On a nonlinear theory for muscle shells: Part II--Application to the beating left ventricle

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.     

Theoretical models in mechanics of the left ventricle

Different rheological concepts and theoretical studies have been recently presented using models of myocardial mechanics. Complex analysis of the mechanical behavior of the left ventricular wall have been developed in order to estimate the local stresses and deformations that occur during the heart cycle as well as the ventricular stroke volume and pressure. Theoretical models have taken into account non-linear and viscoelastic passive properties of the myocardium tissue, when subjected to large deformations, through given strain energy functions or stress-strain relations. Different prolate spheroid geometries have been considered for such thick shell cardiac structure. During the active state of the contraction, the rheological behavior of the fibers has been described using different muscle models and relationships between fiber tension and strain, and activation degree. A forthcoming approach for bridging the gap between the knowledge of the muscle fiber microrheological properties and the study of the mechanical behavior of the entire ventricle, consists in including anisotropic and inhomogeneous effects through fiber direction field.     

Relation between left ventricular cavity pressure and volume and systolic fiber stress and strain in the wall.

Pumping power as delivered by the heart is generated by the cells in the myocardial wall. In the present model study global left-ventricular pump function as expressed in terms of cavity pressure and volume is related to local wall tissue function as expressed in terms of myocardial fiber stress and strain. On the basis of earlier studies in our laboratory, it may be concluded that in the normal left ventricle muscle fiber stress and strain are homogeneously distributed. So, fiber stress and strain may be approximated by single values, being valid for the whole wall. When assuming rotational symmetry and homogeneity of mechanical load in the wall, the dimensionless ratio of muscle fiber stress (sigma f) to left-ventricular pressure (Plv) appears to depend mainly on the dimensionless ratio of cavity volume (Vlv) to wall volume (Vw) and is quite independent of other geometric parameters. A good (+/- 10%) and simple approximation of this relation is sigma f/Plv = 1 + 3 Vlv/Vw. Natural fiber strain is defined by ef = In (lf/lf,ref), where lf,ref indicates fiber length (lf) in a reference situation. Using the principle of conservation of energy for a change in ef, it holds delta ef = (1/3)delta In (1 + 3Vlv/Vw).     

A finite element simulation scheme for biological muscular hydrostats

An explicit finite element scheme is developed for biological muscular hydrostats such as squid tentacles, octopus arms and elephant trunks. The scheme is implemented by embedding muscle fibers in finite elements. In any given element, the fiber orientation can be assigned arbitrarily and multiple muscle directions can be simulated. The mechanical stress in each muscle fiber is the sum of active and passive parts. The active stress is taken to be a function of activation state, muscle fiber shortening velocity and fiber strain; while the passive stress depends only on the strain. This scheme is tested by simulating extension of a squid tentacle during prey capture; our numerical predictions are in close correspondence with existing experimental results. It is shown that the present finite element scheme can successfully simulate more complex behaviors such as torsion of a squid tentacle and the bending behavior of octopus arms or elephant trunks.     

The Evolution and Development of Cephalopod Chambers and Their Shape

The Ammonoidea is a group of extinct cephalopods ideal to study evolution through deep time. The evolution of the planispiral shell and complexly folded septa in ammonoids has been thought to have increased the functional surface area of the chambers permitting enhanced metabolic functions such as: chamber emptying, rate of mineralization and increased growth rates throughout ontogeny. Using nano-computed tomography and synchrotron radiation based micro-computed tomography, we present the first study of ontogenetic changes in surface area to volume ratios in the phragmocone chambers of several phylogenetically distant ammonoids and extant cephalopods. Contrary to the initial hypothesis, ammonoids do not possess a persistently high relative chamber surface area. Instead, the functional surface area of the chambers is higher in earliest ontogeny when compared to Spirula spirula. The higher the functional surface area the quicker the potential emptying rate of the chamber; quicker chamber emptying rates would theoretically permit faster growth. This is supported by the persistently higher siphuncular surface area to chamber volume ratio we collected for the ammonite Amauroceras sp. compared to either S. spirula or nautilids. We demonstrate that the curvature of the surface of the chamber increases with greater septal complexity increasing the potential refilling rates. We further show a unique relationship between ammonoid chamber shape and size that does not exist in S. spirula or nautilids. This view of chamber function also has implications for the evolution of the internal shell of coleoids, relating this event to the decoupling of soft-body growth and shell growth.     

