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COMMAS. 7th MIT Conference. BIOMECHANICS. OF THE HEART. LECTURE 1: Modeling an Organ. DAVID NORDSLETTEN1. 1LECTURER, KINGS COLLEGE ...

BIOMECHANICS OF THE HEART

LECTURE 1: Modeling an Organ DAVID NORDSLETTEN1 1LECTURER, KINGS COLLEGE LONDON

11/20/13   COMMAS NORDSLETTEN, 2013 2013

1   7th MIT COMMAS Conference

LECTURE SUMMARY   • 

mo$va$on    

• 

structure  and  func$on  in  the  heart  

• 

(patho)  physiology  in  the  heart  

• 

a  model’s  role  in  the  heart  

• 

a  con$nuum  interpreta$on  of  the  heart  

• 

cons$tu$ve  laws  for  cardiac  $ssue  

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MOTIVATION Heart  Failure  /  Cardiovascular  Disease  

Remain  the  most  significant  costs  to  healthcare  worldwide    

Over  0.75  million  Living  with  HF  in  UK     Annual  cost  of  £0.75  billion     Significant  Problem  in     Germany  /  EU    

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MOTIVATION Heart  Failure  /  Cardiovascular  Disease  

Remain  the  most  significant  costs  to  healthcare  worldwide    

Over  0.75  million  Living  with  HF  in  UK     Annual  cost  of  £0.75  billion     Significant  Problem  in     Germany  /  EU    

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MOTIVATION Heart  Failure  /  Cardiovascular  Disease  

Remain  the  most  significant  costs  to  healthcare  worldwide    

Treatment  /  Stra$fica$on  Challenges   Diagnos$c  /  Treatment  guidelines  based  on  bulk     measurements     Room  for  Op$miza$on       Development  of  New  Devices,  Drugs   and  treatment  protocols  

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ANATOMY OF THE HEART

NORDSLETTEN, 2013

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ANATOMY OF THE HEART Right   Atrium  

LeV    

Atrium  

Right    

Ventricle   NORDSLETTEN, 2013

LeV    

Ventricle  

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ANATOMY OF THE HEART

3   1   4  

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2  

1  

Aor$c  Valve  

2  

Mitral  Valve  

3  

Tricuspid  Valve    

4  

Pulmonary  Valve  

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Superior   Aorta   Vena  Cava  

ANATOMY OF THE HEART Pulmonary     Artery  

Pulmonary     Veins  

Inferior  

Vena  Cava  

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2µm  

100µm  

STRUCTURE 103  µm  

OF THE HEART 104  µm  

105  µm   NORDSLETTEN, 2013

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2µm  

MICRO  

SCALE MULTI NORDSLETTEN, 2013

100µm  

STRUCTURE 103  µm  

OF THE HEART 104  µm  

MACRO

105  µm   COMMAS

2µm  

100µm  

STRUCTURE 103  µm  

OF THE HEART 104  µm  

105  µm   NORDSLETTEN, 2013

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2µm  

100µm  

STRUCTURE 103  µm  

OF THE HEART 104  µm  

105  µm   NORDSLETTEN, 2013

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2µm  

100µm  

STRUCTURE 103  µm  

OF THE HEART 104  µm  

105  µm   NORDSLETTEN, 2013

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2µm  

100µm  

STRUCTURE 103  µm  

OF THE HEART 104  µm  

105  µm   NORDSLETTEN, 2013

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2µm  

100µm  

STRUCTURE 103  µm  

OF THE HEART 104  µm  

105  µm   NORDSLETTEN, 2013

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FUNCTION OF THE HEART

Katz,  A.    Heart  Physiology,  2004  

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FUNCTION OF THE HEART

Katz,  A.    Heart  Physiology,  2004  

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FUNCTION Sarcoplasmic  Re$culum  

OF THE HEART

Myofibrils  

T-­‐tubules   Mitochondria   Cell  Membrane   NORDSLETTEN, 2013

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FUNCTION OF THE HEART

Bers,  D.  Calcium  Cycling  and  Signaling  in  Cardiac  Myocytes.    Annu.  Rev.  Physiol.,  2008  

