A mathematical analysis of physiological and morphological aspects of wound closure

Geometric control by active mechanics of epithelial gap closure

Epithelial wound healing is one of the most important biological processes occurring during the lifetime of an organism. It is a self-repair mechanism closing wounds or gaps within tissues to restore their functional integrity. In this work we derive a new diffuse interface approach for modelling the gap closure by means of a variational principle in the framework of non-equilibrium thermodynamics. We investigate the interplay between the crawling with lamellipodia protrusions and the supracellular tension exerted by the actomyosin cable on the closure dynamics. These active features are modeled as Korteweg forces into a generalised chemical potential. From an asymptotic analysis, we derive a pressure jump across the gap edge in the sharp interface limit. Moreover, the chemical potential diffuses as a Mullins-Sekerka system, and its interfacial value is given by a Gibbs-Thompson relation for its local potential driven by the curvature-dependent purse-string tension. The finite element simulations show an excellent quantitative agreement between the closure dynamics and the morphology of the edge with respect to existing biological experiments. The resulting force patterns are also in good qualitative agreement with existing traction force microscopy measurements. Our results shed light on the geometrical control of the gap closure dynamics resulting from the active forces that are chemically activated around the gap edge.

A model for 3D deformation and reconstruction of contractile microtissues

Tissue morphogenesis and regeneration are essentially mechanical processes that involve coordination of cellular forces, production and structural remodeling of extracellular matrix (ECM), and cell migration. Discovering the principles of cell-ECM interactions and tissue-scale deformation in mechanically-loaded tissues is instrumental to the development of novel regenerative therapies. The combination of high-throughput three-dimensional (3D) culture systems and experimentally-validated computational models accelerate the study of these principles. In our previous work [E. Mailand, et al., Biophys. J., 2019, 117, 975-986], we showed that prominent surface stresses emerge in constrained fibroblast-populated collagen gels, driving the morphogenesis of fibrous microtissues. Here, we introduce an active material model that allows the embodiment of surface and bulk contractile stresses while maintaining the passive elasticity of the ECM in a 3D setting. Unlike existing models, the stresses are driven by mechanosensing and not by an externally applied signal. The mechanosensing component is incorporated in the model through a direct coupling of the local deformation state with the associated contractile force generation. Further, we propose a finite element implementation to account for large deformations, nonlinear active material response, and surface effects. Simulation results quantitatively capture complex shape changes during tissue formation and as a response to surgical disruption of tissue boundaries, allowing precise calibration of the parameters of the 3D model. The results of this study imply that the organization of the extracellular matrix in the bulk of the tissue may not be a major factor behind the morphogenesis of fibrous tissues at sub-millimeter length scales.

Wound opening in a thin incompressible viscoelastic tissue

We develop a model to investigate analytically and numerically the mechanics of wound opening made in a viscoelastic, isotropic, homogeneous, and incompressive thin tissue. This process occurs just immediately after the wound infliction. Before any active biological action has taken place, the tissue relaxes, and the wound opens mostly due to the initial homeostatic tension of the tissue, its elastic and viscous properties, and the existing friction between the tissue and its substrate. We find that for a circular wound the regimes of deformation are defined by a single adimensional parameter λ, which characterizes the relative importance of viscosity over friction.

A Model for Heterogeneous Brain Tumor Cells that Cause Dizziness

Background::

Various models are used for brain tumors modeling. To our knowledge, no earlier work has been done on modeling the heterogeneous brain tumor that causes dizziness. However, specifying a cell using a model is not new. Unlike all earlier works in this regard, which consider analog or analog-CPU computers in the cell model, the model presented in this work fully utilizes processor computers.

Materials and Methods::

The purpose of this study is to offer a comprehensive approach for heterogeneous modeling of brain tumor cells. The model uses a brain tumor cell in Matlab and Simulink 3D software. Two heterogeneous models are presented for brain tumor cells: the imagebased cell model and computer-generated cell model. The image-based cell model is obtained through the figure altering on X-ray or ocular figures by recognizing the dissimilar states in the tumors.

Results::

The computer-generated cell model works based on locating computer produced aggregate cells into tumors. Some subdivisions for both the image-based and computer-generated cell models are presented as well.

Conclusion::

The positive and negative points of the image-based cell models and computergenerated cell models some scientific advices are presented in this work. Generally, the imagebased cell models could offer analytical facts for each state, but they are expensive and timeconsuming; besides, their performance is heavily influenced by deformation techniques. The computer-generated cell model, on the other hand, provides a higher cost of production and simplicity, but its main contribution is the overall performance and accuracy.

