Intravoxel incoherent motion modeling
Optimization of acquisition, analysis and
tumor tissue characterization
Oscar Jalnefjord
Department of Radiation Physics,
Institute of Clinical Sciences at
Sahlgrenska Academy
University of Gothenburg
Gothenburg, Sweden, 2018
Cover illustration by Oscar Jalnefjord
Intravoxel incoherent motion modeling - Optimization of
acquisition, analysis and tumor tissue characterization
c©2018 Oscar Jalnefjord
oscar.jalnefjord@gu.se
ISBN: 978-91-7833-077-5 (PRINT)
ISBN: 978-91-7833-078-2 (PDF)
http://hdl.handle.net/2077/56355
Printed in Gothenburg, Sweden 2018
Printed by BrandFactory
All models are wrong but some are useful
George E.P. Box (1979)
i
ii
Abstract
Intravoxel incoherent motion (IVIM) analysis provides a means to obtain informa-
tion on diffusion and perfusion from a single MRI sequence. The measurements
are completely noninvasive and the results have been shown to be of interest, for
example, in oncological applications. Although the use of IVIM analysis has in-
creased substantially the last decade, choice of acquisition parameters and analysis
methods are still open questions.
The aim of this thesis was to improve IVIM analysis by optimization of the
image acquisition and parameter estimation methods, and to study the ability of
IVIM parameters to be used for tumor tissue characterization.
With standard model-fitting methods and data quality, IVIM parameter esti-
mation uncertainty is typically high. However, several Bayesian approaches have
been shown to improve parameter quality. In Paper I, these Bayesian approaches
are compared using simulated data and data from a tumor mouse model. The
results emphasize the impact of methodological choices, especially the prior distri-
bution, at typical noise levels.
Quick and robust IVIM examinations are important for clinical adoption, but
consensus regarding methodology is lacking. To address this issue a framework
for protocol optimization is presented in Paper III and a comparison of estimation
methods was done in Paper II. To test the optimization framework, a protocol
for liver examination was generated and tested on simulated data and data from
healthy volunteers resulting in improved IVIM parameter quality. The compared
estimation methods were evaluated on simulated data and data from patients with
liver metastases with similar results for all methods, thereby making the compu-
tationally most effective method preferable.
Studies of tumors using quantitative imaging methods such as IVIM often
only extract an average parameter value from the entire tumor and may thus
miss important information. Paper IV explores the ability of IVIM parameters to
identify tumor subregions of functionally different status using clustering methods.
The obtained subregions were found to have different proliferative status as derived
from histological analysis.
The work presented in this thesis has resulted in improved IVIM acquisition
and analysis methods. It also shows that IVIM has the potential to provide insight
into tumor physiology and be used as a noninvasive imaging biomarker.
Keywords: IVIM, MRI, diffusion, perfusion, cancer
iii
Populärvetenskaplig
sammanfattning
Magnetkameran (MR) spelar en mycket viktig roll inom medicinsk diagnostik med
sin förmåga att avbilda mjukdelar i kroppen. Unikt med MR är också det stora
antalet vävnadsegenskaper som kan avbildas. Bland annat kan man skapa MR-
bilder där pixelvärdet är kopplat till omfattningen av mikroskopisk rörelse hos
vattenmolekyler i en vävnad. I kroppen finns två viktiga typer av sådan rörelse:
diffusion och mikrocirkulation.
Diffusion är slumpmässig rörelse hos vattenmolekyler som beror på deras
rörelseenergi. I vävnad hindras denna rörelse av bland annat cellmembran och
andra mikroskopiska strukturer. Genom att kunna mäta storleken på diffusions-
rörelsen är det därför möjligt att få information om vävnadens mikrostruktur.
Exempelvis kan det ge information om celltäthet och membrangenomtränglighet.
Mikrocirkulation, dvs. blodflödet genom de minsta blodkärlen (kapillärerna),
är mycket viktigt för en vävnad eftersom det är genom den processen som syre
och näringsämnen överlämnas till vävnaden. Eftersom syre och näringsämnen är
nödvändiga för att en vävnad ska fungera kan mätning av mikrocirkulation ge
information om hur en vävnad mår.
Genom att ta flera MR-bilder med varierande känslighet för mikroskopisk vat-
tenrörelse kan man beräkna kvantitativa mått på diffusion och mikrocirkulation i
en vävnad.
Denna avhandling kretsar kring mätning och analys av magnetkamerabilder
känsliga för mikroskopisk vattenrörelse och hur de kan användas för karakterisering
av tumörer. Målet med arbetet var att förbättra bildtagning och analys av denna
typ av bilder.
För en enkel variant på avbildning, mer lämpad för en klinisk situation, togs en
metod fram för optimala val av nivåer på rörelsekänslighet. Olika analysmetoder
jämfördes också för att se vilken som gav mest säkra och stabila mått på diffusion
och mikrocirkulation.
För en något mer tidskrävande variant på avbildning, mer lämpad för forsk-
ningsapplikationer, jämfördes analysmetoder som använder olika antaganden och
som därför kan ge mer stabila mått på diffusion och mikrocirkulation. En av de
studerade analysmetoderna användes sedan ihop med en nyutvecklad metod för
att identifiera delregioner i tumörer med olika vävnadsegenskaper.
Avhandlingsarbetet har resulterat i förbättrad bildtagning och analys för mät-
ning av mikroskopisk vattenrörelse med magnetkamera, särskilt för karakterisering
av tumörer.
iv
List of papers
This thesis is based on the following studies, referred to in the text
by their Roman numerals.
I. Impact of prior distributions and central tendency measures
on Bayesian intravoxel incoherent motion model fitting
Oscar Gustafsson*, Mikael Montelius, Göran Starck, Maria
Ljungberg
Magnetic Resonance in Medicine 2018;79(3):1674-1683
II. Comparison of methods for estimation of the intravoxel
incoherent motion (IVIM) diffusion coefficient (D) and
perfusion fraction (f)
Oscar Jalnefjord, Mats Andersson, Mikael Montelius, Göran Starck,
Anna-Karin Elf, Viktor Johanson, Johanna Svensson, Maria
Ljungberg
Magnetic Resonance Materials in Physics, Biology and Medicine
2018; In press
III. Optimization of b-value schemes for estimation of the
diffusion coefficient (D) and the perfusion fraction (f) with
segmented intravoxel incoherent motion (IVIM) model
fitting
Oscar Jalnefjord, Mikael Montelius, Göran Starck, Maria Ljungberg
Manuscript
IV. Data-driven identification of tumor subregions using
intravoxel incoherent motion
Oscar Jalnefjord, Mikael Montelius, Jonathan Arvidsson, Eva
Forssell-Aronsson, Göran Starck, Maria Ljungberg
Manuscript
Paper I is reprinted with permission of John Wiley & Sons, Inc.
Paper II is Open Access and available under the CC BY license
*Gustafsson was the name of the author until June 2017
v
Related presentations
Improved IVIM model fitting with non-rigid motion
correction
Oscar Gustafsson, Mikael Montelius, Maria Ljungberg
Annual Meeting ISMRM 2015 Toronto, Canada
IVIM reveals higher blood perfusion of liver metastases after oral
intake of salovum
Mikael Montelius,Oscar Gustafsson, Mats Andersson, Eva Forssell-Aronsson,
Ragnar Hultborn, Susanne Ottosson, Göran Carlsson, Stefan Lange, Maria
Ljungberg
Annual Meeting ESMRMB 2015 Edinburgh, UK
An assessment of Bayesian IVIM model fitting
Oscar Gustafsson, Mikael Montelius, Göran Starck, Maria Ljungberg
Annual Meeting ISMRM 2016 Singapore
Can Cramer-Rao Lower Bound be used to find optimal
b-values for IVIM?
