Background and Purpose: This study is related to the design of advanced dermal scaffolds (non-woven fibrous mats) to provide multifunctional properties.

Materials and Methods: The nanofibre fabric from poly (ε-caprolactone) (PCL) solutions in acetone were produced by electrospinning. Based on SEM and its stochastic fibre characterization, equivalent fabric was generated by computer. Fibres network was created by random deposition of elastic straight segments within a representative volume element
(RVE).Elastic and permeability properties determined experimentally and numerically.

Results: The fibres diameter produced fibrous structure depends on solution concentration and electric field, what have been identified by surface response methodology. Effective Young’s modules for fibrous network measured and predicted by numerical model depend on fibres density and fibres diameter. Calculated data for permeability and pore distributionwere compared with models for fibrous media with acceptable error level.

Conclusions: Effective elastic and permeability properties are determined by separate computational models. The scaffold multiscale behaviour is commented with multiphysic design procedure

Ante Agić

BUDIMIR MIJOVIĆ

Milorad NIKITOVIĆ

HRČAK - Portal of Croatian Scientific and Professional Journals

Design of dermal electrospun replacementAbstractBackground and Purpose: This study is related to the design of ad-vanced dermal scaffolds (non-woven fibrous mats) to provide multifunc-tional properties.Materials and Methods: The nanofibre fabric from poly (e-capro-lactone) (PCL) solutions in acetone were produced by electrospinning.Based on SEM and its stochastic fibre characterization, equivalent fabricwas generated by computer. Fibres network was created by random deposi-tion of elastic straight segments within a representative volume element(RVE).Elastic and permeability properties determined experimentally andnumerically.Results: The fibres diameter produced fibrous structure depends on solu-tion concentration and electric field, what have been identified by surfaceresponse methodology. Effective Young’s modules for fibrous network mea-sured and predicted by numerical model depend on fibres density and fibresdiameter. Calculated data for permeability and pore distribution were com-pared with models for fibrous media with acceptable error level.Conclusions: Effective elastic and permeability properties are deter-mined by separate computational models. The scaffold multiscale behav-iour is commented with multiphysic design procedure.INTRODUCTIONHuman skin is the largest organ which protects the body from dis-ease and physical damage. It is composed of two major layers, theepidermis and the dermis (Figure 1). When the skin has been seriouslydamaged through disease or burns, the body cannot act fast enough toproduce necessary replacement cells. For example, burn victims diefrom infection and loss of plasma. Skin grafts were developed as a wayto prevent such consequences. Artificial skin grafts can be made in largequantities and be frozen for storage and shipping, making them avail-able at any time. The artificial skin does not contain immunogeniccells, it is not rejected by the body, and finally it involves reduced reha-bilitation time.This research focuses on the creation of advanced dermal scaffolds(non-woven fibrous mats) to provide a functional three dimensionalsubstrate, with enhanced properties. The purpose of tissue engineeringis to regenerate biological tissues by the use of porous, degradablebiomaterials (scaffolds) to deliver cells, genes, and/or proteins (’bio-logics’). The scaffold will provide temporary mechanical support anddefine tissue shape while delivering 'biologics' from internal pores andthrough release after degradation. The two primary requirements for ascaffold are temporary mechanical support and tissue regenerationANTE AGI]1MILORAD NIKITOVI]2BUDIMIR MIJOVI]31Faculty of Chemical Engineeringand TechnologyUniversity of ZagrebMaruli}ev trg 1910000 Zagreb, Croatia2College for Information Technologies10000 Zagreb, Croatia3Faculty of Textile TechnologyUniversity of ZagrebPrilaz b. Filipovi}a 28a10000 Zagreb, CroatiaCorrespondence:Ante Agi}Faculty of Chemical Engineeringand TechnologyUniversity of ZagrebMaruli}ev trg 1910000 Zagreb, CroatiaE-mail: aagicºfkit.hrKey words: electrospinninig, dermalscaffold, multiscale designReceived August 26, 2009.PERIODICUM BIOLOGORUM UDC 57:61VOL. 112, No 1, 63–68, 2010 CODEN PDBIADISSN 0031-5362Original scientific paperthrough biologic delivery (1). Mechanical support can bedefined through constitutive parameters poroelasticitycontinua. Tissue regeneration is related to mass trans-port characteristics like permeability, but