The effects of medical image processing techniqueson the computational haemodynamics

Ana J. Joao, Alberto M. Gambaruto, Adelia SequeiraCEMAT, Departamento de Matematica

Instituto Superior Tecnico

Lisbon, Portugal

ana.joao@ist.utl.pt, agambar@math.ist.utl.pt, adelia.sequeira@math.ist.utl.pt

Abstract—Many of the diseases affecting the cardiovascular system

include a variety of disorders and conditions that are relatedin part to the haemodynamics, as well as genetic predispo-sition and biochemistry amongst others. With respect to thehaemodynamics, the commonly sought factors are near-wallmechanical properties including wall shear stress (and derivedparameters) and transport phenomena, such as mixing andmass transport. These factors are susceptible to large variationsamongst individuals, and in order to perform accurate clinicalevaluation careful interpretation of patient specific information isrequired. Taking an example of a configuration of the aorto-illiacbifurcation, we examine the effects of image filtering and contrastenhancement on the reconstructed geometry and the resultingcomputed haemodynamics. The algorithms used to quantify theprocessed images are based on pixel intensity variance, peaksignal-to-noise ratio and segmentation. In this study we focus onthe effects of uncertainty in clinically acquired medical imagesto the variability in the reconstructed vessel geometry, and thesubsequent error propagation to the computed haemodynamicswith emphasis on factors related to diseased states.

Index Terms—Medical Image Processing, Image Filtering,Image Enhancement, Heamodynamics, Error Analysis.

I. INTRODUCTION

An emergent demand from the medical community to

investigate vascular diseases through numerical simulations

is motivated by the need for high resolution results to assist

identifying the mechanisms of disease and assist in therapy

selection. This has led to a development of mathematical

models, algorithms and numerical tools to perform patient

specific numerical simulations, in both healthy and diseased

states.

The initiation and progression of a number of threatening

diseases, like aneurysms and atherosclerosis which have been

the focus of substantial studies, are believed to be intimately

tied to the inability of the vasculature to respond satisfactorily

to abnormal conditions [4]. Aneurysms usually appear at

the apex of bifurcations and outer bends of curved arteries

where high stresses are present, while artheroma is often

correlated to slow moving, fluctuating and disturbed flow.

2nd Portuguese Meeting in Bioengineering, February 2012Portuguese chapter of IEEE EMBSRectory of the University of Coimbra

Studies indicate that these cardiovascular diseases form, grow

and rupture in association with local haemodynamics and

lumen structural mechanics such as the wall shear stress and

its temporal and spatial gradients [2]. These parameters are

however insufficient to describe the mechanisms of disease

on their own, which involves a complex biochemical cascade

and signalling pathway that are not yet completely understood

[12].

While the relation of mechanical and biochemical factors

have been keenly studied in relation to disease, permitting

sophisticated mathematical models and highly resolved numer-

ical simulations to be performed, there has conversely been

little work on the associated errors and uncertainties in the

simulations. One of the areas to be addressed involves the

interpretation of medical data acquired in vivo in a clinical

setting that are susceptible to a broad range of error in defin-

ing the reconstructed computational domain and appropriate

boundary conditions. Solely in the task of reconstructing the

geometry for subsequent numerical simulations, the medical

images used are subject to uncertainty due to limited imaging

resolution, artefacts due to imaging modality and patient

movement, as well as the ever present random noise, which

can lead to noticeable differences in the reconstructed vessel

surface definition and in the subsequent computed solution.

It is the aim of this study to demonstrate the need for

care in medical images filtering and enhancement in the

reconstruction procedure prior to the numerical simulations,

identifying approximate error bounds of important haemo-

dynamic factors in relation to different image processing

methodologies. The consequences of filtering, contrast en-

hancement and segmentation of medical images is studied with

the associated differences in the reconstructed vessel geometry.

