Multiscale cartilage biomechanics: technical...

on May 13, 2018http://rsfs.royalsocietypublishing.org/Downloaded from

rsfs.royalsocietypublishing.org

ResearchCite this article: Erdemir A, Bennetts C, Davis

S, Reddy A, Sibole S. 2015 Multiscale cartilage

biomechanics: technical challenges in realizing

a high-throughput modelling and simulation

workflow. Interface Focus 5: 20140081.

http://dx.doi.org/10.1098/rsfs.2014.0081

One contribution of 11 to a theme issue

‘Multiscale modelling in biomechanics:

theoretical, computational and translational

challenges’.

Subject Areas:biomechanics, computational biology,

biophysics

Keywords:cartilage, extracellular, pericellular,

chondrocyte, finite-element analysis,

post-processing

Author for correspondence:Ahmet Erdemir

e-mail: [email protected]

& 2015 The Author(s) Published by the Royal Society. All rights reserved.

Multiscale cartilage biomechanics:technical challenges in realizing ahigh-throughput modelling andsimulation workflow

Ahmet Erdemir1,2, Craig Bennetts1,2, Sean Davis1,2,3, Akhil Reddy1,2,4

and Scott Sibole1,2,5

1Computational Biomodeling (CoBi) Core, and 2Department of Biomedical Engineering, Lerner ResearchInstitute, Cleveland Clinic, Cleveland, OH 44195, USA3Department of Mechanical Engineering, University of Akron, Akron, OH 44325, USA4Weill Cornell Medical College, New York, NY 10065, USA5Human Performance Laboratory, Faculty of Kinesiology, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Understanding the mechanical environment of articular cartilage and chon-

drocytes is of the utmost importance in evaluating tissue damage which is

often related to failure of the fibre architecture and mechanical injury to the

cells. This knowledge also has significant implications for understanding

the mechanobiological response in healthy and diseased cartilage and can

drive the development of intervention strategies, ranging from the design of

tissue-engineered constructs to the establishment of rehabilitation protocols.

Spanning multiple spatial scales, a wide range of biomechanical factors dictate

this mechanical environment. Computational modelling and simulation pro-

vide descriptive and predictive tools to identify multiscale interactions, and

can lead towards a greater comprehension of healthy and diseased cartilage

function, possibly in an individualized manner. Cartilage and chondrocyte

mechanics can be examined in silico, through post-processing or feed-forward

approaches. First, joint–tissue level simulations, typically using the finite-

element method, solve boundary value problems representing the joint

articulation and underlying tissue, which can differentiate the role of compart-

mental joint loading in cartilage contact mechanics and macroscale cartilage

field mechanics. Subsequently, tissue–cell scale simulations, driven by the

macroscale cartilage mechanical field information, can predict chondrocyte

deformation metrics along with the mechanics of the surrounding pericellular

and extracellular matrices. A high-throughput modelling and simulation

framework is necessary to develop models representative of regional and

population-wide variations in cartilage and chondrocyte anatomy and mech-

anical properties, and to conduct large-scale analysis accommodating a

multitude of loading scenarios. However, realization of such a framework is

a daunting task, with technical difficulties hindering the processes of model

development, scale coupling, simulation and interpretation of the results.

This study aims to summarize various strategies to address the technical chal-

lenges of post-processing-based simulations of cartilage and chondrocyte

mechanics with the ultimate goal of establishing the foundations of a high-

throughput multiscale analysis framework. At the joint–tissue scale, rapid

development of regional models of articular contact is possible by automating

the process of generating parametric representations of cartilage boundaries

and depth-dependent zonal delineation with associated constitutive relation-

ships. At the tissue–cell scale, models descriptive of multicellular and

fibrillar architecture of cartilage zones can also be generated in an automated

fashion. Through post-processing, scripts can extract biphasic mechanical

metrics at a desired point in the cartilage to assign loading and boundary con-

ditions to models at the lower spatial scale of cells. Cell deformation metrics

can be extracted from simulation results to provide a simplified description

of individual chondrocyte responses. Simulations at the tissue–cell scale can

http://crossmark.crossref.org/dialog/?doi=10.1098/rsfs.2014.0081&domain=pdf&date_stamp=2015-02-20

mailto:[email protected]

http://rsfs.royalsocietypublishing.org/

rsfs.royalsocietypublishing.orgInterfa

2


be parallelized owing to the loosely coupled nature of the

feed-forward approach. Verification studies illustrated the

necessity of a second-order data passing scheme between

scales and evaluated the role that the microscale represen-

tative volume size plays in appropriately predicting the

mechanical response of the chondrocytes. The tools sum-

marized in this study collectively provide a framework

for high-throughput exploration of cartilage biomecha-

nics, which includes minimally supervised model

generation, and prediction of multiscale biomechanical

metrics across a range of spatial scales, from joint regions

and cartilage zones, down to that of the chondrocytes.

ceFocus

5:20140081

1. Background and motivationThe mechanics of cartilage and chondrocytes is a function of

complicated load sharing between musculoskeletal joints of

the body, individual tissue structures within the joint, and

the organization and mechanical properties of cells and micro-

structure within the tissue (figure 1). The external loads applied

to the body are translated to musculoskeletal joints, which may

also endure large applied muscular forces. To achieve equili-

brium, the passive structures of the joints often must sustain

high stresses. Contact is the modality through which cartilage

sustains load, where different regions of the musculoskeletal

joint may exhibit varying contact force transmission. As one

moves from the intermediate scale of joint compartments to

the lower spatial scale resolution of the tissue, the load trans-

mission at the articular surface is now resolved as a contact

stress field on the cartilage surface. This load distribution is

not only dependent on the total contact force, but it is also

likely to be influenced by the cartilage geometry and properties,

i.e. its thickness, radius of curvature and macroscopic material

properties. Different zones along the depth of the cartilage

(superficial, transitional and deep) exhibit varying stresses,

strains, fluid pressures, etc., depending on the zonal thickness

and underlying material properties, which are dictated by vari-

ations in collagen matrix organization and proteoglycan

content [2]. These zonal loads influence the deformation and

loading of chondrocytes, which are dependent on the relative

mechanical properties of chondrocytes, and pericellular and

extracellular matrices [3]. Chondrocyte shape, size and possibly

their organization affect the loads transmitted to these cells and

to their nuclei, which may cause cell damage and death [4] or

may trigger mechanobiological responses for the regulation of

extracellular matrix integrity [5].

A comprehensive understanding of the mechanical environ-

ment of cartilage and chondrocytes potentially has significant

implications to evaluate the function of this tissue in health

and disease. Such an understanding is necessary as cartilage

bears loads exceeding multiple body weights during the

activities of daily life and the tissue’s resulting mechanical

environment can lead to regulation of its structural integrity.

Mechanical insults not only result in matrix damage, but may

also decrease chondrocyte viability [6]. Chondrocytes also

respond to their mechanical environment, where mechanical

transduction results in biological responses to maintain the

structure of the extracellular environment [5]. Changes in

the multiscale load sharing pathway alter mechanical signals

transmitted to the cells, possibly compromising the mechanobio-

logical function of chondrocytes. When quantified for healthy

populations, the joint–tissue and tissue–cell scale mechanical

environments of cartilage can better elucidate the forcing func-

tions of chondrocyte mechanobiology, i.e. cellular tractions

and osmotic pressure gradients [7]. Such information can be

used as a design specification for tissue engineering of cartilage,

i.e. how construct loading protocols should be implemented to

promote matrix growth and adaptation in cell-seeded con-

structs. Mechanics also has a significant role in chronic and

acute disorders of the cartilage. For example, the mechanical

capacity and function of the cartilage and chondrocytes are

affected during the onset and progression of osteoarthritis [8].

This debilitating musculoskeletal disease is the most common

form of arthritis. Osteoarthritis influences more than 30% of

adults above the age of 65 [9] and can manifest itself in any syno-

vial joint. The knowledge of cartilage and chondrocyte loading

can also indicate when and how trauma or tears may occur

and it may offer an explanation for the formation of chondral

defects. In any of these diseased or injured populations, quanti-

fication of cartilage and chondrocyte mechanics may have

additional value for mechanical risk assessment and for diag-

nostic purposes, and may assist in the evaluation of disease

progression or efficacy of prevention and intervention strategies.

Computational modelling and simulation provides a means

of quantifying the mechanical environment of the cartilage and

chondrocytes [1]. In vivo measurement of cartilage and chon-

drocyte mechanics in human subjects, where the joints are

exposed to loadings of daily activities, is not necessarily

within the reach of the current state of the art. Recent exper-

imentation illustrated the possibility to quantify cartilage and

chondrocyte deformations of fresh cadaveric samples in situ[10] and of animals in vivo [11], through invasive approaches.

Modelling and simulation provide a means for in-depth

exploration of such data through descriptive analysis [12]. In

addition, through predictive simulations, it may be possible

to establish multiscale mechanical interactions and to evaluate

the influence of disease- or intervention-related perturbations

on the mechanics of cartilage and chondrocytes [1].

Musculoskeletal joints exhibit large variations, anatomically

and mechanically, at multiple scales, within an individual

and across populations. Regional cartilage anatomy (curvature,

thickness), stiffness, zonal properties, i.e. delineation of super-

ficial, transitional and deep layers, and fibrillar architecture

[13–16], and chondrocyte organization, shape, size [17,18]

and mechanical properties [19] all vary largely among species

and between individuals and can be altered by injury or

disease. A high-throughput modelling and simulation frame-

work seeks to provide the ability to represent such variations

and conduct individualized analysis to assess this multiscale

mechanical system under various loading conditions, which

can be used for understanding trauma risk, for evaluating the

protective performance of interventions, and for diagnosis

and prognosis through the use of biomechanical markers.

