Unraveling changes in myocardial contractility during human fetal growth: a finite element analysis based on in vivo ultrasound measurements.

Knowledge of normal fetal heart (FH) performance and development is crucial for evaluating and understanding how various congenital heart lesions may modify heart contractility during the gestational period. However, since biomechanical models of FH are still lacking, structural approaches proposed to describe the mechanical behavior of the adult human heart cannot be used to model the evolution of the FH. In this paper, a finite element model of the healthy FH wall is developed to quantify its mechanical properties during the gestational period. An idealized thick-walled ellipsoidal shape was used to model the left ventricle (LV). The diastolic LV geometry was reconstructed from in vivo ultrasound measurements performed on 24 normal FHs between 20 and 37 weeks of gestation. An anisotropic hyperelastic constitutive law describing the mechanical properties of the passive and active myocardium was used. The evolution of the mechanical properties of the normal LV myocardium during fetal growth was obtained by successfully fitting the ejection fraction predicted by the model to in vivo measurements. We found that only the active tension varies significantly during the gestational period, increasing linearly from 20 kPa (at 20 weeks) to 40 kPa (at 37 weeks of gestation). We propose a possible explanation of the increasing force-generating ability of the myocardial tissue during fetal heart development based on a combination of myocyte enlargement, differentiation, and proliferation kinetics.

Quantification of cardiomyocyte contraction based on image correlation analysis.

Quantification of cardiomyocyte contraction is usually obtained by measuring globally cell shortening from the displacement of cell extremities. We developed a correlation-based optical flow method, which correlates the whole-cell temporal pattern with a precise quantification of the intracellular strain wave at the sarcomeres level. A two-dimensional image correlation analysis of cardiomyocytes phase-contrast images was developed to extract local cell deformations from videomicroscopy time-lapse sequences. Test images, synthesized from known intensity displacement fields, were first used to validate the method. Intracellular strain fields were then computed from videomicroscopy time-lapse sequences of single adult and neonatal cardiomyocytes. The propagation of the sarcomeres contraction-relaxation wave during cell contraction has been successfully quantified. The time-varying patterns of intracellular displacement were obtained accurately, even when cardiomyocyte bending occurred in pace with contraction. Interestingly, the characterization of the successive 2D displacement fields show a direct quantification of the variation with time of intracellular strains anywhere in the cell. The proposed method enables a quantitative analysis of cardiomyocyte contraction without requiring wave tracking with the use of fluorescent calcium probes. Thus, our algorithmic approach provides a fast and efficient tool for analyzing the correlation between global cell dynamical behavior and mechanosensitive intracellular processes.

Theoretical analysis of the adaptive contractile behaviour of a single cardiomyocyte cultured on elastic substrates with varying stiffness.

In vivo, cardiomyocytes interact with surrounding extracellular matrix while performing periodically a contractile behaviour, which is the main determinant of heart performance. As extracellular substrates with easily tunable stiffness properties, polyacrylamide gels (PAGs) provide valuable flexible media for studying in vitro the dynamical behaviour of cardiomyocytes responding to stiffness variations of their surrounding environment. We propose in this paper an original mechano-chemical model of the cardiac cell contraction that sheds light on the adaptive response of cardiomyocytes evidenced recently in the experiments of Qin et al. [2007. Dynamical stress characterization and energy evaluation of single cardiac myocyte actuating on flexible substrate. Biochem. Biophys. Res. Commun. 360, 352-356]. The model links the amplitude of the extracellular PAGs strain fields to the spatio-temporal variation of the intracellular stresses in every part of the cell during the sarcomeres contraction-relaxation. In a continuum mechanics framework, we derived a unified description of the sarcomere-length dependence of intracellular active stress and of its control by anisotropic calcium diffusion and autocatalytic calcium release from the sarcoplasmic reticulum. Taking benefit of our previous work on the characterization of mechanical properties of PAGs with varying stiffness, we were thus able to evaluate the active intracellular stress exerted by the cardiomyocyte on flexible PAGs with different and known Young's moduli. Interestingly, we were able to explain the intriguing increase of maximal cellular stress observed experimentally when substrate stiffness is increased. By providing an evaluation of the whole-field cell stresses and strains, this integrative approach of cardiomyocyte contraction provides a reliable basis for further analysis of additional cooperativity and mechanotransduction mechanisms involved in cell contractility regulation, notably in physiological and pathological situations where modifications of cardiac performance are linked to varied stiffness of the cardiomyocytes environment.

