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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 21 — Oct. 8, 2012
  • pp: 23442–23455
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Improved Fourier-based characterization of intracellular fractal features

Joanna Xylas, Kyle P. Quinn, Martin Hunter, and Irene Georgakoudi  »View Author Affiliations


Optics Express, Vol. 20, Issue 21, pp. 23442-23455 (2012)
http://dx.doi.org/10.1364/OE.20.023442


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Abstract

A novel Fourier-based image analysis method for measuring fractal features is presented which can significantly reduce artifacts due to non-fractal edge effects. The technique is broadly applicable to the quantitative characterization of internal morphology (texture) of image features with well-defined borders. In this study, we explore the capacity of this method for quantitative assessment of intracellular fractal morphology of mitochondrial networks in images of normal and diseased (precancerous) epithelial tissues. Using a combination of simulated fractal images and endogenous two-photon excited fluorescence (TPEF) microscopy, our method is shown to more accurately characterize the exponent of the high-frequency power spectral density (PSD) of these images in the presence of artifacts that arise due to cellular and nuclear borders.

© 2012 OSA

1. Introduction

The analysis of biomedical images is critical for detection of abnormalities and disease, but it is often subject to the interpretation of a medical professional. Starting as early as the 1960s, efforts have been made to develop quantitative tools based on automated image analysis algorithms to assist physicians and researchers in characterizing tissue properties [1

1. K. Doi, “Computer-aided diagnosis in medical imaging: historical review, current status and future potential,” Comput. Med. Imaging Graph. 31(4-5), 198–211 (2007). [CrossRef] [PubMed]

]. Optimization and development of these methods is still underway, and more groups are recognizing the utility of these techniques for extracting patterns and information from biomedical images. Uncovering this image information is likely to lead to the discovery of novel and objective diagnostic criteria, improving diagnostic sensitivity and enabling earlier disease detection. In this way, widespread application and optimization of quantitative image analysis techniques has great potential to impact the performance of clinical diagnostics and basic research that relies on interpretation of biomedical images.

Fourier-based techniques have wide-range applications in signal and image assessment and are gaining a more critical role in tissue characterization. For example, these techniques have been implemented to characterize bone structure in computed tomography images [2

2. G. Dougherty and G. M. Henebry, “Fractal signature and lacunarity in the measurement of the texture of trabecular bone in clinical CT images,” Med. Eng. Phys. 23(6), 369–380 (2001). [CrossRef] [PubMed]

] and to detect cardiac arrhythmias in electrocardiogram signals [3

3. H. Gothwal, S. Kedawat, and R. Kumar, “Cardiac arrhythmias detection in an ECG beat signal using fast Fourier transform and artificial neural network,” J. Biomed. Sci. Eng. 4(04), 289–296 (2011). [CrossRef]

]. The squared amplitude of the Fourier transform (FT) is referred to as the power spectral density (PSD). Biomedical images of cells and tissues often exhibit PSDs with inverse power-law frequency dependence (i.e., proportional to k-β, where k is spatial frequency and β is the power-law exponent), which can indicate a scale-invariant (fractal) organization of the imaged features [4

4. D. L. Turcotte, Fractals and Chaos in Geology and Geophysics (Cambridge Univ. Press, 1997).

]. Scale-invariance describes features or patterns that persist over multiple length scales. These features must satisfy conditions of self-similarity; meaning a fractal object is similar to a subset of itself. In special cases, fractal can be considered self-affine if variation in one direction scales differently than variation in another direction [4

4. D. L. Turcotte, Fractals and Chaos in Geology and Geophysics (Cambridge Univ. Press, 1997).

6

6. B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman and Company, 2000).

].

Fractals are present in a wide variety of natural systems [5

5. P. Meakin, Fractals, Scaling, and Growth Far from Equilibrium (Cambridge University Press, 1998).

, 6

6. B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman and Company, 2000).

], including rock strain distributions [7

7. H. S. Wu, “Fractal strain distribution and its implications for cross section balancing,” J. Struct. Geol. 15(12), 1497–1507 (1993). [CrossRef]

], microvascular networks [8

8. Y. Gazit, D. A. Berk, M. Leunig, L. T. Baxter, and R. K. Jain, “Scale-invariant behavior and vascular network formation in normal and tumor tissue,” Phys. Rev. Lett. 75(12), 2428–2431 (1995). [CrossRef] [PubMed]

], and chromatin aggregation [9

9. A. J. Einstein, H. S. Wu, and J. Gil, “Self-Affinity and lacunarity of chromatin texture in benign and malignant breast epithelial cell nuclei,” Phys. Rev. Lett. 80(2), 397–400 (1998). [CrossRef]

]. Numerous groups have reported on the fractal nature of subcellular inhomogeneities and their variation with disease state [9

9. A. J. Einstein, H. S. Wu, and J. Gil, “Self-Affinity and lacunarity of chromatin texture in benign and malignant breast epithelial cell nuclei,” Phys. Rev. Lett. 80(2), 397–400 (1998). [CrossRef]

16

16. K. J. Chalut, J. H. Ostrander, M. G. Giacomelli, and A. Wax, “Light scattering measurements of subcellular structure provide noninvasive early detection of chemotherapy-induced apoptosis,” Cancer Res. 69(3), 1199–1204 (2009). [CrossRef] [PubMed]

]. However, many of these studies are based on light scattering properties of cells and tissues, providing an indirect (and non-singular) determination of cell and tissue morphology. Quantitative characterization of fractal features can thus vary with the particular model assumed for subcellular or tissue density fluctuations; for example, whether these features exhibit von Kármán [11

11. J. M. Schmitt and G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21(16), 1310–1312 (1996). [CrossRef] [PubMed]

, 13

13. M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, V. Gopal, K. Badizadegan, G. D. Stoner, and M. S. Feld, “Tissue self-affinity and polarized light scattering in the born approximation: a new model for precancer detection,” Phys. Rev. Lett. 97(13), 138102 (2006). [CrossRef] [PubMed]

], exponential [10

10. S. A. Kartazayeva, X. Ni, and R. R. Alfano, “Backscattering target detection in a turbid medium by use of circularly and linearly polarized light,” Opt. Lett. 30(10), 1168–1170 (2005). [CrossRef] [PubMed]