Likewise, parametric analysis supported by large-scale unsuper-

vised simulations can allow probabilistic studies (for population

analysis or for uncertainty estimation) [20], inverse analysis

(for estimation of patient-specific mechanical properties) [21]

and virtual prototyping (to design new therapeutic, surgical or

rehabilitative management strategies) [22]. While the use of

modelling and simulation for explorations of cartilage and chon-

drocyte mechanics is appealing, routine investigations are

hampered by many technical challenges associated with the

need to accommodate the nonlinear, multiscale and multiphasic

biomechanical nature of this tissue [1].


body~1 m

jointloads

jointloads

contact

externalloads

bonebone

cartilage

cartilage

cartilageloads

meniscus

ligament

joint~0.1 m

tissue~0.001 m

cell/fibre~0.00001 m

fibrearchitecture

chondrocytemuscleforces

Figure 1. The mechanics of articular cartilage and the chondrocytes are dictated by a multiscale load sharing pathway (illustration based on Halloran et al. [1]).External forces and inertial loads of the body are reflected upon musculoskeletal joints. The passive structures of the joint, e.g. cartilage, ligaments, menisci, bearlarge forces and deformations, which are distributed as a function of the joint’s anatomical architecture and the properties of individual tissues. For the articularcartilage, these loads are mainly in the form of contact, which induce deformations of the superficial, transitional and deep layers along the tissue’s depth andeventually the mechanics of chondrocytes, the cell type found in cartilage. (Online version in colour.)

rsfs.royalsocietypublishing.orgInterface

Focus5:20140081

3


The goal of this paper is to summarize various strategies to

address the technical challenges of realizing a modelling and

simulation workflow to characterize the mechanical environ-

ment of cartilage and chondrocytes. The tissue of interest is

articular cartilage, but the outlined strategies are equally appli-

cable to fibrocartilage and other tissues. The modelling and

simulation strategy is based on a post-processing-type analysis

approach (from higher spatial scales to lower spatial scales),

where mechanical load sharing across joint–tissue and

tissue–cell scales can be differentiated in a feed-forward

manner. Individual components of the workflow are aimed to

address difficulties in model development and its automation,

scale coupling, computational burden and extraction of rel-

evant biomechanical metrics from simulation results.

Immediate utilization of a streamlined in silico framework

for understanding the regional and population-wide mechan-

ical responses of cartilage and chondrocytes may still not be

computationally feasible, particularly when the physical

phenomena need to be represented at a higher fidelity. None-

theless, approaches outlined here lay the foundation of a

high-throughput framework for large-scale analysis of cartilage

mechanics at multiple scales.

2. High-throughput frameworkThe modelling and simulation work presented herein provides

a feed-forward workflow where body-level musculoskeletal

loading is differentiated to tissue-level and cell-scale defor-

mations using multiple models in sequence (figure 2). If one

is interested in a certain contact region of the joint, e.g.

cartilage-to-cartilage contact area of the knee’s medial com-

partment, a model at the joint–tissue scale (spatial lengths of

0.001–0.01 m) can be constructed for prediction of detailed con-

tact mechanics and, more importantly, the internal mechanics of

the cartilage. Such models should incorporate the articulation

between the cartilage surfaces for evaluation of the distribution

of load transmission through the contact interface. These models

will ideally include zonal thickness and material properties of

the cartilage as well to appropriately predict the internal mech-

anical response of the tissue. Driven by the regional contact force

(or articulation kinematics), simulations can predict contact

pressure distributions and the detailed mechanics of the

superficial, transitional and deep zones. In the case of solid

phase analysis, tissue-level mechanical metrics include strains

and stresses. With the addition of the fluid phase, as in biphasic

simulations, they also include fluid flux and pressure. Such

information can be used to drive cell scale models to predict

chondrocyte deformation metrics. For example, one can pick

up any point in the joint–tissue scale model, e.g. midpoint of

the transitional zone, extract the deformation gradient, fluid

pressure and fluid pressure gradient from the solution of the

joint–tissue scale model, and apply these as loading and bound-

ary conditions to a tissue–cell scale model (spatial lengths

of 1026 to 0.0001 m) representative of the cartilage and chondro-

cyte characteristics of that specific region [23]. This process is

equivalent to using a one term Taylor series to approximate

the field of displacements and fluid pressures in the neighbour-

hood of the point of interest. From the results of the cell scale

models, chondrocyte mechanical metrics can be calculated

including the cell aspect ratio change, average effective (von

Mises) stress and strain, and mass exchange rate. Through

appropriate parametrization, streamlined data flow in between

scales, automation of post-processing and effective comput-

ing strategies, a high-throughput modelling and simulation

framework can be realized for prediction of cartilage and

chondrocyte mechanics.

At the spatial of scale of the musculoskeletal joint, simu-

lations using a finite-element (FE) representation of the

complete joint anatomy [24], e.g. the tibiofemoral joint, includ-

ing its substructures such as the ligaments and meniscus,

can provide the necessary information to define loading and

boundary conditions of the aforementioned joint–tissue scale

model. It should be noted that a comprehensive model,

where all structures of the joint are represented, may be used

to bypass the use of the compartmental joint–tissue scale

model [25]. Yet, a detailed representation of zonal anatomy,

requiring a finer level of discretization to accommodate zonal

thicknesses, may be computationally prohibitive to include in

the representation of the entire musculoskeletal joint. Contact

information, e.g. force or articulation kinematics, to drive the

joint–tissue scale model may be estimated from experimen-

tation. Such measurements are possible, either directly on

cadaver specimens by using contact pressure measurement

systems [26], or indirectly on live subjects through the use of

biplanar fluoroscopy by estimating proximity of articulating


regionaljoint loading

regionaljoint–tissue scale

model

zonaltissue–cell scale

model

zonaltissue

mechanicschondrocytemechanics

finite-elementanalysis

finite-elementanalysis

Figure 2. Sequential simulations using models at multiple spatial levels can predict cartilage and chondroycte loads and deformations. Regional joint – tissue scalemodels can provide representations of joint articulation and cartilage at the spatial order of 0.001 – 0.1 m. Zonal tissue – cell scale models allow modelling of cartilageand chondrocytes at the spatial order of 1026 to 0.0001 m. Given joint-level contact loads (or contact kinematics), the joint – tissue scale model of interfacing articularcartilage regions can be solved to obtain mechanical field information (deformation gradients, fluid pressure, etc.) for various layers of the cartilage. For a given point ofinterest in this macroscale tissue representation, the mechanics information can be mapped to a zone-specific tissue – cell scale model as boundary conditions.Subsequently, FE analysis can predict chondrocyte mechanics. The volume of the joint – tissue scale model can be adjusted to include a desired contact region ofthe joint such as cartilage-to-cartilage interaction area within the medial compartment of the knee. Detailed anatomical features of the cartilage’s zonal delineationcan also be modelled (illustrated as layers in different colours). The tissue – cell scale model represents chondrocyte distribution (seven cells are explicitly shown)and anatomical features including the representation of the pericellular matrix (transparent layer around cells). (Online version in colour.)


Focus5:20140081

4


surfaces [27]. In other words, an elaborate model of the joint

can be foregone, and a regional model can be employed

instead, with boundary conditions and loads taken directly

from regional measurements in the experiment.

Extensibility is paramount for a modelling and simulation

framework as the desired utility of the in silico workflow in

future may necessitate replacing models and analysis types

with higher fidelity versions. The feed-forward approach

lends itself to expansion towards higher and lower spatial

scales. At the higher spatial scale, body-level musculoskeletal

models can be used to determine muscular joint loading [28],

which can be processed to drive joint-level FE analysis, e.g.

for muscle force-driven patellofemoral joint contact analysis

[29]. At the lower spatial scale, cell deformation metrics

can be used to drive cellular–subcellular scale models,

i.e. incorporating the cytoskeleton (actin and tubulin) [30], to

understand the mechanical load transmission to intracellular

components, in particular to the nucleus. Similar submodelling

can be performed for the extracellular matrix, for example, to

evaluate microstructural-level loading of collagen fibres,

which may provide the relationship between macroscopic

tissue damage accumulation and microscale fibre failure [31].

At all spatial scales, it is also possible to increase the fidelity

of mechanical representation; in terms of physics (nonlinear

elasticity, multiphasic response) [32], with additional anatom-

ical components, and by using more advanced constitutive

relationships [24,33]. Utilization of open source and freely

accessible software, e.g. PYTHON (for scripting to automate

tasks; https://www.python.org/), SALOME (for geometry and

mesh generation; http://www.salome-platform.org/) and

FEBIO (for FE analysis; http://www.febio.org/) [34], facilitates

future development by any interested parties.

3. Tackling challengesThis section summarizes our recent published and unpub-

lished work to overcome various technical challenges

associated with high-throughput modelling and simulation

in multiscale biomechanics of the cartilage and chondrocytes.

The term throughput is used in a general sense, referring to

the rate at which something, in this case modelling and simu-

lation, can be conducted. The challenges (and the proposed

solutions) span the life cycle of the modelling and simulation

workflow. To enable high-throughput modelling and simu-

lation, the ‘challenge of automated model development’

should be met. Both joint–tissue and tissue–cell scale models

need to be developed on-the-fly and in an unsupervised

manner to facilitate large-scale analysis. Irrespective of being

high-throughput or not, the ‘challenge of appropriate scale

coupling’ exists. If the information exchange between scales

is erroneous, multiscale analysis of cartilage may become a use-

less effort. High-throughput multiscale simulations of cartilage

and chondrocytes also require expedited solutions of large

number of models, i.e. the ‘challenge of computing’, for dif-

ferent joint regions, tissue zones and loading conditions.

Interactive post-processing of simulation results, in order to

understand biomechanics of cartilage and chondrocytes, is

not suitable for high-throughput analysis. Therefore, the ‘chal-

lenge of streamlined extraction of information’ needs to be

addressed. The ‘challenge of accessible modelling and simu-

lation framework’ is readily addressed by using free and

open source software and publicly sharing models and tools

developed as part of the summarized studies (see §5).

