Deliverable 8.4 Whole Heart, Coupled FSI Simulation Report · D8.4 Whole Heart, Coupled FSI...

D8.4 Whole Heart, Coupled FSI Simulation Report MD-Paedigree - FP7-ICT-2011-9 (600932)

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Model Driven Paediatric European Digital Repository

Call identifier: FP7-ICT-2011-9 - Grant agreement no: 600932

Thematic Priority: ICT - ICT-2011.5.2: Virtual Physiological Human

Deliverable 8.4

Whole Heart, Coupled FSI Simulation Report

Due date of delivery: 30th November 2016

Actual submission date: 6th December 2016

Start of the project: 1st March 2013

Ending Date: 30th May 2017

Partner responsible for this deliverable: Siemens Healthcare GmbH

Version: 1.0


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Dissemination Level: Restricted to a group specified by the consortium (including the Commission

Services)

Document Classification

Title Whole-heart coupled Fluid-Structure-Interaction simulation Report

Deliverable D8.4

Reporting Period 4

Authors Tobias Heimann, Viorel Mihalef, Cosmin Nita, Lucian Itu, Tommaso Mansi, Ingmar Voigt

Work Package WP8

Security Restricted to a group specified by the consortium (including the Commission Services)

Nature Report

Keyword(s) FSI, Fluid Structure Interaction, Haemodynamics, Simulation, CFD, Heart Model

Document History

Name Remark Version Date

Viorel Mihalef Structure & intro; some content. V0.1 07/19/2016

Viorel Mihalef Extended content. V0.4 11/18/2016

Ingmar Voigt Added valve modeling refinements section. V0.7 11/22/2016

Viorel Mihalef Finalized content. V0.8 11/28/2016

Tobias Heimann Minor fixes & formatting. V0.9 11/28/2016

Tobias Heimann Incorporated internal reviewer feed-back. V1.0 12/06/2016

List of Contributors

Name Affiliation

Tobias Heimann SHC

Viorel Mihalef SCR

Lucian Itu UTBV

Tommaso Mansi SCR

Ingmar Voigt SHC

Cosmin Nita UTBV

List of reviewers

Name Affiliation

Xavier Pennec INRIA


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Abbreviations

0D Zero-dimensional

3D Three-dimensional

CFD Computational fluid dynamics

EDC End-diastolic volume

ELBM Entropic Lattice-Boltzmann Method

EM Electromechanics

ESV End-systolic volume

FSI Fluid structure interaction

LA Left atrium

LBGK Lattice-Bhatnagar-Gross-Krook

LBM Lattice-Boltzmann Method

LV Left ventricle

LVOT Left ventricular outflow tract

MPR Multiplanar reconstruction

MRI Magnetic resonance imaging

PC-MRI Phase contrast MRI

PV Pressure-volume

SV Stroke volume

TEE Trans-esophageal echocardiogram

TTE Transthoracic echocardiogram

WP Work package


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Table of Contents Executive Summary ................................................................................................................................ 6

1. Introduction ................................................................................................................................... 7

2. Materials and Methods ................................................................................................................... 8

2.1 Electro-mechanical model of the heart and FSI ............................................................................ 8

2.1.1 Overview .................................................................................................................................... 8

2.1.2 EM to CFD info ........................................................................................................................... 9

2.1.3 EM to valves info ....................................................................................................................... 9

2.1.4 FSI to EM info ............................................................................................................................. 9

2.2 Valve model and FSI .................................................................................................................... 10

2.2.1 Overview .................................................................................................................................. 10

2.2.2 Valve Modeling Refinements .................................................................................................. 10

2.2.3 FSI to Valve model ................................................................................................................... 12

2.2.4 Valve model to CFD ................................................................................................................. 13

2.3 CFD model and FSI ....................................................................................................................... 13

2.3.1 Overview .................................................................................................................................. 13

2.3.2 The Lattice-Boltzmann method (LBM) .................................................................................... 14

2.3.3 Turbulence options .................................................................................................................. 16

2.3.4 CFD to EM info ......................................................................................................................... 17

2.3.5 FSI to CFD info .......................................................................................................................... 17

3. Experiments and Results ............................................................................................................... 18

3.1 Validation of 3D CFD one-way and two-way FSI against 3D+time PC-MRI measurements ........ 18

3.2 Comparison of reduced and full two-way FSI computations ...................................................... 26

4. Conclusions .................................................................................................................................. 28

5. Appendix: Additional tests of the FSI CFD model ............................................................................ 29

5.1 Experiment 1: Peristaltic transport ............................................................................................. 29

5.2 Experiment 2: Expanding and contracting vessel ........................................................................ 29

6. References ................................................................................................................................... 31


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Executive Summary

Constant interaction of the blood flow with cardiac walls, valves, and adjacent arteries and veins, can

trigger and influence various cardiac pathologies. Therefore, the haemodynamics model is an important

component of our personalized cardiac model. The approach we have been using so far (presented in

deliverable D8.3) features a limited one-way interaction with the solid parts of the simulation, as geometric

changes by the cardiac walls and valves are directly imposed on the fluid without any feed-back.

The goal of this work package is to build a fluid-structure interaction (FSI) model which ties fluid and solid

parts of the simulation closer together and allows bi-directional feed-back and influences. In this report, we

present the developed methods, the updated modelling and computation pipelines, and an evaluation of

the new approach in direct comparison to its predecessor. Based on the available data, the new two-way

coupled FSI method is able to model the blood flow as measured by PC-MRI more accurately than our

previous approach.


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1. Introduction

We remind that as part of WP 8.3, a three-dimensional haemodynamics model has been created, such that

the imaging-extracted information of cardiac walls and valves were used to drive a Computational fluid

dynamics (CFD) solver. The flow of information was uni-directional, from the cardiac walls (positions and

velocities) to the 3D CFD model, which computed blood velocity and relative fluid pressure information. In

this work package 8.4, we enhance the model to a complete Fluid Structure Interaction (FSI) model, by

allowing bi-directional information exchange between the cardiac walls and valves on one hand, and the

blood fluid on the other. The FSI model is validated by comparison of baseline computations with 4D PC-

MRI patient data.

