Determination of a constitutive relation for passive myocardium: I. A new functional form

The specific aim of this study is to determine a constitutive relation for non-contracting myocardium in terms of a pseudostrain-energy function W whose form is guided by both theory and experiment. We assume that the material symmetry of myocardium is initially and locally transversely-isotropic, and seek a W which depends upon only two coordinate invariant measures of the finite deformation. The specific functional form of such a W is inferred directly from experimental protocols in which one invariant is held constant while the other is varied, and vice versa. On the basis of data from families of these "constant invariant" tests on thin slabs of myocardium taken from the mid-walls of six canine left ventricles, we propose a new polynomial form of W containing only five material parameters.     

Structural three-dimensional constitutive law for the passive myocardium

A three-dimensional constitutive law is proposed for the myocardium. Its formulation is based on a structural approach in which the total strain energy of the tissue is the sum of the strain energies of its constituents: the muscle fibers, the collagen fibers and the fluid matrix which embeds them. The ensuing material law expresses the specific structural and mechanical properties of the tissue, namely, the spatial orientation of the comprising fibers, their waviness in the unstressed state and their stress-strain behavior when stretched. Having assumed specific functional forms for the distribution of the fibers spatial orientation and waviness, the results of biaxial mechanical tests serve for the estimation of the material constants appearing in the constitutive equations. A very good fit is obtained between the measured and the calculated stresses, indicating the suitability of the proposed model for describing the mechanical behavior of the passive myocardium. Moreover, the results provide general conclusions concerning the structural basis for the tissue overall mechanical properties, the main of which is that the collagen matrix, though comprising a relatively small fraction of the whole tissue volume, is the dominant component accounting for its stiffness.     

Biaxial constitutive relations for the passive canine diaphragm.

Samples of the muscular sheet excised from the midcostal region of dog diaphragms were subjected to biaxial loading. That is, stresses in the direction of the muscle fibers and in the direction perpendicular to the fibers in the plane of the sheet were measured at different combinations of strains in the two directions. Stress-strain relations were obtained by fitting equations to these data. In the direction of the muscle fibers, for strains up to 0.7, stress is a modestly nonlinear function of strain and ranges up to approximately 60 g/cm. In the direction perpendicular to the fibers, the sheet is stiffer and more strongly nonlinear. At a strain in the perpendicular direction of approximately 0.35, stress increases abruptly. The stress-strain relation in the muscle direction is consistent with observations of passive muscle shortening in vivo. However, the stiffness in the perpendicular direction is not high enough to explain the observation that strains in the perpendicular direction in vivo are nearly zero. We conclude that, in the passive diaphragm in vivo, stress in the direction perpendicular to the muscle fibers is small.     

Parameter estimation of a monod-type model. Part II: Use of a neural network with associative memories

Summary On the basis of a previous study related with parameter identifiability and sensitivity analysis of a Monod-type model, a parameter estimation method based on Artificial Neural Networks (ANN) with Associative Memories (AMs) is presented. A combination of an iterative procedure and a convergence index given by AMs allows to confirm the nature of relations existing between state variables and parameters which were found in the first part of the study. The convergence criterion is particularly well adapted to showing various influences of state variables on parameter estimation of such a model.     

Parameter estimation for a point-source diffusion-decay morphogen model

Journal of Mathematical Biology - In this paper we present a novel method for finding unknown parameters for an unknown morphogen. We postulate the existence of an unknown morphogen in a given...     

Mechanical properties of the passive myocardium: experiment and a mathematical model

A three-dimensional mathematical model of the rheological properties of a morphofunctional unit of the myocardium has been constructed, which consists of transversal and longitudinal elastic elements and tilted viscoelastic elements hinged without friction. The parameters of the viscosity and elasticity of structural elements of the model do not depend on the deformation value. The model makes it possible to adequately describe the features of the viscoelastic behavior of isolated samples of the passive myocardium and of the myocardium under longitudinal strain. A good agreement between the calculated and experimental data for an intact preparation of the rat myocardium and a preparation with removed cardiomyocytes has been shown.     

