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RESEARCH PAPERS

J Biomech Eng. 1992;114(4):427-434. doi:10.1115/1.2894091.

A theoretical model is presented for the tubular heart of the stage-16 chick embryo (2.3 days of a 21-day incubation period). The model is a thick-walled, pseudoelastic cylindrical shell composed of three isotropic layers: the endocardium, the cardiac jelly, and the myocardium. The analysis is based on a shell theory that accounts for large deformation, material nonlinearity, residual strain, and muscle activation, with material properties inferred from available experimental data. We also measured epicardial strains from recorded motions of microspheres on the primitive right ventricles of stage-16 white Leghorn chick embryos. Relative to end diastole, peak axial and circumferential Lagrange strains occurred near end systole and had similar values. The magnitudes of these strains varied along the longitudinal axis of the heart (-0.16 ± 0.08), being larger near the ends of the primitive right ventricle and smaller near midventricle. The in-plane shear strain was less than 0.05. Comparison of theoretical and experimental strains during the cardiac cycle shows generally good agreement. In addition, the model gives strong stress concentrations in the myocardial layer at end systole.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):435-441. doi:10.1115/1.2894092.

Seven male subjects ran at 3.0 m/s on a motorized treadmill including a force platform under the tread. The subjects ran at each of five treadmill inclinations: + 0.17, +0.077, 0, -0.077, and -0.17 radians. The position of the subjects’ legs were read from ciné films (100 frames/s). Results of the film and force plate analysis generally corroborated the “hanging triangle” hypothesis, which postulates that the angle between the leg and the vertical upon foot strike does not change as the treadmill is tipped up or down. A mathematical model of running, in which the leg is represented as a nonlinear spring, made satisfactory predictions of the way many parameters of running change with the treadmill angle, including the length of the leg at touchdown and liftoff and the peak leg force in the middle of a step. The peak leg force reaches a maximum at a treadmill angle near −0.12 radians, close to the downhill angle where other authors have found a minimum in the rate of oxygen consumption.

Topics: Force , Oxygen , Springs
Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):442-449. doi:10.1115/1.2894093.

The study of lumbar muscle force distribution in response to externally applied loads is based on the introduction of biomechanical models of the lumbar region. The evaluation of such models requires the execution of loading exercises while monitoring the EMG activity of certain lumbar muscles. This work uses muscle activity maps as the major design tool of such exercises, provided that the subject is constrained to an upright erect posture. The maps describe the predicted muscle force for a given combination of externally applied bending moments. A series of shoulder adduction exercises were designed and the EMG signals of eight lumbar muscles were measured while subjects performed the exercises. The results show good agreement between the model predictions and the EMG measurements, especially when the load and the muscle were contralateral to one another.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):450-460. doi:10.1115/1.2894094.

This paper describes a computational method for solving optimal control problems involving large-scale, nonlinear, dynamical systems. Central to the approach is the idea that any optimal control problem can be converted into a standard nonlinear programming problem by parameterizing each control history using a set of nodal points, which then become the variables in the resulting parameter optimization problem. A key feature of the method is that it dispenses with the need to solve the two-point, boundary-value problem derived from the necessary conditions of optimal control theory. Gradient-based methods for solving such problems do not always converge due to computational errors introduced by the highly nonlinear characteristics of the costate variables. Instead, by converting the optimal control problem into a parameter optimization problem, any number of well-developed and proven nonlinear programming algorithms can be used to compute the near-optimal control trajectories. The utility of the parameter optimization approach for solving general optimal control problems for human movement is demonstrated by applying it to a detailed optimal control model for maximum-height human jumping. The validity of the near-optimal control solution is established by comparing it to a solution of the two-point, boundary-value problem derived on the basis of a bang-bang optimal control algorithm. Quantitative comparisons between model and experiment further show that the parameter optimization solution reproduces the major features of a maximum-height, countermovement jump (i.e., trajectories of body-segmental displacements, vertical and fore-aft ground reaction forces, displacement, velocity, and acceleration of the whole-body center of mass, pattern of lower-extremity muscular activity, jump height, and total ground contact time).

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):461-466. doi:10.1115/1.2894095.