Influences of adhesion area and biological sample size on the estimation of Young's modulus and Poisson's ratio assessed by micropipette aspiration technique.

The micropipette aspiration experiment remains a widely used micromanipulation technique for quantifying the mechanical properties of biological samples. Our study extends previous results by investigating the influence of sample size and adhesion area on the mechanical response of compressible thin biological samples. We thus defined a nonlinear relationship between aspirated length, Young's modulus, Poisson's ratio and sample thickness which allowed us to develop an original experimental protocol for simultaneous quantification of the Poisson's ratio and Young's modulus of adherent samples. We first validated our method by characterizing mechanical properties of polyacrylamide gels with tunable stiffness. We then considered application of these results to the quantification of cell elasticity, focusing on the influence of cell adhesion area onto the measured apparent cell stiffness.

Morphological and biomechanical aspects of vulnerable coronary plaque.

Vulnerable plaque morphology has been described by gross pathology and intravascular ultrasound, but morphological criteria cannot fully explain vulnerability, which involves four distinct factors: 1) inflammatory and biological processes; 2) geometry; 3) composition; and 4) hemodynamic stress. These last three aspects underlie the biomechanical study of vulnerable plaque. By virtue of the nature of their evolution, atherosclerotic plaques tend to be excentric, and this is a crucial morphological feature, causing circumferential stress to peak in very specific juxta-luminal locations, where it can exceed the rupture threshold of collagen, the basic constituent of arterial architecture. The lipido-necrotic core covered by a fibrous cap, formed in young plaques, is another morphological feature, which, can also increase and concentrate circumference stress in the juxta-luminal fibrous cap. The larger the lipid core, the thinner the fibrous cap and the greater is the stress. There are also inflammatory processes in such areas, which tend to reduce cap thickness. Ruptures occur when this thickness falls below 65 microns. Heart rate, blood pressure and pulse pressure are all biomechanical factors affecting vulnerable arterial walls, increasing circumferential stress and material fatigue. Vulnerable plaques are almost always associated with positive arterial remodeling. Numerical simulation has shown such so-called compensatory remodeling to be exclusively due to the healthy arc stretching in vulnerable plaques. Positive remodeling is optimal when the healthy arc is around 170 degrees, which keeps the lumen area relatively stable as long as the plaque does not exceed 40% to 50%. This mechanism does not apply to concentric plaques. In conclusion, the mechanism of vulnerable plaque rupture is highly complex and multifactorial. This complexity more or less precludes prediction in individual cases: we are in the realms of chaos theory and acute sensitivity to initial conditions. The greatest caution is therefore required in any attempt to predict rupture from diagnostic imagery, which provides only morphological data on plaque's nature.

The rigidity in fibrin gels as a contributing factor to the dynamics of in vitro vascular cord formation.

While the formation of vascular cords in in vitro angiogenesis assay is commonly used to test the angiogenic properties of many molecular or cellular components, an extensive characterisation of the dynamics of this process is still lacking. Up to now, quantitative studies only focused on the resulting capillary structures characterised through static morphometric approaches. We therefore propose in this paper a rather extensive characterisation aiming to identify different stages in the dynamics of this process, through the investigation of the influence of the rigidity of the fibrin extracellular matrix on the growth of the vascular cords. Using time lapse videomicroscopy, the time evolution of relevant morphodynamical parameters has been considered both at the cell level and at the cell population level. At the cell level, a trajectography analysis of individual cells observed in different locations of the growing network has been conducted and analysed using a random walk model. From image sequence analysis and segmentation i.e. extraction of the boundaries of the lacunae formed through matrix degradation and cell tractions, the evolution of the lacunae surface has been precisely quantified, revealing different phases and transitions in the growth patterns. Our results indicate that the rigidity of the extracellular fibrin matrix strongly influences the different stages, i.e. the dynamics of the angiogenic process. Consequently, optimal rigidity conditions for the formation of stable vascular cord networks could be identified in the context of our experiments.