, 17

17. M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18(4), 948–960 (2001). [CrossRef] [PubMed]

] or stretched exponential spatial correlations [16

16. K. J. Chalut, J. H. Ostrander, M. G. Giacomelli, and A. Wax, “Light scattering measurements of subcellular structure provide noninvasive early detection of chemotherapy-induced apoptosis,” Cancer Res. 69(3), 1199–1204 (2009). [CrossRef] [PubMed]

]. Furthermore, direct analysis of subcellular features is often attained by invasive methods, such as by histological staining [17

17. M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18(4), 948–960 (2001). [CrossRef] [PubMed]

] or electron microscopy [18

18. M. Bartek, X. Wang, W. Wells, K. D. Paulsen, and B. W. Pogue, “Estimation of subcellular particle size histograms with electron microscopy for prediction of optical scattering in breast tissue,” J. Biomed. Opt. 11(6), 064007 (2006). [CrossRef] [PubMed]

], which may alter morphology from its nascent state.

Biomedical images of living cells or tissues obtained via fluorescence microscopy, on the other hand, have the advantage of providing a direct and noninvasive means for determining cell morphology down to submicron length scales. In this study, we characterize images that rely on intrinsic fluorescence from nicotinamide adenine dinucleotide (NADH), which emanates from cell mitochondria [19

19. B. Chance, P. Cohen, F. Jobsis, and B. Schoener, “Intracellular oxidation-reduction states in vivo,” Science 137(3529), 499–508 (1962). [CrossRef] [PubMed]

]. Mitochondria are the main energy-converting organelles of mammalian cells [20

20. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, “Energy conversion: Mitochondria and Chloroplasts,” in Molecular Biology of the Cell (Garland Science, 2002) http://www.ncbi.nlm.nih.gov/books/NBK21063/.

]. Metrics that quantitatively assess the degree of correlation of mitochondrial networks could serve as useful indicators of cellular health status. For example, early work by Hackenbrock showed that mitochondrial networks rapidly become more condensed in liver cells in which oxidative phosphorylation was activated [21

21. C. R. Hackenbrock, “Ultrastructural bases for metabolically linked mechanical activity in mitochondria. II. Electron transport-linked ultrastructural transformations in mitochondria,” J. Cell Biol. 37(2), 345–369 (1968). [CrossRef] [PubMed]

23

23. C. R. Hackenbrock, “Ultrastructural bases for metabolically linked mechanical activity in mitochondria. I. Reversible ultrastructural changes with change in metabolic steady state in isolated liver mitochondria,” J. Cell Biol. 51, 123–137 (1971).

]. More recent work has focused on the thinning and branching of the mitochondrial networks upon the switch from glycolytic to oxidative energy metabolism, which is thought to be a critical indicator of cell growth or differentiation [24

24. H. Mortiboys, K. J. Thomas, W. J. Koopman, S. Klaffke, P. Abou-Sleiman, S. Olpin, N. W. Wood, P. H. Willems, J. A. Smeitink, M. R. Cookson, and O. Bandmann, “Mitochondrial function and morphology are impaired in parkin-mutant fibroblasts,” Ann. Neurol. 64(5), 555–565 (2008). [CrossRef] [PubMed]

, 25

25. R. Rossignol, R. Gilkerson, R. Aggeler, K. Yamagata, S. J. Remington, and R. A. Capaldi, “Energy substrate modulates mitochondrial structure and oxidative capacity in cancer cells,” Cancer Res. 64(3), 985–993 (2004). [CrossRef] [PubMed]

].

For PSD-based analysis, as the measured power-law exponent (β) of an image increases, the fractal metrics we assess indicate the mitochondrial networks become more correlated. Generally, when the PSD spectral content is flat (β = 0), the signal is uncorrelated, white Gaussian noise (Table 1

Table 1. β values and corresponding statistical processes

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). Images that contain features characterized by power exponent values ranging between 0 < β < 2 are considered fractional Gaussian noises (fGNs), whereas fractional Brownian motions (fBMs) are characterized by exponent values between 2 < β < 4 [4

4. D. L. Turcotte, Fractals and Chaos in Geology and Geophysics (Cambridge Univ. Press, 1997).

]. fBMs are distinct from fGNs in that they have long-range correlation (“memory”), which has been described as a statistical relation between the increments of data set values [4

4. D. L. Turcotte, Fractals and Chaos in Geology and Geophysics (Cambridge Univ. Press, 1997).

]. Visually, these long-range correlations result in progressively more clustering as the power-law exponent increases.

In practice, most PSD-based analyses that assess the fractal nature of biomedical images do not account for artifacts that can be produced by edge effects from cellular and nuclear borders [14

14. J. M. Levitt, M. Hunter, C. Mujat, M. McLaughlin-Drubin, K. Münger, and I. Georgakoudi, “Diagnostic cellular organization features extracted from autofluorescence images,” Opt. Lett. 32(22), 3305–3307 (2007). [CrossRef] [PubMed]

, 26

26. J. M. Levitt, M. E. McLaughlin-Drubin, K. Münger, and I. Georgakoudi, “Automated biochemical, morphological, and organizational assessment of precancerous changes from endogenous two-photon fluorescence images,” PLoS ONE 6(9), e24765 (2011). [CrossRef] [PubMed]

]. To overcome these artifacts, some studies have focused on small regions inside cells, which requires sub-image selection and drastically limits resolution and field of view. Others have discussed in a geological context how power spectral estimates of scale-invariant data sets depend on factors other than the fractal characteristics of the given data set, such as sampling, noise, and, edge effects [27

27. T. H. Wilson, “Fractal strain distribution and its implications for cross section balancing: further discussion,” J. Struct. Geol. 19(1), 129–132 (1997). [CrossRef]

].