In this work a patient specific geometry is generated from

medical images obtained in vivo from computed tomography

angiography (CTA). Different image filtering techniques are

tested and the most accurate will be combined with a method

for contrast enhancement (Unsharp Masking [10]). The re-

sults are segmented using a clustering method [7] and a 3-

dimensional computational domain is reconstructed. Compu-

tational haemodynamics is then analysed using OpenFoam.

II. METHODS

In this section the different filtering approaches are

presented, followed by the contrast enhancement and

segmentations methods. The medical image data set used

in this work is obtained using computed tomographyangiography (CTA) and comprising of 266 images in the

axial plane with resolution parameters: 512×512 pixels of

0.78×0.78 mm size, 1.0 mm slice thickness and spacing. The

images presented in this abstract will be for a cropped region

of interest on slice 154 of the stack.

A. Filtering Algorithms

Anisotropic Diffusion

The anisotropic diffusion method, proposed by Perona-

Malik [9], simulates the process of creating a scale-space,

where an image generates a parametrised family of succes-

sively blurred images based on a diffusion process. Each of

the resulting images are used as a convolution between the

image and a 2D isotropic Gaussian filter. The conductance

coefficients are chosen to be a decreasing function of the

signal gradient. This process is a linear and space-invariant

transformation of the initial image.

∂

∂tI(x, y, t) = ∇[c(x, y, t)∇I(x, y, t)] (1)

where I(x, y, t) denotes the image pixel at position (x, y),t refers to the interaction step and c(x, y) is the monotonically

decreasing conductivity function, that depends on the image

gradient magnitude as:

c1(x, y, t) = e−( |∇I(x,y,t)|

β

)2(2)

or

c2(x, y, t) =1

1 +(‖∇I(x,y,t)‖

β

)2 (3)

Forward and Backward Anisotropic Diffusion

The goal of Forward and Backward diffusion is to empha-

sise the extrema, if they indeed represent singularities and are

not results of noise. It can be understood as moving back in

time along the scale space or more generally, reversing the

diffusion process.

Even though we could simply use a inverse linear diffu-

sion, by changing the sign of the conductance coefficient to

negative, this process has proven to be unstable. In order to

avoid this instability a higher gradient value for the inverse

diffusion coefficient is used (β1 < β2) [13]. In this way, when

the singularity exceeds a given threshold it stops affecting the

process. The conductivity function is given by:

c1(x, y, t) = 2e−( |∇I(x,y,t)|

β1

)2− e

−( |∇I(x,y,t)|

β2

)2(4)

Fig. 1. Left column shows the image and the right column shows the imagegradient magnitude. a) Original Image; b) filtered image using anisotropicdiffusion with 8 iterations; c) filtered image using anisotropic diffusion with20 iterations.

or

c2(x, y, t) =2

1 +(‖∇I(x,y,t)‖

β1

)2 − 1

1 +(‖∇I(x,y,t)‖

β2

)2 (5)

where β1 < β2.

Improved adaptive complex diffusion despeckling filter(NCDF)

The main purpose of the adaptive complex diffusion de-

speckling filter is to improve the process of speckle noise

reduction and to improve the preservation of edge and image

features. This is filter is usually applied to Optical coherence

tomography data from the human eye. As opposed to the

majority of nonlinear complex diffusion processes, which use

a constant time step (δt) close to the time step limit of the

convergence of the iterative update process, this algorithm uses

an adaptive time step. The reason behind this approach is based

on the fact that the coefficient of diffusion depends on the

gradient of the image and, because of the noise, this gradient

is high in the initial steps of the diffusion process [1]. The

filtering can be written as:

I(n+1)x,y = I(n)x,y +Δt(n)(D

(n)

x,yΔhI(n)x,y +∇hD

(n)x,y ·∇hI

(n)x,y ) (6)

where Δh and ∇h are respectively the discrete Laplacian and

gradient operators, Δt(n) is the step in time for iteration n,

Fig. 2. Left column shows the image and the right column shows the imagegradient magnitude. a) Original Image; b) filtered image using backwardand forward anisotropic diffusion with 8 iterations; c) filtered image usingbackward and forward anisotropic diffusion with 20 iterations.

and x, y are the indexes for the pixels of image I and

D(n)

x,y =4D

(n)x,y +D

(n)x±1,y +D

(n)x,y±1

8(7)

The adaptive time step is given by:

Δt(n) =1

α

[a+ b exp

{−max

(|Re(∂I

(n)

∂t )|Re(I(n))

)}](8)

where|Re( ∂I(n)

∂t )|Re(I(n))

is the fraction of change of the image at

iteration n, and α, and a, b control the time step through

(a+ b ≤ 1).