Using a continuum model to decipher the mechanics of embryonic tissue spreading from time-lapse image sequences: An approximate Bayesian computation approach

Advanced imaging techniques generate large datasets capable of describing the structure and kinematics of tissue spreading in embryonic development, wound healing, and the progression of many diseases. These datasets can be integrated with mathematical models to infer biomechanical properties of the system, typically identifying an optimal set of parameters for an individual experiment. However, these methods offer little information on the robustness of the fit and are generally ill-suited for statistical tests of multiple experiments. To overcome this limitation and enable efficient use of large datasets in a rigorous experimental design, we use the approximate Bayesian computation rejection algorithm to construct probability density distributions that estimate model parameters for a defined theoretical model and set of experimental data. Here, we demonstrate this method with a 2D Eulerian continuum mechanical model of spreading embryonic tissue. The model is tightly integrated with quantitative image analysis of different sized embryonic tissue explants spreading on extracellular matrix (ECM) and is regulated by a small set of parameters including forces on the free edge, tissue stiffness, strength of cell-ECM adhesions, and active cell shape changes. We find statistically significant trends in key parameters that vary with initial size of the explant, e.g., for larger explants cell-ECM adhesion forces are weaker and free edge forces are stronger. Furthermore, we demonstrate that estimated parameters for one explant can be used to predict the behavior of other similarly sized explants. These predictive methods can be used to guide further experiments to better understand how collective cell migration is regulated during development.

Mathematical Analysis of Glioma Growth in a Murine Model

Five immunocompetent C57BL/6-cBrd/cBrd/Cr (albino C57BL/6) mice were injected with GL261-luc2 cells, a cell line sharing characteristics of human glioblastoma multiforme (GBM). The mice were imaged using magnetic resonance (MR) at five separate time points to characterize growth and development of the tumor. After 25 days, the final tumor volumes of the mice varied from 12 mm³ to 62 mm³, even though mice were inoculated from the same tumor cell line under carefully controlled conditions. We generated hypotheses to explore large variances in final tumor size and tested them with our simple reaction-diffusion model in both a 3-dimensional (3D) finite difference method and a 2-dimensional (2D) level set method. The parameters obtained from a best-fit procedure, designed to yield simulated tumors as close as possible to the observed ones, vary by an order of magnitude between the three mice analyzed in detail. These differences may reflect morphological and biological variability in tumor growth, as well as errors in the mathematical model, perhaps from an oversimplification of the tumor dynamics or nonidentifiability of parameters. Our results generate parameters that match other experimental in vitro and in vivo measurements. Additionally, we calculate wave speed, which matches with other rat and human measurements.

A REALISTIC 3D COMPUTATIONAL MODEL OF THE CLOSURE OF SKIN WOUND WITH INTERRUPTED SUTURES

Wounds or cuts are the most common form of skin injuries. While a shallow wound may heal over time, deep wounds often require clinical interventions such as suturing to ensure the wound closure and timely healing. To date, suturing practices are based on a surgeon's experience and there is no benchmark to what is right or wrong. In the literature, there have been few attempts to characterize wound closure and suture mechanics using simple 2D computational models. In our current work, for the first time, a realistic three-dimensional (3D) computational model of the skin with the two layers, namely the epidermis and dermis, have been developed. A 3D diamond shaped wound with a varying cross-section has been modeled, and interrupted sutures have been placed numerically in multiple steps to close the wound. Nonlinear hyperelastic material properties have been adopted for the skin and a skin pre-stress was applied bi-axially. The force requirements for each suture were estimated numerically using a novel suture pulling technique. The suture forces were found to lie in the range of 0–5 N with a maximum value at the center. Also, the center suture was observed to require an approximately four times pull force compared to the first end suture. All these findings provide important guidelines for suturing. Additionally, the suture force can be approximated as a polynomial function of the displacement. Given a wound geometry, wound depth, skin material properties, skin pre-stress, suture wire material and cross-sectional area, using our computational model, such a relationship can be used to estimate and characterize the suture force requirements accurately. To our knowledge, such a 3D computational model of skin wound closure with interrupted sutures have not been developed till date, and would be indispensable for planning robotic surgeries and improving clinical suturing practices in the future.

