rus
Issue #7-8/2019
O.P.Posnansky
Voxel based topometry: fractal and Euclidean descriptors of diffusion-weighted shape of multi-directional magnetic resonance signal
Voxel based topometry: fractal and Euclidean descriptors of diffusion-weighted shape of multi-directional magnetic resonance signal
Voxel-based surface shape in multi-directional diffusion-weighted magnetic resonance imaging exhibits remarkable topological heterogeneity. Once the outline of an irregular shape is identified and segmented from a digital image, geometrical descriptors can be applied to numerical characterization the irregularity of the shapes surface. In our work the fractal and Euclidean dimensions of volume, area and mean radius for each surface were determined. These indices can be used as a surrogate parameter for determining the roughness of the surface. The relationship between indices from various voxels reveals the existence of distinct groups with significant structural differences which may be caused by underlying biological tissue types. This new method allows for the standardized continuous numerical classification of shape, which is useful for the quantitative analysis of altered surface morphology, e.g. biological tissue inflammatory diseases, aging, and development.
Теги: change of the shape fractal analysis geometry quantification magnetic resonance imaging изменение формы количественная геометрия магнитная резонансная томография фрактальный анализ
Voxel-based surface shape in multi-directional diffusion-weighted magnetic resonance imaging exhibits remarkable topological heterogeneity. Once the outline of an irregular shape is identified and segmented from a digital image, geometrical descriptors can be applied to numerical characterization the irregularity of the shapes surface. In our work the fractal and Euclidean dimensions of volume, area and mean radius for each surface were determined. These indices can be used as a surrogate parameter for determining the roughness of the surface. The relationship between indices from various voxels reveals the existence of distinct groups with significant structural differences which may be caused by underlying biological tissue types. This new method allows for the standardized continuous numerical classification of shape, which is useful for the quantitative analysis of altered surface morphology, e.g. biological tissue inflammatory diseases, aging, and development.
INTRODUCTION
Measurement of diffusion in magnetic resonance imaging (MRI) is one of the most sensitive ways to explore the heterogeneity of biological tissue non-invasively. Small differences in diffusivity within tissue placed in a magnetic field give rise to effects which are well measurable in diffusion weighted MRI. Calculation of the apparent diffusion coefficient (ADC) over the various diffusion gradient directions in MRI allows us to investigate the anisotropic properties of biological tissue in every voxel (or pixel) using the tensor model [1] or spherical harmonics decomposition [2].
The improvements in digital imaging and computer-based analysis, as well as advances in new mathematical methods such as fractal geometry [3], are enabling new approaches to the geometrical description of shape. Fractal analysis is being increasingly applied to a variety of biological questions [4], and has proven useful in the analysis of medical images [5]. With the introduction of fractal geometries in the early 1980s a new parameter, the so-called fractal dimension (FD), was defined. The FD essentially measures the irregularity or roughness of a shape or surface and thus, FD is a quantitative descriptor of the actual geometrical shape [3]. The determination of the FD outlines enables the attribution of numerical values to peculiar qualitative features of spatial structures such as irregularity, morphologic complexity and roughness, which can be used to characterize images of different origins or in different functional and/or pathologic states.
With the aim of contributing to the clarification of the anisotropy and heterogeneity within the voxel, we introduce an alternative blowing compact-surface method leading to the setting the new group of scalar parameters – topological non-integer dimensions [6, 7]. The blowing sphere and ellipsoid are chosen as probing surfaces. Our findings suggest that non-Gaussian distribution of the diffusion signal dominates in the biological tissue and carries more detailed information.
This article describes an automated and thus standardized approach for the quantification of shapes. It is based on a combination of computer-based tools for image analysis and pattern recognition. In this study, images were acquired using magnetic resonance phenomenon, which provides excellent contrast for the recognition of surface structures of ADCs. Outlines of shapes were automatically segmented from digital images using a newly developed algorithm. Geometrical parameters of shape were then prospectively evaluated; this included the determination of the FD, the calculation of shape descriptors such as volume and surface area as well as mean radius. We provide detailed quantitative information on the topological features of shapes, describing for the first time the relationship between the complex geometry of ADCs and FD, which is useful for the automated, standardized and reproducible description of irregular surfaces.
