Simulation of extracellular matrix remodeling by fibroblast cells in soft three-dimensional bioresorbable scaffolds

To culture functional soft tissues and organs in three-dimensional scaffolds, it is essential to elucidate the optimal scaffold mechanical properties. However, mechanoregulated soft tissue remodeling is not well understood. In this study, we hypothesized that individual cells are capable of remodeling extracellular matrix within a short proximity of themselves in order to match the stiffness of the broader surrounding matrix. This theory was implemented in a three-dimensional finite element model to simulate soft tissue remodeling of human fibroblast cells in two collagen–chitosan scaffolds with different mechanical properties. Simulation results closely matched with previously reported experimental data, showing that soft tissue growth in compliant (Scaf-A, 4.30 kPa) and stiff (Scaf-B, 17.03 kPa) scaffolds led to an almost eightfold difference in the resulting stiffnesses after 10 days (8.40 kPa for Scaf-A, 59.25 kPa for Scaf-B). Furthermore, varying the simulated rate for tissue remodeling by \(\pm \)50 % caused unequal changes in the resulting stiffness (+3.6 and \(-\)23 % for Scaf-A, +5 and \(-\)17 % for Scaf-B), and \(\pm \)50 % changes in the assumed upper limit on tissue stiffness only had larger effects on the stiff scaffold (+42 and \(-\)44 % for Scaf-B). These results reinforce the notion that soft tissue remodeling is governed by the stiffness of the surrounding matrix, until meeting an upper limit on tissue stiffness.     

Improved identifiability of myocardial material parameters by an energy-based cost function

Myocardial stiffness is a valuable clinical biomarker for the monitoring and stratification of heart failure (HF). Cardiac finite element models provide a biomechanical framework for the assessment of stiffness through the determination of the myocardial constitutive model parameters. The reported parameter intercorrelations in popular constitutive relations, however, obstruct the unique estimation of material parameters and limit the reliable translation of this stiffness metric to clinical practice. Focusing on the role of the cost function (CF) in parameter identifiability, we investigate the performance of a set of geometric indices (based on displacements, strains, cavity volume, wall thickness and apicobasal dimension of the ventricle) and a novel CF derived from energy conservation. Our results, with a commonly used transversely isotropic material model (proposed by Guccione et al.), demonstrate that a single geometry-based CF is unable to uniquely constrain the parameter space. The energy-based CF, conversely, isolates one of the parameters and in conjunction with one of the geometric metrics provides a unique estimation of the parameter set. This gives rise to a new methodology for estimating myocardial material parameters based on the combination of deformation and energetics analysis. The accuracy of the pipeline is demonstrated in silico, and its robustness in vivo, in a total of 8 clinical data sets (7 HF and one control). The mean identified parameters of the Guccione material law were \(C_1=3000\pm 1700\,\hbox {Pa}\) and \(\alpha =45\pm 25\) (\(b_f=25\pm 14\), \(b_{ft}=11\pm 6\), \(b_{t}=9\pm 5\)) for the HF cases and \(C_1=1700\,\hbox {Pa}\) and \(\alpha =15\) (\(b_f=8\), \(b_{ft}=4\), \(b_{t}=3\)) for the healthy case.     

Generalized quasispecies model on finite metric spaces: isometry groups and spectral properties of evolutionary matrices

The quasispecies model introduced by Eigen in 1971 has close connections with the isometry group of the space of binary sequences relative to the Hamming distance metric. Generalizing this observation we introduce an abstract quasispecies model on a finite metric space X together with a group of isometries \(\Gamma \) acting transitively on X. We show that if the domain of the fitness function has a natural decomposition into the union of tG-orbits, G being a subgroup of \(\Gamma \), then the dominant eigenvalue of the evolutionary matrix satisfies an algebraic equation of degree at most \(t\cdot \mathrm{rk}_{\mathbf {Z}} R\), where R is the orbital ring that is defined in the text. The general theory is illustrated by three detailed examples. In the first two of them the space X is taken to be the metric space of vertices of a regular polytope with the natural “edge” metric, these are the cases of a regular m-gon and of a hyperoctahedron; the final example takes as X the quotient rings \(\mathbf {Z}/p^n\mathbf {Z}\) with p-adic metric.