Oscar Gustafsson, Maria Ljungberg, Göran Starck
Annual Meeting ISMRM 2016 Singapore
Impact of prior distribution and central tendency measure on
Bayesian IVIM model fitting
Oscar Gustafsson, Mikael Montelius, Göran Starck, Maria Ljungberg
Annual Meeting ISMRM 2017 Honolulu, USA
IVIM D and f - Optimal estimation technique and their
potential for tissue differentiation
Oscar Jalnefjord, Mats Andersson, Mikael Montelius, Göran Starck, Anna-
Karin Elf, Viktor Johanson, Johanna Svensson, Maria Ljungberg
Joint Annual Meeting ISMRM-ESMRMB 2018 Paris, France
vi
Abbreviations
ADC Apparent Diffusion Coefficient
ASL Arterial Spin Labeling
CRLB Cramer-Rao Lower Bound
D Diffusion coefficient
D∗ pseudo-Diffusion coefficient
DCE Dynamic Contrast Enhanced
DSC Dynamic Susceptibility Contrast
DWI Diffusion Weighted Image/Imaging
f perfusion fraction
GMM Gaussian Mixture Model/Modeling
IVIM IntraVoxel Incoherent Motion
MRI Magnetic Resonance Imaging
NLLS NonLinear Least Squares
S0 Signal without diffusion weighting
sIVIM simplified IVIM
SNR Signal-to-Noise Ratio
vii
viii
Contents
Introduction 1
Aims 5
Intravoxel incoherent motion (IVIM) 7
The IVIM concept . . . . . . . . . . . . . . . . . . . . . . . 7
The IVIM model . . . . . . . . . . . . . . . . . . . . . . . . 8
Relation to other MR perfusion techniques . . . . . . . . . . 11
The use of IVIM . . . . . . . . . . . . . . . . . . . . . . . . 12
IVIM parameter estimation 15
Maximum likelihood . . . . . . . . . . . . . . . . . . . . . . 15
Least squares . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Segmented fitting . . . . . . . . . . . . . . . . . . . . . . . . 17
Bayesian parameter estimation . . . . . . . . . . . . . . . . 18
Prior distribution . . . . . . . . . . . . . . . . . . . . . 20
Measures of central tendency . . . . . . . . . . . . . . 22
Optimization of b-value schemes for IVIM 25
Objective function . . . . . . . . . . . . . . . . . . . . . . . 25
Expression-based approaches . . . . . . . . . . . . . . . . . 27
Error propagation . . . . . . . . . . . . . . . . . . . . . 27
Cramer-Rao lower bound . . . . . . . . . . . . . . . . 28
Experiment-like approaches . . . . . . . . . . . . . . . . . . 29
Analysis of IVIM tumor parameter maps 33
Methods for intratumor parameter analysis . . . . . . . . . 33
Histogram and texture analysis . . . . . . . . . . . . . 33
Region-based analysis . . . . . . . . . . . . . . . . . . 34
Cluster-based methods . . . . . . . . . . . . . . . . . . . . . 35
ix
Validation of clustering . . . . . . . . . . . . . . . . . . . . . 38
Conclusions 41
Future aspects 43
Acknowledgements 45
References 56
x
Introduction
Magnetic resonance imaging (MRI) can be made sensitive to motion
through the use of magnetic field gradients. If the motion of spins
is coherent on the voxel level, it will introduce a phase offset rela-
tive to stationary tissue. On the other hand, if the motion of spins
is incoherent within a voxel, i.e. if the motion of spins varies within
the voxel, a phase dispersion is introduced that attenuates the sig-
nal in varying degree depending on the motion and how the motion
encoding was performed. One well known incoherent motion is the
self-diffusion of water molecules. The effect of diffusion on the MR
signal was early recognized [1] and was followed by development of a
method to measure it [2]. Later, diffusion sensitizing gradients were
combined with an imaging readout, which made it possible to produce
images that quantified the tissue water diffusion in vivo [3]. However,
in the same paper it was noted that a diffusion weighted image (DWI)
is not only sensitive to diffusion, but also to all other motions that
are incoherent on the voxel scale, especially blood microcirculation
or perfusion. The technique was therefore called intravoxel incoherent
motion (IVIM) imaging. Shortly after the measurement technique was
presented, a way of separately quantifying the diffusion and perfusion
was presented [4]. This quantification approach has later become syn-
onymous to IVIM, while DWI has become the standard name for the
imaging technique.
The main assumption in quantitative IVIM analysis is that the
MR signal originates from two compartments, the intravascular space
and the extravascular space, respectively. The IVIM model that is
most often used includes one parameter for describing the tissue wa-
ter diffusion (D), one for describing the motion of water molecules in
the blood (D∗) and one for the amount of blood in the voxel (f) [4].
The values of these parameters can be estimated by acquiring images
with at least four different diffusion weightings (b-values) followed by
1
fitting the model to the acquired data. The diffusion coefficient D is of
interest since it is affected by e.g. cellularity and membrane permeabil-
ity and therefore provides a means to probe tissue microstructure [5].
The perfusion-related parameters (f and D∗) are not as well-studied,
but perfusion MRI in general and dynamic contrast enhanced (DCE)
MRI in particular is an important diagnostic tool with its sensitivity
to blood supply [6]. If IVIM based perfusion MRI can be used in
at least a subset of all perfusion examinations it would be beneficial
since it is completely noninvasive unlike DCE MRI that requires an
intravenous injection of a contrast agent.
While appearing conceptually simple, IVIM has been shown to be
nontrivial to implement in practice. The estimation of the IVIM pa-
rameters has been found to be problematic, especially for D∗, due to
susceptibility to noise [7]. To improve the quality of parameter esti-
mates, a large number of estimation approaches have been proposed,
including some where only D and f are estimated [8, 9]. Improved
robustness has been achieved through the use of specialized or ad-
vanced estimation approaches, but there is still little or no consensus
and estimation approaches varies between studies, which reduces the
comparability between studies.
In addition to using a specifically designed estimation approach,
choosing optimal b-values has a potential to reduce the uncertainty
of parameter estimates substantially. Methods for experiment design
have been applied successfully to several aspects of diffusion-weighted
imaging including IVIM [10–13]. However, the optimal choice of b-
values may dependent on the estimation approach. Most work on
b-value optimization for IVIM has been focused on a generic simul-
taneous estimation of all model parameters, while optimization of b-
values for other more specialized estimation approaches is mainly un-
explored. For example, estimation limited to D and f is attractive in
a clinical setting due to its short acquisition and processing time, and
robustness, but optimization of b-values for this approach has only
been studied for a special case [14, 15].
With its ability to quantitatively map both diffusion and perfusion,
IVIM has been proposed as an interesting diagnostic tool in several
oncological applications [16]. Tumor tissue characterization based on
quantitative imaging, such as IVIM, has a great potential to, for ex-
ample, improve the understanding of tumor development by repeated
imaging either during growth or after therapy. However, even though
it is well known that tumors can be highly heterogeneous and that
2
quantitative imaging can provide potentially important spatial infor-
mation, parameter values are often averaged across the entire tumor
[17]. To obtain more information, it has been proposed to do sepa-
rate analyses in smaller subregions of the tumor. By doing so, spatial
information is provided in a way that is relatively easy to interpret.
Such subregions can be defined, for example, based on geometrical
properties like distance from the center or based on functional char-
acteristics derived from the parameter maps. With both diffusion and
perfusion parameter maps, IVIM is an interesting candidate for such
a functional definition of tumor subregions.
3
4
Aims
The overall aim of this thesis was to improve intravoxel incoherent
motion (IVIM) analysis, especially for tumor tissue characterization.
The specific aims of the papers included in the thesis were:
• to evaluate the impact of estimation approach both for the full
IVIM model (Paper I) and for the analysis restricted to the pa-
rameters D and f (Paper II)
• to develop and evaluate a framework for optimization of b-value
schemes for DWI data used to estimate the parameters D and
f (Paper III)
• to investigate if clustering based on IVIM parameter maps can
be used for identification of tumor subregions with biological
relevance (Paper IV)
5
6
Intravoxel incoherent motion
(IVIM)
The IVIM concept
The principle for diffusion weighted imaging is that nonstationary
spins gain a phase offset relative to stationary spins due to the dif-
fusion encoding. The obtained phase depends on the trajectory tra-
versed by the spin during the diffusion encoding and may therefore
differ between spins within a voxel. For an incoherent motion, i.e. if
all spins do not move in a single straight line with the same speed, the
result is a phase dispersion and therefore a reduced amplitude of the
signal. The incoherence can be due to motion in different directions
by different spins (spatial incoherence), motion in different directions
by the same spin (temporal incoherence) or a combination of both [4,
18]. The resulting signal attenuation of the diffusion weighted signal
depends on the characteristics of the incoherent motion as well as the
technique used for diffusion encoding. To emphasize that it is motions
that are incoherent on the voxel level, these kinds of motion are called
intravoxel incoherent motion (IVIM).