B. Contrast Enhancement: Unsharp Masking

In the Unsharp masking method, the enhanced image

H(x, y) is obtained from the input image I(x, y) as

H(x, y) = I(x, y) + λF (x, y) (9)

where F (x, y) is the correction signal computed as the output

of linear high-pass filter and λ the positive scaling factor

which controls the contrast enhancement level acquired as

the output image [10].

C. Image Segmentation: Kittler method

The Kittler method [7] is an iterative method that relies on

fitting a Gaussian to the background and to the foreground

pixels in the histogram of the image pixel intensity. The

Fig. 3. Left column shows the image and the right column showsthe image gradient magnitude. a) Original Image; b) filtered image usinganisotropic diffusion with diffusion time of 0.75 seconds; c) filtered imageusing anisotropic diffusion with diffusion time of 2.5 seconds.

new threshold is obtained by solving a quadratic equation,

and the value corresponds to the crossing location of the

two Gaussians. The assumption is that the object and

pixel values are normally distributed. The segmentation is

therefore a constant value of grey-scale for each slice in

the stack. The Kittler method ranked top in the survey of [11].

III. GEOMETRY RECONSTRUCTION

The vessel boundary is obtained using the segmentation

for each slice. This results in a stack of closed contours

that needs to be interpolated and a surface definition defined.

This was performed using an implicit function formulation,

with cubic radial-basis function interpolation. The iso-surface

of the implicit function that defines the vessel surface is

extracted using a marching tetrahedra approach to give an

initial piecewise linear triangulation [5]. These initial surface

definitions are then prepared for numerical simulations by

firstly smoothing the geometries and secondly by extending

the inflow and outflow boundaries to circular cross-sections

[5].

IV. COMPUTATIONAL HAEMODYNAMICS

The numerical simulations were performed using the Open-

FOAM software package which relies on the finite volume

method. The simulations were run for steady-state and time-

varying boundary conditions. These simulation criteria were

chosen to emphasise the use of the proposed methods clearly

Fig. 4. Left column shows the image and the right column shows the imagegradient magnitude. a) Original Image; b) filtered image using anisotropicdiffusion with 8 iterations; c) filtered image using anisotropic diffusion with20 iterations

with steady-state, and demonstrate the relevance in a more

physiological scenario with the unsteady computations. The

schemes used were SIMPLE for the steady state and PISO for

the unsteady computations.

V. RESULTS AND CONCLUSIONS

Results indicate that medical image preprocessing can sig-

nificantly improve the quality of the images and therefore

facilitate vessel extraction. Filtering and contrast enhancement

algorithms prove to remove a certain level of uncertainty in

the segmentation process. Automatic techniques for medical

image preprocessing and geometry reconstruction are impor-

tant to accurately analysing clinical data. Robust schemes are

proposed, reducing the effect of errors in subsequent analysis

and post-processing.

These represent a preliminary investigation of the impact of

uncertainties in medical imaging reconstruction with the goal

of identifying a possible uncertainty bound on the solution

of haemodynamic simulations and parameters that are used

to study disease. Further work is still necessary to quantify

the impact of the medical image uncertainties on the model

boundary definition that each segmentation method responds

to differently.

ACKNOWLEDGMENT

The authors kindly acknowledge the Imagiology Depart-

ment (directed by Prof. J. Campos) of Hospital Santa

Maria, for providing the medical data. This work has been

partially supported by the research centre CEMAT/ IST

through FCT’s funding program, and by the FCT project

UTAustin/CA/0047/2008.

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