A time-dependent phenomenological model for cell mechano-sensing

Adherent cells normally apply forces as a generic means of sensing and responding to the mechanical nature of their surrounding environment. How these forces vary as a function of the extracellular rigidity is critical to understanding the regulatory functions that drive important phenomena such as wound healing or muscle contraction. In recognition of this fact, experiments have been conducted to understand cell rigidity-sensing properties under known conditions of the extracellular environment, opening new possibilities for modeling this active behavior. In this work, we provide a physics-based constitutive model taking into account the main structural components of the cell to reproduce its most significant contractile properties such as the traction forces exerted as a function of time and the extracellular stiffness. This model shows how the interplay between the time-dependent response of the acto-myosin contractile system and the elastic response of the cell components determines the mechano-sensing behavior of single cells.     

Cells Actively Stiffen Fibrin Networks by Generating Contractile Stress

During wound healing and angiogenesis, fibrin serves as a provisional extracellular matrix. We use a model system of fibroblasts embedded in fibrin gels to study how cell-mediated contraction may influence the macroscopic mechanical properties of their extracellular matrix during such processes. We demonstrate by macroscopic shear rheology that the cells increase the elastic modulus of the fibrin gels. Microscopy observations show that this stiffening sets in when the cells spread and apply traction forces on the fibrin fibers. We further show that the stiffening response mimics the effect of an external stress applied by mechanical shear. We propose that stiffening is a consequence of active myosin-driven cell contraction, which provokes a nonlinear elastic response of the fibrin matrix. Cell-induced stiffening is limited to a factor 3 even though fibrin gels can in principle stiffen much more before breaking. We discuss this observation in light of recent models of fibrin gel elasticity, and conclude that the fibroblasts pull out floppy modes, such as thermal bending undulations, from the fibrin network, but do not axially stretch the fibers. Our findings are relevant for understanding the role of matrix contraction by cells during wound healing and cancer development, and may provide design parameters for materials to guide morphogenesis in tissue engineering.     

Cells Actively Stiffen Fibrin Networks by Generating Contractile Stress

During wound healing and angiogenesis, fibrin serves as a provisional extracellular matrix. We use a model system of fibroblasts embedded in fibrin gels to study how cell-mediated contraction may influence the macroscopic mechanical properties of their extracellular matrix during such processes. We demonstrate by macroscopic shear rheology that the cells increase the elastic modulus of the fibrin gels. Microscopy observations show that this stiffening sets in when the cells spread and apply traction forces on the fibrin fibers. We further show that the stiffening response mimics the effect of an external stress applied by mechanical shear. We propose that stiffening is a consequence of active myosin-driven cell contraction, which provokes a nonlinear elastic response of the fibrin matrix. Cell-induced stiffening is limited to a factor 3 even though fibrin gels can in principle stiffen much more before breaking. We discuss this observation in light of recent models of fibrin gel elasticity, and conclude that the fibroblasts pull out floppy modes, such as thermal bending undulations, from the fibrin network, but do not axially stretch the fibers. Our findings are relevant for understanding the role of matrix contraction by cells during wound healing and cancer development, and may provide design parameters for materials to guide morphogenesis in tissue engineering.     

Tissue engineering science: Consequences of cell traction force

Blood and tissue cells mechanically interact with soft tissues and tissue-equivalent reconstituted collagen gels in a variety of situations relevant to biomedicine and biotechnology. A key phenomenon in these interactions is the exertion of traction force by cells on local collagen fibers which typically constitute the solid network of these tissues and gels and impart gross mechanical integrity. Two important consequences of cells exerting traction on such collagen networks are first, when the cells co-ordinate their traction, resulting in cell migration, and second, when their traction is sufficient to deform the network. Such cell-collagen network interactions are coupled in a number of ways. Network deformation, for example, can result in net alignment of collagen fibers, eliciting contact guidance, wherein cells move with bidirectional bias along an axis of fiber alignment, potentially leading to a nonuniform cell distribution. This may govern cell accumulation in wounds and be exploited to control cell infiltration of bioartificial tissues and organs. Another consequence of cell traction is the resultant stress and strain in the network which modulate cell protein and DNA synthesis and differentiation. We summarize, here, relevant mathematical theories which we have used to describe the inherent coupling of cell dynamics and tissue mechanics in cell-populated collagen gels via traction. The development of appropriate models based on these theories, in an effort to understand how events in wound healing govern the rate and extent of wound contraction, and to measure cell traction forces in vitro, are described. Relevant observations and speculation from cell biology and medicine that motivate or serve to critique the assumptions made in the theories and models are also summarized.Abbreviations ECM Extracellular Matrix - FPCL Fibroblast-Populated Collagen Lattice - FPCM Fibroblast-Populated Collagen Microsphere     

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.     

