Alan D. Freed

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.

Detailed Error Analysis for a Fractional Adams Method

We investigate a method for the numerical solution of the nonlinear fractional differential equation D*αy(t)=f(t,y(t)), equipped with initial conditions y(k)(0)=y0(k), k=0,1,...,⌈α⌉−1. Here α may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardsons extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.

On the Solution of Nonlinear Fractional-Order Differential Equations Used in the Modeling of Viscoplasticity

The authors have recently developed a mathematical model for the description of the behavior of viscoplastic materials. The model is based on a nonlinear differential equation of order β, where β is a material constant typically in the range 0 < β < 1. This equation is coupled with a first-order differential equation. In the present paper, we introduce and discuss a numerical scheme for the numerical solution of these equations. The algorithm is based on a PECE-type approach.

Maturational and Adaptive Modulation of Left Ventricular Torsional Biomechanics. Doppler Tissue Imaging Observation From Infancy to Adulthood

Background— Left ventricular (LV) torsional deformation, based in part on the helical myocardial fiber architecture, is an important component of LV systolic and diastolic performance. However, there is no comprehensive study describing its normal development during childhood and adult life. Methods and Results— Forty-five normal subjects (25 children and 20 adults; aged 9 days to 49 years; divided into 5 groups: infants, children, adolescents, and young and middle-age adults) underwent assessment of LV torsion and untwisting rate by Doppler tissue imaging. LV torsion increased with age, primarily owing to augmentation in basal clockwise rotation during childhood and apical counterclockwise rotation during adulthood. Although LV torsion and untwisting overall showed age-related increases, when normalized by LV length, they showed higher values in infancy and middle age. The proportion of untwisting during isovolumic relaxation was lowest in infancy, increased during childhood, and leveled off thereafter, whereas peak untwisting performance (peak untwisting velocity normalized by peak LV torsion) showed a decrease during adulthood. Conclusions— We have shown the maturational process of LV torsion in normal subjects. Net LV torsion increases gradually from infancy to adulthood, but the determinants of this were different in the 2 age groups. The smaller LV isovolumic untwisting recoil during infancy and its decline in adulthood may suggest mechanisms for alterations in diastolic function.

Fractional order viscoelasticity of the aortic valve cusp: an alternative to quasilinear viscoelasticity.

BACKGROUNDnQuasilinear viscoelasticity (QLV) theory has been widely and successfully used to describe the time-dependent response of connective tissues. Difficulties remain, however, particularly in material parameter estimation and sensitivities. In this study, we introduce a new alternative: the fractional order viscoelasticity (FOV) theory, which uses a fractional order integral to describe the relaxation response. FOV implies a fractal-like tissue structure, reflecting the hierarchical arrangement of collagenous tissues.nnnMETHOD OF APPROACHnA one-dimensional (I-D) FOV reduced relaxation function was developed, replacing the QLV box-spectrum function with a fractional relaxation function. A direct-fit, global optimization method was used to estimate material parameters from stress relaxation tests on aortic valve tissue.nnnRESULTSnWe found that for the aortic heart valve, FOV had similar accuracy and better parameter sensitivity than QLV, particularly for the long time constant (tau2). The mean (n = 5) fractional order was 0.29, indicating that the viscoelastic response of the tissue was strongly fractal-like. RESULTS SUMMARY: mean QLV parameters were C = 0.079, tau1 = 0.004, tau2 = 76, and mean FOV parameters were beta = 0.29, tau = 0.076, and rho = 1.84.nnnCONCLUSIONSnFOV can provide valuable new insights into tissue viscoelastic behavior Determining the fractional order can provide a new and sensitive quantitative measure for tissue comparison.

Inverse parameter fitting of biological tissues: a response surface approach.

In this paper, we present the application of a semi-global inverse method for determining material parameters of biological tissues. The approach is based on the successive response surface method, and is illustrated by fitting constitutive parameters to two nonlinear anisotropic constitutive equations, one for aortic sinus and aortic wall, the other for aortic valve tissue. Material test data for the aortic sinus consisted of two independent orthogonal uniaxial tests. Material test data for the aortic valve was obtained from a dynamic inflation test. In each case, a numerical simulation of the experiment was performed and predictions were compared to the real data. For the uniaxial test simulation, the experimental targets were force at a measured displacement. For the inflation test, the experimental targets were the three-dimensional coordinates of material markers at a given pressure. For both sets of tissues, predictions with converged parameters showed excellent agreement with the data, and we found that the method was able to consistently identify model parameters. We believe the method will find wide application in biomedical material characterization and in diagnostic imaging.

An efficient algorithm for the evaluation of convolution integrals

We propose an algorithm for the numerical evaluation of convolution integrals of the form @!0^xk(x-y)f(y,x) dy, for x@?[0,X]. Our method is especially suitable in situations where the fundamental interval [0, X] is very long and the kernel function k is expensive to calculate. Separate versions are provided where the forcing function f is known in advance, and where it must be determined step-by-step along the solution path. These methods are efficient with respect to both run time and memory requirements.

Viscoelastic Model for Lung Parenchyma for Multi-Scale Modeling of Respiratory System, Phase II: Dodecahedral Micro-Model

In the first year of this contractual effort a hypo-elastic constitutive model was developed and shown to have great potential in modeling the elastic response of parenchyma. This model resides at the macroscopic level of the continuum. In this, the second year of our support, an isotropic dodecahedron is employed as an alveolar model. This is a microscopic model for parenchyma. A hopeful outcome is that the linkage between these two scales of modeling will be a source of insight and inspiration that will aid us in the final years activity: creating a viscoelastic model for parenchyma.

Invariant Formulation for Dispersed Transverse Isotropy in Tissues of the Aortic Outflow Tract

We have proposed an efficient, invariant-based alternative to structural constitutive equations that accounts for statistical dispersion of fibers. In contrast to existing models, our new invariant theory easily handles a 3D fiber population with a single mean preferred direction. The invariant theory is based on a novel closed-form ‘splay invariant’ that requires a single parameter in the 2D case, and two parameters in the 3D case. The proposed model is polyconvex, and fits biaxial data for aortic valve tissue better than existing aortic-valve models Billiar and Sacks (2000). A modification in the fiber stress-strain law requires no re-formulation of the constitutive tangent matrix, making the model flexible for different types of soft tissues. Most importantly, the model is computationally expedient in a finite element analysis.

Viscoelastic Model for Lung Parenchyma for Multi-Scale Modeling of Respiratory System Phase I: Hypo-Elastic Model for CFD Implementation

An isotropic constitutive model for the parenchyma of lung has been derived from the theory of hypo-elasticity. The intent is to use it to represent the mechanical response of this soft tissue in sophisticated, computational, fluid-dynamic models of the lung. This demands that the continuum model be accurate, yet simple and effcient. An objective algorithm for its numeric integration is provided. The response of the model is determined for several boundary-value problems whose experiments are used for material characterization. The effective elastic, bulk, and shear moduli, and Poisson’s ratio, as tangent functions, are also derived. The model is characterized against published experimental data for lung. A bridge between this continuum model and a dodecahedral model of alveolar geometry is investigated, with preliminary findings being reported.