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CARDIAC  PUMP  FUNCTION  IS  THE  DIRECT  RESULT  OF  

≈2 BILLION MYOCYTES

responding  to   electrochemical   s$mulus,  

WHICH  ALTER  THEIR  LENGTH  THROUGH  ACTIVE  CONTRACTION  

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FUNCTION OF THE HEART

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FUNCTION OF THE HEART

P.  Kohl,  Imperial  College  London  and  Camelli$      hap://www.camelli$.it/fig3g.html  

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FUNCTION OF THE HEART

Myofiber  Alignment  

Laminar  Sheets  

Pope,  Sands,  Smaill,  Le  Grice.  Three-­‐dimensional  transmural  organiza$on  of  perimysial  collagen  in  the  heart.  AJP  2008  

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PHYSIOLOGY OF THE HEART

NORDSLETTEN, 2013

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PHYSIOLOGY OF THE HEART

Primal  Pictures  Ltd.  

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PHYSIOLOGY OF THE HEART

Primal  Pictures  Ltd.  

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PHYSIOLOGY OF THE HEART

Primal  Pictures  Ltd.  

NORDSLETTEN, 2013

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PHYSIOLOGY OF THE HEART

Primal  Pictures  Ltd.  

NORDSLETTEN, 2013

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PHYSIOLOGY OF THE HEART

Primal  Pictures  Ltd.  

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PHYSIOLOGY OF THE HEART

Fenton  Lab:  Electrophysiology,  Smith  Lab:  Coronary  Perfusion    

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PHYSIOLOGY OF THE HEART

  Movie  Courtesy  of  Efimov  Lab:  Lou  Q,  Li  W,  Efimov  IR.  The  role  of  dynamic  instability  ...  AJP.  (302)  2012  

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PHYSIOLOGY OF THE HEART

  Movie  Courtesy  of  R  Chabiniok  KCL  2013  

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PHYSIOLOGY OF THE HEART

  Movie  Courtesy  of  J  Wong  and  D  Nordsleaen  KCL  2013  

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PHYSIOLOGY OF THE HEART

  Movie  Courtesy  of  J  Wong,  R  Chabiniok  and  D  Nordsleaen  KCL  2013  

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PATHO PHYSIOLOGY OF THE HEART

STRUCTURAL Proteins  (Density,  Expression)   Cell  (Structure,  Organiza$on)   Tissue  (ECM,  Architecture,  Perfusion)   Whole-­‐Organ  (DCM,  HCM,  MA)     NORDSLETTEN, 2013

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PATHO PHYSIOLOGY OF THE HEART

FUNCTIONAL Proteins  (Behavior,  Isoforms)   Cell  (Ion  Concentra$on,  Interac$ons)   Tissue  (Arrhythmia,  S$ffness)   Whole-­‐Organ  (Systolic  /  Diastolic  HF)     NORDSLETTEN, 2013

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PATHO PHYSIOLOGY OF THE HEART

CHRONIC HF a

PROGRESSIVE DETERIORATION

IN HEALTH & FUNCTION OF THE HEART NORDSLETTEN, 2013

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    CTEPH – An Example of HF

PATHO PHYSIOLOGY OF THE HEART

Chronic  Thromboembolic  Pulmonary  Hypertension  

 

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    CTEPH – An Example of HF  

PATHO PHYSIOLOGY OF THE HEART

Chronic  Thromboembolic  Pulmonary  Hypertension    

Early  Stage  

Blockage  -­‐>  Pressure  Overload  -­‐>  RV  Hypertrophy    

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    CTEPH – An Example of HF  

PATHO PHYSIOLOGY OF THE HEART

Chronic  Thromboembolic  Pulmonary  Hypertension    

Mid  Stage  

RV  Dila$on  -­‐>  RV  S$ffening  +  increased  a-­‐myosin  -­‐>  Path  to  Failure  

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    CTEPH – An Example of HF  

PATHO PHYSIOLOGY OF THE HEART

Chronic  Thromboembolic  Pulmonary  Hypertension    

Late  Stage  

RV  Failure  -­‐>  Severe  Anatomical  Changes  -­‐>  Abnormal  Conduc$on  -­‐>  LV  Atrophy  +  Failure  