At a technical level, FSI is comprised of an electromechanics (EM) solid solver and a fluid solver, which are

run simultaneously, and which exchange information at a controlled number of iteration steps. The

pressure field computed by the fluid solver provides normal stress boundary conditions for the solid solver.

Conversely, the myocardium and 3D valve position and instantaneous wall velocity are used to define the

location and velocity of the deforming boundary of the fluid solver domain. The personalisation of the

integrated cardiac model is based on the personalisation of the electro-mechanical and haemodynamic

models built in WP 8.2 and 8.3.

The complete cardiac model is validated against measurements provided by 3D+time MRI. The advanced

hybrid cardiac mesh model integrating anatomical information from cine MRI and 3D echo data created in

WP 8.3 is further extended to build a geometric model of the myocardium, the dynamics of which is

computed by the solid solver. The 3D valve dynamics is integrated using an intermediate 0D valve flow

model which links the dynamic opening state of the valve used in the FSI solver with the kinematic state of

the valve computed by the segmentation. Velocity vector fields from 3D+time MRI data are used for

validating the 3D FSI computations.

An illustration of the computation system is given in Figure 1. The methods and models developed for this

task are presented in section 2. The specific methods used for building and personalizing the

electromechanical model are presented in section 2.1. The valve dynamics modeling is presented as part of

section 2.2. The CFD solver is presented in section 2.3. Finally, the validation of computed velocity fields

versus the MRI measured data is presented in section 3.

Figure 1 Fluid structure interaction system for cardiac haemodynamics computation. The interactions between the electromechanical model, valves and the CFD model are controlled by the FSI interface module.


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2. Materials and Methods

In this section, we provide details concerning the components used in the cardiac FSI computations. An FSI

module acts as coupling interface between the myocardial electromechanical (EM) solver, the valve and the

fluid solvers, and controls the proper exchange of information. At each time step, the FSI module enables

the following interactions:

– the stress load from the fluid is sent to the EM solver

– the myocardial positions and velocities are sent to the CFD solver

– the myocardial positions and valve orifice areas are sent to the valve solver

– the 3D valve positions are sent to the CFD solver

Let us underline that, in comparison to the setup in WP 8.3, the two-way FSI system (comprised of the EM

model, the valves and the blood) has more “freedom”, with the walls responding to intra-cardiac pressures

and the valves responding to pressure forces acting across them.

In the following, we present in detail the interaction of all of these components. As the main elements of

the electromechanical model have already been presented in WP8.2 and WP8.3, its description will be

shorter. At the same time, we provide a more comprehensive description of the valve model, the CFD

model and of various elements of the coupling logic.

2.1 Electro-mechanical model of the heart and FSI

2.1.1 Overview

Our two-way FSI system uses the cardiac electro-mechanical model introduced in WP 8.2, which has been

extended in WP 8.3 (see section 2.2.1) to include a reduced order (0D) haemodynamics model. We recall

that the 0D haemodynamics model accounts for the artery-ventricle-atrium dynamic modules as coupled

physical systems, obeying mass conservation as well as appropriate pressure jumps between modules. The

system uses a two-way interaction logic between the arterial and the atrial systems on one hand, which

receive flow from the valve modules, and the ventricle on the other hand, which receives pressure values

used as boundary conditions from the arterial and atrial systems. The mass conservation is modeled using

an artificial compressibility formulation given by 𝑑𝑉

𝑑𝑡= 𝑞𝑚𝑖𝑡𝑟𝑎𝑙 + 𝑞𝑎𝑜𝑟𝑡𝑖𝑐 − 𝐶

𝑑𝑝

𝑑𝑡, where the pressure and

volume variables are ventricular, and 𝐶 is the artificial compressibility coefficient. Mitral and aortic flows

are treated as independent variables, which are influenced by the pressure drop across the valves, as

dictated by 0D valve models. The 0D valve model part of the EM model assumes that the pressure drop ∆𝑝

due to flow 𝑞 can be written as ∆𝑝 = 𝑅𝑞 + 𝐵𝑞|𝑞| + 𝐿𝑑𝑞

𝑑𝑡 (see WP8.3, 2.2.1). A valve phase function 𝜑 is

defined as a smooth function varying between 0 (closed valve) and 1 (open valve), with dynamics

controlled by the pressure gradient as follows:

𝑑𝜑

𝑑𝑡= {

(1 − 𝜑)𝐾𝑜𝑝𝑒𝑛∆𝑝 , ∆𝑝 > 0

𝜑𝐾𝑐𝑙𝑜𝑠𝑒∆𝑝, ∆𝑝 < 0

This formulation enables the computation of pressure-driven valve dynamics, which makes it amenable to

being part of the two-way FSI system.


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Figure 2 From myocardium tetrahedral mesh (left, white) the endocardial surface is extracted (middle, green) and used to define the boundary of the CFD domain (right, lime).

2.1.2 EM to CFD info

The end-diastolic (at time t = 0) myocardial tetra mesh is provided with LV and LVOT tags which are used to

extract an endocardial surface whose topology never changes over the course of the cardiac cycle. The

endocardial surface boundary edges are separated into the mitral and aortic contours, which are then

tessellated to create the mitral and aortic virtual boundaries (outlets) for the use of the CFD code (Figure

2). Each endocardial mesh vertex is associated to a (closest at t=0) tetra mesh vertex and follows rigidly its

motion over the course of the cardiac cycle. As the endocardial mesh gets updated at any time t, the aortic

and mitral outlet surface barycenters are updated accordingly. The displacement rate of the endocardial

vertices is the velocity information being sent out to FSI, to be further used by the CFD solver.

2.1.3 EM to valves info

Each of the valve boundary vertices is kinematically-linked to a closest myocardial tetra mesh vertex,

computed at end-diastolic time (t = 0). In effect, as the EM mesh moves over time, the new EM mesh

position is imposed as a rigid constraint to the valve boundary vertices. This ensures that the valve

boundary travels along with the mesh as it moves over time. To compute the rest of the valve mesh motion

extra care is needed, as described in section 2.2.

