Multiscale mechanobiology: Coupling models of adhesion kinetics and nonlinear tissue mechanics

The mechanical behavior of tissues at the macroscale is tightly coupled to cellular activity at the microscale. Dermal wound healing is a prominent example of a complex system in which multiscale mechanics regulate restoration of tissue form and function. In cutaneous wound healing, a fibrin matrix is populated by fibroblasts migrating in from a surrounding tissue made mostly out of collagen. Fibroblasts both respond to mechanical cues, such as fiber alignment and stiffness, as well as exert active stresses needed for wound closure.Here, we develop a multiscale model with a two-way coupling between a microscale cell adhesion model and a macroscale tissue mechanics model. Starting from the well-known model of adhesion kinetics proposed by Bell, we extend the formulation to account for nonlinear mechanics of fibrin and collagen and show how this nonlinear response naturally captures stretch-driven mechanosensing. We then embed the new nonlinear adhesion model into a custom finite element implementation of tissue mechanical equilibrium. Strains and stresses at the tissue level are coupled with the solution of the microscale adhesion model at each integration point of the finite element mesh. In addition, solution of the adhesion model is coupled with the active contractile stress of the cell population. The multiscale model successfully captures the mechanical response of biopolymer fibers and gels, contractile stresses generated by fibroblasts, and stress-strain contours observed during wound healing. We anticipate that this framework will not only increase our understanding of how mechanical cues guide cellular behavior in cutaneous wound healing, but will also be helpful in the study of mechanobiology, growth, and remodeling in other tissues.     

Stress and strain localization in stretched collagenous tissues via a multiscale modelling approach

Mechanobiology of cells in soft collagenous tissues is highly affected by both tissue response at the macroscale and stress/strain localization mechanisms due to features at lower scales. In this paper, the macroscale mechanical behaviour of soft collagenous tissues is modelled by a three-level multiscale approach, based on a multi-step homogenisation technique from nanoscale up to the macroscale. Nanoscale effects, related to both intermolecular cross-links and collagen mechanics, are accounted for, together with geometric nonlinearities at the microscale. Moreover, an effective submodelling procedure is conceived in order to evaluate the local stress and strain fields at the microscale, which is around and within cells. Numerical results, obtained by using an incremental finite element formulation and addressing stretched tendinous tissues, prove consistency and accuracy of the model at both macroscale and microscale, confirming also the effectiveness of the multiscale modelling concept for successfully analysing physiopathological processes in biological tissues.     

Micromechanical model of a surrogate for collagenous soft tissues: development, validation and analysis of mesoscale size effects

Aligned, collagenous tissues such as tendons and ligaments are composed primarily of water and type I collagen, organized hierarchically into nanoscale fibrils, microscale fibers and mesoscale fascicles. Force transfer across scales is complex and poorly understood. Since innervation, the vasculature, damage mechanisms and mechanotransduction occur at the microscale and mesoscale, understanding multiscale interactions is of high importance. This study used a physical model in combination with a computational model to isolate and examine the mechanisms of force transfer between scales. A collagen-based surrogate served as the physical model. The surrogate consisted of extruded collagen fibers embedded within a collagen gel matrix. A micromechanical finite element model of the surrogate was validated using tensile test data that were recorded using a custom tensile testing device mounted on a confocal microscope. Results demonstrated that the experimentally measured macroscale strain was not representative of the microscale strain, which was highly inhomogeneous. The micromechanical model, in combination with a macroscopic continuum model, revealed that the microscale inhomogeneity resulted from size effects in the presence of a constrained boundary. A sensitivity study indicated that significant scale effects would be present over a range of physiologically relevant inter-fiber spacing values and matrix material properties. The results indicate that the traditional continuum assumption is not valid for describing the macroscale behavior of the surrogate and that boundary-induced size effects are present.     