depends also oncell material interaction parameters. Defined target pro-perties are attained by distributing specified base materialsat multiple length scales ranging from several nano-meters to millimeters. Material distribution at the multi-ple material scale range (»scaffold microstructure«) hasnot been topic in structural design, but it is realized byfabrication processes. It is desirable to calculate effectivescaffold properties like stiffness and permeability basedon scaffold microstructure, so that the effect of theseproperties on tissue regeneration may better be predict-ed. Design is further complicated by the fact that targetproperties for mechanical support are opposed to thoseassociated with enhanced tissue regeneration, essentiallymass transport properties. Therefore, the best possiblecombination of mechanical support and mass transportrelies on optimizing material distribution within a po-rous microstructure (2). The goal was to determine boundsof single effective properties (elasticity, conductivity, per-meability) that can be attained at a specified porosity andhow these materials should be arranged in three dimen-sional space. These bounds define what properties maytheoretically be attained for a given base scaffold mate-rial. Numerous top-down and bottom-up nanofabrica-tion technologies (such as electrospinning, chemical va-por deposition, self-assembly processes) are available tosynthesize nanomaterials with fine microstructure (3).After decreasing material size into the nanoscale specificsurface area, dramatically increased, surface roughnessleads to superior physiochemical properties (i.e., me-chanical, electrical, catalytic properties, etc.). Studies havedemonstrated that nanostructured materials with cell fa-vorable surface properties may promote greater amountsof specific protein interactions to more efficiently stimu-late new tissue growth compared to conventional materials;this may be one of the main mechanisms why nano-materials are superior to conventional materials. Tissuescaffolds were developed through electrospinning pro-cess which creates a non-woven fibrous construct of highpermeability and proper mechanical integrity similar tothe scale of the extracellular matrix of cells. Electro-spinning is a fascinating processing technique capable ofconverting polymers into interconnected flexible nano-fibrous structures, and it is particularly appealing for fab-ricating large sheets of scaffolds suitable for dermal im-plantation. When electric field increases beyond a criticalvalue – at which repulsive electrical forces overcome thesurface tension of polymer solution (or melt) droplet atthe tip of a the nozzle – a charged jet is ejected (Figure 2).As the jet travels to the collector, it either cools down (incase of the melt) or the solvent evaporates (in case of thesolution) to produce ultrafine fibers in the form of anon-woven nanofabric.The non-woven fabrics of electrospun fibers have uni-que properties, such as dramatically increased specificsurface, excellent mechanical strength, and highly openporous structures. A number of nanofiber assemblieshave been developed through electrospinning by incor-porating functional agents to achieve antibacterial, mag-netic properties. The morphology of electrospun fibersdepends on a number of factors, such as solution proper-ties (e.g., concentration, viscosity, conductivity, surfacetension, etc.), processing conditions (e.g., electrical po-tential, collection distance, etc.), and ambient conditions(e.g., temperature, humidity, etc.). The goal of this paperwas to examine the range of elastic and permeabilityproperties that can be spanned using electrospinning ap-proaches. These architectures can then be tested both invitro to determine how well fabricated scaffold micro-structures can match the computational design, and invivo to determine if these designs will affect tissue regen-eration (4).METHODSElastic propertiesBased on SEM and its stochastic fibre characteriza-tion, equivalent fabric was generated by computer. Fibernetwork was created by random deposition of elasticstraight segments within a representative volume ele-ment (RVE). Nucleation sites were generated randomlyinside a RVE and gave rise to the segment along ran-domly chosen directions. The line between two cross--links associated with the same segment defines fibred64 Period biol, Vol 112, No 1, 2010.A. Agi} et al. Design of dermal electrospun replacementFigure 1. Human skin: cross-section.Figure 2. Electrospinning process.structural units (beam or truss element). Selecting direc-tional vectors from probability distribution function al-lows generation of networks with equivalent structuresas experimental SEM image. Effective mechanical prop-erties