Stiffness-tunable biomaterials provide a good extracellular matrix environment for axon growth and regeneration

Neuronal growth, extension, branching, and formation of neural networks are markedly influenced by the extracellular matrix—a complex network composed of proteins and carbohydrates secreted by cells. In addition to providing physical support for cells, the extracellular matrix also conveys critical mechanical stiffness cues. During the development of the nervous system, extracellular matrix stiffness plays a central role in guiding neuronal growth, particularly in the context of axonal extension, which is crucial for the formation of neural networks. In neural tissue engineering, manipulation of biomaterial stiffness is a promising strategy to provide a permissive environment for the repair and regeneration of injured nervous tissue. Recent research has fine-tuned synthetic biomaterials to fabricate scaffolds that closely replicate the stiffness profiles observed in the nervous system. In this review, we highlight the molecular mechanisms by which extracellular matrix stiffness regulates axonal growth and regeneration. We highlight the progress made in the development of stiffness-tunable biomaterials to emulate in vivo extracellular matrix environments, with an emphasis on their application in neural repair and regeneration, along with a discussion of the current limitations and future prospects. The exploration and optimization of the stiffness-tunable biomaterials has the potential to markedly advance the development of neural tissue engineering.

Systematic Elastostatic Stiffness Model of Over-Constrained Parallel Manipulators Without Additional Constraint Equations

The establishment of an elastostatic stiffness model for over constrained parallel manipulators(PMs),particularly those with over constrained subclosed loops,poses a challenge while ensuring numerical stability.This study addresses this issue by proposing a systematic elastostatic stiffness model based on matrix structural analysis(MSA)and independent displacement coordinates(IDCs)extraction techniques.To begin,the closed-loop PM is transformed into an open-loop PM by eliminating constraints.A subassembly element is then introduced,which considers the flexibility of both rods and joints.This approach helps circumvent the numerical instability typically encountered with traditional constraint equations.The IDCs and analytical constraint equations of nodes constrained by various joints are summarized in the appendix,utilizing multipoint constraint theory and singularity analysis,all unified within a single coordinate frame.Subsequently,the open-loop mechanism is efficiently closed by referencing the constraint equations presented in the appendix,alongside its elastostatic model.The proposed method proves to be both modeling and computationally efficient due to the comprehensive summary of the constraint equations in the Appendix,eliminating the need for additional equations.An example utilizing an over constrained subclosed loops demonstrate the application of the proposed method.In conclusion,the model proposed in this study enriches the theory of elastostatic stiffness modeling of PMs and provides an effective solution for stiffness modeling challenges they present.

New tangent stiffness matrix for geometrically nonlinear analysis of space frames

A three-dimensional beam element is derived based on the principle of stationary total potential energy for geometrically nonlinear analysis of space frames. A new tangent stiffness matrix, which allows for high order effects of element deformations, replaces the conventional incremental secant stiffness matrix. Two deformation stiffness matrices due to the variation of axial force and bending moments are included in the tangent stiffness. They are functions of element deformations and incorporate the coupling among axial, lateral and torsional deformations. A correction matrix is added to the tangent stiffness matrix to make displacement derivatives equivalent to the commutative rotational degrees of freedom. Numerical examples show that the proposed dement is accurate and efficient in predicting the nonlinear behavior, such as axial-torsional and flexural-torsional buckling, of space frames even when fewer elements are used to model a member.