The valve opening phase computed by the 0D valve model is also sent to the FSI interface, to be used by

the valve (3D) model, which is programmed to match the opening phase.

2.1.4 FSI to EM info

The EM model requires stress information as a boundary condition applied to its endocardial surface, and

the FSI interface provides this information from the CFD solver, interpolated at each endocardial triangle

barycenter (Figure 3). Only the normal stress component provided by – 𝑝𝑛 (where n is the surface normal

and p the pressure) is considered, due to the specific intra-ventricular flow regime characterized by small

viscous stresses compared to pressure stresses. The pressure can be written as 𝑝 = �̅� + 𝑝𝑟𝑒𝑙, where �̅� is

the mean intra-ventricular pressure, which is also compted by the 0D haemodynamics pressure model

coupled with the EM model. The pressure 𝑝𝑟𝑒𝑙 is effectively the internal pressure used by the CFD solver,


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which can be considered a relative pressure if one fixes the average outlet pressure to a default relative

value as zero. In practice, the mean intraventricular pressure, as computed by the 3D CFD solver, is

subtracted from all the 3D CFD endocardial values, to which we add back the 0D mean pressure to obtain

the final absolute pressure field on the endocardium.

Figure 3 Electro-mechanical model interaction with the FSI interface.

2.2 Valve model and FSI

2.2.1 Overview

Cardiac valves that are part of a two-way FSI computational system are required, on one hand, to obey the

instantaneous pressure gradients that drive their opening and closing dynamics, as well as behave in a

manner reasonably close to the kinematics obtained from the 3D+time echo data. One way to solve these

two incompatible requirements could be to define valve-specific parameters characterizing their material

properties (including their wall connecting ligaments) and personalize them based on the observed valve

motion and flow information. To minimize the modeling and computational burden of such an approach we

preferred a simplified approach that combines the 0D pressure-driven valve system and the 3D valve

kinematics segmented from echo ground truth. Before we describe this in detail, we briefly give an update

of the anatomical valve modeling workflow.

2.2.2 Valve Modeling Refinements

Compared to our previous deliverable D8.3, the valve modeling capabilities from pediatric TTE echo data

have been improved with a more streamlined and more robust manual initialization method, particularly to

address specific challenges in modeling pediatric TTE echo data, with comparably lower signal-to-noise

ratio.

Particularly, a valve model is now initialized by drawing a line on one of the MPR slices approximately along

the long axis of the aortic root and from LA to LV for the mitral valve. Subsequently, a valve model appears

on screen, which can be rigidly translated, rotated and resized using the mouse input. Also incorporating

the model size is particularly important since the data is in fact pediactric transthoracic echo data, while the

automation was originally trained on adult transesophageal echo data, which means that both valve size

and orientation are expected to differ from the distribution of the TEE training data. This method is much

more convenient and produced more reliable results in contrast to the previous method, which was purely

based on an unintuitive MPR-only adjustment, where no model was visualized to guide the placement. The

sequence of alignment steps is illustrated in Figure 4.


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Figure 4: Manual valve initialization for mitral (left column) and aortic valve (right column). The initialization starts with drawing a line on one of the MPRs approximately along the long axis of the aortic root and from LA to LV for the mitral valve (top row). Subsequently, a valve model appears on screen, which can be rigidly translated, rotated and resized using the mouse input (mid 2

nd row). The resulting pose and size are input to the valve model estimation algorithm to refine the manual initialization

automatically (3rd

row) and track over time. Finally the result is manually adjusted where necessary (bottom row).


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The resulting pose and size are input to the valve model estimation algorithm to refine the manual

initialization automatically. Specifically (1) the image is resampled to match both orientation and size of the

adult TTE data and (2) the input pose and size is also used to constrain the search space to ensure a robust

yet accurate result. Furthermore, due to the significantly different appearance of the data given the smaller

acoustic window in children, the model post-processing component – which ensures spatio-temporal

consistency – was adapted to better cope with potential outliers in landmark estimation, by projecting

landmarks on the surfaces and facilitate subsequent manual refinement.

This manual input is remembered for the particular volume, and used in both tracking and subsequent

detections, as the optimal sequence of cardiac phases had to be determined individually for each processed

volume, as due to the cardiac motion the aortic root sometimes was not fully included into the volume in

some cardiac cycles. Additionally, the fusion with MRI meshes had to be adapted for this fact, since in one

of the cases instead of the typical sequence of systole-diastole, the opposite combination i.e. diastole-

systole had to be modeled from echo data, since otherwise it was not possible to model the aortic root

with sufficient detail.

2.2.3 FSI to Valve model

Once the anatomical valve model from TTE is available, the timing of its opening and closing needs to be

adjusted to fit the simulation. For this, the 0D valve model part of the EM haemodynamics model provides

at every moment in time the opening phase information for each of the valves, in terms of a smooth

function between 0 (closed) and 1 (open). This information is passed on by the FSI interface, along with the

current position of the myocardial tetra mesh. The idea is to dynamically control the segmented valve

kinematics using the 0D phase functions. In Figure 5 we give an example of how the valve phases, as

computed by the 0D valve solver, map to 3D valve mesh sequences over the course of the cardiac cycle. In

practice we start from a sequence of 3D valve meshes (whose phases are also known from echo

measurements), for example aortic valves during systole, and create a smooth valve-mesh-over-systolic-

time function by taking the Fourier transform of the meshes. This function can provide the kinematic 3D

mesh geometry corresponding to any phase, however, the challenge is to have such a mesh that matches

also the freely-moving myocardial mesh.