of nanofiber sheets at the macro scale level can bedetermined using the two dimensional (2D) Timoshen-ko beam-network. The non-load bearing fiber segmentswere removed and trimmed to keep dimensions L ´ L ofthe representative window (Figure 3). A line representa-tive network model was replaced by finite element beammesh. The number of intersections/unit area, probabilitydistribution function and mean lengths were obtainedfrom image analysis of electrospun sheets. We consideredhere tensile stress, and fibers were rigidly bonded to eachother at every fiber-fiber crossing points The problemwas reduced to solution of the linear system of equations¢K£ ·¤u¥ = ¤f¥ (1)where is global displacement vector, ¤f¥ global nodalforce vector, and ¢K£ global stiffness matrix (5). Effectiveelasticity tensor can be determined by homogenizationprocedure by the following formulaCVC x y z dVijklRVEijkl= ∫1( , , ) (2)C x y zijklij kl( , , )=∂∂ ∂Ue ewhere U elastic deformation energy and eij deformationtensor, respectively.Pore network modelMany fibers form a 3D network of pore bodies andpore throats that constitute the microstructure of scaf-fold. In order to calculate permeability of the generatedfiber network we need to construct the pore space of ascaffold by a network in which the bonds and sites repre-sent pore throats and pore bodies, respectively. A grid wasthen imposed on images that closely mimics the topologyof pore space, so that the lines that connect the gridpoints run more or less along pore throats, with the gridpoints being in pore bodies. Pore throats are channelsthat are formed between fibers. Pore bodies are the cavi-ties where pore throats meet each other. Local coordina-tion numbers, connectivity and their statistical distribu-tions, are determined from pore network mesh. Eachthroat is characterized by effective radius and length,both of which follow, in general, some statistical distribu-tions. Since we represent each pore throat effectively as acylindrical pore of radius rij and length lij, the volumetricflow rate qij through the tube from the i-th to the j-thnode is given by (Figure 4)qij =p Dr plij ijij48m(3)where Dpij is the pressure drop along the throat, m is ef-fective viscosity, and lij is the throat’s length.The fluids are assumed incompressible, leading toconservation of volume flux at each node{ }gijij=∑ 0 (4)Period biol, Vol 112, No 1, 2010. 65Design of dermal electrospun replacement A. Agi} et al.Figure 3. The finite element model of fiber net.Figure 4. Pore net model.where qij is the volumetric flow rate through a tube con-necting pore i and j. The sum is over all throats ij that areconnected to pore body i. If we write equation (3) for ev-ery interior node i of the network, then together withequation (4) we obtain a set of simultaneous linear equa-tions for nodal pressures. Subject to specified boundaryconditions, we get matrix equation¢D£ ·¤p¥ = ¤P¥ (5)where D is conductance matrix, ¤p¥ is pressure vector(internal nodes) and ¤P¥ pressure vector on boundaries.The result of solution equation (5) is pressure distribu-tion in the network. Assuming that the flow of a fluid in afiber network is slow enough that the Darcy’s law is ap-plicable, then the flow properties of an anisotropic net-work are characterized by a symmetric and second rankpermeability tensor Kij, defined bynm= − ∇Kpiji, j = 1,2,3 (6)where n is the average fluid velocity vector, m is the vis-cosity of the fluid, and ∇p is the pressure gradient im-posed on the porous medium. To compute the effectivepermeability of a fabric in given direction, a pressure gra-dient is imposed on the network along that direction.OPTIMAL DESIGNThe real issue for creating dermal scaffolds is to derivea microstructure design that will give the desired multi-functional properties. Inverse topology optimization me-thod iteratively constructed the microstructure that pro-vides the desired properties. Based on our previous study(6), we proposed multiphysics designing approach inwhich the first invariant of permeability tensor is maxi-mized as objective function J, with constraints on desiredeffective elasticity tensors, and on porosity topology. Mic-rostructure design for this problem, with the formal opti-mization statement has the following formmax ( )rJ Kiiscaffoldsubject to C Cijklscaffoldijklt et≤ arg (7)1 – r £ porosityr = V/VRVECijklt etarg is the target value of where effective elasticity ten-sor, and VRVE and V denote representative volume elementand volume of the solid material, respectively. Homoge-nization theory is used within an iterative optimizationscheme to compute effective properties as part of objec-tive function. Design approach creates a volumetric 3Ddesign by replicating representative unit cells