Page 1

Personalization of Cubic Hermite Meshes for

Efficient Biomechanical Simulations

Pablo Lamata1, Steven Niederer1, David Barber2, David Norsletten1,

Jack Lee1, Rod Hose2, and Nic Smith1

1Computing Laboratory, University of Oxford, UK

{pablo.lamata,steven.niederer,david.norsletten,

jack.lee,nic.smith}@comlab.ox.ac.uk

2Department of Cardiovascular Science, University of Sheffield, UK

{d.barber,d.r.hose}@sheffield.ac.uk

Abstract. Cubic Hermite meshes provide an efficient representation of

anatomy, and are useful for simulating soft tissue mechanics. However,

their personalization can be a complex, time consuming and labour-

intensive process. This paper presents a method based on image reg-

istration and using an existing template for deriving a patient-specific

cubic Hermite mesh. Its key contribution is a solution to customise a

Hermite continuous description of a shape with the use of a discrete

warping field. Fitting accuracy is first tested and quantified against an

analytical ground truth solution. To then demonstrate its clinical utility,

a generic cubic Hermite heart ventricular model is personalized to the

anatomy of a patient, and its mechanical stability is successfully tested.

The method achieves an easy, fast and accurate personalization of cubic

Hermite meshes, constituting a crucial step for the clinical adoption of

physiological simulations.

1Introduction

Computational physiology provides tools to quantitatively describe physiological

behaviour across a range of time scales and anatomical levels using mathemat-

ical and computational models [1,2]. The heart is arguably the most advanced

current exemplar of this approach [3,4], and ongoing developments now have the

potential to provide a significant impact in the management of cardiovascular

diseases. One of the key challenges in fulfilling this potential is the personaliza-

tion of models to represent the clinical status of a patient. This work focuses on

the efficient and automated development of patient-specific geometrical descrip-

tion of organs for biomechanical simulations.

A computational model requires the geometrical description of the solution do-

main where material constitutive equations are solved. The most popular choice

is linearly interpolated tetrahedral meshes, mainly due to its conceptual simplic-

ity and availability of tools for an automatic mesh generation [5]. Nevertheless,

they introduce significant numerical error in the solution of the displacements in

the soft tissue deformation problem [6]. Alternatively, cubic Hermite meshes are

T. Jiang et al. (Eds.): MICCAI 2010, Part II, LNCS 6362, pp. 380–387, 2010.

c ? Springer-Verlag Berlin Heidelberg 2010

Page 2

Cubic Hermite Personalization381

an efficient representation of the geometrical state of an organ, and a more suit-

able choice for biomechanical simulations compared to tetrahedral meshes [6].

Another important requirement is mesh robustness, related to the convergence

of simulation results in changing physiological conditions. For these reasons, cu-

bic Hermite meshes are a popular choice for the simulation of heart mechanical

deformations [7,8,9]. Nevertheless, construction of these meshes can be a com-

plex, time consuming and labour-intense process. There is thus a need for a fast,

accurate, robust and easy to use cubic Hermite personalization method.

There are two broad approaches for the personalization of geometrical meshes:

construction from segmented images [5,10] or customization from an existing

mesh model [11,10]. Whereas the literature for linear meshes is extensive [5], its

translation to Hermite meshes is not straightforward.The change of interpolation

scheme, from linear to Hermite functions, requires a completely different meshing

strategy, like the adaptation of the Iterative Closest Point proposed in [10]. In a

Hermite mesh shape is interpolated, not only from the 3D Cartesian coordinates

of nodes, but also from the derivatives of shape versus local finite element (FE)

coordinates. This enables a compact representation, but results in a complex

mesh construction and customization. This article presents an image registration

based solution for cubic Hermite mesh personalization.

2Material and Methods

The proposed personalization method combines a fast binary image registration

with a cubic Hermite warping technique. A schematic illustration of the complete

process is provided in Fig.1.

Fig.1. Dataflow designed to generate patient specific cubic Hermite meshes

Page 3

382P. Lamata et al.

2.1 Image Registration

A binary image registration technique using a fast optical flow algorithm pro-

posed in [11] is chosen for its robustness, accuracy and computational efficiency.

It requires a preliminary segmentation of the patient’s anatomy, which in the

case of heart ventricles from static MRI or CT is a quite mature field of research.

Discrete values of the warping field between the two binary images are defined

at nodes of a regular hexahedral grid superimposed on the binary image of the

template shape. The two main parameters of the registration technique are the

spacing between nodes D and the smoothing weighting factor λ (based on a

Tikhonov regularisation of the linear least squares problem). Registration is ini-

tialised by aligning the principal axis of the shapes after a Principal Component

Analysis of the 3D coordinates of each shape.

