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RESEARCH PAPERS

J Biomech Eng. 1991;113(1):1-10. doi:10.1115/1.2894079.

A prototype mathematical model for Brown and Goldstein’s pioneering studies on the LDL receptor mediated pathway for the regulation of the cellular content of cholesterol has been developed in this paper. In order to analyze the essential features of this complex system quantitatively and still reflect the framework of the total system, six important processes are considered in the model. They are: (1A, B) the hydrolysis and synthesis of the LDL receptor; (2) the binding of LDL to its receptors; (3) the hydrolysis of LDL; (4) the storage of cholesteryl esters; (5) the regulation of de novo synthesis of cholesterol; and (6) the efflux of free cholesterol to the external medium. All these processes form a system to let the cells take up enough cholesterol from the external medium for their utilization and yet avoid the excessive accumulation of the lipid within the cells. The validity of the model is tested by showing that it can predict many of experimental curves obtained for human fibroblasts in tissue culture studies. The main purpose of the model is to determine how the free cholesterol level in the cell is related to the external LDL concentration and the regulatory capacity of the cells to adapt to a changing LDL environment. In addition, the model reveals an important behavior of SMC, i.e., for a slowly increasing LDL concentration in the extracellular medium, the rate of intracellular degradation of LDL will first increase and then become saturated. It is proposed based on these results that the saturation of LDL degradation by SMCs and the subsequent increase in subendothelial LDL levels in regions of high macromolecular permeability might play a vital role in the formation of the early foam cell lesion.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):11-20. doi:10.1115/1.2894078.

To obtain more efficient operation of a COBE Model 2997 clinical cell separator using either a Single Stage II (SS II) or a Dual Stage separation chamber, modifications were made to allow complete computer control. Product cell density was detected using an optical sensor and controlled by automatic feedback through a microcomputer interface. Control was accomplished by automatically adjusting the red blood cell (RBC) and plasma product flow rates using a proportional-integral (PI) algorithm. Results show that, using either chamber, the product cell density can be maintained at a preselected value for extended periods of time without operator intervention. This system allowed investigation of optimal operating regions for plateletpheresis and leukapheresis procedures. The effects of centrifuge rpm and controller set point on centrifuge operation were investigated using a second order factorial experimental design. Theoretical significance of model parameters was assessed with the aid of a hindered settling model and simple reasoning about the interface position relative to the collection port. The results suggest that, in either chamber, the optimum operating region for plateletpheresis procedures occurs at moderate controller set points and high centrifuge rpm. The resultant operating efficiency and product purity values are approximately 63 percent and 0.65 respectively in the SS II chamber and approximately 70 percent and 0.70 respectively in the Dual Chamber. In the SS II, the optimum operating region for leukapheresis procedures occurred at high controller set point values for any centrifuge rpm above 1200 with an operating efficiency near 100 percent. However, in the Dual Chamber, the optimum operating region for leukapheresis procedures occurred at high controller set points and high centrifuge rpm’s, again providing an operating efficiency near 100 percent.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):21-26. doi:10.1115/1.2894080.

Using an axisymmetric geometry that retains certain qualitative features of the trachea, we extend one-dimensional modeling of flow in collapsible tubes to include both curved shell effects and, for untethered tubes, wall inertia. A systematic scaling of the finite deformation membrane equations leads to an approximate set which is consistent with the one-dimensional fluid model; axial and normal wall variables are coupled elastically, but only axial inertia is retained. Transverse curvature causes elastic coupling that can give rise to axial wall motion and a flutter instability. The source of instability is the product of a nonzero reference axial curvature with axial tension variation due to axial stretching. The numerical results suggest that this mechanism may be significant even in processes which cannot be assumed one-dimensional.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):27-29. doi:10.1115/1.2894081.

In the present paper pressure changes induced by sudden body acceleration are studied “in vivo” on the dog and compared to the results obtainable with a recently developed mathematical model. A dog was fixed to a movable table, which was accelerated by a compressed air piston for less than 1 s. Acceleration was varied by changing the air pressure in the piston. Pressure was measured during the experiment at different points along the vascular bed. However, only data obtained in the carotid artery and abdominal aorta are presented here. The results demonstrated that impulse body accelerations cause significant pressure peaks in the vessel examined (about + 25 mmHg in the carotid artery with body acceleration of g/2). Moreover, pressure changes are rapidly damped, with a time constant of about 0.1s. From the present results it may be concluded that, according to the prediction of the mathematical model, body accelerations such as those occurring in normal life can induce pressure changes well beyond the normal pressure value.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):30-35. doi:10.1115/1.2894082.

The effects of the stratum corneum and dermis on shear wave propagation along the skin surface was investigated using a mathematical model. The skin was modeled as two distinct viscoelastic layers, one representing the stratum corneum and the other representing the dermis. The layers were supported by a semi-infinite viscoelastic half-space representing the subcutaneous fat. Physical and mechanical properties of the materials in the model were determined from the literature and from our own experimental measurements. Although the stratum corneum is very thin (12-15 microns), results showed that it could have a strong effect on the wave propagation due to its high stiffness relative to the dermis. Results of the analysis are discussed with respect to an experimental procedure used to determine age-related changes in mechanical properties of skin.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):36-41. doi:10.1115/1.2894083.