3.1. Automation of modelling3.1.1. Joint – tissue scale model developmentFor a given contact region at the joint–tissue scale, the articulat-

ing cartilage tissue exhibits variations in its gross geometry,

e.g. thickness [13]. These parameters dictate how contact press-

ures are distributed during load transmission between

interfacing cartilage, e.g. for the hip [35]. Within the tissue,

superficial, transitional and deep zones of the cartilage may

have varying thicknesses and their mechanical responses

depend on the fibrous network of collagen (primarily type

https://www.python.org/


http://www.salome-platform.org/


http://www.febio.org/



joint region of interest

parametric modelling

anatomicalF

bone

bone

wxyz

contact

cartilage

rb

rd

rtrs

region size (w × h × l)zonal surface curvature (r)zonal thickness (t)

discretizationt(x, y, z)

E(x, y, z)v(x, y, z)k(x, y, z)

mechanical

examples

elastic modulus (E)Poisson’s ratio (v)permeability (k)fibre properties (x, b, –)contact force (F)lateral wall boundary conditions

x(x, y, z) b(x, y, z) –(x, y, z)

mesh density

Figure 3. Parametrization can simplify the modelling process at the spatialscale incorporating joint- and tissue-level features. Articular cartilage exhibitslarge variations at this spatial level. The curvature of contact surfaces andboundaries of the superficial, transitional and deep zones can be differentbased on the individual, the musculoskeletal joint and the region of interestwithin the joint. Cartilage also has depth-dependent mechanical properties,dictated by collagen and proteoglycan content, and collagen fibre architec-ture. Through an automated pipeline, the generation of numerous modelsand the execution of simulations under different loading conditions maybe conducted with ease. (Online version in colour.)


Focus5:20140081

5


II), water content (75–80% wet weight), dissolved ions and

aggregating proteoglycans [2]. The material properties are

depth-dependent [36], and as a result of varying fibre align-

ment in each zone, they also exhibit anisotropy [37]. All these

influence how the mechanical deformations and loads

distribute within the tissue.

A streamlined workflow for model generation at the

joint–tissue scale should exploit the parametrization of ana-

tomical and mechanical properties such that any desirable

variations can be represented expeditiously in a minimally

supervised manner, e.g. within an input text file. Our team

has been working on the development of an automated

model generation procedure to represent regional anatomy

and depth-dependent material properties for articulating

regions of cartilage [38,39] (figure 3). PYTHON scripting was

used along with SALOME to automate geometry and mesh

generation and to prepare a model ready to be analysed

using FEBIO, including definitions of material properties

and prescription of loading and boundary conditions. The

user can specify anatomical parameters in a human-readable

configuration file, which includes overall dimensions of the

articulating region of interest, cartilage boundaries and

zonal delineation with ellipsoidal surface approximations

(for all zones, essentially allowing representation of non-

uniform thickness), and mesh density. The execution of

PYTHON scripts generates the model geometry and discretizes

the whole region (into a hexahedral mesh) with appropriate

element, node and surface sets. The user can also prescribe

depth-dependent material properties, e.g. elastic modulus,

Poisson’s ratio, hydraulic permeability, fibre properties [37],

all as a function of FE centroid location. The regional contact

force (to drive the model) and boundary conditions for the

margins of the model, e.g. confined or unconfined, are other

user-specified inputs. In an additional step, execution of

PYTHON scripts allows the generation of a human-readable

model input file that can be used with FEBIO to conduct FE

analysis. Results of such analyses quantify the mechanical

response of the cartilage at a given anatomical and mechanical

state and for preferred loading and boundary conditions, which

can also be fed-forward to tissue–cell scale models for further

simulations. This scripted environment allows generation of a

plethora of models with different characteristics (figure 3),

with minimal user interaction through the automation of

geometry, mesh and model definition processes.

3.1.2. Tissue – cell scale model developmentModels at the tissue–cell scales, when and if generated in an

expedited fashion, can allow large-scale estimation of chondro-

cyte mechanics at any desired location within a given zone of

the cartilage. Diverse anatomical variations exist for chon-

drocyte organization, shape and size across the layers of

cartilage and sometimes within the layers [17]. A pericellular

matrix surrounds the chondrocyte(s) to form a unit called a

chondron. In the superficial zone, the chondrocytes exhibit

flatter, ellipsoidal shapes, with relatively denser cellular distri-

bution [17]. Depending on the body region, e.g. for the ankle

[40], several cells may cluster to form a multicellular chondron.

In the transitional zone, chondrocyte distribution is sparser

[17]. Chondrocytes are nearly spherical and typically form

chondrons with a single cell. In the deep zone, stacking of chon-

drocytes into columnar chondrons is observed [17]. Shape, size

and organization (and variations in each) can influence the way

mechanical loading is transmitted to individual cells, and

therefore the cell’s mechanotransduction. The relative material

properties of chondrocytes, their pericellular environment and

the extracellular matrix are also important to resolve the mech-

anical load sharing problem. The pericellular matrix has a

significant role in the regulation of chondrocytes’ mechanical

environment and their mechanobiological response [41]. The

region is primarily of type VI collagen, which is exclusive

to this zone [41]. The extracellular matrix includes a zone-

dependent fibrous network of collagen of primarily type II

[2] (as described above).

A robust workflow for model development at the tissue–cell

scale can include parametrization of anatomical and mechanical

properties such that rapid and appropriate representations

specific to specimen, and tissue location and depth can be

constructed for prediction of chondrocyte mechanics in a


superficial

transitional

deep

cartilage

bone

PCM

chondrocyte

d

Nc

rcrp

Nv

pericellular matrix (PCM) material (E, v, k; x, b, –) extracellular matrix (EM) material (E, v, k; x, b, –)

chondrocyte material (E, v, k)wall boundary conditions

z

x

anatomical

discretization

mechanical

mesh density

region size (w × h × l)layer thicknesseschondrocyte density (Nv)chondrocyte locationchodrocyte orientationchondrocytes/chondron (Nc)chondrocyte radii (rc)

chondron spacing (d)PCM radii (rp)

w

EM

tissue zone of interest

parametric modelling

examples


Focus5:20140081

6


high-throughput manner. In a recent study, our team developed

an automated model generation procedure to represent cellular

organization, shape and size for zone-specific tissue–cell scale

models [42] (figure 4). PYTHON scripting was used to interact

with SALOME to automate geometry and mesh generation for a

rectangular volume with a distribution of spherical/ellipsoidal

inclusions. This geometrical model is a representative volume

for cellular-level description of the tissue. In a human-readable

file, the user can specify the tissue arrangement (a variable

rectangular volume, with single or multiple stacked layers),

cell shape and size (spherical or ellipsoidal with variable

radii), pericellular layer shape and size (spherical or ellipsoidal

with variable radii), cell arrangement (location and orientation,

clustering of multiple cells within a chondron) and mesh den-

sity. The geometrical properties can be specified explicitly, or

by providing cellular geometry and distribution statistics,

e.g. mean and standard deviations for cell radii, number of

cells per unit volume, and pericellular thickness. The execution

of PYTHON scripts provides discretized (tetrahedral) model

geometry in a human-readable form, including appropriate

element, node and surface sets to subsequently define loading

and boundary conditions. Through additional configuration

parameters, users can assign material properties for the extra-

cellular matrix in a similar fashion as was done for the

cartilage layer properties described for the joint–tissue scale

models. In addition, direction-dependent material properties

of the pericellular matrix can be prescribed, i.e. the region’s

fibre organization aligned tangentially to the chondrocyte sur-

face [43] as a function of element centroid location relative to

the chondrocyte. Finally, the loading and boundary conditions

can be prescribed using deformation gradient and fluid

pressure information (for biphasic case), e.g. from the results

of joint–tissue scale simulations. PYTHON scripting manages

the assembly of this heterogeneous information to generate

a human-readable model input file that can be used with

FEBIO to conduct FE analysis. Simulations can quantify the

mechanical response of individual or clusters of chondrocytes

for a given anatomical and mechanical property set and for

preferred loading and boundary conditions. Similar to the

pipeline provided for the joint–tissue scale modelling, this

scripted environment allows generation of a plethora of

models with varying characteristics individualized for the car-

tilage layer of interest (figure 4), with minimal user interaction

and through the automation of the geometry, mesh and model

definition processes.

Figure 4. Based on statistical descriptors of cellular anatomy and organization,model generation can be automated at the spatial scale of chondrocytes. Zonalcartilage features and chondrocyte properties influence the mechanics of cells.Chondrocyte size, shape and distribution vary largely between individual layersof the cartilage. These properties are also dependent on the location of the car-tilage within the joint and across joints. Cell size, shape and density can beprescribed in a statistical manner to generate a representative volume for adesired zone of interest. Definitions of parameters representing pericellularmatrix anatomy and material properties for chondrocytes, and pericellular andextracellular matrices complete the computational representation of the cartilageat the tissue – cell scale. Tissue-level mechanics information can be mapped onthe boundaries of the representative volume as loading and boundary conditionsto carry out simulations. This streamlined model generation approach provides theopportunity to build models in an expedited fashion. (Online version in colour.)