Global analysis of endothelial cell line proliferation patterns based on nutrient-depletion models: implications for a standardization of cell proliferation assays.

It is known that cell populations growing in different environmental conditions may exhibit different proliferation patterns. However, it is not clear if, despite the diversity of the so-observed patterns, inherent cellular growth characteristics of the population can nevertheless be determined. This study quantifies the proliferative behaviour of the permanent endothelial human cell line, Eahy926, and establishes to which extent the estimation of the cell proliferation rate depends on variations of the experimental protocols. Cell proliferation curves were obtained for cells cultured over 16 days and the influences of cell seeding densities, foetal bovine serum content and frequency of culture medium changes were investigated. Quantitative dynamic modelling was conducted to evaluate the kinetic characteristics of this cell population. We proposed successive models and retained a nutrient-depletion toxicity dependant model, which takes into account the progressive depletion of nutrients, as well as the increase of toxicity in the cell culture medium. This model is shown to provide a very good and robust prediction of the experimental proliferation curves, whatever are the considered frequency of culture medium changes and serum concentrations. Thus, the model enables an intrinsic quantification of the parameters driving in vitro EAhy926 proliferation, including proliferation, nutrient consumption and toxicity increase rates, rather independently of the experiments design. We therefore propose that such models could provide a basis for a standardized quantification of intrinsic cell proliferation kinetics.

A mathematical model for the dynamics of large membrane deformations of isolated fibroblasts.

In this paper we develop and extend a previous model of cell deformations, initially proposed to describe the dynamical behaviour of round-shaped cells such as keratinocytes or leukocytes, in order to take into account cell pseudopodial dynamics with large amplitude membrane deformations such as those observed in fibroblasts. Beyond the simulation (from a quantitative, parametrized model) of the experimentally observed oscillatory cell deformations, a final goal of this work is to underline that a set of common assumptions regarding intracellular actin dynamics and associated cell membrane local motion allows us to describe a wide variety of cell morphologies and protrusive activity. The model proposed describes cell membrane deformations as a consequence of the endogenous cortical actin dynamics where the driving force for large-amplitude cell protrusion is provided by the coupling between F-actin polymerization and contractility of the cortical actomyosin network. Cell membrane movements then result of two competing forces acting on the membrane, namely an intracellular hydrostatic protrusive force counterbalanced by a retraction force exerted by the actin filaments of the cell cortex. Protrusion and retraction forces are moreover modulated by an additional membrane curvature stress. As a first approximation, we start by considering a heterogeneous but stationary distribution of actin along the cell periphery in order to evaluate the possible morphologies that an individual cell might adopt. Then non-stationary actin distributions are considered. The simulated dynamic behaviour of this cytomechanical model not only reproduces the small amplitude rotating waves of deformations of round-shaped cells such as keratinocytes [as proposed in the original model of Alt and Tranquillo (1995, J. Biol. Syst. 3, 905-916)] but is furthermore in very good agreement with the protrusive activity of cells such as fibroblasts, where large amplitude contracting/retracting pseudopods are more or less periodically extended in opposite directions. In addition, the biophysical and biochemical processes taken into account by the cytomechanical model are characterized by well-defined parameters which (for the majority) can be discussed with regard to experimental data obtained in various experimental situations.

Quantification and macroscopic modeling of the nonlinear viscoelastic behavior of strained gels with varying fibrin concentrations.

The mechanical properties of fibrin gels under uniaxial strains have been analyzed for low fibrin concentrations using a free-floating gel device. We were able to quantify the viscous and elastic moduli of gels with fibrin concentration ranging from 0.5 to 3 mg/ml, reporting significant differences of biogels moduli and dynamical response according to fibrin concentration. Furthermore, considering sequences of successively imposed step strains has revealed the strain-hardening properties of fibrin gels for strain amplitude below 5%. This nonlinear viscoelastic behavior of the gels has been precisely analyzed through numerical simulations of the overall gel response to the strain steps sequences. Phenomenological power laws relating the instantaneous and relaxed elasticity moduli to fibrin concentration have been validated, with concentration exponent in the order of 1.2 and 1.0, respectively. This continuous description of strain-dependent mechanical moduli was then used to simulate the biogel behavior when continuously time-varying strains are applied. We discuss how this experimental setup and associated macroscopic modeling of fibrin gels enable a further quantification of cell traction forces and mechanotransduction processes induced by biogel compaction or stretching.