RESEARCH METHODS
65 magnetic resonance images of the human brain, downloaded from public database [8], were acquired on a 3Tesla magnetic resonance scanner using diffusion-weighted spin echo sequence (Fig.1a). 60 diffusion-weighted images (DWI), Sk∈{1,...,60}, with b = 1 000 [sec/mm2], corresponding to the sixty directions, gk∈{1,...,60} = (gxgygz)k∈{1,...,60}, of diffusion gradients distributed homogeneously over the sphere (Fig.1b), were used. Additionally, five images S0 with b=0, were randomly acquired between the diffusion-weighted images to factor out the T2–weighting of the signal. All images were corrected according the procedure presented in Fig.2 [9, 10]. In the Fig.3 the segmented anatomical image of a transverse slice (right side) is shown and the DWI profiles of signal (left side) are output for voxels located in different types of tissues of the brain. The profiles demonstrated a very complex structure, correlated with the anatomical architecture. For these profiles it is possible to reconstruct a diffusion tensor image (DTI), or coefficients of fiber orientation density function (fODF) using Moore-Penrose method [1, 2]:
D = G+s', (1)
where quadratic pseudo-inverse matrices, , and ADC values,
s'T = (In ( S1 / S0) ... In ( Sn / S0) ... In ( S60 / S0)), (2)
are presented respectively. Augmented matrices, B↔aug, used for DTI and for fODF can be found in Table 1. From Eq.(1) in the case of DTI mean diffusivity (MD), , and fractional anisotropy (FA), , can be estimated. Here Tr is a trace of matrices, i.e. .
The ADC profiles recovered with Eq.2 [11] and afterwards interpolated for voxels within and outside the brain can be analyzed by the newly proposed iterative blowing surface method [6, 7]. As a probing surface, the blowing sphere was chosen for the isotropic case (Fig.4a, upper row). For the anisotropic case, an ellipsoid built on eigenvectors with kept ratio between different eigenvalues was selected (Fig.4a, low row). The key point of this method consists in the step-by-step increase of the characteristic scale over all diffusion gradient directions from the minimal to the maximal value. During expansion of the surface in the certain direction the growth was stopped if the scale exceeded the acquired ADC profile boundary in a given direction. In the Fig.4a, the 3D process is depicted with some intermediate steps in plane (in these figures the equatorial cross sections of the ADC profiles are presented).
Fractal dimension was used as a measure for the roughness of ADC surfaces, i.e. the spatial heterogeneity of their outline. Classically, FD is strictly defined only for self-similar objects that exhibit the same structure at different length scales Ri. Different approaches to measure FD exist. Here, the adjusting power method was applied, which works as follows: The outline of ADC for a single voxel was reconstructed (1), the minimum and maximal values of ADC were detected (2), eigenvalues and eigenvectors were calculated (3), sphere and ellipsoid were built and their geometric properties, i.e. volume (V), surface area (S), and mean radius (R), were expressed via power law of the scale Ri (4). The process of building of sphere and ellipsoid was then repeated for different length scales, i.e. from minimal to the maximal in order to cover the whole ADC profile. For perfectly self-similar objects, the plot of log (Ri) versus log (S), log (V), or log (R), is a straight line. In our case, the FD is given by the integer and non-integer values, which is a characteristic for non-perfect fractal behavior. FD is expressed as a tangent of a line in this case.
RESULTS
We calculated the volume V, surface area S and mean radius R during the blowing sphere and ellipsoid iterative scaling process and compared them to the unit measures by adjusting the power FD of the scaling parameter Ri of the surfaces (Eq.3, 4; i is an iteration number). For the case of ellipsoid λ1, λ2, λ3 are the eigenvalues.
M | M = V, S, R ~ RiFD, (3)
V ~ RiFD (λ1 λ2 λ3), (4a)
S ~ RiFD (λ1 λ2 + λ1 λ3 + λ2 λ3), (4b)
R ~ RiFD (λ1 + λ2 + λ3). (4c)
The scaling iterative procedure of FD-power changes (Fig.4b) exhibits three regimes of behaviour: stable regime (plateau) (I), transition and linear collapsing regime (II), and recovered stable regime (plateau) (III). The linear collapsing regime approaches the attractor points corresponding to the random surface (ADC profile). In the stable regime (I) for the volume, surface and mean radius values FD-power was 3, 2 and 1 correspondently. These are the dimensions of the initial, non-corrupted iteratively blown surfaces. Collapsing regime (II) of non-integer topological indices demonstrates geometrical contrast between the sphere and the ellipsoid (Fig.4b). Contrast is clearly depicted in the maps of the collapsed parameters (attractor points of FD-power) which are presented in Fig.5 (c,e,g correspond to the ellipsoid and d,f,h correspond to the sphere). For comparison, we also present fractional anisotropy (FA) and normalized mean diffusivity (MD) maps as well in Fig.5a,b respectively.
DISCUSSION AND CONCLUSIONS
We presented a method to characterize the complex structure of the ADC profiles. The key point of the method comprises the iterative scaling of the blowing surfaces: a sphere and an ellipsoid. With the use of the proposed method, a new set of parameters were mapped. These parameters represent the topology of the ADC profiles and correlate with the anatomical structure of the brain. The method provides a novel approach that may enhance the utility and specificity of the diffusion-weighted MRI to better assess the structural changes that occur during the development and progression of various neuro-pathologies.