The dependence of chamber dynamics on chamber dimensions

One can learn something about the determinants of ventricular dimensions and dynamics from a simple spherical model. We have derived equations showing how isometric pressure, compliance, isometric P-V curves and viscous resistance to wall displacement depend on dimensions of a spherical chamber whose fibers adjust for a "normal" stretch at a particular point in the pump cycle. The derivations show: (a) that isometric pressure at this point is proportional to the logarithm of total chamber volume (cavity plus wall) relative to cavity volume; (b) that compliance at this point is proportional to cavity volume and to total chamber volume relative to wall volume; (c) that the rate of wall displacement relative to the disparity between isometric pressure and actual pressure depends on dimensions like compliance depends on dimensions; and (d) since reciprocal compliance does not increase with wall/cavity ratios as much as isometric pressure at the normal-stretch volume, the P-V curves spread out on either side of the normal-stretch volume as the chamber undergoes adaptive thickening, resulting in disproportionate increases of isometric pressure at low cavity volumes. This tends to increase ejection fraction and reduce cavity volumes relative to stroke volume, and it is partly responsible for the "concentric" character of hypertrophy in response to high systolic pressure.     

Theoretical analysis of the effects of a radial activation wave and twisting motion on the mechanics of the left ventricle

The mechanical effects resulting from the normal transmural delay of electrical depolarization of the myocardium are investigated. An activation sequence having a finite radial propagation velocity is introduced into the equations of ventricular mechanics. The resulting system of coupled integral equations is solved using a perturbation method based on the small ratio of transmural propagation time to cardiac period. Numerical calculations are performed using cavity pressure and volume waveforms characteristic of the canine left ventricle (LV), for both simultaneous and delayed activation of fiber layers. The results show that a finite transmural electrical propagation velocity tends to: (i) equalize the transmural distribution of sarcomere length during systole; (ii) equalize the transmural distribution of fiber external work/vol; and (iii) insignificantly affect myocardial tissue pressure. Calculations are also performed to investigate the mechanical effects resulting from the application of an externally applied moment that prevents LV torsion. Those results are highly dependent on the transmural distribution of sarcomere length in the stress-free reference state (unloaded diastole). When we assume a uniform distribution, then normal torsion acting with normal activation delay tends to: (i) increase the magnitude of fiber strain in the subendocardium and decrease it in the subepicardium; (ii) equalize the transmural distribution of fiber external work/vol; and (iii) lower myocardial tissue pressure. The normally occurring transmural delay of activation tends to lessen endocardial O2 demand, while the normally occurring torsion further lessens that demand and improves O2 supply.     

Dynamic left ventricular elastance: a model for integrating cardiac muscle contraction into ventricular pressure-volume relationships.

To integrate myocardial contractile processes into left ventricular (LV) function, a mathematical model was built. Muscle fiber force was set equal to the product of stiffness and elastic distortion of stiffness elements, i.e., force-bearing cross bridges (XB). Stiffness dynamics arose from recruitment of XB according to the kinetics of myofilament activation and fiber-length changes. Elastic distortion dynamics arose from XB cycling and the rate-of-change of fiber length. Muscle fiber stiffness and distortion dynamics were transformed into LV chamber elastance and volumetric distortion dynamics. LV pressure equaled the product of chamber elastance and volumetric distortion, just as muscle-fiber force equaled the product of muscle-fiber stiffness and lineal elastic distortion. Model validation was in terms of its ability to reproduce cycle-time-dependent LV pressure response, DeltaP(t), to incremental step-like volume changes, DeltaV, in the isolated rat heart. All DeltaP(t), regardless of the time in the cycle at which DeltaP(t) was elicited, consisted of three phases: phase 1, concurrent with the leading edge of DeltaV; phase 2, a brief transient recovery from phase 1; and phase 3, sustained for the duration of systole. Each phase varied with the time in the cycle at which DeltaP(t) was elicited. When the model was fit to the data, cooperative activation was required to sustain systole for longer periods than was possible with Ca(2+) activation alone. The model successfully reproduced all major features of the measured DeltaP(t) responses, and thus serves as a credible indicator of the role of underlying contractile processes in LV function.     