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MODELLING

IN THE HEART

XKCD:  hap://xkcd.com/171/  

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MODELLING

EXPERIMENT

IN THE HEART

Illustra$ng  Observed  Phenomena  

THEORY

Explaining  Observed     Phenomena  

PREDICTION

MAKE NEW PREDICTION

An$cipated  Behavior  (from  Theory)  

REVISE THEORY To  Explain  All  Observa$ons  

(NEW) EXPERIMENT Tes$ng  Predicted  Behavior  

MATCHES  

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EXPERIMENT

Illustra$ng  Observed  Phenomena  

Mathematical MODELLING IN THE HEART

Quantitative THEORY Explaining  Observed     Phenomena  

Quantitative PREDICTION

MAKE NEW Quantitative PREDICTION

An$cipated  Behavior  (from  Theory)  

REVISE Quantitative THEORY To  Explain  All  Observa$ons  

Quantitative (NEW) EXPERIMENT Tes$ng  Predicted  Behavior  

NORDSLETTEN, 2013

MATCHES  

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Mathematical MODELLING IN THE HEART

NORDSLETTEN, 2013

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Mathematical MODELLING

4

VIGUERAS, ROY, COOKSON, LEE, SMITH, NORDSLETTEN

IN THE HEART

4.1. Electrophysiology Problem Modeling electrophysiology in the heart is typically accomplished using the monodomain [20, 21]          Electrophysiology   or bidomain [22, 23, 24, 25, 26] equations which simulate the spread of membrane potential or intra    potential,                    Con$nuum   odel  we of  focus transmembrane   oten$al,   / extracellular respectively. In this m paper, on modeling p the electrophysiology in 3   by   the  domain            cellular   ac$on   nd  , cusing alcium   ynamics   model. Here the heart,   denoted Ω⊂R (withpoten$al   boundaryaδΩ the dmonodomain we  seek a membrane potential u : Ω × I → R and the m−cell model variables v : Ω × I → Rm over some time interval I = [0, T ] satisfying [27],

 

                   

 

           Monodomain  (bidomain)  Equa$ons  

Cm

∂u − ∇ · (D∇u) − Iion (u, v) − Iext ∂t dv − f (t, u, v) dt (D∇u) · n u = u0 ,

v

=

0,

on Ω × I,

(1)

=

0,

on Ω × I,

(2)

=

0,

on δΩ × I,

(3)

=

v0 ,

on Ω × [0]

(4)

Coupled  (strong  /  weak)  to  Tissue  Mechanics  

where D : Ω → R3×3 is the diffusion tensor related to the gap junctions between cells and the Stretch   ac$vated  cIhannels,   adapted  conduc$vity   membrane capacitance. ion (u, v) is the total ionic current (which is a function of the voltage u, the gating variables and the ion concentrations), Iext : Ω × I → R the stimulus current, f is a function governing rate-of-change in the m−cell model variables, and n is the normal to the surface of the NORDSLETTEN, 2013 COMMAS boundary δΩ. The diffusion tensor D is of the form χCσm where σ is the conductivity, Cm is the membrane capacitance and χ is the cell surface to volume ratio. In this paper we have defined −1

1 α∗ = ∗2 ∗2 R V

(2.13)

!

R∗

2r∗ vx∗2 dr∗ ,

0

(2.8) and (2.9) can be written as

Mathematical MODELLING ∂(R V ) ∂R + 2R =0 ∗2

(2.14)







∂x∗

∂t∗

and

 

(2.15)  

 

IN THE HEART

" ∗#   2      Coronary   Perfusion   2 2 ∗ ∂(R∗ V ∗ ) ∂(α∗ R∗ V ∗ ) ∂p 2λν ∗2 ∗ ∂vx + + R = R .    ∗                  Con$nuum   m odel   o f   b lood   fl ow   (1D  ∗ /  3D)   2 ∂t ∂x∗ ∂x∗ ∂r Vo R R∗

 

 

             coupled  to  porous  flow  model  

 By making the transformations R = RR∗ , α = α∗ , and V = Vo V ∗ , (2.13) and (2.15) terms of dimensional as   can   be  written          1D  in Navier-­‐Stokes   /  Dquantities arcy  Equa$ons  

        (2.16)     and     (2.17)    

∂R ∂R R ∂V +V + =0 ∂t ∂x 2 ∂x

V ∂R ∂V 1 ∂p 2ν ∂V + 2 (1 − α) + αV + = ∂t R ∂t ∂x ρ ∂x R

"

∂vx ∂r

#

.