To summarize, by measuring FD, we have shown that profiles with the same area, the same volume or the same radius can exhibit highly heterogeneous outline structures. Indeed, outline structure is the most important parameter in the qualitative morphological classification of ADC. We showed that outline structures exhibited significant differences in our automatically measured parameters. We showed that FD can be used as a surrogate parameter for the determination of the roughness of the ADC outline.
We hypothesize that with this method for the sensitive quantitative analysis of outlines, potential differences in shape and the dynamics of its change can now be numerical evaluated in different physiological and pathophysiological conditions. This will improve our understanding of the role of diffusivity in the development of illness that is accompanied by increased molecular activity, such as inflammatory diseases.
The precise quantitative evaluation of outline shape is of interest in the medical applications, as well as in materials research. Future investigations should seek to precisely evaluate the influence of pathological states on outline surface with a focus on the newly described descriptors to explore the relationships between outline shape and its physiological and pathophysiological conditions. ■
INTRODUCTION
Measurement of diffusion in magnetic resonance imaging (MRI) is one of the most sensitive ways to explore the heterogeneity of biological tissue non-invasively. Small differences in diffusivity within tissue placed in a magnetic field give rise to effects which are well measurable in diffusion weighted MRI. Calculation of the apparent diffusion coefficient (ADC) over the various diffusion gradient directions in MRI allows us to investigate the anisotropic properties of biological tissue in every voxel (or pixel) using the tensor model [1] or spherical harmonics decomposition [2].
The improvements in digital imaging and computer-based analysis, as well as advances in new mathematical methods such as fractal geometry [3], are enabling new approaches to the geometrical description of shape. Fractal analysis is being increasingly applied to a variety of biological questions [4], and has proven useful in the analysis of medical images [5]. With the introduction of fractal geometries in the early 1980s a new parameter, the so-called fractal dimension (FD), was defined. The FD essentially measures the irregularity or roughness of a shape or surface and thus, FD is a quantitative descriptor of the actual geometrical shape [3]. The determination of the FD outlines enables the attribution of numerical values to peculiar qualitative features of spatial structures such as irregularity, morphologic complexity and roughness, which can be used to characterize images of different origins or in different functional and/or pathologic states.
With the aim of contributing to the clarification of the anisotropy and heterogeneity within the voxel, we introduce an alternative blowing compact-surface method leading to the setting the new group of scalar parameters – topological non-integer dimensions [6, 7]. The blowing sphere and ellipsoid are chosen as probing surfaces. Our findings suggest that non-Gaussian distribution of the diffusion signal dominates in the biological tissue and carries more detailed information.
This article describes an automated and thus standardized approach for the quantification of shapes. It is based on a combination of computer-based tools for image analysis and pattern recognition. In this study, images were acquired using magnetic resonance phenomenon, which provides excellent contrast for the recognition of surface structures of ADCs. Outlines of shapes were automatically segmented from digital images using a newly developed algorithm. Geometrical parameters of shape were then prospectively evaluated; this included the determination of the FD, the calculation of shape descriptors such as volume and surface area as well as mean radius. We provide detailed quantitative information on the topological features of shapes, describing for the first time the relationship between the complex geometry of ADCs and FD, which is useful for the automated, standardized and reproducible description of irregular surfaces.
RESEARCH METHODS
65 magnetic resonance images of the human brain, downloaded from public database [8], were acquired on a 3Tesla magnetic resonance scanner using diffusion-weighted spin echo sequence (Fig.1a). 60 diffusion-weighted images (DWI), Sk∈{1,...,60}, with b = 1 000 [sec/mm2], corresponding to the sixty directions, gk∈{1,...,60} = (gxgygz)k∈{1,...,60}, of diffusion gradients distributed homogeneously over the sphere (Fig.1b), were used. Additionally, five images S0 with b=0, were randomly acquired between the diffusion-weighted images to factor out the T2–weighting of the signal. All images were corrected according the procedure presented in Fig.2 [9, 10]. In the Fig.3 the segmented anatomical image of a transverse slice (right side) is shown and the DWI profiles of signal (left side) are output for voxels located in different types of tissues of the brain. The profiles demonstrated a very complex structure, correlated with the anatomical architecture. For these profiles it is possible to reconstruct a diffusion tensor image (DTI), or coefficients of fiber orientation density function (fODF) using Moore-Penrose method [1, 2]:
D = G+s', (1)
where quadratic pseudo-inverse matrices, , and ADC values,
s'T = (In ( S1 / S0) ... In ( Sn / S0) ... In ( S60 / S0)), (2)
are presented respectively. Augmented matrices, B↔aug, used for DTI and for fODF can be found in Table 1. From Eq.(1) in the case of DTI mean diffusivity (MD), , and fractional anisotropy (FA), , can be estimated. Here Tr is a trace of matrices, i.e. .