     

Principal trabecular structural orientation predicted by quantitative ultrasound is strongly correlated with \upmu FEA determined anisotropic apparent stiffness

The microarchitecture and alignment of trabecular bone adapts to the particular mechanical milieu applied to it. Due to this anisotropic mechanical property, measurement orientation has to be taken into consideration when assessing trabecular bone quality and fracture risk prediction. Quantitative ultrasound (QUS) has demonstrated the ability in predicting the principal structural orientation (PSO) of trabecular bone. Although the QUS prediction for PSO is very close to that of \(\upmu \) CT, certain angle differences still exist. It remains unknown whether this angle difference can induce significant differences in mechanical properties or not. The objective of this study was to evaluate the mechanical properties in different PSOs predicted using different methods, QUS and \(\upmu \) CT, thus to investigate the ability of QUS as a means to predict the PSO of trabecular bone noninvasively. By validating the ability of QUS to predict the PSO of trabecular bone, it is beneficial for future QUS applications because QUS measurements in the PSO can provide information more correlated with the mechanical properties than with other orientations. In this study, seven trabecular bone balls from distal bovine femurs were used to generate finite element models based on the 3-dimensional \(\upmu \) CT images. Uniaxial compressive loading was performed on the bone ball models in the finite element analysis (FEA) in six different orientations (three anatomical orientations, two PSOs predicted by QUS and the longest vector of mean intercept length (MIL) tensor calculated by \(\upmu \) CT). The stiffness was calculated based on the reaction force of the bone balls under loading, and the von Mises stress results showed that both the mechanical properties in the PSOs predicted by QUS are significantly higher than the anatomical orientations and comparatively close to the longest vector of MIL tensor. The stiffness in the PSOs predicted by QUS is also highly correlated with the stiffness in the MIL tensor orientation (ATTmax vs. MIL, \(R^{2}\)  = 0.98, \(p<001\) ; UVmax vs. MIL, \(R^{2}\)  = 0.92, \(p<001\) ). These results were validated by in vitro mechanical testing on the bone ball samples. This study demonstrates that the PSO of trabecular bone predicted by QUS has an equally strong apparent stiffness with the orientation predicted by \(\upmu \) CT.     

Correction to: Mechanical factors contributing to the Venus flytrap’s rate-dependent response to stimuli

The sensory hairs of the Venus flytrap (Dionaea muscipula Ellis) detect mechanical stimuli imparted by their prey and fire bursts of electrical signals called action potentials (APs). APs are elicited when the hairs are sufficiently stimulated and two consecutive APs can trigger closure of the trap. Earlier experiments have identified thresholds for the relevant stimulus parameters, namely the angular displacement \(\theta \) and angular velocity \(\omega \). However, these experiments could not trace the deformation of the trigger hair’s sensory cells, which are known to transduce the mechanical stimulus. To understand the kinematics at the cellular level, we investigate the role of two relevant mechanical phenomena: viscoelasticity and intercellular fluid transport using a multi-scale numerical model of the sensory hair. We hypothesize that the combined influence of these two phenomena and \(\omega \) contribute to the flytrap’s rate-dependent response to stimuli. In this study, we firstly perform sustained deflection tests on the hair to estimate the viscoelastic material properties of the tissue. Thereafter, through simulations of hair deflection tests at different loading rates, we were able to establish a multi-scale kinematic link between \(\omega \) and the cell wall stretch \(\delta \). Furthermore, we find that the rate at which \(\delta \) evolves during a stimulus is also proportional to \(\omega \). This suggests that mechanosensitive ion channels, expected to be stretch-activated and localized in the plasma membrane of the sensory cells, could be additionally sensitive to the rate at which stretch is applied.

