眼固体生物力学研究

眼是光转换的视觉器官, 通常考虑将其作为一个生物力学结构. 眼睛是眼内肌和眼外肌产生压力的系统, 具有复杂的内血管系统,产生流体和溶合物的传导系统. 从生物力学角度看, 眼存在固体生物力学、流体生物力学和生物传输等问题. 本综述中, 介绍眼睛有意义的生物固体力学各方面研究成果及尚待解决的问题, 其研究对象为巩膜、角巩膜、角膜、视网膜、筛板以及眼调节和老花. 回顾测量巩膜力学性质和角巩膜应力－应变性质模型的有关研究成果, 然后论述这些研究成果的应用.巩膜力学另一重要应用是对近视的了解, 即眼的轴长过长使长距离光线不能够清晰聚焦于视网膜而引起近视. 角膜生物力学一个显著的应用是预测激光切开剖面手术, 它将使屈光角膜手术达到最优的术后视敏度. 筛板是眼中最具有生物力学研究兴趣的组织之一, 它是跨过巩膜管的多孔的结缔组织. 推导出青光眼的视神经病的力学理论.证据表明围绕着睫状肌的结蹄组织发生变化可能妨碍它的自由收缩的能力, 因此老花的病理学原因可能是多因素的.      

Identification of a constitutive law for trabecular bone samples under remodeling in the framework of irreversible thermodynamics

We construct in the present paper constitutive models for bone remodeling based on micromechanical analyses at the scale of a representative unit cell (RUC) including a porous trabecular microstructure. The time evolution of the microstructure is simulated as a surface remodeling process by relating the surface growth remodeling velocity to a surface driving force incorporating a (surface) Eshelby tensor. Adopting the framework of irreversible thermodynamics, a 2D constitutive model based on the setting up of the free energy density and a dissipation potential is identified from FE simulations performed over a unit cell representative of the trabecular architecture obtained from real bone microstructures. The static and evolutive effective properties of bone at the scale of the RUC are obtained by combining a methodology for the evaluation of the average kinematic and static variables over a prototype unit cell and numerical simulations with controlled imposed first gradient rates. The formulated effective growth constitutive law at the scale of the homogenized set of trabeculae within the RUC is of viscoplastic type and relates the average growth strain rate to the homogenized stress tensor. The postulated model includes a power law function of an effective stress chosen to depend on the first and second stress invariants. The model coefficients are calibrated from a set of virtual testing performed over the RUC subjected to a sequence of loadings. Numerical simulations show that overall bone growth does not show any growth kinematic hardening. The obtained results quantify the strength and importance of different types of external loads (uniaxial tension, simple shear, and biaxial loading) on the overall remodeling process and the development of elastic deformations within the RUC.     

反演在骨生长方程参数识别中的应用

基于BFGS算法,根据自行设计的动物实验,得到在不同应力环境中,快速生长期大鼠股骨生 长与重建过程中骨密度实测数据,采用反演数值方法,获取了骨生长方程中随时间变化的 生物参数B和K. 通过正演验证,表明所建立的基于BFGS算法的参数识别方法具有较好的 稳定性和较高的识别精度,能够反演出比较切合实际的生物参数数值. 研究表明,反问 题方法是解决生命现象不可测性和未知性问题的有效手段,把反演方法应用到骨生长与重建 等生命现象的研究中,可确定、修正预设的数学模式,为数值量化骨适应生物模型的建立提 供了一条可行的途径.     

Stress amplification and plastic flow for spheroidal shells: Applications to myopia

Amplification of effective stress and plastic strain rates after yielding are derived for the anisotropic stress fields of the prolate and oblate spheroidal shells as models of the myopic and hyperopic eyes. Dimensionless closed-form results are presented for arbitrary axis ratio with both constant shell thickness and variable shell thickness, using the constant scleral mass assumption. The results show that the myopic and hyperopic eyes are susceptible to failure by plastic yielding at the equator and pole, respectively, for high intraocular pressures. Experimental data from the equatorial zone of rabbit sclera shows scleral yielding and plastic flow for intraocular pressures greater than 32 mm Hg. Practical applications include glaucoma and pathological myopia.     

