Complex models and computational methods in statistics

Estimation Approaches for the Apparent
Diffusion Coefficient in Rice-Distributed
MR Signals

Stefano Baraldo, Francesca Ieva, Luca Mainardi, and Anna Maria Paganoni

Abstract The Apparent Diffusion Coefficient (ADC) is often considered in the
differential diagnosis of tumors, since the analysis of a field of ADCs on a particular
region of the body allows to identify regional necrosis. This quantity can be
estimated from magnitude signals obtained in diffusion Magnetic Resonance (MR),
but in some situations, like total body MRs, it is possible to repeat only few
measurements on the same patient, thus providing a limited amount of data for
the estimation of ADCs. In this work we consider a Rician distributed magnitude
signal with an exponential dependence on the so-called b-value. Different pixelwise
estimators for the ADC, both frequentist and Bayesian, are proposed and compared
by a simulation study, focusing on issues caused by low signal-to-noise ratios and
small sample sizes.

1 Introduction

Diffusion magnetic resonance (MR) is as an important tool in clinical research, as
it allows to characterize some properties of biological tissues. When tumor areas
are analyzed using this technique, it can be observed that the diffusion tensor,
estimated from the magnetic MR magnitude signal, has reduced values in lesions
with respect to surrounding physiological tissues, allowing to identify pathological
areas or necrosis. When the tissue region of interest can be considered as isotropic
the Apparent Diffusion Coefficient (ADC) is sufficient to characterize the diffusion

S. Baraldo ( ) F. Ieva A.M. Paganoni
MOX - Department of Mathematics, Politecnico di Milano, Milan, Italy
e-mail: stefano1.baraldo@mail.polimi.it; francesca.ieva@mail.polimi.it;
anna.paganoni@polimi.it
L. Mainardi
Department of Bioengineering, Politecnico di Milano, Milan, Italy
e-mail: luca.mainardi@polimi.it

M. Grigoletto et al. (eds.), Complex Models and Computational Methods in Statistics, 13
Contributions to Statistics, DOI 10.1007/978-88-470-2871-5 2,
© Springer-Verlag Italia 2013

14 S. Baraldo et al.

properties of the tissue, and it is usually estimated from the exponential decay of the
signal with respect to the b-value, the MR acquisition parameter. The assumption of
isotropy is common and reasonable in various cases, like breast and prostate cancer
(see, for example, [7, 10]).
In many practical situations it may not be possible to collect more than few
measures at different b-values, limiting the accuracy of the estimation. A reduction
in the total number of measures necessary to achieve a certain accuracy is convenient
in term of costs and allows to keep the patient involved in the MR procedure for a
shorter amount of time (the experience may be unpleasant, especially when total
body MR must be performed). The purpose of this work is to compare different
frequentist and Bayesian approaches to the estimation of the ADC, underlining their
statistical properties and computational issues.

2 Rice-Distributed Diffusion MR Signals

2.1 The Rice Distribution

The random variables we deal with derive from the complex signal w D wr C iwi
measured in diffusion MR. It is usual to assume that both wr and wi are affected
by a Gaussian noise with equal, constant variance, i.e. wr N. cos.#/; 2 / and
2 C
wi N. sin.#/; /, with 2 R and # 2 Œ0; 2 /. The quantity at hand is the
modulus M of this signal, which has then a Rice (or Rician) distribution, that we
will denote as M Rice. ; 2 /. The density of this random variable has the form

2 m m2 C 2 m Á
fM .mj ; /D 2
e 2 2 I0 2
I.0;C1/ .m/; (1)

where I0 is the zeroth-order modified Bessel function of the first kind (see [1]).
Using the series expression of I0 , it is possible p deduce a different, equivalent
to
definition of a Rician random variable as M D R, where R is a noncentral 2
2
variable that can be expressed as a mixture of .2P C 2/ distributions with P
Poisson. 2 =2 2 /. This formulation becomes particularly useful for sampling from
a Rice distribution, as it allows an easy implementation of a Gibbs sampler.

