“The common descent of biological shape description and special functions”. Gielis J, Caratelli D, de Jong van Coevorden M, Ricci PE page 119 (2018).
Abstract: Gielis transformations, with their origin in botany, are used to define square waves and trigonometric functions of higher order. They are rewritten in terms of Chebyshev polynomials. The origin of both, a uniform descriptor and the origin of orthogonal polynomials, can be traced back to a letter of Guido Grandi to Leibniz in 1713 on the mathematical description of the shape of flowers. In this way geometrical description and analytical tools are seamlessly combined.
Keywords: H1 Book chapter; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1007/978-3-319-75647-9_10
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“Comparison of dwarf bamboos (Indocalamus sp.) leaf parameters to determine relationship between spatial density of plants and total leaf area per plant”. Shi P-J, Xu Q, Sandhu HS, Gielis J, Ding Y-L, Li H-R, Dong X-B, Ecology and evolution 5, 4578 (2015). http://doi.org/10.1002/ECE3.1728
Abstract: The relationship between spatial density and size of plants is an important topic in plant ecology. The self-thinning rule suggests a −3/2 power between average biomass and density or a −1/2 power between stand yield and density. However, the self-thinning rule based on total leaf area per plant and density of plants has been neglected presumably because of the lack of a method that can accurately estimate the total leaf area per plant. We aimed to find the relationship between spatial density of plants and total leaf area per plant. We also attempted to provide a novel model for accurately describing the leaf shape of bamboos. We proposed a simplified Gielis equation with only two parameters to describe the leaf shape of bamboos one model parameter represented the overall ratio of leaf width to leaf length. Using this method, we compared some leaf parameters (leaf shape, number of leaves per plant, ratio of total leaf weight to aboveground weight per plant, and total leaf area per plant) of four bamboo species of genus Indocalamus Nakai (I. pedalis (Keng) P.C. Keng, I. pumilus Q.H. Dai and C.F. Keng, I. barbatus McClure, and I. victorialis P.C. Keng). We also explored the possible correlation between spatial density and total leaf area per plant using log-linear regression. We found that the simplified Gielis equation fit the leaf shape of four bamboo species very well. Although all these four species belonged to the same genus, there were still significant differences in leaf shape. Significant differences also existed in leaf area per plant, ratio of leaf weight to aboveground weight per plant, and leaf length. In addition, we found that the total leaf area per plant decreased with increased spatial density. Therefore, we directly demonstrated the self-thinning rule to improve light interception.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1002/ECE3.1728
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“Cytokinin dynamics in cell suspension cultures of Bambusa balcooa Roxburgh using UPLC-ESI/MS/MS”. Van den Akker S, Bormans P, Peeters H, Gielis J, Prinsen E page 539 (2012).
Keywords: H3 Book chapter; Engineering sciences. Technology; Integrated Molecular Plant Physiology Research (IMPRES); Sustainable Energy, Air and Water Technology (DuEL)
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“Design of electroporation process in irregularly shaped multicellular systems”. Mescia L, Chiapperino MA, Bia P, Lamacchia CM, Gielis J, Caratelli D, Electronics (Basel) 8, 37 (2019). http://doi.org/10.3390/ELECTRONICS8010037
Abstract: Electroporation technique is widely used in biotechnology and medicine for the transport of various molecules through the membranes of biological cells. Different mathematical models of electroporation have been proposed in the literature to study pore formation in plasma and nuclear membranes. These studies are mainly based on models using a single isolated cell with a canonical shape. In this work, a spacetime (x,y,t) multiphysics model based on quasi-static Maxwells equations and nonlinear Smoluchowskis equation has been developed to investigate the electroporation phenomenon induced by pulsed electric field in multicellular systems having irregularly shape. The dielectric dispersion of the cell compartments such as nuclear and plasmatic membranes, cytoplasm, nucleoplasm and external medium have been incorporated into the numerical algorithm, too. Moreover, the irregular cell shapes have been modeled by using the Gielis transformations.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.3390/ELECTRONICS8010037
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“Design of irregularly shaped lens antennas including supershaped feed”. Mescia L, Lamacchia CM, Chiapperino MA, Bia P, Gielis J, Caratelli D, Progress in Electromagnetic Research Symposium (PIERS)
T2 –, 2019 PhotonIcs &, Electromagnetics Research Symposium –, Spring (PIERS-Spring), 17-20 June, 2019, Rome, Italy , 169 (2019). http://doi.org/10.1109/PIERS-SPRING46901.2019.9017900
Abstract: A new class of irregularly shaped dielectric lens antennas with a supershaped microstrip antenna feeder is presented and detailed in this work. The surface of the lens antenna and the feeder shape have been modelled by using the three and two-dimensional Gielis formula, respectively. The antenna design has been carried out by integrating an home-made software tool with the CST Microwave Studio®. The radiation properties of the whole antenna system have been evaluated using a dedicated high-frequency technique based on the tube tracing approximation. Moreover, the effects due to the multiple internal reflections have been properly modeled. The proposed model was applied to study unusual and complex lens antenna systems with the aim to design special radiation characteristics.