This is ensured by first creating a correspondence map between the valve base vertices and the myocardial

mesh, such that the base vertices have hard kinematic contraints as prescribed by their corresponding myo

vertices. Second, the valve vertex positions are defined using a linear combination between a rigid

transform (barycentric translation) and the transform given by the (myo constrained) base vertex

kinematics. The two transforms are weighted using the relative distances from any valve vertex to its

corresponding rim and base valve vertices. This ensures that the rim opening area/phase always matches

the prescribed one from 0D, while the valve also follows the “live” heart motion.


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Figure 5 Aortic and mitral 3D valves are controlled by 0D opening phase functions whose dynamics is governed by pressure gradient forces.

2.2.4 Valve model to CFD

The 3D valve position and velocity is sent to the FSI, for use by the CFD solver, which uses them as moving

surfaces along which the no-slip condition (valve velocity and fluid velocity coincide at contact points) holds

true (Figure 6).

Figure 6 The valve module sets current opening from 0D valve info, and sends 3D position and velocity info to the CFD solver.

2.3 CFD model and FSI

2.3.1 Overview

Our flow model relies on the Lattice Boltzmann Method (LBM), which describes physics of fluid flow at a mesoscopic scale by taking into account molecular interactions between flow particles. The LBM provides ultimately the same solution as the Navier-Stokes based solvers [Chen1998], but it is “naturally” highly parallelizable, which was considered a necessary feature to enable an efficient computation of the two-way FSI problem. In particular, our CFD solver used in the one-way FSI WP 8.3 computations was second-order accurate in velocity computation but only first order accurate in pressure computation. This means that reasonable convergence for velocities, which were the metric of interest in WP 8.3, requires lower-resolution grids compared to what is required to obtain accurate pressure, which could ultimately prove to


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be more relevant clinically. As a consequence, the high-resolution grids necessary for providing accurate pressure values, which are one of the essential pieces of information in the two-way FSI computations, turn out to be unfeasible to compute with WP 8.3 CFD solver in terms of computational time.

As a solution, a Siemens in-house LBM CFD solver, which has been successfully tested against commercial

solvers (CFX, FLUENT) and proved to be accurate for low to medium Reynolds number flow in complex (but

static) geometries, was chosen for its efficiency and accuracy. The extensions done as part of WP 8.4 were

the following:

Handling of moving outer boundaries (i.e. cardiac walls)

Handling of moving internal surfaces (i.e. valves)

Handling of moving outlets and inlets

Handling of turbulent flow

In the following, we provide a short description of the LBM solver mechanism and implementation.

2.3.2 The Lattice-Boltzmann method (LBM)

LBM models the physics of the interaction of fluid particles at a meso-scale level using a mathematical model the Boltzmann equation:

𝜕𝑓

𝜕𝑡+ 𝐮 ⋅ ∇𝑓 = Ω(𝑓)

(1)

Here 𝑓 = 𝑓(𝐮, 𝐱, 𝑡) is a probability density function and it gives the probability of a particle to have the

velocity 𝐮 and to be at position 𝐱 at time t. The right hand side of eq. 1 is known as the collision operator

and accounts for the contribution of the collision between particles. For the numerical implementation of

LBM, eq.1 is written in a discrete form:

𝜕𝑓𝑖

𝜕𝑡+ 𝐜i ⋅ ∇fi = Ω(𝑓i), 𝑓𝑖 = 𝑓𝑖(𝐱, 𝑡)

(2)

where 𝑓𝑖 is a discrete representation of 𝑓 with respect to the variable 𝐮, more specifically instead of a

single function 𝑓 that depends on 𝐮, 𝐱, and t, there are a finite number of 𝑓𝑖 functions that depend on just 𝐱

and t. The discrete velocities 𝐜𝑖 are associated to a lattice structure as displayed in Figure 7, each velocity 𝐜𝑖

corresponds to a link connecting a node 𝐱 in the grid with a neighbouring node 𝐱 + 𝐜𝑖. The most commonly

used lattice structure for 3D fluid computations contains 15, 19 or 27 links.

Figure 7: 15-velocity lattice structure.


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The macroscopic velocity 𝐮 and pressure 𝑃 of the fluid are related to the density functions 𝑓𝑖 as follows:

𝑃 = 𝜌𝑐𝑠2 = ∑ 𝑓𝑖𝑐𝑠

2

𝑁

𝑖=0

(3)

𝐮 =1

𝜌∑ 𝐜𝑖𝑓𝑖

𝑁

𝑖=0

(4)

Where 𝑐𝑠 is the non-dimensional speed of sound and is 1/√3 for the 15, 19 and 27 velocity lattice

structures.

Eq. 2 is solved using an explicit time discretization scheme on two steps:

1. Collision

𝑓𝑖(𝐱, 𝑡 + Δ𝑡) = 𝑓𝑖(𝐱, 𝑡) − Ωi,j(𝑓𝑖(𝐱, 𝑡) − 𝑓𝑖𝑒𝑞(𝐱, 𝑡)) (5)

2. Propagation

𝑓𝑖(𝐱 + 𝐜𝑖 , 𝑡 + Δ𝑡) = 𝑓𝑖(𝐱, +Δ𝑡) (6)

The collision term Ω𝑖,𝑗(𝑓𝑖(𝐱, 𝑡) − 𝑓𝑖𝑒𝑞(𝐱, 𝑡)) is formulated as a relaxation towards a thermodynamic

equilibrium where Ω is termed the collision matrix and contains the relaxation factors. In this work the

multiple-relaxation-time (MRT) collision operator [D’Humieres2002] is employed.

Moving Boundary Treatment

For lattice nodes located near a boundary, i.e. where there is a neighbouring node located outside the fluid

region, there are unknown 𝑓𝑖 values that are required to complete the propagation step (eq. 6). The most

commonly used way to compute the unknown distributions is the bounce-back approach: the unknown 𝑓𝑖

values i.e. the values that are propagated from a solid node are set to the value corresponding to the

opposite lattice direction. This is equivalent to reversing the velocity of a particle colliding with the wall.

