Computational Modelling of the Trabecular Bone Network from the GB-µFEA Technique Gustavo Peixoto de Oliveira, Dr. Waldir Leite Roque, Dr.
[email protected]
*
Graduate Program in Computational and Mathematical Modelling Center of Informatics Federal University of Paraíba
João Pessoa, March 10th, 2016 @PGMEC-UFPB
Outline • • • • • •
Introduction µFEA Overview Mathematical Formulation Some Simulations Future Research Questions
Introduction
Motivation
Hip fractures
•
Men 310%
Women 240%
From: http://www.iofbonehealth.org/epidemiology
Bone fragility • • • • • •
Osteoporosis in elderly people Public health issue High costs of treatment Socio-economical impact Possible to grade? How to model?
Bone architecture (hierarchical view)
Donelly, Clin Orthop Relat Res (2011) 469:2128–2138
Focus: microscale cortical bone + trabecular bone
• •
Stages of degradation Bone mass loss
Bone cell population and remodelling
• •
Integrity of the skeletal system maintained
by continuous remodelling Roles of the Basic Multicellular Units
Osteoclasts: removers of “old” bone
(compromised mechanical integrity)
Osteoblasts: synthesizers of “new” bone
(bone matrix mineralization)
Osteocytes: network “designers”
(remodelling control)
Check this out —> http://youtu.be/inqWoakkiTc
Oftadeh et al., J. of Biomech. Eng. (2015) 137(1)
doi: 10.1115/1.4029176
Trabecular bone as load-bearing struts Forth Road Bridge, Scotland
• •
Ability to rapidly adapt to the mechanical loading environment Optimizes its mass and structure to bear high loads with as little bone tissue as possible
“Trabecular Bone: Light as a Feather, Stiff as a Board” Christiansen, J. of Biomech. Eng. (2015) 137, Editorial
From: http://billharvey.typepad.com/
The Mechanical Competence Parameter • •
Measure the load-bearing capacity of trabecular bone is crucial for assessing fracture risk A proposed MCP to assess the characterization of trabecular bone fragility from high spatial resolution images connectivity
morphometry
MCP tortuosity
elastic properties
MCP M CP = 0.52BV /T V trabecular volume fraction
EP CV = 12 (#I
0.49EP CV + 0.51Ez Euler-Poincaré characteristic
#B + #H)
#I
number of objects (isolated parts)
#B
number of tunnels (redundant connections)
#H
number of enclosed cavities (holes)
Roque et al., IEEE Transactions on Biomed. Eng. (2013), (60),5
Young modulus of elasticity
Ez =
✏z
⌧+z =
LG,z LE,z
0.48⌧+z vertical tortuosity
isotropy + 1D
Normalized MCP • •
MCP was verified to be similar with both µCT and MRI A normalized MCP per sample M CPN,k =
M CPk M CPmin,k M CPmax,k M CPmin,k ,
spectrum 1: low fracture risk 0: high fracture risk 0 M CPN,k 1 Roque & Bayarri (2015),
DOI 10.1007/978-3-319-13407-9_11
k = 1, 2, . . . , nk
Image acquisition µCT / MRI images 3D reconstruction
(Brief mathematics of images) I(0, 0)
I(0, N )
continuous signal
I(x, y)
discretization (pixels)
I(m, n) m = 1, 2, . . . , M n = 1, 2, . . . , N
I(M, 0)
I(M, N )
color spaces I(M, N ) ⇥ C
H S V
Image segmentation and binarization Segmentation: generic process by which an image is subdivided into regions or constituent objects (Solomon & Breckon, Wiley, 2011) z
⌦ =
z ⌦bone
[
z ⌦marrow ,
z = 1, 2, . . . , Z
I(m, n, C)z ; C = {0, 1} binarization
region of interest (ROI)
µFEA Overview
Aspects of µFEA • •
•
• •
Micro-Finite Element analysis (µFEA) have been widely used to derive bone mechanical properties Under loading condition, can be used to derive stiffness, strength of the bone, as well as the stresses and strains in the bone tissue Some assumptions over the years:
µFE better predicts bone stiffness and strength?
µFE better reflects bone fracture risk? Several validation tests with different imaging techniques Other applications: effect of diseases, treatments, activities and growth
Van Ritbergen, Journal of Biomechanics 48 (2015) 832–841
Two main approaches •
•
Geometry-based (GB): defines a geometric model comprised of curves and surfaces that is finally discretized into finite elements Voxel-based (VB): each group of voxels (base unit of 3D imaging) is directly converted into hexahedral elements weaknesses GB VB
•
•
strengths
complex geometries may require extreme computational cost contact problems require smoothing techniques
• •
•
possibility of creating smooth surfaces simulates any kind of interface direct conversion of elements
Bocaccio et al, Int. J. Biol. Sci. (2011), 7
Example: VB x GB VB brick elements
GB tetrahedral elements
Clinic Rev Bone Miner Metab (2016) 14:26–37
3D reconstruction + FE meshing trabecular bone region
unhealthy individual
healthy individual
⌦U bone
⌦H bone
(What about the full FE meshing?)