For in vivo MR imaging there are two major types of IVIM’s that
affect the diffusion weighted signal. These are tissue water diffusion
and blood microcirculation.
Diffusion is a random motion that is driven by the kinetic energy of
the particles, in this case water molecules. During a typical encoding
time of tens of milliseconds a water molecule collides and changes
directions on the order of 1010 times [19]. The motion of the molecules
can therefore be seen as a random walk with multiple steps taken
during the diffusion encoding, with different trajectories for different
water molecules (left illustration in Figure 1). In tissue, the diffusion
7
can be hindered or restricted by structures such as cell membranes and
organelles, which reduces the distance covered by the water molecules.
The reduced magnitude of the diffusion can therefore be used to probe
tissue microstructure through diffusion weighted MRI.
Microcirculation is the flow of blood in the capillaries, also referred
to as perfusion in this thesis. While the motion of water molecules due
to blood flow is not necessarily an incoherent motion, it is reasonable
to assume that the capillaries contained in a typical voxel (side 2-3
mm) are not perfectly aligned but rather arranged in many directions
(right illustration in Figure 1). The variable orientation of capillaries
is a source of spatial incoherence, which makes the microcirculation
an IVIM. Depending on the capillary architecture, blood velocity and
duration of the diffusion encoding, microcirculation can also give rise
to temporal incoherence [4]. In such case, microcirculation has an
equivalent effect on the MR signal as diffusion and has therefore been
called pseudo-diffusion. Assuming temporal incoherence simplifies the
signal modeling (see below) and is therefore used in most studies, but
it is still unclear in what measurement situations that the assumption
holds. The assumption has, for example, been shown to be violated in
the brain [18] and the liver [20]. However, the commonly used model
usually fits well to data and the major implication that this violation
on the assumption has is therefore to make the interpretation of some,
but not all, model parameters less straight forward. One can, for ex-
ample, end up with parameter estimates that are highly dependent on
the choice of b-values [18]. Nevertheless, if appropriate measurement
methods and analysis are used, the effect of blood microcirculation on
the diffusion weighted signal enables extraction of perfusion informa-
tion from DWI regardless of capillary architecture [4].
The IVIM model
The overall assumption for IVIM analysis is that the MR signal orig-
inates from two compartments: the intravascular space and the ex-
travascular space, which have individual response to diffusion encoding
[4]. The two compartment are also referred to as the perfusion com-
partment and the diffusion compartment, respectively, in this thesis
based on the most influential IVIM in the particular compartment. To
separate the effects of diffusion and perfusion on the diffusion weighted
MR signal a model can be formulated as follows:
8
Diffusion Microcirculation
Figure 1: Illustration of the two major sources of intravoxel incoherent mo-
tion in living tissue: water diffusion and microcirculation
0 200 400 600 800
0
0.5
1
b-value [s/mm2]
S
ig
n
a
l
[a
.u
.]
Extravascular
Intravascular
Total
Figure 2: An example IVIM signal vs. b-value curve. The curves show the
signal from each compartment and their sum, which is what is measurable.
Data was generated using Equation 3 with parameter values: S0 = 1, D = 1
µm2/ms, D∗ = 10 µm2/ms, f = 0.2
9
Sb = S0 ((1− f)FD + fFP ) (1)
where Sb is the signal at b-value b, S0 is the signal without diffusion
weighting, f is the signal fraction originating from the intravascular
space (referred to as perfusion fraction), and FD and FP are functions
that describe how the signal is attenuated by the diffusion weighting
in the extravascular and intravascular spaces respectively. In most
cases, the signal decay due to diffusion weighting is stronger in the
perfusion compartment. This results in a signal vs. b curve where the
initial slope is dominated by perfusion effects, and the latter parts by
diffusion effects (Fig. 2).
For weak to intermediate magnitudes of the diffusion weighting (b-
values less than approximately 1000 s/mm2), the signal attenuation
in the extravascular space is well approximated by a monoexponential
function [19]. Equation 1 can therefore be written as:
Sb = S0
(
(1− f) e−bD + fFP
)
(2)
where D is the diffusion coefficient in the extravascular space. This is
the form that was originally proposed by le Bihan et al. [4]. More re-
cent work has extended that model to include the effects of diffusional
variance by the use of e.g. the kurtosis model [16]. However, studies
of such extensions are beyond the scope of this thesis.
Remaining to be determined in Equation 2 is the effect of diffusion
weighting on the intravascular space (FP ). However, since temporal
incoherence is not necessarily a valid assumption it can be a non-trivial
task to describe FP . The two special cases of no temporal incoherence
and full temporal incoherence was described by le Bihan in the original
paper. The intermediate case is yet an open topic for research where
some attempts have been made that warrants further research [20, 21].
In the case of full temporal incoherence, it is assumed that the cap-
illary architecture and the blood flow is such that the water molecules
in the blood changes direction a sufficient number of times during the
diffusion encoding [4]. The motion of water molecules can then be
approximated by a random walk similar to the case of diffusion and
FP can be described by a monoexponential function. This results in
the following IVIM model:
Sb = S0
(
(1− f) e−bD + fe−bD
∗
)
(3)
10
where D∗ describes the motion of the blood water. Due to the pseudo-
diffusion like motion of microcirculation under these circumstances,
D∗ is referred to as the pseudo-diffusion coefficient. Even though the
assumption of temporal incoherence cannot be said to hold in general,
and has been proved invalid in some studies [18, 20], the biexponential
model is currently the by far mostly used IVIM model.
An alternative approach is to avoid the b-value range where FP
is not well known. Specifically, one can exploit the stronger signal
attenuation in the perfusion compartment and choose to use b-values
where FP is either 1 (b = 0) or of negligible size (b > bthr, where bthr
is some threshold b-value). This gives the simplified IVIM (sIVIM)
model:
Sb = S0
(
(1− f) e−bD + fδ(b)
)
(4)
where δ(x) is the discrete delta function, i.e. δ(x = 0) = 1 and
δ(x 6= 0) = 0 [22–24]. Since the low b-values > 0 are avoided, no
information can be obtain regarding the blood flow by e.g. D∗, but f is
still possible to estimate if S(0) is measured. Note that this simplified
IVIM model does not attempt to describe the signal attenuation in the
full range of b-values, but rather to provide a simple model that can
be used for estimation of at least a subset of the IVIM parameters.
Relation to other MR perfusion techniques
IVIM is just one of several techniques that can be used to measure per-
fusion with MRI. The more commonly used techniques are dynamic
contrast enhancement (DCE), dynamic susceptibility contrast (DSC)
and arterial spin labeling (ASL). The techniques have different ben-
efits, but a major advantage for IVIM and ASL is that they do not
depend on an injection of a contrast agent. This is especially beneficial
for patients with renal dysfunction and for small children [16].
Since diffusion weighting sensitizes the MR images to motion, the
perfusion information attainable via IVIM analysis is related to the
blood that is flowing [25]. This means that IVIM in principle can
be used to quantify blood flow and volume of flowing blood. In fact,
under some very specific conditions, it is possible to translate the IVIM
perfusion parameters f and the product f×D∗ into measures of blood
volume and blood flow, respectively [25].
11
Several studies have been conducted with the aim to study the re-
lationship between IVIM derived measures of blood volume and blood
flow, and corresponding measures obtained from the other MR perfu-
sion techniques, but the results are inconclusive [26]. Multiple studies
have found correlations between f and blood volume derived from
DSC and DCE [27–30], but contradicting results exist as well [30–32].
Some studies have compared f ×D∗ with measures of blood flow from
DSC [27] and ASL [30, 33], but the data available is scarse and mainly
contradicting. The relationship between IVIM perfusion parameters
and measures of blood volume and blood flow from other MR perfusion
techniques is thus still an open field of research [26].
There are several possible explanations for the currently poor agree-
ment between perfusion parameters derived from IVIM and other MR
techniques. On the IVIM side, there are difficulties associated with
both f and D∗. f is a signal fraction, so to get something similar to
a blood volume, one needs to compensate for relaxation (T1 and T2)
and possibly also other factors such as contamination by cerebrospinal
fluid [34, 35]. D∗ on the other hand is given in absolute units, but
is often strongly affected by noise [7] and is only easily interpreted
under certain conditions (temporal incoherence) [18]. Different MR
perfusion techniques can also be sensitive to different kinds of blood
flow which complicates the comparison further [26]. Future studies on
this topic should try to compensate for these confounding factors in
order to facilitate more direct comparisons.