Designing Freeform Lenses for Intensity and Phase Control of Coherent Light with Help from Geometry and Mass Transport

We study here the problem of determining a system of two refractive interfaces transforming a plane wavefront of a given shape and radiation intensity into a coherent output plane wavefront with prescribed output position, shape and intensity. Such interfaces can be refracting surfaces of two different lenses or of one lens. In geometrical optics approximation, the analytic formulation of this problem in both cases requires construction of maps with controlled Jacobian. Though this Jacobian can be expressed as a second order partial differential equation of Monge-Ampère type for a scalar function defining one of the refracting surfaces, its analysis is not straightforward. In this paper we use a geometric approach for reformulating the problem in certain associated measures and defining weak solutions. Existence and uniqueness of weak solutions in Lipschitz classes for both cases are established by variational methods. Our results show, in particular, that two types of interfaces exist in each case for the same data: one of these types always consists of two interfaces, one of which is concave or convex and the second convex or concave, while the interfaces of the second type may be neither convex nor concave. The availability of a design with convex/concave lenses is particularly important for fabrication. The truly geometric nature of this problem permits its statement and investigation in \mathbb R^N+1, N \geqq 1{\mathbb {R}^{N+1},\, N \geqq 1} .     

On the location of flame edge in Shadowgraph pictures of spherical flames: a theoretical and experimental study

In this paper, a theoretical model based on geometrical optics has been developed to analyze the light intensity pattern of Shadowgraph pictures of spherical flames. The theoretical results have been compared with experimental measurement of light intensity profiles across the flame front using commercially available image processing software. These results are in good agreement. The theory predicts that the sudden change of light intensity from dark to bright does not coincide with the flame edge unless the flame thickness is negligible. Experimental results agree very well with the theoretical predictions.     

Residual stresses in soft tissue as a consequence of growth and remodeling: application to an arterial geometry

We develop a thermodynamically consistent model for growth and remodeling in elastic arteries. The model is specialized to a cylindrical geometry, strain energy of the Holzapfel–Gasser–Ogden type and remodeling of the collagen fiber angle. A numerical method for calculating the evolution of the adaptation process is developed. For a particular choice of the thermodynamic forces of growth and remodeling (configurational forces), it is shown that an almost homogeneous transmural axial and tangential stress distribution is obtained. Residual stresses develop during this adaption process and these resemble what is found in experiments and by parameter identification methods.     

Velocity-Gradient Dynamics in Turbulence: Effect of Viscosity and Forcing

The restricted Euler equation is a promising but incomplete model for velocity-gradient dynamics in turbulent flows. While it captures many of the geometric features of the vorticity vector and the strain rate tensor, viscous and anisotropic pressure Hessian effects are not accounted for satisfactorily. Inadequate viscous-effect modeling causes velocity gradients to diverge in finite time, rendering the restricted Euler model unsuitable for practical applications. We perform a Lagrangian frame analysis to comprehend fully the physics of the viscous relaxation time scale and propose a variable time-scale model that can adequately account for deformation history. Most importantly, the finite-time singularity (divergence of velocity gradients) problem is fully resolved with the present model. We also model the effects of forcing that is used in numerical simulations to sustain stationary isotropic turbulence. Detailed comparison of the new model with DNS data reveals good agreement.     