A morphoelastic model for dermal wound closure

We develop a model of wound healing in the framework of finite elasticity, focussing our attention on the processes of growth and contraction in the dermal layer of the skin. The dermal tissue is treated as a hyperelastic cylinder that surrounds the wound and is subject to symmetric deformations. By considering the initial recoil that is observed upon the application of a circular wound, we estimate the degree of residual tension in the skin and build an evolution law for mechanosensitive growth of the dermal tissue. Contraction of the wound is governed by a phenomenological law in which radial pressure is prescribed at the wound edge. The model reproduces three main phases of the healing process. Initially, the wound recoils due to residual stress in the surrounding tissue; the wound then heals as a result of contraction and growth; and finally, healing slows as contraction and growth decrease. Over a longer time period, the surrounding tissue remodels, returning to the residually stressed state. We identify the steady state growth profile associated with this remodelled state. The model is then used to predict the outcome of rewounding experiments designed to quantify the amount of stress in the tissue, and also to simulate the application of pressure treatments.     

A self-consistent cell flux expression for simultaneous chemotaxis and contact guidance in tissues

During wound healing, both chemotaxis and contact guidance can contribute to the migration of blood and tissue cells to the wound. In order to understand the wound healing process, we must thus understand how cells respond to both these simultaneous directional cues, which are not necessarily coaligned. Although chemotaxis and contact guidance have been studied individually, the interaction between them has not been addressed. We extend a stochastic cell movement model, developed by Dickinson and Tranquillo (1995) [6] for individual cues, for simultaneous chemotaxis and contact guidance by a two-parameter perturbation analysis in terms of the two associated cues, a chemotactic factor gradient and aligned tissue fibers. We present results from analysis of the first-order perturbation, which includes the cell flux expression heuristically proposed by others, but reveals paradoxical results for other indices of cell movement, such as the mean-squared displacement. We then present second-order perturbation results that resolve these paradoxical results. Finally, we relate these results to a continuum mechanical model developed by Barocas and Tranquillo (1997) [3] that predicts fiber alignment due to cell traction induced tissue contraction. Received: 30 April 1999 / Revised version: 30 October 1999 / Published online: 14 September 2000     

Numerical modelling of the angiogenesis process in wound contraction

Angiogenesis consists of the growth of new blood vessels from the pre-existing vasculature. This phenomenon takes place in several biological processes, including wound healing. In this work, we present a mathematical model of angiogenesis applied to skin wound healing. The developed model includes biological (capillaries and fibroblasts), chemical (oxygen and angiogenic growth factor concentrations) and mechanical factors (cell traction forces and extracellular matrix deformation) that influence the evolution of the healing process. A novelty from previous works, apart from the coupling of angiogenesis and wound contraction, is the more realistic modelling of skin as a hyperelastic material. Large deformations are addressed using an updated Lagrangian approach. The coupled non-linear model is solved with the finite element method, and the process is studied over two wound geometries (circular and elliptical) of the same area. The results indicate that the elliptical wound vascularizes two days earlier than the circular wound but that they experience a similar contraction level, reducing its size by 25 %.     

Intercellular Stress Reconstitution from Traction Force Data

Cells migrate collectively during development, wound healing, and cancer metastasis. Recently, a method has been developed to recover intercellular stress in monolayers from measured traction forces upon the substrate. To calculate stress maps in two dimensions, the cell sheet was assumed to behave like an elastic material, and it remains unclear to what extent this assumption is valid. In this study, we simulate our recently developed model for collective cell migration, and compute intercellular stress maps using the method employed in the experiments. We also compute these maps using a method that does not depend on the traction forces or material properties. The two independently obtained stress patterns agree well for the parameters we have probed and provide a verification of the validity of the experimental method.     

Theoretical models of wound healing: past successes and future challenges

The complex biology of wound healing is an area in which theoretical modelling has already made a significant impact. In this review article, the authors describe the key features of wound healing biology, divided into four components: epidermal wound healing, remodelling of the dermal extracellular matrix, wound contraction, and angiogenesis. Within each of these categories, previous modelling work is described, and the authors identify what they regard as the main challenges for future theoretical work.     

