Spatio-temporal tumour model for analysis and mechanism of action of intracellular drug accumulation

We have developed a one-dimensional tumour simulator to describe the biodistribution of chemotherapeutic drugs to a tumoral lesion and the tumour cell’s response to therapy. A three-compartment model is used for drug dynamics within the tumour. The first compartment represents the extracellular space in which cells move, the second corresponds to the intracellular fluid space (including cell membrane) which is in direct equilibrium with the extracellular space, and the third is a non-exchangeable compartment that represents sequestered drug which is trapped in the nucleus to damage the cellular DNA, directly triggering cell death. Analytical and numerical techniques (Finite Element Method) are used to describe the tumour’s response to therapy and the effect of parameter variation on the drug concentration profiles in the three compartments.     

Mesenchymal stem cell durotaxis depends on substrate stiffness gradient strength

Mesenchymal stem cells (MSCs) respond to the elasticity of their environment, which varies between and within tissues. Stiffness gradients within tissues can result from pathological conditions, but also occur through normal variation, such as in muscle. MSC migration can be directed by shallow stiffness gradients before differentiating. Gradients with fine control over substrate compliance – both in range and rate of change (strength) – are needed to better understand mechanical regulation of MSC migration in normal and diseased states. We describe polyacrylamide stiffness gradient fabrication using three distinct systems, generating stiffness gradients of physiological (1 Pa/μm), pathological (10 Pa/μm), and step change (≥ 100Pa/μm) strength. All gradients spanned a range of physiologically relevant elastic moduli for soft tissues (1–12 kPa). MSCs migrated to the stiffest region on each gradient. Time-lapse microscopy revealed that migration velocity correlated directly with gradient strength. Directed migration was reduced in the presence of the contractile agonist lysophosphatidic acid (LPA) and cytoskeleton-perturbing drugs nocodazole and cytochalasin. LPA- and nocodazole-treated cells remained spread and protrusive on the substrate, while cytochalasin-treated cells did not. Nocodazole-treated cells spread in a similar manner to untreated cells, but exhibited greatly diminished traction forces. These data suggest that a functional actin cytoskeleton is required for migration whereas microtubules are required for directed migration. The data also imply that, in vivo, MSCs may preferentially accumulate in regions of high elastic modulus and make a greater contribution to tissue repairs in these locations.     

A hybrid computational model for collective cell durotaxis

Collective cell migration is regulated by a complex set of mechanical interactions and cellular mechanisms. Collective migration emerges from mechanisms occurring at single cell level, involving processes like contraction, polymerization and depolymerization, of cell–cell interactions and of cell–substrate adhesion. Here, we present a computational framework which simulates the dynamics of this emergent behavior conditioned by substrates with stiffness gradients. The computational model reproduces the cell’s ability to move toward the stiffer part of the substrate, process known as durotaxis. It combines the continuous formulation of truss elements and a particle-based approach to simulate the dynamics of cell–matrix adhesions and cell–cell interactions. Using this hybrid approach, researchers can quickly create a quantitative model to understand the regulatory role of different mechanical conditions on the dynamics of collective cell migration. Our model shows that durotaxis occurs due to the ability of cells to deform the substrate more in the part of lower stiffness than in the stiffer part. This effect explains why cell collective movement is more effective than single cell movement in stiffness gradient conditions. In addition, we numerically evaluate how gradient stiffness properties, cell monolayer size and force transmission between cells and extracellular matrix are crucial in regulating durotaxis.     

A Numerical Simulation of the Diffusion of Near Infrared Light Through Brain Tissue

A numerical simulation of the transport of light energy in the near infrared region of the spectrum through human brain tissue is presented. This simulation models the use of near infrared spectroscopic techniques to quantify the levels of oxygen present in brain tissue. Successful application of the technique requires knowledge of the optical pathlength in the tissue, and it is the goal of this simulation to quantify the relationship of the optical pathlength and the oxygenation state of the tissue. Both implicit and explicit finite element schemes for unstructured grids are implemented and discussed. Several application simulations using three tissue grids of varying degrees of physiological accuracy are then conducted, and figures for the optical pathlength of light through the tissue at varying levels of oxygenation are computed. These results are then used to develop a quantitative relationship between the pathlength and the absorption parameter for the three tissue models which we explore.     