空气流动增加压力型整体体积描记箱内压力变化的空气动力学研究

动物实验中,计算小动物呼吸容量如潮气量时通常遵循Boyle定律。而小动物在压力型体积描记箱内呼吸时,箱内气体处于不停的流动状态。为了研究气体流动本身对描记箱内压力是否产生影响,本研究采用空气动力学理论推导出压力型体积描记箱内压力变化的近似公式,并设计了三个实验对理论公式进行证明。首先,往体积描记箱内注入0.1mL、0.2mL和0.4mL空气,结果显示,描记箱内压力迅速上升至一峰值然后下降为一水平的基线压力值。3种容量的空气注入产生的压力变化最大值(峰压)及压力保持平稳时的压力变化值(基线压)分别为:(0.828±0.004)cmH2O、(0.684±0.003)cmH2O;(1.650±0.010)cmH2O、(1.350±0.007)cmH2O;(3.280±0.014)cmH2O、(2.658±0.011)cmH2O,峰压均显著高于基线压。其次,往体积描记箱内注入相同容量、但通气频率不同的空气,结果显示,随着注入空气的频率增加,描记箱内压力变化幅度亦显著增加。最后,用小动物呼吸机和微量移液器分别往描记箱内注入相同容量、相同频率但流量-时间函数不同的气体,结果显示,呼吸机引起的体积描记箱内峰压及基线压均显著高于微量移液器。以上结果表明,空气流动本身在压力型体积描记箱内引起压力变化;空气流动越快,压力变化幅度也越大;流动空气的流量是时间的不同函数时,压力变化也将显著不同。因此要精确地计算小动物呼吸容量变化,需要使用更多的空气动力学原理。     

A model for predicting growth responses in plants to changes in external water potential: Zea mays primary roots

In response to osmotic step changes, three distinct phases have been noted in the growth response of Zea mays primary roots. They are cessation or slowing of growth over a period of 15–20 minutes, tissue contraction, and a damped oscillatory return to nearly normal growth rate, all within a period of about one hour. A system model of the tissue response is presented to explain such behavior and to serve in a predictive capacity to govern future experiments.It is supposed that for turgor pressure in excess of a cell wall yield threshold, plastic flow is the major component of wall deformation, and that when turgor falls below yield threshold, elastic deformation is dominant. The equations of the model describe growth rate as a function of time in terms of the following properties; plastic flow, elastic deformation, permeability to water, and solute uptake. They are derived from basic equations of feedback interactions between internal osmotic pressure and growth rate, and between wall softening, turgor and growth rate.The model predicts oscillatory growth rate regulation, and phase and amplitude relationships between turgor pressure and growth rate. The simplest model which accounts for all observations is that of biphasic deformation, two modes of wall softening, and a dual feedback system involving osmotic and yield threshold control of growth rate.It should be noted that to predict the time course of turgor pressure, osmotic pressure, yield pressure, and growth rate, two initial conditions and six system parameter values are sufficient. So far only the initial values of growth rate and its derivative can be obtained for Zea mays primary roots. However, values for wall softening and hardening coefficients (including the strain and turgor independent component), plastic extensibility, water permeability and dilution rate coefficients have not been obtained as yet for Zea roots. Values for some of these parameters have been obtained for other roots, coleoptiles, and giant algal cells.Lest the reader despair, it should be pointed out that experimental observations coupled with simulation studies will help establish restricted ranges of values that the system parameters might assume. These can then be compared with known values in the literature and values experimentally obtained in the future.     

Elastomechanical characterization of brain tissues.

The fluid-induced changes in the intracranial pressure which have important clinical implications are believed to be largely determined by the elastomechanical properties of the brain tissues. To define and evaluate the elastomechanical characteristics of the brain tissues a nonlinear hyperelastic hollow spherical shell has been employed to model the craniospinal complex for its fluid-induced intracranial pressure volume changes. The strain energy function proposed by Hart-Smith has been used to derive the constitutive equations. In 10 dogs, fluid has been infused in the lateral ventricle of the brain. The resulting changes in the ventricular fluid pressure (VFP) and the epidural pressure (EDP) have been recorded. The plot of pressure as a function of volume increases first, reaches a maximum, decreases, reaches a minimum and increases monotonously. The values of maximum and minimum pressures (pv max and pv min) due to fluid infusion are found to be, respectively, 42.4 +/- 15.4 mmHg and 33.1 +/- 12.2 mmHg. The pressure achieved the maximum and minimum values with infusion of 0.19 +/- 0.09 ml and 0.51 +/- 0.15 ml of fluid, respectively. The elastomechanical parameters of the Hart-Smith function that characterize the brain tissues have been evaluated by matching the experimentally obtained pressure-volume curves with the corresponding model generated curves. It is found that the agreement between the experimentally obtained pressure-volume curves and the corresponding Hart-Smith profile is satisfactory at a high inflation level but less so at the lower inflation level.     