We present a new theoretically motivated experimental approach for identifying the functional form of a constitutive relation for any nonlinear, anisotropic pseudoelastic biological membrane. The utility of this approach is illustrated by identifying, from biaxial data, a new constitutive relation for excised ventricular epicardium. Values of the associated material parameters are calculated and compared for right and left ventricular specimens. Based on our findings, we suggest that there are no significant differences in the biomechanical behavior of epicardium excised from the right and left ventricular free walls of canine hearts.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):467-472. doi:10.1115/1.2894096.

The process of freezing in healthy lung tissue and in tumors in the lung during cryosurgery was modeled using one-dimensional close form techniques and finite difference techniques to determine the temperature profiles and the propagation of the freezing interface in the tissue. A thermal phenomenon was observed during freezing of lung tumors embedded in healthy tissue, (a) the freezing interface suddenly accelerates at the transition between the tumor and the healthy lung, (b) the frozen tumor temperature drops to low values once the freezing interface moves into the healthy lung, and (c) the outer boundary temperature has a point of sharp inflection corresponding to the time at which the tumor is completely frozen.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):473-481. doi:10.1115/1.2894097.

Several three-dimensional vascular models have been developed to study the effects of adding equations for large blood vessels to the traditional bioheat transfer equation of Pennes when simulating tissue temperature distributions. These vascular models include “transiting” vessels, “supplying” arteries, and “draining” veins, for all of which the mean temperature of the blood in the vessels is calculated along their lengths. For the supplying arteries this spatially variable temperature is then used as the arterial temperature in the bioheat transfer equation. The different vascular models produce significantly different locations for both the maximum tumor and the maximum normal tissue temperatures for a given power deposition pattern. However, all of the vascular models predict essentially the same cold regions in the same locations in tumors: one set at the tumors’ corners and another around the inlets of the large blood vessels to the tumor. Several different power deposition patterns have been simulated in an attempt to eliminate these cold regions; uniform power in the tumor, annular power in the tumor, preheating of the blood in the vessels while they are traversing the normal tissue, and an “optimal” power pattern which combines the best features of the above approaches. Although the calculations indicate that optimal power deposition patterns (which improve the temperature distributions) exist for all of the vascular models, none of the heating patterns studied eliminated all of the cold regions. Vasodilation in the normal tissue is also simulated to see its effects on the temperature fields. This technique can raise the temperatures around the inlet of the large blood vessles to the tumor (due to the higher power deposition rates possible), but on the other hand, normal tissue vasodilation makes the temperatures at the tumor corners slightly colder.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):482-489. doi:10.1115/1.2894098.

A model-free deconvolution method is proposed for evaluating the frequency distribution function of organ transit times. The deconvolution is treated as a nonlinear constrained optimization problem and it is solved by using a modified constrained variable metric approach. The only constraint implemented in the solution is that all the discrete transport function values are not allowed to become negative. The method is tested on model mathematical systems of known analytical transport functions. The tests are performed on systems that included noise in both the input and output functions. The criteria of successful deconvolution are the reconvolution error and, most importantly, the deviation of the computed transport function from the known analytical one. The proposed method is then applied, as a pilot experiment, to biological data obtained from an isolated, perfused rabbit lung preparation contained within a plethysmograph. The results indicate that this type of deconvolution produces stable estimates which faithfully follow the analytical function while negating the need to assume either any functional form for the behavior of the transport function or any educated initial guess of its values.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):490-496. doi:10.1115/1.2894099.

The behavior of nonlinear pulsatile flow of incompressible blood contained in an elastic tube is examined. The theory takes into account the nonlinear convective terms of the Navier-Stokes equations. The motion of the arterial wall is characterized by a set of linearized differential equations. The region bounded by the flexible arterial wall is mapped into a fixed area in which numerical discretization takes place. The finite element method (Galerkin weighted residual approach) is used for the solution of this nonlinear system. The results obtained are pressure distribution, velocity profile, flow rate and wall displacements along the elastic tube (20 cm long).

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):497-503. doi:10.1115/1.2894101.