Eigenstrain as a mechanical set-point of cells

Cell contraction regulates how cells sense their mechanical environment. We sought to identify the set-point of cell contraction, also referred to as tensional homeostasis. In this work, bovine aortic endothelial cells (BAECs), cultured on substrates with different stiffness, were characterized using traction force microscopy (TFM). Numerical models were developed to provide insights into the mechanics of cell–substrate interactions. Cell contraction was modeled as eigenstrain which could induce isometric cell contraction without external forces. The predicted traction stresses matched well with TFM measurements. Furthermore, our numerical model provided cell stress and displacement maps for inspecting the fundamental regulating mechanism of cell mechanosensing. We showed that cell spread area, traction force on a substrate, as well as the average stress of a cell were increased in response to a stiffer substrate. However, the cell average strain, which is cell type-specific, was kept at the same level regardless of the substrate stiffness. This indicated that the cell average strain is the tensional homeostasis that each type of cell tries to maintain. Furthermore, cell contraction in terms of eigenstrain was found to be the same for both BAECs and fibroblast cells in different mechanical environments. This implied a potential mechanical set-point across different cell types. Our results suggest that additional measurements of contractility might be useful for monitoring cell mechanosensing as well as dynamic remodeling of the extracellular matrix (ECM). This work could help to advance the understanding of the cell-ECM relationship, leading to better regenerative strategies.     

Roles of lactoferrin on skin wound healing

Skin wound healing is a complex biological process that requires the regulation of different cell types, including immune cells, keratinocytes, fibroblasts, and endothelial cells. It consists of 5 stages: hemostasis, inflammation, granulation tissue formation, re-epithelialization, and wound remodeling. While inflammation is essential for successful wound healing, prolonged or excess inflammation can result in nonhealing chronic wounds. Lactoferrin, an iron-binding glycoprotein secreted from glandular epithelial cells into body fluids, promotes skin wound healing by enhancing the initial inflammatory phase. Lactoferrin also exhibits anti-inflammatory activity that neutralizes overabundant immune response. Accumulating evidence suggests that lactoferrin directly promotes both the formation of granulation tissue and re-epithelialization. Lactoferrin stimulates the proliferation and migration of fibroblasts and keratinocytes and enhances the synthesis of extracellular matrix components, such as collagen and hyaluronan. In an in vitro model of wound contraction, lactoferrin promoted fibroblast-mediated collagen gel contraction. These observations indicate that lactoferrin supports multiple biological processes involved in wound healing.     

The mathematical modelling of cell kinetics in corneal epithelial wound healing

This paper considers the comparison of experimental spatial and temporal data of mitotic rates measured during corneal epithelial wound healing (CEWH) of a rat model with the predictions of a computer modelling framework. We begin by briefly showing that previous models, used in the study of corneal epithelial wound healing speeds, are inadequate for the study of cell kinetics. We proceed to formulate a new modelling framework more suited to such a study. This framework is simulated in its simplest form, and the results from this motivate a new realisation of the modelling framework, including a caricature of age structuring. Finally, a model with a simple representation of juxtacrine signalling is considered. The final model captures many, though not all, of the trends of the experimental data. This paper thus lays a foundation for the modelling of the cell kinetics of corneal epithelial wound healing, and yields valuable insight regarding the important mechanisms a model should consider in order to reproduce the observed experimental trends.     

Finite element analysis of the effects of focal adhesion mechanical properties and substrate stiffness on cell migration

The attachment of cells to the extracellular matrix (ECM) is achieved by the specific binding of cell-surface receptors to ligands present in the ECM. These interactions are important for many biological processes, including cell migration, cancer development, and wound healing. Our objective was to develop a computational model to investigate how focal adhesion mechanical properties, substrate stiffness, and intracellular stresses affect cell-matrix interactions during cell migration on a flat substrate. In our model, the cell-substrate traction was proportional to the bound receptor concentration, relative velocity between the cell and substrate, and the cell-substrate friction coefficient. Simulation results showed that even if the receptor number and ligand density were fixed, the mechanical properties of the focal adhesions still affected cell-ECM interactions. In fact, the cell-substrate traction was biphasic with respect to the friction coefficient, a parameter that can be used to quantify focal adhesion properties. In contrast, the cell speed was a monotonically decreasing function with respect to this parameter. Furthermore, tractions showed greater increases when the maximum intracellular stress was increased from 400 to 600Pa than when substrate stiffness was increased from 0.5 to 100kPa. This mathematical model is able to quantify the effects of focal adhesion mechanical properties, extracellular stiffness, and intracellular stresses on cell-ECM interactions, and should be beneficial to research in cancer development.     