Loading effect of fibroblast-myocyte coupling on resting potential, impulse propagation, and repolarization: insights from a microstructure model

The numerous nonmyocytes present within the myocardium may establish electrical connections with myocytes through gap junctions, formed naturally or as a result of a cell therapy. The strength of the coupling and its potential impact on action potential characteristics and conduction are not well understood. This study used computer simulation to investigate the load-induced electrophysiological consequences of the coupling of myocytes with fibroblasts, where the fibroblast resting potential, density, distribution, and coupling strength were varied. Conduction velocity (CV), upstroke velocity, and action potential duration (APD) were analyzed for longitudinal and transverse impulse propagation in a two-dimensional microstructure tissue model, developed to represent a monolayer culture of cardiac cells covered by a layer of fibroblasts. The results show that 1) at weak coupling (<0.25 nS), the myocyte resting potential was elevated, leading to CV up to 5% faster than control; 2) at intermediate coupling, the myocyte resting potential elevation saturated, whereas the current flowing from the myocyte to the fibroblast progressively slowed down both CV and upstroke velocity; 3) at strong couplings (>8 nS), all of the effects saturated; and 4) APD at 90% repolarization was usually prolonged by 0-20 ms (up to 60-80 ms for high fibroblast density and coupling) by the coupling to fibroblasts. The changes in APD depended on the fibroblast resting potential. This complex, coupling-dependent interaction of fibroblast and myocytes also has relevance to the integration of other nonmyocytes in the heart, such as those used in cellular therapies.     

Roles of subcellular Na+ channel distributions in the mechanism of cardiac conduction

The gap junction and voltage-gated Na⁺ channel play an important role in the action potential propagation. The purpose of this study was to elucidate the roles of subcellular Na⁺ channel distribution in action potential propagation. To achieve this, we constructed the myocardial strand model, which can calculate the current via intercellular cleft (electric-field mechanism) together with gap-junctional current (gap-junctional mechanism). We conducted simulations of action potential propagation in a myofiber model where cardiomyocytes were electrically coupled with gap junctions alone or with both the gap junctions and the electric field mechanism. Then we found that the action potential propagation was greatly affected by the subcellular distribution of Na⁺ channels in the presence of the electric field mechanism. The presence of Na⁺ channels in the lateral membrane was important to ensure the stability of propagation under conditions of reduced gap-junctional coupling. In the poorly coupled tissue with sufficient Na⁺ channels in the lateral membrane, the slowing of action potential propagation resulted from the periodic and intermittent dysfunction of the electric field mechanism. The changes in the subcellular Na⁺ channel distribution might be in part responsible for the homeostatic excitation propagation in the diseased heart.     

Nonequilibrium Arrhythmic States and Transitions in a Mathematical Model for Diffuse Fibrosis in Human Cardiac Tissue

We present a comprehensive numerical study of spiral-and scroll-wave dynamics in a state-of-the-art mathematical model for human ventricular tissue with fiber rotation, transmural heterogeneity, myocytes, and fibroblasts. Our mathematical model introduces fibroblasts randomly, to mimic diffuse fibrosis, in the ten Tusscher-Noble-Noble-Panfilov (TNNP) model for human ventricular tissue; the passive fibroblasts in our model do not exhibit an action potential in the absence of coupling with myocytes; and we allow for a coupling between nearby myocytes and fibroblasts. Our study of a single myocyte-fibroblast (MF) composite, with a single myocyte coupled to fibroblasts via a gap-junctional conductance , reveals five qualitatively different responses for this composite. Our investigations of two-dimensional domains with a random distribution of fibroblasts in a myocyte background reveal that, as the percentage of fibroblasts increases, the conduction velocity of a plane wave decreases until there is conduction failure. If we consider spiral-wave dynamics in such a medium we find, in two dimensions, a variety of nonequilibrium states, temporally periodic, quasiperiodic, chaotic, and quiescent, and an intricate sequence of transitions between them; we also study the analogous sequence of transitions for three-dimensional scroll waves in a three-dimensional version of our mathematical model that includes both fiber rotation and transmural heterogeneity. We thus elucidate random-fibrosis-induced nonequilibrium transitions, which lead to conduction block for spiral waves in two dimensions and scroll waves in three dimensions. We explore possible experimental implications of our mathematical and numerical studies for plane-, spiral-, and scroll-wave dynamics in cardiac tissue with fibrosis.     