A mesoscopic stochastic mechanism of cytosolic calcium oscillations

Based on a model of intracellular calcium (Ca(2+)) oscillation with self-modulation of inositol 1,4,5-trisphosphate signal, the mesoscopic stochastic differential equations for the intracellular Ca(2+) oscillations are theoretically derived by using the chemical Langevin equation method. The effects of the finite biochemical reaction molecule number on both simple and complex cytosolic Ca(2+) oscillations are numerically studied. In the case of simple intracellular Ca(2+) oscillation, it is found that, with the increase of molecule number, the coherence resonance or autonomous resonance phenomena can occur for some external stimulation parameter values. In the cases of complex cytosolic Ca(2+) oscillations, each extremum of concentration of cytosolic Ca(2+) oscillations corresponds to a peak in the histogram of Ca(2+) concentration, and the most probability appeared during the bursting plateau level for bursting, but at the largest minimum of Ca(2+) concentration for chaos. For quasi-periodicity, however, there are only two peaks in the histogram of Ca(2+) concentration, and the most probability is located at low concentration state.     

A model for the oscillatory motion of single dynein molecules

Axonemal dynein is the molecular motor responsible for the rhythmic beating of eukaryotic cilia and flagella. An individual axonemal dynein molecule is capable of both unidirectional and oscillatory motion along a microtubule (Nature 393 (1998) 711). We propose a model which links the physical motion of a two-headed dynein molecule to its ATP hydrolysis cycle, and which exhibits both processive and oscillatory behaviors. A mathematical analysis of the model is used to make experimentally testable predictions.     

An integro-partial differential equation for modeling biofluids flow in fractured biomaterials

A novel mathematical model in the framework of a nonlinear integro-partial differential equation governing biofluids flow in fractured biomaterials is proposed, solved, verified, and evaluated. A semi-analytical solution is derived for the equation, verified by a mass-lumped Galerkin finite element method (FEM), and calibrated with two in vitro experimental datasets. The solution process uses separation of variables and results in explicit expression involving complete and incomplete beta functions. The proposed semi-analytical model shows reasonable agreements with the finite element simulator as well as with two in vitro experimental time series and can be successfully used to simulate biofluids (e.g. water, blood, oil, etc.) flow in natural and synthetic porous biomaterials.     

A new cell-based FE model for the mechanics of embryonic epithelia

In order to overcome a significant stiffening artefact associated with current finite element (FE) models for the mechanics of embryonic epithelia, two new FE formulations were developed. Cell–cell interfacial tensions γ are represented by constant-force rod elements as in previous models. However, the viscosity of the cytoplasm with its embedded organelles and filament networks is modeled using viscous triangular elements, it is modeled using either radial and circumferential dashpots or an orthogonal dashpot system rather than the viscous triangular elements typical of previous two-dimensional FE models. The models are tested against tissue (epithelium) stretching because it gives rise to significant changes in cell shape and against cell sorting because it involves high rates of cell rearrangement. The orthogonal dashpot system is found to capture cell size and shape effects well, give the model cells characteristics that are consistent with those of real cells, provide high computational efficiency and hold promise for future three-dimensional analyses.     

Subchondral bone microarchitecture and failure mechanism under compression: A finite element study

Subchondral bone (SCB) microdamage is commonly observed in traumatic joint injuries and has been strongly associated with post-traumatic osteoarthritis (PTOA). Knowledge of the three-dimensional stress and strain distribution within the SCB tissue helps to understand the mechanism of SCB failure, and may lead to an improved understanding of mechanisms of PTOA initiation, prevention and treatment. In this study, we used high-resolution micro-computed tomography (µCT)-based finite element (FE) modelling of cartilage-bone to evaluate the failure mechanism and the locations of SCB tissue at high-risk of initial failure under compression. The µCT images of five cartilage-bone specimens with an average SCB thickness of 1.23 ± 0.20 mm were used to develop five µCT-based FE models. The FE models were analysed under axial compressions of approximately 30 MPa applied to the cartilage surface while the bone edges were constrained. Strain and stress-based failure criteria were then applied to evaluate the failure mechanism of the SCB tissue under excessive compression through articular cartilage. µCT-based FE models predicted two locations in the SCB at high-risk of initial failure: (1) the interface of the calcified-uncalcified cartilage due to excessive tension, and (2) the trabecular bone beneath the subchondral plate due to excessive compression. µCT-based FE models of cartilage-bone enabled us to quantify the distribution of the applied compression which was transferred through the articular cartilage to its underlying SCB, and to investigate the mechanism and the mode of SCB tissue failure. Ultimately, the results will help to understand the mechanism of injury formation in relation to PTOA.     