2.2 Rice Exponential Regression

Diffusion MR aims at computing the diffusion tensor field on a portion of tissue,
and this is achieved by analyzing the influence of water diffusion on the measured
signal, under different experimental settings. In particular, the classical model for
relating the magnitude signal to the acquisition parameters and the 3-dimensional
diffusion tensor D is the Stejskal–Tanner equation

Estimation Approaches for ADC in MR Signals 15

g D 0 exp. gT Dgb/; (2)

where g is the “real” intensity signal we want to measure, 0 is the signal at b D 0
and the vector g 2 R3 is the applied magnetic gradient. The b-value is a function of
other acquisition settings, which we will omit since their description and discussion
is beyond the scope of this article. See, for example, [3] for an overview on MR
techniques, including diffusion MR, and a discussion of various issues and recent
advances in this field.
In general, even in the ideal noiseless case, at least six observations are needed
to determine the components of the symmetric, positive definite diffusion tensor
D, by varying the direction g of the magnetic field gradient. However, if the tissue
under study can be considered as isotropic, the diffusion tensor has the simpler form
D D ˛I , where ˛ is the ADC, a scalar parameter, and I is the identity matrix. This
reduces model (2) to the following

D 0 exp. ˛b/ (3)

for any vector g (in the following, we will omit it for ease of notation).
Equation (3) describes pointwise the phenomenon on the tissue region of interest.
In this study we consider the pixels of a diffusion MR sequence of images as
independent and focus on the estimation problem for a single point in space. We
do not consider a spatial modeling for the ADC field: although it could be a useful
way to filter noise and to capture underlying tissue structures; on the other hand, for
diagnostic purposes it may be preferable to submit to the physician an estimate that
has not been artificially smoothed.

3 Estimation Methods

In this section we present different methods for the estimation of ˛, the unknown
parameter of interest. We consider a sample of signal intensities on a single
pixel Mi Rice. 0 e ˛bi ; 2 /, i D 1; : : : ; n, and their respective realizations m D
m1 ; : : : ; mn at b-values b D b1 ; : : : ; bn .
The dispersion parameter 2 is usually measured over regions where almost pure
noise is observed, and used as a known parameter in the subsequent estimates. This
estimate of 2 is considered as reliable, since it can be based on a very large number
of pixels, so we will consider the case of known dispersion parameter.
We consider nonlinear least squares, maximum likelihood and three Bayesian
point estimators. In the case of a simple Rice. ; 2 / random variable an iterative
method of moments estimator has been proposed in [2], but this technique has
no straightforward extension to the case of covariate-dependent , while moment
equations would be difficult to invert in the considered case. Moreover, under
the model assumptions presented in Sect. 2 a decoupling of noise and signal in


the fashion of signal detection theory could not be pursued (see [4] for a brief
presentation and a Bayesian implementation of the SDT classical scheme).
The different estimation methods for the couple . 0 ; ˛/ presented here will be
tested under different signal-to-noise ratios (SNRs) = in Sect. 4.

3.1 Nonlinear Regression

A standard approach for the estimation of 0 and ˛ is to solve a nonlinear least
squares problem, which is equivalent to approximating Mi D 0 exp. ˛bi / C "i for
i D 1; : : : ; n, where "i are iid, zero mean, Gaussian noise terms. The estimators O 0
LS

and ˛ are deﬁned as
O LS

n
X
˛bi 2
. O 0 ; ˛ LS / D argmin
LS
O .mi 0e / ;
. 0 ;˛/ i D1

for 0; ˛ > 0, which is equivalent to the solution of the following equations
( Pn Pn
2˛bi ˛bi
0 i D1 e D i D1 mi e ;
Pn 2˛bi
Pn ˛bi
(4)
0 i D1 bi e D i D1 m i bi e :

The approximation to a nonlinear regression model is inconsistent with the phe-
nomenon under study, most evidently for the fact that in this case the noise term is
symmetric and it can assume real values. This inconsistency is negligible for high
SNR values, since a Rice. ; 2 / distribution in this case approaches a N. 0 ; 2 /,
but becomes important with “intermediate” and low SNRs. In [9], the behavior of
the Rice distribution with ﬁxing D 1 and varying is examined, observing that
normality can be considered a good approximation at about = > 2:64, but the
sample variance approaches 2 only for SNR values greater than 5:19. Even for
pixels with high SNRs at b D 0, for large b-values the real signal could reach
the same order of magnitude of noise, depending on the unknown value of ˛, and
this could lead to very biased estimates. However, the least squares approach is
computationally simpler and quicker to carry out, since it can be seen from (4)
that 0 can be expressed as a function of ˛, thus requiring just a one-dimensional
optimization to compute the estimates.