Quantitative study of dynamic behavior of cell monolayers during in vitro wound healing by optical flow analysis.

BACKGROUND: In vitro wound healing assays are experimental models commonly used to analyze cell behavior during the migration process. A new approach is proposed for the quantification of cell motility based on an optical flow method. METHODS: We assumed that cell-population dynamics can be defined by an a priori affine-motion model. Identified model parameters are used as motion descriptors quantifying both elementary and complex cell movements, either at the wound margins or within the cell monolayer. RESULTS: When compared with the estimation of cell motility calculated from wound area temporal variation, it allows a more detailed and precise characterization of cell population movements. Comparative analysis of normal and cancerous cell lines revealed that typical measured velocities were about 2 microm/h and 7 microm/h for L929 and HeLa cells, respectively, at the beginning of the wound closure. The quantification of the effect of Hoechst 33342 on cell dynamics showed a similar behavior for control and stained cells within 20 h after wound scratching, but then a decreased velocity of stained cells. CONCLUSIONS: The results demonstrate that this approach can be used to gain new insights into the dynamic changes induced by the extracellular environment and by anticancer drugs.

Mechanical signalling and angiogenesis. The integration of cell-extracellular matrix couplings.

In vitro angiogenesis assays have shown that the couplings between fibrin gel and cell traction forces trigger biogel pre-patterning, consisting, in the formation of lacunae which evolve toward capillary-like structures (CLS) networks. Depending on the experimental conditions (number of seeded cells, gel elasticity,...), this pre-patterning can be enhanced or inhibited. A theoretical model based on a description of the cell-biogel biochemical and mechanical interactions is proposed as a basis for understanding how integrating these interactions can lead to the pre-patterning of the biogel. We showed that the critical parameter values corresponding to the bifurcation of the model solutions correspond to threshold values of the experimental variables. Furthermore, simulations of the mechanocellular model give rise to dynamic remodelling patterns of the biogel which are in good agreement both with the lacunae morphologies and with the time and space scales derived from the in vitro angiogenesis assays. Special attention has been paid in the simulations to cell proteolytic activity and to the amplitude of cell traction forces. We finally discussed how modelling guided experiments can be inferred from these results.

Characterization of cell deformation and migration using a parametric estimation of image motion.

This paper deals with the spatio-temporal analysis of two-dimensional deformation and motion of cells from time series of digitized video images. A parametric motion approach based on an affine model has been proposed for the quantitative characterization of cellular movements in different experimental areas of cellular biology including spontaneous cell deformation, cell mitosis, individual cell migration and collective migration of cell populations as cell monolayer. The accuracy and robustness of the affine model parameter estimation, which is based on a multiresolution algorithm, has been established from synthesized image sequences. A major interest of our approach is to follow with time the evolution of a few number of parameters characteristic of cellular motion and deformation. From the time-varying eigenvalues of the affine model square matrix, a precise quantification of the cell pseudopodial activity, as well as of cell division has been performed. For migrating cells, the motion quantification confirms that cell body deformation has a leading role in controlling nucleus displacement, the nucleus itself undergoing a larger rotational motion. At the cell population level, image motion analysis of in vitro wound healing experiments quantifies the heterogeneous cell populations dynamics.

Standardization of a method for characterizing low-concentration biogels: elastic properties of low-concentration agarose gels.

Low-concentration biogels, which provide an extracellular matrix for cells in vitro, are involved in a number of important cell biological phenomena, such as cell motility and cell differentiation. In order to characterize soft tissues, which collapse under their own weight, we developed and standardized a new experimental device that enabled us to analyze the mechanical properties of floating biogels with low concentrations, i.e., with values ranging from 2 g/L to 5 g/L. In order to validate this approach, the mechanical responses of free floating agarose gel samples submitted to compression as well as stretching tests were quantified. The values of the Young's moduli, measured in the range of 1000 to 10,000 Pa, are compared to the values obtained from other experimental techniques. Our results showed indeed that the values we obtained with our device closely match those obtained independently by performing compression tests on an Instron device. Thus, the floating gel technique is a useful tool first to characterize and then to model soft tissues that are used in biological science to study the interaction between cell and extracellular matrix.