     

A Discrete Velocity Kinetic Model with Food Metric: Chemotaxis Traveling Waves

We introduce a mesoscopic scale chemotaxis model for traveling wave phenomena which is induced by food metric. The organisms of this simplified kinetic model have two discrete velocity modes, \(\pm s\) and a constant tumbling rate. The main feature of the model is that the speed of organisms is constant \(s\,{>}\,0\) with respect to the food metric, not the Euclidean metric. The uniqueness and the existence of the traveling wave solution of the model are obtained. Unlike the classical logarithmic model case there exist traveling waves under super-linear consumption rates and infinite population pulse-type traveling waves are obtained. Numerical simulations are also provided.     

Inverse remodelling algorithm identifies habitual manual activities of primates based on metacarpal bone architecture

Previously, a micro-finite element (micro-FE)-based inverse remodelling method was presented in the literature that reconstructs the loading history of a bone based on its architecture alone. Despite promising preliminary results, it remains unclear whether this method is sensitive enough to detect differences of bone loading related to pathologies or habitual activities. The goal of this study was to test the sensitivity of the inverse remodelling method by predicting joint loading histories of metacarpal bones of species with similar anatomy but clearly distinct habitual hand use. Three groups of habitual hand use were defined using the most representative primate species: manipulation (human), suspensory locomotion (orangutan), and knuckle-walking locomotion (bonobo, chimpanzee, gorilla). Nine to ten micro-computed tomography scans of each species (\(n=48\) in total) were used to create micro-FE models of the metacarpal head region. The most probable joint loading history was predicted by optimally scaling six load cases representing joint postures ranging from \(-\,75^{\circ }\) (extension) to \(+\,75^{\circ }\) (flexion). Predicted mean joint load directions were significantly different between knuckle-walking and non-knuckle-walking groups (\(p<0.05\)) and in line with expected primary hand postures. Mean joint load magnitudes tended to be larger in species using their hands for locomotion compared to species using them for manipulation. In conclusion, this study shows that the micro-FE-based inverse remodelling method is sensitive enough to detect differences of joint loading related to habitual manual activities of primates and might, therefore, be useful for palaeoanthropologists to reconstruct the behaviour of extinct species and for biomedical applications such as detecting pathological joint loading.

     

A thermomechanical framework for reconciling the effects of ultraviolet radiation exposure time and wavelength on connective tissue elasticity

Augmentation of the mechanical properties of connective tissue using ultraviolet (UV) radiation—by targeting collagen cross-linking in the tissue at predetermined UV exposure time \((t)\) and wavelength \((\lambda )\) —has been proposed as a therapeutic method for supporting the treatment for structural-related injuries and pathologies. However, the effects of \(\lambda \) and \(t\) on the tissue elasticity, namely elastic modulus \((E)\) and modulus of resilience \((u_\mathrm{Y})\) , are not entirely clear. We present a thermomechanical framework to reconcile the \(t\) - and \(\lambda \) -related effects on \(E\) and \(u_\mathrm{Y}\) . The framework addresses (1) an energy transfer model to describe the dependence of the absorbed UV photon energy, \(\xi \) , per unit mass of the tissue on \(t\) and \(\lambda \) , (2) an intervening thermodynamic shear-related parameter, \(G\) , to quantify the extent of UV-induced cross-linking in the tissue, (3) a threshold model for the \(G\) versus \(\xi \) relationship, characterized by   \(t_\mathrm{C}\) —the critical \(t\) underpinning the association of \(\xi \) with \(G\) —and (4) the role of \(G\) in the tissue elasticity. We hypothesized that \(G\) regulates \(E\) (UV-stiffening hypothesis) and \(u_\mathrm{Y}\) (UV-resilience hypothesis). The framework was evaluated with the support from data derived from tensile testing on isolated ligament fascicles, treated with two levels of \(\lambda \) (365 and 254 nm) and three levels of \(t\) (15, 30 and 60 min). Predictions from the energy transfer model corroborated the findings from a two-factor analysis of variance of the effects of \(t\) and \(\lambda \) treatments. Student’s t test revealed positive change in \(E\) and \(u_\mathrm{Y}\) with increases in \(G\) —the findings lend support to the hypotheses, implicating the implicit dependence of UV-induced cross-links on \(t\) and \(\lambda \) for directing tissue stiffness and resilience. From a practical perspective, the study is a step in the direction to establish a UV irradiation treatment protocol for effective control of exogenous cross-linking in connective tissues.     