Keywords: P1 Proceeding; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1109/PIERS-SPRING46901.2019.9017900
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“Diatom frustule morphogenesis and function : a multidisciplinary survey”. De Tommasi E, Gielis J, Rogato A, Marine Genomics 35, 1 (2017). http://doi.org/10.1016/J.MARGEN.2017.07.001
Abstract: Diatoms represent the major component of phytoplankton and are responsible for about 2025% of global primary production. Hundreds of millions of years of evolution led to tens of thousands of species differing in dimensions and morphologies. In particular, diatom porous silica cell walls, the frustules, are characterized by an extraordinary, species-specific diversity. It is of great interest, among the marine biologists and geneticists community, to shed light on the origin and evolutionary advantage of this variability of dimensions, geometries and pore distributions. In the present article the main reported data related to frustule morphogenesis and functionalities with contributions from fundamental biology, genetics, mathematics, geometry and physics are reviewed.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1016/J.MARGEN.2017.07.001
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“The Dirichlet problem for the Laplace equation in supershaped annuli”. Caratelli D, Gielis J, Tavkhelidze I, Ricci PE, Boundary value problems , 113 (2013). http://doi.org/10.1186/1687-2770-2013-113
Abstract: The Dirichlet problem for the Laplace equation in normal-polar annuli is addressed by using a suitable Fourier-like technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by the so-called superformula introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1186/1687-2770-2013-113
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Gielis J (2021) Double helix of phyllotaxis : analysis of the geometric model of plant morphogenesis, by Boris Rozin. 139–140
Keywords: Review; Sustainable Energy, Air and Water Technology (DuEL)
Impact Factor: 4.25
DOI: 10.1086/714470
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“Editorial: leaf functional traits : ecological and evolutionary implications”. Niklas KJ, Shi P, Gielis J, Schrader J, Niinemets U, Frontiers in plant science 14, 1169558 (2023). http://doi.org/10.3389/FPLS.2023.1169558
Keywords: Editorial; Sustainable Energy, Air and Water Technology (DuEL)
Impact Factor: 5.6
DOI: 10.3389/FPLS.2023.1169558
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“Electromagnetic characterization of supershaped lens antennas for high-frequency applications”. Bia P, Caratelli D, Mescia L, Gielis J page 1679 (2013).
Keywords: H1 Book chapter; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
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“Electromagnetic mathematical modeling of 3D supershaped dielectric lens antennas”. Mescia L, Bia P, Caratelli D, Chiapperino MA, Stukach O, Gielis J, Mathematical problems in engineering: theory, methods, and applications , 8130160 (2016). http://doi.org/10.1155/2016/8130160
Abstract: The electromagnetic analysis of a special class of 3D dielectric lens antennas is described in detail. This new class of lens antennas has a geometrical shape defined by the three-dimensional extension of Gielis formula. The analytical description of the lens shape allows the development of a dedicated semianalytical hybrid modeling approach based on geometrical tube tracing and physical optic. In order to increase the accuracy of the model, the multiple reflections occurring within the lens are also taken into account.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1155/2016/8130160
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“Electromagnetic modeling and design of a novel class of complementary split‐ring resonators”. Martínez-Dueñas EJR, de Jong van Coevorden CM, Stukach OV, Panokin NV, Gielis J, Caratelli D, International journal of RF and microwave computer-aided engineering 29, e21582 (2019). http://doi.org/10.1002/MMCE.21582
Abstract: This research study reports the assessment of complementary split ring resonators based on Gielis transformation as basic elements for the design of high‐performance microwave components in printed technology. From the electromagnetic simulation of said structures, suitable equivalent circuit models are extracted and analyzed. Physical prototypes are fabricated and tested for design validation. The obtained results confirm that the adoption of supershaped geometries enables the synthesis of very compact scalable microwave filters.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1002/MMCE.21582
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“Er bestaan geen absurde, irrationele, onregelmatige of onderling niet-onmeetbare meetkundige getallen”. Gielis J, Wiskunde en onderwijs 47, 23 (2021)
Keywords: A2 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
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“Fourier-Hankel solution of the Robin problem for the Helmholtz equation in supershaped annular domains”. Caratelli D, Gielis J, Tavkhelidze I, Ricci PE, Boundary value problems , 253 (2013). http://doi.org/10.1186/1687-2770-2013-253
Abstract: The Robin problem for the Helmholtz equation in normal-polar annuli is addressed by using a suitable Fourier-Hankel series technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by the so-called superformula introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1186/1687-2770-2013-253
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“Fourier-like solution of the Dirichlet problem for the Laplace Equation in k-type Gielis domains”. Caratelli D, Gielis J, Ricci PE, Journal of pure and applied mathematics : advances and applications 5, 99 (2011)
Abstract: The interior and exterior Dirichlet problems for the Laplace equation in k-type Gielis domains are analytically addressed by using a suitable Fourier-like technique. A dedicated numerical procedure based on the computer-aided algebra tool Mathematica© is developed in order to validate the proposed approach. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained. Computed results are found to be in good agreement with theoretical findings on Fourier series expansion presented by Carleson.