• •
see also Metzger (2015), 137; 011006-1
•
multimaterial meshing mechanobiology of
bone/marrow interface:
- homogeneous fluid hypothesis
- FSI simulations to appear…
Behind the scenes… Image processing: • multithresholding • “de-islanding” (hole removal) • filtering • smoothing 3D Meshing: • CGAL • iso2Mesh • adaptivity
d Tbone
:=
S Jb
d j=1 Tj ,
d Tp
\
d Tq
6= ; d = 2, 3
(Effective area) @⌦bone,top
@⌦bone @⌦bone,bot Aef f = A(@⌦bone )
2 A(Tbone )
Aef f = PJb 1 = j=1 2 ||(x3
=
P Jb
2 T j=1 j =
x1 )j ⇥ (x2
x1 )j ||
Mathematical Formulation
Classical displacement problem
“Determine the distribution of displacements, strains and stresses in the interior of an elastic body in equilibrium when body forces are given and the distribution of the displacements are prescribed over the surface of the body.” (Saad, 2011) u(x@⌦bone ) = g(x@⌦bone )
FEBio software Nonlinear finite element solver that is specifically designed for biomechanical applications Open-source and multi-platform (Windows, OS-X and Linux platforms available) Maintenance: Original: Musculoskeletal Research Laboratories - University of Utah
Current: University of Utah / Musculoskeletal Biomechanics Laboratory - Columbia University
Weak form for solid mechanics Virtual work equation W = W d v f t
R
⌦bone
: d d⌦
R
⌦bone
: virtual work : Cauchy's stress tensor : virtual rate of deformation tensor : virtual velocity : body force : traction force
f · v d⌦
R
@⌦bone
t · v d@⌦ = 0
Linearization and discrete form In FEBio, virtual work is solved using an incremental-iterative strategy based on Newton’s method, which requires linearization W ( k , v) + D W ( k , v) · u = 0 directional derivative
: deformation
k : trial function
For more details on the derivation, see: Maas, Steve A., et al. "FEBio: finite elements for biomechanics."
J. of biomechanical eng. 134.1 (2012): 011005. Bonet, J., and Wood, R. D., 1997, Nonlinear Continuum Mechanics for Finite Element Analysis,
Cambridge University Press, Cambridge, NY.
à la Ritz position, displacement deform. grad. tensor real and virtual velocity real rate of deformation virtual rate of deformation linear strain tensor
Na
element shape functions
rNa =
@Na grad. init. config. @X
l.h.s. e
W ( , Na v a ) = R f · (Na va ) dTj Tj
R
: ( va ⌦ rNa ) dTj t · (Na va ) d@Tj = 0, @Tj
TRj
W e ( , Na v a ) = v a ·
⇣R
: (u ⌦ v) ⌘ u · v Tj
rNa dTj
internal
e
W ( , Na v a ) = v a ·
e (Ta
R
Tj
f Na dTj
j = 1, 2, . . . , J1 R
@Tj
tNa d@Tj = 0
external
e Fa )
= va ·
⌘
e Ra
r.h.s. D W e ( , Na va )[Nb ub ] = D( va · (Tea va · D(Tea Fea ))[Nb ub ] = va · Keab ub tangent stiffness matrix
Element assembly
e Kab
Fea ))[Nb ub ] =
= Kec,ab + Ke ,ab + Kep,ab constitutive
component
stress
arbitrary
virtual
velocity
vT Ku =
vT R ) Ku =
R
external
forces
Linear system solution K(xk )u = xk =
R(xk ), k (X),
xk+1 = xk + u P X= Na Xa
iterative quasi-Newton BFGS method
Constitutive equations linear elasticity =c:✏ c ✏
: Cauchy's stress tensor : elasticity tensor : small strain tensor
nonlinear neo-Hookean solid =
µ J (b T
b = FF b F µ,
I) + J (ln J)I J = det F
: left Cauchy-Green deformation tensor : deformation gradient tensor : Lamé’s constants
uniaxial extension ≈ linear for
small strains
1: linear
2: neo-Hookean
3: Mooney-Rivlin From: Wikipedia (rheology tests)
Febio theory manual C. W. Macosko, 1994, Rheology: principles,
measurement and applications,
VCH Publishers, p. 47
Some Simulations
Bone compression test Material elastic properties Young's modulus (E)= 10GPa Poisson ratio (ν)= 0.3 Imposed strain = 1%
µ=
E 2(1+⌫) ,
=
E⌫ (1+⌫)(1 2⌫)
u(@⌦bone,top ) =
u z ez
Bayarri et al, 2010, IntechOpen
u(@⌦bone,bot ) = 0
Computational simulations
Outside…
Some considerations • • • •
Warm-up results Limited computational resources
Thousands —> millions of elements Verification and validation MCP parameters
Future Research
State-of-art and future research •
@CI validate MCP through µFE and establish MCPn-based biomarkers
•
@world
•
mechanobiology of the interaction between bone and marrow
•
FSI simulations & poroelastic theory
•
rigorous validation of µFE methods
•
standardisation of biomarkers for fracture risk
•
influence of age, osteoporosis status and other variables on bone fracture
•
biomaterial development and scaffold engineering
•
intersticial transport of drugs for bone treatment See cited refs.
Thanks for listening!
QUESTIONS?
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