R

The above derivation eliminates vr , the radial component of velocity. However, it Coupled   to  Tissue  Mechanics  &  Hemodynamics   requires the assumption that v is solely a function of the radial coordinate r. This is x

Addi$ve   tress  to  Man echanics,   Deforma$on-­‐altered   Porosity   equivalent to Sspecifying axial velocity profile. Once a profile is determined, α and flow  linked  to  Hemodynamics  in  Aor$c  Sinus   the Coronary   viscous term NORDSLETTEN, 2013

(2.18)

2ν R

"

∂vx ∂r

#

R

COMMAS

demonstrating the energy preservation of the method. Finally, the method is test showing both convergence and stability for complex non-linear coupled mechani 1.1. Model problem

Mathematical MODELLING

In this paper, we focus on the coupling of a Navier–Poisson fluid and a qua Problems 1 and 2, respectively. Though the paper focuses on these models, the sch mechanical systems. The linking of these problems is enforced via Problem 3, ens opposite traction. The fluid and solid will be represented geometrically by the domains X Xi ! Rd " I; i ¼ 1; 2 is a moving domain which alters shape through the time inter ary of each domain, Ci, is treated to be at least Lipshitz continuous and is partition and to subdomains of the boundary, r   C refer          the          CNeumann, on$nuum  Dirichlet model  oand f  3D  Coupling blood  flow   C C C are coupled about C :¼ C1 ¼ C2 .

IN THE HEART

 

 

                       

 

 

 

 

 Ventricular  Blood  Flow  

ALE  Navier-­‐Stokes  Equa$ons*  

           1        (Navier–Stokes     Problem Equations). Consider flow over X1. Let v and p be the ve                *Arbitrary  Lagrangian-­‐Eulerian   satisfy, @v þ rx & ðqvv ' lrx v þ pIÞ ¼ f @t rx & v ¼ 0 in X1 ;

q

1

in X1 ;

v ¼ g D1 on CD1 ; ðlrx v ' pIÞ & n ¼ g N1 on CN1 ; vð&; 0Þ ¼ v 0 in X1 ð0Þ;

where viscosity, q the density, (I)jk: = djk, nPthe outward boundary norma Coupled   to  lTthe issue   Mechanics   &  Coronary   erfusion  

gradient operator, is the contribution momentum of body forces, and g D1 ¼ g D1 Fluid-­‐Solid   Interac$on   on  f1Endocardial   /  atrial  to walls   and Neumann data.flow  in  Aor$c  Sinus   Hemodynamics   linked  boundary to  Coronary   NORDSLETTEN, 2013

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Problem 2 (Quasi-Static Finite Elasticity). Consider finite elasticity mechanics over pressure state variables, which satisfy,

Mathematical MODELLING IN THE HEART

SOLID MECHANICS IN THE HEART

 

 

                   

 

3.2  

 

 

3

 Tissue  Mechanics  

Finite Elasticity Weak Form  

                 Con$nuum  model  of  3D  Tissue  Deforma$on  

In the previous section 3.1.4, the equations governing the motion of a body were derive   in      the            Lagrangian    Cauchy’s   Equa$on   (Nonlinear   Mechanics)   Taken framework discussed in section 2.2.1, the law (Cauchy’s fir law) may be written as, @t (⇢vJ )

J (F

T

r⌘ ) ·

fJ

=

0,

on ⌦0 ⇥ I.

