Orientation Density Function Research Articles

A transverse isotropic constitutive model for the aortic valve tissue incorporating rate-dependency and fibre dispersion: Application to biaxial deformation.

This paper presents a continuum-based transverse isotropic model incorporating rate-dependency and fibre dispersion, applied to the planar biaxial deformation of aortic valve (AV) specimens under various stretch rates. The rate dependency of the mechanical behaviour of the AV tissue under biaxial deformation, the (pseudo-) invariants of the right Cauchy-Green deformation-rate tensor Ċ associated with fibre dispersion, and a new fibre orientation density function motivated by fibre kinematics are presented for the first time. It is shown that the model captures the experimentally observed deformation of the specimens, and characterises a shear-thinning behaviour associated with the dissipative (viscous) kinematics of the matrix and the fibres. The application of the model for predicting the deformation behaviour of the AV under physiological rates is illustrated and an example of the predicted σ-λ curves is presented. While the development of the model was principally motivated by the AV biomechanics requisites, the comprehensive theoretical approach employed in the study renders the model suitable for application to other fibrous soft tissues that possess similar rate-dependent and structural attributes.

Deformation and Fabric in Compacted Clay Soils

Hydro-mechanical anisotropy of clay soils in response to deformation or deposition history is related to the micromechanics of plate-like clay particles and their orientations. In this letter, we examine the relationship between microstructure, deformation and moisture content in kaolin clay using a technique based on neutron scattering. This technique allows for the direct characterisation of the microstructure within representative samples using traditional measures such as the orientation density function and soil fabric tensor. From this information, evidence for a simple relationship between components of the deviatoric strain tensor and the deviatoric fabric tensor emerged. This relationship provides a physical basis for future anisotropic constitutive models based on the micromechanics of these materials.

Modeling white matter microstructure with fiber ball imaging

Fiber ball imaging (FBI) provides a means of calculating the fiber orientation density function (fODF) in white matter from diffusion MRI (dMRI) data obtained over a spherical shell with a b-value of about 4000 s/mm2 or higher. By supplementing this FBI-derived fODF with dMRI data acquired for two lower b-value shells, it is shown that several microstructural parameters may be estimated, including the axonal water fraction (AWF) and the intrinsic intra-axonal diffusivity. This fiber ball white matter (FBWM) modeling method is demonstrated for dMRI data acquired from healthy volunteers, and the results are compared with those of the white matter tract integrity (WMTI) method. Both the AWF and the intra-axonal diffusivity obtained with FBWM are found to be significantly larger than for WMTI, with the FBWM values for the intra-axonal diffusivity being more consistent with recent results obtained using isotropic diffusion weighting. An important practical advantage of FBWM is that the only nonlinear fitting required is the minimization of a cost function with just a single free parameter, which facilitates the implementation of efficient and robust numerical routines.

Over length quantification of the multiaxial mechanical properties of the ascending, descending and abdominal aorta using Digital Image Correlation.

In this paper, we hypothesize that the biaxial mechanical properties of the aorta may be dependent on arterial location. To demonstrate any possible position-related difference, our study analyzed and compared the biaxial mechanical properties of the ascending thoracic aorta, descending thoracic aorta and infrarenal abdominal aorta stemming from the same porcine subjects, and reported values of constitutive parameters for well-known strain energy functions, showing how these mechanical properties are affected by location along the aorta. When comparing ascending thoracic aorta, descending thoracic aorta and infrarenal abdominal aorta, abdominal tissues were found to be stiffer and highly anisotropic. We found that the aorta changed from a more isotropic to a more anisotropic tissue and became progressively less compliant and stiffer with the distance to the heart. We observed substantial differences in the anisotropy parameter between aortic samples where abdominal samples were more anisotropic and nonlinear than the thoracic samples. The phenomenological model was not able to capture the passive biaxial properties of each specific porcine aorta over a wide range of biaxial deformations, showing the best prediction root mean square error ε=0.2621 for ascending thoracic samples and, especially, the worst for the infrarenal abdominal samples ε=0.3780. The micro-structured model with Bingham orientation density function was able to better predict biaxial deformations (ε=0.1372 for ascending thoracic aorta samples). The root mean square error of the micro-structural model and the micro-structured model with von Mises orientation density function were similar for all positions.

