

_{cf}inside the sample (see Fig. 2). Prior to constructing the data set for the present analysis, speckle in each of the OCT images was reduced by a 4×4 moving average filter (corresponding to approximately 1 coherence length). In the curve fitting, Eq. (4) was expanded with an offset i

_{0}and multiplier A; i

_{0}was fixed at the average noise level,

*z*

_{cf}was fixed at its pre-set position and

*z*

_{R}was fixed at n×2×z

_{0}= 195 μm (α=2). A and μ

_{t}are free running parameters in the fit. Standard deviation of the OCT data was used for weighting. We expect this method to be close to what will be possible for clinical images. Figure 5 shows an average A-scan of sample B1 and the best fit to the data (red curve). From this fit, μ

_{t}= 3.92 ± 0.15 mm

^{-1}which corresponds well to the attenuation coefficient measured using dynamic focusing (μ

_{t}= 3.69 ± 0.37 mm

^{-1}). To further illustrate the distinction between specular and diffuse reflection in

*h*(

*z*), the blue line shows a fit using

*z*

_{R}fixed at 98 μm (α=1) which gives μ

_{t}= 3.09 ± 0.17 mm

^{-1}. As expected, α=2 provides the better fit. Other examples are shown in Fig. 2(b).

_{t}are smaller than the c.i. obtained for dynamic focusing in Section 3. In the present data, 50 additional A-scans were used for averaging, which would maximally reduce the s.d. of the OCT data by √50 ≈ 7 for fully developed speckles. Recall that the size of the measurement errors used for weighting determines the sensitivity of

*χ*

^{2}to small changes in the fit parameter, which is expressed in the magnitude of the confidence intervals. Nevertheless, the 95% c.i. may not represent a reliable estimate of the accuracy with which the attenuation coefficient can be determined, because of possible dependence of the fitted μ

_{t}on the focus position.

^{2}> 0.8 and W’>0.8 (except for sample A1 which had R

^{2}> 0.7). Since all points of an average A-scan were used in the curve fitting, the runs-test did not yield valid results (Section 2), therefore the ‘reasonability’ of the best fit values, R

^{2}and W’ were used as the only goodness-of-fit criteria.

_{t}measurements regardless of the focus position is calculated by averaging the fixed focus μ

_{t}over z

_{cf}; the result is shown in the right panel of Fig. 6. For all samples, the s.d. in μ

_{t}was < 0.8 mm

^{-1}. The correspondence with the dynamic focusing μ

_{t}’s shown in Fig. 6 is excellent. Moreover, t-tests showed that the means of the measured distributions of μt were all significantly different from each other (p<0.002). This accuracy is expected to be sufficient for discrimination between tissue structures, e.g., for calcified and lipid lesions of atherosclerotic tissue [0]. The systematic underestimation of μ

_{t}when the focus is located near the sample boundary can partly be caused by the influence of the sample boundary itself, or may point to under-parameterization of our model. However, from the fits no statistical evidence was found that shows Eq. (5) to be inappropriate for describing the OCT signal in the fixed focus geometry, for this range of scattering coefficients and our setup.

## 6. Discussion and conclusion

_{t}< 6 mm

^{-1}) by fitting a model describing the OCT signal to a region of interest in the OCT 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. On the other hand, in Ref. [8

**12**, 249–259 (2004). [CrossRef] [PubMed]

_{t}range. For larger μ

_{t}the analysis should be repeated, and this will be the subject of further research. It is important to note that choices in the fitting procedure (e.g., size of speckle reduction kernel, number of A-scans for averaging) influence the confidence intervals of the best fit values. We choose these parameters in accordance with what we expect to be relevant for clinical images; increasing kernel size or number of A-scans would lead to unacceptable loss of resolution or image speed.

12. T.G. van Leeuwen, D.J. Faber, and M.C. Aalders, IEEE J. Sel. Top. Quantum Electron. **9**, 227–233 (2003). [CrossRef]

*et al*[7

**17**, 484–490 (2000). [CrossRef]

^{-1}. The precision of the individual measurements (i.e., defined as 95% confidence intervals in this paper) is much higher and in general less then 10%, comparable to for example [11

11. A.I. Kholodnykh, I.Y. Petrova, M. Motamedi, and R.O. Esenaliev, “Accurate measurement of total attenuation coefficient of thin tissue with optical coherence tomography”, IEEE J. Sel. Top. Quantum Electron. **9**, 210–221 (2003) [CrossRef]

17. J.M. Schmitt, A. Knüttel, and R.F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt **32**(30), 6032. [PubMed]

### Clinical implications

*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 OCT image alone [18

18. F.J. van der Meer, D.J. Faber, D.M. Baraznji Sassoon, M.C. Aalders, G. Pasterkamp, and T.G. van Leeuwen, “Localized measurement of optical attenuation coefficients of atherosclerotic plaque constituents by quantitative optical coherence tomography”, *submitted to IEEE transactions on Medical Imaging, 2004*..