Experimental study on tissue phantoms to understand the effect of injury and suturing on human skin mechanical properties

Skin injuries are the most common type of injuries occurring in day-to-day life. A skin injury usually manifests itself in the form of a wound or a cut. While a shallow wound may heal by itself within a short time, deep wounds require surgical interventions such as suturing for timely healing. To date, suturing practices are based on a surgeon's experience and may vary widely from one situation to another. Understanding the mechanics of wound closure and suturing of the skin is crucial to improve clinical suturing practices and also to plan automated robotic surgeries. In the literature, phenomenological two-dimensional computational skin models have been developed to study the mechanics of wound closure. Additionally, the effect of skin pre-stress (due to the natural tension of the skin) on wound closure mechanics has been studied. However, in most of these analyses, idealistic two-dimensional skin geometries, materials and loads have been assumed, which are far from reality, and would clearly generate inaccurate quantitative results. In this work, for the first time, a biofidelic human skin tissue phantom was developed using a two-part silicone material. A wound was created on the phantom material and sutures were placed to close the wound. Uniaxial mechanical tests were carried out on the phantom specimens to study the effect of varying wound size, quantity, suture and pre-stress on the mechanical behavior of human skin. Also, the average mechanical behavior of the human skin surrogate was characterized using hyperelastic material models, in the presence of a wound and sutures. To date, such a robust experimental study on the effect of injury and sutures on human skin mechanics has not been attempted. The results of this novel investigation will provide important guidelines for surgical planning and validation of results from computational models in the future.

Mathematical modelling of angiogenesis using continuous cell-based models

In this work, we develop a mathematical formalism based on a 3D in vitro model that is used to simulate the early stages of angiogenesis. The model treats cells as individual entities that are migrating as a result of chemotaxis and durotaxis. The phenotypes used here are endothelial cells that can be distinguished into stalk and tip (leading) cells. The model takes into account the dynamic interaction and interchange between both phenotypes. Next to the cells, the model takes into account several proteins such as vascular endothelial growth factor, delta-like ligand 4, urokinase plasminogen activator and matrix metalloproteinase, which are computed through the solution of a system of reaction–diffusion equations. The method used in the present study is classified into the hybrid approaches. The present study, implemented in three spatial dimensions, demonstrates the feasibility of the approach that is qualitatively confirmed by experimental results.

A mathematical model for lymphangiogenesis in normal and diabetic wounds

Several studies suggest that one possible cause of impaired wound healing is failed or insufficient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis) very few have been proposed for the regeneration of the lymphatic network. Moreover, lymphangiogenesis is markedly distinct from angiogenesis, occurring at different times and in a different manner. Here a model of five ordinary differential equations is presented to describe the formation of lymphatic capillaries following a skin wound. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from experimental and clinical data. The system is then solved numerically and the results are compared with the available biological literature. Finally, a parameter sensitivity analysis of the model is taken as a starting point for suggesting new therapeutic approaches targeting the enhancement of lymphangiogenesis in diabetic wounds. The work provides a deeper understanding of the phenomenon in question, clarifying the main factors involved. In particular, the balance between TGF-β and VEGF levels, rather than their absolute values, is identified as crucial to effective lymphangiogenesis. In addition, the results indicate lowering the macrophage-mediated activation of TGF-β and increasing the basal lymphatic endothelial cell growth rate, inter alia, as potential treatments. It is hoped the findings of this paper may be considered in the development of future experiments investigating novel lymphangiogenic therapies.

A Multiscale Computational Framework to Understand Vascular Adaptation

The failure rate for vascular interventions (vein bypass grafting, arterial angioplasty/stenting) remains unacceptably high. Over the past two decades, researchers have applied a wide variety of approaches to investigate the primary failure mechanisms, neointimal hyperplasia and aberrant remodeling of the wall, in an effort to identify novel therapeutic strategies. Despite incremental progress, specific cause/effect linkages among the primary drivers of the pathology, (hemodynamic factors, inflammatory biochemical mediators, cellular effectors) and vascular occlusive phenotype remain lacking. We propose a multiscale computational framework of vascular adaptation to develop a bridge between theory and experimental observation and to provide a method for the systematic testing of relevant clinical hypotheses. Cornerstone to our model is a feedback mechanism between environmental conditions and dynamic tissue plasticity described at the cellular level with an agent based model. Our implementation (i) is modular, (ii) starts from basic mechano-biology principle at the cell level and (iii) facilitates the agile development of the model.

