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TRABAJO DE GRADO
MÉTODOS ROBUSTOS DE DECONVOLUCIÓN PARA LA ESTIMACIÓN DE PARÁMETROS DESCRIPTORES DE
PERFUSIÓN CEREBRAL POR RESONANCIA MAGNÉTICA*
(*) Título original: “Deconvolution Robustness for Local Perfusion Parameters Estimation in Magnetic Resonance Imaging (MRI)”
Juan Pablo SANTAMARIA Noviembre de 2004
i
A José Octavio Santamaría y Beatriz Chavarro de Santamaría…
ii
© 2003 Juan Pablo Santamaría, Todos los Derechos Reservados, All rights reserved
iii
RESUMEN
El presente trabajo tuvo como objetivo general comparar la robustez de dos
algoritmos para la estimación de parámetros de perfusión en imágenes de
resonancia magnética (RM). Uno de los parámetros que comúnmente se quiere
determinar en la práctica clínica, es el flujo sanguíneo cerebral ya que es un
valioso indicador de la viabilidad del tejido en un paciente. Para la estimación
del flujo, los dos algoritmos utilizan un modelo fisiológico común pero difieren
en el método matemático utilizado. A lo largo del proceso, se hace necesaria la
utilización de un operador matemático denominado deconvolución.
Generalmente, para estimar los parámetros de perfusión, se utiliza como
operador de deconvolución un método de descomposición en valores
singulares (SVD). Sin embargo, en éste trabajo se utilizó también un método
autorregresivo de promedio móvil (ARMA) en diferentes condiciones de ruido y
finalmente se presentaron los resultados obtenidos.
Palabras clave: Imágenes por Resonancia Magnética, deconvolución, ARMA,
SVD, estimación de flujo sanguíneo cerebral.
iv
ABSTRACT
In clinical practice, it is frequently desired to estimate the cerebral blood flow
since it is a valuable indicator of the patient’s tissular viability. The purpose of
this study was to determine the robustness of two algorithms in blood flow
estimation using MR imaging. To identify this perfusion parameter, both
algorithms use the same physiological model but they differ in the mathematical
procedure. The required mathematical operator is called deconvolution and one
of the algorithms uses a singular value decomposition (SVD) method, whereas
the other one applies an Auto-Regressive Moving Average (ARMA) model. The
performances of both deconvolution methods are exposed here, under different
noisy environments, in cerebral blood flow estimation.
Keywords: Magnetic Resonance imaging, deconvolution, ARMA, SVD,
cerebral blood flow estimation.
v
ARTÍCULO 23 DE LA RESOLUCIÓN No. 13 DE JUNIO DE 1946
"La universidad no se hace responsable de los conceptos emitidos por sus
alumnos en sus proyectos de grado.
Sólo vela porque no se publique nada contrario al dogma y la moral católica y
porque los trabajos no contengan ataques o polémicas puramente personales.
Antes bien, que se vea en ellos el anhelo de buscar la verdad y la justicia".
vi
PONTIFICIA UNIVERSIDAD JAVERIANA
FACULTAD DE INGENIERÍA
CARRERA DE INGENIERÍA ELECTRÓNICA RECTOR MAGNIFICO: R.P. GERARDO REMOLINA S.J. DECANO ACADÉMICO: Ing. ROBERTO ENRIQUE MONTOYA VILLA DECANO DEL MEDIO UNIVERSITARIO: R.P. ANTONIO JOSÉ SARMIENTO NOVA S.J. DIRECTOR DE CARRERA: Ing. JUAN CARLOS GIRALDO CARVAJAL DIRECTORES DEL PROYECTO: MARLENE WIART, BRUNO NEYRAN ASESORES DEL PROYECTO: Ing. CARLOS PARRA, Ing. JAIRO HURTADO
vii
Información acerca de los directores del proyecto:
Bruno Neyran Maestro de conferencias
Investigador del laboratorio CREATIS
Lyon - Francia
Marlène Wiart Investigadora del laboratorio CREATIS
Hospital Neurocardiológico de Lyon
Lyon - Francia
CREATIS Centre de Recherche et d'Applications en Traitement de l'Image et du Signal
CREATIS es una unidad de investigación (UMR 5515) del centro nacional de la
investigación científica de Francia, CNRS. Esta unidad está afiliada al INSERM,
y es común al Instituto Nacional de Ciencias Aplicadas (INSA de Lyon) y a la
Universidad Claude Bernard-Lyon1 (UCBL). El campo de investigación de
CREATIS es el procesamiento de imágenes y sistemas aplicados a la
medicina.
Para mayor información consultar en Internet, las siguientes direcciones:
www.creatis.insa-lyon.fr
www.insa-lyon.fr
viii
Preliminares Este trabajo fue elaborado durante el segundo semestre de 2003 como una
pasantía de investigación en el laboratorio CREATIS de Lyon -Francia-. Esta
labor me generó interesantes experiencias y pensando en que puedan servir de
motivador para quien desee seguir un plan similar, o simplemente le interese el
tema de los intercambios académicos, plasmo aquí algunas de las que me
trajeron mayores satisfacciones.
A comienzos del año 2003 me encontraba cursando octavo semestre de
ingeniería electrónica en la Universidad Javeriana, y un día cualquiera un aviso
colocado en una de las carteleras de la Facultad, llamó mi atención, se trataba
de un afiche que invitaba a los estudiantes a realizar un intercambio académico
con el INSA1 de Lyon, gracias a un convenio entre la facultad de ingeniería de
la Javeriana y esa Institución. Era la primera experiencia del convenio que se
acaba de concretar, como resultado del ingente esfuerzo del decano
académico de nuestra Facultad, ingeniero Roberto Montoya. Una vez
informado sobre los trámites necesarios, decidí aplicar como estudiante de
intercambio al departamento de Eléctrica del INSA de Lyon. Semanas más
tarde, recibí la buena noticia de que había sido admitido en esa institución.
Cuando llegué al INSA, me enteré de la posibilidad que tenían los estudiantes
de quinto año del departamento de Eléctrica para realizar un programa
académico que se conoce con el nombre de “doble inscripción”. Esta
modalidad consiste en hacer simultáneamente el quinto año de Ingeniería
ix
1 Los Institutos de ciencias aplicadas (Instituts de sciences appliquées) I.N.S.A. están conformados por una red de establecimientos académicos públicos del gobierno francés localizados en cinco grandes metrópolis regionales : Lyon, Rennes, Rouen, Strasbourg y Toulouse.
Eléctrica y un DEA2. Para ingresar al DEA, el estudiante debe presentar una
entrevista, un examen de admisión y por último elegir un tema de investigación
que debe desarrollarse en los laboratorios asociados a la Institución. Después
de tocar muchas puertas, y gracias a la ayuda de una persona que conocí allá,
Marcela Hernández Hoyos, pude contactar a dos investigadores -Marlène Wiart
y Bruno Neyran- quienes venían desarrollando un tema que resultó de mi
interés. Una vez elegido el tema de investigación, el grupo CREATIS me abrió
sus puertas y de esta manera pude matricularme como doble inscrito en quinto
año de Ingeniería y en DEA. El tema elegido era sobre el estudio de métodos
para el cálculo de descriptores de irrigación sanguínea por imágenes de RM.
Una vez inscrito, debía cursar y aprobar los módulos generales y específicos
del DEA, tales como el de tratamiento de imágenes y señales, movimiento,
resonancia magnética, estimación, decisión y problemas inversos, entre otros.
Para cuando estaba terminando la parte experimental del trabajo de
investigación, debía comenzar a redactar el informe final y en ese momento
sólo tenía dos opciones con relación al idioma: hacerlo en francés o en inglés.