3.2. Scale couplingMultiscale analysis requires the transfer of mechanical infor-

mation from joint–tissue scale simulations (where cartilage

mechanics is quantified) to tissue–cell scale models (where

chondrocyte mechanics can be predicted). This data exchange

requires a multitude of assumptions. A fundamental assump-

tion is the required consistency of the tissue mechanical

behaviour between the joint–tissue and tissue–cell scales. At

the higher spatial scale, the cartilage may be represented by

homogenized material properties [24]. This constitutive

response should be equivalent to the average response of the

tissue–cell scale model, which represents the cartilage in more

detail, including the delineation of chondrocytes, pericellular

matrix and the extracellular matrix. Multiple strategies exist to

enforce this constraint. For example, the homogenized tissue

properties (of the joint–tissue scale model) and the properties


1.0

0.1 s 100 s0

2 × 10–5 2 × 10–3

0.4

100 × 100 × 100 µm

t = 5.0 s

chondrons

deformation time history

effective strain

fluid flux magnitude (mm s–1)

control casefirst-order

approximationsecond-order

approximation

control case

joint–tissue scale model

tissue–cell scale modelfirst-order

approximationsecond-order

approximation

deformation

cartilage

impermeablerigid wall

point ofinterest

Figure 5. Predictions of chondrocyte mechanics can be influenced by the assumptions of translating mechanical information obtained from joint – tissue scalemodels to drive models with cellular inclusions. A biphasic cartilage model with homogeneous material properties was used to simulate mechanics of thetissue under non-uniform loading and asymmetric permeability conditions [23]. Information on solid and fluid phase responses at the point of interest wasused to drive a tissue – cell scale model including 11 spherical chondrocytes with each surrounded by a pericellular matrix. A first-order approximation used defor-mation gradient, and fluid pressure and its gradient to prescribe boundary conditions. A second-order approximation used additional terms, including thedeformation and fluid pressure Laplacians. A control case was also simulated to obtain a reference solution. For this purpose, a region with the same cellularinclusions was explicitly represented in the joint – tissue scale model and macroscopic and cell scale solutions were carried out simultaneously. Reduced orderof data passing resulted in the loss of predictive capacity, both for mechanical metrics descriptive of the solid phase (effective strain) and the fluid phase(fluid flux magnitude). Additional quantitative details on the impact of data passing assumption during the prediction of chondrocyte mechanics can befound in Sibole et al. [23]. (Online version in colour.)


Focus5:20140081

7


of the extracellular matrix (of the tissue–cell scale model) can be

treated as the same. When the volume fraction of the extracellu-

lar matrix is large, the contribution of the pericellular matrix and

chondrocytes to the tissue scale response can be neglected. We

have used this assumption in our previous studies where

94.2% of the tissue–cell scale model was the extracellular

matrix (for an 11 cell model of the transitional zone of cartilage)

[23,25]. Alternatively, homogenization based on spherical

inclusions [44] can be adapted to estimate joint–tissue scale

macroscopic properties of the cartilage as an analytical function

of tissue–cell scale constituent properties and their volume frac-

tion. Finally, computational homogenization strategies can be

employed [45], where average mechanical response of a repre-

sentative volume element (in this case, the tissue–cell scale

model) can be calculated from simulations of uniform loading

cases, and a surrogate macroscopic constitutive law can be

fitted to determine homogenized material properties for the

joint–tissue scale model. This approach may be beneficial for

complicated and dense cellular arrangements [17] and for

explicit fibre-based modelling of the tissue matrix microstructu-

rally [31]. The gross impact of tissue heterogeneity and the

contributions of cells and fibres on tissue behaviour can be

incorporated within the continuum mechanics framework.

Another assumption when transferring information from

the higher spatial scale to the lower one is related to the level

of approximation of loading and deformation metrics that

need to be provided as boundary conditions for the tissue–

cell scale model. One option is to overlay the boundary of the

tissue–cell scale model within the joint–tissue scale model

geometry and linearly interpolate the solution of the macro-

scopic scale as displacement (and fluid pressure if necessary)

boundary conditions [46,47]. While this approach does not

require any further assumptions for the required detail of

data exchange, it may be cumbersome to do so, particularly

for large numbers of cellular-level analyses. Alternatively, one

can use loading and deformation information from a given

point within the cartilage of the joint–tissue scale model to

quantify chondrocyte deformations [25]. If the size of the

tissue–cell scale model is negligible when compared with

the characteristic size of the corresponding geometry in the

joint–tissue scale model, a first-order approximation may be

sufficient, i.e. using the deformation gradient at a given macro-

scale point to set boundary conditions of the cell scale model.

Unfortunately, for cartilage and chondrocytes, decoupling of

length scales can be problematic, potentially requiring a

second-order approximation for data exchange [45], i.e. using

the deformation gradient, which is a measure of local defor-

mations (shape change) at a given point, and deformation

Laplacian, which provides the spatial derivatives of the

deformation gradient. In a recent study, our team evaluated

the role of data passing assumptions in the characterization of

biphasic chondrocyte mechanics [23]. A 1 � 1 � 1 mm homo-

geneous and isotropic cartilage tissue underwent non-

uniform compression (imitating a curved indenter, 0–10%

nominal strain across the boundary; figure 5). Along the loading

axis, asymmetric permeability conditions were prescribed

(permeable at the indentation site, impermeable at the fixed

bottom); lateral walls of the model were confined and imper-

meable. Full loading was applied in 0.1 s with a relaxation

duration up to 100 s. The biphasic response of the model was

obtained using FEBIO. Loading and deformation informa-

tion at the centre point of this model was extracted to drive a


1.22

1.2238

1.24

number of cells

chondrocyte shape factor at steady state

2 1536

1189

47

10

5 14

131

12increasing representative volume

1 5 10 15 19

Figure 6. The tissue – cell scale model’s representative volume size may havean influence on the predictions of chondrocyte mechanics. Chondrocytes ofknee cartilage’s transitional zone were modelled by varying the representativevolume size. The cell density was kept constant at reported levels. The timehistory of cartilage mechanics was mapped on the representative volumesfrom the solution of a homogeneous biphasic cartilage model subjected tonon-uniform loading and asymmetric permeability conditions. A second-order data passing scheme was used. Chondrocyte shape factors wereobtained from biphasic simulations at steady state. The results indicatedthe role of representative volume size for consistent prediction of chondro-cytes’ mechanical environment. The solution for the 11-cell model ishighlighted. The edge length for this representative volume corresponds tothe commonly used 100 mm. While the illustrated differences in cellshape factor may not have significance in relation to other errors associatedwith modelling assumptions, the use of other cell scale biomechanical metricsand the exploration of different zones of cartilage or different types of tissueswarrant execution of similar analyses. (Online version in colour.)


Focus5:20140081

8


tissue–cell scale model. The lower scale model was a 100 mm �100 mm � 100 mm representative volume with spherical cellu-

lar inclusions representative of the transitional layer of the

cartilage [48] (11 cells within the volume; chondrocyte radius

of 5 mm and pericellular thickness of 2.5 mm; figure 5). Chon-

drocyte, pericellular matrix and extracellular matrix material

properties were homogeneous, isotropic and biphasic. Two

types of data passing schemes were employed to test the

required fidelity when prescribing tissue–cell scale boundary

conditions. First, a first-order Taylor series expansion was

used to map macroscale deformation gradient, and fluid

pressure and its gradient to the boundaries of the tissue–cell

scale model. Then, a second-order Taylor series expansion,

with additional terms containing the deformation and fluid

pressure Laplacians, was used. Results of both these cases

were compared against the predictions of a control model,

where the joint–tissue scale model had a region at its centre

explicitly incorporating the chondrocyte and pericellular

matrix representation exactly as in the separate tissue–cell

scale model. While the results were not necessarily compared

against the behaviour of actual cells, this investigation revealed

that data passing assumptions may lead to additional misre-

presentation of the chondrocytes’ mechanical environment, at

least for the case presented here (figure 5). It is advised to use

a higher-order data passing scheme or one that relies on interp-

olation (albeit with the increased difficulty for high-throughput

analysis). It should also be noted that for each additional term

included in a Taylor series approach, the order of the required

tensor increases by one. In three dimensions, this will contribute

3nþ 1 additional coefficients to be calculated for both the solid

and fluid phases, where n is the order of the series.

Another challenge for cell scale analysis has been the selec-

tion of an appropriate volume for the tissue–cell scale model.

Ideally, the representative volume should be small to allow

the scale separation assumption in that the characteristic

lengths of higher and lower scales differ by orders of magni-

tude. The representative volume should also be large enough

to include characteristic geometrical features of the region’s cel-

lular organization. Traditionally, computational simulations of

chondrocytes used models with an edge length of 100 mm [49],

originated from the study by Guilak & Mow [46]. This model

and its successors placed a single chondron at the centre of

the modelling volume, commonly overlooking the actual cell

density of the cartilage zone. One has multiple options when

building models which keep the volume ratio of the chondro-

cytes and extracellular matrix the same. Cartilage can be

represented at the tissue–cell scale using a single cell model

with a small overall model volume or by placing multiple

cells in larger volumes. Ideally, all these models should result

in similar chondrocyte mechanics predictions. Yet, with the

increased size of the representative volume, mechanical

interactions between the cells can be investigated and the influ-

ence of a cell’s location within the neighbourhood of the point

of interest can be explored. Our ongoing work explored the

influence of representative volume size on the prediction of

chondrocyte loading and deformation [50]. Tissue–cell scale

models representative of chondrocyte density of the cartilage’s

transitional zone were developed, starting with representative

volumes containing a single cell, through 20 cells (figure 6).

The model edge length varied between 30 and 120 mm to

keep cell density constant. The cells were randomly distributed

within the representative volume maintaining a prescribed

minimum distance between cells and the volume boundaries.

For each level of cellular representation, multiple models

were generated and solved providing mechanics information

for at least 130 chondrocytes in total. Chondrocytes were

spherical, each with a radius of 5 mm and a pericellular thick-

ness of 2.5 mm. Biphasic simulations were conducted with

loading and boundary conditions reflecting the deformation

gradient and fluid pressure (and their derivatives based on a

second-order data passing scheme) obtained from macro-

level simulations. The macroscale joint–tissue model was

loaded by a non-uniform surface displacement with asym-

metric fluid behaviour (described above) [23]. The average

predicted cell metrics (and their variability) became more con-

sistent as the representative volume size included larger

number of cells, e.g. for chondrocyte shape factor (defined in

[51]; figure 6). Differences can be attributed to the variability

of cell location within the representative volume and/or cell-

to-cell mechanical interactions. Simulations were completed

approximately 10 times faster at smaller representative

volumes with lower cell numbers. However, for this specific

illustrative case, distribution of cell mechanical metrics was

better captured as the volume of interest increased—possibly

providing location-dependent variability of cell mechanics

and cell-to-cell mechanical interactions.

3.3. ComputingSimulations both at the joint–tissue scale and the tissue–cell

scale can be inherently computationally intensive. Geometri-

cal and material nonlinearity, consideration of multiple


joint–tissue scaleeffective strain

0.2

0

parametric analysis at joint–tissue scale

M conditions

N conditions

M × N interactions

P points in tissue space

effective strain (cartilage)

effe

ctiv

e st

rain

(ch

ondr

ocyt

e)

0.4

00 0.2

+ + +...