2.2 Cubic Hermite Mesh Warping

A cubic Hermite mesh is a set of 3D FE that uses Hermite interpolation func-

tions. Mesh nodes have both coordinate values and derivatives (single, double

and triple cross derivatives) in order to encode a C1shape. Let u(ξ1,ξ2,ξ3) be

a shape defined in the Cartesian space as a function of the material coordinates

ξ. The four 1D Hermite interpolation basis functions are described in (1), and

interpolation in a line element u(ξ) is given by a linear combination of these four

basis functions (2). This scheme can be extended to 3D as illustrated in Fig.2,

u(ξ1,ξ2,ξ3) is then interpolated from a total of 192 variables in each element.

ψ0

1(ξ) = 1 − 3ξ2+ 2ξ3; ψ0

ψ1

2(ξ) = ξ2(3 − 2ξ)

2(ξ) = ξ2(ξ − 1)

1(ξ)du/dξ|1+ ψ1

1(ξ) = ξ(ξ − 1)2; ψ1

1(ξ)u1+ ψ0

(1)

u(ξ) = ψ0

2(ξ)u2+ ψ1

2(ξ)du/dξ|2

(2)

Warping a cubic Hermite mesh therefore implies calculation of a deformed state

of its three coordinate fields (x,y,z). The total number of degrees of freedom is

therefore Ndof = 3 × 8 × n = 24n, being n the number of nodes of the mesh.

Note that this number will be slightly smaller if the mesh has collapsed elements,

and slightly bigger if there are discontinuities modelled with different versions

of node values. For further details about these meshes see [10].

The solution for the warping of a cubic Hermite mesh is the core contribution

of this work. This third step in Fig.1 calculates the optimal description of a

warped shape with Hermite interpolation functions. This is built on three basic

concepts. First, it is important to realise that the warping of a FE requires know-

ing the warping field throughout the complete domain, and not only in the local

neighbourhood of mesh nodes. Second, the solution uses a FEM technique for

finding an optimal representation of a field in a domain, i.e. it uses a variational

formulation of the problem based on a dot product of functions. And third, a

numerical technique is required to handle continuous domains with computer

discrete representations, i.e. a numerical integration method is used to calculate

the dot product of functions.

Page 4

Cubic Hermite Personalization383

(a)(b)

Fig.2. Cubic Hermite meshes. (a) A two element mesh, showing the vectors corre-

sponding to the 7 derivatives at each node; (b) template of truncated left and right

ventricles of the heart. This complex shape is represented with only 112 cubic elements

(cubes with black lines) and 183 nodes.

The central idea is that warping of a cubic Hermite mesh is an addition of a FE

description of a warping field. It requires finding the adequate FE description,

with a set of Ndof variables as described before, of the warping field. And this

description is then added to the corresponding Ndof nodal variables describing

the 3 coordinate fields, i.e. the shape. Let us define a variational problem for

finding the C1continuous function g that approximates w, one of the three

components of the warping field W in the domain defined by the cubic Hermite

template ΘT. Let g be formulated as a linear combination of the set of Ndof/3

basis functions φTof the mesh. Note that each φTis a combination of 3D Hermite

basis from adjacent elements in order to enforce the continuity of the function

and its derivatives. Let us introduce the definition of a dot product of functions:

?

i

Finding g becomes the problem of finding the ci coefficients (or nodal values)

which satisfy a set of Ndof/3 equations, one for each basis φT:

(g,φj) =

Θ

g·φj=

?

Θ

?

ciφi·φj=

?

i

ci

?

Θ

φi·φj

(3)

(g,φj) = (w,φj)∀φj

?

Mx = b

(4)

i

ci

?

ΘT

φi·φj=

?

ΘT

w·φj

(5)

(6)

where vector x is the set of Ndof/3 coefficients ci, and integrals required to

calculate matrix M and vector b are computed using Gaussian Quadrature.

Calculi of vector b uses order 4, and cubic interpolation of the warping field is

used to calculate the data (deformation field) at Gauss Points. The linear system

of the fitting process requires the solution of a sparse matrix system. This matrix

is symmetric and positive definite, allowing the use of fast, low-memory solvers

such as Conjugate Gradients.

Page 5

384P. Lamata et al.