periodi-cally in 3D space. Multiphysical optimization concept isschematicaly presented in Figure 5.EXPERIMENTAL PART OF THE STUDYPoly(e-caprolactone) (PCL) solutions (Sigma–Aldrich,mol. weight = 65,000) in acetone were produced by stir-ring the solvent and particulate polymer at 50 °C untilthe solid was fully dissolved. Electrospun PCL fiberswere produced using a 20 ml/h solution flow rate, a10–20 kV potential between the needle tip and 10–20 cmcollector distance. Samples were collected at room tem-perature and certain humidity (30% RH). After deposi-tion, the samples were exposed to a vacuum at room tem-perature for 8 h to ensure that residual solvent levels werenot significant. Characterization of the tissue scaffoldsincluded image analysis, tensile tests, and porosimetrymeasurements to demonstrate enhanced properties andpreferred morphologies.RESULTSExperimental data for fiber diameter produced fi-brous structure depending on solution concentration andelectric field have been identified. Based on surface re-sponse methodology (7) the coefficient Aij in second-or-der model (eq. 8) was determined by multiple regressionanalysis using experimental data for electrospinning PCLsolution.Fiber diameter D is given byD = A00 + A01x + A10y + A02x2 + A20y2 + A11xy (8)where are x and y PCL concentration and electric field,respectively.66 Period biol, Vol 112, No 1, 2010.A. Agi} et al. Design of dermal electrospun replacementFigure 5. The multiphysics optimization scheme.The contour plots suggest that the lower concentra-tion together with electric field increase gives lower fiberdiameter. The stationary point is minimum values of theresponse or the smallest fiber diameter. The surface con-tour plots of these parameters outline the optimum con-dition for electrospinning (Figure 6).Computational model for elastic property was match-ed with the experimental stress–strain curve for a PCLelectrospun mesh. The tensile properties of electrospunfabrics were measured as a function of fiber diameter anddensity, and compared to those predicted using a mathe-matical model for fibrous networks. The resultant best fitfor the effective Young’s modulus relative value was foundto depend on fiber density and fiber diameter, which isshown in Figure 7.By porosimetry method for electrospun fabric distri-bution pores related average pore diameters were deter-mined (Figure 8). The distribution has the form of Gam-ma distribution (8).Pore net model was determined for a given fiber (9).Using demo version of Pore-Cor software (10) the per-meability of fibre net was determined as function of po-rosity (Figure 9). The calculated data were comparedwith permeability models for fibrous media (11) withaceptable error level.DISCUSSIONExperimental results and related mathematical modelsreveal that the elastic and transport properties of fibrousmeshes are sensitive to the microstructural architecture.Processing parameters on one way influence process sta-bility in one manner and microstructure space in theother. Future study will focus on the use of scaffolds infunctional tissue engineering scenarios, with the applica-tion of tissue growth phenomena.REFERENCES1. METCALFE A D, FERGUSON M W J 2007 Bioengineering skinusing mechanism of regeneration and repair. Biomaterials 28: 5100–5113Period biol, Vol 112, No 1, 2010. 67Design of dermal electrospun replacement A. Agi} et al.Figure 6. Fiber diameter dependence on solution concentration andelectric field.Figure 7. Modulus dependence on fiber density and diameter.1.4POREDISTRIBUTI ON1.6 1.8 2AVERAGE PORE DIAMETER micron2.2 2.4 2.6 2.801020304050607080Figure 8. Pore distributrion.0.4 0.5 0.6 0.7 0.8 0.9 1-2–1.5–1–0.500.511.52GEBARTLOGDIM.PERMEABILITYPOROSITY %OUR CALCULATIONFigure 9. Permeability dependence on porosity.2. PORTER B, ZAUEL R, STOCKMAN H, GULDBERG R, FYH-RIE D 2005 3-D computational modeling of media flow through scaf-folds in a perfusion bioreactor. J Biomechanics 38: 543–5493. HOLLISTER S J, LIN C Y 2007 Computational design of tissue en-gineering scaffolds. Comp. methods in app. mechanics and engineering196: 2991–29984. SENGERS B G, TAYLOR M, PLEASE C P, OREFFO R O C 2007Computational modelling of cell spreading and tissue regenerationin porous scaffolds. Biomaterials 28: 1926–19405. ZIENKIEWICZ O 2005 The Finite Element Method: Its basic andFundamentals, Elsevier, New York.6. AGIC A 2008 Multiscale mechanical phenomena in electrospuncarbon nanotube composites. J Appl Polymer Science 108: 1191– 12007. MYERS R H, MONTGOMERY D C 1995 Response surface meth-odology: process and product optimization using designed experi-ments. Wiley-Interscience, New York.8. EICHHORN S J, SAMPSON W W 2005 Statistical geometry ofpores and statistics of porous nanofibrous assemblies. J Royal SocietyInterface 2: 309–3189. GHASSEMZADEH J, SAHIMI M 2004 Pore network simulationof fluid imbibition into paper during coating: II. Characterization ofpaper’s morphology and computation of its effective permeabilitytensor. Chemical Engineering Science 59: 2265–8010. Pore-Cor Software, Environmental and Fluid Modelling Group,University of Plymouth, U.K.11. GEBART B R 1992 Permeability of unidirectional reinforcements ofRTM. J Composite Mater 26: 1100–3368 Period biol, Vol 112, No 1, 2010.A. Agi} et al. Design of dermal electrospun replacement