急性高血压引起血管重建的数值模拟

通过建立血管重建率方程对急性高血压引起的血管零应力状态重建进行模拟。为体现高血压血管重建在时间和空间上的不均匀性，在重建率方程中引入生长因子，对零应力状态血管的重建进程进行有效控制。以6只正常大鼠腹主动脉的重建为例，对该动脉段由急性高血压引起的重建进行数值模拟。结果表明，所模拟的6只大鼠急性高血压引起的腹主动脉重建都具有相类似的重建趋势和特征，说明所提出的血管重建方法对 急性高血压诱发血管重建的预测具有一定的普遍性。同时，由于在分析模型中引进反映重建非均匀性的生长因子，使得所模拟的重建进程能较好地符合急性高血压血管重建的基本生理病理特征。     

A multiscale approach of nonlinear composites under finite deformation: Experimental characterization and numerical modeling

The present paper is devoted to the study of the mechanical behavior of an ethylene propylene diene monomer (EPDM) rubber reinforced by polypropylene (PP) particles, revealed as compressible. The hyperlastic behavior of this blend has been characterized under cyclic uni-axial tensile tests. The experimental results show a significant effect of the fraction of (PP) particles (5%, 10%, 25% and 30% by weight) on the macroscopic behavior of the composite. In order to model this behavior, we first develop and implement a micromechanically-based nonlinear model for hyperelastic composites. The approach is based on the second order homogenization method proposed by Ponte Castaneda and Tiberio (2000) and for which suitable energy densities are adopted for the matrix and the inclusions phases, both assumed as compressible. We then proceed to the model verification by comparison with Finite Element simulations on a unit cell. Finally, we propose an extension of the model in order to take into account damage due to voids growth phenomena. The comparison of the multiscale damage model predictions with the experimental data obtained on the EPDM/PP composite indicates a very good agreement.     

Growth and remodeling of the left ventricle: A case study of myocardial infarction and surgical ventricular restoration

Cardiac growth and remodeling in the form of chamber dilation and wall thinning are typical hallmarks of infarct-induced heart failure. Over time, the infarct region stiffens, the remaining muscle takes over function, and the chamber weakens and dilates. Current therapies seek to attenuate these effects by removing the infarct region or by providing structural support to the ventricular wall. However, the underlying mechanisms of these therapies are unclear, and the results remain suboptimal. Here we show that myocardial infarction induces pronounced regional and transmural variations in cardiac form. We introduce a mechanistic growth model capable of predicting structural alterations in response to mechanical overload. Under a uniform loading, this model predicts non-uniform growth. Using this model, we simulate growth in a patient-specific left ventricle. We compare two cases, growth in an infarcted heart, pre-operative, and growth in the same heart, after the infarct was surgically excluded, post-operative. Our results suggest that removing the infarct and creating a left ventricle with homogeneous mechanical properties does not necessarily reduce the driving forces for growth and remodeling. These preliminary findings agree conceptually with clinical observations.     

Refractive index of benzene and methyl derivatives: temperature and wavelength dependencies

Measurements of the refractive index from 288 to 318 K at five fixed wavelengths, from 656.3 to 404.7 nm, are reported for benzene, toluene (methylbenzene), o-xylene (1,2-dimethylbenzene), m-xylene (1,3-dimethylbenzene), and p-xylene (1,4-dimethylbenzene). We also report the temperature and wavelength dependencies of the refractive index obtained from a least-squares routine. The agreement between the measured and calculated refractive indices lies within the experimental uncertainty.     

On the behavior of fine mud suspensions

Flows of natural mud-water mixtures are of great interest for industrial and civil engineering. But there is still no general agreement about the methods for determining the main rheological characteristics of these systems. We propose here an accurate rheological study of some natural mud-water mixtures. We first discuss the possible effects of changing various parameters such as temperature, pH, electrolyte concentration, solid concentration, clay type. The behavior of these muds appears to be very sensitive to most of these parameters and to be hardly predictable from a knowledge of their components. Then, we show that a Herschel-Bulkley model fits very well steady flow experimental data for a very large range of shear rates. We also suggest physical explanations of this model in agreement with our observations of behavior changes when some parameters change. The yield stress value of this model provides a good estimation of real yield stress which is a key parameter for mixture behavior. These considerations are very useful to characterize, predict, and compare various mud flows.     