3Results

3.1Analytical Workbench for Accuracy and Robustness Analysis

Accuracy and sensitivity to parameters are analysed using a virtual workbench

with known analytical solutions. Three experiments study the accuracy of (1)

proposed warping scheme, step 3 in Fig.1, (2) the binary image registration, step

2 in Fig.1, and (3) the concatenation of the two processes. Template cylinders are

built with dimensions 30, 10 and 5mm in length, outer and internal radius, and

with 24, 81 and 192 elements. Warped versions of these cylinders are generated

under two known warping fields W1and W2, see Fig.3. Shape error is calculated

in each element as the integral of the RMS error between warped coordinates

field and their ground truth. An order 5 Gaussian Quadrature volume-weighted

integration of error is used, which is independent of the mesh discretisation

resolution and topology.

The warping step is analysed by using a perfect solution of the registration

step, obtained by sampling the ground truth warping fields. This experiment is

repeated for W1and W2, for the three cylinder mesh resolutions and for a range

of node spacings in the discretisation of the warping field (SW, with 9 values from

1 to 5 mm in steps of 0.5mm). Average shape error is 5.5e−3and 5.5e−2mm2

for W1 and W2 respectively, and the dependence with the two factors (mesh

resolution and SW) is illustrated in Fig.4.

The registration step is analysed by comparison of obtained warping fields

with their analytical expression. Binary images are generated from the Ground

Truth shapes. Registration is performed for each W1and W2, for a total of 49

binary image resolutions (from 0.2mm to 5mm in steps of 0.1mm), and for 9

values of node spacing (D from 2 to 10 voxels). The smoothing coefficient λ is

set automatically by an empirical theorem proposed in [11]. Average registration

volume-weighted error is 1.86 and 0.69mm RMS for W1 and W2 respectively,

results are shown in Fig.5.

(a) Template

(b) W1 = (x3,x3,1)

(c) W2 = (sin(z),0,0)

Fig.3. GroundTruth. All three shapes (template, customised by W1 and customised

by W2) are created with three resolutions (24, 81 and 192 elements).

Page 6

Cubic Hermite Personalization 385

012345

0

0.02

0.04

0.06

0.08

0.1

0.12

SW (mm)

(a)

Shape error (mm RMS)

W1

W2

050 100150200

0

0.02

0.04

0.06

0.08

0.1

0.12

# Elements

(b)

Shape error (mm RMS)

W1

W2

Fig.4. Error introduced in the warping of cubic Hermite meshes using a discrete ver-

sion of the ideal warping fields W1 and W2. (a) Dependence on sampling resolution,

(SW), averaging results with three mesh resolutions. (b) Dependence on mesh resolu-

tion, averaging results with nine sampling resolutions.

012345

0

1

2

3

4

Binary image voxel length (mm)

(a)

Shape error (mm RMS)

W1

W2

02468 1012

0

1

2

3

4

Node spacing (voxels)

(b)

Shape error (mm RMS)

W1

W2

Fig.5. Registration error (volume-weighted integral) for warping fields W1 and W2.

(a) Dependence on image resolution, averaging results with 9 node spacings. (b) De-

pendence on node spacing, averaging results with 49 image resolutions.

The previous two steps are concatenated, and template cylinder meshes are

warped with the result of binary image registrations. The mesh resolution of 81

elements is chosen for this experiment, since higher resolutions did not introduce

any significant improvement in accuracy, see Fig.4b. The analysis is repeated for

W1and W2, for all 49 binary image resolutions (from 0.2mm to 5mm in steps of

0.1mm), and for 9 node spacing (from 2 to 10 voxels). The average shape error

for W1and W2is 1.90 and 0.702mm RMS respectively, an increment of roughly a

1.5% with respect to the registration error. The dependence on binary resolution

and node spacing is the same as reported for the registration step, see Fig.5.

3.2Clinical Case

A patient specific cubic Hermite mesh of heart ventricles is constructed following

the process of Fig.1. The template mesh chosen is the result of fitting a mesh to

the anatomy of a first patient following the methodology described in [10] and

Page 7

386P. Lamata et al.

illustrated in Fig.2. A T1 MRI heart study of a second patient (0.88x0.88x0.75

mm voxel resolution) is manually segmented and truncated just underneath of

the opening of valve planes after vertical alignment. The agreement between

segmentation and resulting cubic Hermite mesh is shown in Fig.6, and the av-

erage distance between the two surfaces is measured as 1.32mm. The process

from image segmentation to mesh fitting for this biventricular dataset of 903

voxels requires about one minute. Mechanical stability of personalised geometry

is successfully tested by simulating a passive inflation and an isochronous active

contraction of the heart following the methods described in [7].