Nonlinear hierarchical multiscale modeling of cortical bone considering its nanoscale microstructure

We have used a hierarchical multiscale modeling scheme for the analysis of cortical bone considering it as a nanocomposite. This scheme consists of definition of two boundary value problems, one for macroscale, and another for microscale. The coupling between these scales is done by using the homogenization technique. At every material point in which the constitutive model is needed, a microscale boundary value problem is defined using a macroscopic kinematical quantity and solved. Using the described scheme, we have studied elastic properties of cortical bone considering its nanoscale microstructural constituents with various mineral volume fractions. Since the microstructure of bone consists of mineral platelet with nanometer size embedded in a protein matrix, it is similar to the microstructure of soft matrix nanocomposites reinforced with hard nanostructures. Considering a representative volume element (RVE) of the microstructure of bone as the microscale problem in our hierarchical multiscale modeling scheme, the global behavior of bone is obtained under various macroscopic loading conditions. This scheme may be suitable for modeling arbitrary bone geometries subjected to a variety of loading conditions. Using the presented method, mechanical properties of cortical bone including elastic moduli and Poisson's ratios in two major directions and shear modulus is obtained for different mineral volume fractions.     

Multiscale characterization and micromechanical modeling of crop stem materials

An essential prerequisite for the efficient biomechanical tailoring of crops is to accurately relate mechanical behavior to compositional and morphological properties across different length scales. In this article, we develop a multiscale approach to predict macroscale stiffness and strength properties of crop stem materials from their hierarchical microstructure. We first discuss the experimental multiscale characterization based on microimaging (micro-CT, light microscopy, transmission electron microscopy) and chemical analysis, with a particular focus on oat stems. We then derive in detail a general micromechanics-based model of macroscale stiffness and strength. We specify our model for oats and validate it against a series of bending experiments that we conducted with oat stem samples. In the context of biomechanical tailoring, we demonstrate that our model can predict the effects of genetic modifications of microscale composition and morphology on macroscale mechanical properties of thale cress that is available in the literature.

     

A mathematical model of electrotonic interactions between ventricular myocytes and fibroblasts

Functional intercellular coupling has been demonstrated among networks of cardiac fibroblasts, as well as between fibroblasts and atrial or ventricular myocytes. In this study, the consequences of these interactions were examined by implementing the ten Tusscher model of the human ventricular action potential, and coupling it to our electrophysiological models for mammalian ventricular fibroblasts. Our simulations reveal significant electrophysiological consequences of coupling between 1 and 4 fibroblasts to a single ventricular myocyte. These include alterations in plateau height and/or action potential duration (APD) and changes in underlying ionic currents. Two series of simulations were carried out. First, fibroblasts were modeled as a spherical cell with a capacitance of 6.3 pF and an ohmic membrane resistance of 10.7 G Omega. When these "passive" fibroblasts were coupled to a myocyte, they caused slight prolongation of APD with no changes in the plateau, threshold for firing, or rate of initial depolarization. In contrast, when the same myocyte-fibroblast complexes were modeled after addition of the time- and voltage-gated K(+) currents that are expressed in fibroblasts, much more pronounced effects were observed: the plateau height of the action potential was reduced and the APD shortened significantly. In addition, each fibroblast exhibited significant electrotonic depolarizations in response to each myocyte action potential and the resting potential of the fibroblasts closely approximated the resting potential of the coupled ventricular myocyte.     