Single-crystal Ih ice surfaces unveil connection between macroscopic and molecular structure

Physics and chemistry of ice surfaces are not only of fundamental interest but also have important impacts on biological and environmental processes. As ice surfaces-particularly the two prism faces-come under greater scrutiny, it is increasingly important to connect the macroscopic faces with the molecular-level structure. The microscopic structure of the ubiquitous ice Ih crystal is well-known. It consists of stacked layers of chair-form hexagonal rings referred to as molecular hexagons. Crystallographic unit cells can be assembled into a regular right hexagonal prism. The bases are labeled crystallographic hexagons. The two hexagons are rotated 30° with respect to each other. The linkage between the familiar macroscopic shape of hexagonal snowflakes and either hexagon is not obvious per se. This report presents experimental data directly connecting the macroscopic shape of ice crystals and the microscopic hexagons. Large ice single crystals were used to fabricate samples with the basal, primary prism, or secondary prism faces exposed at the surface. In each case, the same sample was used to capture both a macroscopic etch pit image and an electron backscatter diffraction (EBSD) orientation density function (ODF) plot. Direct comparison of the etch pit image and the ODF plot compellingly connects the macroscopic etch pit hexagonal profile to the crystallographic hexagon. The most stable face at the ice-water interface is the smallest area face at the ice-vapor interface. A model based on the molecular structure of the prism faces accounts for this switch.

Quantifying the brain's sheet structure with normalized convolution.

The hypothesis that brain pathways form 2D sheet-like structures layered in 3D as "pages of a book" has been a topic of debate in the recent literature. This hypothesis was mainly supported by a qualitative evaluation of "path neighborhoods" reconstructed with diffusion MRI (dMRI) tractography. Notwithstanding the potentially important implications of the sheet structure hypothesis for our understanding of brain structure and development, it is still considered controversial by many for lack of quantitative analysis. A means to quantify sheet structure is therefore necessary to reliably investigate its occurrence in the brain. Previous work has proposed the Lie bracket as a quantitative indicator of sheet structure, which could be computed by reconstructing path neighborhoods from the peak orientations of dMRI orientation density functions. Robust estimation of the Lie bracket, however, is challenging due to high noise levels and missing peak orientations. We propose a novel method to estimate the Lie bracket that does not involve the reconstruction of path neighborhoods with tractography. This method requires the computation of derivatives of the fiber peak orientations, for which we adopt an approach called normalized convolution. With simulations and experimental data we show that the new approach is more robust with respect to missing peaks and noise. We also demonstrate that the method is able to quantify to what extent sheet structure is supported for dMRI data of different species, acquired with different scanners, diffusion weightings, dMRI sampling schemes, and spatial resolutions. The proposed method can also be used with directional data derived from other techniques than dMRI, which will facilitate further validation of the existence of sheet structure.

Construction of weighted crystallographic orientations capturing a given orientation density function

To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initially kernel estimated orientation density function and in particular its non-negativity, or to preserve the unbiased estimates of the first Fourier coefficients up to a given finite order. Methods are presented how to construct a much smaller set of weighted orientations such that their kernel density estimate approximates the initial estimate. To preserve its overall shape the de la Vallee Poussin kernel is applied as it is the only known non-negative kernel with a finite Fourier series expansion avoiding truncation errors. If the first Fourier coefficients are to be preserved the Dirichlet kernel applies as it is the only kernel providing unbiased estimates of the Fourier coefficients up to any given finite order. The weights are determined numerically by resolving a least squares or a maximum likelihood problem. Due to the linearity of kernel density estimation and the Fourier transform the approaches in spatial and spectral domain are related to each other in a unique complementary way. For an exemplary practical application we use a large EBSD dataset of about 80.000 orientations from a recrystallized low alloyed Zirconium sheet. Our methods reduce the size of the dataset by about \(99.75\,\%\) to the order of 200 weighted orientations supporting a secondary approximate distribution with a volume portion of crystallites oriented differently than initially of less than \(10\,\%\).