Cualquiera de las opciones representaba para mí un desafío, básicamente por
dos motivos: de una parte, ninguna de las anteriores es mi lengua materna, y
por otro lado, era mi primera experiencia en redactar un documento extenso,
práctica que había abandonado desde la época del colegio. Finalmente opté
por escribir éste trabajo en inglés porque soy consciente de la importancia del
dominio de la lengua anglosajona, especialmente en nuestro medio, pues en
los últimos semestres de la carrera y sobre todo a lo largo de éste trabajo, pude
constatar que la mayoría de los artículos científicos y estudios están escritos en
inglés. Por ende, no preocuparme por perfeccionar ésta lengua era
simplemente un lujo que no podía darme, máxime cuando la importancia del
inglés no solo se ve reflejada en el contexto de la ingeniería, sino en casi todos
los campos del conocimiento humano, imponiéndose poco a poco como un
estándar de comunicación universal. 2 El DEA es el diploma francés oficial orientado hacia la investigación. Es requisito para los estudiantes que deseen realizar un doctorado. Con la reforma LMD (Licenciatura, Maestría y Doctorado), a partir del
x
Realizar el trabajo de grado en ésta lengua, implicó un esfuerzo adicional, ya
que me obligó a familiarizarme con nuevos términos técnicos en inglés,
manejar expresiones que introducían ecuaciones matemáticas, emplear
sinónimos con el fin de no repetir palabras, etc. Sin embargo, hoy por hoy,
cuando me detengo y miro hacia atrás, ya no veo todo aquello como un trabajo
adicional, sino más bien como la doble oportunidad que tuve de aprender,
circunstancia que conjuntamente con el hecho de haber cumplido las dos
metas, me produce una gran satisfacción.
Finalmente, he decidido no asumir el rol de traductor y presentar éste escrito en
el idioma en el cual fue escrito el original y muy probablemente con los
innumerables errores gramaticales, propios del principiante. Por ésta razón,
ruego al lector su benevolencia y espero no distraiga su atención del tema
central, pues no quise que mi ignorancia lingüística fuera, ella misma, excusa
para no enfrentar el reto científico.
xiaño académico 2004-05 el DEA se convierte en una maestría de investigación.
Agradecimientos
Son pocas las oportunidades que he buscado para agradecerles a esas
personas que siempre han estado a mi lado, a aquellos que me han dado
absolutamente todo lo que he necesitado para llegar hasta acá. Esas personas
son mis padres, a ellos es que dedico éste trabajo de grado e igualmente les
doy infinitas gracias.
Debo también reconocer la ayuda brindada por toda mi familia, especialmente
la de mi hermana Camila, por su tiempo y conocimiento en los temas
lingüísticos. A Liliam, Ángela y a Julián, igualmente les agradezco por haberme
apoyado emocionalmente.
En Lyon, sin la colaboración de Marcela Hernández, estoy convencido de que
éste trabajo no habría sido posible, ya que fue ella la que me abrió muy
amablemente, las puertas de CREATIS. Gracias a los responsables del
proyecto: Marlène Wiart quien fue enormemente cordial a lo largo de mi estadía
en Lyon y muy paciente al resolver mis dudas sobre el IRM de perfusión. Y
Bruno Neyran, quien hizo que temas inicialmente incomprensibles para mí,
fueran claros al final. A Leonardo Florez le agradezco por las necesarias horas
de esparcimiento en las que me enseñó a disfrutar del buen cine. Les doy
gracias también a Isabelle Magnin y a Olivier Basset por su colaboración y
flexibilidad en la parte administrativa para que yo pudiera cumplir con mis
compromisos en Colombia.
En la Universidad Javeriana, agradezco en primer lugar a Javier Coronado, con
su ayuda y buena voluntad, pude encontrar el camino que me llevó a la
solución de múltiples problemas. Con Carlos Parra estoy endeudado
igualmente: primero porque fue una persona supremamente accesible y xii
amable desde el primer momento en que me acerqué a su oficina y en segundo
lugar, porque fue una persona clave en la homologación de mi trabajo de
grado. Por otro lado, a Jairo Hurtado, le debo el haberme introducido al área de
análisis de señales y junto con Carolina Soto, les agradezco el haber dedicado
de su tiempo, en la revisión de éste documento.
xiii
Contents
ABSTRACT..................................................................................................................... v
List of abbreviations.................................................................................................. xvi
List of figures .............................................................................................................. xvii
INTRODUCTION ............................................................................................................1
2. THEORETICAL BACKGROUND ...........................................................................4 2.1. Deconvolution in blood flow estimation using MR perfusion imaging ......................... 4
2.1.1. Singular Value decomposition (SVD).............................................................................. 8 2.1.1.1. General Singular Value decomposition (SVD)....................................................... 8
2.1.1.2. Singular Value decomposition adaptive threshold (aSVD)............................... 9
2.1.2. Auto Regressive Moving Average (ARMA) .................................................................. 12 2.1.2. Auto Regressive Moving Average (ARMA) .................................................................. 12
2.2. Biological system modeling ................................................................................................... 15 2.2.1. The human body model as a dynamic system ........................................................... 15 2.2.2. The Stewart-Hamilton model ........................................................................................... 18
2.3. Sample size selection in statistical experiments .............................................................. 23
3. METHODS ................................................................................................................26 3.1. Monte–Carlo simulations......................................................................................................... 27
3.1.2. Noisy MR signals simulation procedure ...................................................................... 28 3.1.4. Deconvolution performance criteria.............................................................................. 30 3.1.5. ARMA vs. SVD..................................................................................................................... 31 3.1.6. SVD vs. adaptive threshold SVD .................................................................................... 33
3.2. Perfusion MRI in stroke patients ........................................................................................... 34
4. RESULTS AND DISCUSSION ..............................................................................35 4.1. Monte-Carlo simulations.......................................................................................................... 35
4.1.2. SVD and aSVD deconvolution......................................................................................... 42 4.2. Perfusion MRI in stroke patients ........................................................................................... 43
5. CONCLUSIONS AND FURTHER WORK ...........................................................46
xiv
5.1. CONCLUSIONS .......................................................................................................................... 46 5.2. FURTHER WORK ....................................................................................................................... 48
REFERENCES .............................................................................................................49
xv
List of abbreviations
MRI: Magnetic resonance imaging
SVD: Singular value decomposition
aSVD: Adaptive singular value decomposition
PSVD: Singular value decomposition percentage threshold
ARMA: Auto-regressive moving average
rCBF: Regional cerebral blood flow
rCBV: Regional cerebral blood volume
rMBF: Regional myocardial blood flow
rMBV: Regional myocardial blood volume
MTT: Mean transit time
SI: Signal intensity
TE: Echo time
ROI: Region of interest
LTI: Linear time Invariant
AIF: Arterial input function
SNR: Signal-to-noise ratio
MPE: Mean percentage error
SSD: Sample standard deviation
SD: Standard deviation
iff: If and only if
xvi
List of figures Figure 01. A typical RM brain grey-level image................................................ 2
Figure 02. The expected function (a) and the one obtained with SVD (b)........ 9
Figure 03. The Gaussian distribution (a) and its second derivative (b) ............10
Figure 04. Adaptive SVD algorithm ..................................................................11
Figure 05. Discrete time system - ARMA model representation.......................12
Figure 06. The blood circulation as a dynamic system.....................................16
Figure 07. Perfusion black-model approach.....................................................17
Figure 08. Parameter estimation procedure using MR perfusion imaging........32
Figure 09. Example of pixel or ROI selection in MR images. ...........................32
Figure 10. ARMA vs. SVD perfusion simulation strategy. ................................32
Figure 11. SVD vs. aSVD perfusion simulation strategy. .................................33
Figure 12. rCBF estimation without random noise using ARMA and the SVD
deconvolution (PSVD = 30%). ..........................................................................35
Figure 13. Influence of Psvd threshold selection in flow estimation without
random noise in SVD deconvolution method. ................................................37
Figure 14. Standard deviation comparison for ARMA and SVD deconvolution
techniques, for R=1Hz, SNRtis= 18 dB and SNRaif = 15 dB. ........................37
Figure 15. This figure illustrates the SNR shift sensitivity in flow estimation
using both deconvolution methods for a CBV=2% and a sampling rate of 1Hz.