Coupling microscale vegetation–soil water and macroscale vegetation–precipitation feedbacks in semiarid ecosystems

At macroscale, land–atmosphere exchange of energy and water in semiarid zones such as the Sahel constitutes a strong positive feedback between vegetation density and precipitation. At microscale, however, additional positive feedbacks between hydrology and vegetation such as increase of infiltration due to increase of vegetation, have been reported and have a large impact on vegetation distribution and spatial pattern formation. If both macroscale and microscale positive feedbacks are present in the same region, it is reasonable to assume that these feedback mechanisms are connected. In this study, we develop and analyse a soil‐vegetation‐atmosphere model coupling large‐scale evapotranspiration–precipitation feedback with a model of microscale vegetation–hydrology feedback to study the integration of these nonlinearities at disparate scales. From our results, two important conclusions can be drawn: (1) it is important to account for spatially explicit vegetation dynamics at the microscale in climate models (the strength of the precipitation feedback increased up to 35% by accounting for these microscale dynamics); (2) studies on resilience of ecosystems to climate change should always be cast within a framework of possible large‐scale atmospheric feedback mechanism (substantial changes in vegetation resilience resulted from incorporating macroscale precipitation feedback). Analysis of full‐coupled modelling shows that both type of feedbacks markedly influence each other and that they should both be accounted for in climate change models.     

Myoblast proliferation and syncytial fusion both depend on connexin43 function in transfected skeletal muscle primary cultures

Muscles are formed by fusion of individual postmitotic myoblasts to form multinucleated syncytial myotubes. The process requires a well-coordinated transition from proliferation, through migratory alignment and cycle exit, to breakdown of apposed membranes. Connexin43 protein and cell-cycle inhibitor levels are correlated, and gap junction blockers can delay muscle regeneration, so a coordinating role for gap junctions has been proposed. Here, wild-type and dominant-negative connexin43 variants (wtCx43, dnCx43) were introduced into rat myoblasts in primary culture through pIRES-eGFP constructs that made transfected cells fluoresce. GFP-positive cells and vitally-stained nuclei were counted on successive days to reveal differences in proliferation, and myotubes were counted to reveal differences in fusion. Individual transfected cells were injected with Cascade Blue, which permeates gap junctions, mixed with FITC-dextran, which requires cytoplasmic continuity to enter neighbouring cells. Myoblasts transfected with wtCx43 showed more gap-junctional coupling than GFP-only controls, began fusion sooner as judged by the incidence of cytoplasmic coupling, and formed more myotubes. Myoblasts transfected with dnCx43 remained proliferative for longer than either GFP-only or wtCx43 myoblasts, showed less coupling, and underwent little fusion into myotubes. These results highlight the critical role of gap-junctional coupling in myotube formation.     

The reliability of the ankle-brachial index in the Atherosclerosis Risk in Communities (ARIC) study and the NHLBI Family Heart Study (FHS)

Background

Organ transplantation is presently often the only available option to repair a damaged heart. As heart donors are scarce, engineering of cardiac grafts from autologous skeletal myoblasts is a promising novel therapeutic strategy. The functionality of skeletal muscle cells in the heart milieu is, however, limited because of their inability to integrate electrically and mechanically into the myocardium. Therefore, in pursuit of improved cardiac integration of skeletal muscle grafts we sought to modify primary skeletal myoblasts by overexpression of the main gap-junctional protein connexin 43 and to study electrical coupling of connexin 43 overexpressing myoblasts to cardiac myocytes in vitro.

Methods

To create an efficient means for overexpression of connexin 43 in skeletal myoblasts we constructed a bicistronic retroviral vector MLV-CX43-EGFP expressing the human connexin 43 cDNA and the marker EGFP gene. This vector was employed to transduce primary rat skeletal myoblasts in optimised conditions involving a concomitant use of the retrovirus immobilising protein RetroNectin^® and the polycation transduction enhancer Transfectam^®. The EGFP-positive transduced cells were then enriched by flow cytometry.