The Matrix Stiffness and Physical Confinement of Hydrogel Microchannel Jointly Induce the Mesenchymal-Amoeboid Transition for Cancer Cell Migration

The migration mode transition of cancer cell enhances its invasive capability and the drug resistance,where physical confinement of cell microenvironment has been revealed to induce the mesenchymal-amoeboid transition(MAT).However,most existing studies are performed in PDMS microchannels,of which the stiffness is much higher than that of most mammalian tissues.Therefore,the amoeboid migration transition observed in these studies is actually induced by the synergistic effect of matrix stiffness and confinement.Since the stiffness of cell microenvironment has been reported to influence the cell migration in 2D substrate,the decoupling of stiffness and confinement effects is thus in need for elucidating the underlying mechanism of MAT.However,it is technically challenging to construct microchannels with physiologically relevant stiffness and channel size,where existing microchannel platforms with physiological relevance stiffness are all with>10μm channel width.Such size is too wide to mimic the physical confinement that migrating cancer cells confront in vivo,and also larger than the width of PDMS channel,in which the MAT of cancer cell was observed.Therefore,an in vitro cell migration platform,which could mimic both stiffness and confinement of the native physical microenvironment during cancer metastasis,could profoundly contribute to researches on cancer cell migration and cellular mechanotransduction.In this paper,we overcome the limitations of engineering soft materials in microscale by combining the collagen-alginate hydrogel with photolithography.This enables us to improve the accuracy of molded microchannel,and thus successfully construct a 3D microchannel platform,which matches the stiffness and width ranges of native environmental confinement that migrating cancer cells confront in vivo.The stiffness(0.3~20 kPa),confinement(channel width:3.5~14μm)and the adhesion ligand density of the microchannel can be tuned independently.Interestingly,using this platform,we observed that the migration speed of cancer cell is influenced by the synergistic effect of channel stiffness and width,and the increasing stiffness reverses the effect of channel width on the migration speed of cancer cells.In addition,MAT has a strong correlation with the channel stiffness.These findings make us reconsider the widely accepted hypothesis:physical confinement can induce MAT.Actually,this transition can only occur in stiff confined microenvironment not in soft one.For soft microchannels,the compliance of the channel walls could cause little cell/nucleus deformation,and the MAT could not be induced.To further investigate the mechanism of MAT,we developed a computational model to simulate the effect of nucleus deformation on MAT.With the model,we found that deforming the cell nuclear by decreasing the nucleus stiffness will reduce the cellmigration speed.This implies that nuclear stiffness plays an important role in the regulation of cancer migration speed and thus MAT in microchannels.The effect of channel stiffness on MAT and migration speed as observed in our experiment could partially explain previous findings reported in the literature,where the increasing matrix stiffness of tumor microenvironment promotes cancer metastasis.Our observations thus highlight the critical role of cell nuclear deformation not only in MAT,but also in regulating cellular mechanotransduction and cell-ECM interactions.This developed platform is capable of mimicking the native physical microenvironment during metastasis,providing a powerful tool for high-throughput screening applications and investigating the interaction between cancer migration and biophysical microenvironment.

Analysis of the hull girder vibration by dynamic stiffness matrix method

Dynamic stiffness matrix method is applied to compute vibration of hull girder in this paper. This method can not only simplify the computational model, but also get much higher frequencies and responses accurately. The analytical expressions of dynamic stiffness matrix of a Timoshenko beam for transverse vibration are presented in this paper. All effects of rotatory inertia and shear deformation are taken into account in the formulation. The resulting dynamic stiffness matrix combined with the Wittrick-Williams algorithm is used to compute natural frequencies and mode shapes of the 299,500 DWT VLCC, and then the vibrational responses are solved by the mode superposition method. The computational results are compared with those obtained from other approximate methods and experiment, and it indicates that the method is accurate and efficient.

Analysis of Vibration of the Euler-Bernoulli Pipe Conveying Fluid by Dynamic Stiffness Method and Transfer Matrix

The dynamic stiffness method and Transfer method is applied to study the vibration characteristics of the Euler-Bernoulli pipe conveying fluid in this paper. According to the dynamics equation of the pipe conveying fluid, the element dynamic stiffness is established. The vibration characteristic of the single-span pipe is analyzed under two kinds of boundary conditions. The results compared with the literature, which has a good consistency. Based on this method, natural frequency and the critical speed of the two types of multi-span pipe are deserved. This paper shows that the dynamic stiffness method and transfer matrix is an effective method to deal with the vibration problem of pipe conveying fluid.