The use of IVIM
After some initial interest [36], perfusion evaluation by IVIM analysis
did not gain much attention until the start of the current decade (Fig.
3). After Luciani et al. showed that IVIM analysis could be used
for diagnostic purposes of liver cirrhosis [37], the IVIM field gained
renewed attention [38] and the number of publications has since in-
creased exponentially (Fig. 3).
During this decade, IVIM analysis has been applied successfully
in various organs, often with oncological applications [16, 26]. It has,
for example, been shown that f can be helpful in grading gliomas [29]
and that both D and f provide important information for stroke as-
sessment [39, 40]. In addition to applications related to liver cirrhosis
[31, 37], IVIM has also been used in the liver for, e.g., grading of hep-
atocellular carcinomas [41]. IVIM parameters have also been used for
12
19
90
19
95
20
00
20
05
20
10
20
15
0
50
100
150
*
#
P
u
b
li
sh
e
d
p
a
p
e
rs
Figure 3: Number of IVIM papers published each year since the first in 1986.
The data was obtained from a PubMed search with search string: "intravoxel
incoherent motion" OR "IVIM". *Note that the number of published papers
in 2018 is only until July 16. An extrapolated number for 2018 is 220
differentiation of common malignant pancreas tumors with promising
results [42]. Another promising application of IVIM is breast cancer,
where a difference in both D and f between benign and malignant
tumors has been shown [43].
13
14
IVIM parameter estimation
This chapter is mainly related to Papers I and II.
In Paper I, the impact of methodological choices related to Bayesian es-
timation of IVIM parameters based on the biexponential model were studied.
Estimation with different prior distributions and central tendency measures
used in previous studies were compared.
In Paper II, methods for estimation of D and f were compared. Seg-
mented model fitting was evaluated among with fitting of the sIVIM model
based on either least squares or Bayesian methods with different central ten-
dency measures.
Estimation of model parameters is typically done by fitting the specific
model to a set of data. Therefore, parameter estimation and model fit-
ting have equivalent meaning and will be used interchangeably in this
text. Parameter estimation is often based on the maximum-likelihood
method and in particular using the least-squares criterion, although
other methods such as Bayesian estimation can be useful in some ap-
plications.
Maximum likelihood
Parameter estimates based on the maximum-likelihood method are
obtained by finding the parameter values that maximizes the likeli-
hood function, where the likelihood function describes the probability
of the data given the set of model parameters. To formulate the likeli-
hood function, one needs to choose a signal model and a noise model.
For the work presented in this thesis, the signal model is one of the
versions of the IVIM model (Eq. 3 or 4). Since magnitude images are
used in general for DWI, the noise distribution is often assumed to be
Rician [44]. To simplify model fitting, the signal-to-noise ratio (SNR)
15
is typically assumed to be high enough such that the Rician distribu-
tion can be approximated by a Gaussian one. For a single data point,
this gives:
P (S(b)|θ, σ) =
1
√
2piσ2
exp
[
−
(S(b)− Sb (θ))
2
2σ2
]
(5)
where P (S(b)|θ, σ) is the probability density function describing how
the measured value S(b) varies around the predicted value Sb given
the signal model parameters θ = [S0, D, f,D∗] and the noise variance
σ2 [45]. Furthermore, if data points in a measurement series are also
(conditionally) independent, the likelihood function is given by:
L(θ) = P (S|θ, σ) =
N∏
i=1
P (S(bi)|θ, σ) =
=
(
2piσ2
)−N/2
exp
[
−
∑N
i=1 (S(bi)− Sbi (θ))
2
2σ2
]
(6)
where N is the number of measurements [45]. It is clear that max-
imization of the likelihood function with respect to the signal model
parameters is achieved by minimizing the sum in the exponent. The
maximum-likelihood estimate, given these specific noise assumptions,
is therefore given by the so-called least-squares estimate.
Least squares
Parameter estimation based on least-squares fitting is by far the most
commonly used. For linear models, closed-form solutions exist and
the method is therefore very robust for such models. On the other
hand, for nonlinear models, such as the IVIM model, closed-form so-
lutions do not exist and iterative optimization methods have to be
used. This leads to increased numerical complexity and decreased
robustness since multiple local maxima may exist in the likelihood
function.
Nonlinear least squares (NLLS) fitting of the biexponential IVIM
model (Eq. 3) is very commonly used and may appear attractive
since few explicit assumptions are made and it is easy to perform with
routines available in most software packages. However, it was early
shown to be susceptible to noise [46], a result that has been confirmed
16
several times since then [7–9]. The possibly most important reason for
the noise sensitivity of the biexponential model is its flexibility. An
over-ability to fit well to data becomes problematic if the measured
signal at some key b-value is strongly influenced by noise or some
artefact. Furthermore, if the perfusion fraction is small and the noise
level is high, data may be very well described by a monoexponential
model. Unless D and D∗ are constrained to not overlap, D∗ may
take the role of D while f takes extremely high values or D∗ and D
take similar values while f can take any value in the allowed range.
Given its limited robustness, NLLS for fitting of the biexponential
IVIM model should therefore be avoided unless the data is of very
high quality, especially for tissues with a low perfusion fraction.
The simplified IVIM model (Eq. 4) can also be fit using NLLS [22].
The model is less flexible than the biexponential model and is therefore
less susceptible to noise as shown in Paper II and previously by others
[9]. Nevertheless, it is a nonlinear model with multiple parameters
and therefore requires iterative estimation methods, which may be
computationally expensive compared with other methods.
Segmented fitting
To increase the robustness of the IVIM parameter estimates, a step-
wise procedure was proposed that is often referred to as segmented or
asymptotic model fitting [46]. The rationale behind the method is that
one can typically assume that the diffusion weighting attenuates the
signal from the perfusion compartment more strongly than that from
the diffusion compartment. The result is that the term FP in Equation
2 can be assumed to be of negligible size at a sufficiently high b-value,
equivalent to the argument used to formulate the sIVIM model (Eq.
4). If the biexponential IVIM model is used, this assumption is often
expressed as D  D∗. If the assumption holds (FP ≈ 0) and only b-
values above the chosen threshold are used, the signal model simplifies
to:
Sb = S0 (1− f) e
−bD = Ae−bD (7)
This monoexponential model is fitted to get an estimate of D. The
estimate of D can then be fixed in a NLLS fit where all b-values are
used and the remaining IVIM parameters are estimated. Alternatively,
if images with the b-value b = 0 have been acquired, they may be used
together with the extrapolated y-axis intercept A to get an estimate
17
of f as:
f = 1−A/S0 (8)
where S0 has been set to the measured value S(0). Both D and f can
then be fixed in the NLLS fit where D∗ is estimated. If f is estimated
through the extrapolation step, the last step with estimation of D∗
might be skipped while information on both diffusion and perfusion is
still obtained. This reduces scan time since fewer b-values are needed
as well as computational time since the NLLS step for D∗ is avoided.
In fact, a special case of the segment fitting approach with only three
b-values was the one proposed for estimation of D and f in the orig-
inal IVIM paper [4]. When only three b-values are used, closed form
solutions exist for both D and f as shown in Paper III.
The segmented fitting has been compared with the simultaneous
least-squares estimation of all IVIM parameters from the biexponential
IVIM model in several studies [9, 47–50]. It is clear from these results
that the segmented approach is preferable, with the caveat that a
proper threshold b-value must be chosen.
In Paper II, segmented fitting for estimation of D and f was com-
pared with fitting the sIVIM model (Eq. 4) using either least squares
or Bayesian methods. The estimation approaches were compared re-
garding estimation variability and bias as well as ability to differentiate
between tumor and healthy liver. Apart from some minor differences,
it was found that all approaches produced very similar results (see
example in Figure 4). The major difference between the approaches
is instead that the segmented fitting is associated with a substantially
lower computational cost and is easier to implement due to the simple
stepwise procedure. In fact, segmented fitting could easily be fit into
a clinical workflow where parameter maps are generated directly as an
extension to the reconstruction. The conclusion from Paper II is there-
fore that the segmented fitting approach is preferable for estimation
of D and f only.