Modelling biological gel contraction by cells: mechanocellular formulation and cell traction force quantification.

Traction forces developed by most cell types play a significant role in the spatial organisation of biological tissues. However, due to the complexity of cell-extracellular matrix interactions, these forces are quantitatively difficult to estimate without explicitly considering cell properties and extracellular mechanical matrix responses. Recent experimental devices elaborated for measuring cell traction on extracellular matrix use cell deposits on a piece of gel placed between one fixed and one moving holder. We formulate here a mathematical model describing the dynamic behaviour of the cell-gel medium in such devices. This model is based on a mechanical force balance quantification of the gel visco-elastic response to the traction forces exerted by the diffusing cells. Thus, we theoretically analyzed and simulated the displacement of the free moving boundary of the system under various conditions for cells and gel concentrations. This model is then used as the theoretical basis of an experimental device where endothelial cells are seeded on a rectangular biogel of fibrin cast between two floating holders, one fixed and the other linked to a force sensor. From a comparison of displacement of the gel moving boundary simulated by the model and the experimental data recorded from the moving holder displacement, the magnitude of the traction forces exerted by the endothelial cell on the fibrin gel was estimated for different experimental situations. Different analytical expressions for the cell traction term are proposed and the corresponding force quantifications are compared to the traction force measurements reported for various kind of cells with the use of similar or different experimental devices.

In vitro angiogenesis is modulated by the mechanical properties of fibrin gels and is related to alpha(v)beta3 integrin localization.

This study deals with the role of the mechanical properties of matrices in in vitro angiogenesis. The ability of rigid fibrinogen matrices with fibrin gels to promote capillarylike structures was compared. The role of the mechanical properties of the fibrin gels was assessed by varying concentration of the fibrin gels. When the concentration of fibrin gels was decreased from 2 mg/ml to 0.5 mg/ml, the capillarylike network increased. On rigid fibrinogen matrices, capillarylike structures were not formed. The extent of the capillarylike network formed on fibrin gels having the lowest concentration depended on the number of cells seeded. The dynamic analysis of capillarylike network formation permitted a direct visualization of a progressive stretching of the 0.5 mg/ml fibrin gels. This stretching was not observed when fibrin concentration increases. This analysis shows that 10 h after seeding, a prearrangement of cells into ringlike structures was observed. These ringlike structures grew in size. Between 16 and 24 h after seeding, the capillarylike structures were formed at the junction of two ringlike structures. Analysis of the alpha(v)beta3 integrin localization demonstrates that cell adhesion to fibrinogen is mediated through the alpha(v)beta3 integrin localized into adhesion plaques. Conversely, cell adhesion to fibrin shows a diffuse and dot-contact distribution. We suggest that the balance of the stresses between the tractions exerted by the cells and the resistance of the fibrin gels triggers an angiogenic signal into the intracellular compartment. This signal could be associated with modification in the alpha(v)beta3 integrin distribution.

In vitro dynamics of chromatin organization and migration.

The organization of eukaryotic chromatin is not static but changes as a function of cell status during processes such as proliferation, differentiation, and migration. DNA quantification has not been used extensively to investigate chromatin dynamics in combination with cellular migration. In this context, an optimized DNA-specific, nonperturbant method has been developed for studying chromatin organization, using the fluorescent vital bisbenzimidazole probe Hoechst 33342: this property has been described by Hamori et al. (1980). Computer-assisted image analysis was used to follow migratory activity and chromatin organization of L929 fibroblasts during in vitro wound healing. Cell movements were analyzed using an optical flow technique, which consists in the calculation of the velocity field of cells and nuclear movements in the frame. This system allows the correlation of cell migration and position in the cell cycle. It makes it possible to study chromatin dynamics using a quantitative analysis of nuclear differentiation reorganization (nuclear texture) and to correlate this with migration characteristics. The present system would be of interest for studying cell-extracellular matrix interactions using differing substrates, and also the migratory response to chemotactic factors. Such a model is a prerequisite for gaining better understanding of drug action.

A mathematical model of glioma growth: the effect of extent of surgical resection.