An Active Contraction Model of Valvular Interstitial Cells

Valvular interstitial cells (VICs) play a critical role in the pathophysiology of heart valves. The objective of this study was to investigate the roles of different subcellular structures of the VICs, especially α-SMA stress fibers, to the VIC mechanical responses under different mechanical loading conditions and activation states. We modeled the VIC as a continuum with two distinct domains: the cell nucleus and cytoplasm. The nucleus was modeled as an incompressible neo-Hookean material while the cytoplasm was modeled as a mixture of two solid phases: the basal, isotropic cytoskeleton phase and α-SMA stress fiber phase, which exhibit some orientations at each point described by an orientation density function. The α-SMA stress fibers also exhibit passive elastic and active contractile responses in the direction of the orientations. We developed VIC mechanical model, which integrated the data from two experiments: micropipette aspiration (MA) and atomic force microscopy (AFM) of the aortic VIC (AVIC) and pulmonary VIC (PVIC) that each exhibits different expression levels of the α-SMA and contraction strength. In the MA experiment, VICs are in inactivated states while in AFM experiment, VICs are in activated states, exhibiting higher level of α-SMA expression and contraction. Thus, using our model in conjunction with the experimental data, we investigated how the expression level and active contraction of the α-SMA fibers affect the mechanical responses of the VICs. We implemented our model on the finite element method. We determined a ∼10-fold difference in the contractions strength between the AVICs and PVICs, implying that not only the expression level of the α-SMA fibers but also the contraction level increases from PVICs compared to AVICs. The higher expression of the α-SMA fibers may facilitate more efficient contraction and increased ability to synthesize collagen and other ECM components.

Microstructural quantification of collagen fiber orientations and its integration in constitutive modeling of the porcine carotid artery

BackgroundMechanical characteristics of vascular tissue may play a role in different arterial pathologies, which, amongst others, requires robust constitutive descriptions to capture the vessel wall’s anisotropic and non-linear properties.Specifically, the complex 3D network of collagen and its interaction with other structural elements has a dominating effect of arterial properties at higher stress levels.The aim of this study is to collect quantitative collagen organization as well as mechanical properties to facilitate structural constitutive models for the porcine carotid artery.This helps the understanding of the mechanics of swine carotid arteries, being a standard in clinical hypothesis testing, in endovascular preclinical trials for example. MethodPorcine common carotid arteries (n=10) were harvested and used to (i) characterize the collagen fiber organization with polarized light microscopy, and (ii) the biaxial mechanical properties by inflation testing.The collagen organization was quantified by the Bingham orientation density function (ODF), which in turn was integrated in a structural constitutive model of the vessel wall.A one-layered and thick-walled model was used to estimate mechanical constitutive parameters by least-square fitting the recorded in vitro inflation test results.Finally, uniaxial data published elsewhere were used to validate the mean collagen organization described by the Bingham ODF. ResultsThick collagen fibers, i.e.the most mechanically relevant structure, in the common carotid artery are dispersed around the circumferential direction.In addition, almost all samples showed two distinct families of collagen fibers at different elevation, but not azimuthal, angles.Collagen fiber organization could be accurately represented by the Bingham ODF (κ1,2,3=[13.5,0.0,25.2] and κ1,2,3=[14.7,0.0,26.6]; average error of about 5%), and their integration into a structural constitutive model captured the inflation characteristics of individual carotid artery samples.Specifically, only four mechanical parameters were required to reasonably (average error from 14% to 38%) cover the experimental data over a wide range of axial and circumferential stretches.However, it was critical to account for fibrilar links between thick collagen fibers.Finally, the mean Bingham ODF provide also good approximation to uniaxial experimental data. ConclusionsThe applied structural constitutive model, based on individually measured collagen orientation densities, was able to capture the biaxial properties of the common carotid artery. Since the model required coupling amongst thick collagen fibers, the collagen fiber orientations measured from polarized light microscopy, alone, seem to be insufficient structural information. Alternatively, a larger dispersion of collagen fiber orientations, that is likely to arise from analyzing larger wall sections, could have had a similar effect, i.e. could have avoided coupling amongst thick collagen fibers. Statement of SignificanceThe applied structural constitutive model, based on individually measured collagen orientation densities, was able to capture the biaxial and uniaxial properties of the common carotid artery. Since the model required coupling amongst thick collagen fibers, an effective orientation density that accounts for cross-links between the main collagen fibers has been porposed. The model provides a good approximation to the experimental data.