English

Design of dermal electrospun replacement

Hrčak - Portal of scientific journals of Croatia

Design of dermal electrospun replacementAbstractBackground and Purpose: This study is related to the design of ad-vanced dermal scaffolds (non-woven fibrous mats) to provide multifunc-tional properties.Materials and Methods: The nanofibre fabric from poly (e-capro-lactone) (PCL) solutions in acetone were produced by electrospinning.Based on SEM and its stochastic fibre characterization, equivalent fabricwas generated by computer. Fibres network was created by random deposi-tion of elastic straight segments within a representative volume element(RVE).Elastic and permeability properties determined experimentally andnumerically.Results: The fibres diameter produced fibrous structure depends on solu-tion concentration and electric field, what have been identified by surfaceresponse methodology. Effective Young’s modules for fibrous network mea-sured and predicted by numerical model depend on fibres density and fibresdiameter. Calculated data for permeability and pore distribution were com-pared with models for fibrous media with acceptable error level.Conclusions: Effective elastic and permeability properties are deter-mined by separate computational models. The scaffold multiscale behav-iour is commented with multiphysic design procedure.INTRODUCTIONHuman skin is the largest organ which protects the body from dis-ease and physical damage. It is composed of two major layers, theepidermis and the dermis (Figure 1). When the skin has been seriouslydamaged through disease or burns, the body cannot act fast enough toproduce necessary replacement cells. For example, burn victims diefrom infection and loss of plasma. Skin grafts were developed as a wayto prevent such consequences. Artificial skin grafts can be made in largequantities and be frozen for storage and shipping, making them avail-able at any time. The artificial skin does not contain immunogeniccells, it is not rejected by the body, and finally it involves reduced reha-bilitation time.This research focuses on the creation of advanced dermal scaffolds(non-woven fibrous mats) to provide a functional three dimensionalsubstrate, with enhanced properties. The purpose of tissue engineeringis to regenerate biological tissues by the use of porous, degradablebiomaterials (scaffolds) to deliver cells, genes, and/or proteins (’bio-logics’). The scaffold will provide temporary mechanical support anddefine tissue shape while delivering 'biologics' from internal pores andthrough release after degradation. The two primary requirements for ascaffold are temporary mechanical support and tissue regenerationANTE AGI]1MILORAD NIKITOVI]2BUDIMIR MIJOVI]31Faculty of Chemical Engineeringand TechnologyUniversity of ZagrebMaruli}ev trg 1910000 Zagreb, Croatia2College for Information Technologies10000 Zagreb, Croatia3Faculty of Textile TechnologyUniversity of ZagrebPrilaz b. Filipovi}a 28a10000 Zagreb, CroatiaCorrespondence:Ante Agi}Faculty of Chemical Engineeringand TechnologyUniversity of ZagrebMaruli}ev trg 1910000 Zagreb, CroatiaE-mail: aagicºfkit.hrKey words: electrospinninig, dermalscaffold, multiscale designReceived August 26, 2009.PERIODICUM BIOLOGORUM UDC 57:61VOL. 112, No 1, 63–68, 2010 CODEN PDBIADISSN 0031-5362Original scientific paperthrough biologic delivery (1). Mechanical support can bedefined through constitutive parameters poroelasticitycontinua. Tissue regeneration is related to mass trans-port characteristics like permeability, but depends also oncell material interaction parameters. Defined target pro-perties are attained by distributing specified base materialsat multiple length scales ranging from several nano-meters to millimeters. Material distribution at the multi-ple material scale range (»scaffold microstructure«) hasnot been topic in structural design, but it is realized byfabrication processes. It is desirable to calculate effectivescaffold properties like stiffness and permeability basedon scaffold microstructure, so that the effect of theseproperties on tissue regeneration may better be predict-ed. Design is further complicated by the fact that targetproperties for