Investigation of the pressure-volume relationship for the eyeball loaded by a thin rod

The static problem of loading the eyeball by a thin cylindrical cornea-deforming rod is solved for obtaining data on the mechanical properties of the eye in clinical examination. The analysis is carried out within the framework of a simple model which treats the cornea as a soft shell and takes into account the elastic properties of the sclera and tissues surrounding it by introducing an elastic constraint between the intraocular pressure and the scleral part of the intraocular volume. The dependence of the intraocular volume on the pressure and the load applied, as well as on the elastic characteristics of the system, is investigated. The problem of relationship between the intraocular pressure and the load at a constant volume is considered. It is shown that loading the same eye by thin rods and flat stamps of different weights makes it possible to find in clinical examination the individual elastic characteristics of the eyeball and the pressure in the unloaded eye.     

A hierarchical model for rate-dependent polycrystals

A hierarchical model of a polycrystalline aggregate of rigid viscoplastic grains is formulated, and a robust and efficient computational algorithm for its solution is proposed. The polycrystalline aggregate is modeled as a binary tree. The leaves of the binary tree represent grains, and higher tree nodes represent increasingly larger sub-aggregates of grains. The root of the tree represents the entire polycrystalline aggregate. Velocity and traction continuity are enforced across the interface between the children of each non-leaf node in the binary tree. The hierarchical model explicitly models intergranular interactions but is nevertheless comparable in computational effort to the mean field models of polycrystal plasticity. Simulations of tensile, compressive, torsional, and plane strain deformation of copper lead to predictions in good agreement with experiments, and highlight the interconnection between grain deformations and intergranular constraints. It is inferred from the results that a hybrid mean field/hierarchical model represents a computationally efficient methodology to simulate polycrystal deformation while accounting for intergranular interactions.     

A theoretical model for surface bone remodeling under electromagnetic loads

A theoretical model for surface bone remodeling under electromagnetic loads is proposed in this paper. In the model, surface bone remodeling is assumed to be related to growth factors. Growth factors in latent form in osteocytes are released to the bone fluid after the osteocytes are absorbed by osteoclasts, and then regulate the bone formation process. At the same time, environmental loadings can influence the generation of growth factors. This paper shows how surface bone remodeling is triggered under the influence of growth factors. Based on this hypothesis, a computational model is established that simulates the bone coupling remodeling process, including internal and surface bone remodeling. The effects of various loadings, including electrical and magnetic loadings, are simulated and compared. The interactions between internal and surface bone remodeling are investigated via the numerical method. The results indicate that an electromagnetic field can strongly influence the bone remodeling process and that the remodeling process will be altered after surface bone remodeling is triggered, compared to the sole effect of the internal remodeling process.     

Crystallization of alkanes under quiescent and shearing conditions

We report molecular dynamics simulation of crystallization of model alkane systems conducted under constant pressure conditions. We have studied crystallization of n-eicosane (C₂₀H₄₂) and n-hexacontane (C₆₀ H₁₂₂) under quiescent and shearing conditions. We find preshearing before subjecting the melt to quiescent crystallization enhances the crystallization of higher molecular weight hexacontane, whereas, for low molecular weight eicosane, no significant change can be detected. For both alkanes applying steady planar shear significantly speeds up the crystallization. The crystal growth rate increases with the shear rate. However, we find that the critical shear rate above which the crystallization is enhanced, is inversely proportional to the size of the chains. In all cases the Weissenberg numbers of the sheared systems are moderate. We estimate them to be in the range of 0.01–10. Our quiescent simulations for eicosane predict crystallization temperature and lattice parameters of the crystalline phase in good agreement with experimental measurements. We have compared an order parameter used in the simulations against one analogous to that used in dilatometry experiments. Using this order parameter as a measure of crystallinity we predict the crystal growth rate of n-eicosane to be a maximum at ∼300 K in good agreement with experiments. Fitting crystallization growth data to Avrami's model we have calculated Avrami growth functions and exponents for many cases. For quiescent crystallization of n-eicosane we found the Avrami exponent calculated using our order parameter for defining degree of crystallinity, agrees well with that obtained in the experiments. For C60 the crystallization process is very slow at quiescent conditions; however preshearing enhances the crystal growth.     

