Comparison of tissue dispersion measurement techniques based on optical coherence tomography

Christos Photiou; Costas Pitris

doi:10.1117/1.JBO.24.4.046003

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25 April 2019 Comparison of tissue dispersion measurement techniques based on optical coherence tomography

Christos Photiou; Costas Pitris

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J. of Biomedical Optics, 24(4), 046003 (2019). https://doi.org/10.1117/1.JBO.24.4.046003

Abstract

The effects of dispersion on optical coherence tomography (OCT) images have long been documented. The imbalance of spectral broadening, caused by dispersion mismatches in the two arms of the OCT interferometer, can result in significant resolution degradation. Efforts to correct this phenomenon have resulted in improved image quality using various techniques. However, dispersion is also present and varies in tissues. As a result, group velocity dispersion (GVD) can be used to detect changes in tissues and provide useful information for diagnosis. Several methods can be utilized to measure the GVD from OCT images: (i) the degradation of the point spread function (PSF), (ii) the shift (walk-off) between images taken at different wavelengths, (iii) the changes in the second derivative of the spectral phase, as well as two new methods, which do not require a reflector and are applicable in intact tissues, i.e., using (iv) the speckle degradation, and (v) the speckle cross correlation. A systematic, experimental, evaluation of these methods is presented to elucidate the capabilities, the limitations, and the accuracy of each technique when attempting to estimate the GVD in scattering samples. The most precise values were obtained from the estimation of the PSF degradation, whereas using the phase derivative method was only applicable to minimally scattering samples. Speckle broadening appears to be the most robust method for tissue GVD measurements.

1. Introduction

Dispersion, a result of wavelength-dependent index of refraction variations, causes pulse-width broadening with detrimental effects in many pulsed-laser applications. It is also considered to be one of the major causes of resolution degradation in optical coherence tomography (OCT) imaging and, thus much effort has been expended over the years to develop techniques to counteract its effects. As a result, several dispersion compensation methods have been developed. Conventional methods rely on placing the right amount of dispersion balancing material in one interferometer arm of the OCT setup to counteract the effects of the dispersion in the other arm. Initially, dispersion balancing was achieved by a fused-silica prism pair is inserted with faces contacted and index matched to form a variable-thickness window in the reference arm. However, this method is usually only practical for second-order dispersion.¹^–³ Grating-based phase delay scanners can also be used for second-order dispersion compensation. Using the rapid-scanning optical delay, one can adjust dispersion by displacing the diffraction grating from the focal plane of the lens to achieve transform-limited interferogram profiles.⁴ Dual optical fiber stretchers can also be used for dispersion compensation while affording some degree of tunability. Using two fiber stretchers made up of different fiber types, an all-fiber tunable dispersion compensator can be implemented with the delay and the dispersion in the two arms of the interferometer can be adjusted independently. However, these approaches require bulky equipment and high voltage so they are not easy to use.⁵ Recently, a fiber-stretching-based dispersion compensator has been combined with a grating-based, time domain OCT system to compensate for second- and third-order dispersion, but the system becomes increasingly complicated.⁶ Numerical dispersion compensation, based on the use of the fractional Fourier transform (FT), is also possible providing new perspectives on the nature and role of group-velocity dispersion in Fourier domain OCT.⁷

Even after an OCT interferometer is optimized, dispersion differences are still present due to the dispersive properties of the material of tissues that are imaged. Since this dispersion is specific to the sample that is causing the effect, the concept of extracting useful material information from the OCT signal has emerged.⁸^,⁹ The idea of using dispersion, as a source of contrast, is not new. For example, there have already been reports, in the literature, of using the dispersion of biomolecules to quantify their concentration. For example, the dispersion of hemoglobin was used to extract the concentration of hemoglobin in intact red blood cell.¹⁰ The relation between dispersion and biochemical composition was further demonstrated using quantitative dispersion microscopy, which has confirmed that the dispersion of live HeLa cells agrees well with the dispersion measured for pure proteins solutions.¹¹ Variations in the dispersion of different types of normal skin have also been identified in vivo with coherent reflection measurements of different skin types.¹² Given the dramatic changes in cellular biochemistry caused by cancer,¹³ which are discernible by other optical techniques such as Raman spectroscopy,¹⁴ it is highly likely that dispersion can also be used as a contrast mechanism in OCT imaging of early cancer and result in more accurate disease diagnosis. However, if dispersion is to be used for quantitative measurements, it is very important to know the capabilities and limitations of the different methods of measuring dispersion with OCT. In this paper, we compare different techniques for estimating the group velocity dispersion (GVD) from OCT images, in order to evaluate their accuracy and applicability to highly scattering media such as tissues.