An active contraction model of the valvular interstitial cell

An active contraction model of the valvular interstitial cell

Using simulations with realistic fibrous network geometry to determine the achievable ranges of macroscopic mechanical behaviors of elastomeric scaffolds

Event Abstract Back to Event Using simulations with realistic fibrous network geometry to determine the achievable ranges of macroscopic mechanical behaviors of elastomeric scaffolds Michael S Sacks1, 2 and Greg Rodin2 1 The University of Texas at Austin, Biomedical Engineering, United States 2 the University of Texas at Austin, Department of Aerospace and Mechanical Engineering, United States Introduction: Tissue engineers seek to produce tissues that function biologically and mechanically as well as the native tissues they are built to replace. This macroscopic behavior is strongly influenced by the evolution of the complex microstructural geometry of the underlying scaffold. Elastomeric scaffolds manufactured by the electrospinning process produces layers of long-fiber networks. It has been found that knowledge of the fiber properties, fiber density, and orientation density function are sufficient to predict the bulk mechanical behavior. However, the micro- and meso-scale mechanisms responsible for the behavior remain unclear. Initial simulations based on experimentally derived geometry suggest that the load induced emergence of long fiber paths may be a mechanism contributing to the observed bulk mechanical behavior. Methods: The purpose of this study was to develop and utilize computational models, based on realistic fiber geometry, to understand the mechanisms that translate scaffold fiber network structure into tissue simulating function. We also explored the range of macroscopic material behaviors that are achievable from the domain of producible microstructural geometries and elastomeric fiber properties, both essential first steps in developing the ability to design engineered tissues that mimic native tissue behavior. Results: We performed 3-D, non-linear micromechanical simulations of layers of long-fiber networks, generated using a random walk procedure that mimics the scaffold structure. Using these realistic fiber networks, we are able to investigate the mechanisms responsible for the stress-strain curve in ways not possible previously. For example, by independently varying the initial geometric anisotropy and the initial tortuosity, we separated the effects of each of these variables on the nonlinearity of the stress-straincurve. Discussion: Insights such as these inform macroscale material models that can be used to guide the design of scaffolds and the selection of manufacturing parameters so that the resulting engineered tissues mimic the non-linear mechanical behavior of the native tissues. NIH R01 HL68816 Keywords: modeling, Scaffold, microstructure, 3D scaffold Conference: 10th World Biomaterials Congress, Montréal, Canada, 17 May - 22 May, 2016. Presentation Type: Poster Topic: Computational modeling in biomaterials science and engineering Citation: Sacks M and Rodin G (2016). Using simulations with realistic fibrous network geometry to determine the achievable ranges of macroscopic mechanical behaviors of elastomeric scaffolds. Front. Bioeng. Biotechnol. Conference Abstract: 10th World Biomaterials Congress. doi: 10.3389/conf.FBIOE.2016.01.03023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Mar 2016; Published Online: 30 Mar 2016. Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Michael S Sacks Greg Rodin Google Michael S Sacks Greg Rodin Google Scholar Michael S Sacks Greg Rodin PubMed Michael S Sacks Greg Rodin Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Inverse Funk Transform of Diffusion MRI Signal as an Estimate for Fiber Orientation Density Function

Inverse Funk Transform of Diffusion MRI Signal as an Estimate for Fiber Orientation Density Function

Macroscopic modeling of anisotropic magnetostriction and magnetization in soft ferromagnetic materials

Magnetic and magnetoelastic properties of soft ferromagnetic materials, used as laminated sheets, are sensitive to manufacturing processes such as rolling, cutting and coating. One of the effects of these processes is to induce an anisotropic behavior of materials. Therefore, an anhysteretic magnetostriction and magnetization calculation taking into account the anisotropy effect at macroscopic scale is presented. This model is based on the expression and then the minimization of the total energy in order to determine magnetization and magnetostriction at equilibrium. The total energy to minimize depends on energy terms identified from measurements of the magnetization and magnetostriction at a scale large enough to neglect the heterogeneity due to grains. Therefore, this approach attempts to reproduce ferromagnetic polycrystal behavior at macroscopic without knowing texture (Orientation Density Function) nor grain properties.