The objective of our study is to develop an automated, PSD-based analysis for the quantitative characterization of internal morphology (texture) of image features with well-defined borders that overcomes the limitations introduced by the presence of these borders, in order to develop more robust diagnostic biomarkers of organizational attributes. We present a mechanistic computational study in which we evaluate simulated images of scale-invariant (fractal) patterns combined with controlled cell-shaped features to show that these cell-shaped features and image background have predictable effects on PSD power-law decays. Based on this evaluation, we quantify the manner in which the presence of cell- and nuclear-shaped features hinders the ability of PSD-based methods to accurately assess intracellular patterns. To overcome this limitation, we have developed a digital object cloning (DOC) method for use prior to traditional PSD analysis. We characterize simulated images that contain structures and textures common to epithelial tissues and compare the ability of PSD-based methods to accurately quantify intracellular texture both with and without DOC pre-processing. Finally, to showcase the diagnostic utility of this improved technique, we apply it to characterize experimentally acquired autofluorescence images from engineered normal and pre-cancerous epithelial tissues

2. Methods

2.1 Simulations of scale-invariant images

To test the ability of PSD-based approaches to sense changes in fractal scaling, simulated fractal images, S(r), of size N × N and which have power spectral density that scales as a power-law with pixel values represented by the vector r, r=xi^+yj^, are generated in MATLAB based on Voss’s inverse Fourier filtering method [28

28. R. F. Voss, in Fundamental Algorithms for Computer Graphics, edited by R. A. Earnshaw (Springer-Verlag, Berlin, 1985).

]. In order to construct a fractal image using this approach, white noise, W(r), is generated with a mean of 0 and a standard deviation of 1 with the randn function in MATLAB. The discrete Fourier transform, F(k), as a function of radial spatial frequency, k, wherek=kx+ky and for W(r) is taken as:

F(k)=1N2x=0N1y=0N1W(r)e2πi(k·r)N.
(1)

Next, the phase, Φ(k), of the Fourier transform of W(r) is computed by dividing the Fourier transform (real and imaginary parts) by its magnitude. For F(k):

Φ(k)=F(k)|F(k)|.
(2)

The phase is filtered with a transfer function, T(k), to produce the random function,

Φ'(k)=T(k)Φ(k),
(3)

which is retransformed into the spatial domain to produce a fractal image, W'(r), where

W(r)=1N2kx=0N1ky=0N1Φ'(k)e2πi(k·r)N.
(4)

To generate inverse power-law dependent PSD scaling from W'(r) requires that:

T(k)=kβ/2,
(5)

where β is the exponent of the power-law decay and for β values ranging from 0 to 4 [28

28. R. F. Voss, in Fundamental Algorithms for Computer Graphics, edited by R. A. Earnshaw (Springer-Verlag, Berlin, 1985).

]. For each generated image, Eq. (6) is fit to the radial (angularly-averaged) power spectral density (PSD):

R(k)=Akβ,
(6)

2.2 Engineered tissue constructs and TPEF data acquisition

Organotypic rafts cultured with normal human foreskin keratinocytes (HFKs) allow for generation of multilayered tissues that mimic the gradient of differentiation of native epithelium and are grown as described in detail previously [29

29. C. Meyers, T. J. Mayer, and M. A. Ozbun, “Synthesis of infectious human papillomavirus type 18 in differentiating epithelium transfected with viral DNA,” J. Virol. 71(10), 7381–7386 (1997). [PubMed]

]. Briefly, keratinocytes are plated on top of a neutralized bovine type I collagen (~4mg/ml) matrix with dermal fibroblasts, and raised to an air-liquid interface. The tissues are provided nutrients from below to simulate natural nutrient delivery by a vasculature bed for 10 days, at which point they are imaged. For pre-cancerous tissues, human-papillomavirus (HPV-) immortalized epithelial cells are used in place of normal HFKs as detailed previously [14

14. J. M. Levitt, M. Hunter, C. Mujat, M. McLaughlin-Drubin, K. Münger, and I. Georgakoudi, “Diagnostic cellular organization features extracted from autofluorescence images,” Opt. Lett. 32(22), 3305–3307 (2007). [CrossRef] [PubMed]

]. TPEF images are acquired on a Leica TCS SP2 confocal microscope (Wetzlar, Germany) equipped with a Ti:sapphire laser (Spectra Physics, Mountain View, CA). Samples are placed on glass coverslips, excited with 755nm (TPEF) light and imaged using a 63x/1.2 NA water immersion objective, which yields 8-bit, 512 x 512 pixel images of 238 x 238 μm2, in approximately 1 s. TPEF images are acquired by a non-descanned PMT with a filter cube containing a 700 nm short pass filter (Chroma SPC700bp) a dichroic mirror (Chroma 495dcxr), and an emitter bandpass filter centered at 460 nm (Chroma 460bp40). This excitation and emission filter combination allows collection of fluorescence signal emanating primarily from mitochondrial NADH [14

14. J. M. Levitt, M. Hunter, C. Mujat, M. McLaughlin-Drubin, K. Münger, and I. Georgakoudi, “Diagnostic cellular organization features extracted from autofluorescence images,” Opt. Lett. 32(22), 3305–3307 (2007). [CrossRef] [PubMed]

, 30

30. W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. U.S.A. 100(12), 7075–7080 (2003). [CrossRef] [PubMed]

, 31

31. W. L. Rice, D. L. Kaplan, and I. Georgakoudi, “Two-photon microscopy for non-invasive, quantitative monitoring of stem cell differentiation,” PLoS ONE 5(4), e10075 (2010). [CrossRef] [PubMed]

]. Five regions from three different tissue constructs for the diseased and healthy tissues are assessed with PSD analysis.

2.3 Creation of simulated cell objects (SCOs)

In the high frequency limit, the PSD of a Jinc function is known to have oscillations that decay as an inverse power-law with a power exponent equal to 3. In order to examine the possible interference of the behavior of a Jinc function with the measurement of intracellular texture from cellular images, fractal images with known power-law decay values are digitally applied to the circular regions of the simulated images. To explore the sensitivity of this analysis to cell morphology and nuclear structures, we fit the PSD of the simulated images with Eq. (6) and the exponent of the fit is compared with input exponents of the fractal that has been digitally applied. To weaken contributions from background signal and edge effects, background is set to the average value of the intracircular signal when indicated in the text.