Keywords: A1 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
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“The general case of cutting GML bodies : the geometrical solution”. Gielis J, Caratelli D, Tavkhelidze I page 397 (2020).
Keywords: H1 Book chapter; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1007/978-3-030-56323-3_31
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“The general case of cutting of Generalized Möbius-Listing surfaces and bodies”. Gielis J, Tavkhelidze I, 4Open 3, 7 (2020). http://doi.org/10.1051/FOPEN/2020007
Abstract: The original motivation to study Generalized Möbius-Listing GML surfaces and bodies was the observation that the solution of boundary value problems greatly depends on the domains. Since around 2010 GML’s were merged with (continuous) Gielis Transformations, which provide a unifying description of geometrical shapes, as a generalization of the Pythagorean Theorem. The resulting geometrical objects can be used for modeling a wide range of natural shapes and phenomena. The cutting of GML bodies and surfaces, with the Möbius strip as one special case, is related to the field of knots and links, and classifications were obtained for GML with cross sectional symmetry of 2, 3, 4, 5 and 6. The general case of cutting GML bodies and surfaces, in particular the number of ways of cutting, could be solved by reducing the 3D problem to planar geometry. This also unveiled a range of connections with topology, combinatorics, elasticity theory and theoretical physics.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1051/FOPEN/2020007
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Gielis J (2017) The geometrical beauty of plants. 229 p
Keywords: MA3 Book as author; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.2991/978-94-6239-151-2
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“A geometrical model for testing bilateral symmetry of bamboo leaf with a simplified Gielis equation”. Lin S, Zhang L, Reddy GVP, Hui C, Gielis J, Ding Y, Shi P, Ecology and evolution 6, 6798 (2016). http://doi.org/10.1002/ECE3.2407
Abstract: The size and shape of plant leaves change with growth, and an accurate description of leaf shape is crucial for describing plant morphogenesis and development. Bilateral symmetry, which has been widely observed but poorly examined, occurs in both dicot and monocot leaves, including all nominated bamboo species (approximately 1,300 species), of which at least 500 are found in China. Although there are apparent differences in leaf size among bamboo species due to genetic and environmental profiles, bamboo leaves have bilateral symmetry with parallel venation and appear similar across species. Here, we investigate whether the shape of bamboo leaves can be accurately described by a simplified Gielis equation, which consists of only two parameters (leaf length and shape) and produces a perfect bilateral shape. To test the applicability of this equation and the occurrence of bilateral symmetry, we first measured the leaf length of 42 bamboo species, examining >500 leaves per species. We then scanned 30 leaves per species that had approximately the same length as the median leaf length for that species. The leaf-shape data from scanned profiles were fitted to the simplified Gielis equation. Results confirmed that the equation fits the leaf-shape data extremely well, with the coefficients of determination being 0.995 on average. We further demonstrated the bilateral symmetry of bamboo leaves, with a clearly defined leaf-shape parameter of all 42 bamboo species investigated ranging from 0.02 to 0.1. This results in a simple and reliable tool for precise determination of bamboo species, with applications in forestry, ecology, and taxonomy.
Keywords: A1 Journal article; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1002/ECE3.2407
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“Gielis transformations for the audiovisual geometry database”. Chapman D, Gielis J, Symmetry : culture and science 32, 177 (2021). http://doi.org/10.26830/SYMMETRY_2021_2_177
Abstract: This publication introduces the audiovisual geometry database with Gielis transformations as initial records for a prototype of the database. A concise overview is given of the rationale behind the database and studying wave phenomena with Gielis transformations. First results on a form of timbral polyphony observed in Gielis curves and future work are briefly discussed.