_{t}.

^{-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 μ

_{t}with an accuracy of about 0.8 mm

^{-1}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.

## Acknowledgments

## References and links

1. | D. Huang, E. A. Swanson, C.P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C.A. Puliafito, and J.G. Fujimoto, “Optical coherence tomography,” Science |

2. | N. Nassif, B. Cense, B.H. Park, S.H. Yun, T.C. Chen, B.E. Bouma, G.J. Tearney, and J.F. de Boer, “In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography,”Opt. Lett , |

3. | A. Unterhuber, B. Povazay, B. Hermann, H. Sattmann, W. Drexler, V. Yakovlev, G. Tempea, C. Schubert, E. M. Anger, P. K. Ahnelt, M. Stur, J. E. Morgan, A. Cowey, G. Jung, T. Le, and A. Stingl, “Compact, low-cost TiAl 2 O 3 laser for in vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett, |

4. | J. M. Schmitt, A. Knüttel, M. Yadlowsky, and M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. |

5. | R. O. Esenaliev, K. V. Larin, I. V. Larina, and M. Motamedi, “Noninvasive monitoring of glucose concentration with optical coherence tomography,” Opt. Lett. |

6. | A.I. Kholodnykh, I.Y. Petrova, K.V. Larin, M. Motamedi, and R.O. Esenaliev, “Precision of Measurement of Tissue Optical Properties with Optical Coherence Tomography”, Appl. Opt. |

7. | L. Thrane, H. T. Yura, and P. E. Andersen, “Analysis of optical coherence tomography systems based on the extended HuygensFresnel principle,” J. Opt. Soc. Am. A |

8. | D. Levitz, L. Thrane, M.H. Frosz, P.E. Andersen, C.B. Andersen, S. Andersson-Engels, J. Valanciunaite, J. Swartling, and P.R. Hansen, “Determination of optical scattering properties of highly-scattering media in optical coherence tomography images,” Opt.Express |

9. | A. Knuettel, S. Bonev, and W. Knaak, “New method for evaluation of |

10. | J. A. Izatt, M.R. Hee, G.M. Owen, E.A. Swanson, and J.G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. |

11. | A.I. Kholodnykh, I.Y. Petrova, M. Motamedi, and R.O. Esenaliev, “Accurate measurement of total attenuation coefficient of thin tissue with optical coherence tomography”, IEEE J. Sel. Top. Quantum Electron. |

12. | T.G. van Leeuwen, D.J. Faber, and M.C. Aalders, IEEE J. Sel. Top. Quantum Electron. |

13. | W. H. Press, “Numerical Recipes” (Cambridge University Press, Cambridge,1986). |

14. | D.G. Altman, “Practical statistics for medical research” (Chapman&Hall, London,1991). |

15. | J. Swartling, J. S. Dam, and S. Andersson-Engels, “Comparison of Spatially and Temporally Resolved Diffuse-Reflectance Measurement Systems for Determination of Biomedical Optical Properties” Appl. Opt. |

16. | G.A. Massey and A.E. Siegman, “Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,” Appl. Opt. |

17. | J.M. Schmitt, A. Knüttel, and R.F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt |

18. | F.J. van der Meer, D.J. Faber, D.M. Baraznji Sassoon, M.C. Aalders, G. Pasterkamp, and T.G. van Leeuwen, “Localized measurement of optical attenuation coefficients of atherosclerotic plaque constituents by quantitative optical coherence tomography”, |

**OCIS Codes**

(110.4500) Imaging systems : Optical coherence tomography

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(170.4500) Medical optics and biotechnology : Optical coherence tomography

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 28, 2004

Revised Manuscript: August 30, 2004

Published: September 20, 2004

**Citation**

Dirk Faber, Freek van der Meer, Maurice Aalders, and Ton van Leeuwen, "Quantitative measurement of attenuation coefficients of weakly scattering media using optical coherence tomography," Opt. Express **12**, 4353-4365 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-19-4353