(3.2

As discussed, the Cauchy stress for cardiac materials is typically defined in terms of th ˆ As a result equation 3.29, as written, requires the displacement as we isochoric strain C.

as the velocity (though one may be trivially related to the other through di↵erentiation

While, in the discrete context, some formulations require the solution be computed f

both variables, treat thePvelocity as an unknown with the aim to define the require Coupled   to  All  Pwe hysical   henomena  

displacement in terms of only velocity components. Assuming that the body moves und some Dirichlet / traction conditions,

NORDSLETTEN, 2013

v(·, t) = g(·, t) on and initial conditions,

D 0 ,

J · (F

T

N N ) = t(·, t) on COMMAS 0 ,

(3.2

Mathematical MODELLING IN THE HEART

NORDSLETTEN, 2013

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MECHANICS

OF HEART TISSUE

Overview

Aim  is  to  represent  cardiac  $ssue  as  a  con$nuum     and  model  $ssue  response  to  loads  

 

CARDIAC TISSUE AS  AN   MODELED  

ANISOTROPIC HYPERELASTIC MATERIAL NORDSLETTEN, 2013

Anisotropy  due  to  inherent   structure  of  the  myocardium   Hyperelas$city  is  due  to  the   elas$c  response  of  the   myocardium*    

COMMAS

ofilaments stacked within the myocyte to form functional

myocytes aligned end-to-end to build myofibers, myofibers

sheets stacked to form the tissue walls of the heart. The

MECHANICS

all leads to varying force response depending on orientation structures [85]. Thus it is critical for a continuumOF model to HEART Modeling Structural Anisotropy

TISSUE

  Fiber  Coordinate  Frame   heart, this is achieved by defining the continuous fields fˆ, sˆ Construct   fields  represen$ng   direc$ons    -­‐  fiber   rt (which, in the reference frame, is denoted ⌦0 ). The field  -­‐  sheet   en in Figure 2.1, which denotes the orientation of myofibers  -­‐  sheet   normal   At every point orthogonal to the fiber field is the sheet field,   on in which myofibers are aligned. Lastly – and also mutually   ons – is the sheet normal field, n ˆ , denoting the direction in Forms  an  Orthonormal  Transforma$on   ogether.Enabling   Usingthe   these fields, we dmay define mapping   of  global   irec$ons   to  a     the orthonormal local  microstructural  direc$on  

Q = (fˆ, sˆ, n ˆ ),

(2.1)

crostructure directions into their global equivalents. For NORDSLETTEN, 2013 eˆ1 , eˆ2 , and eˆ3 to be the usual base vectors in R3 and let

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MECHANICS Passive Tissue Properties

OF HEART TISSUE

     

Stress-­‐Shear  Response    

Shows  significant  dependence  on   local  $ssue  microstructure  

Costa,  KD.    Trans  ASME,  1996.  

NORDSLETTEN, 2013

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mained unchanged for a given sample. Asymmetry of shear opposites sides of the block), indicating that there was properties was commonly observed in initial tests. This was some change in sheet orientation, up to a maximum of due, in part, to residual shear strain caused by small relative 30°, across each block. displacements of upper and lower surfaces of the specimen Results from a representative shear test, in which during mounting. Residual shear displacement was estimated from initial test results, and the lower platform was four cycles of sinusoidal shear displacement with an offset to correct for this. In all cases, the offset needed to amplitude of 40% were applied in the NF mode, are minimize residual shear displacement was !10% of the sam- presented in Fig. 4. The viscoelastic properties of this ple thickness. On completion of sinusoidal testing, separate myocardial specimen are evident in the stress-strain step tests were performed in X and Y directions. A rapid 50% hysteresis and in the stress relaxation behavior after shear displacement was imposed, and the resultant forces 50% step shear displacement (Fig. 4, inset). Correwere recorded for 300 s. These protocols were carried out for sponding viscoelastic behavior was observed in all all three samples from each heart, with samples mounted in specimens. Most   passive   mechanical   models   random order, in one of the different orientations (I, II,are   andbased  on  Dokos*   In sinusoidal tests, stress was always greater on III) shown in   Fig. 1B. A schematic representation of the six different modes of shear deformation achieved by imposing X initial displacement in positive and negative directions and Y shear displacements in the three specimen orienta- than in subsequent cycles. After the first cycle, stressstrain loops were reproducible (Fig. 4). This softening tions (I, II, and III) is given in Fig. 2.