.......................................................................................................................38
Figure 16.Comparison of SNR shift sensitivity in blood flow estimation for both
deconvolution methods. CBV=2%, SNRtis= 18 dB and SNRaif = 15 dB. ......39
Figure 17. Comparison of tissular SNR shift sensitivity in blood flow estimation
for both deconvolution methods. Rs = 0.5 Hz and SNRaif = 15 dB................41
Figure 18. Example of oscillation index calculation of four different residue
functions from the adaptive SVD algorithm....................................................42
xvii
Figure 19. Example of arterial and tissular concentrations calculated from SI
functions for patient 6.....................................................................................43
figure 20. Estimated residue function using the ARMA and SVD deconvolution
model for patient 6. ........................................................................................44
Figure 21. Comparison of identified cerebral blood flow in 19 stroke patients
using ARMA and SVD deconvolution.............................................................45
xviii
INTRODUCTION
Cerebral and cardiovascular diseases caused by the obstruction of blood
vessels, are one of the major causes of death in our modern society. The
medical community has determined this class of disorders as an important
public health threat.
Magnetic resonance imaging (MRI) technology has been widely used in clinical
practice as a non-invasive technique to visualize the inside of the human body
and to detect health-threatening situations in patients. In many cases, it is
required to image the patient flowing blood, with the purpose to detect
anomalies in the circulatory system. Perfusion3 MR imaging is a suitable
method for this task, in which a contrast agent is introduced into the blood
stream and then, it is digitally imaged as it travels across the tissue. MR
perfusion imaging is especially a good choice for diagnosis and follow-up
studies of several injuries in the arterial system. The different grey scale slices
of a MR human brain image is shown in figure 01.
The study field for new concepts and applications within MRI medicine is today
a vast and dynamic area, as it was proved last year by Paul C. Lauterbur4 and
Sir Peter Mansfield5, 2003 physiology/medicine Nobel Prize winners, for their
discoveries concerning magnetic resonance imaging.
3 The term “perfusion”, from Latin perfusus, is defined as the act of perfusing; specifically: the pumping of a fluid through an organ or tissue. 4 University of Illinois Urbana, IL, USA.
1
5 University of Nottingham, School of Physics and Astronomy Nottingham, United Kingdom
When thrombosis is detected, it is also often desired to evaluate the regional
cerebral blood flow (rCBF). This perfusion parameter quantifies the blood flow
at the capillary level where the delivery of nutrients takes place. Its value is an
indicator of the tissue viability, which is useful in taking appropriate treatment
decisions. However, in clinical practice there is still a need for robust diagnostic
tools for determining perfusion parameters in environments affected by random
electrical noise. The rCBF can be estimated using a circulatory system
modeling, in which at a certain moment it is necessary to use a mathematical
operator, called deconvolution.
figure 01. A typical RM brain grey-level image
The purpose of this study was to determine the robustness of two deconvolution
techniques in rCBF estimation: the singular value decomposition (SVD) and the
Auto-Regressive Moving Average (ARMA) model. The SVD method is
commonly used in the literature whereas the ARMA model is being recently
applied at the CREATIS laboratory in some specific tests. Preceding
experiments used the ARMA method for experimental data from an isolated pig
heart preparation due the similarities between the pig and human’s heart. After
some studies, satisfactory results were obtained so ARMA was proposed as an
appropriate method for regional myocardial blood flow (rMBF) estimation. On
the other hand, it is known that the order of magnitude of the blood flow in the
human brain is less than the rMBF. Therefore, it is desired to compare both
deconvolution methods in rCBF estimation under different noisy environments,
to see their behavior in the cerebral context.
2
Each algorithm was tested on 150 data sets in two distinct cerebral blood
volumes, and error measurements were defined in order to compare the
performance of the different methods. Computer simulations here were not
intended to substitute practical data but were only applied as a first approach
before analyzing the clinical samples.
3
2. THEORETICAL BACKGROUND
In the following chapter a review on the basic theory required to understand the
current work is presented. From a physical point of view, some concepts such
as the magnetic resonance technology are not entirely covered here since its
understanding is not indispensable to comprehend the development of this
work. The reader can refer to Edelman’s work [11] for a deeper description of
the MR basis and its clinical applications.
2.1. Deconvolution in blood flow estimation using MR perfusion imaging
One way of estimating the human blood flow is by using MR perfusion imaging
and a mathematical model of the circulatory system. The main process to
visualize the inside of the human body in order to measure the perfusion
parameters can be summarized as follows:
I. The patient receives an intravascular injection of a contrast product that
has specials physical properties and allows an easier visualization of the
blood flow when the tissue is affected by a magnetic field. This magnetic
field is created inside the cube of the MRI machine which can vary from
0.5 Tesla to 2 Tesla.
II. A finite sequence of images is taken to detect the variations of the signal
intensity functions (SI) over the time.
III. Once the sequence of digital images is obtained, the concentrations
signals of the contrast product are calculated from the SI functions using a
logarithmic relation.
4
IV. A mathematical operator called deconvolution is used between the arterial
and tissular concentration in order to calculate a particular signal named
residue function.
V. The blood flow is calculated evaluating the residue function in zero.
This five-step process may sound a little complex to the reader in the beginning;
however it is not crucial to understand all its steps as this procedure is
explained and discussed in more detail in the following sections. For the
moment, it is just necessary to keep in mind that somehow a mathematical
operator called deconvolution must be used (step V) in order to estimate the
rCBF. However, to understand what a deconvolution method it is first necessary
to introduce the convolution integral.
In signal processing, the convolution operator is a very useful mathematical
procedure when describing linear systems. The convolution product, usually
denoted or , between two signals X(t) and Y(t) is defined, in continuous
time, as:
⊗ ∗
)()()();( tYtXtYtXnConvolutio ⊗= Where:
τττ dtYXtYtX )()()()( −=⊗ ∫+∞
∞− (1)
However, the convolution of those two functions X and Y can be defined as
well, over a finite interval [ ]21 , tt as follows:
τττ dtYXtYtXt
t
)()()()(2
1
−=⊗ ∫ (2)
Usually, this product leads to another signal. Let Z be the generated signal:
5
)()()( tYtXtZ ⊗=
One way of dealing with those signals is using digital processors because of its
versatility and execution time. This is usually known as digital signal processing.
However, the computers of today do not work with analog signals so the
discrete-time form of the previous equations must be used. Therefore, the
convolution of X and Y in discrete time is defined as:
∑ −=k
knYkXnZ )()()( (3)
Considering now, the function Z only over a finite sample interval
∑=
−=N
kknYkXnZ
1)()()(ˆ (4)
The expression bellow generates N equations:
)1()1()1()2()0()1(
)1()()21()2()11()1()1()()1(ˆ1
NYXYXYX
NYNXYXYXkYkXZN
k
−++−+=
−++−+−=−=∑=
L
L
∑=
−+++=−=N
kNYNXYXYXkYkXZ
1)2()()0()2()1()1()2()()2(ˆ L
∑=
++−+−=−=N
k
YNXNYXNYXkNYkXNZ1
)0()()2()2()1()1()()()(ˆ L
M
For simplication, these equations can be considered as a set of simultaneous
equations that can be expressed in its matricial representation:
6
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−−
=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
)(
)2()1(
)0()2()1(
)2()0()1()1()1()0(
)(ˆ
)2(ˆ)1(ˆ
NX
XX
YNYNY
NYYYNYYY
NZ
Z
Z
M
L
OM
L
L
M (5)
The equation bellow can be compactly denoted as:
XYZ =ˆ (6)
Now, independently if working with the integral or matricial form of this operator,
it can be stated in general terms that:
YXYXnConvolutioZ ⊗== ; (7)
Supposing that Z and Y are known an inverse mathematical process could be
defined to calculate an approximation6 of X. This inverse operator is called
deconvolution and it could be used as follows:
YZionDeconvolutXYXZIf ;≅⇒⊗= (8)
Or equivalently:
XZionDeconvolutYYXZIf ;≅⇒⊗= (9)
In the following section two different deconvolution methods -ARMA and SVD-
and its applications in MR imaging are presented. It is important to keep in mind
that the convolution can be treated from a matricial point of view.