3.2 Maximum Likelihood

The maximum likelihood approach allows to take into account the asymmetry of
the signal distribution, always providing admissible values of the parameters. The
objective function is the log-likelihood


n
X
2 2 ˛bi 2
l. 0 ; ˛jm; b; / D log L. 0 ; ˛jm; b; /D log fMi .mi j 0 e ; /
i D1
n n Ä Â Ã
1 X 2 2˛bi
X mi 0e
˛bi
/ 2 0e C log I0 2
;
2 i D1 i D1

where fMi is the Rice density (1), for i D 1; : : : ; n. The ML estimator is then
2
. O 0 ; ˛ ML / D argmax l. 0 ; ˛jm; b;
ML
O /;
. 0 ;˛/

for 0 ; ˛ > 0.
0
Looking for stationary points of l and using the fact that I0 .x/ D I1 .x/, we obtain
the following estimating equations
8 mi 0 e ˛bi
ˆ
ˆ Pn 2˛bi
Pn I1 . 2 /mi ˛bi
ˆ
< 0 i D1 e D i D1 mi 0 e ˛bi
e ;
I0 . 2 /
e ˛bi
ˆ
ˆ Pn P n
m
I1 . i 0 2 /mi
ˆ
: 0 i D1 bi e 2˛bi
D i D1 bi e ˛bi
:
mi 0 e ˛bi
I0 . 2 /

Notice that these score equations differ from (4) only for the Bessel functions ratios
˛bi ˛bi
I1 . mi 0 e2 /=I0 . mi 0 e2 /, which multiplies the observations mi . In particular, this
factor decreases the values of observations, since 0 < I1 .x/=I0 .x/ < 1 for x > 0,
and increases asymptotically to 1 for large SNRs, so that the score equations tend
to (4).
As shown in [8], the maximum likelihood estimator for obtained from
an iid sample M1 ; : : : ; Mn Rice. ; 2 / and known 2 becomes exactly 0
when the moment estimator for EŒM 2 D 2 C 2 2 becomes inadmissible, i.e. when
Pn 2
i D1 Mi =n 2 2 Ä 0, even if the real value of is larger than 0. The case of
Rice exponential regression suffers from a similar problem in a nontrivial way, and
would require 2 to be estimated with the other parameters to keep parameter values
coherent with the model. Here we will not address this problem, but efforts in this
direction are currently in progress.

3.3 Bayesian Approaches

We consider also three different estimators based on a Bayesian posterior distribu-
tion: its mean, its median, and its mode. To allow an easy implementation using
BUGS code, we introduce a slightly different formulation of the model.
If M Rice. ; 2 /, then R D M 2 = 2 has noncentral 2 distribution with 2
degrees of freedom and noncentrality parameter D 2 =.2 2 /. Be now R1 ; : : : ; Rn


the random sample considered, with Ri D Mi2 = 2 and Mi Rice. 0 e ˛bi ; 2 /
for i D 1; : : : ; n, and let r D .r1 ; : : : ; rn / be the observations from this sample. Let
. 0 / and .˛/ be the prior distributions of the two unknown parameters, while the
2
density of each Ri will be denoted as fRi .ri /, with parameter i D 0 e 2˛bi =2 2 .
The joint posterior distribution of 0 and ˛ is then
n
Y
2
p. 0 ; ˛jr; b; // fRi .ri j i / . 0 / .˛/:
i D1

As anticipated in Sect. 2, a noncentral 2 distribution of noncentrality can be
sampled as a mixture of 2 .2P C 2/ with P P. /. This allows an easy BUGS
implementation of these estimators.

4 Simulation Study

We compared five estimators for ˛—least squares (LS), maximum likelihood (ML)
and posterior mean (PMe), median (PMd) and mode (PMo)—in terms of mean
and mean square error. For the two frequentist approaches, ranges for the possible
parameter values have been chosen, considering 0 2 Œ0:1; 10 and ˛ 2 Œ0:1; 5,
while the fixed parameter 2 has been taken always equal to 1. For the Bayesian
point estimators we chose uninformative, uniform priors, with the same support as
the ranges chosen for LS and ML. The first two estimators have been computed
with R 2.12.2 (see [6]), using built-in optimization functions: optimize for the
one-dimensional minimization required in LS and optim, using the L-BFGS-B
method, for the likelihood maximization, with startup values . 0start ; ˛start / D
.1; 1/. Bayesian posterior distributions have been computed using a Gibbs sampler
implemented in JAGS (see [5]). In particular, the following model code (valid for
any program supporting BUGS-type language) was used:
model f
for ( i in 1: n )f
l am bda [ i ]< ( nu0 nu0 ) exp ( 2 a l p h a b [ i ] ) / ( 2 si gm a si gm a )
p [ i ] ˜ d p o i s ( l am bda [ i ] )
k [ i ]< 2 p [ i ]+ 2
M[ i ] ˜ d c h i s q r ( k [ i ] )
g
alpha ˜ dunif (0 .1 ,5 )
nu0 ˜ d u n i f ( 0 . 1 , 1 0 )
g