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### References

- D. Huang, E. A. Swanson, C.P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C.A. Puliafito, J.G. Fujimoto, �??Optical coherence tomography,�?? Science 254, 1178-1181 (1991). [CrossRef] [PubMed]
- N. Nassif, B. Cense, B.H. Park, S.H. Yun, T.C. Chen, B.E. Bouma, G.J. Tearney, J.F. de Boer, �??In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography,�?? Opt. Lett, 29, 480-482 (2004). [CrossRef] [PubMed]
- A. Unterhuber, B. Povazay, B. Hermann, H. Sattmann, W. Drexler, V. Yakovlev, G. Tempea, C. Schubert, E. M. Anger, P. K. Ahnelt, M. Stur, J. E. Morgan, A. Cowey, G. Jung, T. Le, A. Stingl, �??Compact, low-cost TiAl 2 O 3 laser for in vivo ultrahigh-resolution optical coherence tomography,�?? Opt. Lett, 28, 905-907 (2003). [CrossRef]
- J. M. Schmitt, A. Knüttel, M. Yadlowsky, and M. A. Eckhaus, �??Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,�?? Phys. Med. Biol. 39, 1705-1720 (1994). [CrossRef] [PubMed]
- R. O. Esenaliev, K. V. Larin, I. V. Larina, and M. Motamedi, �??Noninvasive monitoring of glucose concentration with optical coherence tomography,�?? Opt. Lett. 26, 992-994 (2001). [CrossRef]
- A.I. Kholodnykh, I.Y. Petrova, K.V. Larin, M. Motamedi, R.O. Esenaliev, �??Precision of Measurement of Tissue Optical Properties with Optical Coherence Tomography�??, Appl. Opt. 42, 3027-3037 (2003). [CrossRef] [PubMed]
- L. Thrane, H. T. Yura, and P. E. Andersen, �??Analysis of optical coherence tomography systems based on the extended HuygensFresnel principle,�?? J. Opt. Soc. Am. A 17, 484-490 (2000). [CrossRef]
- D. Levitz, L. Thrane, M.H. Frosz, P.E. Andersen, C.B. Andersen, S. Andersson-Engels, J. Valanciunaite, J. Swartling, P.R. Hansen, �??Determination of optical scattering properties of highly-scattering media in optical coherence tomography images,�?? Opt. Express 12, 249-259 (2004). [CrossRef] [PubMed]
- A. Knuettel, S. Bonev, W. Knaak, �??New method for evaluation of in vivo scattering and refractive index properties obtained with optical coherence tomography,�?? J. Biomed. Opt. 9, 232-273 (2004).
- J. A. Izatt, M.R. Hee, G.M. Owen, E.A. Swanson, J.G. ujimoto, �??Optical coherence microscopy in scattering media,�?? Opt. Lett. 19, 590-592 (1994). [CrossRef] [PubMed]
- A.I.Kholodnykh, I.Y. Petrova, M. Motamedi, R.O. Esenaliev, �??Accurate measurement of total attenuation coefficient of thin tissue with optical coherence tomography�??, IEEE J. Sel. Top. Quantum Electron. 9, 210- 221 (2003) [CrossRef]
- T.G. van Leeuwen, D.J. Faber, M.C. Aalders, IEEE J. Sel. Top. Quantum Electron. 9, 227-233 (2003). [CrossRef]
- W. H. Press, �??Numerical Recipes�?? (Cambridge University Press, Cambridge, 1986).
- D.G. Altman, �??Practical statistics for medical research�?? (Chapman&Hall, London, 1991).
- J. Swartling, J. S. Dam, and S. Andersson-Engels, �??Comparison of Spatially and Temporally Resolved Diffuse-Reflectance Measurement Systems for Determination of Biomedical Optical Properties�?? Appl. Opt. 42, 4612-4620 (2003). [CrossRef] [PubMed]
- G.A. Massey, A.E. Siegman, �??Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,�?? Appl. Opt. 8, 975-978 (1969) [CrossRef] [PubMed]
- J.M. Schmitt, A. Knüttel, R.F. Bonner, �??Measurement of optical properties of biological tissues by low-coherence reflectometry,�?? Appl. Opt 32(30), 6032. [PubMed]
- F.J. van der Meer, D.J. Faber, D.M. Baraznji Sassoon, M.C. Aalders, G. Pasterkamp, T.G. van Leeuwen, �??Localized measurement of optical attenuation coefficients of atherosclerotic plaque constituents by quantitative optical coherence tomography�??, submitted to IEEE transactions on Medical Imaging, 2004..

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