7
6 The term of approximation is used here, since most of the time the signals found in practice are blurred with random noise, making it difficult to calculate of an exact solution.
2.1.1. Singular Value decomposition (SVD)
2.1.1.1. General Singular Value decomposition (SVD) The singular value decomposition is a widely used technique to solve ill-
conditioned problems with several applications, (e.g., in image compression,
watermarking, image filtering). The general SVD statement can be expressed
as:
Every real matrix A can be decomposed into a product of three matrices of the
form: TVSU=Α (10)
Where, U and V are orthogonal matrices.
On the other hand, S is a diagonal matrix whose elements are the singular
values of the original matrix. Therefore:
),...,,( 21 rdiagS σσσ= (11) With:
0...21 ≥≥≥≥ rσσσ
Consequently, the inverse of A can be expressed as:
TUWV=Α−1 (12)
Where W is also a diagonal matrix of elements i
i sw 1
=
Depending on the signal-to-noise ratio of the signal intensity function, a
tolerance threshold PSVD is set. The values of wi corresponding to values where
si is less than PSVD are set to zero. Typically the threshold is given as a
percentage of the greater singular value 1σ . Generally, in MR perfusion imaging
8
the threshold value varies between 20 and 30% and the general principle is the
higher the noise, the higher the PSVD.
2.1.1.2. Singular Value decomposition adaptive threshold (aSVD) When the SVD algorithm is used to calculate the residue function it is common
to obtain a graph of the function with some non-expected oscillations. In theory,
the residue function is monotone decreasing and therefore those oscillations do
not have a physical sense (see figure 02). Liu et al. [4] showed that the PSVD
selection had a significant influence in the shape of the residue function and an
apparently inaccuracy in the rCBF estimation. The residue function is a signal
from which the blood flow is calculated and it is the one mentioned on the 4th
step in section 2.1.
R(t) R(t)
t
(a) (b)
figure 02. The expected residue function (a) and the one obtained with SVD (b)
An oscillation index O, was proposed by Østergaard et al. [1] to measure the
distortion in the residue function as follows:
⎟⎠
⎞⎜⎝
⎛−+−−
⋅= ∑
−
=
1
2max
)2()1(2)(1 L
kkfkfkf
fLO (13)
9
Where f is the scaled estimated residue function, fmax is the maximum amplitude
of f, and L is the number of sample points.
This oscillation index may be viewed as the discrete form of the convolution
between the residue function and the second derivative of a Gaussian
distribution which is shown in figure 03. The use of this digital filter, which is the
same as calculating the sum of the absolute value of the second derivative of
f(t), quantifies the change in the curvature of the function over the time.
figure 03. The Gaussian distribution (a) and its second derivative (b)
As it was said before, a threshold is typically chosen between 20 and 30% in
rCBF estimation, depending on the SNR of the digital image. However, it is
theoretically possible to design an algorithm in which an optimum PSVD is
chosen automatically. The main adaptive SVD algorithm that was used in this
work is shown in figure 04. Again, it can be summarized in a step-by-step
procedure as follows:
I. The lower value of the PSVD threshold is initialized. 10
II. The residue function is calculated using the SVD deconvolution method.
III. The oscillation index O of the correspondent residue function is calculated.
IV. The PSVD threshold value is incremented.
V. The process is repeated from step II until the upper value of the PSVD
threshold is reached.
VI. The residue function which has the minimum oscillation index O is selected
and the rCBF is calculated using Omin.
figure 04. Adaptive SVD algorithm
Select R(t) with minimum O
aSVD
Initialize PSVD
Calculate R(t)
Measure Oscillation
End?
Calculate minimun O
Y
N
Inc Psvd
Calculate perfusion parameters
11
2.1.2. Auto Regressive Moving Average (ARMA)
ARMA is the second method of deconvolution used in this work to estimate the
rCBF. As is was noted earlier, to calculate the blood flow it is necessary to
calculate a specific function denoted r(t) -more specifically the discrete form r(n)
because the sequence of images is discrete-. The estimation of this residue
function r(n) can be treated as a spectral parametric analysis problem. One way
of solving this kind of problems is presented here from the system theory point
of view and showing its basic mathematical statements.
It is known that when a linear time-invariant (LTI) system is excited by the
Dirac's delta function, the output is the transfer function of the system.
figure 05. Discrete time ARMA model representation
The discrete time filtering theory states that the impulse response of a linear
system h(n) can be modeled using one of the following frequency responses:
I. The “zeros model” or moving average (MA):
)()( zBzH = (14)
12
II. The “poles model” or autoregressive (AR):
)(1)(zA
zH = (15)
III. And the “zeros and poles” model (ARMA):
)()()(
zAzBzH = (16)
This last model (III) is the one used and as its name suggest, it is presented as
the ratio of poles and zeros in the Z-transform. Therefore equation (16)
becomes:
∑
∑
=
−
=
−
⋅−
⋅= N
l
ll
M
k
kk
za
zbzH
1
0
1)( (17)
The aim of the ARMA technique modeling is to find the transfer function H(z),
whose impulse response h(n) approximates the scaled tissue response r(n),
such that the sum of the squared error, denoted e, is minimum:
[ ]∑ −= 2)()( nrnhe (18)
Considering the system illustrated in figure 05, and using the ARMA model, the
output y(n) is computed as:
∑∑==
−⋅+−⋅−=M
kk
N
ll knxblnyany
01
][][][ (19)
13
This mathematical statement generates a set of simultaneous equations:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
−−++−+−+−−−−−−−−=−
−++++−−−−−−=−++−++−−−−−−−=
)1()2()1()1()3()2()1(
)1()0()1()1()1()0()1()()1()0()()2()1()0(
1
021
1021
1021
mNxbNxbNxbnNyaNyaNyaNy
mxbxbxbnyayayaymxbxbxbnyayayay
m
n
mn
mn
L
L
M
LL
LL
(20)
This can be compactly expressed, in matrix notation as
Θ= AY (21)
where Y represents the output matrix, A is the input-output matrix and Θ is the
ARMA parameter matrix (coefficients ak and bk).
Therefore, this general ARMA-model matricial representation is written as
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
−
−
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
−−−−−−−−−−−−−−−−
−−−−−−−−−−
=
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
−−
n
m
a
ab
b
nNyNyNymNxNxNxnNyNyNymNxNxNx
nyyymxxxnyyymxxx
nyyymxxx
NyNy
yy
M
M
LL
LL
MLLLM
LL
LL
LL
M 1
0
)1()3()2()1()2()1()2()4()3()2()3()2(
)2()0()1()2()1()2()1()1()0()1()0()1(
)()2()1()()1()0(
)1()2(
)1()0(
To solve the linear equations system Θ= AY , an estimation square error is
defined as
TeeSE =
where Θ−= AYe
14
The minimization of the error by the least mean square process gives the
coefficients ai and bi, and consequently the searched solution.
( )[ ])(minargˆ Θ=Θ LMSQ (22)
Where 2)( Θ−=Θ AYQLMS
To minimize this equation the SE must be differentiated with respect to Θ and
equated to zero.
0=Θ∂
∂ SE (23)
Solving equation (23) :
YAAA TT 1)(ˆ −=Θ (24) Where represents the unknown ARMA parameters and the solution of the
deconvolution problem needed to estimate the rCBF.
Θ
2.2. Biological system modeling
2.2.1. The human body model as a dynamic system The physiological behavior of some organs can be modeled using the dynamic
systems theory. The heart, veins, arteries and capillaries may be treated as a
biological system which can be mathematically represented, where variables
and parameters interact regularly. This representation provides a powerful tool
to estimate variables when some parameters are unknown.