Optimally conductive networks in randomly dispersed CNT:graphene hybrids

A predictive model is proposed that quantitatively describes the synergistic behavior of the electrical conductivities of CNTs and graphene in CNT:graphene hybrids. The number of CNT-to-CNT, graphene-to-graphene, and graphene-to-CNT contacts is calculated assuming a random distribution of CNTs and graphene particles in the hybrids and using an orientation density function. Calculations reveal that the total number of contacts reaches a maximum at a specific composition and depends on the particle sizes of the graphene and CNTs. The hybrids, prepared using inkjet printing, are distinguished by higher electrical conductivities than that of 100% CNT or graphene at certain composition ratios. These experimental results provide strong evidence that this approach involving constituent element contacts is suitable for investigating the properties of particulate hybrid materials.

Fiber ball imaging

By modeling axons as thin cylinders, it is shown that the inverse Funk transform of the diffusion MRI (dMRI) signal intensity obtained on a spherical shell in q-space gives an estimate for a fiber orientation density function (fODF), where the accuracy improves with increasing b-value provided the signal-to-noise ratio is sufficient. The method is similar to q-ball imaging, except that the Funk transform of q-ball imaging is replaced by its inverse. We call this new approach fiber ball imaging. The fiber ball method is demonstrated for healthy human brain, and fODF estimates are compared to diffusion orientation distribution function (dODF) approximations obtained with q-ball imaging. The fODFs are seen to have sharper features than the dODFs, reflecting an enhancement of the higher degree angular frequencies. The inverse Funk transform of the dMRI signal intensity data provides a simple and direct method of estimating a fODF. In addition, fiber ball imaging leads to an estimate for the ratio of the fraction of MRI visible water confined to the intra-axonal space divided by the square root of the intra-axonal diffusivity. This technique may be useful for white matter fiber tractography, as well as other types of microstructural modeling of brain tissue.

Quantifying texture evolution during hot rolling of AZ31 Twin Roll Cast strip

Multi-pass rolling experiments with an AZ31 Twin Roll Cast (TRC) alloy were performed on an industrial scaled four-high rolling mill. Within the rolling with an intermediate annealing the evolution of texture was investigated. To quantify the extent of preferred crystallographic orientation experimental X-ray pole figures were measured after different process steps and analyzed using the free and open Matlab® toolbox MTEX for texture analysis. The development of the fiber texture was observed and analyzed in dependence on rolling conditions. In the initial state the specimen exhibits a texture composed of a weak basal texture and a cast texture with {0001}-planes oriented across the rolling direction. During the following rolling process a fiber texture was developed. The expected strength increment of the fiber texture was quantitatively confirmed in terms of volume portions of the orientation density function around the fiber and in terms of the canonical parameters of fitted pseudo Bingham distributions. On the results of this work a model for prediction of the texture evolution during the strip rolling of magnesium in the examined parameter range was developed.

Dilatometric and microstructural response of variant selection during α′ transformation

Variant selection during displacive transformation is popularly monitored by assessing the pole figure or orientation density function of the bulk texture or pole figures from single prior austenite grains. It can also be done by measuring the physical orientation of α′ (bainite or marteniste) plates in the microstructure. It is known that when variants of α′ form randomly, i.e. there is no variant selection, the shear strain associated with each α′ plate gets canceled and transformation strain becomes equal to the volume strain. Under this condition, orientation of sheaves of α′ also shows random distribution. In this work, it has been shown how transformation strain and orientation of sheaves change when variant selection occurs. Mathematical models have already been developed to calculate the transformation strain under various loading conditions; extensive experiments have been conducted during the course of this work to show how the nature of externally applied stress influences the transformation strain. In addition to this, it has also been shown that the variant selection might leave its trace in the microstructure by affecting the alignment of the sheaves under externally applied stress and strain. Mathematical models have been developed to show that variant selection has negligible influence on physical orientation of α′ plates when transformation occurs under external stress, but prior plastic deformation can dictate the orientation to a great extent.