Keywords: A1 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.26830/SYMMETRY_2021_2_177
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“De kleine boerderij : twee bijzondere tuinkamers”. Vermander C, De Wael J, Gielis J, Groencontact 45, 14 (2019)
Keywords: A2 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
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“Lamé-Gielis curves in biology and geometry”. Gielis J, Shi P, Beirinckx B, Caratelli D, Ricci PE, (2021)
Keywords: P3 Proceeding; Sustainable Energy, Air and Water Technology (DuEL)
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“Leaf area-length allometry and its implications in leaf shape evolution”. Shi P, Liu M, Ratkowsky DA, Gielis J, Su J, Yu X, Wang P, Zhang L, Lin Z, Schrader J, Trees: structure and function 33, 1073 (2019). http://doi.org/10.1007/S00468-019-01843-4
Abstract: According to Thompson’s principle of similarity, the area of an object should be proportional to its length squared. However, leaf area–length data of some plants have been demonstrated not to follow the principle of similarity. We explore the reasons why the leaf area–length allometry deviates from the principle of similarity and examine whether there is a general model describing the relationship among leaf area, width and length. We sampled more than 11,800 leaves from six classes of woody and herbaceous plants and tested the leaf area–length allometry. We compared six mathematical models based on root-mean-square error as the measure of goodness-of-fit. The best supported model described a proportional relationship between leaf area and the product of leaf width and length (i.e., the Montgomery model). We found that the extent to which the leaf area–length allometry deviates from the principle of similarity depends upon the extent of variation of the ratio of leaf width to length. Estimates of the parameter of the Montgomery model ranged between 1/2, which corresponds to a triangular leaf with leaf length as its height and leaf width as its base, and π/4, which corresponds to an elliptical leaf with leaf length as its major axis and leaf width as its minor axis, for the six classes of plants. The narrow range in practice of the Montgomery parameter implies an evolutionary stability for the leaf area of large-leaved plants despite the fact that leaf shapes of these plants are rather different.
Keywords: A1 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1007/S00468-019-01843-4
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“The Mӧbius phenomenon in Generalized Mӧbius-Listing bodies with cross sections of odd and even polygons”. Gielis J, Tavkhelidze I, Sn –, 1512-0066 34, 23 (2020)
Keywords: A3 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
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“The Möbius phenomenon in Generalized Möbius-Listing surfaces and bodies, and Arnold's Cat phenomenon”. Gielis J, Ricci PE, Tavkhelidze I, Advanced Studies : Euro-Tbilisi Mathematical Journal 14, 17 (2021). http://doi.org/10.3251/ASETMJ/1932200812
Abstract: Möbius bands have been studied extensively, mainly in topology. Generalized Möbius-Listing surfaces and bodies providing a full geometrical generalization, is a quite new field, motivated originally by solutions of boundary value problems. Analogous to cutting of the original Möbius band, for this class of surfaces and bodies, results have been obtained when cutting such bodies or surfaces. In general, cutting leads to interlinked and intertwined different surfaces or bodies, resulting in very complex systems. However, under certain conditions, the result of cutting can be a single surface or body, which reduces complexity considerably. Our research is motivated by this reduction of complexity. In the study of cutting Generalized Möbius-Listing bodies with polygons as cross section, the conditions under which a single body results, displaying the Möbius phenomenon of a one-sided body, have been determined for even and odd polygons. These conditions are based on congruence and rotational symmetry of the resulting cross sections after cutting, and on the knife cutting the origin. The Möbius phenomenon is important, since the process of cutting (or separation of zones in a GML body in general) then results in a single body, not in different, intertwined domains. In all previous works it was assumed that the cross section of the GML bodies is constant, but the main result of this paper is that it is sufficient that only one cross section on the whole GML structure meets the conditions for the Möbius phenomenon to occur. Several examples are given to illustrate this.
Keywords: A1 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.3251/ASETMJ/1932200812
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“Modeling in mathematics : proceedings of the second Tbilisi-Salerno workshop on modeling in mathematics”. Gielis J, Ricci PE, Tavkhelidze I page 185 p. (2017).