2. Theory and Methods

There are several methods described in the literature on how to measure GVD from the OCT signals or images. They depend on resolution degradation, image feature walk-off shift, and phase differences (Fig. 1). In addition, two recently developed methods for dispersion estimation in scattering tissues, without the need of a reflector or a strong distinct scatterer, were also included.⁸^,¹⁵ Each is described further in the following sections along with the mathematical framework for the GVD calculations for each case. Since the index of refraction of the material was needed for some of the calculations, this was measured using the method described by Tearney et al.¹⁶

Fig. 1

Effects of dispersion: (a) resolution degradation, (b) walk-off shift, and (c) phase difference.

2.1.

Resolution Degradation

Dispersion causes degradation of the image point spread function (PSF) and, therefore, of the image resolution. The resolution degradation is evident in the pulse width of the broadened Gaussian resolution envelope $t_{d}$ . If the width of the original envelope $t_{τ}$ and the sample thickness $L$ are known, or can be measured, the GVD can be estimated. The original Gaussian resolution envelope width is related to the pulse FWHM by

Eq. (1)

t_{τ} = \frac{t_{FWHM}}{1.665}

and increases with sample thickness

L

according to¹⁷^,¹⁸

Eq. (2)

t_{d}^{2} = t_{τ}^{2} [1 + {(\frac{L \cdot GVD}{t_{τ}^{2}})}^{2}] .

Thus the GVD is given by

Eq. (3)

GVD = \sqrt{\frac{t_{d}^{2} t_{τ}^{2} - t_{τ}^{4}}{L^{2}}} .

Experimentally, the original envelope width

t_{τ}

can be measured from a free-space portion of the reflector (i.e., not covered by the sample) and the degraded width

t_{d}

from the reflector peak behind the tissue. The sample thickness is measured as the distance between the sample top surface and the plane of the free-space portion of the reflector.

2.2.

Walk-Off Shift

When dispersion is present, beams at different wavelengths perceive different path lengths. The result is in an apparent shift in the OCT images taken at different center wavelengths, an effect called “walk-off.” The mathematical relationship between the shift $Δ z$ and GVD is derived below starting with

Eq. (4)

GVD = \frac{1}{c} \frac{Δ n}{Δ ω},

where

c

is the speed of light,

n

is the index of refraction, and

ω

is the optical frequency. The differential walk-off

Δ z

between two spectrally separated independent source spectra as a function of the index of refraction is related to the difference in index of refraction by

Eq. (5)

Δ n = \frac{Δ z}{L},

where

L

is the sample thickness. Thus the GVD can be written as

Eq. (6)

GVD = \frac{Δ z}{c L Δ ω} .

Given that

ω = 2 π f = \frac{2 π c}{λ} \Rightarrow Δ ω = \frac{2 π c Δ λ}{λ_{0}^{2}},

Eq. (7)

GVD = \frac{Δ z λ_{0}^{2}}{2 π c^{2} L Δ λ},

where

Δ z

is the differential walk-off in the images acquired from two sources,

Δ λ

is the source bandwidth,

λ_{0}

is the center wavelength, and

L

is the sample thickness.¹⁸^,¹⁹

For the experimental verification, a single broad source can be used. After acquisition of the interferometric signal, the spectrum is split into two by multiplication with two shifted Gaussian envelopes. Subsequently, two OCT images are formed, corresponding to the two different spectra, from which the walk-off is measured. This approach results in images with reduced resolution but it assures that the OCT images are acquired at exactly the same location and are, therefore, comparable.

2.3.

Phase Difference

In spectral interferometry, dispersion can be estimated from the interference spectrum produced by two time-delayed beams. The interferometric signal, for a given time or phase delay, can be expressed as

Eq. (8)

I (ω) = {| E_{0} (ω) |}^{2} + {| E (ω) |}^{2} + f (ω) \exp (i ω τ) + f * (ω) \exp (- i ω τ) .