mechanical support are opposed to thoseassociated with enhanced tissue regeneration, essentiallymass transport properties. Therefore, the best possiblecombination of mechanical support and mass transportrelies on optimizing material distribution within a po-rous microstructure (2). The goal was to determine boundsof single effective properties (elasticity, conductivity, per-meability) that can be attained at a specified porosity andhow these materials should be arranged in three dimen-sional space. These bounds define what properties maytheoretically be attained for a given base scaffold mate-rial. Numerous top-down and bottom-up nanofabrica-tion technologies (such as electrospinning, chemical va-por deposition, self-assembly processes) are available tosynthesize nanomaterials with fine microstructure (3).After decreasing material size into the nanoscale specificsurface area, dramatically increased, surface roughnessleads to superior physiochemical properties (i.e., me-chanical, electrical, catalytic properties, etc.). Studies havedemonstrated that nanostructured materials with cell fa-vorable surface properties may promote greater amountsof specific protein interactions to more efficiently stimu-late new tissue growth compared to conventional materials;this may be one of the main mechanisms why nano-materials are superior to conventional materials. Tissuescaffolds were developed through electrospinning pro-cess which creates a non-woven fibrous construct of highpermeability and proper mechanical integrity similar tothe scale of the extracellular matrix of cells. Electro-spinning is a fascinating processing technique capable ofconverting polymers into interconnected flexible nano-fibrous structures, and it is particularly appealing for fab-ricating large sheets of scaffolds suitable for dermal im-plantation. When electric field increases beyond a criticalvalue – at which repulsive electrical forces overcome thesurface tension of polymer solution (or melt) droplet atthe tip of a the nozzle – a charged jet is ejected (Figure 2).As the jet travels to the collector, it either cools down (incase of the melt) or the solvent evaporates (in case of thesolution) to produce ultrafine fibers in the form of anon-woven nanofabric.The non-woven fabrics of electrospun fibers have uni-que properties, such as dramatically increased specificsurface, excellent mechanical strength, and highly openporous structures. A number of nanofiber assemblieshave been developed through electrospinning by incor-porating functional agents to achieve antibacterial, mag-netic properties. The morphology of electrospun fibersdepends on a number of factors, such as solution proper-ties (e.g., concentration, viscosity, conductivity, surfacetension, etc.), processing conditions (e.g., electrical po-tential, collection distance, etc.), and ambient conditions(e.g., temperature, humidity, etc.). The goal of this paperwas to examine the range of elastic and permeabilityproperties that can be spanned using electrospinning ap-proaches. These architectures can then be tested both invitro to determine how well fabricated scaffold micro-structures can match the computational design, and invivo to determine if these designs will affect tissue regen-eration (4).METHODSElastic propertiesBased on SEM and its stochastic fibre characteriza-tion, equivalent fabric was generated by computer. Fibernetwork was created by random deposition of elasticstraight segments within a representative volume ele-ment (RVE). Nucleation sites were generated randomlyinside a RVE and gave rise to the segment along ran-domly chosen directions. The line between two cross--links associated with the same segment defines fibred64 Period biol, Vol 112, No 1, 2010.A. Agi} et al. Design of dermal electrospun replacementFigure 1. Human skin: cross-section.Figure 2. Electrospinning process.structural units (beam or truss element). Selecting direc-tional vectors from probability distribution function al-lows generation of networks with equivalent structuresas experimental SEM image. Effective mechanical prop-erties of nanofiber sheets at the macro scale level can bedetermined using the two dimensional (2D) Timoshen-ko beam-network. The non-load bearing fiber segmentswere removed and trimmed to keep dimensions L ´ L ofthe representative window (Figure 3). A line representa-tive network model was replaced by finite element beammesh. The number of intersections/unit area, probabilitydistribution function and mean lengths were obtainedfrom image analysis of electrospun sheets. We consideredhere tensile stress, and fibers were rigidly bonded to eachother at every fiber-fiber crossing points The problemwas reduced to solution of the linear system of equations¢K£ ·¤u¥ = ¤f¥ (1)where is global displacement vector, ¤f¥ global nodalforce vector, and ¢K£ global stiffness matrix (5). Effectiveelasticity tensor can be determined by homogenizationprocedure by the following formulaCVC x y z dVijklRVEijkl= ∫1 ( , , ) (2)C x y zijklij kl( , , )=∂∂ ∂Ue ewhere U elastic deformation energy and eij deformationtensor, respectively.Pore network modelMany fibers form a 3D network of pore bodies andpore throats that constitute the microstructure of scaf-fold. In order to calculate permeability of the generatedfiber network we need to construct the pore space of ascaffold by a network in which the bonds and sites repre-sent pore throats and pore bodies, respectively. A grid wasthen imposed on images that closely mimics the topologyof pore space, so that the lines that connect the gridpoints run more or less along pore throats, with the gridpoints being in pore bodies. Pore throats are channelsthat are formed between fibers. Pore bodies are the cavi-ties where pore throats meet each other. Local coordina-tion numbers, connectivity and their statistical distribu-tions, are determined from pore network mesh. Eachthroat is characterized by effective radius and length,both of which follow, in general, some statistical distribu-tions. Since we represent each pore throat effectively as acylindrical pore of radius rij and length lij, the volumetricflow rate qij through the tube from the i-th to the j-thnode is given by (Figure 4)qij =p Dr plij ijij48m(3)where Dpij is the pressure drop along the throat, m is ef-fective viscosity, and lij is the throat’s length.The fluids are assumed incompressible, leading toconservation of volume flux at each node{ }gijij=∑ 0 (4)Period biol, Vol 112, No 1, 2010. 65Design of dermal electrospun replacement A. Agi} et al.Figure 3. The finite element model of fiber net.Figure 4. Pore net model.where qij is the volumetric flow rate through a tube con-necting pore i and j. The sum is over all throats ij that areconnected to pore body i. If we write equation (3) for ev-ery interior node i of the network, then together withequation (4) we obtain a set of simultaneous linear equa-tions for nodal pressures. Subject to specified boundaryconditions, we get matrix equation¢D£ ·¤p¥ = ¤P¥ (5)where D is conductance matrix, ¤p¥ is pressure vector(internal nodes) and ¤P¥ pressure vector on boundaries.The result of solution equation (5) is pressure distribu-tion in the network. Assuming that the flow of a fluid in afiber network is slow enough that the Darcy’s law is ap-plicable, then the flow properties of an anisotropic net-work are characterized by a symmetric and second rankpermeability tensor Kij, defined bynm= − ∇Kpiji, j = 1,2,3 (6)where n is the average fluid velocity vector, m is the vis-cosity of the fluid, and ∇p is the pressure gradient im-posed on the porous medium. To compute the effectivepermeability of a fabric in given direction, a pressure gra-dient is imposed on the network along that direction.OPTIMAL DESIGNThe real issue for creating dermal scaffolds is to derivea microstructure design that will give the desired multi-functional properties. Inverse topology optimization me-thod iteratively constructed the microstructure that pro-vides the desired properties. Based on our previous study(6), we proposed multiphysics designing approach inwhich the first invariant of permeability tensor is maxi-mized as objective function J, with constraints on desiredeffective elasticity tensors, and on porosity topology. Mic-rostructure design for this problem, with the formal opti-mization statement has the following formmax ( )rJ Kiiscaffoldsubject to C Cijklscaffoldijklt et≤ arg (7)1 – r £ porosityr = V/VRVECijklt etarg is the target value of where effective elasticity ten-sor, and VRVE and V denote representative volume elementand volume of the solid material, respectively. Homoge-nization theory is used within an iterative optimizationscheme to compute effective properties as part of objec-tive function. Design approach creates a volumetric 3Ddesign by replicating representative unit cells periodi-cally in 3D space. Multiphysical optimization concept isschematicaly presented in Figure 5.EXPERIMENTAL PART OF THE STUDYPoly(e-caprolactone) (PCL) solutions (Sigma–Aldrich,mol. weight = 65,000) in acetone were produced by stir-ring the solvent and particulate polymer at 50 °C untilthe solid was fully dissolved. Electrospun PCL fiberswere produced using a 20 ml/h solution flow rate, a10–20 kV potential between the needle tip and 10–20 cmcollector distance. Samples were collected at room tem-perature and certain humidity (30% RH). After deposi-tion, the samples were exposed to a vacuum at room tem-perature for 8 h to ensure that residual solvent