Estimation of the effective yield properties of human trabecular bone using nonlinear micro-finite element analyses

Micro-finite element (\(\upmu \)FE) analyses are often used to determine the apparent mechanical properties of trabecular bone volumes. Yet, these apparent properties depend strongly on the applied boundary conditions (BCs) for the limited size of volumes that can be obtained from human bones. To attenuate the influence of the BCs, we computed the yield properties of samples loaded via a surrounding layer of trabecular bone (“embedded configuration”). Thirteen cubic volumes (10.6 mm side length) were collected from \(\upmu \)CT reconstructions of human vertebrae and femora and converted into \(\upmu \)FE models. An isotropic elasto-plastic material model was chosen for bone tissue, and nonlinear \(\upmu \)FE analyses of six uniaxial, shear, and multi-axial load cases were simulated to determine the yield properties of a subregion (5.3 mm side length) of each volume. Three BCs were tested. Kinematic uniform BCs (KUBCs: each boundary node is constrained with uniform displacements) and periodicity-compatible mixed uniform BCs (PMUBCs: each boundary node is constrained with a uniform combination of displacements and tractions mimicking the periodic BCs for an orthotropic material) were directly applied to the subregions, while the embedded configuration was achieved by applying PMUBCs on the larger volumes instead. Yield stresses and strains, and element damage at yield were finally compared across BCs. Our findings indicate that yield strains do not depend on the BCs. However, KUBCs significantly overestimate yield stresses obtained in the embedded configuration (+43.1 ± 27.9%). PMUBCs underestimate (?10.0 ± 11.2%), but not significantly, yield stresses in the embedded situation. Similarly, KUBCs lead to higher damage levels than PMUBCs (+51.0 ± 16.9%) and embedded configurations (+48.4 ± 15.0%). PMUBCs are better suited for reproducing the loading conditions in subregions of the trabecular bone and deliver a fair estimation of their effective (asymptotic) yield properties.     

Stability of a Stochastic Model of an SIR Epidemic with Vaccination

We prove almost sure exponential stability for the disease-free equilibrium of a stochastic differential equations model of an SIR epidemic with vaccination. The model allows for vertical transmission. The stochastic perturbation is associated with the force of infection and is such that the total population size remains constant in time. We prove almost sure positivity of solutions. The main result concerns especially the smaller values of the diffusion parameter, and describes the stability in terms of an analogue \(\mathcal{R}_\sigma\) of the basic reproduction number \(\mathcal{R}_0\) of the underlying deterministic model, with \(\mathcal{R}_\sigma \le \mathcal{R}_0\). We prove that the disease-free equilibrium is almost sure exponentially stable if \(\mathcal{R}_\sigma <1\).     

Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions

We introduce a mathematical model of the in vivo progression of Alzheimer’s disease with focus on the role of prions in memory impairment. Our model consists of differential equations that describe the dynamic formation of \(\upbeta \) -amyloid plaques based on the concentrations of A \(\upbeta \)  oligomers, PrP^C proteins, and the A \(\upbeta \) - \(\times \) -PrP^Ccomplex, which are hypothesized to be responsible for synaptic toxicity. We prove the well-posedness of the model and provided stability results for its unique equilibrium, when the polymerization rate of \(\upbeta \) -amyloid is constant and also when it is described by a power law.     

On a three–dimensional model for the description of the passive characteristics of skeletal muscle tissue

In this work, a three–dimensional model was developed to describe the passive mechanical behaviour of anisotropic skeletal muscle tissue. To validate the model, orientation–dependent axial (\(0^\circ\), \(45^\circ\), \(90^\circ\)) and semi–confined compression experiments (mode I, II, III) were performed on soleus muscle tissue from rabbits. In the latter experiments, specimen deformation is prescribed in the loading direction and prevented in an additional spatial direction, fibre compression at \(0^\circ\) (mode I), fibre elongation at \(90^\circ\) (mode II) and a neutral state of the fibres at \(90^\circ\) where their length is kept constant (mode III). Overall, the model can adequately describe the mechanical behaviour with a relatively small number of model parameters. The stiffest tissue response during orientation–dependent axial compression (\(-\,7.7\,\pm \,1.3\) kPa) occurs when the fibres are oriented perpendicular to the loading direction (\(90^\circ\)) and are thus stretched during loading. Semi–confined compression experiments yielded the stiffest tissue (\(-\,36.7\,\pm \,11.2\) kPa) in mode II when the muscle fibres are stretched. The extensive data set collected in this study allows to study the different error measures depending on the deformation state or the combination of deformation states.