Understanding cornea epithelial stem cells and stem cell deficiency: Lessons learned using vertebrate model systems

Animal models have contributed greatly to our understanding of human diseases. Here, we focus on cornea epithelial stem cell (CESC) deficiency (commonly called limbal stem cell deficiency, LSCD). Corneal development, homeostasis and wound healing are supported by specific stem cells, that include the CESCs. Damage to or loss of these cells results in blindness and other debilitating ocular conditions. Here we describe the contributions from several vertebrate models toward understanding CESCs and LSCD treatments. These include both mammalian models, as well as two aquatic models, Zebrafish and the amphibian, Xenopus. Pioneering developments have been made using stem cell transplants to restore normal vision in patients with LSCD, but questions still remain about the basic biology of CESCs, including their precise cell lineages and behavior in the cornea. We describe various cell lineage tracing studies to follow their patterns of division, and the fates of their progeny during development, homeostasis, and wound healing. In addition, we present some preliminary results using the Xenopus model system. Ultimately, a more thorough understanding of these cornea cells will advance our knowledge of stem cell biology and lead to better cornea disease therapeutics.     

Mechanical modelling of cell/ECM and cell/cell interactions during the contraction of a fibroblast-populated collagen microsphere: theory and model simulation

The cell-derived forces generated during wound healing may be beneficial in reducing the wound size by contraction, but are also detrimental because of the high mechanical stresses in and around the scar that can cause pain, disfigurement and loss of tissue function. The fibroblasts seeded collagen matrix is regarded as an in vitro model for this process and as a suitable way to study these mechanical aspects which are poorly understood. It is proposed here, to improve the continuum theory of Murray-Oster by assuming that more than one control system may be operative in wound contraction regulation. In particular, it is suggested that the wound contraction mechanism is not exclusively due to cell/ECM interaction forces but rather that both ECM/cell and the cell/cell interactions operate together in such process.     

Strain tunes proteolytic degradation and diffusive transport in fibrin networks

Proteolytic degradation of fibrin, the major structural component in blood clots, is critical both during normal wound healing and in the treatment of ischemic stroke and myocardial infarction. Fibrin-containing clots experience substantial strain due to platelet contraction, fluid shear, and mechanical stress at the wound site. However, little is understood about how mechanical forces may influence fibrin dissolution. We used video microscopy to image strained fibrin clots as they were degraded by plasmin, a major fibrinolytic enzyme. Applied strain causes up to 10-fold reduction in the rate of fibrin degradation. Analysis of our data supports a quantitative model in which the decrease in fibrin proteolysis rates with strain stems from slower transport of plasmin into the clot. We performed fluorescence recovery after photobleaching (FRAP) measurements to further probe the effect of strain on diffusive transport. We find that diffusivity perpendicular to the strain axis decreases with increasing strain, while diffusivity along the strain axis remains unchanged. Our results suggest that the properties of the fibrin network have evolved to protect mechanically loaded fibrin from degradation, consistent with its function in wound healing. The pronounced effect of strain upon diffusivity and proteolytic susceptibility within fibrin networks offers a potentially useful means of guiding cell growth and morphology in fibrin-based biomaterials.     

Keratinocyte growth factor signalling: a mathematical model of dermal-epidermal interaction in epidermal wound healing

A wealth of growth factors are known to regulate the various cell functions involved in the repair process. An understanding of their therapeutic value is essential to achieve improved wound healing. Keratinocyte growth factor (KGF) seems to have a unique role as a mediator of mesenchymal-epithelial interactions: it originates from mesenchymal cells, yet acts exclusively on epithelial cells. In this paper, we study KGF's role in epidermal wound healing, since its production is substantially up-regulated after injury. We begin by modelling the dermal-epidermal signalling mechanism of KGF to investigate how this extra production affects the signal range. We then incorporate the effect of KGF on cell proliferation, and using travelling wave analysis we obtain an approximation for the rate of healing. Our modelling shows that the large up-regulation of KGF post-wounding extends the KGF signal range but is above optimal for the rate of wound closure. We predict that other functions of KGF may be more important than its role as a mitogen for the healing process.     

Exogenous ATP causes the contraction of intact fibroblasts in vitro

The contraction of intact fibroblasts was investigated in vitro. The addition of ATP to the cells lead to a rapid, reversible reduction of cell area, which appeared to be a contraction. The change of cell area was measured using image analysis. ATP (15 mM) elicited the maximal contraction, with a 50% reduction in area achieved within 90 s. Adenosine was a partial agonist for the contraction. The contraction was dependent on control of the environmental calcium; cells in a calcium- and magnesium-free environment underwent spontaneous contraction. Replenishing the calcium and magnesium lead to stability of the cells. Since fibroblast contraction is involved in wound healing at many sites in the body, this system provides a physiological model for the direct investigation of fibroblasts with intact cell membranes, and allows for the testing of drugs which may influence wound healing in vivo.