Bayesian parameter estimation
A more general way to incorporate prior knowledge into the model
fitting procedure is to take a Bayesian perspective. Instead of simply
finding the parameter values that maximizes the likelihood function,
the posterior parameter distribution is derived through Bayes’ rule and
used to obtain the parameter estimates. According to Bayes’ rule, the
18
Figure 4: b = 0 image and IVIM parameter maps of a patient with a liver
metastasis. The parameter maps are based on all estimation approaches
studied in Paper II. The manual delineation of the tumor is shown as a
dashed line (red in b = 0 image and black in parameter maps). Note that
the parameter maps based on different estimation approaches are almost
indistinguishable. Subtle differences can be seen, for example, in the upper
left region of the D maps. Adapted from Paper II, Figure 3 with data from
another patient
19
posterior distribution is given by:
P (θ|S) ∝ P (S|θ)P (θ) (9)
where P (θ|S) is the posterior distribution and P (θ) is the prior dis-
tribution [45].
Bayesian model fitting was early proposed as a robust alterative
to NLLS for IVIM parameter estimation [51]. However, the compu-
tational cost associated with Bayesian methods and perhaps also the
lower familiarity with such methods have resulted in limited use. Due
to the continued development of computer hardware, the computa-
tional cost is currently less of a problem, but the lack of availability of
simple Bayesian methods in standard software packages still limits the
use1. One reason for the high computational cost is that analytical
forms of the posterior distribution only exist for a few specific com-
binations of likelihood function and prior distribution, but in these
cases the Bayesian estimation problem is essentially equivalent to a
maximum likelihood estimation [52]. However, use of these specific
prior distributions is not necessarily the most preferable.
Prior distribution
The prior distribution is used to describe what can be known about
the model parameters without seeing the data of the specific exper-
iment. Such information is, for example, that some parameters are
nonnegative or even constrained to a closed interval, such as f that is
defined to be in the range 0 – 1. However, one may also include less
well defined information such as a belief that D is typically around 1
µm2/ms. By doing so, the parameter estimates may become less sus-
ceptible to noise, but they may on the other hand also become biased.
Choice of prior distribution for Bayesian IVIM parameter estimation
has varied between studies [8, 51, 53–57]. The impact of this choice
has recently been studied both for prior distributions that act on each
voxel independently (Paper I) as well as for prior distributions that
take into account information from other voxels [57]. The results in
both studies have shown substantial effects on parameter estimates
due to the choice of prior distribution and also emphasize the need to
challenge proposed methods by testing them in new situations.
1To partially remedy this issue and enable the use of Bayesian methods to a
larger group, the MATLAB code used for Bayesian IVIM model fitting in Paper I
has been published online:
www.mathworks.com/matlabcentral/fileexchange/65579-ivim-model-fitting
20
In Paper I, three prior distributions previously used for Bayesian
IVIM model fitting of the biexponential IVIM model were compared.
The three prior distributions were all uniform for f and S0, but either
uniform [8, 56], reciprocal [51] or lognormal [53] for D and D∗. The
uniform distribution implies that any given interval of parameter val-
ues is equally likely on a linear scale. For example, a uniform prior
gives equal probability for values in the range 1 - 2 as for values in
the range 2 - 3. The reciprocal distribution has the equivalent mean-
ing on a log-scale. This means that the reciprocal prior, for example,
gives equal probability for values in the range 1 - 10 as for values in
the range 10 - 100. The lognormal priors used in Paper I were nor-
mal distributions on the log-scale with relatively high variance. All
investigated priors can therefore be described as non-informative or
weakly informative since no or only very little information is provided
by the priors. This is in contrast to, for example, a lognormal prior
with low variance implying a strong belief for a specific narrow range
of parameter values. The comparison in Paper I showed that the re-
ciprocal prior tended to dominate the posterior distribution at SNR
levels below 40 (Fig. 5). At SNR levels around 20, which are typical
levels for IVIM, at least in applications outside the brain, the param-
eter estimates were therefore strongly biased (Fig. 6). Worth noting
is that the first study on Bayesian IVIM model fitting used the recip-
rocal priors [51], but only tested for SNR levels of 40 or higher where
such a prior is less dominant and this negative effect was therefore
not observable. The other two prior distributions, i.e. uniform and
lognormal, produced more similar results except for estimates of D∗,
where the spuriously high values that are typically produced by noise
corruption were strongly suppressed by the use of lognormal priors.
The resulting D∗ parameter maps were of subjectively high quality al-
though it should be noted that it is unclear what the magnitude of the
bias on the obtained parameters was (Fig. 6). The conclusion from
Paper I was that the reciprocal prior needs unrealistically high SNR
to be useful, while the other two compared priors have comparable
performance. The lognormal prior necessitates choice of distribution
parameters (mean and variance) and is therefore somewhat more sub-
jective, but appears to have slightly lower demands on noise level for
estimation of D∗. The uniform prior on the other hand, has no distri-
bution parameters and may therefore be regarded as more objective,
but appears to be somewhat more susceptible to noise which is demon-
strated especially for D∗. The specific choice of prior distribution for
21
Figure 5: Posterior distribution for D based on simulated data with SNR
= 20 for different prior distributions (uniform, reciprocal or lognormal).
The posterior distribution is strongly dominated by the reciprocal prior,
but mainly unaffected by the other priors. The data was scaled to improve
visibility. Adapted from Paper I, Figure 2 with permission from John Wiley
& Sons, Inc.
Bayesian IVIM model fitting should be made with these aspects in
mind and the optimal choice typically depends on the specific con-
text.
Only the uniform prior was used in Paper II. Due to the reduced
flexibility of the sIVIM model compared with the biexponential IVIM
model, it was believed that the choice of prior distribution would be
less important.
Measures of central tendency
To arrive in a parameter estimate, the posterior distribution must be
summarized by some central tendency measure. A simple approach
is to find the parameter values that maximize the joint posterior dis-
tribution similar to maximum likelihood estimation [52]. However,
for Bayesian IVIM parameter estimation it has been more common
to find the marginal posterior distributions, i.e. P (D|S), P (f |S) and
P (D∗|S). From these marginal posterior distributions, parameter es-
timates have been obtained by calculation of the mean [53, 55] or the
mode, i.e. the parameter value where the marginal posterior distribu-
tion has its maximum [8, 51].
In Papers I and II parameter estimates based on the mean or the
mode were compared for the biexponential IVIM model (Eq. 3) and
the sIVIM model (Eq. 4), respectively. The results showed that the
choice of central tendency measure has an impact on the resulting pa-
rameter estimates, but that it is of limited magnitude (Figs. 4 and 6).
The most prominent differences could be seen when the true parame-
22
Figure 6: IVIM parameter maps (color) of a neuroendocrine tumor of the
GOT1 tumor model [58] overlaid on the b = 0 image (grayscale). The param-
eter maps were obtained with different combinations of prior distributions
(uniform, reciprocal or lognormal on D and D∗, and uniform on f and S0)
and central tendency measures (mean or mode). The reciprocal prior distri-
bution produced parameter maps that were strongly affected by the prior.
Adapted from Paper I, Figure 3 with permission from John Wiley & Sons,
Inc.
ter value was close to a parameter boundary such as for small values
of f . In those cases, the posterior distribution is often strongly skewed
which gives a noticeable difference between the mean and the mode,
and the typical result is a positive bias of parameter estimates based
on the mean. In Paper I it was also shown that for the biexponential
IVIM model, the choice of central tendency measure is not as critical
as the choice of prior distribution, but still may impact the results in
some specific cases, e.g. estimation of D∗ with a uniform or lognormal
prior (Fig. 6).
23
24
Optimization of b-value
schemes for IVIM
This chapter is mainly related to Paper III.
In Paper III, a framework for optimization of b-value schemes was pre-
sented and evaluated. The framework was used to create a b-value scheme
for examination of healthy liver, which was compared with linearly distributed
b-values.
A key aspect of an IVIM experiment is the choice of b-values. The
optimal choice depends of factors such as specific model, expected
parameter values, noise level and purpose of analysis. A commonly
used strategy is to define an objective measure to describe how good
a b-value scheme is and then maximize this measure by finding the
optimal b-values. The definition should take into account all aspects
that are considered important and combine them into a single figure of
merit. The way that the goodness of a b-value scheme is summarized
into a single number is typically referred to as the objective function.