A minimal mechanics model for mechanosensing of substrate rigidity gradient in durotaxis

Durotaxis refers to the phenomenon in which cells can sense the spatial gradient of the substrate rigidity in the process of cell migration. A conceptual two-part theory consisting of the focal adhesion force generation and mechanotransduction has been proposed previously by Lo et al. to explain the mechanism underlying durotaxis. In the present work, we are concerned with the first part of the theory: how exactly is the larger focal adhesion force generated in the part of the cell adhering to the stiffer region of the substrate? Using a simple elasticity model and by assuming the cell adheres to the substrate continuously underneath the whole cell body, we show that the mechanics principle of static equilibrium alone is sufficient to account for the generation of the larger traction stress on the stiffer region of the substrate. We believe that our model presents a simple mechanistic understanding of mechanosensing of substrate stiffness gradient at the cellular scale, which can be incorporated in more sophisticated mechanobiochemical models to address complex problems in mechanobiology and bioengineering.     

A stepwise model system for limb regeneration

The amphibian limb is a model that has provided numerous insights into the principles and mechanisms of tissue and organ regeneration. While later stages of limb regeneration share mechanisms of growth control and patterning with limb development, the formation of a regeneration blastema is controlled by early events that are unique to regeneration. In this study, we present a stepwise experimental system based on induction of limb regeneration from skin wounds that will allow the identification and functional analysis of the molecules controlling this early, critical stage of regeneration. If a nerve is deviated to a skin wound on the side of a limb, an ectopic blastema is induced. If a piece of skin is grafted from the contralateral side of the limb to the wound site concomitantly with nerve deviation, the ectopic blastema continues to grow and forms an ectopic limb. Our analysis of dermal cell migration, contribution, and proliferation indicates that ectopic blastemas are equivalent to blastemas that form in response to limb amputation. Signals from nerves are required to induce formation of both ectopic and normal blastemas, and the diversity of positional information provided by blastema cells derived from opposite sides of the limb induces outgrowth and pattern formation. Hence, this novel and convenient stepwise model allows for the discovery of necessary and sufficient signals and conditions that control blastema formation, growth, and pattern formation during limb regeneration.     

A mathematical model for pattern formation of glioma cells outside the tumor spheroid core

Glioblastoma is the most common and the most aggressive type of brain cancer. The median survival time from the time of diagnosis is approximately one year. Invasion of glioma cells from the core tumor into the surrounding brain tissue is a major reason for treatment failure: these migrating cells are not eliminated in surgical resection and cause tumor recurrence. Variations are seen in number of invading cells, and in the extent and patterns of migration. Cells can migrate diffusely and can also be seen as clusters of cells distinct from the main tumor mass. This kind of clustering is also evident in vitro using 3D spheroid models of glioma invasion. This has been reported for U87 cells stably expressing the constitutively active EGFRVIII mutant receptor, often seen expressed in glioblastoma. In this case the cells migrate as clusters rather than as single cells migrating in a radial pattern seen in control wild type U87 cells. Several models have been suggested to explain the different modes of migration, but none of them, so far, has explored the important role of cell–cell adhesion. The present paper develops a mathematical model which includes the role of adhesion and provides an explanation for the various patterns of cell migration. It is shown that, depending on adhesion, haptotactic, and chemotactic parameters, the migration patterns exhibit a gradual shift from branching to dispersion, as has been reported experimentally.     

Subject specific finite element mesh generation of the pelvis from biplanar x-ray images: application to 120 clinical cases

Several Finite Element (FE) models of the pelvis have been developed to comprehensively assess the onset of pathologies and for clinical and industrial applications. However, because of the difficulties associated with the creation of subject-specific FE mesh from CT scan and MR images, most of the existing models rely on the data of one given individual. Moreover, although several fast and robust methods have been developed for automatically generating tetrahedral meshes of arbitrary geometries, hexahedral meshes are still preferred today because of their distinct advantages but their generation remains an open challenge. Recently, approaches have been proposed for fast 3D reconstruction of bones based on X-ray imaging. In this study, we adapted such an approach for the fast and automatic generation of all-hexahedral subject-specific FE models of the pelvis based on the elastic registration of a generic mesh to the subject-specific target in conjunction with element regularity and quality correction. The technique was successfully tested on a database of 120 3D reconstructions of pelvises from biplanar X-ray images. For each patient, a full hexahedral subject-specific FE mesh was generated with an accurate surface representation.     