Keywords: ME3 Book as editor; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.2991/978-94-6239-261-8
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“Modeling of electroporation induced by pulsed electric fields in irregularly shaped cells”. Mescia L, Chiapperino MA, Bia P, Gielis J, Caratelli D, IEEE transactions on biomedical engineering 65, 414 (2018). http://doi.org/10.1109/TBME.2017.2771943
Abstract: During the past decades, the poration of cell membrane induced by pulsed electric fields has been widely investigated. Since the basic mechanisms of this process have not yet been fully clarified, many research activities are focused on the development of suitable theoretical and numerical models. To this end, a nonlinear, nonlocal, dispersive, and space-time numerical algorithm has been developed and adopted to evaluate the transmembrane voltage and pore density along the perimeter of realistic irregularly shaped cells. The presented model is based on the Maxwell's equations and the asymptotic Smoluchowski's equation describing the pore dynamics. The dielectric dispersion of the media forming the cell has been modeled by using a general multirelaxation Debye-based formulation. The irregular shape of the cell is described by using the Gielis' superformula. Different test cases pertaining to red blood cells, muscular cells, cell in mitosis phase, and cancer-like cell have been investigated. For each type of cell, the influence of the relevant shape, the dielectric properties, and the external electric pulse characteristics on the electroporation process has been analyzed. The numerical results demonstrate that the proposed model is an efficient numerical tool to study the electroporation problem in arbitrary-shaped cells.
Keywords: A1 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1109/TBME.2017.2771943
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“Multiphysics modelling of membrane electroporation in irregularly shaped cells”. Mescia L, Chiapperino MA, Bia P, Lamacchia CM, Gielis J, Caratelli D, Progress in Electromagnetic Research Symposium (PIERS)
T2 –, 2019 PhotonIcs &, Electromagnetics Research Symposium –, Spring (PIERS-Spring), 17-20 June 2019, Rome, Italy , 2992 (2019). http://doi.org/10.1109/PIERS-SPRING46901.2019.9017428
Abstract: Electroporation is a non-thermal electromagnetic phenomenon widely used in medical diseases treatment. Different mathematical models of electroporation have been proposed in literature to study pore evolution in biological membranes. This paper presents a nonlinear dispersive multiphysic model of electroporation in irregular shaped biological cells in which the spatial and temporal evolution of the pores size is taken into account. The model solves Maxwell and asymptotic Smoluchowski equations and it describes the dielectric dispersion of cell media using a Debye-based relationship. Furthermore, the irregular cell shape has been modeled using the Gielis superformula. Taking into account the cell in mitosis phase, the electroporation process has been studied comparing the numerical results pertaining the model with variable pore radius with those in which the pore radius is supposed constant. The numerical analysis has been performed exposing the biological cell to a rectangular electric pulse having duration of 10 μs. The obtained numerical results highlight considerable differences between the two different models underling the need to include into the numerical algorithm the differential equation modeling the spatial and time evolution of the pores size.
Keywords: P1 Proceeding; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1109/PIERS-SPRING46901.2019.9017428
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“Nonlinear dispersive model of electroporation for irregular nucleated cells”. Chiapperino MA, Bia P, Caratelli D, Gielis J, Mescia L, Dermol-Cerne J, Miklavcic D, Bioelectromagnetics 40, 331 (2019). http://doi.org/10.1002/BEM.22197
Abstract: In this work, the electroporation phenomenon induced by pulsed electric field on different nucleated biological cells is studied. A nonlinear, non-local, dispersive, and space-time multiphysics model based on Maxwell's and asymptotic Smoluchowski's equations has been developed to calculate the transmembrane voltage and pore density on both plasma and nuclear membrane perimeters. The irregular cell shape has been modeled by incorporating in the numerical algorithm the analytical functions pertaining to Gielis curves. The dielectric dispersion of the cell media has been modeled considering the multi-relaxation Debye-based relationship. Two different irregular nucleated cells have been investigated and their response has been studied applying both the dispersive and non-dispersive models. By a comparison of the obtained results, differences can be highlighted confirming the need to make use of the dispersive model to effectively investigate the cell response in terms of transmembrane voltages, pore densities, and electroporation opening angle, especially when irregular cell shapes and short electric pulses are considered. Bioelectromagnetics. 2019;40:331-342. (c) 2019 Wiley Periodicals, Inc.
Keywords: A1 Journal article; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.1002/BEM.22197
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“A note about generalized forms of the Gielis formula”. Gielis J, Natalini P, Ricci PE page 107 (2017).
Abstract: We generalize the Gielis Superformula by extending the R. Chacon approach, but avoiding the use of Jacobi elliptic functions. The obtained results are extended to the three-dimensional case. Several new shapes are derived by using the computer algebra system Mathematica(C).
Keywords: H1 Book chapter; Engineering sciences. Technology; Sustainable Energy, Air and Water Technology (DuEL)
DOI: 10.2991/978-94-6239-261-8_8
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