+ + +...

+ + +...

+ + +...

+ + +...

parametric analysis at tissue–cell scale

parametric analysis at both scales

feed-forward multiscale analysis

Figure 7. Feed-forward modelling approach decouples joint – tissue andtissue – cell scale simulations and facilitates adaptation of various computingstrategies in order to expedite solutions. Probabilistic analysis, sensitivity studiesand uncertainty quantification can be conducted through the parallelization ofparametric simulations at individual scales or both (top three panels). Simu-lations can also be performed in parallel when one post-processes joint –tissue scale results to obtain chondrocyte response at a large number ofpoints within the cartilage (bottom panel). For example, FE analysis of theknee joint quantified femoral and tibial cartilage deformations under a compres-sive load of one body weight [25]. For each element centroid, a nonlinearlyelastic model of a tissue – cell scale model was solved. Utilization of parallel pro-cessing in a high-performance computing resource reduced the solution time foralmost 8000 models to less than a day, which would otherwise take more than70 days when using a single computer. (Online version in colour.)


Focus5:20140081

9


material phases, and large mesh sizes may all contribute to

the added cost of conducting simulations at the single spatial

scale. Multiscale analysis further complicates this issue as

multiples of models need to be solved for quantification

of cartilage and chondrocyte mechanics. Nonetheless, the

post-processing-based feed-forward approach to multiscale

simulations provides various opportunities to tackle the

computing burden. From a user’s perspective, multiple simu-

lation strategies may exist (figure 7): parametric studies at the

joint–tissue scale, parametric studies at the tissue–cell scale,

a combination of both, and lastly large-scale simulations at

the tissue–cell scale for characterization of chondrocyte

mechanical environment for a given solution of a joint–

tissue scale model. In the first use case, one may be interested

in the cartilage mechanics for a set of loading and boundary

conditions or in the sensitivity of the cartilage mechanical

environment to variations in anatomical and mechanical

properties. Similarly, as an example of the second use case,

one may be interested in chondrocyte mechanics as a function

of macroscale deformations or owing to variations in ana-

tomical and mechanical properties of the chondrocytes,

pericellular matrix and extracellular matrix. These analyses

can be conducted in conjunction to evaluate the changes in

multiscale load sharing path as model parameters at all

scales are perturbed simultaneously (third use case). All

these use cases can lead into probabilistic analysis [20],

i.e. for assessment of population variations, sensitivity studies

and uncertainty quantification. The final use case illustrates

the multiscale feed-forward analysis approach. First, a

joint–tissue scale model is solved. At each point of interest

in that model, e.g. all locations across all zones of the carti-

lage, tissue–cell scale models are simulated to map

chondrocyte mechanical metrics on the topology of the

macroscale tissue anatomy. This may require the solution

of hundreds to thousands of tissue–cell scale models, all

driven by the results of a single joint–tissue scale model. In

all the aforementioned cases, simulations are loosely coupled,

i.e. minimal or no communication is required between

models of the same scale. Massively scalable parallel comput-

ing strategies can be employed by distributing simulation

workload to independent computing nodes in order to expe-

dite the whole solution process. Albeit various computational

strategies, case-specific model simplifications may still be

necessary to balance the requirements of a high-throughput

simulation workflow and scientific accuracy.

In a previous study, an approach to tackle large-scale simu-

lations through the use of loosely coupled computing strategies

was presented for the prediction of chondrocyte deformations

in tibial and femoral cartilage [25]. First, a tibiofemoral joint

model was solved to obtain tissue-level deformation metrics

for the cartilage during the application of a compressive load

with a magnitude of one body weight. This model incorpor-

ated the anatomical detail of the joint articulation and the

passive tissue structures, e.g. ligaments, and a homogeni-

zed representation of tissue material properties, i.e. apparent

properties at the macroscale. Nonlinearly elastic FE analysis

provided deformation gradients at each element centroid

within the transitional layers of femoral and tibial cartilage.

For each element centroid, a tissue–cell scale model was

solved to obtain a joint-level distribution of chondrocyte

deformations within femoral and tibial cartilage. The tissue–

cell scale model was a 100 � 100 � 100 mm representative

volume with spherical cellular inclusions and homogeneous

and isotropic elastic properties for chondrocytes, pericellular

matrix and extracellular matrix. The cellular organization rep-

resented the transitional zone [48]: 11 cells within the volume;



Focus5:20140081

10


chondrocyte radius of 5 mm and pericellular thickness of

2.5 mm. The boundary conditions of the model reflected defor-

mations based on a first-order data passing scheme. This

analysis required the solution of 7822 tissue–cell scale

models (same model simulated with different boundary con-

ditions extracted from 7822 locations in tibial and femoral

cartilage). Exploiting the parallel nature of feed-forward

analysis, these models were solved in a high-performance com-

puting environment (multiple compute nodes using 101

threads on Glenn Cluster of the Ohio Supercomputer Centre),

using FEBIO for FE analysis. At the time, Glenn Cluster pro-

vided up to 9572 compute cores, ranging in frequencies of

2.4–2.6 GHz, offering a peak performance of more than 75 ter-

aflops. Through parallel processing, solution of 7822 models

resulted in a wall-clock time of 19 h. If using a single CPU,

estimated wall-clock time would be 1735.1 h (72.3 days), indi-

cating a prohibitive computing burden. A study of scalability

was not performed, but because each tissue-scale model is

entirely independent, the limiting factor should only be the

initial distribution of problems to computer nodes, and like-

wise, the pooling of results after simulations have completed.

Some network overhead is expected but it is ideally low, and

thus scaling to many thousands of nodes is highly feasible.

The feed-forward analysis provided the possibility for this

loosely coupled parallel processing. In turn, timely prediction

of chondrocyte mechanics for large geometrical regions of

femoral and tibial cartilage was attainable (figure 7).

For cartilage and chondrocyte mechanics, nonlinear simu-

lations incorporating the biphasic or multiphasic nature of

the tissue and cell behaviour may also increase the compu-

tational cost. It should be noted that FE analysis software

provides capabilities for multithreading (on a single comput-

ing node) [52–54] and parallel processing (on a number of

distributed computing nodes) [53,54], during the generation

of system stiffness matrices and/or the solution of linearized

system equations. The use of these built-in computing strat-

egies may expedite single simulation of a given joint–tissue

or tissue–cell scale model. FEBIO, the FE analysis package

used in our modelling and simulation workflow, provides

multithreading on shared memory computing architectures

[52]. By relying on PARDISO, a shared-memory multiproces-

sing parallel direct sparse solver for linear systems of

equations [55], solution times can be reduced on a given

compute node equipped with multiple processors.

3.4. Post-processingFE analyses, both at the joint–tissue and tissue–cell scales,

result in a large set of variables to evaluate the mechanical

environment of the cartilage and chondrocytes. However, it

may be challenging to interpret the raw simulation results

without post-processing, where desirable biomechanical

metrics within the context of research question or clinical pur-

pose can be calculated. Similarly, movement of information

from joint–tissue simulations down to tissue–cell scale

models requires preparation of the raw simulation results of

the higher spatial scale in a form amenable to use in the

models of the lower spatial scale. In this section, post-proces-

sing solutions relying on PYTHON scripting are provided for

simulations conducted using freely accessible FEBIO software.

At the spatial scale of joint–tissue, interpretation of FE

analysis results, e.g. for individual zones of the cartilage, is

possible by using principal stress and strain (solid phase)

and fluid flux and pressure (fluid phase) distributions,

which are common for the evaluation of biphasic mechanics.

For multiscale analysis, more detailed information about the

mechanical loading and deformation is needed. One may

want to solve a tissue–cell scale model associated with a

point in the macroscale model, e.g. at an element centroid.

The complete mechanical state of the point of interest needs

to be mapped to the tissue–cell scale model as boundary con-

ditions. For biphasic simulations for example, the time

histories of the deformation gradient (and possibly the defor-

mation Laplacian) and the fluid pressure (and its derivatives)

will be necessary for data passing (also see discussion on

scale coupling above). Such information is not readily avail-

able in FE analysis results and it needs to be derived from

nodal displacement and fluid pressure information. To facili-

tate this procedure, PYTHON scripts were developed to parse

through FEBIO output files to extract raw nodal and element

results for desired sets, e.g. elements for which multiscale

analysis would be performed [23]. Using the FE isopara-

metric formulation, the nodal displacements and fluid

pressures can be used to compute time histories of defor-

mation gradient, fluid pressure and their spatial derivatives

for desired points of the joint–tissue scale models. This infor-

mation can subsequently be provided to tissue–cell scale

models in a streamlined human-readable form. For illus-

tration of joint–tissue scale post-processing, the solution of

a 1 � 1 � 1 mm homogeneous and isotropic cartilage tissue

was explored (figure 8). A summary of the model and simu-

lation conditions is provided here, and further details are

available elsewhere [23]. The cartilage model underwent

non-uniform compression, essentially representing the influ-

ence of a curved indenter (0–10% nominal strain across the

boundary). The biphasic response was sought, with asym-

metric permeability conditions along the loading axis

(permeable at the indentation site, impermeable at the fixed

bottom), and with confined and impermeable walls. Full

loading was applied in 0.1 s with a relaxation duration up

to 100 s. The time histories of compressive strain and fluid

pressure for the centroid of the element at the centre of the

tissue are shown in figure 8.

At the spatial scale of tissue–cell, interpretation of FE

analysis results, e.g. for the chondrocytes, pericellular

matrix and extracellular matrix, is also possible through the

evaluation of principal stress and strain (solid phase) and

fluid flux and pressure (fluid phase) distributions. This infor-

mation can provide an understanding of nearly focal stresses

and strains on chondrocyte surfaces and within, which may

indicate potential failure and mechanical signalling mechan-

isms. Yet, an understanding of multi-modal loading and

overall deformations of individual cells and/or populations

of cells may be necessary. Metrics such as cell height and

width, aspect ratio and volume (undeformed and deformed)

have commonly been used for describing experimental data

[11] and simulation results [56], with the intention to associ-

ate cell mechanics with cell damage and mechanical stimuli.