MECHANICS

Passive Tissue Properties

OF HEART TISSUE

Stress-­‐Shear  Response    

Shows  significant  dependence  on   local  $ssue  microstructure    

Fig. 2. Six possible modes of simple shear defined with respect to FSN material coordinates. Shear deformation is commonly characterized by specifying 2 coordinate axes: the first denotes (is normal to) the face that is translated by the shear, and the second is the direction in which that face is shifted. Thus NS shear represents translation of the N face in the S direction.

AJP-Heart Circ Physiol • VOL

283 • DECEMBER 2002 •

www.ajpheart.org

Dokos  et  al,  Shear  Proper$es  of  Passive  Ventricular  Myocardium,  Am  J  Physiol,  2002  

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MECHANICS

OF HEART TISSUE

Passive Tissue Properties

Most  passive  mechanical  models  are  based  on  Dokos*   Modelling of passive myocardium  

Stress-­‐Shear  Response    

16

Shows  significant  dependence  on   local  $ssue  microstructure  

346 (fs)

14 (fn)

shear stress (kPa)

12 10 8 6

(sf )

4

(sn)

2

0

(nf ), (ns) 0.1

0.2

0.3 0.4 amount of shear

0.5

0.6

Figure 6. Fit of the model (5.39) with the final term omitted (full curves) to the experimental dat Holzapfel  &  Ogden,  Phil  Trans  Royal  Soc,  2009   (circles) for the loading curves from figure 2: (nf)–(ns) and mean of the loading curves for (fs) an (fn) and for (sf) and (sn). The material parameters used are given in COMMAS table 1. NORDSLETTEN, 2013

NORDSLETTEN, 2013

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MECHANICS Ac$ve  Tissue  Proper$es  

OF HEART TISSUE

Ac$ve  response  of  $ssue  based  on  known  observa$ons   predominantly  at  the  cellular  level*    

Ac$ve  Tension  Modulated  by  Mechanical  Factors   i.e.  Stretch,  stretch-­‐rate  

Hunter,  P.  Modelling  the  Mechanical  Proper$es  of  Cardiac  Muscle.    Prog.  Biophys.  &  Mol.  Biol.,  1998  

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MECHANICS Ac$ve  Tissue  Proper$es  

OF HEART TISSUE

Ac$ve  response  of  $ssue  based  on  known  observa$ons   predominantly  at  the  cellular  level*    

Ac$ve  Tension  Modulated  by  Mechanical  Factors   i.e.  Stretch,  stretch-­‐rate  

  Challenging  to  Quan$fy  Experimentally  

Nordsleaen,  D.  IJNMBME  2012  

NORDSLETTEN, 2013

COMMAS

Positions in the unstressed heart, denoted by X is motion critical forheart a continuum model he of the in the Lagrangian frameto where we d a bijective mapping function L : ⌦ ⇥ I ! R (w l points (or particles). The unstressed reference domain of0 a rectangular coordinate frame (see figure the ??). physical pos Thiscartesian mapping function defines MECHANICS art, denoted by X 2 ⌦0 , then moves in time according to ˆ OF HEART TISSUE y defining the continuous fields f , s ˆ Review of Kinematics t 2 I = [0, T ], so that d L : ⌦0 ⇥ I ! R (where d = 3 in the case of the heart) [?].   cetheframe, isposition denoted field es physical the body given time ⌦t := L(⌦0 t at any 0 ). ⌦The Deforma$on   of  a  Bofody   ⌦ LAGRANGIAN  MAPPING  

denotes the orientation of myofibers more of the b ⌦t or, := L(⌦ (2.2) 0 , t), precisely, the physical position OMAIN   al to PHYSICAL   the Dfiber field is the sheet field, may be mapped to one (and Dis OMAIN   al positionwhich of theREFERENCE   body defined by all coordinates x 2 Rdonly one) corr e(and aligned. – and also mutually only one)Lastly corresponding point in the reference domain ⌦0 . l field, n ˆ , denoting the direction in d ⌦t)} | 9 X 2 ⌦0 , su t := {x 2 R(2.3) Rd | 9 X 2 ⌦0 , such that x = L(X, lds, we may define the orthonormal