Modeling of growth and remodeling in soft biological tissues with multiple constituents

Among the various important characteristics of biological tissues is their ability to grow and remodel. It is well-known that one of the primary triggers behind the growth and remodeling process is changes in the mechanical environment, for instance changes in stress, strain, etc. These mechanisms of mechanotransduction are the driving force behind many changes in structure and function including growth and remodeling. The purpose of this article is to formulate better constitutive equations for the stress in tissues with multiple constituents undergoing growth and remodeling. This is a very complex problem and is of tremendous importance. Here, we do the modeling from a mechanics point of view, utilizing the theory of natural configurations coupled with population dynamics to accurately model the production and removal of the different constituents that comprise the tissue. This is accomplished by deriving a generalized McKendrick equation for growth and remodeling and has the advantage of directly including the age distribution of constituents into the model. The population distribution function is then used to determine the stress in the tissue.     

Perspectives on biomechanical growth and remodeling mechanisms in glaucoma

Glaucoma is a blinding diseases in which damage to the axons results in loss of retinal ganglion cells. Experimental evidence indicates that chronic intraocular pressure elevation initiates axonal insult at the level of the lamina cribrosa. The lamina cribrosa is a porous collagen structure through which the axons pass on their path from the retina to the brain. Recent experimental studies revealed the extensive structural changes of the lamina cribrosa and its surrounding tissues during the development and progression of glaucoma. In this perspective paper we review the experimental evidence for growth and remodeling mechanisms in glaucoma including adaptation of tissue anisotropy, tissue thickening/thinning, tissue elongation/shortening and tissue migration. We discuss the existing predictive computational approaches that try to elucidate the potential biomechanical basis of theses growth and remodeling mechanisms and highlight open questions, challenges, and avenues for further development.     

On Recent Progress in Modelling and Simulations of Multi-scale Transfer of Mass,Momentum and Particles in Bio-medical Applications

We present a short overview of some of our most recent work that combines the mathematical modeling, advanced computer simulations and state-of-the-art experimental techniques of physical transport phenomena in various bio-medical applications. In the first example, we tackle predictions of complex blood flow patterns in the patient-specific vascular system (carotid artery bifurcation) and transfer of the so-called “bad” cholesterol (low-density lipoprotein, LDL) within the multi-layered artery wall. This two-way coupling between the blood flow and corresponding mass transfer of LDL within the artery wall is essential for predictions of regions where atherosclerosis can develop. It is demonstrated that a recently developed mathematical model, which takes into account the complex multi-layer arterial-wall structure, produced LDL profiles within the artery wall in good agreement with in-vivo experiments in rabbits, and it can be used for predictions of locations where the initial stage of development of atherosclerosis may take place. The second example includes a combination of pulsating blood flow and medical drug delivery and deposition controlled by external magnetic field gradients in the patient specific carotid artery bifurcation. The results of numerical simulations are compared with own PIV (Particle Image Velocimetry) and MRI (Magnetic Resonance Imaging) in the PDMS (silicon-based organic polymer) phantom. A very good agreement between simulations and experiments is obtained for different stages of the pulsating cycle. Application of the magnetic drug targeting resulted in an increase of up to ten fold in the efficiency of local deposition of the medical drug at desired locations. Finally, the LES (Large Eddy Simulation) of the aerosol distribution within the human respiratory system that includes up to eight bronchial generations is performed. A very good agreement between simulations and MRV (Magnetic Resonance Velocimetry) measurements is obtained. Magnetic steering of aerosols towards the left or right part of lungs proved to be possible, which can open new strategies for medical treatment of respiratory diseases.