Dynamic stiffness matrix of a poroelastic multi-layered site and its Green's functions

Few studies of wave propagation in layered saturated soils have been reported in the literature.In this paper,a general solution of the equation of wave motion in saturated soils,based on one kind of practical Blot's equation, was deduced by introducing wave potentials.Then exact dynamic-stiffness matrices for a poroelastic soil layer and half- space were derived,which extended Wolf's theory for an elastic layered site to the case of poroelasticity,thus resolving a fundamental problem in the field of wave propagation and soil-structure interaction in a poroelastic layered soil site.By using the integral transform method,Green's functions of horizontal and vertical uniformly distributed loads in a poroelastic layered soil site were given.Finally,the theory was verified by numerical examples and dynamic responses by comparing three different soil sites.This study has the following advantages:all parameters in the dynamic-stiffness matrices have explicitly physical meanings and the thickness of the sub-layers does not affect the precision of the calculation which is very convenient for engineering applications.The present theory can degenerate into Wolf's theory and yields numerical results approaching those for an ideal elastic layered site when porosity tends to zero.

An Application of the Modified Shear Lag Model to Study the Influence of Thermal Residual Stresses on the Stiffness and Yield Strength of Short Fiber Reinforced Metal Matrix Composites

The modified shear lag model proposed recently was applied to calculate thermal residual stresses and subsequent stress distributions under tensile and compressive loadings. The expressions for the elastic moduli and the yield strengths under tensile and compressive loadings were derived which take account of thermal residual stresses. The asymmetries in the elastic modulus and the yield strength were interpreted using the derived expressions and the obtained results of the stress calculations. The model predictions have exhibited good agreements with the experimental results and also with the other theoretical predictions

Dynamic Change of Matrix Stiffness Switches Astrocyte Phenotype in Three Dimensions

Background Damage to the central nervous system(CNS)usually leads to the activation of astrocytes,followed by glial scar formation.For years,glial scar has been thought as a major obstacle for successful axon regeneration.However,increasing evidence suggests a beneficial role for this scar tissue as part of the endogenous local immune regulation and repair process.Surprisingly,in contrast to scars in other tissues,glial scars(mainly consist of reactive astrocytes)in both rat cortex and spinal cord were recently found to be significantly softer than healthy CNS tissues.Naive astrocytes have been found to change their phenotype to reactive astrocytes and gradually into scar-forming astrocytes,upregulating the astrocyte marker glial fibrillary acidic protein(GFAP),vimentin,and inflammatory proteins in almost all known brain disorders.Such phenotype transformation process has been widely thought unidirectional or irreversible.However,recent research revealed the environment-dependent plasticity of astrocyte phenotypes,with reactive astrocytes could revert in retrograde to naive astrocytes in proper microenvironment.In consideration of the important roles of mechanical cues in CNS and the unique softening behavior of glial scars,it is of great interesting to study the effects of dynamic changes of matrix stiffness on astrocyte phenotypic switch.Materials&methods Primary astrocyes were isolated from the cortex of SpragueDawley(SD)rats at PI.After cultured for 2 weeks,astrocytes were encapsulated into a set of three-dimensional(3D)hybrid hydrogel system composed of type I collagen and alginate.Immunofluorescence and Western blot expression analysis were applied for characterizting cell responses to different and dynamically changed matrix stiffness.A molecular dynamics model was developed for simulation.Results&discussion In this work,we established an in-vitro model to study the effects of dynamic changes of matrix stiffness on astrocyte phenotypic switchings in 3D.To simulate native cellular environment,we fabricated a set of hybrid hydrogel system composed of type I collagen and alginate.The stiffness of the hybrid hydrogels was demonstrated to be dynamically changed by adding calcium chloride or sodium citrate to crosslink or decrosslink alginate,respectively.Using 3D culture models,we showed that the decrease of matrix stiffness could promote astrocyte activation,with upregulated GFAP and IL-1β.In addition,3D cultured astrocytes spread greater with decreasing matrix stiffness.Moreover,we surprisingly found that astrocyte phenotype could be switched by dynamically changing matrix stiffness.Specifically,matrix stiffening reverted the activation of astrocytes,whereas matrix softening induced astrocyte activation.We further demonstrated that matrix stiffness-induced astrocyte activation was mediated through cytoskeletal tension and YAP protein.To some extent,YAP inhibition enhanced the responses of astrocytes to matrix stiffness.These may guide researchersto re-examine the role of matrix stiffness in reactive astrogliosis in vivo,and inspire the development of novel therapeutic approach for reducing glial scar following injury,enabling axonal regrowth and improving functional recovery by exploiting the benefits of mechanobiology studies.Conclusions Taken together,our results clarify the effects of matrix stiffness and its dynamic changes on phenotypic swtich of astrocytes in three dimensions and reveal environmental factors that regulate astrocytic phenotype transformation process,which may provide potential therapeutic approach for CNS injury.