of ⌦ ⇢ Rd ⇥ I as a subset of space and time denoting the

We may also choose to think of ⌦ ⇢ Rd ⇥ I as t tissue at di↵erent stages through the deformation. The NORDSLETTEN, 2013 COMMAS (2.1) occupied the heart tissue at di↵eren agrangianvolume motion of the body. Theby kinematic displacement Nordsleaen,  D.  Prog.  Biophys.  Mol.  Biol,  2009.    

Positions in the unstressed denoted by X tion, we consider the motion of the heart in the heart, Lagrangian frame where we is[0,motion critical forheart a continuum model he of the in the Lagrangian frameto where we T ], so that d o motion of individual points (or particles). The unstressed reference domain a bijective mapping function L : ⌦ ⇥ I ! R (w 0 l points (or particles). The unstressed domain of ⌦t := L(⌦reference , t), ( 0 s defined by ⌦0 in a rectangular cartesian coordinate frame (see figure ??) a rectangular cartesian coordinate frame (see figure the ??). physical pos This mapping function defines precisely, the physical positionby of X the2body is defined coordinates ne the unstressed heart, denoted ⌦0 , then movesbyinall time accordingxto2 art, denoted by X 2 ⌦0 , then moves in time according to d ˆ may be mapped to one (and only one) corresponding point in the reference OF HEART TISSUE mapping function L : ⌦ ⇥ I ! R (where d = 3 in the case of the heart) dom [?] y defining the continuous fields f , s ˆ Review of Kinematics t 2 I = [0, T ], so that 0 d L : ⌦0 ⇥ I ! R (where d = 3 in the case of the heart) [?].   ing function defines the physical position of the body ⌦t at any given time cetheframe, isposition denoted field es physical the body given time ⌦t := L(⌦0 t at any 0 ). ⌦The Deforma$on   of  a  Bof   ⌦ dody ( T ], so that ⌦t := {x 2 R | 9 X 2 ⌦0 , such that x = L(X, t)}

MECHANICS

denotes the orientation of0,myofibers ⌦t := L(⌦ t), (2.2) or, more precisely, the physical position of the b ⌦t := L(⌦0 , t), (2.2) d y also choose to think of ⌦ ⇢ R ⇥ I as a subset of space and time denotingd al to the fiber field isofthe sheetis defined field, by all dcoordinates x 2 R ecisely, the physical position the body mayistissue be mapped tostages onethrough (and one) corr aloccupied positionwhich of the defined all coordinates x 2 R only by the body heart at by di↵erent the deformation. bealigned. mapped to Lastly one (and only one)also corresponding point in the reference domain e(and – and mutually corresponding point in theofreference g, L, only thus one) the Lagrangian motion the body.domain The kinematic displacem ⌦defines . 0 d body, u : ⌦ ⇥ I ! R is thenthe defined by the di↵erence between the position 0 l field, n ˆ , denoting direction in d d t)} ⌦t := {x 2 R | 9 X 2 ⌦0 , ⌦ such:= that{x x =2L(X, (2.3) d R | 9 X 2 ⌦ , su t in the reference | 9physical X 2 ⌦0 ,domain such that x= t)} (2.3)domain, i.e.0 nRthe ⌦ and itsL(X, position lds, we may define the orthonormal Displacement  

so choose dto think of ⌦ ⇢u(X, Rd ⇥t)I:= asL(X, a subset ofXspace and time denoting the( t) of ⌦ ⇢ R ⇥ I as a subset of space and time denoting the d We may also choose to think of ⌦ ⇢ R ⇥ I as upied byNordsleaen,   the heart tissue at di↵erent stages through the deformation. The D.  Prog.  Biophys.  Mol.  Biol,  2009.     t tissue at udi↵erent stages through the deformation. The simplicity, = x X. 2013 Lagrangian motion of the body. L, thusNORDSLETTEN, defines the The kinematicCOMMAS displacement (2.1) occupied the heart tissue at di↵eren agrangianvolume motiond of the body. Theby kinematic displacement y, u : ⌦0 ⇥ I ! R is then defined by the di↵erence between the position of a