As it can be seen from the model code, uniform prior distributions have been
chosen, with supports equal to the search ranges for LS and ML. 10,000 Gibbs
sampling iterations have been run for each different sample, with a thinning of 10,
and standard diagnostics revealed a good behavior of the generated chains.


We chose b-values in a typical range for diffusion MR machine settings, i.e.
from 0 to 1; 000 s=mm2 , on equally spaced grids of n D 5, 10, 15, 20, 25, 30
points. Different simulations have been run with parameter values 0 D 2; 4; 8,
which represent a low, an intermediate and a high SNR, and ˛ D 0:7; 1; 3, typical
low, intermediate, and high physiological values of ADC.
It must be reported that the ML estimator, in cases of low SNR, reached the
boundaries of the optimization region in various simulations. In the combination
n D 5, 0 D 2, ˛ D 3 only 45% of the simulations gave ML estimates that converged
to a value inside the predeﬁned ranges of parameters search, while in the other
cases this number oscillated around 70% when ˛ D 1 or 100% when ˛ D 0:7. These
degenerate results have been removed for the computation of bias and variance.
Figure 1 displays the decaying exponential curves we aim to estimate in the
nine different combinations of 0 and ˛, along with a horizontal line at level ,
to represent the order of magnitude of noise with respect to the signal. The quality
of estimates depends both on the SNR at b D 0 and on the ADC, as will be clear
from simulations.
Figure 2 shows the behavior of bias for the estimators of ˛ with different
sample sizes n. For what concerns the frequentist estimators (LS and ML), there
is no uniform ordering through the considered values of n when the signal decays
slowly (˛ D 0:7), but in the other cases, when noise is stronger along the curve, the
maximum likelihood estimate is always less biased than the least squares one; notice
also that the least squares estimates do not seem to have a decreasing bias when n
increases among the considered values. Concerning the three Bayesian estimators,
no striking differences arise among them, while with respect to the frequentist
estimators in many cases they have comparable or higher bias, with the exception
of the “worst case” 0 D 2, ˛ D 3, where they are uniformly more accurate.
From what concerns variance, analyzed in Fig. 3, the LS estimator shows almost
always the best performance, excepted for low sample sizes when ˛ D 3. The other
estimators have similar performances and behaviors at different sample sizes n, with
ML and PMe having strikingly higher variance in some noisy cases. As expected,
variance notably decreases for all estimators at increasing n in most combinations
of parameters, but with very low SNR ( 0 D 2) the only one showing empirical
convergence of variance to 0 is LS.
An overall index of estimator performance can be evaluated by the mean square
error (MSE). Since the MSE is the sum of square bias and variance, the orders of
magnitude of these two characteristics assume an important role. As it can be seen
from Fig. 4, the LS estimator has the lowest MSE when ˛ D 0:7; 1, but exhibits the
worst performances in the critical cases of high ADC, where Bayesian estimators
seem to work better.
Results for 0 are not detailed here, but it is worth mentioning that, since it is
necessary to estimate the two parameters jointly, the precisions and accuracies of
their estimators are mutually inﬂuenced. Anyway, estimators for 0 show a more
classical behavior: the LS estimator is in all cases less accurate but more precise
(high bias and low variance), and the consistency of all estimators is evident when
increasing n. The summary plots for the MSE of the estimators for 0 can be seen
in Fig. 5.


nu0=2, alpha=0.7 nu0=2, alpha=1 nu0=2, alpha=3
8 8 8

6 6 6

4 4 4

2 2 2

0 0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
b b b
8 8 8

6 6 6

4 4 4

2 2 2

0 0 0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
b b b
8 8 8

6 6 6

4 4 4

2 2 2

0 0 0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
b b b

Fig. 1 Stejskal–Tanner model in simulation parameter combinations. b-values are expressed in
1; 000 s/mm2