ELASTO-PLASTIC STIFFNESS MATRIX OF ROCK MASS ELEMENT WITH SOFT CLAY STRATA AND ITS APPLICATION

The fault element is used to handle soft clay strata in a rock mass.The formulas or clasto-plastic stiffnessmatrix for the fault element are derived using the constitutive relationship between plastic increment stress andstrain.A numerical example of a circular tunnel with soft clay strata in the rock medium are examined.

Measuring the flexibility matrix of an eagle’s flight feather and a method to estimate the stiffness distribution

Flight feathers stand out with extraordinary mechanical properties for flight because they are lightweight but stiff enough.Their elasticity has great effects on the aerodynamics, resulting in aeroelasticity.Our primary task is to figure out the stiffness distribution of the feather to study the aeroelastic effects.The feather shaft is simplified as a beam, and the flexibility matrix of an eagle flight feather is tested.A numerical method is proposed to estimate the stiffness distributions along the shaft length based on an optimal Broyden–Fletcher–Goldfarb–Shanno(BFGS) method with global convergence.An analysis of the compressive behavior of the shaft based on the beam model shows a good fit with experimental results.The stiffness distribution of the shaft is finally presented using a 5 th order polynomial.

THE GENERATION OF NON－LINEAR STIFFNESS MATRIX OF TRIANGLE ELEMENT WHENCONSIDERING BOTH THE BENDING AND IN－PLANEME MBRANE FORCES

Using Stricklin Melhod￣[5],we have this paper has derived the formulas for the ge-neration of non-linear element stiffness matrix of a triangle element when considering both the bending and the in-plane membrane forces. A computer programme for the calculation of large deflection and inner forces of shallow shells is designed on theseformulas. The central deflection curve computed by this programme is compared with other pertaining results.

Matrix Stiffness Controls Cardiac Fibroblast Differentiation Through Regulating YAP via AT1R