In this paper, a numerical simulation of steady laminar and turbulent flow in a twodimensional model for the total artificial heart is presented. A trileaflet polyurethane valve was simulated at the outflow orifice while the inflow orifice had a trileaflet or a flap valve. The finite analytic numerical method was employed to obtain solutions to the governing equations in the Cartesian coordinates. The closure for turbulence model was achieved by employing the k-ε-E model. The SIMPLER algorithm was used to solve the problem in primitive variables. The numerical solutions of the simulated model show that regions of relative stasis and trapped vortices were smaller within the ventricular chamber with the flap valve at the inflow orifice than that with the trileaflet valve. The predicted Reynolds stresses distal to the inflow valve within the ventricular chamber were also found to be smaller with the flap valve than with the trileaflet valve. These results also suggest a correlation between high turbulent stresses and the presence of thrombus in the vicinity of the valves in the total artificial hearts. The computed velocity vectors and turbulent stresses were comparable with previously reported in vitro measurements in artificial heart chambers. Analysis of the numerical solutions suggests that geometries similar to the flap valve (or a tilting disk valve) results in a better flow dynamics within the total artificial heart chamber compared to a trileaflet valve.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):504-511. doi:10.1115/1.2894102.

Three-dimensional flows through canine femoral bifurcation models were predicted under physiological flow conditions by solving numerically the time-dependent threedimensional Navier-stokes equations. In the calculations, two models were assumed for the blood, those of (a) a Newtonian fluid, and (b) a non-Newtonian fluid obeying the power law. The blood vessel wall was assumed to be rigid this being the only approximation to the prediction model. The numerical procedure utilized a finite volume approach on a finite element mesh to discretize the equations, and the code used (ASTEC) incorporated the SIMPLE velocity-pressure algorithm in performing the calculations. The predicted velocity profiles were in good qualitative agreement with the in vivo measurements recently obtained by Jones et al. [1]. The non-Newtonian effects on the bifurcation flow field were also investigated, and no great differences in velocity profiles were observed. This indicated that the non-Newtonian characteristics of the blood might not be an important factor in determining the general flow patterns for these bifurcations, but could have local significance. Current work involves modeling wall distensibility in an empirically valid manner. Predictions accommodating these will permit a true quantitative comparison with experiment.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):512-514. doi:10.1115/1.2894103.

A non-Newtonian constitutive equation for blood has been introduced in this paper. Using this equation, blood flow attributes such as velocity profiles, flowrate, pressure gradient, and wall shear stress in both straight and stenotic (constricted) tubes have been examined. Results showed that compared with Newtonian flow at the same flowrate, the non-Newtonian normally features larger pressure gradient, higher wall shear stress, and different velocity profile, especially in stenotic tube. In addition, the non-Newtonian stenotic flow appears to be more stable than Newtonian flow.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):515-520. doi:10.1115/1.2894104.

Wall shear stress estimates from laminar boundary layer theory were found to agree fairly well with the magnitude of shear stress levels along coronary artery constrictions obtained from solutions of the Navier Stokes equations for both steady and pulsatile flow. The relatively simple method can be used for in vivo estimates of wall shear stress in constrictions by using a vessel shape function determined from a coronary angiogram, along with a knowledge of the flow rate.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):521-526. doi:10.1115/1.2894105.

Fluid dynamic properties of Dacron vascular grafts were studied under controlled steady-flow conditions over a Reynolds number range of 800 to 4500. Knitted and woven grafts having nominal diameters of 6 mm and 10 mm were studied. Thermal anemometry was used to measure centerline velocity at the downstream end of the graft; pressure drop across the graft was also measured. Transition from laminar flow to turbulent flow was observed, and turbulence intensity and turbulent stresses (Reynolds normal stresses) were measured in the turbulent regime. Knitted grafts were found to have greater pressure drop than the woven grafts, and one sample was found to have a critical Reynolds number (Rc ) of less than one-half the value of Rc for a smooth-walled tube.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):527-532. doi:10.1115/1.2894106.