Physical invariant strain energy function for passive myocardium

Principal axis formulations are regularly used in isotropic elasticity, but they are not often used in dealing with anisotropic problems. In this paper, based on a principal axis technique, we develop a physical invariant orthotropic constitutive equation for incompressible solids, where it contains only a one variable (general) function. The corresponding strain energy function depends on six invariants that have immediate physical interpretation. These invariants are useful in facilitating an experiment to obtain a specific constitutive equation for a particular type of materials. The explicit appearance of the classical ground-state constants in the constitutive equation simplifies the calculation for their admissible values. A specific constitutive model is proposed for passive myocardium, and the model fits reasonably well with existing simple shear and biaxial experimental data. It is also able to predict a set of data from a simple shear experiment.     

Shear properties of passive ventricular myocardium

We examined the shear properties of passive ventricular myocardium in six pig hearts. Samples (3 x 3 x 3 mm) were cut from adjacent regions of the lateral left ventricular midwall, with sides aligned with the principal material axes. Four cycles of sinusoidal simple shear (maximum shear displacements of 0.1-0.5) were applied separately to each specimen in two orthogonal directions. Resulting forces along the three axes were measured. Three specimens from each heart were tested in different orientations to cover all six modes of simple shear deformation. Passive myocardium has nonlinear viscoelastic shear properties with reproducible, directionally dependent softening as strain is increased. Shear properties were clearly anisotropic with respect to the three principal material directions: passive ventricular myocardium is least resistant to simple shear displacements imposed in the plane of the myocardial layers and most resistant to shear deformations that produce extension of the myocyte axis. Comparison of results for the six different shear modes suggests that simple shear deformation is resisted by elastic elements aligned with the microstructural axes of the tissue.     

Experimentally tractable,pseudo-elastic constitutive law for biomembranes: II. Application

This is the second paper of a 2 part series. In the first, a theoretical framework is developed that provides 3 enhancements for the constitutive theory of biomembranes. 1) Covariance amongst response terms is minimized, 2) the pseudo-strain-energy potential is redefined so as to be measurable, and 3) the stress response is decomposed into 2 parts--a hyperelastic part plus a hypoelastic-stress-residual. This paper illustrates the experimental advantages of this novel constitutive theory via analysis of biaxial test data obtained from chemically treated bovine pericardium.     

A cell-based constitutive relation for bio-artificial tissues

By using a combination of continuum and statistical mechanics we derive an integral constitutive relation for bio-artificial tissue models consisting of a monodisperse population of cells in a uniform collagenous matrix. This constitutive relation quantitatively models the dependence of tissue stress on deformation history, and makes explicit the separate contribution of cells and matrix to the mechanical behavior of the composite tissue. Thus microscopic cell mechanical properties can be deduced via this theory from measurements of macroscopic tissue properties. A central feature of the constitutive relation is the appearance of "anisotropy tensors" that embody the effects of cell orientation on tissue mechanics. The theory assumes that the tissues are stable over the observation time, and does not in its present form allow for cell migration, reorientation, or internal remodeling. We have compared the predictions of the theory to uniaxial relaxation tests on fibroblast-populated collagen matrices (FPMs) and find that the experimental results generally support the theory and yield values of fibroblast contractile force and stiffness roughly an order of magnitude smaller than, and viscosity comparable to, the corresponding properties of active skeletal muscle. The method used here to derive the tissue constitutive equation permits more sophisticated cell models to be used in developing more accurate representations of tissue properties.     

Parameter estimation techniques for interaction and redistribution models: a predator-prey example

Summary The use of parameter estimation techniques for partial differential equations is illustrated using a predatorprey model. Whereas ecologists have often estimated parameters in models, they have not previously been able to do so for models that describe interactions in heterogeneous environments. The techniques we describe for partial differential equations will be generally useful for models of interacting species in spatially complex environments and for models that include the movement of organisms. We demonstrate our methods using field data from a ladybird beetle (Coccinella septempunctata) and aphid (Uroleucon nigrotuberculatum) interaction. Our parameter estimation algorithms can be employed to identify models that explain better than 80% of the observed variance in aphid and ladybird densities. Such parameter estimation techniques can bridge the gap between detail-rich experimental studies and abstract mathematical models. By relating the particular bestfit models identified from our experimental data to other information on Coccinella behavior, we conclude that a term describing local taxis of ladybirds towards prey (aphids in this case) is needed in the model.     

