Bastien Milani^{1}, Jean-Baptiste Ledoux^{2}, and Menno Pruijm^{3}

We explain why diffusion MRI should always be acquired when blood has maximum velocity in the organ of interest. We first give some theoretical arguments to support this hypothesis. We then demonstrate it with numerical experiments and with in-vivo experiments.

The IVIM model, proposed by Le Bihan in 1988, suggests that the trace-weighted images measured in a diffusion-MRI experiment follows a bi-exponential decay as a function of the b-values:

I_{trace}(b) = PF·e^{-b·Dpseudo} + (1-PF)·e^{-b·Dpure} (1)

A voxel-by-voxel fitting procedure yields three coefficients characterizing each voxel: the perfusion fraction (PF) which corresponds to the volume fraction (between 0 and 1) occupied by blood in the voxel, Dpseudo which is the pseudo-diffusion-coefficient of blood related to blood-velocity in capillaries, and Dpure which is the diffusion-coefficient of water protons diffusing passively in the tissues. This model has been intensively used to quantify the perfusion fraction in different organs with different protocols. However, results are very heterogeneous and lack reproducibility. No standard values for the perfusion fraction has therefore been accepted yet.

Fitting data to a multi-exponential model is
known to be an ill-posed problem which is exacerbated by noise ^{1}. For bi-exponential models such as in Eq. (1),
it has been demonstrated that the ratio of the two decay-constants (i.e. D_{pseudo}/D_{pure})
must exceed a certain noise-dependent threshold to correctly measure these
parameters and, conversely, that no algorithm can lead to any reliable results
if this ratio is not large enough ^{1}. This result implies that the blood velocity
(related to D_{pseudo})
must be maximized during the image acquisition to obtain more accurate
measurements. In this study, we numerically and experimentally tested this
hypothesis for the application of kidney diffusion-MRI.

Numerical experiment: a least square
fitting procedure following a bi-exponential model was applied to a simulated
decay curve with realistic parameters (PF = 0.2, Dpure = 2·10^{-3}
mm^{2}/s) and additive noises with an amplitude of 7% (in agreement
with our MRI measurements). The experiment was repeated 10’000 times for
different values of D_{pseudo}: from 4·10^{-3} to 22·10^{-3}
mm^{2}/s in steps of 2·10^{-3} mm^{2}/s. This lead to 10’000 simulated PF-measurements
for each of ten different decay-constant-ratios (D_{pseudo}/D_{pure}) that ranged from 2
to 11. To compare the ten distributions of the PF measurements, the histograms
were quantified with three markers:

-the mean PF of all trials with values smaller that 0.6 (marker A),

-the percent of trials with PF smaller than 0.1 (marker B),

-the percent of trials with PF bigger than 0.6 (marker C).

In vivo experiment: 5 healthy volunteers and one
chronic kidney disease (ckd)-patient with solitary kidney (11 kidneys in total)
where scanned on a 3-T MRI scanner (Magnetom Prisma, Siemens Medical Systems, Erlangen,
Germany). The protocol was directly inspired by Wittsack et al^{2}. Phase-contrast flow imaging was performed on one
kidney-artery to determine the delays t_{max} and t_{min} after
the R-wave for which the blood velocity in the renal artery was maximal (v_{max})
resp. minimal (v_{min}). DWI was performed with electrocardiogram- and respiratory-gating
(liver-dome pencil-beam-navigator) on
one coronal plane and the following parameters: 10 b-values, bi-polar diffusion
gradients, orthogonal diffusion-directions, echo-planar readout, TE=65ms, TR=220ms, matrix=192x192,
FoV=400mm, 4 averages. Two different acquisitions were performed at the two
different time-points of the cardiac cycle i.e. t_{max} and t_{min}.
For
each pair of acquisitions, a radiologist with 20 years experience evaluated the
cortico-medullary-differentiation and quality of the PF-maps. The
PF-map-histogram of each kidney-parenchyma was quantified with the three
previously defined markers to assess the similarity with the numerical
experiment.

1.Istratov, A.A. and O.F. Vyvenko, Exponential analysis in physical phenomena. Review of Scientific Instruments, 1999. 70(2): p. 1233-1257.

2.Wittsack, H.J., et al., Temporally resolved electrocardiogram-triggered diffusion-weighted imaging of the human kidney: correlation between intravoxel incoherent motion parameters and renal blood flow at different time points of the cardiac cycle. Invest Radiol, 2012. 47(4): p. 226-30.

Three example of PF-histograms
produced by the bi-exponential fitting of simulated data with PF = 0.2. The
higher is the decay-constant-ratio D_{pseudo}/D_{pure}, the
more precise is the PF-measurement. We observe three pathological behavior of
the fit when this ratio decreases: the mean value of the trials with reliable PF
(<0.6) decreases, and there is a tendency of the PF to be stacked to very
small values (< 0.1) as well as very high value (> 0.6). We identify these
three pathologies with the measurement of the three previously defined markers.

Evaluation
of the three markers Marker A, Marker B and Marker C on the PF-histograms
corresponding to 10 different numerical experiments with ten different values
of the decay-constant-ratio.

Example
of the PF-maps of a healthy volunteer acquired at different blood velocities. A
and E are PF-map acquired at maximal blood-velocity. The corresponding
histograms are shown on B and F. C and G are PF-map acquired at minimal blood-velocity.
The corresponding histograms are shown on D and H.

Example
of the PF-maps of a CKD-patient with solitary kidney acquired at different
blood velocities. A is the PF-map acquired at maximal blood-velocity. The
corresponding histogram is shown on B. C is the PF-map acquired at minimal
blood-velocity. The corresponding histogram is shown on D. We note the strong
difference of perfusion fraction between cortex and medulla appearing on A.
This difference disappears totally on C.

Evaluation
of the three markers Marker A, Marker B and Marker C on the PF-histograms of
the 11 kidney-parenchymas. We observe an agreement with the numerical
experiments: marker A has a higher value when blood velocity is maximal (i.e.
when D_{pseudo}/D_{pure} is maximal) and marker B and C have a
higher value when blood velocity is minimal (i.e. when D_{pseudo}/D_{pure}
is minimal).