[IEEE 2012 IEEE 2nd Portuguese Meeting in Bioengineering (ENBENG) - Coimbra, Portugal...
Transcript of [IEEE 2012 IEEE 2nd Portuguese Meeting in Bioengineering (ENBENG) - Coimbra, Portugal...
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
[email protected], [email protected], [email protected]
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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