The zero-stress state of a vein is, like that of an artery, not a closed cylindrical tube, but is a series of segments whose cross-sections are open sectors. An opening angle of each sector is defined as the angle subtended between two radii joining the midpoint of the inner wall to the tips of the inner wall. Data on the opening angles (mean ± standard deviation) of the veins and vena cava of the rat are presented. For the superior vena cava and subclavian, jugular, facial, renal, common iliac, saphenous, and plantar veins, the opening angle varies in the range of 25 to 75 deg. The inferior vena cava (below the heart), however, has noncircular, nonaxisymmetric cross-sections, a curved axis, and a rapid longitudinal variation of its “diameter;” its zero-stress state is not circular sectors; but the opening angle is still a useful characterization. The mean opening angle of the interior vena cava varies in the range of 40 to 150 deg in the thoracic portion, and 75 to 130 deg in the abdominal portion, with the larger values occurring about the middle of each portion. There are considerable length, diameter reductions, and wall thickening of the vena cava from the homeostatic state to the no-load state in vitro. Physically, the zero-stress state is the basis of the stress analysis of blood vessels. The change of opening angle is a convenient parameter to characterize any nonuniform remodeling of the vessel wall due to changes in physical stress or chemical environment. Change of zerostress state influences the compliance and collapsibility of the viens, their pressure-volume and pressure-flow relationships, the waterfall phenomenon, and the tone in the vascular smooth muscles if the homeostatic stress is changed.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):42-55. doi:10.1115/1.2894084.

The equatorial region of the canine left ventricle was modeled as a thick-walled cylinder consisting of an incompressible hyperelastic material with homogeneous exponential properties. The anisotropic properties of the passive myocardium were assumed to be locally transversely isotropic with respect to a fiber axis whose orientation varied linearly across the wall. Simultaneous inflation, extension, and torsion were applied to the cylinder to produce epicardial strains that were measured previously in the potassium-arrested dog heart. Residual stress in the unloaded state was included by considering the stress-free configuration to be a warped cylindrical arc. In the special case of isotropic material properties, torsion and residual stress both significantly reduced the high circumferential stress peaks predicted at the endocardium by previous models. However, a resultant axial force and moment were necessary to cause the observed epicardial deformations. Therefore, the anisotropic material parameters were found that minimized these resultants and allowed the prescribed displacements to occur subject to the known ventricular pressure loads. The global minimum solution of this parameter optimization problem indicated that the stiffness of passive myocardium (defined for a 20 percent equibiaxial extension) would be 2.4 to 6.6 times greater in the fiber direction than in the transverse plane for a broad range of assumed fiber angle distributions and residual stresses. This agrees with the results of biaxial tissue testing. The predicted transmural distributions of fiber stress were relatively flat with slight peaks in the subepicardium, and the fiber strain profiles agreed closely with experimentally observed sarcomere length distributions. The results indicate that torsion, residual stress and material anisotropy associated with the fiber architecture all can act to reduce endocardial stress gradients in the passive left ventricle.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):56-62. doi:10.1115/1.2894085.

This paper presents a theory for studies of the large-strain behavior of biological shells composed of layers of incompressible, orthotropic tissue, possibly muscle, of arbitrary orientation. The intrinsic equations of the laminated-shell theory, expressed in lines-of-curvature coordinates, account for large membrane [O(1)] and moderately large bending and transverse shear strains [O(0.3)], nonlinear material properties, and transverse normal stress and strain. An expansion is derived for a general two-dimensional strain-energy density function, which includes residual stress and muscle activation through a shifting zero-stress configuration. Strain-displacement relations are given for the special case of axisymmetric deformation of shells of revolution with torsion.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):63-71. doi:10.1115/1.2894086.

This paper specializes the nonlinear laminated-muscle-shell theory developed in Part I to cylindrical geometry and computes stresses in arteries and the beating left ventricle. The theory accounts for large strain, material nonlinearity, thick-shell effects, torsion, muscle activation, and residual strain. First, comparison with elasticity solutions for pressurized arteries shows that the accuracy of the shell theory increases as transmural stress gradients and the shell thickness decrease. Residual strain reduces the stress gradients, lowering the error in the predicted peak stress in thick-walled arteries (R/t = 2.8) from about 30 to 10 percent. Second, the canine left ventricle is modeled as a thick-walled laminated cylinder with an internal pressure. Each layer is composed of transversely isotropic muscle with a fiber orientation based on anatomical data. Using a single pseudostrain-energy density function (with time-varying coefficients) for passive and active myocardium, the model predicts strain distributions that agree fairly well with published experimental measurements. The results also show that the peak fiber stress occurs subendocardially near the beginning of ejection and that residual strains significantly alter stress gradients within each lamina, but the magnitude of the peak fiber stress changes by less than 20 percent.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):72-78. doi:10.1115/1.2894087.