     

Modulation of the cAMP Response by G\alpha _i and G\beta \gamma : A Computational Study of G Protein Signaling in Immune Cells

Cyclic AMP is important for the resolution of inflammation, as it promotes anti-inflammatory signaling in several immune cell lines. In this paper, we present an immune cell specific model of the cAMP signaling cascade, paying close attention to the specific isoforms of adenylyl cyclase (AC) and phosphodiesterase that control cAMP production and degradation, respectively, in these cells. The model describes the role that G protein subunits, including G \(\alpha _s\) , G \(\alpha _i\) , and G \(\beta \gamma \) , have in regulating cAMP production. Previously, G \(\alpha _i\) activation has been shown to increase the level of cAMP in certain immune cell types. This increase in cAMP is thought to be mediated by \(\beta \gamma \) subunits which are released upon G \(\alpha \) activation and can directly stimulate specific isoforms of AC. We conduct numerical experiments in order to explore the mechanisms through which G \(\alpha _i\) activation can increase cAMP production. An important conclusion of our analysis is that the relative abundance of different G protein subunits is an essential determinant of the cAMP profile in immune cells. In particular, our model predicts that limited availability of \(\beta \gamma \) subunits may both \((i)\) enable immune cells to link inflammatory G \(\alpha _i\) signaling to anti-inflammatory cAMP production thereby creating a balanced immune response to stimulation with low concentrations of PGE2, and \((ii)\) prohibit robust anti-inflammatory cAMP signaling in response to stimulation with high concentrations of PGE2.     

Optimal foraging behavior with an explicit consideration of within-individual behavioral variation: an example of predation

Animal behavior is flexible, and the same individual can exhibit variable expressions under the equivalent ecological situations (i.e., within-individual behavioral variation). This study examines the evolution of within-individual behavioral variation using an individual-based model. A common predation scenario is considered where a predator spends a period h to handle and consume a captured prey. The model assumes the handling time of the predator to be a random variable. The average and within-individual variance of handling time are described by \(\mu _h\) and \(\sigma _h^2\), respectively, where each individual has its own unique \(\mu _h\) and \(\sigma _h^2\). Using a genetic algorithm, the evolution of \(\sigma _h^2\) is traced. The results show that natural selection acts on both \(\mu _h\) and \(\sigma _h^2\), and the optimal behavioral variation depends on the density of prey. In particular, individuals with high behavioral variance \(\sigma _h^2\) are more likely selected when prey density is low. Individual based modeling can be a useful tool for studying the ultimate significance of within-individual behavioral variation and generating empirically testable predictions. The mechanisms of the evolution of within-individual behavioral variation and their ecological implications are discussed.     

A Mathematical Model for the Macrophage Response to Respiratory Viral Infection in Normal and Asthmatic Conditions

Respiratory viral infections are common in the general population and one of the most important causes of asthma aggravation and exacerbation. Despite many studies, it is not well understood how viral infections cause more severe symptoms and exacerbations in asthmatics. We develop a mathematical model of two types of macrophages that play complementary roles in fighting viral infection: classically \((\hbox {CA}\)-\(\hbox {M}\Phi )\) and alternatively activated macrophages \((\hbox {AA}\)-\(\hbox {M}\Phi )\). \(\hbox {CA}\)-\(\hbox {M}\Phi \) destroy infected cells and tissues to remove viruses, while \(\hbox {AA}\)-\(\hbox {M}\Phi \) repair damaged tissues. We show that a higher viral load or longer duration of infection provokes a stronger immune response from the macrophage system. By adjusting the parameters, we model the differences in response to respiratory viral infection in normal and asthmatic subjects and show how this skews the system toward a response that generates more severe symptoms in asthmatic patients.     