Parameter relation rows: a query system for protein structure function relationships

In the course of molecular modeling or mutant prediction one often wants quick answers to questions such as: 'Are there any residues in a beta-strand that point into an internal cavity, and are highly mutable?' 'Are there large polar residues in a helix that make a contact with a hydrophobic residue in a sheet, and don't make the maximal number of hydrogen bonds?' or 'Which hydrophobic residues are in a helix with a large hydrophobic moment, and make a contact with a co-factor, but at the same time still have a large accessible surface?'. I describe here a method to get answers to these kinds of questions in a very quick and easy manner. The method described is partly based on the principles used in the design of relational databases, and its mode of operation is similar to the query methods used in a relational database environment. Although designed for aiding in molecular modeling, its applicability is much more general. The method has been implemented as part of a large molecular modeling package which copes with the numerous problems in systematic handling of protein structures, e.g. residue numbering. This also implies that many normal tools such as graphical analyses, I/O facilities, etc. are available on-line.     

Parameter estimation for a mathematical model of the cell cycle in frog eggs.

Parameter values for a kinetic model of the nuclear replication-division cycle in frog eggs are estimated by fitting solutions of the kinetic equations (nonlinear ordinary differential equations) to a suite of experimental observations. A set of optimal parameter values is found by minimizing an objective function defined as the orthogonal distance between the data and the model. The differential equations are solved by LSODAR and the objective function is minimized by ODRPACK. The optimal parameter values are close to the "guesstimates" of the modelers who first studied this problem. These tools are sufficiently general to attack more complicated problems, where guesstimation is impractical or unreliable.     

Mechanical properties of passive myocardium: experiment and mathematical model

We present here a 3D mathematical model of rheological properties of a morphofunctional unit of myocardium as example of biological tissue. The model consists of longitudinal and transverse elastic elements and inclined viscoelastic elements connected pivotally without friction. The parameters of viscosity and elasticity of the structural elements of the model do not depend on the magnitude of deformation. The model adequately describes nonlinear viscoelastic behavior of isolated samples of passive myocardium both in static condition and under dynamic loading. The simulation data fit the experiment very well both for intact rat papillary muscle and for a decellularized specimen.     

A generic constitutive model for the passive porcine coronary artery

Constitutive models describing the arterial mechanical behavior are important in the development of catheterization products, to be used in arteries with a specific radius. To prove the possible existence of a constitutive model that, provided with a generic set of material and geometric parameters, is able to predict the radius-specific mechanical behavior of a coronary artery, the passive pressure–inner radius (P–r _i ) and pressure–axial force change (P–ΔF _z ) relations of seven porcine left anterior descending coronary arteries were measured in an in-vitro set-up and fitted with the model of Driessen et al. in J Biomech Eng 127(3):494–503 (2005), Biomech Model Mechanobiol 7(2):93–103 (2008). Additionally, the collagen volume fraction, physiological axial pre-stretch, and wall thickness to inner radius ratio at physiological loading were determined for each artery. From this, two generic parameter sets, each comprising four material and three geometric parameters, were obtained. These generic sets were used to compute the deformation of each tested artery using a single radius measurement at physiological loading as an artery-specific input. Artery-specific P–r _i and P–ΔF _z relations were predicted with an accuracy of 32 μm (2.3%) and 6 mN (29% relative to ΔF _z -range) on average compared to the relations measured in-vitro. It was concluded that the constitutive model provided with the generic parameters found in this study can well predict artery-specific mechanical behavior.     

Estimating material parameters of a structurally based constitutive relation for skin mechanics

This paper presents a structurally based modeling framework to characterize the structure–function relation in skin tissues, based upon biaxial tensile experiments performed in vitro on porcine skin. Equi-axial deformations were imposed by stretching circular skin specimens uniformly along twelve directions, and the resultant loads at the membrane attachment points were measured. Displacement fields at each deformation step were tracked using an image 2D cross-correlation technique. A modeling framework was developed to simulate the experiments, whereby measured forces were applied to finite element models that were created to represent the geometry and structure of the tissue samples. Parameters of a structurally based constitutive relation were then identified using nonlinear optimization. Results showed that the ground matrix stiffness ranged from 5 to 32 kPa, fiber orientation mean from 2 to 13^◦ from the torso midline, fiber undulation mean from 1.04 to 1.34 and collagen fiber stiffness from 48 to 366 MPa. It was concluded that the objective function was highly sensitive to the mean orientation and that a priori information about fiber orientation mean was important for the reliable identification of constitutive parameters.