Such descriptors, however, are not readily available in raw

simulation results, and automated procedures are necessary

if these variables need to be evaluated for hundreds of cells

and for a large number of simulation conditions. Motivated

by this need, PYTHON scripts were developed to parse FEBIO

output files to extract raw results for nodes, elements and sur-

faces representing each chondrocyte [23,51]. The analysis

scripts are capable of calculating the volume of each cell,


100

0

1.8

chondrons

00 20 40 60 80 100

volu

me

aver

aged

eff

ectiv

ech

ondr

ocyt

e st

rain

chon

droc

yte

mas

s ex

chan

gera

te (

pgs–1

)

time (s)

0 20 40 60 80 100time (s)

100 × 100 × 100 µm

tissue–cell scale model

Figure 9. Post-processing of tissue – cell scale simulations can provide descrip-tors of chondrocytes’ multiphase mechanical behaviour in a usable form. Suchinformation is not readily available in raw results. In this case, time histories ofeffective chondrocyte strain and mass exchange rate are shown for a total of11 cells. The tissue – cell scale model is representative of the transitionalzone of the knee cartilage. Cartilage mechanics was mapped on this represen-tative volume from the solution of a homogeneous biphasic cartilage modelsubjected to non-uniform loading and asymmetric permeability conditions.A second-order data passing scheme was used. (Online version in colour.)

1.0

0.1 s 100 s

12

0

deformation time historydeformation

cartilage

impermeablerigid wall

0

–0.14

point ofinterest

joint–tissue scale model

time (s)0 20 40 60 80 100

time (s)0 20 40 60 80 100

stra

in a

long

load

ing

dire

ctio

nfl

uid

pres

sure

(M

Pa)

Figure 8. If one is interested in multiphasic mechanics of cartilage at thejoint – tissue scale, post-processing of simulation results can providecommon descriptors of the tissue’s mechanical environment at any desiredlocation. In this case, time histories of compressive strain and fluid pressurewere extracted for a point at the centre of a cartilage region (1 � 1 �1 mm). A non-uniform displacement profile was prescribed at the top.Impermeable walls confined the tissue. (Online version in colour.)


Focus5:20140081

11


and using volume-weighted averaging, the effective stress

and strain, and maximum shear strain. The net mass

exchange rate for each chondrocyte is also calculated by the

surface integral of fluid flux. This metric, obtained from

biphasic analysis, can be an indicator of cell-to-cell signalling.

More common cellular deformation metrics are calculated by

first finding the mass moment inertia of each cell and then

finding the eigenvalues of this tensor, which represent the

principal moments of inertia [57]. These eigenvalues are pro-

cessed to calculate axis lengths of the inertial ellipsoid, which

in turn provide a shape factor from their ratio [51]. Further

processing provides chondrocyte height, width and depth,

and therefore aspect ratio [51]. From the results of biphasic

FE analysis, time histories of all these cell scale metrics are

available in a human-readable form. Post-processing capabili-

ties at the tissue–cell scale were illustrated using the results of

a 100 � 100 � 100 mm representative volume with spherical

cellular inclusions and homogeneous and isotropic biphasic

properties for chondrocytes, pericellular matrices and

extracellular matrix (figure 9). The cellular organization

represented the transitional zone [48]: 11 cells within the

volume; chondrocyte radius of 5 mm and pericellular thickness

of 2.5 mm. The boundary conditions of the model reflected

deformation gradient and fluid pressure (and their derivatives

based on a second-order data passing scheme) at the centre of a

joint–tissue scale model described previously in this section.

Extracellular matrix properties were kept the same as the

macroscopic properties of the joint–tissue scale. Time histories

of various biomechanical loading and deformation metrics for

each chondrocyte are provided in figure 9.

4. Concluding remarksThe vision and initial realization of a high-throughput mod-

elling and simulation workflow for large-scale analysis of



Focus5:20140081

12


cartilage and chondrocyte mechanics was provided in this

document. Various strategies to realize individual com-

ponents of the modelling framework were implemented

with a specific focus on the automation of developing

models at the joint–tissue and tissue–cell scales that are

capable of incorporating subject-specific and regional vari-

ations in anatomical and mechanical parameters. Some of

the technical challenges associated with multiscale analysis

were addressed. In separate studies, the requirement for

higher-order data passing from the joint–tissue scale down

to the tissue–cell scale was emphasized, and the appropriate-

ness of the selection of the tissue–cell scale model size was

confirmed. From a usability perspective, the need for stream-

lined extraction of important biomechanical metrics, for the

evaluation of cartilage and chondrocyte mechanics and for

feed-forward movement of the data through the spatial

scales was met. Computing strategies, in particular the use

of parallel processing, were presented as a means to exploit

the loosely coupled nature of post-processing-based multi-

scale analysis to conduct thousands of simulations in an

expedited fashion.

The outlined modelling and simulation approach focused

on a feed-forward analysis strategy relying on the sequential

solution and post-processing of computational models at the

joint–tissue scale and then at the tissue–cell scale. This

approach was found to be suitable for solving the load shar-

ing problem starting from the higher spatial scales, and

determining the loads and deformations at various cartilage

layers and at the level of individual chondrocytes. A signifi-

cant advantage of this feed-forward approach is related to

the reduced demand on computing as macro- and microscale

models need to be solved only once for a desired loading con-

dition at preferred locations. This advantage fundamentally

relies on the assumption that tissue representations at multiple

scales are mechanically consistent, i.e. they provide the same

overall constitutive behaviour. It may not provide the struc-

ture–function relationships, which can be accomplished

through a feedback loop, i.e. by computational homogeniz-

ation that relies on iterative solutions of lower scale models

during the solution process of the model at the higher spatial

scale [45]. It should be noted that such a feedback loop is not

necessary to obtain the mechanical environment of cartilage

and chondrocytes for a given tissue state. It will be necessary

when constitutive properties change as a result of tissue

growth and damage [31]. Nonetheless, the tools presented

in this study can be adapted for such purposes. It is worth

mentioning that this study did not provide an understanding

of error propagation when one uses the proposed multiscale

modelling workflow. Owing to the feed-forward nature of

the simulations, it is anticipated that the errors will accumu-

late in an additive manner as one moves from the higher

spatial scale of the joint down to the scale of chondrocytes.

The components of the modelling and simulation workflow

were prototyped using PYTHON scripting, for pre-processing

of the models using user-specified parameters and for post-

processing of the simulation results. The integrated and

scriptable computer-aided design package SALOME was used

in an unsupervised manner for generation of models. Likewise,

FEBIO, FE analysis software for biomechanics [34], was used to

conduct simulations. These readily available, open source and

freely accessible pieces of software provided the means for

quick prototyping. Nonetheless, it is possible to integrate

model development, simulation and post-processing tools

into a turn-key modelling and simulation platform. Some desir-

able features for the expansion of this framework are noted as

follows. Geometrical model generation procedures can use

explicitly defined individual anatomy, e.g. from image segmen-

tation, for tissue representation at the macroscale, and for cell

representation at the microscale. For models at the joint–

tissue scale, the representation of additional components, e.g.

meniscus, will be necessary when the desired contact region

of interest becomes wider. For models at the tissue–cell scale,

incorporation of cell membrane may be needed to investigate

the influence this structure’s permeability on the mechanical

behaviour of the chondrocyte. Robust physics formulations pro-

vided in FEBIO [32] can also allow the modelling and simulation

workflow to include multiphasic behaviour of the cartilage and

chondrocytes beyond biphasic analysis, to incorporate the mech-

anical effects of charged particles and proteoglycans, namely

osmotic pressure gradients and swelling.

The capacity to explore cartilage mechanics for any joint

of interest, at any region of contact within the joint, for any

zone within the tissue and under different loading condi-

tions when supported by high-performance computing can

empower large-scale analysis of cartilage and chondrocyte

function in health and disease. Population responses can be

quantified based on probabilistic analysis and in-depth sensi-

tivity studies can be performed to understand the role of

biomechanical markers in cartilage function. It is anticipated

that this framework will be used to understand cartilage

and chondrocyte mechanics during ageing. Ageing mani-

fests anatomical and mechanical changes at multiple scales

[13,16,19,58], which in turn perturb the mechanical envi-

ronment and therefore associated biological responses to

mechanical stimuli. Similarly, this modelling and simulation

workflow will likely be a significant toolset to evaluate the

complicated mechanical state of the tissue and cells during

the onset and progression of osteoarthritis.

5. Access to data, models and scriptsVarious scripts, models, data and simulation results can be

found at the project site, https://simtk.org/home/j2c, in

the downloads section.

Acknowledgements. This study was conducted at the Cleveland Clinicand was previously presented at the Seventh World Congress of Bio-mechanics, 6–11 July 2014, Boston, MA, USA [59]. The documentprovides a synthesis of our group’s published and unpublishedwork on in silico mechanics of cartilage and chondrocytes. Theauthors acknowledge Simbios, NIH National Center for BiomedicalComputing at Stanford University, http://simbios.stanford.edu/(project hosting); Ohio Super Computer Center, https://www.osc.edu/ and Extreme Science and Engineering Discovery Environment,https://www.xsede.org/ (high-performance computing); FEBIO,http://www.febio.org/ (FE analysis software) [34]; SALOME, http://www.salome-platform.org/ (integrated software for computer-aided design); PYTHON, https://www.python.org/ (scripting). Theauthors are also grateful for in-depth discussions with JasonHalloran and Snehal Chokhandre (Cleveland Clinic), Cees Oomensand Rene van Donkelaar (Eindhoven University of Technology),Jeff Weiss and Steve Maas (University of Utah), and Farshid Guilak(Duke University).