𝑓𝑖 = 𝑓𝑖′ (7) where 𝒄𝑖 = −𝒄𝑖′ are two opposite lattice directions. In this work an interpolated bounce-back scheme

[Bouzidi2001] was employed that can be used for curved moving walls:

𝑓𝑖′(𝐱, 𝑡 + Δ𝑡) = 2𝑞𝑓𝑖(𝐱, 𝑡 + Δ𝑡) + (1 − 𝑞)𝑓𝑖(𝐱 − 𝐜𝑖 , 𝑡 + Δ𝑡) + 2𝑤𝑖𝐜𝑖′𝐮𝑤, 𝑞 < 0.5 (8)

𝑓𝑖′(𝐱, 𝑡 + Δ𝑡) =1

2𝑞𝑓𝑖(𝐱, 𝑡 + Δ𝑡) +

2𝑞 − 1

2𝑞𝑓𝑖′(𝐱, 𝑡 + Δ𝑡) +

1

𝑞𝑤𝑖𝐜𝑖′𝐮𝑤 , 𝑞 ≥ 0.5

(9)

where 𝐮𝑤 is the wall velocity and 𝑞 is a factor between 0 and 1 that accounts for the exact position of the

wall between two lattice nodes.

For the outflow boundary, i.e. wherever the flow leaves the domain, the velocity is usually unknown and

the pressure is specified. In this case the non-equilibrium extrapolation method [Zhao-Li 2002] is employed.

The non-equilibrium extrapolation method replaces all the 𝑓𝑖 values at the boundary using information

extrapolated from neighbouring locations:

𝑓𝑖(𝐱, 𝑡 + Δ𝑡) = 𝑓𝑖𝑒𝑞(𝐱, 𝑡) + (1 − Ω𝑖,𝑗)𝑓𝑖

𝑛𝑒𝑞(𝐱𝑛𝑒𝑖𝑔ℎ , 𝑡) (10)


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Here 𝑓𝑖𝑛𝑒𝑞

= 𝑓𝑖 − 𝑓𝑖𝑒𝑞

is the non-equilibrium part of the distribution functions and 𝐱𝑛𝑒𝑖𝑔ℎ is a neighbouring

fluid node located along the boundary surface normal 𝐧𝑏 such that (𝐱𝑛𝑒𝑖𝑔ℎ − 𝐱) × 𝐧 = 0. The 𝑓𝑖 values at

𝐱𝑛𝑒𝑖𝑔ℎ are found by means of spatial interpolation.

2.3.3 Turbulence options

Cardiac flow may enter physical regimes characterized by high Reynolds numbers, which require turbulence

modeling to be added to the regular LBM description. We tested two implementations of turbulence, the

Smagorinsky and the Entropic Collision models.

Smagorinsky sub-grid model

The Smagorinsky model is a well established approach for simulating turbulent flows. It consists of

modeling sub-grid scale effects as an additional viscosity 𝜈𝑡 called turbulent viscosity. The turbulent

viscosity is computed as follows:

𝜈𝑡(𝐱, 𝑡) = 𝐶𝑠2Δ𝑥2|𝑆̅(𝐱, 𝑡)|𝑆𝑖𝑗(𝐱, 𝑡) (11)

Where 𝐶𝑠 is the Smagorinsky constant and is usually set to 0.1 - 0.2 and 𝑆𝑖𝑗 is the strain rate tensor. The

turbulent viscosity 𝜈𝑡 is computed at each time-step and for each lattice location and is added to the

dynamic viscosity 𝜈. The relaxation parameter 1/𝜏 which is used in the LBM collision operation becomes:

𝜏 =1

𝑐𝑠2(𝜈 + 𝜈𝑡) + 0.5

(12)

When the flow is laminar, the turbulent viscosity 𝜈𝑡 will be close to zero. Otherwise, in the regions where

strain rates (i.e. velocity gradients) are high, the turbulent viscosity will increase and will stabilize the

simulation by increasing the total viscosity.

Entropic Collision Model

One of the most important limitations of the LBM is numerical instability that appears when viscosity is too

low. Avoiding instability when simulating large Reynolds numbers (corresponding to low viscosities) using

the classic LBGK collision model is only possible with unpractically high grid resolution, and the Smagorinsky

sub-grid model was discovered to not be stable enough for the requirements of our computations.

An approach to improve the stability is the entropic Lattice-Boltzmann method (ELBM) [Ansumali2003]

which is based on the Boltzmann’s H theorem and it consists of changing the collision model so that it

satisfies the second law of thermodynamics.

The equilibrium distribution function is changed as follows:

𝑓𝑒𝑞(𝐱, 𝑡) = 𝑤𝑖𝜌 ∏ (2 − √1 + 3𝑢𝑑2) (

2𝑢𝑑 + √1 + 3𝑢𝑑

1 − 𝑢𝑑)

𝑒𝑖𝑑𝐷

𝑑=1

(13)

And the collision operation is changed as follows:

𝑓𝑖(𝐱 + 𝐞𝑖𝛿𝑡, 𝑡 + 𝛿𝑡) = 𝑓𝑖(𝐱, 𝑡) + 𝛼𝛽(𝑓𝑖𝑒𝑞(𝐱, 𝑡) − 𝑓𝑖(𝐱, 𝑡)) (14)


17

where 𝛽 =2

𝜏 is the equivalent of the LBGK relaxation time and is related to the fluid viscosity. The

additional factor 𝛼 is specific to the ELBM model and it is computed for each lattice node at each iteration

by solving a non-linear equation:

𝐻(𝑓) = 𝐻(𝑓 + 𝛼(𝑓𝑒𝑞 − 𝑓)) (15)

𝐻(𝑓) = ∑ 𝑓𝑖 ln (𝑓𝑖

𝑤𝑖)

𝑁

𝑖=0

(16)

Equation 15 is usually solved iteratively using a Newton method. If the factor 𝛼 is 2 then the LBGK collision

is recovered, the simulation is stabilized by automatically tuning the 𝛼 parameter. This allows the

simulation to run at very high Reynolds numbers without becoming unstable or requiring large grid

resolution. The ELBM model was in the end used for the simulations reported here.