Results

More than four-fold overexpression of connexin 43 in the transduced skeletal myoblasts, compared with non-transduced cells, was shown by Western blotting. Functionality of the overexpressed connexin 43 was demonstrated by microinjection of a fluorescent dye showing enhanced gap-junctional intercellular transfer in connexin 43 transduced myoblasts compared with transfer in non-transduced myoblasts. Rat cardiac myocytes were cultured in multielectrode array culture dishes together with connexin 43/EGFP transduced skeletal myoblasts, control non-transduced skeletal myoblasts or alone. Extracellular field action potential activation rates in the co-cultures of connexin 43 transduced skeletal myoblasts with cardiac myocytes were significantly higher than in the co-cultures of non-transduced skeletal myoblasts with cardiac myocytes and similar to the rates in pure cultures of cardiac myocytes.

Conclusion

The observed elevated field action potential activation rate in the co-cultures of cardiac myocytes with connexin 43 transduced skeletal myoblasts indicates enhanced cell-to-cell electrical coupling due to overexpression of connexin 43 in skeletal myoblasts. This study suggests that retroviral connexin 43 transduction can be employed to augment engineering of the electrocompetent cardiac grafts from patients' own skeletal myoblasts.     

Multiscale tissue engineering for liver reconstruction

The liver is a target of in vitro tissue engineering despite its capability to regenerate in vivo. The construction of liver tissues in vitro remains challenging. In this review, conventional 3D cultures of hepatocytes are first discussed. Recent advances in the 3D culturing of liver cells are then summarized in the context of in vitro liver tissue reconstruction at the micro- and macroscales. The application of microfluidics technology to liver tissue engineering has been introduced as a bottom-up approach performed at the microscale, whereas whole-organ bioengineering technology was introduced as a top-down approach performed at the macroscale. Mesoscale approaches are also discussed in considering the integration of micro- and macroscale approaches. Multiple parallel multiscale liver tissue engineering studies are ongoing; however, no tissue-engineered liver that is appropriate for clinical use has yet been realized. The integration of multiscale tissue engineering studies is essential for further understanding of liver reconstruction strategies.     

Multiscale tissue engineering for liver reconstruction

The liver is a target of in vitro tissue engineering despite its capability to regenerate in vivo. The construction of liver tissues in vitro remains challenging. In this review, conventional 3D cultures of hepatocytes are first discussed. Recent advances in the 3D culturing of liver cells are then summarized in the context of in vitro liver tissue reconstruction at the micro- and macroscales. The application of microfluidics technology to liver tissue engineering has been introduced as a bottom-up approach performed at the microscale, whereas whole-organ bioengineering technology was introduced as a top-down approach performed at the macroscale. Mesoscale approaches are also discussed in considering the integration of micro- and macroscale approaches. Multiple parallel multiscale liver tissue engineering studies are ongoing; however, no tissue-engineered liver that is appropriate for clinical use has yet been realized. The integration of multiscale tissue engineering studies is essential for further understanding of liver reconstruction strategies.     

A Mathematical Model of the Electrophysiological Alterations in Rat Ventricular Myocytes in Type-I Diabetes

Our mathematical model of the rat ventricular myocyte (Pandit et al., 2001) was utilized to explore the ionic mechanism(s) that underlie the altered electrophysiological characteristics associated with the short-term model of streptozotocin-induced, type-I diabetes. The simulations show that the observed reductions in the Ca²⁺-independent transient outward K⁺ current (I_t) and the steady-state outward K⁺ current (I_ss), along with slowed inactivation of the L-type Ca²⁺ current (I_CaL), can result in the prolongation of the action potential duration, a well-known experimental finding. In addition, the model demonstrates that the slowed reactivation kinetics of I_t in diabetic myocytes can account for the more pronounced rate-dependent action potential duration prolongation in diabetes, and that a decrease in the electrogenic Na⁺-K⁺ pump current (I_NaK) results in a small depolarization in the resting membrane potential (V_rest). This depolarization reduces the availability of the Na⁺ channels (I_Na), thereby resulting in a slower upstroke (dV/dt_max) of the diabetic action potential. Additional simulations suggest that a reduction in the magnitude of I_CaL, in combination with impaired sarcoplasmic reticulum uptake can lead to a decreased sarcoplasmic reticulum Ca²⁺ load. These factors contribute to characteristic abnormal [Ca²⁺]_i homeostasis (reduced peak systolic value and rate of decay) in myocytes from diabetic animals. In combination, these simulation results provide novel information and integrative insights concerning plausible ionic mechanisms for the observed changes in cardiac repolarization and excitation-contraction coupling in rat ventricular myocytes in the setting of streptozotocin-induced, type-I diabetes.     