The ADC profiles recovered with Eq.2 [11] and afterwards interpolated for voxels within and outside the brain can be analyzed by the newly proposed iterative blowing surface method [6, 7]. As a probing surface, the blowing sphere was chosen for the isotropic case (Fig.4a, upper row). For the anisotropic case, an ellipsoid built on eigenvectors with kept ratio between different eigenvalues was selected (Fig.4a, low row). The key point of this method consists in the step-by-step increase of the characteristic scale over all diffusion gradient directions from the minimal to the maximal value. During expansion of the surface in the certain direction the growth was stopped if the scale exceeded the acquired ADC profile boundary in a given direction. In the Fig.4a, the 3D process is depicted with some intermediate steps in plane (in these figures the equatorial cross sections of the ADC profiles are presented).
Fractal dimension was used as a measure for the roughness of ADC surfaces, i.e. the spatial heterogeneity of their outline. Classically, FD is strictly defined only for self-similar objects that exhibit the same structure at different length scales Ri. Different approaches to measure FD exist. Here, the adjusting power method was applied, which works as follows: The outline of ADC for a single voxel was reconstructed (1), the minimum and maximal values of ADC were detected (2), eigenvalues and eigenvectors were calculated (3), sphere and ellipsoid were built and their geometric properties, i.e. volume (V), surface area (S), and mean radius (R), were expressed via power law of the scale Ri (4). The process of building of sphere and ellipsoid was then repeated for different length scales, i.e. from minimal to the maximal in order to cover the whole ADC profile. For perfectly self-similar objects, the plot of log (Ri) versus log (S), log (V), or log (R), is a straight line. In our case, the FD is given by the integer and non-integer values, which is a characteristic for non-perfect fractal behavior. FD is expressed as a tangent of a line in this case.
RESULTS
We calculated the volume V, surface area S and mean radius R during the blowing sphere and ellipsoid iterative scaling process and compared them to the unit measures by adjusting the power FD of the scaling parameter Ri of the surfaces (Eq.3, 4; i is an iteration number). For the case of ellipsoid λ1, λ2, λ3 are the eigenvalues.
M | M = V, S, R ~ RiFD, (3)
V ~ RiFD (λ1 λ2 λ3), (4a)
S ~ RiFD (λ1 λ2 + λ1 λ3 + λ2 λ3), (4b)
R ~ RiFD (λ1 + λ2 + λ3). (4c)
The scaling iterative procedure of FD-power changes (Fig.4b) exhibits three regimes of behaviour: stable regime (plateau) (I), transition and linear collapsing regime (II), and recovered stable regime (plateau) (III). The linear collapsing regime approaches the attractor points corresponding to the random surface (ADC profile). In the stable regime (I) for the volume, surface and mean radius values FD-power was 3, 2 and 1 correspondently. These are the dimensions of the initial, non-corrupted iteratively blown surfaces. Collapsing regime (II) of non-integer topological indices demonstrates geometrical contrast between the sphere and the ellipsoid (Fig.4b). Contrast is clearly depicted in the maps of the collapsed parameters (attractor points of FD-power) which are presented in Fig.5 (c,e,g correspond to the ellipsoid and d,f,h correspond to the sphere). For comparison, we also present fractional anisotropy (FA) and normalized mean diffusivity (MD) maps as well in Fig.5a,b respectively.
DISCUSSION AND CONCLUSIONS
We presented a method to characterize the complex structure of the ADC profiles. The key point of the method comprises the iterative scaling of the blowing surfaces: a sphere and an ellipsoid. With the use of the proposed method, a new set of parameters were mapped. These parameters represent the topology of the ADC profiles and correlate with the anatomical structure of the brain. The method provides a novel approach that may enhance the utility and specificity of the diffusion-weighted MRI to better assess the structural changes that occur during the development and progression of various neuro-pathologies.
To summarize, by measuring FD, we have shown that profiles with the same area, the same volume or the same radius can exhibit highly heterogeneous outline structures. Indeed, outline structure is the most important parameter in the qualitative morphological classification of ADC. We showed that outline structures exhibited significant differences in our automatically measured parameters. We showed that FD can be used as a surrogate parameter for the determination of the roughness of the ADC outline.
We hypothesize that with this method for the sensitive quantitative analysis of outlines, potential differences in shape and the dynamics of its change can now be numerical evaluated in different physiological and pathophysiological conditions. This will improve our understanding of the role of diffusivity in the development of illness that is accompanied by increased molecular activity, such as inflammatory diseases.
The precise quantitative evaluation of outline shape is of interest in the medical applications, as well as in materials research. Future investigations should seek to precisely evaluate the influence of pathological states on outline surface with a focus on the newly described descriptors to explore the relationships between outline shape and its physiological and pathophysiological conditions. ■
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