Numerical Analysis of An Anastomotic Device

The explicit dynamic finite element method was utilized to investigate the deformation behaviour of a woven wire mesh tubular device that is used in a side-to-side anastomotic procedure for achieving gastrointestinal anastomosis. The numerical model was initially verified by comparison to experimental results that were obtained using a specialized testing mechanism. Once validated, the finite element model (FEM) was parameterized to ascertain the influence of several device parameters on its deformation behaviour. The importance of these parameters, as related to its optimal design for use in minimally invasive surgery (MIS), was subsequently ascertained and discussed.     

A model of muscle contraction based on the Langevin equation with actomyosin potentials

We propose a muscle contraction model that is essentially a model of the motion of myosin motors as described by a Langevin equation. This model involves one-dimensional numerical calculations wherein the total force is the sum of a viscous force proportional to the myosin head velocity, a white Gaussian noise produced by random forces and other potential forces originating from the actomyosin structure and intra-molecular charges. We calculate the velocity of a single myosin on an actin filament to be 4.9–49 μm/s, depending on the viscosity between the actomyosin molecules. A myosin filament with a hundred myosin heads is used to simulate the contractions of a half-sarcomere within the skeletal muscle. The force response due to a quick release in the isometric contraction is simulated using a process wherein crossbridges are changed forcibly from one state to another. In contrast, the force response to a quick stretch is simulated using purely mechanical characteristics. We simulate the force–velocity relation and energy efficiency in the isotonic contraction and adenosine triphosphate consumption. The simulation results are in good agreement with the experimental results. We show that the Langevin equation for the actomyosin potentials can be modified statistically to become an existing muscle model that uses Maxwell elements.     

A Leap-frog Algorithm for Stochastic Dynamics

Abstract

A third-order algorithm for stochastic dynamics (SD) simulations is proposed, identical to the powerful molecular dynamics leap-frog algorithm in the limit of infinitely small friction coefficient γ. It belongs to the class of SD algorithms, in which the integration time step Δt is not limited by the condition Δt ≤ γ^?1, but only by the properties of the systematic force. It is shown how constraints, such as bond length or bond angle constraints, can be incorporated in the computational scheme. It is argued that the third-order Verlet-type SD algorithm proposed earlier may be simplified without loosing its third-order accuracy. The leap-frog SD algorithm is proven to be equivalent to the verlet-type SD algorithm. Both these SD algorithms are slightly more economical on computer storage than the Beeman-type SD algorithm.     

Thermo-Mechanical Analysis of Restored Molar Tooth using Finite Element Analysis

The aim of the study is to find most optimum combination of crown material and adhesive to avoid loosening and thereby failure of restored tooth. This study describes the Thermo-Mechanical analysis of restored molar tooth crown for determination of the stress levels due to thermal and mechanical loads on restored molar tooth. The potential use of the 3-D model was demonstrated and analyzed using different materials for crown. Thermal strain, stress and deformation were measured at hot and cold conditions in ANSYS and correlated with analytical calculation and existing experimental data for model validation and optimization. It is concluded that amongst various material porcelain crown with composite resin adhesive cement closely simulates the behavior of natural crown and should ideally result into long lasting restoration.     

On a probabilistic model for the numerical estimation of nocturnal migration of birds

The study of nocturnal bird migration by cone methods of observation has a century-long history but has continued to be used up to the present. To describe the flux and estimate the number of passing birds a probabilistic model is proposed. This model is based on the concept of dynamic Poisson ensemble of points in appropriate phase space and has two parameters. One is scalar and the other one is functional. We constructed consistent estimations of these parameters and discuss their use for the numerical estimation of the flux of birds observed in a narrow light cone generated by the bright lunar disk and formed by an open angle of telescope. Selection on the same type of birds was suggested as the necessary condition for the model application. Ground speed of each bird was introduced into the model as a new but obligatory value determining the quantification of the flux of bird.