Objective function
The objective functions used for experiment design, such as choice of
b-value, are typically based on some measure of estimation variability
for the parameters of interest. Note that in such case, the objec-
tive function should be minimized rather than maximized in order to
obtain a b-value scheme that minimizes the estimation variability. A
simple objective function for analysis of the biexponential IVIM model
25
could be:
O =
∑
p∈{D,f,D∗}
cpσp (10)
where σp is the estimation variability of the parameter p and cp is a
scaling constant used to set the σp’s on the same scale. A common
choice is to set cp = 1/p, which turns Equation 10 into a sum of
coefficients of variation if the σp’s are standard deviations, but the
cp’s can in principle be set to any values of choice. Although S0 is a
parameter of the IVIM model it is usually not included in the sum in
an objective function like this since it is typically not of interest.
The exact type of estimation variability and how to combine them
when multiple model parameters are of interest such as in the case
of IVIM, is a subjective choice. In Paper III and some other studies
on optimization of b-value schemes for IVIM [12, 50, 59], a sum of
coefficients of variation (as in Eq. 10) was used, but other choices
are reasonable as well. Some examples, used for experiment design
in diffusion MRI, are weighted sums of estimation variance [11] and
median absolute percentage deviation [14]. As discussed in Paper
III, no measure can be considered generally superior, but the choice
will have an impact on the obtained b-value scheme. It is therefore
important to consider the strengths and weaknesses of the potential
measures when designing the objective function and make a choice
that fit well to the particular application of interest.
Most often it is not enough to provide knowledge about the model
and typical model parameter values to the objective function, but re-
strictions on the experiment parameter, i.e. the b-values, and indirect
effects related to them may also be needed. An important example of
such restrictions for IVIM is that b-values are nonnegative, and for the
sIVIM model it is also necessary to have both b = 0 and at least two
different b-values in a higher interval above the threshold. One may
also take into account the increased echo time, and thereby reduced
SNR, for higher b-values that is caused by limited gradient perfor-
mance as done in Paper III and previously by others [11, 13]. With
simplified models, some intervals of b-values may be poorly described
by the model. Examples are low b-values for the sIVIM model and
high b-values for all models with only a monoexponential term for the
diffusion compartment. Such unfavorable b-values should be avoided
in order not to introduce a potential estimation bias. Furthermore, if
there are any additional assumptions for the model to be valid, it is
important to design the optimization such that these assumptions are
26
not violated. It is, for example, often assumed that the SNR is suffi-
cient for a Gaussian noise approximation, which is not necessarily true
at high b-values. To take all this into account, one can perform a con-
strained optimization or add penalty terms to the objective function
such that b-value schemes that violate some assumption or restric-
tion are associated with a large penalty. Typically a combination of
both is used, i.e. constrained optimization of an objective function
with penalty terms. The exact implementation depends on what is
considered most suitable for the particular optimization.
To evaluate the objective function it is necessary to be able to cal-
culate all of its components. There exist two general ways of obtaining
these values; either by expressions derived from theory, which are fast
and relatively simple to calculate, or by experiment-like approaches
such as simulations, which are slower, but more closely related to the
actual measurement procedure.
Expression-based approaches
Expression-based approaches typically use some theory to derive an
expression of the estimation variance based on the model, experiment
design and noise properties. For diffusion MRI, the most commonly
used approaches are error propagation and Cramer-Rao lower bound
[10–13, 59–63].
Error propagation
If closed form solutions exist for all model parameters of interest, it
is possible to use error propagation to find the variance of parameter
estimates. For independent measurements, the error propagation is
given by:
σ2θj =
N∑
i=1
σ2i
(
∂θj
∂Sbi
)2
(11)
where σ2θj is the estimation variance of parameter θj and σ
2
i is the
variance of the noise of the ith measurement [45].
Error propagation has, for example, been used to find optimal b-
values for estimation of the apparent diffusion coefficient (ADC) based
on the monoexponential diffusion model [10, 61]. In Paper III, it is
shown that error propagation can be used to obtain optimal b-values
27
for estimation of D and f based on segmented model fitting and a
protocol containing three unique b-values.
Cramer-Rao lower bound
When the number of measurements is higher than the number of pa-
rameters in the model, parameter estimation is typically done by least
squares. In this case, the standard method for error propagation pre-
sented above cannot be used. Instead a similar method called Cramer-
Rao lower bound (CRLB) can be used. The CRLB’s are theoretical
lower bounds of the parameter estimation variances. Minimizing these
bounds has the potential to result in lower estimation variability. The
CRLB’s are given by the diagonal elements of the inverse Fisher matrix
as:
σ2θj ≥ (F )
−1
jj (12)
where the elements of the Fisher matrix are given by [45]:
Fjk = −E
[
∂2
∂θj∂θk
logL (θ)
]
(13)
If the likelihood function L(θ) is based on a Gaussian noise model
(Eq. 6), the expression simplifies to the following [11]:
Fjk =
1
σ2
N∑
i=1
∂Sbi(θ)
∂θj
∂Sbi(θ)
∂θk
(14)
CRLB based methods have been used for optimization of acquisi-
tion parameters for several applications of diffusion MRI, e.g., kurtosis
imaging [64], filter exchange imaging [13] and biexponential IVIM [12,
50, 59].
In Paper III, a CRLB-based method was used to find optimal b-
value schemes for estimation ofD and f based on segmented model fit-
ting, without the limitation of only three unique b-values. Specifically,
a b-value scheme designed for imaging of the liver was optimized. Al-
though additional b-values were allowed, the optimized b-value scheme
contained no more than three unique b-values. These three b-values
were instead repeated if more acquisitions were allowed. The optimal
ratio of repetitions was found to be approximately 1:2:2 for a scheme
containing b-values 0, 200 and 800 s/mm2. This should be contrasted
to the ratio 1:3 that is considered optimal for the monoexponential
diffusion model and b-values 0 and 1000 s/mm2 [10], and the ratio
28
1:2:2:1 for the biexponential IVIM model (Eq. 3) and b-values 0, 15,
100 and 1000 s/mm2 [65]. Optimal signal averaging is thus highly
dependent on the model that is used.
When compared with linearly distributed b-values, simulations in
Paper III showed that the estimation variability could be reduced by
about 20 % by the use of an optimized b-value scheme. These results
were confirmed in vivo using data from healthy volunteers (see example
in Figure 7). A critical issue when D and f are to be estimated is
to find the threshold b-value where the signal contribution from the
perfusion compartment can be considered negligible. Notably, the
results in Paper III showed that the decrease in estimation variability
could be seen regardless of if the threshold was chosen properly or not.
The observation in Paper III that the optimized b-value scheme
contained only three unique b-values is in line with what has been
reported previously by others [12, 59, 63], i.e. that optimization based
on CRLB tends to produce b-value sets where the number of unique b-
values equals the number of parameters included in the model. If three
b-values were considered sufficient for the estimation of D and f based
on segmented model fitting, closed form expressions exist for both
parameters, and it would thus be possible to use optimization methods
based on error propagation. The result would be simpler expressions
for estimation variability, which can be interpreted more easily, and
less complicated optimization since the number of free parameters is
reduced.
Experiment-like approaches
The major drawback of expression-based approaches for experiment
optimization is that some aspects can be hard to incorporate into a
closed form expression. A more straight-forward approach is to com-
pare direct measures of the quality of results from measurements done
with different experiment design. However, if these experiments would
include actual MR scanning only a very limited number of designs can
be tested [12]. A faster and cheaper way to generate data is to do com-
puter simulations. Simulation-based approaches are attractive in the
sense that they are easily understood and closely resemble the actual
experiment that is to be optimized. To obtain the error terms included
in the objective function, two steps are used: generation of noisy data
based on some signal model, noise distribution and experiment design,
and parameter estimation based on the noisy data with the model of
29
choice, which is not necessarily the same as was used for data gener-
ation. This simplicity is in contrast to expression-based approaches
where the effect of these two steps must be described explicitly.
Simulation-based approaches have been used in IVIM applications
to find optimal b-values for the biexponential IVIM model in various
organs [7] and to find an optimal intermediate b-value in a three-b-
value examination used to estimate D and f [14, 15]. However, the
computational cost is significant to evaluate the objective function,
especially if closed form expressions for parameters do not exist. This
has resulted in somewhat simplified optimization approaches. To re-
duce the number of free parameters in the optimization Lemke et al.
chose to build the b-value schemes incrementally by adding additional
b-values one-by-one [7]. Doing so may produce b-value schemes that
are suboptimal as a whole, since the added b-value is only optimal
given the already fixed b-values. While et al. and Meeus et al. chose
to only optimize the intermediate b-value, which enabled an exhaus-
tive search, but the upper b-value was fixed in the optimization [14,
15]. The obtained results are therefore dependent on the preset upper
b-value. The use of simulation-based approaches for experiment de-
sign is interesting and should be further studied, but there is a need
for improvements on computational efficiency or availability of more
computational power such that fewer simplifications need to be made.