Influence of dynamic gap junction resistance on impulse propagation in ventricular myocardium: a computer simulation study

The gap junction connecting cardiac myocytes is voltage and time dependent. This simulation study investigated the effects of dynamic gap junctions on both the shape and conduction velocity of a propagating action potential. The dynamic gap junction model is based on that described by Vogel and Weingart (J. Physiol. (Lond.). 1998, 510:177-189) for the voltage- and time-dependent conductance changes measured in cell pairs. The model assumes that the conductive gap junction channels have four conformational states. The gap junction model was used to couple 300 cells in a linear strand with membrane dynamics of the cells defined by the Luo-Rudy I model. The results show that, when the cells are tightly coupled (6700 channels), little change occurs in the gap junction resistance during propagation. Thus, for tight coupling, there are negligible differences in the waveshape and propagation velocity when comparing the dynamic and static gap junction representations. For poor coupling (85 channels), the gap junction resistance increases 33 MOmega during propagation. This transient change in resistance resulted in increased transjunctional conduction delays, changes in action potential upstroke, and block of conduction at a lower junction resting resistance relative to a static gap junction model. The results suggest that the dynamics of the gap junction enhance cellular decoupling as a possible protective mechanism of isolating injured cells from their neighbors.     

Deriving Macroscopic Myocardial Conductivities by Homogenization of Microscopic Models

We derive the values for the intracellular and extracellular conductivities needed for bidomain simulations of cardiac electrophysiology using homogenization of partial differential equations. In our model, cardiac myocytes are rectangular prisms and gap junctions appear in a distributed manner as flux boundary conditions for Laplace’s equation. Using directly measurable microproperties such as cellular dimensions and end-to-end and side-to-side gap junction coupling strengths, we inexpensively obtain effective conductivities close to those given by simulations with a detailed cyto-architecture (Stinstra et al. in Ann. Biomed. Eng. 33:1743–1751, 2005). This model provides a convenient framework for studying the effect on conductivities of aligned vs. brick-like arrangements of cells and the effect of different distributions of gap junctions along the myocyte membranes.     

Tools for studying and modulating (cardiac muscle) cell mechanics and mechanosensing across the scales

Cardiomyocytes generate force for the contraction of the heart to pump blood into the lungs and body. At the same time, they are exquisitely tuned to the mechanical environment and react to e.g. changes in cell and extracellular matrix stiffness or altered stretching due to reduced ejection fraction in heart disease, by adapting their cytoskeleton, force generation and cell mechanics. Both mechanical sensing and cell mechanical adaptations are multiscale processes. Receptor interactions with the extracellular matrix at the nanoscale will lead to clustering of receptors and modification of the cytoskeleton. This in turn alters mechanosensing, force generation, cell and nuclear stiffness and viscoelasticity at the microscale. Further, this affects cell shape, orientation, maturation and tissue integration at the microscale to macroscale. A variety of tools have been developed and adapted to measure cardiomyocyte receptor-ligand interactions and forces or mechanics at the different ranges, resulting in a wealth of new information about cardiomyocyte mechanobiology. Here, we take stock at the different tools for exploring cardiomyocyte mechanosensing and cell mechanics at the different scales from the nanoscale to microscale and macroscale.