2.4 Identification of cell and nuclear borders for cloning TPEF images

2.5 Creation of simulated fractal images with defined borders

Using the mask determined as described above, fractal images with known PSD power-law decay values (β = 0-4) are digitally applied to the intracellular regions according to this mask. The digital application of a fractal image is repeated four times, yielding a total of 60 simulated images with known intracellular texture (we start with five images from each of the three main epithelial tissue layers). To assess the contributions from the image background, a black background or the average value of the intracellular foreground is digitally applied to background regions. Power-law exponents are determined from fits to the power-law region of the radial PSD for all synthesized images. Measured exponents are then compared to input exponents from fractals that are simulated and digitally applied to the intracellular regions.

2.6 Digital object cloning (DOC)

2.7 Statistical testing and β error calculation

The correlation between measured and input β values for TPEF images is assessed via the Tukey Honestly Significant Different (HSD) test in JMP statistical software. Average β-error is calculated as the average of the absolute value of the difference between the input β value and the measured β value for each simulated image for β’s between 0 and 2.

3. Results

3.1 Simulated cell objects (SCOs) hinder the ability of PSD-based methods to accurately assess intracellular patterns

To demonstrate the baseline sensitivity of the PSD-based approach for recovering power-law exponents (β) that describe the fractal character of images, we evaluate the PSD of square fractal images generated as described in Methods. Figure 1
Fig. 1 (a) Power spectral density of generated scale invariant images of varying fractal character with fits (shown in panels b-f) for β = 0, 1, 2, 3, 4.
displays angularly-averaged radial PSDs (in log-log scale, Panel (a)) that correspond to simulated fractal images (Panels (b-f)), which become more clustered in gross-appearance with increasing power exponents (β). Measured power exponents, determined by fitting Eq. (6), match input exponents used in fractal simulations with negligible associated error (R2 > 0.99).

3.2 PSD-based characterization of simulated images containing epithelial structures

To assess whether PSD-based techniques are sensitive to differences in subcellular texture in addition to cell morphology, fractals of varying β exponents are digitally applied to the intracellular regions of thresholded TPEF images from each tissue layer (Fig. 4
Fig. 4 Nomograms of measured β parameters for superficial (a), parabasal (b), and basal (c) epithelial layers of normal tissue. Plotted are input versus measured power-law exponents from linear fits of the radially-sampled PSDs for simulated images of varying input β parameters and background. Representative images with black background (green, solid lines), with the average value of the foreground in the background (blue, dashed lines) and images that have been clone stamped (red, dotted lines).
). Figure 4 displays nomograms comparing the measured power-law exponents from fits to the PSDs of these simulated images to the input β exponents used in image generation. By digitally applying either a black background (high contrast, strong edge intensity) or the mean value of the image foreground (low contrast, weak edge intensity) to the background region, we are able to explore the effects of image feature edges on β exponent sensitivity. We find there is limited sensitivity of PSD-based methods to quantify intracellular texture in images containing a black background (Fig. 4(a-c), green lines) by observing that the measured power-law exponents of the fits varied marginally and non-linearly with input β exponents (Fig. 3). The average β-error, the average difference between βinput and βM for 0 < β < 2, for each tissue layer with a black background is 1.20 ± 0.74, 1.28 ± 0.75, and 1.63 ± 0.78 for the superficial, para-basal, and basal layers, respectively. By including the average value of the foreground in the background of the simulated images (Fig. 4(a-c); blue dashed lines), the sensitivity to input β values improves and average β-error is calculated to be 0.70 ± 0.32, 0.51 ± 0.32, and 0.69 ± 0.42 for the superficial, para-basal, and basal layers, respectively. However, in spite of this improvement, measured β values from the average value of the foreground in the background do not correspond linearly to input β values and are overestimated for β values lower than 3 and underestimated for β values greater than 3 for all tissue layers.

In order to obtain reliable sensitivity to a range of β parameters, a method to improve PSD sensitivity to input β values in the presence of cell features is necessary. Sensitivity to the correct β value depends on input β values, cell size, presence of a nucleus, and image background, which suggests that image features other than intracellular texture are contributing to the measurement of accurate power-law decay values. To recover β values that represent only the fractal nature of the intracellular signal, a digital object cloning (DOC) method is applied. DOC improves the accuracy of PSD-based characterization of intracellular structure. With DOC, fractal variation from simulated images can be recovered for β exponents that are less than the measured β exponents from the respective TPEF image masks (Fig. 4(a-c); red dotted lines). This limitation is tissue layer specific according to the measured β values from the image masks (Fig. 3). With DOC, relative average β-error decreases significantly to 0.06 ± 0.04, 0.05 ± 0.03, and 0.04 ± 0.04 for the superficial, para-basal, and basal layers, respectively.

3.3 DOC improves the accuracy of PSD-based characterization of intracellular texture allowing separation of normal and pre-cancerous tissues based on subcellular structure

After isolating cell features via thresholding, the DOC technique is applied to the original TPEF images from normal epithelial tissues. After DOC, the superficial, para-basal and basal layers (Fig. 5(a)
Fig. 5 (a) Representative original and DOC-corrected TPEF images of superficial, para-basal, and basal layers of epithelial tissues made with healthy human foreskin keratinocytes (HFK) and HPV-transfected keratinocytes with the average β values for 5 different fields for each group displayed under each image with standard deviations. (b) β values from each HFK (green) and HPV (red) tissue sorted by tissue layer, before and after DOC.
) have respective β values of 0.67 ± 0.13, 0. 69 ± 0.19, and 0.82 ± 0.14. β values between the layers are not statistically different (p ≥ 0.995).

DOC and subsequent PSD-analysis is applied to HPV-immortalized epithelial tissues (Fig. 5(a)). β values of original HPV tissue layers, shown in Fig. 5(a), are respectively 2.04 ± 0.20, 2.05 ± 0.08, and 1.98 ± 0.20 for the superficial, para-basal, and basal layers. After DOC, these values are 1.72 ± 0.19, 1.47 ± 0.15, and 1.53 ± 0.20 and are not statistically different from each other (p ≥ 0.227).