30
Figure 7: Voxelwise mean and standard deviation (std) of IVIM parameter
maps from four repeated examinations of a healthy volunteer. The median
std in the right part of the liver (colored region) with an optimized/linear
b-value scheme was 0.115/0.147 µm2/ms and 5.1/5.8 % for D and f , re-
spectively. The variability of D and f was thus reduced by 22 and 12 %,
respectively, by the use of the optimized b-value scheme in this example.
Adapted from Paper III, Figure S14
31
32
Analysis of IVIM tumor
parameter maps
This chapter is mainly related to Paper IV and partly to Paper II.
In Paper II, the ability of D and f to differentiate between tumor and
healthy liver was studied.
In Paper IV, clustering based on Gaussian mixture models with IVIM
parameter maps as input were used to identify tumor subregions. The iden-
tified subregions were compared with maps of proliferative activity derived
from histological analysis.
Quantitative imaging, such as IVIM, shows great potential for tumor
characterization [66]. By mapping physiologically relevant parameters
it is possible to gain information on potentially interesting spatial vari-
ations within the tumor. Even so, most studies on tumors utilizing
quantitative imaging only analyze parameter values that are averaged
across the entire tumor [17]. To account for intratumor heterogeneity
a couple of methods have been developed. These are mainly based
on either describing the distribution of parameter values within the
tumor, i.e. histogram or texture analysis, or partitioning the tumor
into smaller regions, which are analyzed separately.
Methods for intratumor parameter analysis
Histogram and texture analysis
Histogram and texture analysis are methods based on extracting math-
ematical descriptions of the distribution of data. In histogram analysis
no spatial information is considered, instead all voxel data from a tu-
mor is pooled and a number of descriptives are used to summarize the
33
distribution of values. Typical descriptives include measures of cen-
tral tendency (mean, median, mode), spread (variance, interquartile
range), higher order metrics (skewness, kurtosis) and others (quan-
tiles). In texture analysis, the variation between neighboring voxels
is considered. Typical descriptives in this case are local and angular
versions of correlation and entropy.
Histogram and texture analysis has shown great promise in many
applications [67]. They are easy to understand methodologically and
straight-forward to apply. However, higher order descriptives such
as skewness and kurtosis are not obviously translated to a specific
biological state, thereby resulting in a lack of interpretability [67, 68].
Region-based analysis
Instead of describing the distribution of parameter values within the
tumor, one can choose to divide the tumor into subregions such that
each subregion is relatively homogeneous and easy to describe [17].
One such alternative is to divide the tumor based on prior assump-
tions on its structure. A number of concentric ring-shaped regions with
different distance from the center of the tumor can, for example, be
defined [69, 70]. This approach can give insight to tumor physiology
in different parts and provides a method that is easy to understand
and describe. However, unless the tumor growth and development is
purely radial, some degree of lost sensitivity due to averaging is inher-
ent. Mismatch between rings and actual borders between functional
regions is also a potential issue since statistics derived from nearby
rings may become highly correlated, which complicates subsequent
analysis [70].
Another alternative is to divide the tumor based on prior assump-
tions on the parameter values [71], for example, by setting some thresh-
old parameter value to divide voxels into two groups. A similar ap-
proach, which is somewhat more data-driven, is to base such a thresh-
old on the distribution of data, for example, a specific quantile. It
is, however, less obvious why such a threshold would be biologically
relevant.
A more fully data-driven approach is to partition the voxels within
the tumor based on some kind of clustering [72]. Such an approach
avoids the need for predefined thresholds or metrics. It can handle
fairly complicated distributions and can in many cases easily be trans-
lated into a multiparametric situation in contrast to simple threshold-
34
based approaches where multidimensional thresholds may be non-
trivial to define. It should, however, be noted that this kind of ap-
proach, where identification of tumor subregions is derived from the
data itself, is primarily applicable to studies of new therapies, tumors,
MRI methods or others, with relatively small sample sizes and with
little or no a priori knowledge available. For a clinical setting where
single patients are of interest, e.g., to provide personalized treatment
evaluation, it is likely better to have predefined classification models
trained and validated on large cohorts.
Cluster-based methods
The typical approach for cluster-based methods for identification of
tumor subregions is to pool all voxel data from all tumors in all sub-
jects and then perform a cluster analysis on the complete data set, as
illustrated in Figure 8. The clustering result can then be transformed
back to the individual tumors by identifying the cluster membership
of each individual voxel.
Most studies aiming towards identification of tumor subregions
based on clustering have used either k-means clustering or Gaussian
mixture models. The k-means clustering is a hard clustering, i.e. vox-
els are only allowed to belong to a single cluster. k-means clustering
is computationally efficient and the most commonly used clustering
algorithm for identification of tumor subregions [72–80], but can be
hampered if the distribution of data does not show very distinct clus-
ters. In some recent studies, Gaussian mixture models (GMM) have
been used [81–84]. GMM is, in contrast to k-means, a soft clustering
algorithm, which means that voxels are assigned probabilities of be-
longing to each cluster. The probability for voxel i to belong to cluster
k is given by:
P (xi = k|yi) =
P (yi|xi = k)P (xi = k)
∑
k P (yi|xi = k)P (xi = k)
(15)
where yi is a vector of data, that is D, f and D∗ for data from the
biexponential IVIM model, and xi is the (unknown) class of voxel i
[85].
P (yi|xi = k) =
1
√
2pid|Σk|
exp
[
−
1
2
(yi − µk)
T Σ−1k (yi − µk)
]
(16)
35
where µk and Σk are the mean vector and covariance matrix of cluster
k, respectively, d is the number of dimensions, i.e. three when yi
contains D, f and D∗, and T indicates transpose [85]. P (xi = k) = φk
is the prior probability of belonging to cluster k. GMM can be used
in cases of overlapping clusters and naturally handles parameters that
are on different scales.
Much attention has been focused on clustering based on vascular
properties by the use of parameters or time courses from DCE MRI
[75–81, 84, 86, 87]. However, some studies have also used ADC and T2
[72–74, 78, 88]. Both diffusion and perfusion MRI can provide infor-
mation important for tumor evaluation [89]. Since the two techniques
relate to fundamentally different biological processes, their informa-
tion is likely complementary and therefore highly suitable as input to
a multidimensional clustering algorithm. One example of the comple-
mentary nature of the diffusion and perfusion information was shown
in Paper II where differentiation between tumor and healthy liver was
substantially improved by the combined use ofD and f compared with
use of D or f alone. IVIM is a strong candidate for providing diffusion
and perfusion information to a clustering algorithm, not only because
there is no need for an exogenous contrast agent, but also since only
a single imaging sequence is used. This means that the different pa-
rameter maps for a given tumor are all in the same geometrical space,
which is needed for the clustering analysis. On the other hand, if,
for example, DCE parameters were to be combined with, e.g., ADC,
a nontrivial co-registration would be needed. To further complement
the IVIM parameters, a possibility could be to use T2-extended IVIM
[34]. In addition to D, f and D∗, it also provides T2 relaxation times
for both compartments and compensates for relaxation effects on f .
In paper IV, clustering was performed with the GMM algorithm
and input data containing D, f and D∗ to obtain subregions in neu-
roendocrine tumors of the GOT1 tumor model [58]. Based on the ap-
parent good quality of parameter maps obtained from Bayesian IVIM
model fitting with lognormal prior parameter distributions seen in Pa-
per I, that approach for IVIM parameter estimation was chosen. To
better conform to distributions approximated by a sum of Gaussian
distributions, the log transform was applied to all IVIM parameters
before clustering. This transformation is not permitted by all methods
for IVIM model fitting due to the possibility of parameter estimates
equal to zero, e.g. f obtained from segmented fitting. However, in Pa-
per IV it served as an important preprocessing step. When the trans-
36
Common histogram
GMM fit overlaid
Individual histograms ...
Parameter maps ...
Tumors
Cluster maps ...
Figure 8: Flowchart showing how tumor subregions can be identified by
Gaussian mixture modeling (GMM) based on data from parameter maps.