Fig.6. Shape personalization result. Comparison of the isosurface of the binary manual

segmentation (red wireframe) to the Cubic Hermite mesh (white solid).

4 Discussion

Cubic Hermite mesh warping requires the addition of an adequate representation

of the warping field in nodal values and derivatives. The proposed solution finds

an optimal description of this field, leading to reasonable accurate results.

Results show the importance of the registration step, which is limited by

image resolution. Interpolation errors also become significant when the topology

of the mesh is not able to represent the warped shape. These cases require an

interpolation order higher than cubic or higher element refinement, such as case

W2. An inherent limitation of proposed approach is that binary registration only

aligns the surfaces of models, interpolating the warping inside. This is a valid

approach for vessels and computational fluid dynamics [11], and preliminary

results suggest that this is also adequate for the flat walls of the heart.

The proposed method is its simpler and more robust compared to the state of

the art alternatives, the ”host mesh” technique [10] or the mesh generation by

fitting Hermite surfaces from a linear scaffold [10]. Proposed method is fast and

requires minimal user interaction. In comparison, the two alternative methods

greatly depend on users expertise, and it can take hours of manual interaction

and fitting to get satisfactory results. A second advantage is that, because it

uses a voxelized description of the shape and not a selection of pairs of control

points and degrees of freedom as required in a ”host mesh” technique, it is

immune to subjective and sometimes arbitrary selection of parameters. Finally,

warping an existing high quality template mesh under a smoothness constraint

is a reasonable warranty of simulation stability in the resulting mesh. This does

not occur using a mesh generation from an arbitrary initial linear mesh [10].

Further experiments with higher number of cases are nevertheless required to

Page 8

Cubic Hermite Personalization 387

generalise and accept these desirable properties. Future works will also address

the characterization and development of a metric of mechanical stability of cubic

Hermite meshes, a metric to be optimised during the personalization process.

5 Conclusion

Proposed method achieves an easy, fast and accurate personalization of cubic

Hermite meshes, constituting a crucial step for the clinical adoption of biome-

chanical physiological simulations.

References

1. Hunter, P.J., Crampin, E.J., Nielsen, P.M.F.: Bioinformatics, multiscale modeling

and the IUPS Physiome Project. Brief Bioinform 9(4), 333–343 (2008)

2. Lee, J., Niederer, S., Nordsletten, D., Le Grice, I., Smail, B., Kay, D., Smith, N.:

Coupling contraction, excitation, ventricular and coronary blood flow across scale

and physics in the heart. Phil. Trans. R Soc. A 367(1896), 2311–2331 (2009)

3. Smith, N.P., Nickerson, D.P., Crampin, E.J., Hunter, P.J.: Multiscale computa-

tional modelling of the heart. Acta Numerica 13(1), 371–431 (2004)

4. Bassingthwaighte, J., Hunter, P., Noble, D.: The Cardiac Physiome: perspectives

for the future. Experimental Physiology 94(5), 597–605 (2009)

5. L¨ ohner, R.: Automatic unstructured grid generators. Finite Elem. Anal. Des. 25(1-

2), 111–134 (1997)

6. Pathmanathan, P., Whiteley, J.P., Gavaghan, D.J.: A comparison of numerical

methods used for finite element modelling of soft tissue deformation. The Journal

of Strain Analysis for Engineering Design 44(5), 391–406 (2009)

7. Niederer, S.A., Smith, N.P.: The role of the Frank Starling law in the transduction

of cellular work to whole organ pump function: A computational modeling analysis.

PLoS Comput. Biol. 5(4), e1000371 (2009)

8. Kerckhoffs, R.C., McCulloch, A.D., Omens, J.H., Mulligan, L.J.: Effects of biven-

tricular pacing and scar size in a computational model of the failing heart with left

bundle branch block. Medical Image Analysis 13(2), 362–369 (2009)

9. Wang, V.Y., Lam, H., Ennis, D.B., Cowan, B.R., Young, A.A., Nash, M.P.: Mod-

elling passive diastolic mechanics with quantitative mri of cardiac structure and

function. Medical Image Analysis 13(5), 773–784 (2009)

10. Fernandez, J., Mithraratne, P., Thrupp, S., Tawhai, M., Hunter, P.: Anatomically

based geometric modelling of the musculo-skeletal system and other organs. Biome-

chanics and Modeling in Mechanobiology 2, 139–155 (2004)

11. Barber, D., Oubel, E., Frangi, A., Hose, D.: Efficient computational fluid dynamics

mesh generation by image registration. MedIA 11(6), 648–662 (2007)