To visualize the effects of DOC across the tissue groups and layers, we plot the measured β values from TPEF images of each layer of five normal (Fig. 5(b), green) and five HPV (Fig. 5(b), red) tissues before and after DOC is applied. Before DOC is applied, there is not a significant difference between β values measured from HPV tissues layers and β values measured from the basal layer of normal tissue (p ≥ 0.963), which is most likely due to similarities in morphology and cell size. Remarkably, DOC corrects for tissue layer-dependent changes in β values for normal tissues by lessening the scale-dependent artifacts due to feature edges as well as image background. In this way, DOC enables targeted quantification of the fractal nature of subcellular structure, which is significantly different between the normal and the HPV-infected tissues layers after DOC is applied (p < 0.001). Although we find significant difference between tissue groups with a relatively small sample size, additional studies with larger samples sizes may provide insight into the diagnostic significance. In this way, DOC in combination with PSD-based methods shows great utility for biomedical image analysis, specifically with the goal of correlating subcellular structure to tissue health and disease status.

4. Discussion

This study demonstrates that Fourier-based analysis, relying on the PSD, is well suited to the characterization of high-resolution biomedical images for the assessment of intracellular structure. We specifically show that quantification of the β exponent of fractal biological features, such as mitochondrial organization, can be employed as an indicator of health and disease status. Assessment of mitochondrial structure may help elucidate mechanisms of normal and diseased tissue development, specifically in the context of cell metabolism. However, biomedical images generally contain a mix of fractal organization (e.g., intracellular morphology) and scale-dependent features (e.g., quasi-circular cellular and nuclear borders). The presence of nuclear and cell boundaries introduces edge-effect errors in PSD-based estimates of fractal character, which is clear when comparing pure fractal images (Fig. 1) with fractal images confined to circular shapes (Fig. 2). The images containing both scale-dependent and scale-invariant features require further processing to enable accurate characterization of the intracellular fractal patterns (Fig. 4). After standard thresholding-based processing of these images, fractal patterns with power-law decays greater than the power-law decay of the fractal pattern’s confining borders cannot be accurately measured due to edge artifacts. However, an additional DOC step allows reasonably accurate recovery of subcellular fractal organizational features.

4.1 DOC enables direct comparison of intracellular scale invariance among images with different large-scale features

Boundary effects due to overall cellular and nuclear shape, must be minimized in order to accurately quantify the fractal nature of intracellular structures. Decreasing the change in intensity at the edges of the cells by thresholding and applying the average intensity value of the foreground to the background (2-fold decrease in β-error) or thresholding and DOC (26-fold decrease in β-error) allows for more accurate PSD-based fractal characterization for β < 2. Despite the application of DOC pre-processing techniques to overcome edge artifacts, there is a drop-off in sensitivity to scale-invariant patterns characterized by power-law exponents greater than 2 compared to pure scale-invariant patterns (Fig. 4). Specifically, this occurs because our DOC method becomes less efficient at weakening edges within progressively more clustered images (β > 2), which have large deviations in local average intensity and are thus, more susceptible to DOC-induced edge artifacts (Fig. 1(e and f)).

Fortunately, the variation of intracellular organization observed from our analysis of experimentally-acquired TPEF images of engineered epithelial tissue occurs over a range of β values where PSD methods are highly accurate following DOC (β ≤ 2). In this regime, DOC reduces the contribution from the non-fractal features, allowing direct comparison of subcellular scale-invariance between images that contain large-scale, non-fractal variation. This is useful when evaluating different epithelial tissue layers that have varying characteristic cell sizes.

4.2 Comparison with other techniques

It is particularly advantageous to use DOC pre-processing to reduce edge effects because DOC does not sacrifice signal resolution, as with, for example, Fourier ‘windowing’ techniques [35

35. F. I. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978). [CrossRef]

]. Another Fourier-based technique developed with the goal of circumventing the limitation of Fourier analysis to rectangular images has been applied to the fractal analysis of nuclear chromatin distribution in benign and malignant breast cells [9

9. A. J. Einstein, H. S. Wu, and J. Gil, “Self-Affinity and lacunarity of chromatin texture in benign and malignant breast epithelial cell nuclei,” Phys. Rev. Lett. 80(2), 397–400 (1998). [CrossRef]

]. This method fills the non-fractal region surrounding nuclear features using an iterative algorithm that terminates when the background region has similar statistical properties to the inner fractal region. For this method, it is difficult to obtain the background properties similar to the foreground, with 10% of samples unable to converge to the appropriate solution. Furthermore, the accuracy of the technique was evaluated using a single β value (β = 1.5), and it is unknown whether the method can perform as well with other β values.

4.3 Diagnostic utility of combined DOC and PSD methods

In the absence of edge effects, the β value represents the character of the fractal component of the image, which, in the context of this analysis, physically represents the organization of the mitochondria. We target mitochondrial organization by assessing images of NADH autofluorescence and by our choice of a fitting range that represents image structures on the size order of mitochondrial networks (< 10μm) [14

14. J. M. Levitt, M. Hunter, C. Mujat, M. McLaughlin-Drubin, K. Münger, and I. Georgakoudi, “Diagnostic cellular organization features extracted from autofluorescence images,” Opt. Lett. 32(22), 3305–3307 (2007). [CrossRef] [PubMed]

]. We find an increase in β values measured from the analysis of precancerous (HPV) tissues, which reflects more correlated mitochondrial networks, as compared to the mitochondrial networks from normal tissues [38

38. D. L. Turcotte, “Fractals in petrology,” Lithos 65(3-4), 261–271 (2002). [CrossRef]

].

4.4 Disease progression and cell metabolism are related to mitochondrial autofluorescence patterns

5. Conclusion

Fractals have become valuable descriptors of natural processes. Thus, there is a need for accessible analytical techniques whose limitations are fully understood, and at the same time, have the ability to accurately quantify scale-invariance found in nature. We find that the accuracy of the PSD-based techniques to characterize changes in fractal features within epithelial tissues is limited by the presence of nuclear and cell borders and image background signal. We develop a DOC technique that corrects for measurement inaccuracies induced by these edge effects, which results in a 26-fold decrease in measurement error and enables us to more accurately characterize naturally occurring mitochondrial fluorescence. By assessing the organization of mitochondrial networks, we are able to differentiate normal and precancerous epithelial tissues. We infer that these differences in organization are due to alterations in cell metabolic preferences. This technique could potentially be employed to correct other medical images with similar scale-dependent limitations, such as tissue histology or x-rays. Together, DOC and the PSD method have the ability to quantify the organization of a wide range of irregularly-shaped biomedical images and, more importantly, to isolate features of possible diagnostic significance. This could have applications in the development of automated disease detection software and computer vision.