Optimization of microporous palm shell activated carbon production for flue gas desulphurization: experimental and statistical studies

Optimizing the production of microporous activated carbon from waste palm shell was done by applying experimental design methodology. The product, palm shell activated carbon was tested for removal of SO2 gas from flue gas. The activated carbon production was mathematically described as a function of parameters such as flow rate, activation time and activation temperature of carbonization. These parameters were modeled using response surface methodology. The experiments were carried out as a central composite design consisting of 32 experiments. Quadratic models were developed for surface area, total pore volume, and microporosity in term of micropore fraction. The models were used to obtain the optimum process condition for the production of microporous palm shell activated carbon useful for SO2 removal. The optimized palm shell activated carbon with surface area of 973 m(2)/g, total pore volume of 0.78 cc/g and micropore fraction of 70.5% showed an excellent agreement with the amount predicted by the statistical analysis. Palm shell activated carbon with higher surface area and microporosity fraction showed good adsorption affinity for SO2 removal.     

Theoretical study of flow,pressure and solute concentration in double membrane systems

A model system consisting of two rigidly held membranes in series was investigated through the application of the Kedem and Katchalsky thermodynamic single membrane flow equations. This analysis results in predictions of the steady state flow properties as well as values for the solute concentration and pressure of the internal compartment when the system is under the influence of a constant solute concentration or hydrostatic pressure gradient. It is demonstrated that although the flow properties and internal compartment pressure are complicated functions of the membrane permeability coefficients and driving gradient across the system, the relationships are greatly simplified by the explicit appearance of the internal compartment steady state solute concentration in the equations. It is shown that the steady state volume flow rate depends on the absolute value of the solute concentration in the external compartments, as well as the solute concentration gradient across the system. The properties of non-linear dependence of volume flow on concentration gradient, and rectification of volume flow are discussed and shown to be independent properties of the system. For the system under the influence of a solute concentration gradient, the internal compartment pressure can be greater or less than the ambient pressure, and depends mainly on the order in which the membranes are encountered by the volume flow. These properties are qualitatively correlated with certain available experimental observations in biological systems.     

The buckling of spherical liposomes

In the classical "first approximation" theory of thin-shell structures, the constitutive relations for a generic shell element--i.e. the elastic relations between the bending moments and membrane stresses and the corresponding changes in curvature and strain, respectively-are written as if an element of the shell is flat, although in reality it is curved. In this theory it is believed that discrepancies on account of the use of "flat" constitutive relations will be negligible provided the ratio shell-radius/thickness is of sufficiently large order. In the study of drawing of narrow, cylindrical "tethers" from liposomes it has been known for many years that it is necessary to use instead a constitutive law which explicitly describes a curved element in order to make sense of the mechanics; and indeed such tethers are generally of "thick-walled" proportions. In this paper we show that the proper constitutive relations for a curved element must also be used in the study, by means of shell equations, of the buckling of initially spherical thin-walled giant liposomes under exterior pressure: these involve the inclusion of what we call the "Mkappa" terms, which are not present in the standard "first-approximation" theory. We obtain analytical expressions for both the bifurcation buckling pressure and the slope of the post-buckling path, in terms of the dimensions and elastic constants of the lipid bi-layer, and also the initial state of bending moment in the vesicle. We explain physically how the initial bending moment can affect the bifurcation pressure, whereas it cannot in "first-approximation" theory. We use these results to map the conditions under which the vesicle buckles into an oblate, as distinct from a prolate ("rugby-ball") shape. Some of our results were obtained long ago by the use of energy methods; but our aim here has been to identify precisely what is lacking in "first-approximation" theory in relation to liposomes, and so to put the "shell equations" approach onto a firm footing in mechanics.     

Protein loading,elution, and resolution behavior in a novel device that integrates ultrafiltration and chromatographic separation

Hollow fiber membranes and chromatographic resin beads are commonly employed in a variety of bioseparation processes. A new class of integrated separation devices is being studied in which the shell side of a hollow fiber device is filled with adsorbents/chromatographic resin beads. Such devices and the corresponding separation methods integrate feed broth clarification by the microfiltration/ultrafiltration membrane with bioproduct purification by the shell-side resin beads either as an adsorbent or as beads in elution chromatography. A mathematical model has been developed for the prediction of the chromatographic behavior of such an integrated device. Simulations have been done to study the effects of axial dispersion, feed flow rate, water permeation rate, fiber packing density, and void fraction. Numerical solutions were obtained by solving the governing equations. This model can reasonably describe the concentration profiles as well as the breakthrough and elution behaviors in the integrated device.     