The parameter data from each tumor is pooled to form a common parameter
distribution. By fitting a GMM to the common parameter distribution the
probability of belonging to a given class can be calculated for each voxel,
which can be used to obtain maps of the clustering results. Adapted from
Paper IV
37
formed IVIM parameter distributions were displayed in histograms,
two relatively distinct clusters could be seen which both were well
approximated by Gaussian distributions. This was confirmed by the
cluster analysis which regarded two as the optimal number of classes.
That number is specific for this study and in future work it is impor-
tant to determine it for each specific situation.
It has previously been observed that inclusion of information from
neighboring voxels can enhance the quality of the tumor subregions
obtained from clustering [78]. Therefore, cluster analysis with a ver-
sion of GMM that was altered to included information from neigh-
boring voxels was also conducted in Paper IV. However, this only
improved the quality marginally while the complexity of the analysis
and computation time increased substantially. Whether spatial infor-
mation should be included in the clustering or not is thus still an open
question where the optimal choice likely depends on several factors
including data quality, imaging techniques and tumor type.
Validation of clustering
When cluster-based methods are used it is important to note that they
are solely driven by the data that is provided and the resulting subre-
gions do not necessarily have any biological relevance. It is therefore
necessary to evaluate the resulting subregions relative to some other
aspect that is considered relevant for the particular tumor and situa-
tion. In most previous studies, the clustering results have either been
related to some tissue characteristics from histological analysis [72, 73,
78] or evaluated as a predictor or measure for treatment effect [74, 87,
88].
Comparison of subregions obtained from cluster analysis with tis-
sue characteristics derived from histological analysis provides a pos-
sibility to get a biological understanding of the obtained subregions.
Since the characteristics of the clusters, i.e. their centers, are not
known before the clustering analysis, it is preferable if the specific
choice of histological parameters can be made after the clustering re-
sults are available. By doing so, it is possible to have hypotheses re-
garding the biological meaning of the clusters and choose staining tech-
niques to obtain relevant histological parameters accordingly. More-
over, ideally the clusters should be compared with combinations of
histological parameters or some histological parameter that is not di-
rectly comparable with a single MR parameter. The reasoning behind
38
this strategy is that the clustering is based on combinations of MR
derived parameters and therefore the histological counterpart should
be chosen to reflect the clustering results rather than any individual
MR derived parameter. One such example that has been employed
several times is to classify regions as viable tumor or necrosis based
on hematoxylin and eosin staining, sometimes along with other com-
plementary stainings, to compare with the subregions obtained from
cluster analysis [72, 73, 78].
In Paper IV, the two obtained clusters were characterized by ei-
ther high perfusion and low diffusion or low perfusion and somewhat
higher diffusion. Based on the assumptions that the level of perfusion
indicates availability of oxygen and nutrients and that measures of tis-
sue water diffusion relates to cell density, it was hypothesized that the
obtained subregions had different levels of proliferative activity. To
assess the proliferative activity, index maps were derived from histo-
logical sections subject to Ki-67 staining. When the cluster maps were
compared with the proliferation index maps a high degree of agree-
ment could be seen, both regarding size and spatial similarity. The
results thus confirm that IVIM based clustering has the potential to
provide biologically relevant tumor subregions, but also more gener-
ally that IVIM parameters contain information potentially important
for tumor tissue evaluation.
If the tumors are imaged at multiple time points, which are all in-
cluded in the cluster analysis, it is possible to follow the development of
subregions over time [76]. This could, for example, be of interest after
some treatment has been given. It is then possible to extract longitu-
dinal information such as size, average parameter value or even more
complicated metrics from each subregion separately. The extracted
information can then potentially be used as a measure or predictor
for treatment outcome. Such a comparison with treatment effect is
often the objective of a study, but ideally a histological analysis is also
performed in order to improve the interpretability of the results [72,
73, 76, 88].
39
40
Conclusions
This thesis provides several methodological improvements related to
intravoxel incoherent motion (IVIM) analysis. Specifically, it was
shown that:
• choice of prior distribution and central tendency measure influ-
ence the performance of Bayesian IVIM model fitting and can
have a large impact on the parameter estimates (Paper I). For
analysis restricted to the parameters D and f the segmented
IVIM model fitting is preferable (Paper II)
• optimization of b-value schemes has the ability to reduce esti-
mation uncertainty of D and f substantially. Optimization of
b-value schemes was enabled by development of a framework spe-
cific for estimation of D and f . The obtained optimized b-value
schemes contained three unique b-values (Paper III)
• clustering using Gaussian mixture models based on IVIM param-
eter maps is able to identify tumor subregions, which correspond
well with proliferative activity (Paper IV)
Altogether, the results of this thesis emphasize the importance of op-
timization of measurement and analysis methods for IVIM, and show
that IVIM has a great potential to provide imaging biomarkers in e.g.
cancer imaging.
41
42
Future aspects
IVIM is currently transitioning into clinical practice with analysis tools
becoming available on several commercial platforms [90, 91]. How-
ever, if IVIM is to become a routine procedure in the clinic, consensus
on imaging protocols and parameter estimation approaches is needed.
Otherwise, interpretation and comparisons of parameter maps may
be obscured by methodological differences. The results of this thesis
and other work on IVIM methodology can serve as a basis for such a
consensus.
During the last years, major methodological extensions have been
proposed for IVIM, both regarding modeling and methods for diffusion
encoding. To obtain more information and to better describe the dif-
fusion weighted signal, it has been suggested to use an extended model
for the diffusion compartment [92]. In that particular study, the kur-
tosis model was used, but any model that describes the curvature of
the signal-vs-b curve would in principle apply [93]. More advanced
methods for diffusion encoding, such as variable flow encoding, have
also been proposed to better characterize the perfusion compartment
[18, 20]. Including IVIM in the paradigm of multidimensional diffu-
sion MRI has the potential to improve information output as well as
robustness for IVIM analysis [94]. However, the increased complexity
regarding both modeling and diffusion encoding necessitates studies
on what approaches that actually provide important additional infor-
mation and under which conditions. Optimization of acquisition pa-
rameters will be required and development of specialized estimation
approaches may also be needed.
Future efforts should also be aimed at improved understanding of
the origin of the IVIM signal. The physical origin can, for example,
be studied by the use of complementary MRI techniques, while the
biological interpretation can be strengthened by comparison with, for
example, histological samples. Interesting results were shown in a re-
43
cent study where ASL with variable post labeling delay was combined
with diffusion weighting to study how the blood signal is affected by
diffusion weighting at different levels in the vessel tree [95]. In an-
other study, the perfusion fraction was shown to be correlated with
microvessel density as derived from histological analysis [96]. With the
results of studies like these an increased understanding of the origin of
the signal can be gained and the relation of IVIM to other perfusion
techniques can be better understood. This can help in future devel-
opment of the IVIM technique and to determine in which applications
IVIM will give clinically relevant information.
44
Acknowledgements
During this time I have had help and support by many. I would like
to express my gratitude especially to:
Maria Ljungberg, my supervisor. Thank you for letting me explore
my ideas and for carefully getting me back in the right direction when
I got lost. Your considerate mentorship has helped me many times
Göran Starck, my co-supervisor. Thank you for challenging my
ideas and for sharing your long-lasting experience of MR
Mikael Montelius and Jonathan Arvidsson, my roommates since
the start. Thank you for all discussions about science, being a PhD
student and everything else
Other co-authors, especially Ylva Lilja and Mats Andersson. Thank
you for your contributions. It has been a pleasure to work with you
Former and current members of the MR physics group: Kerstin,
Maja, Åsa, Linnéa, Stefanie, Frida, Yosef, Daniel, Christian, Nicolas,
Jens, Christian, Emilia, Lukas. Thank you for filling our little white
house with joy
Radiographers and radiologist at the MR department. Thank you
for creating a nice atmosphere, always sharing your skills and for mak-
ing me, as an MR physicist, feel valuable
MR administration and animal handling personnel. Thank you for
your help in all the practical issues
Fellow PhD students and permanent staff at the department of
Radiation Physics, and everyone else in the building. Thank you for
always making me feel welcome
Emma, my wife. Thank you for your endless encouragements and
support
45
The work presented in this thesis was supported by grants from the
Swedish Cancer Society, the King Gustav V Jubilee Clinic Cancer
Research Foundation, the Swedish Research Council and the Assar
Gabrielsson Cancer Research Foundation.
46
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