levels werenot significant. Characterization of the tissue scaffoldsincluded image analysis, tensile tests, and porosimetrymeasurements to demonstrate enhanced properties andpreferred morphologies.RESULTSExperimental data for fiber diameter produced fi-brous structure depending on solution concentration andelectric field have been identified. Based on surface re-sponse methodology (7) the coefficient Aij in second-or-der model (eq. 8) was determined by multiple regressionanalysis using experimental data for electrospinning PCLsolution.Fiber diameter D is given byD = A00 + A01x + A10y + A02x2 + A20y2 + A11xy (8)where are x and y PCL concentration and electric field,respectively.66 Period biol, Vol 112, No 1, 2010.A. Agi} et al. Design of dermal electrospun replacementFigure 5. The multiphysics optimization scheme.The contour plots suggest that the lower concentra-tion together with electric field increase gives lower fiberdiameter. The stationary point is minimum values of theresponse or the smallest fiber diameter. The surface con-tour plots of these parameters outline the optimum con-dition for electrospinning (Figure 6).Computational model for elastic property was match-ed with the experimental stress–strain curve for a PCLelectrospun mesh. The tensile properties of electrospunfabrics were measured as a function of fiber diameter anddensity, and compared to those predicted using a mathe-matical model for fibrous networks. The resultant best fitfor the effective Young’s modulus relative value was foundto depend on fiber density and fiber diameter, which isshown in Figure 7.By porosimetry method for electrospun fabric distri-bution pores related average pore diameters were deter-mined (Figure 8). The distribution has the form of Gam-ma distribution (8).Pore net model was determined for a given fiber (9).Using demo version of Pore-Cor software (10) the per-meability of fibre net was determined as function of po-rosity (Figure 9). The calculated data were comparedwith permeability models for fibrous media (11) withaceptable error level.DISCUSSIONExperimental results and related mathematical modelsreveal that the elastic and transport properties of fibrousmeshes are sensitive to the microstructural architecture.Processing parameters on one way influence process sta-bility in one manner and microstructure space in theother. Future study will focus on the use of scaffolds infunctional tissue engineering scenarios, with the applica-tion of tissue growth phenomena.REFERENCES1. METCALFE A D, FERGUSON M W J 2007 Bioengineering skinusing mechanism of regeneration and repair. Biomaterials 28: 5100–5113Period biol, Vol 112, No 1, 2010. 67Design of dermal electrospun replacement A. Agi} et al.Figure 6. Fiber diameter dependence on solution concentration andelectric field.Figure 7. Modulus dependence on fiber density and diameter.1.4POREDISTRIBUTI ON1.6 1.8 2AVERAGE PORE DIAMETER micron2.2 2.4 2.6 2.801020304050607080Figure 8. Pore distributrion.0.4 0.5 0.6 0.7 0.8 0.9 1-2–1.5–1–0.500.511.52GEBARTLOGDIM.PERMEABILITYPOROSITY %OUR CALCULATIONFigure 9. Permeability dependence on porosity.2. PORTER B, ZAUEL R, STOCKMAN H, GULDBERG R, FYH-RIE D 2005 3-D computational modeling of media flow through scaf-folds in a perfusion bioreactor. J Biomechanics 38: 543–5493. HOLLISTER S J, LIN C Y 2007 Computational design of tissue en-gineering scaffolds. Comp. methods in app. mechanics and engineering196: 2991–29984. SENGERS B G, TAYLOR M, PLEASE C P, OREFFO R O C 2007Computational modelling of cell spreading and tissue regenerationin porous scaffolds. Biomaterials 28: 1926–19405. ZIENKIEWICZ O 2005 The Finite Element Method: Its basic andFundamentals, Elsevier, New York.6. AGIC A 2008 Multiscale mechanical phenomena in electrospuncarbon nanotube composites. J Appl Polymer Science 108: 1191– 12007. MYERS R H, MONTGOMERY D C 1995 Response surface meth-odology: process and product optimization using designed experi-ments. Wiley-Interscience, New York.8. EICHHORN S J, SAMPSON W W 2005 Statistical geometry ofpores and statistics of porous nanofibrous assemblies. J Royal SocietyInterface 2: 309–3189. GHASSEMZADEH J, SAHIMI M 2004 Pore network simulationof fluid imbibition into paper during coating: II. Characterization ofpaper’s morphology and computation of its effective permeabilitytensor. Chemical Engineering Science 59: 2265–8010. Pore-Cor Software, Environmental and Fluid Modelling Group,University of Plymouth, U.K.11. GEBART B R 1992 Permeability of unidirectional reinforcements ofRTM. J Composite Mater 26: 1100–3368 Period biol, Vol 112, No 1, 2010.A. Agi} et al. Design of dermal electrospun replacement

Design of dermal electrospun replacement

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