An elastic model of the arterial system has been used in which a specially designed pumping unit simulated the heart action. Physiological pressures and normal geometry, size, and flow distribution together with the normal cardiac output and use of prosthetic heart valves are the features of the system. Atherosclerosis was simulated by introducing blockages of known cross-section at specific sites of predilection. It has been shown that, for some specific occlusion magnitude in the left or right subclavian, or in the brachycephalic arteries, the stagnant no blood flow condition will appear in the left vertebral, or the right vertebral, or right common carotid, or the right internal carotid arteries. For larger occlusions the blood flow in these arteries reverses its direction, i.e., the “steal syndrome” appears. It is shown that besides the known single steal syndrome there exists also a double steal syndrome, i.e., blood reverses its flow direction simultaneously in two arteries, both on the right side of the arterial system. This blood is taken from the circle of Willis, which at the same time is significantly supplemented by the increased blood flow through the other arteries leading into the circle of Willis.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):533-538. doi:10.1115/1.2894107.

Leukocyte plugging of capillaries in vivo was measured in rat spinotrapezius muscle. The plug durations, leukocyte and capillary dimensions, and arteriolar pressure at the plug sites were applied to the mechanical model of Needham and Hochmuth (1990) to estimate the leukocyte viscosities. The viscosity distribution of 389 cells was lognormal with a median value of 232 Poise. 3.1 percent of the cells were apparently activated and displayed viscosities greater than 3000 Poise. The median viscosity suggests that inactivated leukocytes have a minimal effect on blood flow, but that leukocyte activation may result in significant increases in microvascular flow resistance.

Commentary by Dr. Valentin Fuster

ERRATA

TECHNICAL BRIEFS

J Biomech Eng. 1992;114(4):539-542. doi:10.1115/1.2894108.

The microvascular organization and thermal equilibration of the primary and secondary arteries and veins that comprise the bleed off circulation to the muscle fibers from the parent countercurrent supply artery and veins are analyzed. The blood perfusion heat source term in the tissue energy equation is shown to be related to this vascular organization and to undergo a fundamental change in behavior as one proceeds from the more peripheral tissue, where the perfusion term is proportional to the Ta - Tv difference in the parent supply vessels, to the deeper tissue layers where the bleed off vessels themselves form a branching countercurrent system for each muscle tissue cylinder and the venous return temperature can vary between the local tissue temperature and Ta . The consequences of this change in behavior are examined for the Weinbaum-Jiji bioheat equation and a modified expression for the effective conductivity of perfused tissue is derived for countercurrent bleed off exchange.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):542-546. doi:10.1115/1.2894109.

The Distribution-Moment Model of skeletal muscle, which has been enhanced recently to make possible the calculation of chemical energy release (Ė) and heat production (Ḣ) rates [1], is applied to isometric muscle. Under steady-state isometric conditions the model predicts a simple relation between the energy rates and the muscle length, namely

(Ė/Ėmax) = (Ḣ/Ḣmax) = [1 + Bα(Λ)]/[1 + B]
, where Λ is the ratio of muscle length to the “optimal” length at which maximal isometric tension is produced, and α(Λ) is a function numerically equal to the ratio of the tetanic isometric force to its maximum value. The single dimensionless constant in this relation, B, can be calculated from model parameters characterizing muscle dynamics at the optimum length, and has a value near unity for frog sartorius at 0°C. The predicted behavior is shown to agree reasonably well with experimental measurements of heat production and phosphocreatine (PCr) hydrolysis. The model relates the isometric energy rates to PCr hydrolysis in (1) cross-bridge interactions, and (2) calcium pumping into the sarcoplasmic reticulum.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):546-549. doi:10.1115/1.2894110.

The motion of two rigid circular cylinders in contact immersed in an incompressible Newtonian fluid in a channel is examined numerically in the zero Reynolds number limit, for various values of the cylinder radius/channel width ratio. Analyses of the time courses of the lateral position and the orientation of the doublet showed that, depending on the initial condition and the doublet/channel size ratio, the doublet exhibit one of the three types of motion: a continuous rotation in the same direction during a period, and a rotation changing its direction at every half period with a large or a small variation of the orientation.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1992;114(4):549-552. doi:10.1115/1.2894111.

A finite element model that simulates indentation of the cornea was generated to examine the feasibility of using indentation data to determine mechanical properties. A layered model which included geometric nonlinearities was necessary to characterize the indentation process. Results indicate that a secant modulus could be determined by measuring indenter force and contact area.

Commentary by Dr. Valentin Fuster

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