2.3.4 CFD to EM info

The fluid stress tensor definition is 𝜎 = −𝑝𝐼 + 𝜇(∇𝑈 + ∇𝑈𝑇)/2, where p is the fluid pressure, U is the

velocity and 𝜇 is the blood dynamic viscosity coefficient. Due to the relatively high intracardiac pressure

stress relative to viscous stress, we may safely omit the contribution of the latter, and approximate the

stress tensor as 𝜎 = −𝑝𝐼. Thus, as also explained earlier in Sec. 2.1.4, the CFD relative pressure field

passed on to the FSI interface is processed as follows:

averaged inside the ventricle and subtracted from all points to ensure zero mean

extrapolated across the endocardial surface using a second order accurate extrapolation method

interpolated at the endocardial triangle barycenter locations.

2.3.5 FSI to CFD info

At every moment, FSI provides location and velocity of the endocardial surface to the CFD solver, including

the virtual outlets (mitral and aortic regions), and also the valve surface location and velocity (Figure 8). The

interpolated bounce-back rule described earlier is used to enforce the no-slip condition at these interfaces.

Figure 8 CFD and FSI information flow.


18

3. Experiments and Results

In this section, we examine the differences between two-way and one-way FSI models. Moreover, we

present several validation studies against PC-MRI velocity measurements, conducted using the two-way FSI

model on 4 patients, 3 from OPBG (13, 21 and 31) and one from GOSH (71). At the moment of writing,

these were all datasets that had all the acquisitions and quality required for FSI validation: MRI (cardiac

walls segmentation), PC-MRI (flow), and echo (valve segmentation). One more patient (OPBG33) had

complete data available, however the quality of the echo data was insufficient for a successful

segmentation of the valves.

3.1 Validation of 3D CFD one-way and two-way FSI against 3D+time PC-MRI measurements

In the following, we provide an in-depth validation of the CFD results from one- and two-way FSI

simulations against PC-MRI velocity measurements. To this end, we use the 7-segment map defined in WP

8.3 and evaluate the average of velocities inside each of the defined segments for the various types of CFD

and for PC-MRI at the moments of peak systolic flow and peak diastolic flow. We compare the magnitudes

for all these CFD velocities and their angles with the corresponding PC-MRI averaged velocities. For better

qualitative understanding, we also visualize peak-systolic and peak-diastolic 7-segment average velocities

(CFD in red, PC-MRI in white), using the same scale for all velocity vectors, such that their length in the

images is proportional to their real magnitude.

Based on the 7-segment velocities, we derived a momentum error function that computes the difference

between PC-MRI and CFD momentum. The error function is normalized by the intra-cardiac volume for

easier longitudinal data comparison. The absolute momentum error function is defined as 𝐸𝑟𝑟𝑎𝑏𝑠 =

∫ 𝜌‖𝑣𝑃𝐶𝑀𝑅𝐼 − 𝑣𝐶𝐹𝐷‖ 𝑑𝑣𝑜𝑙 , while the normalized version is 𝐸𝑟𝑟𝑛𝑜𝑟𝑚 = ∫ 𝜌‖𝑣𝑃𝐶𝑀𝑅𝐼 − 𝑣𝐶𝐹𝐷‖ 𝑑𝑣𝑜𝑙/

∫ 𝜌𝑑𝑣𝑜𝑙. One may use various discretizations of the control integration domain to obtain various measures

of this error. We prefer to use the 7-segment map as it arguably provides a more meaningful

characterization of the goodness-of-match between the flows than a bulk or a more randomly chosen

discretization of the ventricular volume. The discretized version of 𝐸𝑟𝑟𝑛𝑜𝑟𝑚 that we compute is 𝐸𝑟𝑟𝑛𝑜𝑟𝑚 ≈1

𝑣𝑜𝑙(𝐿𝑉)∑ ‖𝑣𝑃𝐶𝑀𝑅𝐼 − 𝑣𝐶𝐹𝐷‖ ∗ 𝑣𝑜𝑙(𝑉𝑖)𝑉𝑖 , where 𝑉𝑖 is each of the segments of the 7-segment map, and

𝑣𝑜𝑙(𝐿𝑉) is the intra-cardiac volume. In Table 1 below and in Figure 9 we provide the computed values for

𝐸𝑟𝑟𝑛𝑜𝑟𝑚, measured in mm/s, for the four cases. The lower the value of 𝐸𝑟𝑟𝑛𝑜𝑟𝑚 is, the better the match.

Table 1 Normalized momentum error for the peak systolic and peak diastolic stages of the four cases. The units are mm/s.

Case 1 Way-SYS 2 Way-SYS 1 Way-DIA 2 Way-DIA

OPBG 15 89 63 73 48

OPBG 21 87 68 77 73

OPBG 31 61 48 51 56

GOSH 71 188 160 166 155


19

Figure 9 Systolic and diastolic peak data for normalized momentum error indicator. The x-axis shows case ID numbers (15,21,31 and 71), the y-axis shows the function Err_norm (see text). The lower the value, the better the match is between PC-MRI data and CFD computations.

We note that in 7 out of the 8 comparisons (4 cases, systole and diastole) the 2-way approach provided a

better momentum match than the 1-way approach. This observation should be thoroughly scrutinized

before concluding that it indicates that 2-way FSI is more accurate as a CFD algorithm, due to various

mitigating factors, as follows:

1-way computations use a set of endocardial meshes segmented at all cycle times, while 2-way

computations use endocardium meshes computed by EM from the ED mesh, and there are various

local geometric differences between the meshes; this is an expected geometric mismatch effect as

the EM personalization is not constrained by the local endocardial geometry.

There are differences between the above meshes even globally, as the intra-ventricular volume

curves, while having a good match for SV, do have some (small) differences in EDV and ESV, as well

as in the curve shapes themselves. This is also expected due to limitations of the EM

personalization and also due to the effect of the 2-way FSI, which implies changes in the

endocardial meshes at any time step.