Acknowledgments

This work was supported by American Cancer Society Research Scholar Grant RSG-09-174-01-CCE, NIBIB/NIH Grant R01 EB007542, and NIAMS/NIH Grant F32 AR061933.  The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

References and links

1.

K. Doi, “Computer-aided diagnosis in medical imaging: historical review, current status and future potential,” Comput. Med. Imaging Graph. 31(4-5), 198–211 (2007). [CrossRef] [PubMed]

2.

G. Dougherty and G. M. Henebry, “Fractal signature and lacunarity in the measurement of the texture of trabecular bone in clinical CT images,” Med. Eng. Phys. 23(6), 369–380 (2001). [CrossRef] [PubMed]

3.

H. Gothwal, S. Kedawat, and R. Kumar, “Cardiac arrhythmias detection in an ECG beat signal using fast Fourier transform and artificial neural network,” J. Biomed. Sci. Eng. 4(04), 289–296 (2011). [CrossRef]

4.

D. L. Turcotte, Fractals and Chaos in Geology and Geophysics (Cambridge Univ. Press, 1997).

5.

P. Meakin, Fractals, Scaling, and Growth Far from Equilibrium (Cambridge University Press, 1998).

6.

B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman and Company, 2000).

7.

H. S. Wu, “Fractal strain distribution and its implications for cross section balancing,” J. Struct. Geol. 15(12), 1497–1507 (1993). [CrossRef]

8.

Y. Gazit, D. A. Berk, M. Leunig, L. T. Baxter, and R. K. Jain, “Scale-invariant behavior and vascular network formation in normal and tumor tissue,” Phys. Rev. Lett. 75(12), 2428–2431 (1995). [CrossRef] [PubMed]

9.

A. J. Einstein, H. S. Wu, and J. Gil, “Self-Affinity and lacunarity of chromatin texture in benign and malignant breast epithelial cell nuclei,” Phys. Rev. Lett. 80(2), 397–400 (1998). [CrossRef]

10.

S. A. Kartazayeva, X. Ni, and R. R. Alfano, “Backscattering target detection in a turbid medium by use of circularly and linearly polarized light,” Opt. Lett. 30(10), 1168–1170 (2005). [CrossRef] [PubMed]

11.

J. M. Schmitt and G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21(16), 1310–1312 (1996). [CrossRef] [PubMed]

12.

A. C. Sullivan, J. P. Hunt, and A. L. Oldenburg, “Fractal analysis for classification of breast carcinoma in optical coherence tomography,” J. Biomed. Opt. 16(6), 066010 (2011). [CrossRef] [PubMed]

13.

M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, V. Gopal, K. Badizadegan, G. D. Stoner, and M. S. Feld, “Tissue self-affinity and polarized light scattering in the born approximation: a new model for precancer detection,” Phys. Rev. Lett. 97(13), 138102 (2006). [CrossRef] [PubMed]

14.

J. M. Levitt, M. Hunter, C. Mujat, M. McLaughlin-Drubin, K. Münger, and I. Georgakoudi, “Diagnostic cellular organization features extracted from autofluorescence images,” Opt. Lett. 32(22), 3305–3307 (2007). [CrossRef] [PubMed]

15.

J. D. Rogers, I. R. Capoğlu, and V. Backman, “Nonscalar elastic light scattering from continuous random media in the Born approximation,” Opt. Lett. 34(12), 1891–1893 (2009). [CrossRef] [PubMed]

16.

K. J. Chalut, J. H. Ostrander, M. G. Giacomelli, and A. Wax, “Light scattering measurements of subcellular structure provide noninvasive early detection of chemotherapy-induced apoptosis,” Cancer Res. 69(3), 1199–1204 (2009). [CrossRef] [PubMed]

17.

M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18(4), 948–960 (2001). [CrossRef] [PubMed]

18.

M. Bartek, X. Wang, W. Wells, K. D. Paulsen, and B. W. Pogue, “Estimation of subcellular particle size histograms with electron microscopy for prediction of optical scattering in breast tissue,” J. Biomed. Opt. 11(6), 064007 (2006). [CrossRef] [PubMed]

19.

B. Chance, P. Cohen, F. Jobsis, and B. Schoener, “Intracellular oxidation-reduction states in vivo,” Science 137(3529), 499–508 (1962). [CrossRef] [PubMed]

20.

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, “Energy conversion: Mitochondria and Chloroplasts,” in Molecular Biology of the Cell (Garland Science, 2002) http://www.ncbi.nlm.nih.gov/books/NBK21063/.

21.

C. R. Hackenbrock, “Ultrastructural bases for metabolically linked mechanical activity in mitochondria. II. Electron transport-linked ultrastructural transformations in mitochondria,” J. Cell Biol. 37(2), 345–369 (1968). [CrossRef] [PubMed]

22.

C. R. Hackenbrock, T. G. Rehn, E. C. Weinbach, and J. J. Lemasters, “Oxidative phosphorylation and ultrastructural transformation in mitochondria in the intact ascites tumor cell,” J. Cell Biol. 30, 269–297 (1966). [CrossRef] [PubMed]

23.

C. R. Hackenbrock, “Ultrastructural bases for metabolically linked mechanical activity in mitochondria. I. Reversible ultrastructural changes with change in metabolic steady state in isolated liver mitochondria,” J. Cell Biol. 51, 123–137 (1971).

24.

H. Mortiboys, K. J. Thomas, W. J. Koopman, S. Klaffke, P. Abou-Sleiman, S. Olpin, N. W. Wood, P. H. Willems, J. A. Smeitink, M. R. Cookson, and O. Bandmann, “Mitochondrial function and morphology are impaired in parkin-mutant fibroblasts,” Ann. Neurol. 64(5), 555–565 (2008). [CrossRef] [PubMed]

25.

R. Rossignol, R. Gilkerson, R. Aggeler, K. Yamagata, S. J. Remington, and R. A. Capaldi, “Energy substrate modulates mitochondrial structure and oxidative capacity in cancer cells,” Cancer Res. 64(3), 985–993 (2004). [CrossRef] [PubMed]

26.