Funding statement. The research activities were supported by theNational Institutes of Health, in particular by the National Instituteof Biomedical Imaging and Bioengineering (R01EB009643; PrincipalInvestigator: A.E., https://simtk.org/home/j2c) and partially bythe National Institute of General Medical Sciences (R01GM104139;Principal Investigator: A.E., https://simtk.org/home/openknee).

https://simtk.org/home/j2c


http://simbios.stanford.edu/

http://simbios.stanford.edu/

https://www.osc.edu/



https://www.xsede.org/

https://www.xsede.org/










https://simtk.org/home/openknee

https://simtk.org/home/openknee


13


References


Focus5:20140081

1. Halloran JP, Sibole S, van Donkelaar CC, vanTurnhout MC, Oomens CWJ, Weiss JA, Guilak F,Erdemir A. 2012 Multiscale mechanics of articularcartilage: potentials and challenges of couplingmusculoskeletal, joint, and microscalecomputational models. Ann. Biomed. Eng. 40,2456 – 2474. (doi:10.1007/s10439-012-0598-0)

2. Heinegard D, Oldberg A. 1989 Structure and biologyof cartilage and bone matrix noncollagenousmacromolecules. FASEB J. 3, 2042 – 2051.

3. Michalek AJ, Iatridis JC. 2007 A numerical study todetermine pericellular matrix modulus and evaluateits effects on the micromechanical environment ofchondrocytes. J. Biomech. 40, 1405 – 1409. (doi:10.1016/j.jbiomech.2006.05.025)

4. Torzilli PA, Deng X-H, Ramcharan M. 2006 Effect ofcompressive strain on cell viability in staticallyloaded articular cartilage. Biomech. Model.Mechanobiol. 5, 123 – 132. (doi:10.1007/s10237-006-0030-5)

5. Grodzinsky AJ, Levenston ME, Jin M, Frank EH.2000 Cartilage tissue remodeling in response tomechanical forces. Annu. Rev. Biomed. Eng. 2,691 – 713. (doi:10.1146/annurev.bioeng.2.1.691)

6. Kurz B, Lemke AK, Fay J, Pufe T, Grodzinsky AJ,Schunke M. 2005 Pathomechanisms of cartilagedestruction by mechanical injury. Ann. Anat. 187,473 – 485. (doi:10.1016/j.aanat.2005.07.003)

7. O’Conor CJ, Case N, Guilak F. 2013 Mechanicalregulation of chondrogenesis. Stem Cell Res. Ther. 4,61. (doi:10.1186/scrt210)

8. Martin JA, Brown TD, Heiner AD, Buckwalter JA.2004 Chondrocyte senescence, joint loading andosteoarthritis. Clin. Orthop. S96 – S103. (doi:10.1097/01.blo.0000143818.74887.b1)

9. Lawrence RC et al. 2008 Estimates of the prevalenceof arthritis and other rheumatic conditions in theUnited States. II. Arthritis Rheum. 58, 26 – 35.(doi:10.1002/art.23176)

10. Han S-K, Colarusso P, Herzog W. 2009 Confocalmicroscopy indentation system for studying in situchondrocyte mechanics. Med. Eng. Phys. 31, 1038 –1042. (doi:10.1016/j.medengphy.2009.05.013)

11. Abusara Z, Seerattan R, Leumann A, Thompson R,Herzog W. 2011 A novel method for determiningarticular cartilage chondrocyte mechanics in vivo.J. Biomech. 44, 930 – 934. (doi:10.1016/j.jbiomech.2010.11.031)

12. Moo EK, Han SK, Federico S, Sibole SC, Jinha A, AbuOsman NA, Pingguan-Murphy B, Herzog W. 2014Extracellular matrix integrity affects the mechanicalbehaviour of in situ chondrocytes undercompression. J. Biomech. 47, 1004 – 1013. (doi:10.1016/j.jbiomech.2014.01.003)

13. Hudelmaier M, Glaser C, Hohe J, Englmeier KH,Reiser M, Putz R, Eckstein F. 2001 Age-relatedchanges in the morphology and deformationalbehavior of knee joint cartilage. Arthritis Rheum.44, 2556 – 2561. (doi:10.1002/1529-0131(200111)44:11,2556::AID-ART436.3.0.CO;2-U)

14. McLeod MA, Wilusz RE, Guilak F. 2013 Depth-dependent anisotropy of the micromechanicalproperties of the extracellular and pericellularmatrices of articular cartilage evaluated via atomicforce microscopy. J. Biomech. 46, 586 – 592. (doi:10.1016/j.jbiomech.2012.09.003)

15. Deneweth JM, Newman KE, Sylvia SM, McLean SG,Arruda EM. 2013 Heterogeneity of tibial plateaucartilage in response to a physiological compressivestrain rate. J. Orthop. Res. 31, 370 – 375. (doi:10.1002/jor.22226)

16. Temple MM, Bae WC, Chen MQ, Lotz M, Amiel D,Coutts RD, Sah RL. 2007 Age- and site-associatedbiomechanical weakening of human articularcartilage of the femoral condyle. OsteoarthritisCartilage 15, 1042 – 1052. (doi:10.1016/j.joca.2007.03.005)

17. Quinn TM, Hunziker EB, Hauselmann H-J. 2005Variation of cell and matrix morphologies inarticular cartilage among locations in the adulthuman knee. Osteoarthritis Cartilage 13, 672 – 678.(doi:10.1016/j.joca.2005.04.011)

18. Quinn TM, Hauselmann H-J, Shintani N, Hunziker EB.2013 Cell and matrix morphology in articular cartilagefrom adult human knee and ankle joints suggestsdepth-associated adaptations to biomechanical andanatomical roles. Osteoarthritis Cartilage 21,1904 – 1912. (doi:10.1016/j.joca.2013.09.011)

19. Steklov N, Srivastava A, Sung KLP, Chen PC, LotzMK, D’Lima DD. 2009 Aging-related differences inchondrocyte viscoelastic properties. Mol. Cell.Biomech. 6, 113 – 119. (doi:10.3970/mcb.2009.006.113)

20. Laz PJ, Browne M. 2010 A review of probabilisticanalysis in orthopaedic biomechanics. Proc. Inst.Mech. Eng. H 224, 927 – 943. (doi:10.1243/09544119JEIM739)

21. Halloran JP, Erdemir A. 2011 Adaptive surrogatemodeling for expedited estimation of nonlineartissue properties through inverse finite elementanalysis. Ann. Biomed. Eng. 39, 2388 – 2397.(doi:10.1007/s10439-011-0317-2)

22. Causin P, Sacco R, Verri M. 2013 A multiscaleapproach in the computational modeling of thebiophysical environment in artificial cartilage tissueregeneration. Biomech. Model. Mechanobiol. 12,763 – 780. (doi:10.1007/s10237-012-0440-5)

23. Sibole SC, Maas S, Halloran JP, Weiss JA, Erdemir A.2013 Evaluation of a post-processing approach formultiscale analysis of biphasic mechanics ofchondrocytes. Comput. Methods Biomech. Biomed.Eng. 16, 1112 – 1126. (doi:10.1080/10255842.2013.809711)

24. Henak CR, Anderson AE, Weiss JA. 2013 Subject-specific analysis of joint contact mechanics:application to the study of osteoarthritis andsurgical planning. J. Biomech. Eng. 135, 021003.(doi:10.1115/1.4023386)

25. Sibole SC, Erdemir A. 2012 Chondrocytedeformations as a function of tibiofemoral joint

loading predicted by a generalized high-throughputpipeline of multi-scale simulations. PLoS ONE 7,e37538. (doi:10.1371/journal.pone.0037538)

26. Rudert MJ, Ellis BJ, Henak CR, Stroud NJ, PedersonDR, Weiss JA, Brown TD. 2014 A new sensor formeasurement of dynamic contact stress in the hip.J. Biomech. Eng. 136, 035001. (doi:10.1115/1.4026103)

27. Liu F, Kozanek M, Hosseini A, Van de Velde SK, GillTJ, Rubash HE, Li G. 2010 In vivo tibiofemoralcartilage deformation during the stance phase ofgait. J. Biomech. 43, 658 – 665. (doi:10.1016/j.jbiomech.2009.10.028)

28. Erdemir A, McLean S, Herzog W, van den Bogert AJ.2007 Model-based estimation of muscle forcesexerted during movements. Clin. Biomech. 22,131 – 154. (doi:10.1016/j.clinbiomech.2006.09.005)

29. Besier TF, Gold GE, Beaupre GS, Delp SL. 2005 Amodeling framework to estimate patellofemoraljoint cartilage stress in vivo. Med. Sci. Sports Exerc.37, 1924 – 1930. (doi:10.1249/01.mss.0000176686.18683.64)

30. Dowling EP, Ronan W, McGarry JP. 2013Computational investigation of in situ chondrocytedeformation and actin cytoskeleton remodellingunder physiological loading. Acta Biomater. 9,5943 – 5955. (doi:10.1016/j.actbio.2012.12.021)

31. Hadi MF, Sander EA, Barocas VH. 2012 Multiscalemodel predicts tissue-level failure from collagenfiber-level damage. J. Biomech. Eng. 134, 091005.(doi:10.1115/1.4007097)

32. Ateshian GA, Maas S, Weiss JA. 2013 Multiphasicfinite element framework for modeling hydratedmixtures with multiple neutral and charged solutes.J. Biomech. Eng. 135, 111001. (doi:10.1115/1.4024823)

33. Wilson W, van Donkelaar CC, van Rietbergen R,Huiskes R. 2005 The role of computational modelsin the search for the mechanical behavior anddamage mechanisms of articular cartilage. Med.Eng. Phys. 27, 810 – 826. (doi:10.1016/j.medengphy.2005.03.004)

34. Maas SA, Ellis BJ, Ateshian GA, Weiss JA. 2012FEBio: finite elements for biomechanics. J. Biomech.Eng. 134, 011005. (doi:10.1115/1.4005694)

35. Li J, Stewart TD, Jin Z, Wilcox RK, Fisher J. 2013 Theinfluence of size, clearance, cartilage properties,thickness and hemiarthroplasty on the contactmechanics of the hip joint with biphasic layers.J. Biomech. 46, 1641 – 1647. (doi:10.1016/j.jbiomech.2013.04.009)