1

0 0
0

−5
−2
log(bias2)

log(bias2)

log(bias2)
−1

−10
−2 −4

−15
−3
−6

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
n n n


−2 0
−2

−4 −2
log(bias2)

log(bias2)

log(bias2)
−4
−4
−6
−6
−6

−8
−8 −8

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
n n n

−1
−6
−6
−2
−7
−8 −3
log(bias2)

log(bias2)

log(bias2)

−8
−4
−10
−9
−5
−10
−12 −6
−11
−7

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30

n n n

Fig. 2 Bias of estimators for ˛. Bold lines: solid D LS, dashed D ML; slim lines: solid D PMe,
dashed D PMd, dotted D PMo


0.5 0.5
0.5

0.0
0.0 0.0

−0.5
−0.5
−0.5

log(variance)

log(variance)
log(variance)

−1.0
−1.0
−1.0
−1.5
−1.5
−2.0 −1.5
−2.0
−2.5 −2.0
−2.5
−3.0
5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
n n n

−1.0 −0.5 0.5

−1.0 0.0
−1.5

−0.5
log(variance)

log(variance)

log(variance)
−1.5
−2.0

−1.0
−2.0
−2.5
−1.5
−2.5
−3.0
−2.0
−3.0

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
n n n

−3.0
−0.5
−3.2
−3.0
−3.4 −1.0
log(variance)

log(variance)

log(variance)

−3.6
−3.5
−3.8 −1.5

−4.0
−4.0 −2.0
−4.2

−4.4

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
n n n

Fig. 3 Variance of estimators for ˛. Bold lines: solid D LS, dashed D ML; slim lines: solid D
PMe, dashed D PMd, dotted D PMo



1.0
1.5
1.0
0.5
1.0

0.0 0.5 0.5
log(MSE)

log(MSE)

log(MSE)
−0.5 0.0
0.0
−1.0 −0.5
−0.5
−1.5 −1.0

−1.5
−2.0 −1.0

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
n n n

−0.5
1.0
−0.5
−1.0
0.5
−1.0
−1.5
log(MSE)

log(MSE)

log(MSE)
−1.5 0.0
−2.0

−2.5 −2.0 −0.5

−3.0 −2.5
−1.0
5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
n n n


−3.0

−0.5
−3.0

−3.5
log(MSE)

log(MSE)

log(MSE)

−1.0
−3.5

−4.0

−4.0 −1.5

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
n n n

Fig. 4 MSE of estimators for ˛. Bold lines: solid D LS, dashed D ML; slim lines: solid D PMe,


0.0 0.0
0.0

−0.5 −0.5
−0.5
log(MSE)

log(MSE)

log(MSE)
−1.0 −1.0
−1.0

−1.5 −1.5
−1.5

−2.0
5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
n n n

0.2

−0.5 0.0
−0.5
−0.2

log(MSE)
log(MSE)

log(MSE)

−0.4
−1.0 −1.0
−0.6

−0.8
−1.5 −1.5
−1.0

5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
n n n
−0.2
−0.4
−0.3
−0.6 −0.5
−0.4
−0.8
log(MSE)
log(MSE)

log(MSE)

−0.5
−1.0 −1.0
−0.6
−1.2
−0.7
−1.4 −1.5
−0.8
−1.6
−0.9
5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30
n n n

Fig. 5 MSE of estimators for 0 . Bold lines: solid D LS, dashed D ML; slim lines: solid D PMe,


5 Conclusions

In this work, we proposed different methods for estimating pixelwise the ADC
from diffusion MR signals, following the Rice noise model and the Stejskal–Tanner
equation for magnitude decay. The presented estimators exhibit different features
that should be taken into account when approaching real data. The least squares
approach is the fastest and has low variance, but becomes less accurate when the
conditional signal distribution at different b-values is more distant from normality.
The maximum likelihood estimator is slightly slower, requiring a nonlinear maxi-
mization on two variables, and has the lowest bias in many cases, but, as pointed
out before, it may diverge with samples from noisy signals. Bayesian estimators are
the most expensive in terms of computational costs and may require further tuning
for improving their performances; they are the best in terms of mean square error at
the high ADC here tested and offer the advantage of providing the whole posterior
distribution for inferential purposes, while inferential tools regarding LS and ML
should rely, at present time, on normal approximations, which may not be reliable
with low sample sizes and SNRs. Future studies will focus on inferential aspects,
while extending in efﬁcient ways these estimation methods to full MR images. The
simultaneous estimation of the dispersion parameter 2 will also be developed and
tested, requiring some added computational effort to estimation algorithms.

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