We have developed a mathematical model based on proliferation and infiltration of neoplastic cells that allows predictions to be made concerning the life expectancies following various extents of surgical resection of gliomas of all grades of malignancy. The key model parameters are the growth rate and the diffusion rate. These rates were initially derived from analysis of a case of recurrent anaplastic astrocytoma treated by chemotherapies. Numerical simulations allow us to estimate what would have happened to that patient if various extents of surgical resection, rather than chemotherapies, had been used. In each case, the shell of the infiltrating tumour that remains after 'gross total removal' or even a maximal excision continues to grow and regenerates the tumour mass remarkably rapidly. By developing a model that allows the growth and diffusion rates to define the distribution of cells at the time of diagnosis, and then varying these rates by about 50%, we created a hypothetical tumour patient population whose survival times show good agreement with the results recently reported by Kreth for treatments of glioblastomas. Tenfold decreases in the rates of growth and diffusion mimic the results reported by many other investigators with more slowly growing gliomas. Thus, the model quantitatively supports the ideas that (i) gliomas infiltrate so diffusely that they cannot be cured by resection alone, surgical or radiological, no matter how extensive that may be; (ii) the more extensive the resection, regardless of the degree of malignancy of the glioma, the greater the life expectancy; and (iii) measurements of the two rates, growth and diffusion, may be able to predict survival rates better than the current histological estimates of the type and grade of gliomas.

[Dynamic characterization of the growth of brain tumors based on image sequences of nuclear magnetic resonance]. / Caractérisation dynamique de la croissance de tumeurs cérébrales à partir de séquences d'images obtenues par résonance magnétique nucléaire.

The evolution of a high grade glioblastoma of a patient undergoing radio-therapy has been analysed by considering a mathematical model which simulates the brain tumour growth within a two-dimensional domain defined by the brain and ventricles geometry. This simulated behaviour was compared with morphological data obtained from successive nuclear magnetic resonance scans of the patient. The model parameters include the proliferation rate and the diffusion coefficient of the tumour cells as well as their sensitivity to the irradiation. They were estimated using optimisation techniques to minimise the distance between simulated tumour area and scan data from different brain sections. The relevance of this quantitative estimation for the prognosis and for the consideration of additional parameters in the pre and post therapeutic evaluation of glioma is discussed.

From passive diffusion to active cellular migration in mathematical models of tumour invasion.

Mathematical models of tumour invasion appear as interesting tools for connecting the information extracted from medical imaging techniques and the large amount of data collected at the cellular and molecular levels. Most of the recent studies have used stochastic models of cell translocation for the comparison of computer simulations with histological solid tumour sections in order to discriminate and characterise expansive growth and active cell movements during host tissue invasion. This paper describes how a deterministic approach based on reaction-diffusion models and their generalisation in the mechano-chemical framework developed in the study of biological morphogenesis can be an alternative for analysing tumour morphological patterns. We support these considerations by reviewing two studies. In the first example, successful comparison of simulated brain tumour growth with a time sequence of computerised tomography (CT) scans leads to a quantification of the clinical parameters describing the invasion process and the therapy. The second example considers minimal hypotheses relating cell motility and cell traction forces. Using this model, we can simulate the bifurcation from an homogeneous distribution of cells at the tumour surface toward a nonhomogeneous density pattern which could characterise a pre-invasive stage at the tumour-host tissue interface.

A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth.

During the past two decades computerized tomography (CT) and magnetic resonance imaging (MRI) have permitted the detection of tumours at much earlier stages in their development than was previously possible. In spite of this earlier diagnosis the effects of earlier and more extensive treatments have been difficult to document. This failure has led to an increasing awareness of the importance of infiltration of glioma cells into surrounding grossly normal brain tissue such that recurrence still occurs. In this paper a simple mathematical model for the proliferation and infiltration of such tumours is introduced, based in part on quantitative image analysis of histological sections of a human brain glioma and especially on cross-sectional area/volume measurements of serial CT images while the patient was undergoing chemotherapy. The model parameters were estimated using optimization techniques to give the best fit of the simulated tumour area to the CT scan data. Numerical solution of the model on a two-dimensional domain, which took into account the geometry of the brain and its natural barriers to diffusion, was used to determine the effect of chemotherapy on the spatio-temporal growth of the tumour.