Cell Volume Regulation in the Proximal Tubule of Rat Kidney

We developed a dynamic model of a rat proximal convoluted tubule cell in order to investigate cell volume regulation mechanisms in this nephron segment. We examined whether regulatory volume decrease (RVD), which follows exposure to a hyposmotic peritubular solution, can be achieved solely via stimulation of basolateral K\(^+\) and \(\hbox {Cl}^-\) channels and \(\hbox {Na}^+\)–\(\hbox {HCO}_3^-\) cotransporters. We also determined whether regulatory volume increase (RVI), which follows exposure to a hyperosmotic peritubular solution under certain conditions, may be accomplished by activating basolateral \(\hbox {Na}^+\)/H\(^+\) exchangers. Model predictions were in good agreement with experimental observations in mouse proximal tubule cells assuming that a 10% increase in cell volume induces a fourfold increase in the expression of basolateral K\(^+\) and \(\hbox {Cl}^-\) channels and \(\hbox {Na}^+\)–\(\hbox {HCO}_3^-\) cotransporters. Our results also suggest that in response to a hyposmotic challenge and subsequent cell swelling, \(\hbox {Na}^+\)–\(\hbox {HCO}^-_3\) cotransporters are more efficient than basolateral K\(^+\) and \(\hbox {Cl}^-\) channels at lowering intracellular osmolality and reducing cell volume. Moreover, both RVD and RVI are predicted to stabilize net transcellular \(\hbox {Na}^+\) reabsorption, that is, to limit the net \(\hbox {Na}^+\) flux decrease during a hyposmotic challenge or the net \(\hbox {Na}^+\) flux increase during a hyperosmotic challenge.     

Cellular mechanosensitivity to substrate stiffness decreases with increasing dissimilarity to cell stiffness

Computational modelling has received increasing attention to investigate multi-scale coupled problems in micro-heterogeneous biological structures such as cells. In the current study, we investigated for a single cell the effects of (1) different cell-substrate attachment (2) and different substrate modulus \(\textit{E}_\mathrm{s}\) on intracellular deformations. A fibroblast was geometrically reconstructed from confocal micrographs. Finite element models of the cell on a planar substrate were developed. Intracellular deformations due to substrate stretch of \(\lambda =1.1\), were assessed for: (1) cell-substrate attachment implemented as full basal contact (FC) and 124 focal adhesions (FA), respectively, and \(\textit{E}_\mathrm{s}\,=\,\)140 KPa and (2) \(\textit{E}_\mathrm{s}\,=\,10\), 140, 1000, and 10,000 KPa, respectively, and FA attachment. The largest strains in cytosol, nucleus and cell membrane were higher for FC (1.35\(\text {e}^{-2}\), 0.235\(\text {e}^{-2}\) and 0.6\(\text {e}^{-2}\)) than for FA attachment (0.0952\(\text {e}^{-2}\), 0.0472\(\text {e}^{-2}\) and 0.05\(\text {e}^{-2}\)). For increasing \(\textit{E}_\mathrm{s}\), the largest maximum principal strain was 4.4\(\text {e}^{-4}\), 5\(\text {e}^{-4}\), 5.3\(\text {e}^{-4}\) and 5.3\(\text {e}^{-4}\) in the membrane, 9.5\(\text {e}^{-4}\), 1.1\(\text {e}^{-4}\), 1.2\(\text {e}^{-3}\) and 1.2\(\text {e}^{-3}\) in the cytosol, and 4.5\(\text {e}^{-4}\), 5.3\(\text {e}^{-4}\), 5.7\(\text {e}^{-4}\) and 5.7\(\text {e}^{-4}\) in the nucleus. The results show (1) the importance of representing FA in cell models and (2) higher cellular mechanical sensitivity for substrate stiffness changes in the range of cell stiffness. The latter indicates that matching substrate stiffness to cell stiffness, and moderate variation of the former is very effective for controlled variation of cell deformation. The developed methodology is useful for parametric studies on cellular mechanics to obtain quantitative data of subcellular strains and stresses that cannot easily be measured experimentally.