There is concern among researchers whether the passive muscle properties, characterized by purely passive material testing procedures, are an appropriate representation of the actual passive component of the muscle. This aspect is of particular importance in the biomechanical analysis of heart muscle response where it is generally agreed that the so-called parallel elasticity cannot be ignored as is done justifiably in the analysis of skeletal muscle response. In the present article, a method of quantifying the passive elasticity in contracting muscle bundles is presented. The method consists of imposing isometric transients (such as the quick-stretch or quick release) on a muscle bundle during the contraction phase and observing the differences in decayed force levels between a normal twitch and that of a perturbed twitch. The proposed method provides a means of obtaining useful passive properties from contracting muscle bundles and circumvents the difficulty of having to characterize muscle properties from separate experiments on quiescent muscle bundles.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):79-84. doi:10.1115/1.2894088.

A continuum model is presented that relates the trunk parameters of loading, geometry, and muscle structure to the necessary conditions of static equilibrium. Linear theory for stress-strain behavior is used to describe an elephant trunk for an incremental displacement as the animal slowly lifts a weight at the trunk tip. With this analysis and experimental values for the trunk parameters, the apparent trunk stiffness Ea is estimated for the living animal. For an Asian elephant with a maximum compression strain of 33 percent, Ea is of the order of 106 N/m2 . The continuum model is quite general and may be applied to similar nonskeletal appendages and bodies of other animals.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):85-93. doi:10.1115/1.2894089.

Two intrinsic (scapholunate and lunotriquetral) and two extrinsic (radiolunate and radiocapitate) wrist ligaments were studied at high and low elongation rates (1 and 100 mm/min). Statistically significant differences among all four ligaments were noted for the viscoelastic and elastic components of stress versus strain for the fully recoverable strain and early permanent deformation stress for all ligaments. Intrinsic ligaments became permanently deformed at statistically significantly higher strain levels than the extrinsic ligaments and accept larger permanent deformation at strain levels below evident fiber failure. Ultimate strength data demonstrated significant rate dependency for stress and strain for all ligaments. Intrinsic ligaments failed statistically greater stress and strain levels than the extrinsic group. Some clinical implications of these findings are discussed.

Commentary by Dr. Valentin Fuster
J Biomech Eng. 1991;113(1):94-103. doi:10.1115/1.2894090.

On the basis of earlier reported data on the in vitro kinematics of passive knee-joint motions of four knee specimens, the length changes of ligament fiber bundles were determined by using the points of insertion on the tibia and femur. The kinematic data and the insertions of the ligaments were obtained by using Roentgenstereophotogrammetry. Different fiber bundles of the anterior and posterior cruciate ligaments and the medial and lateral collateral ligaments were identified. On the basis of an assumption for the maximal strain of each ligament fiber bundle during the experiments, the minimal recruitment length and the probability of recruitment were defined and determined. The motions covered the range from extension to 95 degrees flexion and the loading conditions included internal or external moments of 3 Nm and anterior or posterior forces of 30N. The ligament length and recruitment patterns were found to be consistent for some ligament bundles and less consistent for other ligament bundles. The most posterior bundle of each ligament was recruited in extension and the lower flexion angles, whereas the anterior bundle was recruited for the higher flexion angles. External rotation generally recruited the collateral ligaments, while internal rotation recruited the cruciate ligaments. However, the anterior bundle of the posterior cruciate ligament was recruited with external rotation at the higher flexion angles. At the lower flexion angles, the anterior cruciate and the lateral collateral ligaments were recruited with an anterior force. The recruitment of the posterior cruciate ligament with a posterior force showed that neither its most anterior nor its most posterior bundle was recruited at the lower flexion angles. Hence, the posterior restraint must have been provided by the intermediate fiber bundles, which were not considered in the experiment. At the higher flexion angles, the anterior bundles of the anterior cruciate ligament and the posterior cruciate ligament were found to be recruited with anterior and posterior forces, respectively. The minimal recruitment length and the recruitment probability of ligament fiber bundles are useful parameters for the evaluation of ligament length changes in those experiments where no other method can be used to determine the zero strain lengths, ligament strains and tensions.

Commentary by Dr. Valentin Fuster

ERRATA

TECHNICAL BRIEFS

J Biomech Eng. 1991;113(1):104-107. doi:10.1115/1.2894075.

Small interalveolar holes within the lung are called pores of Kohn. Some researchers have correlated enlarged pore size with diseases, e.g. emphysema, that are characterized by tissue destruction. Mathematical models of the pressures generated in closed, fluid-filled and open, fluid-lined pores demonstrate that pressures capable of rupturing lung tissue can be developed in a pore due to the surface tension and shape of the air-liquid interface. Pore enlargement accompanied by tissue destruction is presented as a possible mechanism for the disease process observed during aging and the development of emphysema in the lung.

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster

BOOK REVIEWS

J Biomech Eng. 1991;113(1):109. doi:10.1115/1.2894077.
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Abstract
Topics: Fluids
Commentary by Dr. Valentin Fuster

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