Segmentation errors in the endocardial walls are larger in any other cardiac stages compared to

ED, which is the starting point for EM personalization and 2-way FSI computations; one may

further examine such errors through registration and mesh-to-mesh distance metrics.

Valve segmentation is smoothed-out while in use by 2-way FSI computations; it is unclear how to

best quantify the range of changes due to this, or if it is indeed needed.

PC-MRI data presents both noise and segmentation artifacts due to limitations in data quality and

the segmentation algorithms limitations.

Nevertheless, the uncertainties introduced by these factors cannot diminish the visibly positive effect that

the 2-way FSI algorithmic approach has on reducing the normalized momentum error.


20

In the next few pages, we provide a closer look at the individual case results and comment as necessary.

We remind that the two-way computations are driven by the electro-mechanical (EM) model which was

personalized (as described in WP 8.2) to match the volume curve of the kinematic segmentation. Besides

the 1-way FSI results, we present CFD results obtained from two kinds of two-way FSI computations:

full two-way, as described in Section 2; descriptions of these results bear the identifier “2-way”

reduced two-way , which passes only the mean (bulk) intra-ventricular pressure from fluid to solid

rather than the full pressure; descriptions of these results bear the identifier “reduced 2-way”.

The rationale for performing the two different computations was to observe if the simpler bulk pressure

model, which is faster to compute and has lighter constraints on the stability of the discretized CFD

solution, could also be used as an alternative option to the full two-way FSI. Results for both variants are

compared with each other in Section 3.2.

Patient OPBG15

This is a smaller child heart with relatively small volume indices: EDV = 58 ml, ESV = 32 ml, SV = 26 ml (see

Figure 10). Blood flow velocities and angles are analyzed in Figure 11, while Figure 12 and Figure 13

visualize the flow for the individual segments.

For this case, 2-way FSI has clearly more accurate angles, most importantly for the systolic aortic root

velocity (see Figure 12). An explanation could be that the 1-way valve segmentation at peak systole creates

an excentric jet that may be filtered out by the 2-way valve handling algorithm. The velocity magnitudes

are also more accurate in the 2-way FSI computations.

Figure 10 Volume curve from MRI segmentation.


21

Figure 11 Comparison of regional variation of computed velocities versus MRI measurements at peak flow moments of systole and diastole – the seven regions form the abscissa. The two top graphs show the vector magnitudes, while the bottom ones show the angles (in degrees) between MRI and CFD vectors.

Figure 12 Segment-average velocities for CFD (red) and PC-MRI (white) at peak systole. From left to right: 1-way, reduced 2-way, and 2-way.

Figure 13 Segment-average velocities for CFD (red) and PC-MRI (white) at peak diastole. From left to right: 1-way, reduced 2-way, and 2-way.


22

Patient OPBG021

This is a hypertrophic heart with a normal stroke volume: EDV = 274 ml, ESV = 180 ml, SV = 94 ml (see



For this case, the aortic root peak systolic velocity is more accurate in 1-way FSI, however the rest of the

segments are doing slightly better in 2-way FSI, which biases the global scores toward 2-way FSI for both

peak systole and diastole.


Figure 15 Comparison of regional variation of computed velocities versus MR measurements at peak flow moments of systole and diastole – the seven regions form the abscissa. The two top graphs show the vector magnitudes, while the bottom ones show the angles (in degrees) between MRI and CFD vectors.


23

Figure 16 Segment-average velocities for CFD (red) and PC-MRI (white) at peak systole. From left to right: 1-way, reduced 2-way, and 2-way.


Patient OPBG31

This is a small child heart with volume indices EDV = 46 ml, ESV = 28 ml, SV = 18 ml (see Figure 18). Blood

flow velocities and angles are analyzed in Figure 19, while Figure 20 and Figure 21 visualize the flow for the

individual segments.

For this case, the aortic root peak systolic velocity is more accurate in 1-way FSI, however the rest of the

segments are doing slightly better in the systolic 2-way FSI, especially angle-wise, which biases the global

scores toward 2-way FSI for peak systole. The diastolic segmental velocities for 1-way and 2-way FSI are

very close to one another, with a slight edge for 1-way FSI, especially in segments 6 and 7, which biases the

global diastolic error score in favor of 1-way FSI.



24


Figure 20 Segment-average velocities for CFD (red) and PC-MRI (white) at peak sysstole. From left to right: 1-way, reduced 2-way, and 2-way.



25

Patient GOSH071

This is a slightly enlarged heart, with volume indices given by EDV = 168 ml, ESV = 70 ml, SV = 98 ml (see



While segments 6 and 7 peak systolic velocity are more accurate in 1-way FSI for this case, the rest of the

segments are doing better in 2-way FSI, which biases the global scores toward 2-way FSI for peak systole.

Especially peak diastolic velocities have large differences in angles with the PC-MRI ones, due to reduced

data quality that led to a less accurate segmentation of the PC-MRI flow.




26

Figure 24 Segment-average velocities for CFD (red) and PC-MRI (white) at peak sysstole. From left to right: 1-way, reduced 2-way, and 2-way.


3.2 Comparison of reduced and full two-way FSI computations

Below, we give a brief comparison of the haemodynamics properties of the reduced and the full 2-way FSI,

based on the data from OPBG21. We note that the other cases tell a similar story.

Average LV pressure changes little between both EM-based computations (Figure 26). Please recall

that, at the same time, we have no absolute pressure values for the 1-way computations for

comparison, which gives the 2-way FSI computations an advantage.

During systole, full two-way FSI activated fluid pressure will push back the endocardial walls,

therefore generating a weaker contraction, while at the same time losing some internal pressure

resisting the endocardial wall contraction. Indeed, this behavior is observed for OPBG21, as we

have: ESV_reduced= 180.4 ml while ESV_full = 181.4 ml, and PeakSysPressure_reduced = 133

mmHg while PeakSysPressure_full = 132 mmHg (Figure 26 and Figure 27).