Interrogating a multifactorial model of breast conserving therapy with clinical data

Most women with early stage breast cancer do not require removal of the entire breast to treat their cancer; instead, up to 70% of women can be effectively and safely treated by breast conserving therapy (BCT) with surgical removal of the tumor only (lumpectomy) followed by radiation treatment of the remaining breast tissue. Unfortunately, the final contour and cosmesis of the treated breast is suboptimal in approximately 30% of patients. The ability to accurately predict breast contour after BCT for breast cancer could significantly improve patient decision-making regarding the choice of surgery for breast cancer. Our overall hypothesis is that the complex interplay among mechanical forces due to gravity, breast tissue constitutive law distribution, inflammation induced by radiotherapy and internal stress generated by the healing process play a dominant role in determining the success or failure of lumpectomy in preserving the breast contour and cosmesis. We have shown here from a first patient study that even in the idealistic situation of excellent cosmetic outcome this problem requires multiscale modeling. We propose a method to decide which component of the model works best for each phase of healing and what parameters should be considered dominant and patient specific. This patient study is part of a clinical trial registered on ClinicalTrial.gov, identifier NCT02310711.

Early stages of bone fracture healing: formation of a fibrin-collagen scaffold in the fracture hematoma

This work is concerned with the sequence of events taking place during the first stages of bone fracture healing, from bone breakup until the formation of early fibrous callus (EFC). The latter provides a scaffold over which subsequent remodeling processes will eventually result in successful bone repair. Specifically, some mathematical models are proposed to estimate the time required for (1) the formation immediately after fracture of a fibrin clot, described in terms of a phase transition in a polymerization process, and (2) the onset of EFC which is produced when fibroblasts arising from differentiation of chemotactically recruited mesenchymal stem cells remodel a previous fibrin clot by releasing a collagen matrix over it. An attempt has been made to keep models as simple as possible, so that a explicit dependence of the estimates obtained on relevant biochemical parameters involved is obtained.

An agent-based model approach to multi-phase life-cycle for contact inhibited, anchorage dependent cells

Cellular agent-based models are a technique that can be easily adapted to describe nuances of a particular cell type. Within we have concentrated on the cellular particularities of the human Endothelial Cell, explicitly the effects both of anchorage dependency and of heightened scaffold binding on the total confluence time of a system. By expansion of a discrete, homogeneous, asynchronous cellular model to account for several states per cell (phases within a cell's life); we accommodate and track dependencies of confluence time and population dynamics on these factors. Increasing the total motility time, analogous to weakening the binding between lattice and cell, affects the system in unique ways from increasing the average cellular velocity; each degree of freedom allows for control over the time length the system achieves logistic growth and confluence. These additional factors may allow for greater control over behaviors of the system. Examinations of system's dependence on both seed state velocity and binding are also enclosed.

Modeling the effects of systemic mediators on the inflammatory phase of wound healing

The normal wound healing response is characterized by a progression from clot formation, to an inflammatory phase, to a repair phase, and finally, to remodeling. In many chronic wounds there is an extended inflammatory phase that stops this progression. In order to understand the inflammatory phase in more detail, we developed an ordinary differential equation model that accounts for two systemic mediators that are known to modulate this phase, estrogen (a protective hormone during wound healing) and cortisol (a hormone elevated after trauma that slows healing). This model describes the interactions in the wound between wound debris, pathogens, neutrophils and macrophages and the modulation of these interactions by estrogen and cortisol. A collection of parameter sets, which qualitatively match published data on the dynamics of wound healing, was chosen using Latin Hypercube Sampling. This collection of parameter sets represents normal healing in the population as a whole better than one single parameter set. Including the effects of estrogen and cortisol is a necessary step to creating a patient specific model that accounts for gender and trauma. Utilization of math modeling techniques to better understand the wound healing inflammatory phase could lead to new therapeutic strategies for the treatment of chronic wounds. This inflammatory phase model will later become the inflammatory subsystem of our full wound healing model, which includes fibroblast activity, collagen accumulation and remodeling.

A biomechanical model of wound contraction and scar formation

We propose a biomechanical model for investigating wound contraction mechanism and resulting scarring. Extracellular matrix is modeled as fiber-reinforced anisotropic soft tissue, with its elastic properties dynamically changing with the density and orientation of collagen fibers. Collagen fibers are deposited by fibroblasts infiltrating the wound space, and are dynamically aligned with both migrating fibroblasts and tissue residing tension field. Our new 2D hybrid agent-based model provides a comprehensive platform for examining the mechanobiology in wound contraction and scar formation. Simulation results are consistent with experimental observations and are able to reveal the effects of wound morphology and mechanical environment on contraction patterns. Our model results support the hypothesis that scar formation is the product of collagen fiber synthesis and alignment in the presence of the tensile stress field generated by a wound contraction process.