J. M. Levitt, M. E. McLaughlin-Drubin, K. Münger, and I. Georgakoudi, “Automated biochemical, morphological, and organizational assessment of precancerous changes from endogenous two-photon fluorescence images,” PLoS ONE 6(9), e24765 (2011). [CrossRef] [PubMed]

27.

T. H. Wilson, “Fractal strain distribution and its implications for cross section balancing: further discussion,” J. Struct. Geol. 19(1), 129–132 (1997). [CrossRef]

28.

R. F. Voss, in Fundamental Algorithms for Computer Graphics, edited by R. A. Earnshaw (Springer-Verlag, Berlin, 1985).

29.

C. Meyers, T. J. Mayer, and M. A. Ozbun, “Synthesis of infectious human papillomavirus type 18 in differentiating epithelium transfected with viral DNA,” J. Virol. 71(10), 7381–7386 (1997). [PubMed]

30.

W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. U.S.A. 100(12), 7075–7080 (2003). [CrossRef] [PubMed]

31.

W. L. Rice, D. L. Kaplan, and I. Georgakoudi, “Two-photon microscopy for non-invasive, quantitative monitoring of stem cell differentiation,” PLoS ONE 5(4), e10075 (2010). [CrossRef] [PubMed]

32.

J. Xylas, A. Alt-Holland, J. Garlick, M. Hunter, and I. Georgakoudi, “Intrinsic optical biomarkers associated with the invasive potential of tumor cells in engineered tissue models,” Biomed. Opt. Express 1(5), 1387–1400 (2010). [CrossRef] [PubMed]

33.

R. E. Blahut, Theory of Remote Image Formation (Cambridge University Press, 2004).

34.

K. P. Quinn, E. Bellas, N. Fourligas, K. Lee, D. L. Kaplan, and I. Georgakoudi, “Characterization of metabolic changes associated with the functional development of 3D engineered tissues by non-invasive, dynamic measurement of individual cell redox ratios,” Biomaterials 33(21), 5341–5348 (2012). [CrossRef] [PubMed]

35.

F. I. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978). [CrossRef]

36.

R. Lopes and N. Betrouni, “Fractal and multifractal analysis: A review,” Med. Image Anal. 13(4), 634–649 (2009). [CrossRef] [PubMed]

37.

F. Normant and C. Tricot, “Method for evaluating the fractal dimension of curves using convex hulls,” Phys. Rev. A 43(12), 6518–6525 (1991). [CrossRef] [PubMed]

38.

D. L. Turcotte, “Fractals in petrology,” Lithos 65(3-4), 261–271 (2002). [CrossRef]

39.

O. Warburg, F. Wind, and E. Negelein, “The metabolism of tumor in the body,” J. Gen. Physiol. 8(6), 519–530 (1927). [CrossRef] [PubMed]

OCIS Codes
(070.5010) Fourier optics and signal processing : Pattern recognition
(100.2960) Image processing : Image analysis
(170.3880) Medical optics and biotechnology : Medical and biological imaging
(180.4315) Microscopy : Nonlinear microscopy

ToC Category:
Image Processing

History
Original Manuscript: August 1, 2012
Revised Manuscript: September 13, 2012
Manuscript Accepted: September 15, 2012
Published: September 27, 2012

Citation
Joanna Xylas, Kyle P. Quinn, Martin Hunter, and Irene Georgakoudi, "Improved Fourier-based characterization of intracellular fractal features," Opt. Express 20, 23442-23455 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23442