Another advantage of the dynamic systems theory is that various -apparently
different- physical parameters (e.g., temperature, flow, concentration and
voltage) can be handled without modifying the equation’s essence. In recent 15
years, the study field for new concepts such as the biochemical circuit theory
analysis is today a fundamental field of interest for many laboratories.
The physical elements treated in this medical context, are the perfusion
parameters, i.e., regional cerebral blood flow rCBF, regional cerebral blood
volume rCBV, mean transit time MTT. However, the same parameters and
relationships can be deduced and applied in a different area such as the
myocardial circulatory system. In the published literature, the notation is quite
similar except for that the letter M (myocardial) replaces the letter C (cerebral),
in the perfusion parameter abbreviations.
figure 06. Simplified blood circulation scheme
The human circulatory system is divided into two main parts with the heart
acting as a double pump. This organ pumps a special kind of fluid, which is the
blood. The across variable in this case, the blood flow, leaves the left side of the
pump (heart) and travels through arteries which gradually divide into capillaries.
In the capillaries, the nutrients are released to the body cells. The blood then
travels in veins back to the right side of the pump, and the whole process
begins again. The human simplified blood circulation process is shown in figure 06.
To determine if a specific human tissue is correctly irrigated, the artery tissue
exchange process must be analyzed. To this aim, several approaches have
16
been developed. One of the most common ways in which this kind of process is
analytically described is the black-box model which is shown in figure 07. This
model is not only used to describe electrical circuits, communication systems
but it is also seen as suitable mathematical approach to physiological processes
such as the circulatory system. Usually, the black-model description is used to
predict the output of a system for a particular input when the transfer function is
known.
Supposing that the input of a certain system H is X, the output Y can be
calculated as:
HXY ⊗= (25)
Where ⊗ denotes the convolution operator.
figure 07. Perfusion black-model
approach.
If X and Y represent respectively the arterial and tissular concentration of the
system with transfer function H, the equation (25) may be rewritten as:
( ) )()( thtCtC arterytissue ⊗= (26)
However, in MR perfusion imaging the transfer function of human organs is
rarely known, therefore the convolution approach is not directly used. However,
using the MR images it is possible to indirectly measure the arterial and tissular
concentrations. Therefore, knowing Ctissue(t) and Cartery(t) it is possible to
17
calculate an approximation of the transfer function h(t) using one of the
deconvolution methods explained in the previous section. The estimation of h(t)
as it will be explained later, will lead to the estimation of the residue function
and consequently to the rCBF measure.
The biodynamic system has the following components:
• Parameters: the rCBF and rCBV.
• Dependent variables: the arterial, tissular and venous concentration,
denoted Ca, Ct and Cv respectively.
• Independent variable: the time.
2.2.2. The Stewart-Hamilton model
The Stewart-Hamilton model applied to MRI acquisition techniques can help
estimate perfusion parameters such as the regional cerebral blood flow rCBF,
mean transit time MTT or regional cerebral blood volume rCBV, describing the
tissue condition.
The concentration of tracer within a tissular volume, at a given time t, during the
passage of a bolus injection of a contrast agent, is given by:
( ) )()( tCtRrCBFtC arterytissue ⊗⋅⋅= ρ (27)
Where ρ denotes the tissue density (i.e., the tissue mass per unit volume),
Cartery is the arterial concentration and R(t) is the residue function and
represents the tissular retention of the contrast agent.
18
If this biodynamic system has a single fluid storage element hence, the residue
function can be treated as the response of a first-order circuit. The general
solution for this differential equation is of the form:
τt
eKtR−
⋅=)( (28)
In MR perfusion imaging, K is equal to one and τ is usually replaced by the
mean transit time constant, denoted MTT. This value is considered as the mean
time taken by the blood to pass trough the system, from the arterial entrance to
the vascular exit.
Let
)()( tRrCBFtr ⋅= (29) Where r(t) is a scaled residue function.
Therefore, one way to know the rCBF is evaluating the scaled residue in zero:
)0(rrCBF = (30)
Assuming the tissular density being close to the water density, i.e.,
mlg10 =≈ ρρ ,
Then the equation (27) becomes:
( ) )()( tCtrtC arterytissue ⊗= (31)
Once the regional blood flow is estimated, the unknown perfusion parameters,
can be computed from the central volume theorem (Stewart, 1984; Meier and
Zierler, 1954):
19
MTTrCBVrCBF = (32)
where rCBV and MTT are the regional cerebral blood and mean transit time,
respectively and
∫∞
=0
)( dttRMTT (33)
However, the concentration is measured from the MR digital images, therefore,
the continuous time form of the previous equations is not considered. In
consequence, the Stewart-Hamilton model becomes a discrete convolution:
( ) ( ) ( )ikriCkCk
iarterytissue −⋅= ∑
=0 (34)
Moreover, the discrete convolution can be viewed as a matricial product:
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
)(....
)1()0(
)0()1()(
0)0()1(00)0(
)(...
)1()0(
Nr
rr
CNCNC
CCC
NC
CC
arteryarteryartery
arteryartery
artery
tissue
tissue
tissue
LL
LLLLLLL
ML
LL
Where N is the number of samples taken in the RM sequence.
This product can be compactly denoted as:
ΓΑ=Τ (35) Where is unknown. Γ
Hence, the deconvolution process becomes a linear algebra problem,
specifically, a matrix inversion problem:
20
ΤΑ=Γ −1 (36)
However, this process, when working with noisy signals, could sometimes lead
to determinants close to zero, the matrix A is said to be “ill-conditioned”, and
consequently, unexpected results may be obtained. For this reason, a robust
deconvolution method such like the ones exposed in section 2.1 must be used.
Moreover, the concentration of the contrast agent is not directly measured in
clinical practice; instead, MR images are used to determine signal intensity (SI)
variations over time, which are related with the change in the spin-spin
relaxation -also known as the transverse relaxation ( )*2R∆ -. The SI functions are
obtained by selecting a pixel or a region of interest (ROI) in the MR gray-scale
digital images.
Previous studies have shown the existence of a linear relation between the
measured concentration C(t) and . *2R∆
( ) *2RtC ∆α (37)
On the other hand, the transverse relaxation and the signal intensity have an
exponential dependence. The main relation can be written as follows:
( )0
)(ln1S
tSITE
KtC ⋅−= (38)
Where TE is the sequence echo time, SI(t) is the signal intensity over time and
S0 is the baseline MR intensity. Consequently, by measuring SItissue and SIaif the
regional blood flow can be estimated using a deconvolution technique and
evaluating the expression r(t) at zero. The main process is summarized in
figure 08.
21
r(t)
Bolus Injection Contrast Agent
MR perfusion imaging
SIaif (t) SItissue (t)
SI to concentration conversion
Caif (t) Ctissue (t)
DECONVOLUTION
SI to concentration conversion
r(t) : Scaled residue function rCBF= r(0)
figure 08. Parameter estimation procedure using MR perfusion imaging
22
2.3. Sample size selection in statistical experiments
In the present work, it is desired to know which of the two deconvolution
methods is more appropriate to cerebral blood flow estimation. The first part of
the experiment consisted in simulating the MR signals using the MATLAB
software. However, the first problem appeared when it was necessary to
choose the number of samples in order to obtain statistical data. The statistical
data was needed to make the mathematical comparison of the ARMA, SVD and
aSVD deconvolution methods.
Nevertheless, the sample size selection is not an arbitrary decision; it is
necessary to know the maximum error allowed in the clinical context and to
have a general idea of estimation theory to determine it. Demonstrations and
more detailed information on the principles in population size selection can be
found in the probability and statistics literature [10]. A basic review on the theory
is presented here to help to non-familiarized readers, since its
misunderstanding is usually a source of inaccuracies in system-modeling
simulations.