Cardiac fibrosis is a common pathway to heart injury and failure,where continued activation of cardiac fibroblasts(CFs)during myocardium damage causes excessive deposition of the extracellular matrix and thus increases matrix stiffness.Increasing evidence has shown that stiff matrix plays an important positive role in promoting CF differentiation and cardiac fibrosis,with several signaling factors medicating CF mechanotransduction already identified.However,key moleculesthat perceive matrix stiffness to regulate CF differentiation remain to be fully defined.Recently,Hippo pathway transcriptional coactivators,i.e.,Yes-associated protein(YAP)and transcriptional coactivator with PDZ-binding motif(TAZ),have been found to work as mechanical signal transductors.Importantly,it has shown that YAP plays important roles in various types of fibrosis.Despite these findings,the role of YAP in CF mechanotransduction and cardiac fibrosis still remains elusive.Moreover,several several types of GPCRs have also been found to enable cells to sense mechanical cues,however,the relationship between these GPCRs and YAP in cell mechanotransduction is still not clear.Our recent work demonstrated that blocking of angiotensin II type 1 receptor(AT1R,the first GPCRs found to be mechanosensors)with losartan significantly inhibited the differentiation of CFs to myofibroblasts induced by stiff substrate.Taken these findings into account,we speculate that YAP may work as an important downstream signaling molecule of AT1R in mediating matrix stiffness-induced CF differentiation.In this work,we first characterized the expression of YAP in normal control(NC)and myocardial infarct(Ml)tissues of rats by using immunohistochemistry,immunofluorescence and Western blot analysis.We then investigated the role of YAP in matrix stiffness-induced CF differentiation in vitro by culturing CFs on mechanically tunable gelatin hydrogels.Finally,we explored the relationship between YAP and AT1R in CF mechanotransduction by selective transfection and inhibition experiments.The expression of YAP andα-SMA in cultured CFs were evaluated with immunofluorescence staining,Western blot and real-time quantitative PCR analysis.Immunohistochemical analysis revealed that both YAP andα-SMA significantly increased in Ml tissue compared with NC tissue.The expression and nuclear localization of YAP increased in CFs cultured on stiff matrix.YAP-deficient CFs cultured on soft and stiff matrix both showed decreased expression ofα-SMA.Meanwhile,YAP-overexpressing CFs cultured on soft and stiff matrix both showed increased expression ofα-SMA.Blocking of AT1R decreased the expression levels ofα-SMA and YAP and thus affected the responses of CFs to matrix stiffness.To sum up,our results identified an important role of YAP in mediating matrix stiffness-induced CF differentiation and also established the YAP pathway as an important signaling branch downstream of AT1R in CF mechanotransduction.This study may help to better understand the mechanism of fibrotic mechanotransduction and inspire the development of new approaches for treating cardiac fibrosis.

Matrix Stiffness-Induced Transcriptome Alterations and Regulatory Mechanisms Revealed by RNA-Seq in Endothelial Cells

Changes in vascular stiffness are associated with the development and progression of many diseases, especially in cardiovascular disease. However, the effect of vascular stiffness on the endothelial cells (ECs) is not fully understood. Therefore, this study aims to determine the gene expression changes of ECs cultured on the matrices with different stiffness (1 kPa and 40 kPa, respectively) by RNA-seq, thereby broadening the knowledge between mechanics and biology. We obtained 1775 differentially expressed genes (DEGs) by RNA-seq, with 450 up-regulated and 1325 down-regulated DEGs in ECs cultured on soft matrix (1 kPa) compared to those cultured on stiff matrix (40 kPa). After that, we performed a series of functional enrichment analyses based on DEGs and found that DEGs were enriched in many signaling pathways like adhesion junction. Furthermore, transcription factor (TF) target gene prediction analysis and protein-protein interaction (PPI) analysis were also conducted. We found that mechanotransduction signaling related TFs such as BRD4 are involved in. And in the PPI analysis, some genes encoding extracellular matrix proteins such as fibronectin 1 (FN1) were identified as the hub genes. In order to confirm the RNA-seq results, we performed real-time qPCR analysis on the genes of interest, including FN1, collagen α2 (IV) chain, matrix metalloproteinase-14 and integrin α5, and found that the expression levels of all these genes were down-regulated on soft matrix, suggesting that soft matrix caused by pathological conditions may directly attenuate vascular barrier function. This study offers the insights about the effects of physical stimulation on cells, paving a way for vascular tissue engineering, regenerative medicine, disease modeling and therapies.

Finite element model for arch bridge vibration dynamics considering effect of suspender length adjustment on geometry stiffness matrix

In this paper, we established a finite element (FEM) model to analyze the dynamic characteristics of arch bridges. In this model, the effects of adjustment to the length of a suspender on its geometry stiffness matrix are stressed. The FEM equations of mechanics characteristics, natural frequency and main mode are set up based on the first order matrix perturbation theory. Applicantion of the proposed model to analyze a real arch bridge proved the improvement in the simulation precision of dynamical characteristics of the arch bridge by considering the effects of suspender length variation.