Positions in the unstressed heart, denoted by X u(X, t) := L(X, Xu(X, t) := L(X, t) Xto (2.4) ( is critical fort)a continuum model a bijective mapping function L : ⌦0 ⇥ I ! Rd (w implicity, u = x X. This mapping function defines the physical pos MECHANICS y defining the continuous fields fˆOF , sˆ HEART TISSUE Review of= Kinematics t 2 I [0, T ], so that nt Deformation Tensor Gradient Tensor     ce frame, is denoted ⌦0 ). The field ⌦t := L(⌦0

or, ormation F , characterizes gradient tensor, the transformation F , characterizes of vectors, the transformation areas, of vectors, ar denotes the orientation of myofibers more precisely, the physical position of the b ,umes suchunder as or, L. a Mathematically, mapping, such as theL.deformation Mathematically, gradient the deformation grad

to(inthe fiber field is(in the sheet field, dient salsimply the the spatial reference gradient coordinate the system, reference X) of coordinate L,(and i.e. system, X) ofcorr L, i which may be mapped to one only one)

e aligned. Lastly – and also mutually ⌦0 . @ui @ui F := = rX u L+ = I, rX (u Fij+=X) = r +X iju. + I, Fij (2.5) = + ij . ( X (u + X) l field, n ˆ , denoting [email protected] in d @Xj j ⌦t := {x 2 R | 9 X 2 ⌦0 , su lds, we may define orthonormal Deforma$on   Gradient  the Tensor  

As is the all conservation Kronecker delta. principles As allin conservation the continuum principles settinginare the continuum setting Defines  how  the  Deforma$on  field  varies  in  the  Reference  domain  

er eirareas variation and volumes, under mapping their is fundamental. under To elucidate is fundamental. We may alsovariation choose to mapping think of ⌦ ⇢ Rd ⇥ToI elucid as elationships, vector dX consider =X X the By thedX fundamental = X 2 Xtheorem offundamental theorem 1 . vector 1 . By the NORDSLETTEN, 20132 COMMAS

(2.1) volume occupied by the heart tissue at di↵eren

1 in the unstressed heart, denoted by X Positions chain integration parts, Calculus, usingthe equation chain rule, (2.5)integration and (2.7), the by parts, relation using e is for a (1 continuum model to = Calculus, Fcritical (X 1 ) the · dX + rule, ⇠)rXby(r L(X + ⇠dX) · dX) · dX d⇠ X 1 d ) between a thebijective deformed0 vector, dx = L(Xbetween ) function L(X the) deformed and dXLmay vector, be expressed, dx =IL(X ) RL(X mapping :⌦ ⇥ ! (w 0 Z Z Z @ @ (2.6) This mapping function physical dx = L(X ) L(X )= L(X +dx ⇠dX) = L(X d⇠ =defines ) MECHANICS L(X r L(X ) =the + ⇠dX) L(X · dX+d⇠ ⇠dX) pos d⇠ = @⇠ @⇠ Z 1 Z ⇤ y defining the continuous fields fˆOF , sˆ HEART TISSUE Z Review of= Kinematics t 2 I [0, T ], so that (X 1 ) ·    dX = rXL(X ( rX · dX) d⇠ =r ) ·L(X dX 1 + ⇠dX) ⇠r (r L(X· dX = +r ⇠dX) L(X · (2.7) dX) ) · dX · dX d⇠ ⇠r 0 ce frame, is denoted ⌦0 ).Z The field ⌦t :=Z L(⌦0 2

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= F (X ) d· dX + (1 d ⇠)r (r L(X= F + (X ⇠dX) ) · ·dX dX) +· dX(1d⇠ ⇠)r denotes the orientation of myofibers f infinitesimal vectors, i.e. R✏ = {y 2 R | kyk  ✏, ✏