The present experiment is very similar to the situation of political polls, designed
for example, to determine who is the most likely to be president between two
candidates. As it is known, polls are not error free and most of the time, the
number of votes counted after the elections, is not exactly the same as the
predicted by the official survey. However polls are used world wide since they
are useful not only in politics but in economy, sociology and many other fields. A
mayor issue in a survey is the selection of the population size, in order to obtain
a specific error that can be measured. In other words, it is wanted to know how
confident the poll is.
Of course, different fields require different precision, so the first thing that
should be known is the minimum number that is significant for a specific
23
context. In the politician election example, suppose that we a sample size of
100 000 random citizens is selected in the poll. Results show candidate A with
30% of the votes and candidate B with 40%, now suppose that more precision
is wanted, so the sample size is doubled. The second poll now shows candidate
A with 29% and candidate B with 43% of the votes. It is clear that it was not
worth to double the sample size since both polls are giving almost the same
information, i.e., candidate B is more likely to win the elections. However, time
and money spent on the second poll was almost certainly greater than the one
on the first poll. Therefore, a difference of 1± vote is not crucial, except on the
special case where candidates tie. However in the medical context, a difference
of ml/s in cerebral blood flow assessment can be crucial, especially if the life
of the patient depends on the correct estimation of this perfusion parameter.
1±
Consequently, mathematicians have developed techniques to calculate the
sample size of an experiment to obtain a certain degree of precision, in other
words, a confidence interval. However, the population size selection is
dependent of the standard deviation and most of the time, this parameter is
unknown. Therefore, it is necessary to select an arbitrary sample size (N) in the
beginning (habitually N>30) and then calculate the sample standard deviation
(SSD). However for simplicity, the SSD will be just denoted from now on, as the
standard deviation (SD). Assuming a normal distribution:
( )
N
xxSSDSD
N
ii∑
=
−=≅ 1
2
(39)
Where x is the arithmetic mean of the set of samples.
Once, the SD of the population sample is calculated, N can be determined with
the following theorem:
24
“We can be %100)1( ⋅−α confident that the error will be less than a specified
amount e when the sample size is: 2
2/ ⎟⎠⎞
⎜⎝⎛ ⋅
=e
SDZN α (40)
Where is the value of the standard normal distribution leaving an area of 2/αZ
2/α to the right. ”
25
3. METHODS
With the increasing software and hardware advances, computer simulation is
today seen as an extremely powerful and useful tool in modern science
research. Uncountable experiments are performed using random computer
simulations in many different fields such as economy, acoustics, weather
forecasting, sociology, medicine, engineering, etc.
The main purpose of these simulations is to recreate a virtual environment in
which an experiment is done. The same action in the same experiment
conditions is often repeated an elevated number of times in order to analyze its
statistical properties. This kind of statistical sampling techniques using
computers simulations is frequently known in the scientific literature, as the
Monte-Carlo methods. There are several historical explanations of the use of
this expression. In general, it is believed this term was first used by the Polish-
American mathematician, Stanisław Marcin Ulam. The Monte-Carlo area
(Principality of Monaco) is well known for its gambling and casinos and
apparently, the phrase was given in honor of Ulam’s uncle who was a gambler.
In the present work, the name of Monte-Carlo simulations was used to describe
the first part of the deconvolution experiment in which computer tools were
applied. A basic description of the Monte-Carlo methods used in this text is
presented in this section.
26
3.1. Monte–Carlo simulations
In order to assess the performance of the different deconvolution techniques,
realistic MR signals were generated. The statistical data sample of the Monte
Carlo simulation was the estimated regional blood flow, which was calculated
from noisy concentration functions. These simulated signals were considered as
a suitable approximation of what is typically measured in patients.
3.1.1. Range of simulated perfusion parameters
Representative rCBF values were selected from the scientific literature [1]. The
rCBF range varying from 5 to 35 ml/100g/min in 5 ml/100g/min increments was
chosen for a volume rCBV = 2%, corresponding to healthy white matter.
Afterwards, the rCBF was used to calculate the MTT characteristic values,
applying the central volume theorem (32).
The same process was used for rCBV=4%, corresponding to healthy gray
matter, except for that the range varied from 10 to 70 ml/100g/min, in 10
ml/100g/min increments. These values are summarized in the following table.
MTT (s) 24 12 8 6 4,8 4 3,43
rCBF(ml/100g/min) RBV=2% 5 10 15 20 25 30 35
rCBF(ml/100g/min) RBV=4% 10 20 30 40 50 60 70
table 1. Simulated perfusion parameters. Mean transit time, regional blood flow
and regional blood volume.
27
3.1.2. Noisy MR signals simulation procedure
In order to simulate each of the rCBF in both blood volumes values (i.e., rCBV=
2% and rCBV= 4%), the following procedure was applied:
First, the simulated residue functions were generated as:
simMTTt
sim etR−
=)( (41)
for each of the 14 different rCBF values.
The arterial input function (AIF) was simulated as a gamma-variate function
)()( tuetCtCAIFt
peakaif ⋅⋅⎟⎠⎞
⎜⎝⎛⋅==
+−
αταα
τ (42)
where u(t) is the step function (due to causality in the modeled system).
The concentration tissue response C(t) was then obtained by convolving the
AIF with each residue function -equation (31)-. The signal intensity functions
(SI) were calculated assuming an exponential relationship between C(t) and the
pixel gray-level, as it was exposed in equation (38). Solving this equation, for
SI(t):
TEtCkeStSI )(
0)( −= (43) where SI(t) is the signal intensity function over the time, S0 is the baseline MR
image intensity, k is a proportionality factor and TE is the echo time.
Additive white Gaussian noise was then used to impose the SI function samples
with SNRtissue = 6 or 18 dB and SNRAIF = 15 dB, measured from our set of
patient data. The lower SNR in tissue corresponds to one pixel and higher SNR
28
to a typical region of interest (ROI). It is very important to notice, that the noise
can not be directly added to the concentration functions since the existence of a
non linear relationship between C(t) and the simulated signal intensities.
To simplify the notation, let the function 'Ψ be the noisy version of : Ψ
( ) ( )tt noisyΨ=Ψ'
Once the noisy SI functions were generated, arterial and tissue concentrations
were calculated using equation (38), which is the inverse of equation (43).
Therefore, the arterial concentration over the time is:
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⋅−=
0
'ln1'
StSI
TEKtC arterial
arterial (44)
And similarly, the tissular concentration was calculated as:
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⋅−=
0
'ln1'
StSI
TEKtC tissular
tissular (45)
From the moment that the concentration signals were calculated, the scaled
residue function was computed, applying the Stewart Hamilton model and one
of the deconvolution methods:
( ) ( )[ ])(',')(')(')('' tCtCionDeconvoluttrtCtrtC arterialtissulararterialtissular =⇒⊗= (46)
Finally, the blood flow was calculated as
)0('rrCBFestimated = (47)
29
Every deconvolution method was tested in the 7 different blood flows values
and the number of replicates for each one, denoted N, was of 150.
3.1.4. Deconvolution performance criteria
The mean of the simulated samples, for each of the estimated rCBF, was
calculated and subsequently compared with the true values (see table 1).
To evaluate the performance of the deconvolution techniques, the percentage
error (PE) and the standard deviation was calculated for different rCBF and
blood volumes. The standard deviation for a discrete variable made up of N observations is the positive square root of the variance and is defined as:
( )
N
xxSD
N
ii∑
=
−= 1
2
where x is the arithmetic mean of the set of samples and N=150.
The mean percentage error, denoted MPE, was defined as follows:
∑=
=N
iiPE
NMPE
1
1
where N is the number of simulated flows and,
true
trueidentified
rCBFrCBFrCBF
PE−
⋅= 100 (48)
30
However, the deconvolution performance was not only assessed in terms of its
error and standard deviation. Sensibility to SNR shifts or to sample rate was
also considered in the analysis.