A theory for deformation dependent evolution of continuous fibre distribution applicable to collagen remodelling

This work explores the implications of a constitutive framework that introduces a particular remodelling process for materials having a continuous distribution of fibre directions. It is assumed that the matrix constituent is unaffected and that fibres remodel by a process of dissolution and reassembly. Dissolution results from a time dependent chemical kinetic process that is influenced by fibre stretch. Two possibilities for the reassembly of the fibres are considered: (1) fibres reassemble in a stress-free state or (2) fibres assemble in a pre-strained state matching that of the surrounding matrix. The composite is treated as a non-linear hyperelastic material. The constitutive framework accounts for the time dependent redistribution of fibre orientation and density. A directional fibre orientation density function is defined that combines fibre dissolution, reassembly and their dependence on deformation history. Expressions for the time dependence of this function are developed for the two fibre reassembly assumptions and different deformation histories. Numerical results for uniaxial extension are presented in the form of polar plots that illustrate fibre orientation density at different times and fibre density versus time plots that show evolution at different orientations.

Beyond fractional anisotropy: Extraction of bundle-specific structural metrics from crossing fiber models

Diffusion MRI (dMRI) measurements are used for inferring the microstructural properties of white matter and to reconstruct fiber pathways. Very often voxels contain complex fiber configurations comprising multiple bundles, rendering the simple diffusion tensor model unsuitable. Multi-compartment models deliver a convenient parameterization of the underlying complex fiber architecture, but pose challenges for fitting and model selection. Spherical deconvolution, in contrast, very economically produces a fiber orientation density function (fODF) without any explicit model assumptions. Since, however, the fODF is represented by spherical harmonics, a direct interpretation of the model parameters is impossible. Based on the fact that the fODF can often be interpreted as superposition of multiple peaks, each associated to one relatively coherent fiber population (bundle), we offer a solution that seeks to combine the advantages of both approaches: first the fiber configuration is modeled as fODF represented by spherical harmonics and then each of the peaks is parameterized separately in order to characterize the underlying bundle. In this work, the fODF peaks are approximated by Bingham distributions, capturing first and second-order statistics of the fiber orientations, from which we derive metrics for the parametric quantification of fiber bundles. We propose meaningful relationships between these measures and the underlying microstructural properties. We focus on metrics derived directly from properties of the Bingham distribution, such as peak length, peak direction, peak spread, integral over the peak, as well as a metric derived from the comparison of the largest peaks, which probes the complexity of the underlying microstructure. We compare these metrics to the conventionally used fractional anisotropy (FA) and show how they may help to increase the specificity of the characterization of microstructural properties. While metrics relying on the first moments of the Bingham distributions provide relatively robust results, second-order metrics representing the peak spread are only meaningful, if the SNR is very high and no fiber crossings are present in the voxel.

Quantifying texture evolution during hot rolling of magnesium Twin Roll Cast strip

Multi-pass rolling experiments with an AZ31 Twin Roll Cast (TRC) alloy were performed on an industrial scaled four-high rolling mill. During rolling with an intermediate annealing the evolution of texture and microstructure was investigated. To quantify the extent of preferred crystallographic orientation experimental X-ray pole figures were measured after different process steps and analyzed using the free and open Matlab® toolbox mtex for texture analysis. The evolution of both the fiber texture and the microstructure was observed and analyzed in dependence on rolling conditions. In the initial state the specimen exhibits a texture composed of a weak basal texture and a cast texture with {0001}-planes oriented across the rolling direction. During the following rolling process a fiber texture evolved. The expected strength increment of the fiber texture was quantitatively confirmed in terms of volume portions of the orientation density function around the fiber and in terms of the canonical parameters of fitted pseudo Bingham distributions. Based on these results a model to predict the texture evolution during the strip rolling of magnesium in the examined parameter range was developed.