The delay here is due to index of refraction variations and appears as a phase shift between different wavelengths. Hence, the GVD can be estimated from the phase changes of the spectrum of the OCT signals. Note that

Eq. (9)

f (ω) = I {f (t)} = | E_{0}^{*} (ω) E (ω) | \exp [i Δ φ (ω)]

includes the phase information on the spectral phase difference as

Δ φ (ω) = \arg [f (ω)]

. To derive the phase difference

Δ φ (ω)

, the inverse FT of

I (ω)

is performed and the following relationship is obtained:

Eq. (10)

I^{- 1} [I (ω)] = E_{0}^{*} (- t) \otimes E_{0} (t) + E^{*} (- t) \otimes E (t) + f (t - τ) + f {(- t - τ)}^{*} .

Then an FT is applied to the component

f (t - τ)

to transfer it back to the spectral domain and the complex amplitude becomes

Eq. (11)

f (ω) = | E_{0} (ω) | | E (ω) | \exp [i Δ φ (ω) + ω τ] .

The phase of this complex amplitude minus the linear delay part

ω τ

yields to the spectral phase difference between the two beams. Finally, the phase frequency first derivative yields the group delay

Eq. (12)

GD (ω) = - \frac{\partial (Δ φ)}{\partial (ω)}

and, finally, the GVD is given by the second derivative of the spectrum phase as²⁰^–²²

Eq. (13)

GVD (ω) = - \frac{1}{L} \frac{\partial^{2} (Δ φ)}{\partial^{2} (ω)} .

Experimentally, the spectrum from a single reflector, positioned below the sample, can be extracted by isolating the signal of that particular peak from the real part of the FT of the interferogram (i.e., multiplying the real A-scan with a Gaussian envelope centered at the peak), and then applying an inverse FT to get the spectrum. The second derivative of that spectrum provides the GVD as a function of wavelength. To compare this solution with the results of the previous methods, the average GVD can be calculated.

2.4.

Using the Speckle Degradation to Estimate the GVD

The methods described above are very difficult to apply in vivo and are limited only to samples where strong and distinct reflectors are present. A developed method based on speckle width degradation does not require such reflectors and can thus be applied in vivo.

This method uses the dispersion induced change in the speckle size to estimate the image PSF broadening and then calculate the GVD.⁸ However, speckle variations are difficult to estimate due to the randomness of the speckle signal. To do so, a portion of an OCT image that contains speckle, at depth $z$ from the sample surface, denoted as $i_{s} (z)$ , is related to a similar portion of the OCT image from the sample surface, i.e., $i_{s} (0)$ , by a depth-dependent speckle-degrading impulse response $sdf (z)$ such that

Eq. (14)

i_{s} (z) = sdf (z) * i_{s} (0),

where * is the convolution of the two terms and

z

is the depth, which takes the values of

z = d_{0}

,

2 d_{0}, \dots, L

,

d_{0}

being the system resolution. To estimate the impulse response

sdf (z)

, a Wiener-type minimization is used.²³ For that purpose, the following least mean square error function

ε (z)

is defined

Eq. (15)

ε (z) = E {{| i_{s} (z) - sdf (z) * i_{s} (0) |}^{2}},

where

E

denotes expectation. Minimizing the error function

ε (z)

, using a Wiener deconvolution approach, results in an estimate for

sdf (z)

. In analogy to

sdf (z)

, there exists another impulse response

rdf (z)

of a similar form, which describes the dispersion-induced degradation of the system resolution. For calculating the GVD, it is not necessary to explicitly derive the rdf since only its width is required, which can be estimated from

Eq. (16)

d_{rdf} (z) = d_{o} \frac{d_{sdf} (z)}{d_{sdf} (0)},

where

d_{0}

is the system resolution and

d_{sdf} (z)

is the width of the sdf at depth

z

. The result of the convolution of the rdf with the OCT image is a degraded image with resolution width

d_{d} (z)

given by

Eq. (17)

d_{d} (z) = \sqrt{{(d_{0})}^{2} + {[d_{rdf} (z)]}^{2}},

since the convolution of two Gaussians, the psf and the rdf, results also in a Gaussian with a width that is the root mean square (rms) of the widths of the two original functions. Given this width

d_{d}

, the GVD can be calculated using Eq. (3).

2.5.

Using the Cross Correlation of Speckle to Estimate the GVD

Another approach to estimate the GVD, without the need for strong distinct reflectors, is the use of the speckle cross correlation to estimate the walk-off shift between OCT images at different wavelengths. The walk-off shift can be estimated from the cross correlation of A-scans from corresponding regions of the two half-spectrum images at different center wavelengths created as described in Sec. 2.2 [Fig. 2(a)]. The cross correlation of corresponding A-scans is calculated and the first peak, after the zero lag, in the cross correlation is detected [Figs. 2(b) and 2(c)]. The walk-off shift is estimated from the distance of the peak from the zero lag location [Fig. 2(c)] and the GVD is calculated using the Eq. (7).