3.1.5. ARMA vs. SVD
The first part of the simulation was the comparison of the ARMA and the SVD
deconvolution in the cerebral perfusion context. Similar studies have been done
in the myocardial environment [2], so different blood flow ranges are found in
this project. Moreover, the sample rate was included as an additional issue in
the analysis and it was tested in different noisy environments. The different
simulation environments are summarized in figure 10. For the ARMA
deconvolution algorithm, a first poles-model order and a second order for the
zeros-model was used. On the other hand, a fixed 30% threshold was first
chosen for the SVD method.
Two distinct tissular signal-to-noise ratios were used, in order to simulate an
approximated realistic random noise when using single pixel or ROI selection.
The pixel and ROI SNR are two simulated levels of noise and they recreate an
approximation of the different real noise environments depending on whether
the user selects a single pixel or a group of pixels. An example of ROI or pixel
selection in perfusion MR imaging is illustrated in figure 09. To calculate the
rCBF, the user must first select an option, denoted Op, from two possibilities:
ROI or P. Depending on the selected Op, the area of interest ASelected in the MR
image, will be a set of pixels S or a single pixel P:
⎪⎩
⎪⎨
⎧
==
=⎭⎬⎫
⎩⎨⎧
== =
POpifPixelP
ROIOpifPixelSA
x
N
jj
SelectedU
1
31
figure 09. Example of pixel or ROI selection in MR images. The user can select
a set of pixels (ROI) or a single pixel (P) in rCBF estimation
(e.g., ROI= Pixel 1+Pixel 2+ Pixel3 or P=Pixel 4)
figure 10. ARMA vs. SVD perfusion simulation strategy.
32
3.1.6. SVD vs. adaptive threshold SVD
The adaptive threshold SVD deconvolution (aSVD) performance was tested and
compared with the non-adaptive method for a sampling rate of 1 Hz in two
different noisy environments, as shown in the following scheme:
figure 11. SVD vs. aSVD perfusion
simulation strategy.
The percentage threshold was first set at 30 % and then the oscillation index
was calculated. The aSVD algorithm consisted in systematically changing the
PSVD in the range [20% , 40%] in 4 % increments in order to recalculate the
respectively oscillation index of the residue functions. Afterwards, all the
oscillation indexes were compared and the minimum was selected. The residue
function having this minimum oscillation index is then used to calculate the
desired perfusion parameters. The reader can refer to section 2.1.1.2. to obtain
the detailed description of the adaptive SVD algorithm.
33
3.2. Perfusion MRI in stroke patients
Patients and image acquisition
Nineteen patients (age range, 32-94) with symptoms of acute hemispheric
stroke were retrospectively included in the present study, observing all
respective medical and ethical regulations.
MRI studies were obtained within 6 hours from symptom onset using a 1.5-T
Magnetom Vision whole body MR imager (Siemens, Erlangen, Germany).
Perfusion-weighted MRI was performed with a T2*-weighted gradient-echo
echo-planar imaging sequence, using the bolus passage of contrast agent
(repetition time: 2000 ms; echo time: 60ms; 7 slices; slice thickness: 5 mm;
interslice gap: 0.5 mm; field of view: 240 mm; matrix 128 x 128 voxels; 30
measurements obtained at intervals of 2s).
The contrast injection (15 ml of Gd-DTPA) was performed after the third scan,
using a power injector at a rate of 5 ml/s via access through anantecubital vein;
the bolus of contrast medium was followed by a 15-ml bolus of saline solution at
the same injection rate.
Data post-processing
Perfusion data were transferred onto an independent work-station and analyzed
using homemade programs written in Matlab language (Math Works, Natick,
MA). To obtain the SI functions, tissular and arterial ROIs were manually
selected and converted to concentration signals applying the logarithmic
relationship (43). The cerebral blood flow estimation was then calculated using
the ARMA and SVD deconvolution techniques (46).
34
4. RESULTS AND DISCUSSION 4.1. Monte-Carlo simulations
4.1.1. ARMA vs. SVD
Sampling Rate: 1 Hz (1 sample per second)
Non-noisy data
In the case of non-noisy data for a PSVD threshold of 30%, the SVD estimated
flow was underestimated. The mean percentage error was –34%. In contrast,
using the ARMA deconvolution, the true and identified cerebral blood flows
were identical for both blood volume values hence the percentage error was
0%. The rCBF estimation results are shown in the following figure:
0
5
10
15
20
25
30
35
40
5 10 15 20 25 30 35True CBF [ml/100 g/min]
Iden
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00 g
/min
]
ARMASVD
figure 12. rCBF estimation without random noise using ARMA
and the SVD deconvolution (PSVD = 30%).
35
The estimation in a non-noisy environment was used to show the influence of
the Psvd percentage threshold selection in blood flow estimation. The
underestimation in the SVD method was due to the elimination of the lower
singular values that contain non-noisy information in the matrix. Those elements
are not supposed to be removed without the presence of random noise.
Obviously, if a small (close to zero) Psvd percentage threshold had been
chosen, both results would have been the same. The variation effect of this
parameter is shown in figure 13. Note that the appropriate Psvd for a high SNR
tends to zero.
Noisy data in ROI simulation
For SNRtis = 18 dB and SNRaif = 15 dB the identified rCBF was underestimated
in the ARMA simulation, with a mean percentage error of –6.9% and a mean
standard deviation of 5.8. In SVD, there was also an underestimation; however,
the absolute value of the mean percentage error was greater than the ARMA
one. On the other hand, the standard deviation of the SVD was less than the
ARMA one. This phenomenon was valid for both blood volumes as illustrated in
figure 14.
Noisy data in pixel simulation
When the tissular SNR was decreased to 6 dB without changing the arterial
signal-to-noise ratio (SNRaif = 15 dB) the calculated data was, in this case,
overestimated using the ARMA model with a mean percentage error of +86%.
On the contrary, the identified CBF with the SVD deconvolution technique
remained underestimated with a mean percentage error of -34.5 %.
36
01020304050607080
10 20 30 40 50 60 70True CBF (ml/100g/min)
Iden
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BF
(ml/1
00g/
min
)
Psvd=5%Psvd=30%Psvd=80%True CBF
figure 13. Influence of Psvd threshold selection in flow estimation
without random noise in SVD deconvolution method.
figure 14. Standard deviation comparison for ARMA and SVD.
(R=1Hz, SNRtis= 18 dB and SNRaif = 15 dB.
37
SNR tis = 6dB SNR aif= 15dB
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5 10 15 20 25 30 35True CBF [ml/100 g/min]
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SVDTrue
SNR tis = 18dB SNR aif = 15 dB
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5 10 15 20 25 30 35True CBF [ml/100 g/min]
Iden
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ARMASVDTrue
figure 15. This figure illustrates the SNR shift sensitivity in flow
estimation using both deconvolution methods (rCBV=2% and
sampling rate of 1Hz.)
38
Sampling Rate: Rs= 0.5 Hz (1 sample every 2 seconds)
Non-noisy data
When the sample rate was halved, there was no significant change in the rCBF
estimation. The performance without the presence of random noise for Rs= 0.5
Hz, was similar for both techniques. That means a zero mean percentage error
(MPE) using ARMA and a variable MPE in the SVD deconvolution, depending
on the selected threshold:
%00 →≈ SVDSVD PifMPE
Noisy data in ROI simulation
Sensitivity to sample rate shift:
The estimated blood values that were first underestimated in the ROI SNR, for
Rs=1 Hz in both deconvolution methods changed to overestimation, when the
sample rate was halved to Rs=0.5 Hz.
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60
5 10 15 20 25 30 35
True CBF [ml/100g/min]
Iden
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[ml/1
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]
ARMA 1HzARMA 0,5HzSVD 1HzSVD 0,5HzTRUE
figure 16.Comparison of SNR shift sensitivity in blood flow estimation for both
deconvolution methods. CBV=2%, SNRtis= 18 dB and SNRaif = 15 dB.