36. Chen SS, Falcovitz YH, Schneiderman R, MaroudasA, Sah RL. 2001 Depth-dependent compressiveproperties of normal aged human femoral headarticular cartilage: relationship to fixed chargedensity. Osteoarthritis Cartilage 9, 561 – 569.(doi:10.1053/joca.2001.0424)

37. Ateshian GA, Rajan V, Chahine NO, Canal CE, HungCT. 2009 Modeling the matrix of articular cartilageusing a continuous fiber angular distribution

http://dx.doi.org/10.1007/s10439-012-0598-0

http://dx.doi.org/10.1016/j.jbiomech.2006.05.025


http://dx.doi.org/10.1007/s10237-006-0030-5

http://dx.doi.org/10.1007/s10237-006-0030-5

http://dx.doi.org/10.1146/annurev.bioeng.2.1.691

http://dx.doi.org/10.1016/j.aanat.2005.07.003

http://dx.doi.org/10.1186/scrt210

http://dx.doi.org/10.1097/01.blo.0000143818.74887.b1

http://dx.doi.org/10.1097/01.blo.0000143818.74887.b1

http://dx.doi.org/10.1002/art.23176

http://dx.doi.org/10.1016/j.medengphy.2009.05.013





http://dx.doi.org/10.1002/1529-0131(200111)44:11%3C2556::AID-ART436%3E3.0.CO;2-U









http://dx.doi.org/10.1002/jor.22226

http://dx.doi.org/10.1002/jor.22226

http://dx.doi.org/10.1016/j.joca.2007.03.005




http://dx.doi.org/10.3970/mcb.2009.006.113

http://dx.doi.org/10.3970/mcb.2009.006.113

http://dx.doi.org/10.1243/09544119JEIM739

http://dx.doi.org/10.1243/09544119JEIM739

http://dx.doi.org/10.1007/s10439-011-0317-2

http://dx.doi.org/10.1007/s10237-012-0440-5

http://dx.doi.org/10.1080/10255842.2013.809711

http://dx.doi.org/10.1080/10255842.2013.809711

http://dx.doi.org/10.1115/1.4023386

http://dx.doi.org/10.1371/journal.pone.0037538

http://dx.doi.org/10.1115/1.4026103

http://dx.doi.org/10.1115/1.4026103



http://dx.doi.org/10.1016/j.clinbiomech.2006.09.005

http://dx.doi.org/10.1249/01.mss.0000176686.18683.64

http://dx.doi.org/10.1249/01.mss.0000176686.18683.64

http://dx.doi.org/10.1016/j.actbio.2012.12.021

http://dx.doi.org/10.1115/1.4007097

http://dx.doi.org/10.1115/1.4024823

http://dx.doi.org/10.1115/1.4024823



http://dx.doi.org/10.1115/1.4005694



http://dx.doi.org/10.1053/joca.2001.0424



Focus5:20140081

14


predicts many observed phenomena. J. Biomech.Eng. 131, 061003. (doi:10.1115/1.3118773)

38. Reddy A, Erdemir A. 2014 A modeling andsimulation pipeline for the prediction of localizedcartilage mechanics: towards a multiscaleanalysis framework for joint, tissue, and cellscales. In 7th World Congress of Biomechanics,Boston, MA, USA, 6 – 11 July 2014.

39. Erdemir A, Reddy A. 2014 Parametric modeling forin silico explorations of cartilage and chondrocytemechanics. In 12th Int. Symp. on Computer Methodsin Biomechanics and Biomedical Engineering,Amsterdam, The Netherlands, 13 – 15 October 2014.

40. Schumacher BL, Su J-L, Lindley KM, Kuettner KE,Cole AA. 2002 Horizontally oriented clusters ofmultiple chondrons in the superficial zone of ankle,but not knee articular cartilage. Anat. Rec. 266,241 – 248. (doi:10.1002/ar.10063)

41. Wilusz RE, Sanchez-Adams J, Guilak F. 2014 Thestructure and function of the pericellular matrix ofarticular cartilage. Matrix Biol. J. 39C, 25 – 32.(doi:10.1016/j.matbio.2014.08.009)

42. Bennetts CJ, Sibole S, Erdemir A. 2015 Automatedgeneration of tissue-specific three-dimensionalfinite element meshes containing ellipsoidal cellularinclusions. Comput. Methods Biomech. Biomed. Eng. 18,1293– 1304. (doi:10.1080/10255842.2014.900545)

43. Julkunen P, Wilson W, Jurvelin JS, Korhonen RK.2009 Composition of the pericellular matrixmodulates the deformation behaviour ofchondrocytes in articular cartilage under staticloading. Med. Biol. Eng. Comput. 47, 1281 – 1290.(doi:10.1007/s11517-009-0547-8)

44. Wu JZ, Herzog W. 2002 Elastic anisotropy ofarticular cartilage is associated with the

microstructures of collagen fibers and chondrocytes.J. Biomech. 35, 931 – 942. (doi:10.1016/S0021-9290(02)00050-7)

45. Kouznetsova VG, Geers MGD, Brekelmans WAM.2004 Multi-scale second-order computationalhomogenization of multi-phase materials: a nestedfinite element solution strategy. Comput. MethodsAppl. Mech. Eng. 193, 5525 – 5550. (doi:10.1016/j.cma.2003.12.073)

46. Guilak F, Mow VC. 2000 The mechanicalenvironment of the chondrocyte: a biphasic finiteelement model of cell – matrix interactions inarticular cartilage. J. Biomech. 33, 1663 – 1673.(doi:10.1016/S0021-9290(00)00105-6)

47. Moo EK, Herzog W, Han SK, Abu Osman NA,Pingguan-Murphy B, Federico S. 2012 Mechanicalbehaviour of in situ chondrocytes subjected todifferent loading rates: a finite element study.Biomech. Model. Mechanobiol. 11, 983 – 993.(doi:10.1007/s10237-011-0367-2)

48. Hunziker EB, Quinn TM, Hauselmann H-J. 2002Quantitative structural organization of normal adulthuman articular cartilage. Osteoarthritis Cartilage10, 564 – 572. (doi:10.1053/joca.2002.0814)

49. Chahine NO, Hung CT, Ateshian GA. 2007 In situmeasurements of chondrocyte deformation undertransient loading. Eur. Cell Mater. 13, 100 – 111;discussion 111.

50. Davis S, Erdemir A. 2014 Convergence ofrepresentative volume elements for prediction ofchondrocyte mechanics in multiscale simulations ofcartilage. In 7th World Congress of Biomechanics,Boston, MA, USA, 6 – 11 July 2014.

51. Sibole SC, Maas S, Halloran JP, Weiss JA, Erdemir A.In press. Evaluation of a post-processing approach

for multiscale analysis of biphasic mechanics ofchondrocytes, doi:10.1080/10255842.2013.809711.Comput. Methods Biomech. Biomed. Eng. (doi:10.1080/10255842.2013.869043)

52. Maas S, Rawlins D, Weiss J, Ateshian G. 2013 FEBio,finite elements for biomechanics, user’s manual,v. 1.8. Musculoskeletal Research Laboratories,University of Utah.

53. COMSOL. 2013 Introduction to COMSOLMultiphysics, v. 4.4. Burlington, MA.

54. Dassault Systemes Simulia. 2013 Abaqus AnalysisUser’s Guide, v. 6.13. Johnston, RI.

55. Schenk O, Gartner K. 2004 Solving unsymmetricsparse systems of linear equations with PARDISO.Future Gen. Comput. Syst. 20, 475 – 487. (doi:10.1016/j.future.2003.07.011)

56. Korhonen RK, Julkunen P, Wilson W, Herzog W.2008 Importance of collagen orientation and depth-dependent fixed charge densities of cartilage onmechanical behavior of chondrocytes. J. Biomech.Eng. 130, 021003. (doi:10.1115/1.2898725)

57. Peeters EAG, Bouten CVC, Oomens CWJ, Bader DL,Snoeckx LHEH, Baaijens FPT. 2004 Anisotropic, three-dimensional deformation of single attached cellsunder compression. Ann. Biomed. Eng. 32, 1443 –1452. (doi:10.1114/B:ABME.0000042231.59230.72)

58. Horton WE, Bennion P, Yang L. 2006 Cellular,molecular, and matrix changes in cartilage duringaging and osteoarthritis. J. Musculoskelet. NeuronalInteract. 6, 379 – 381.

59. Erdemir A. 2014 Multiscale cartilage biomechanics:technical challenges in realizing a high-throughputmodeling and simulation workflow. In 7th WorldCongress of Biomechanics, Boston, MA, USA, 6 – 11July 2014.

http://dx.doi.org/10.1115/1.3118773

http://dx.doi.org/10.1002/ar.10063

http://dx.doi.org/10.1016/j.matbio.2014.08.009

http://dx.doi.org/10.1080/10255842.2014.900545

http://dx.doi.org/10.1007/s11517-009-0547-8

http://dx.doi.org/10.1016/S0021-9290(02)00050-7

http://dx.doi.org/10.1016/S0021-9290(02)00050-7

http://dx.doi.org/10.1016/j.cma.2003.12.073

http://dx.doi.org/10.1016/j.cma.2003.12.073

http://dx.doi.org/10.1016/S0021-9290(00)00105-6

http://dx.doi.org/10.1007/s10237-011-0367-2

http://dx.doi.org/10.1053/joca.2002.0814

http://dx.doi.org/10.1080/10255842.2013.809711

http://dx.doi.org/10.1080/10255842.2013.869043

http://dx.doi.org/10.1080/10255842.2013.869043

http://dx.doi.org/10.1016/j.future.2003.07.011

http://dx.doi.org/10.1016/j.future.2003.07.011

http://dx.doi.org/10.1115/1.2898725

http://dx.doi.org/10.1114/B:ABME.0000042231.59230.72


Multiscale cartilage biomechanics: technical...

Documents

Transcript of Multiscale cartilage biomechanics: technical...