THE DYNAMIC STIFFNESS MATRIX OF THE FINITE ANNULAR PLATE ELEMENT

The dynamic deformation of harmonic vibration is used as the shape functions of the finite annular plate element, and sonic integration difficulties related to the Bessel's functions are solved in this paper. Then the dynamic stiffness matrix of the finite annular plate element is established in closed form and checked by the direct stiffness method. The paper has given wide convcrage for decomposing the dynamic matrix into the power series of frequency square. By utilizing the axial symmetry of annular elements, the modes with different numbers of nodal diameters at s separately treated. Thus some terse and complete results are obtained as the foundation of structural characteristic analysis and dynamic response compulation.

Effect of extracellular matrix stiffness on the progression of renal fibrosis

Medical University,Shenyang 110122,China.The mechanical microenvironment of cells is recently reported to associate with the progression of renal fibrosis.However,the mechanism has not been fully elucidated.In the present study,based on the precision measurement of atom force microscope(AFM),hydrogel substrates of various stiffness were prepared to simulate the hardness of extracellular matrix(ECM)in normal and fibrotic kidney tissue.Renal mesangial cells were then cultured on the hydrogel matrix with different hardness.

New Formula for Geometric Stiffness Matrix Calculation

The standard formula for geometric stiffness matrix calculation, which is convenient for most engineering applications, is seen to be unsatisfactory for large strains because of poor accuracy, low convergence rate, and stability. For very large compressions, the tangent stiffness in the direction of the compression can even become negative, which can be regarded as physical nonsense. So in many cases rubber materials exposed to great compression cannot be analyzed, or the analysis could lead to very poor convergence. Problems with the standard geometric stiffness matrix can even occur with a small strain in the case of plastic yielding, which eventuates even greater practical problems. The authors demonstrate that amore precisional approach would not lead to such strange and theoretically unjustified results. An improved formula that would eliminate the disadvantages mentioned above and leads to higher convergence rate and more robust computations is suggested in this paper. The new formula can be derived from the principle of virtual work using a modified Green-Lagrange strain tensor, or from equilibrium conditions where in the choice of a specific strain measure is not needed for the geometric stiffness derivation (which can also be used for derivation of geometric stiffness of a rigid truss member). The new formula has been verified in practice with many calculations and implemented in the RFEM and SCIA Engineer programs. The advantages of the new formula in comparison with the standard formula are shown using several examples.

A New Stiffness Matrix for a 2D-Beam Element with a Transverse Opening

Transverse opening in a beam has a reducing effect of the beam stiffness which will cause a significant increase in beam deflection in the region on the opening. In this paper, a new stiffness matrix for a beam element with transverse opening including the effect of shear deformation has been derived. The strain energy principle is used in the derivation process of the stiffness matrix and the fixed-end force vector for the case of a concentrated or a uniformly distributed load is also derived. The accuracy of the obtained results based on the derived stiffness matrix is examined through comparison with that of the finite element method using Abaqus package and a previous study which show a good agreement with high accuracy.

Calculation Method of Non-Linear Stiffness Matrix in Spline Finite Element Method

According to the analytical characteristics of cubic B spline function,such as compactness and discontinuous smoothness,two methods,Global integral method and Assembling method are proposed to calculate the non-linear stiffness matrix in the spline finite element method.The Global integral method transforms the integral of the product of spline functions into the sum of boundary data of the product by integral by parts.The Assembling method uses the compactness characteristics of the spline function,reduces the n+3 order stiffness matrix in unit interval to the 4 order matrix,and then obtains the global matrix by assembling the small ones.The two proposed methods are clear in theory and easy to be realized by computer program,which can effectively improve the efficiency of the algorithm for nonlinear spline element method.