39
Specifically, the effect in the ARMA performance for a tissular SNR=18dB and
an arterial SNR=15dB, for this sample rate shift, was similar to the simulation for
Rs=1 Hz when the SNR was decreased. That means that the ARMA model
passed from underestimation to overestimation when the sample rate was
halved. For Rs=0.5 Hz the mean percentage estimation error was of 70%. This
variation represents a mean relative increment of 0.4 in the initially identified
flow with a sample rate of 1Hz. The ARMA standard variation increased in
average 50%.
The SVD deconvolution passed from underestimation to overestimation as well,
when the sample rate was changed to 0.5 Hz. This change represented a mean
increment of 0.5 into the estimated flows for Rs=1 Hz. The standard variation
for the SVD method increased in average 60 % in both blood volumes after the
sample rate shift.
Noisy data in pixel simulation
Sensitivity to SNR shift:
When the random noise was increased (SNRtis was changed from ROI to pixel
level) the ARMA deconvolution passed from an MPE overestimation of 68% to
135%. Its mean standard deviation increased as well from 7.5 to 15.6 in other
words there was a standard deviation increment of 108%.
On the other hand, SVD was less sensitive to the SNR shift, when the tissular
signal-to-noise ratio was decreased to SNRtis=6dB as seen in the following
figure. The MPE passed from a mean percentage error of 32% to a MPE=35%
with this SNR change. The mean SVD standard deviation passed from 3.7 to 5.
40
0
10
20
30
40
50
60
70
80
5 10 15 20 25 30 35
True CBF [ml/100g/min]
Iden
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min
]
ARMA 18dBARMA 6dBSVD 18dBSVD 6dBTrue
figure 17. Comparison of tissular SNR shift sensitivity in blood flow
estimation for both deconvolution methods. Rs = 0.5 Hz and SNRaif =
15 dB.
41
4.1.2. SVD and aSVD deconvolution For a percentage threshold PSVD = 30 %, the performance of the SVD method
was not enhanced in terms of its mean percentage error, when the adaptive
threshold deconvolution was used. The estimated flow was still underestimated
in both blood volumes in the same proportions. For pixel and ROI simulations,
neither the MPE nor the standard deviation was improved for both cerebral
blood volumes. The reason for this lack of improvement was probably because
the selected PSVD was appropriate for the S/N ratio and therefore, the optimal
oscillation index was already chosen in the SVD deconvolution.
figure 18. Example of oscillation index calculation of four different residue
functions from the adaptive SVD algorithm. 42
Under these noisy conditions, the main difference found between the basic SVD
and aSVD deconvolution was in terms of its execution time. It is important to
keep in mind that the aSVD algorithm needed to test 5 different matrix
thresholds, recalculating at each time by deconvolution, the estimated residue
function. Therefore aSVD was much more expensive in terms of execution time
than SVD.
4.2. Perfusion MRI in stroke patients
figure 19. Example of arterial and tissular concentrations calculated from SI
functions for patient 6.
43
Due to the lack of a reference method for the assessment of perfusion, it was
not possible to determine if there was under or overestimation for each case.
However, this part of the experiment was useful to determine the orders of
magnitude of the rCBF in clinical practice and to assess the relationship of the
ARMA and SVD deconvolution.
A set of perfusion signals from one of the patients is shown in figure 21 and its
corresponding identified residue functions using both deconvolution methods
are presented in the following figure.
figure 20. Estimated residue function using the ARMA and SVD deconvolution
model for patient 6.
Note that the residue function falls monotonically to zero in ARMA while the
SVD function presents fluctuations over the time. The blood flow range in the 18
patients varied from 84 to 370 ml/min/100g in ARMA and from 43 to 300
ml/min/100g using SVD.
44
The clinically acquired human MRI data were coherent with the simulations
since the estimated blood flow applying ARMA was in average 1.21 times
greater than the SVD method. All the identified rCBF are summarized in the
following figure:
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100
150
200
250
300
350
400
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Patient
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00g]
ARMASVD
figure 21. Comparison of identified cerebral blood flow in 19 stroke patients
using ARMA and SVD deconvolution.
Note that the real values of the regional cerebral blood flow are not shown in
this figure. To obtain the true values it would be necessary to use a reference
method such as the PET7 scan.
45
7 Positron Emission Tomography (PET)
5. CONCLUSIONS AND FURTHER WORK 5.1. CONCLUSIONS
The performances of the ARMA and SVD deconvolution method have been
compared in ROI and pixel SNR, using two different sample rates. Monte-Carlo
simulations and real clinical RM images were used during this work to
determine the strengths and weaknesses of both algorithms. The simulation
data was blurred with a realistic additive noise procedure and the results
obtained were coherent with the flow estimation using the clinical data.
Satisfactory results in blood flow estimation using the ARMA model in previous
studies encouraged the present work to apply this model into the cerebral
context. The preceding tests [2] used experimental data from an isolated pig
heart preparation due the similarities between the pig and human’s heart.
However, the estimation performance in the present circumstances, i.e., under
less than myocardial flow conditions, was not the expected. The difference of
the orders of magnitude between the brain and heart’s flow changed the
expected behavior of the ARMA method. However, each deconvolution method
had its troubles depending on the application environment. This is why it is not
desirable to search for the perfect deconvolution method, without knowing the
context. Instead, the main question should be reformulated as:
Under which circumstances is it better to apply one algorithm instead of
another?
Or, which is the most appropriate algorithm for some specific situation?
46
The ARMA deconvolution was generally closer to the true blood flow value than
the SVD, when working with the higher sample rate and noise ratio. This signal
to noise ratio corresponded respectively, to 18 dB and 15 dB for tissular and
arterial concentration, i.e., a ROI selection. Therefore it is advisable to work with
the ARMA model when working under these circumstances. Oppositely, it can
be stated that ARMA estimation is not advisable when the blood flow is not
greater than 100 ml/100g/minute with under-sampling (i.e., a sample rate of 0,5
Hz). This leads SI functions to be highly susceptible by random noise and
therefore poor precision is found on regional cerebral blood flow identification.
The standard deviation in ARMA was generally greater that the SVD one.
However, when the sample rate was halved, the increment of the SVD standard
deviation was greater than the one of ARMA.
On the other hand, the singular values decomposition technique, showed to be
less sensitive to SNR shifts. In others words, even if the SVD estimator was
biased when the sample rate was of 1Hz it remained biased on the same
proportions when the sample rate was of 0.5 Hz. Consequently, SVD is a
suitable technique when the sample rate needs to be changed and if ROI or
pixel selection is simultaneously being used.
The adaptive threshold deconvolution can be a useful method when the order of
magnitude of the signal-to-noise ratio is unknown. Otherwise, this model is not
advisable, since its execution time was much greater than the SVD
deconvolution.
The simulations suggested that there is no difference in blood flow estimation in
the two simulated volumes. In other words, the behavior of both deconvolution
methods was very similar when dealing with healthy cerebral white or gray
matter.
47
5.2. FURTHER WORK
The results from this study showed the influence of random noise in blood flow
estimation and it was seen how strategies such as the ROI selection could
optimize the rCBF identification. Further research could include regularization
solutions to solve these kinds of discrete ill-posed problems. Different
regularization strategies would be compared in order to determine the most
appropriate technique for this MR perfusion-imaging context.
Knowing that the image of the patient is not completely motionless, it would be
interesting to quantify the influence of image registration in the correct blood
flow estimation.
The integration of both deconvolution methods into the software developed by
the Cardiotools Creatis team is highly conceivable. Additionally, it would be
attractive to prepare a synergetic deconvolution method. This fusion algorithm
would automatically select the strengths of ARMA and SVD deconvolution, in
order to produce a greater effect than the sum of their individual performances.
In this study, the ARMA deconvolution was applied using the first and second
order for the poles and zeros models respectively. However, it would be
interesting to see whether the rCBF estimation could be or not improved by
changing the zeros model order.
48
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