Systems-based approaches toward wound healing

Wound healing in the pediatric patient is of utmost clinical and social importance because hypertrophic scarring can have aesthetic and psychological sequelae, from early childhood to late adolescence. Wound healing is a well-orchestrated reparative response affecting the damaged tissue at the cellular, tissue, organ, and system scales. Although tremendous progress has been made toward understanding wound healing at the individual temporal and spatial scales, its effects across the scales remain severely understudied and poorly understood. Here, we discuss the critical need for systems-based computational modeling of wound healing across the scales, from short-term to long-term and from small to large. We illustrate the state of the art in systems modeling by means of three key signaling mechanisms: oxygen tension-regulating angiogenesis and revascularization; transforming growth factor-β (TGF-β) kinetics controlling collagen deposition; and mechanical stretch stimulating cellular mitosis and extracellular matrix (ECM) remodeling. The complex network of biochemical and biomechanical signaling mechanisms and the multiscale character of the healing process make systems modeling an integral tool in exploring personalized strategies for wound repair. A better mechanistic understanding of wound healing in the pediatric patient could open new avenues in treating children with skin disorders such as birth defects, skin cancer, wounds, and burn injuries.

Using a continuum model to predict closure time of gaps in intestinal epithelial cell layers

A two-dimensional continuum model of collective cell migration is used to predict the closure of gaps in intestinal epithelial cell layers. The model assumes that cell migration is governed by lamellipodia formation, cell-cell adhesion, and cell-substrate adhesion. Model predictions of the gap edge position and complete gap closure time are compared with experimental measures from cell layer scratch assays (also called scratch wound assays). The goal of the study is to combine experimental observations with mathematical descriptions of cell motion to identify effects of gap shape and area on closure time and to propose a method that uses a simple measure (e.g., area) to predict overall gap closure time early in the closure process. Gap closure time is shown to increase linearly with increasing gap area; however, gaps of equal areas but different aspect ratios differ greatly in healing time. Previous methods that calculate overall healing time according to the absolute or percent change in gap area assume that the gap area changes at a constant rate and typically underestimate gap closure time. In this study, data from scratch assays suggest that the rate of change of area is proportional to the first power or square root power of area.

Mathematical modeling of collagen turnover in biological tissue

We present a theoretical and computational model for collagen turnover in soft biological tissues. Driven by alterations in the mechanical environment, collagen fiber bundles may undergo important chronic changes, characterized primarily by alterations in collagen synthesis and degradation rates. In particular, hypertension triggers an increase in tropocollagen synthesis and a decrease in collagen degradation, which lead to the well-documented overall increase in collagen content. These changes are the result of a cascade of events, initiated mainly by the endothelial and smooth muscle cells. Here, we represent these events collectively in terms of two internal variables, the concentration of growth factor TGF-β and tissue inhibitors of metalloproteinases TIMP. The upregulation of TGF-β increases the collagen density. The upregulation of TIMP also increases the collagen density through decreasing matrix metalloproteinase MMP. We establish a mathematical theory for mechanically-induced collagen turnover and introduce a computational algorithm for its robust and efficient solution. We demonstrate that our model can accurately predict the experimentally observed collagen increase in response to hypertension reported in literature. Ultimately, the model can serve as a valuable tool to predict the chronic adaptation of collagen content to restore the homeostatic equilibrium state in vessels with arbitrary micro-structure and geometry.

A phenomenological model for chemico-mechanically induced cell shape changes during migration and cell-cell contacts

A phenomenological model for the evolution of shape transition of cells is considered. These transitions are determined by the emission of growth-factors, as well as mechanical interaction if cells are subjected to hard impingement. The originality of this model necessitates a formal treatment of the mathematical model, as well as the presentation of elementary cases in order to illustrate the consistence of the model. We will also show some small-scale relevant applications.

In silico design of treatment strategies in wound healing and bone fracture healing

Wound and bone fracture healing are natural repair processes initiated by trauma. Over the last decade, many mathematical models have been established to investigate the healing processes in silico, in addition to ongoing experimental work. In recent days, the focus of the mathematical models has shifted from simulation of the healing process towards simulation of the impaired healing process and the in silico design of treatment strategies. This review describes the most important causes of failure of the wound and bone fracture healing processes and the experimental models and methods used to investigate and treat these impaired healing cases. Furthermore, the mathematical models that are described address these impaired healing cases and investigate various therapeutic scenarios in silico. Examples are provided to illustrate the potential of these in silico experiments. Finally, limitations of the models and the need for and ability of these models to capture patient specificity and variability are discussed.