During diastole, full two-way FSI endocardial wall motion is less constraining to the fluid than in

reduced two-way FSI. This is observed with a small difference between the end diastolic volumes,

EDV_reduced = 270.9 ml while EDV_full = 270.4ml, while diastolic pressures are also insignificantly

different (Figure 26 and Figure 27).

As a consequence, we observe that SV_full is smaller than SV_reduced, however the difference is

statistically insignificant, about 1.6 ml (Figure 27).


27

Based on the data collected so far, one could draw the conclusion that it is possible to use the reduced two-

way FSI and obtain results that are within the same tolerance from PC-MRI measurements as the full two-

way approach. This would certainly come with computational efficiency advatages. More data would be

welcome to further support this conclusion.

Figure 26 Average LV volume pressure values for reduced and full two-way computations for OPBG21.

Figure 27 LV volume difference over time (x-axis, in millisecons) between reduced and full two-way FSI computations for OPBG21. As the units on the vertical axis indicate, differences are rather minor.


28

4. Conclusions

We have presented a new two-way FSI computational algorithm and compared it with the one-way FSI

algorithm developed in WP 8.3, analyzing how well both of them match PC-MRI data for a number of

patients in the study. At a qualitative level, we observed that the slingshot dynamics described in WP 8.3 is

observed in both types of CFD simulations (one-way and two-way), as well as in the PC-MRI data. Also

qualitatively, we noted that large segmental velocities match well in both magnitude and direction, while

for lower magnitudes the match accuracy is reduced.

Quantitatively, we developed a novel index, the normalized momentum error, which proved useful in

characterizing the global goodness of fit between the 7-segment velocities from CFD and PC-MRI. Although

based on limited data, the index indicates that the two-way FSI approach provides a better fit with PC-MRI

compared to one-way FSI. Furthermore, we noted that a simplified and more efficient two-way FSI version

can provide reasonable results as well and should be evaluated further once more data becomes available.

When taking technological complexity into account, it is worth noting that the two-way FSI approach

involves less geometric fusion effort between MRI and echo segmentations, which may introduce errors if

data has low quality. Therefore, two-way FSI can be more robust. On the other hand, the one-way FSI

approach does not require the sophisticated myocardial electromechanics technology, so it is easier to set

up. However, it does not provide absolute pressure information like the two-way algorithm. If clinicians

identify a strong need for intra-cardiac absolute pressures, that could be a deciding factor for the choice of

FSI algorithm in the future.


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5. Appendix: Additional tests of the FSI CFD model

In this section, we present two verification tests of the FSI CFD model, which compare results of our two-

way FSI solver to analytical solutions. Both tests were developed for the FP7-ICT-2013-10 project

Cardioproof (which is based on the same FSI solver) and are presented here for completeness.

5.1 Experiment 1: Peristaltic transport

The experiment consists of imposing a time-periodic, wave-like motion to a vessel geometry and in verifying the cross-sectional flow at any time step (as originally described in [Connigton2009]). The wall deformation is changing both in time and space (along the axial direction) and the flow variation at every time step is computed analytically to depend on the wall movement as follows:

Θ =3𝜙2

2 + 𝜙

Figure 28 outlines the favourable results obtained through our numerical simulations. We performed computations with a total of five values for 𝜙 and compared the measured flowrates against the analytical solution. Our experiments show excellent agreement with the analytical solution.

Figure 28 Cross-sectional flow variation with the peristaltic amplitude. An excellent match with theory was obtained.

5.2 Experiment 2: Expanding and contracting vessel

This experiment validates our implementation of mass-conservation in the context of varying inlet and

outlet boundary flows. The geometry was generated synthetically by applying a deformation function to a

straight cylinder. It is given by the following equations:

𝑥 = 𝑥

𝑦 = 𝑅(𝑥, 𝑡) cos 𝜃

𝑧 = 𝑅(𝑥, 𝑡) sin 𝜃

R(x, t) = R0 + Rmax sin(2πt) sin(πx)


30

where R(x, t) is the vessel radius. In our simulations R0 = 0.5 mm, Rmax = 0.3 mm and the length of the

vessel was 2 mm. The time period was one second.

Figure 29 Time variation of the geometry of the expanding and contracting vessel.

We performed computations with the above described setup and compared the flow rates against the

analytical solutions. Since the fluid is incompressible, the flow rate should be equal to the volume change.

The volume change was calculated analytically from the surface equations describing the geometry. We

computed the flow rates on planes located at the inlet and outlet (Figure 30). It is visible that the simulated

flow rates follow the analytical curve almost perfectly. Furthermore, the total flow rate (computed through

a center cross section) was within 2% of the total analytical volume change.

Figure 30 Flow rates vs time.


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6. References [Ansumali2003] S. Ansumali, I.V. Karlin, H.C. Öttinger: Minimal entropic kinetic models for

hydrodynamics. Europhysics Letters, 63(6), p.798, 2003.

[Bouzidi2001] M.H. Bouzidi, M. Firdaouss, P. Lallemand: Momentum transfer of a Boltzmann-

lattice fluid with boundaries. Physics of Fluids (1994-present), 13(11), pp.3452-3459,

2001.

[Chen1998] S. Chen and G.D. Doolen: Lattice Boltzmann method for fluid flows. Annual review

of fluid mechanics, 30(1), pp.329-364, 1998.

[Connigton2009] K. Connington, Q. Kang, H. Viswanathan, A. Abdel-Fatah, S. Chen: Peristaltic particle

transport using the lattice Boltzmann method. Physics of Fluids (1994-present),

21(5), 053301, 2009.

[D’Humieres2002] D. d'Humières: Multiple–relaxation–time lattice Boltzmann models in three

dimensions. Philosophical Transactions of the Royal Society of London A:

Mathematical, Physical and Engineering Sciences, 360(1792), pp.437-451, 2002.

[Zhao-Li2002] G. Zhao-Li, Z. Chu-Guang, S. Bao-Chang: Non-equilibrium extrapolation method for

velocity and pressure boundary conditions in the lattice Boltzmann

method. Chinese Physics, 11(4), p.366, 2002.

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