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References

  1. K. Doi, “Computer-aided diagnosis in medical imaging: historical review, current status and future potential,” Comput. Med. Imaging Graph. 31(4-5), 198–211 (2007). [CrossRef] [PubMed]
  2. G. Dougherty and G. M. Henebry, “Fractal signature and lacunarity in the measurement of the texture of trabecular bone in clinical CT images,” Med. Eng. Phys. 23(6), 369–380 (2001). [CrossRef] [PubMed]
  3. H. Gothwal, S. Kedawat, and R. Kumar, “Cardiac arrhythmias detection in an ECG beat signal using fast Fourier transform and artificial neural network,” J. Biomed. Sci. Eng. 4(04), 289–296 (2011). [CrossRef]
  4. D. L. Turcotte, Fractals and Chaos in Geology and Geophysics (Cambridge Univ. Press, 1997).
  5. P. Meakin, Fractals, Scaling, and Growth Far from Equilibrium (Cambridge University Press, 1998).
  6. B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman and Company, 2000).
  7. H. S. Wu, “Fractal strain distribution and its implications for cross section balancing,” J. Struct. Geol. 15(12), 1497–1507 (1993). [CrossRef]
  8. Y. Gazit, D. A. Berk, M. Leunig, L. T. Baxter, and R. K. Jain, “Scale-invariant behavior and vascular network formation in normal and tumor tissue,” Phys. Rev. Lett. 75(12), 2428–2431 (1995). [CrossRef] [PubMed]
  9. A. J. Einstein, H. S. Wu, and J. Gil, “Self-Affinity and lacunarity of chromatin texture in benign and malignant breast epithelial cell nuclei,” Phys. Rev. Lett. 80(2), 397–400 (1998). [CrossRef]
  10. S. A. Kartazayeva, X. Ni, and R. R. Alfano, “Backscattering target detection in a turbid medium by use of circularly and linearly polarized light,” Opt. Lett. 30(10), 1168–1170 (2005). [CrossRef] [PubMed]
  11. J. M. Schmitt and G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21(16), 1310–1312 (1996). [CrossRef] [PubMed]
  12. A. C. Sullivan, J. P. Hunt, and A. L. Oldenburg, “Fractal analysis for classification of breast carcinoma in optical coherence tomography,” J. Biomed. Opt. 16(6), 066010 (2011). [CrossRef] [PubMed]
  13. M. Hunter, V. Backman, G. Popescu, M. Kalashnikov, C. W. Boone, A. Wax, V. Gopal, K. Badizadegan, G. D. Stoner, and M. S. Feld, “Tissue self-affinity and polarized light scattering in the born approximation: a new model for precancer detection,” Phys. Rev. Lett. 97(13), 138102 (2006). [CrossRef] [PubMed]
  14. J. M. Levitt, M. Hunter, C. Mujat, M. McLaughlin-Drubin, K. Münger, and I. Georgakoudi, “Diagnostic cellular organization features extracted from autofluorescence images,” Opt. Lett. 32(22), 3305–3307 (2007). [CrossRef] [PubMed]
  15. J. D. Rogers, I. R. Capo?lu, and V. Backman, “Nonscalar elastic light scattering from continuous random media in the Born approximation,” Opt. Lett. 34(12), 1891–1893 (2009). [CrossRef] [PubMed]
  16. K. J. Chalut, J. H. Ostrander, M. G. Giacomelli, and A. Wax, “Light scattering measurements of subcellular structure provide noninvasive early detection of chemotherapy-induced apoptosis,” Cancer Res. 69(3), 1199–1204 (2009). [CrossRef] [PubMed]
  17. M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18(4), 948–960 (2001). [CrossRef] [PubMed]
  18. M. Bartek, X. Wang, W. Wells, K. D. Paulsen, and B. W. Pogue, “Estimation of subcellular particle size histograms with electron microscopy for prediction of optical scattering in breast tissue,” J. Biomed. Opt. 11(6), 064007 (2006). [CrossRef] [PubMed]
  19. B. Chance, P. Cohen, F. Jobsis, and B. Schoener, “Intracellular oxidation-reduction states in vivo,” Science 137(3529), 499–508 (1962). [CrossRef] [PubMed]
  20. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, “Energy conversion: Mitochondria and Chloroplasts,” in Molecular Biology of the Cell (Garland Science, 2002) http://www.ncbi.nlm.nih.gov/books/NBK21063/ .
  21. C. R. Hackenbrock, “Ultrastructural bases for metabolically linked mechanical activity in mitochondria. II. Electron transport-linked ultrastructural transformations in mitochondria,” J. Cell Biol. 37(2), 345–369 (1968). [CrossRef] [PubMed]
  22. C. R. Hackenbrock, T. G. Rehn, E. C. Weinbach, and J. J. Lemasters, “Oxidative phosphorylation and ultrastructural transformation in mitochondria in the intact ascites tumor cell,” J. Cell Biol. 30, 269–297 (1966). [CrossRef] [PubMed]
  23. C. R. Hackenbrock, “Ultrastructural bases for metabolically linked mechanical activity in mitochondria. I. Reversible ultrastructural changes with change in metabolic steady state in isolated liver mitochondria,” J. Cell Biol. 51, 123–137 (1971).
  24. H. Mortiboys, K. J. Thomas, W. J. Koopman, S. Klaffke, P. Abou-Sleiman, S. Olpin, N. W. Wood, P. H. Willems, J. A. Smeitink, M. R. Cookson, and O. Bandmann, “Mitochondrial function and morphology are impaired in parkin-mutant fibroblasts,” Ann. Neurol. 64(5), 555–565 (2008). [CrossRef] [PubMed]
  25. R. Rossignol, R. Gilkerson, R. Aggeler, K. Yamagata, S. J. Remington, and R. A. Capaldi, “Energy substrate modulates mitochondrial structure and oxidative capacity in cancer cells,” Cancer Res. 64(3), 985–993 (2004). [CrossRef] [PubMed]
  26. J. M. Levitt, M. E. McLaughlin-Drubin, K. Münger, and I. Georgakoudi, “Automated biochemical, morphological, and organizational assessment of precancerous changes from endogenous two-photon fluorescence images,” PLoS ONE 6(9), e24765 (2011). [CrossRef] [PubMed]
  27. T. H. Wilson, “Fractal strain distribution and its implications for cross section balancing: further discussion,” J. Struct. Geol. 19(1), 129–132 (1997). [CrossRef]
  28. R. F. Voss, in Fundamental Algorithms for Computer Graphics, edited by R. A. Earnshaw (Springer-Verlag, Berlin, 1985).
  29. C. Meyers, T. J. Mayer, and M. A. Ozbun, “Synthesis of infectious human papillomavirus type 18 in differentiating epithelium transfected with viral DNA,” J. Virol. 71(10), 7381–7386 (1997). [PubMed]
  30. W. R. Zipfel, R. M. Williams, R. Christie, A. Y. Nikitin, B. T. Hyman, and W. W. Webb, “Live tissue intrinsic emission microscopy using multiphoton-excited native fluorescence and second harmonic generation,” Proc. Natl. Acad. Sci. U.S.A. 100(12), 7075–7080 (2003). [CrossRef] [PubMed]
  31. W. L. Rice, D. L. Kaplan, and I. Georgakoudi, “Two-photon microscopy for non-invasive, quantitative monitoring of stem cell differentiation,” PLoS ONE 5(4), e10075 (2010). [CrossRef] [PubMed]
  32. J. Xylas, A. Alt-Holland, J. Garlick, M. Hunter, and I. Georgakoudi, “Intrinsic optical biomarkers associated with the invasive potential of tumor cells in engineered tissue models,” Biomed. Opt. Express 1(5), 1387–1400 (2010). [CrossRef] [PubMed]
  33. R. E. Blahut, Theory of Remote Image Formation (Cambridge University Press, 2004).
  34. K. P. Quinn, E. Bellas, N. Fourligas, K. Lee, D. L. Kaplan, and I. Georgakoudi, “Characterization of metabolic changes associated with the functional development of 3D engineered tissues by non-invasive, dynamic measurement of individual cell redox ratios,” Biomaterials 33(21), 5341–5348 (2012). [CrossRef] [PubMed]
  35. F. I. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978). [CrossRef]
  36. R. Lopes and N. Betrouni, “Fractal and multifractal analysis: A review,” Med. Image Anal. 13(4), 634–649 (2009). [CrossRef] [PubMed]
  37. F. Normant and C. Tricot, “Method for evaluating the fractal dimension of curves using convex hulls,” Phys. Rev. A 43(12), 6518–6525 (1991). [CrossRef] [PubMed]
  38. D. L. Turcotte, “Fractals in petrology,” Lithos 65(3-4), 261–271 (2002). [CrossRef]
  39. O. Warburg, F. Wind, and E. Negelein, “The metabolism of tumor in the body,” J. Gen. Physiol. 8(6), 519–530 (1927). [CrossRef] [PubMed]

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