Fig. 2

(a) Images reconstructed from the half spectra (red and green), (b) corresponding A-scans from the two half spectra images (red and green) indicating the lag, in which there is a correlation peak after the 0 lag, and (c) the cross correlation of the corresponding A-scans.

3. Experimental Methods

For the experimental verification and comparison of the various GVD estimation methods, a swept source OCT system, with a center wavelength of 1300 nm and a resolution of $12 μ m$ in air, was used. Samples of various glasses, a collagen gel (2 g of collagen in 10 ml of water let to solidify and then cut into slices) as well as fresh ex vivo sections of porcine muscle and porcine adipose tissue were used. All the samples were placed over a reflector, which served as a reference for the actual thickness and system resolution measurements, and eight images ( $5 mm \times 4 mm$ ) were acquired from different regions for each type of sample.

The glass samples were chosen so that they spanned a wide range of GVD values. Their well-characterized properties were used to verify the validity of the OCT techniques. The glass dimensions were 12.5-mm diameter and 2-mm thickness in all cases. The biological samples were chosen based on their scattering properties, ranging from minimal scattering (collagen) to very scattering (adipose tissue). They were sliced manually so their thickness varied between 1.5 to 2.5 mm. Fortunately, since the actual thickness was measured for each lateral location (see Sec. 4.1), the thickness variation was included in the GVD calculation and did not affect the comparison. In Fig. 3, a list of all the samples with photographs as well as the values of their index of refraction and GVD, obtained from the literature, is provided.

Fig. 3

Samples used for GVD estimation with photographs and references values for n and GVD.²⁴^–²⁷

The OCT data were processed in MATLAB^TM. Initially, an automated algorithm detected the top surface of the sample as well as the mirror peak location in free-space and below the tissue. The tissue thickness, index of refraction, and mirror reflection Gaussian widths were subsequently calculated assuming a Gaussian envelope shape. From those measurements, the GVD was estimated by combining the values from 250 A-scans in each image. The GVD of each sample was taken as the median of those 250 values. The standard deviation of the GVD values obtained from the eight images of each sample, as well as the mean error between the estimated and measured GVD were used as indicators of the precision and accuracy of each of the different methods described above. Furthermore, the standard deviation of the GVD measurements of single A-scans within an image was used as a measure of each technique’s robustness. Given that value, the minimum number of measurements that must be averaged in order to get a GVD estimate with an error $E$ of 10% or less with a confidence level $α$ of 95% was calculated by

Eq. (18)

n = {(\frac{Z_{α / 2} σ}{E})}^{2},

where

α = 0.05

,

E = 0.1

,

σ

is the standard deviation, and

Z_{α / 2} = 1.96

is the critical value of the normal distribution at

α / 2

. The results are compared in the following sections, to evaluate the precision, accuracy, and robustness of each method.

4. Experimental Results

4.1.

Imaging Results

Figure 4 shows typical OCT images of the samples utilized in this study. Using the reflector location (blue lines) as reference, the sample thickness and the group index of refraction, at each lateral location, were calculated using the technique described in the literature,¹⁶ i.e.,

Eq. (19)

n = \frac{L + L^{'}}{L} .

The signal-to-noise-ratio (SNR), defined as the maximum sample intensity divided by the rms noise of the background, was an average of 70 dB with a 7.4-dB standard deviation for the original images and 68 dB with a 6.5-dB standard deviation for the half-spectrum images. One concern might be that the walk-off shift estimation might be affected by the lower SNR. However, a closer look at the relationship between the error in the GVD estimation and SNR reveals that there is no correlation between the two, at least not in the range of 60 to 80 dB (Fig. 5).

Fig. 4

Typical OCT images used in this study: (a) KBr glass, (b) collagen gel, (c) porcine muscle, and (d) porcine adipose tissue, over a reflector. $L$ is the actual sample thickness, from top surface (green) to the level of the reflector (blue). $L$ ’ is the path-length difference, relative to air, because of the sample.

Fig. 5

Scatter plot of percentage error versus SNR showing no correlation between increasing SNR and reducing percentage error.