A differentiable,periodic function for pulsatile cardiac output based on heart rate and stroke volume

Many mathematical models of human hemodynamics, particularly those which describe pressure and flow pulses throughout the circulatory system, require as specified input a modeling function which describes cardiac output in terms of volume per unit time. To be realistic, this cardiac output function should capture, to the greatest extent possible, all relevant features observed in measured physical data. For model analysis purposes, it is also highly desirable to have a model function that is continuous, differentiable, and periodic. This paper addresses both classes of needs by developing such a function. Physically, the present function provides an accurate model for flow into the ascending aorta. It is completely specified by a minimal number of standard input parameters associated with left ventricle dynamics, including heart rate, mean cardiac output, and an estimation of the peak-to-mean flow ratio. Analytically, it can be expressed as a product of two continuous, differentiable and periodic factors. Further, the Fourier expansion of this model function is shown to be a finite Fourier series, and explicit closed-form expressions are given for the non-zero coefficients in this series.     

An anatomically detailed and personalizable head injury model: Significance of brain and white matter tract morphological variability on strain

Finite element head (FE) models are important numerical tools to study head injuries and develop protection systems. The generation of anatomically accurate and subject-specific head models with conforming hexahedral meshes remains a significant challenge. The focus of this study is to present two developmental works: first, an anatomically detailed FE head model with conforming hexahedral meshes that has smooth interfaces between the brain and the cerebrospinal fluid, embedded with white matter (WM) fiber tracts; second, a morphing approach for subject-specific head model generation via a new hierarchical image registration pipeline integrating Demons and Dramms deformable registration algorithms. The performance of the head model is evaluated by comparing model predictions with experimental data of brain–skull relative motion, brain strain, and intracranial pressure. To demonstrate the applicability of the head model and the pipeline, six subject-specific head models of largely varying intracranial volume and shape are generated, incorporated with subject-specific WM fiber tracts. DICE similarity coefficients for cranial, brain mask, local brain regions, and lateral ventricles are calculated to evaluate personalization accuracy, demonstrating the efficiency of the pipeline in generating detailed subject-specific head models achieving satisfactory element quality without further mesh repairing. The six head models are then subjected to the same concussive loading to study the sensitivity of brain strain to inter-subject variability of the brain and WM fiber morphology. The simulation results show significant differences in maximum principal strain and axonal strain in local brain regions (one-way ANOVA test, p < 0.001), as well as their locations also vary among the subjects, demonstrating the need to further investigate the significance of subject-specific models. The techniques developed in this study may contribute to better evaluation of individual brain injury and the development of individualized head protection systems in the future. This study also contains general aspects the research community may find useful: on the use of experimental brain strain close to or at injury level for head model validation; the hierarchical image registration pipeline can be used to morph other head models, such as smoothed-voxel models.

     

Deviations from Michaelis-Menten kinetics. Computation of the probabilities of obtaining complex curves from simple kinetic schemes.

1. It is possible to calculate the intrinsic probability associated with any curve shape that is allowed for rational functions of given degree when the coefficients are independent or dependent random variables with known probability distributions. 2. Computations of such probabilities are described when the coefficients of the rational function are generated according to several probability distribution functions and in particular when rate constants are varied randomly for several simple model mechanisms. 3. It is concluded that each molecular mechanism is associated with a specific set of curve-shape probabilities, and this could be of value in discriminating between model mechanisms. 4. It is shown how a computer program can be used to estimate the probability of new complexities such as extra inflexions and turning points as the degree of rate equations increases. 5. The probability of 3 : 3 rate equations giving 2 : 2 curve shapes is discussed for unrestricted coefficients and also for the substrate-modifier mechanisms. 6. The probability associated with the numerical values of coefficients in rate equations is also calculated for this mechanism, and a possible method for determining the approximate magnitude of product-release steps is given. 7. The computer programs used in the computations have been deposited as Supplement SUP 50113 (21 pages) with the British Library Lending Division, Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem, J. (1978) 169, 5.