Magnetic Resonance Imaging 110 (2024) 35–42

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Magnetic Resonance Imaging
journal homepage: www.elsevier.com/locate/mri

A mathematical model for temporal cerebral blood flow response toacetazolamide evaluated in patients with Moyamoya disease
Markus Fahlström a, b, *, Joao M. Sousa a, b, Teodor Svedung Wettervik c, Johan Berglund a, b,
Per Enblad c, Anders Lewén c, Johan Wikström daMolecular Imaging and Medical Physics, Department of Surgical Sciences, Uppsala University, Uppsala, Sweden
Medical Physics, Uppsala University Hospital, Uppsala, Sweden
Neurosurgery, Department of Medical Sciences, Uppsala University, Uppsala, SwedendNeuroradiology, Department of Surgical Sciences, Uppsala University, Uppsala, Swedenbc

A R T I C L E I N F O

A B S T R A C T

Keywords:
Acetazolamide
Cerebral blood flow
Cerebrovascular reserve
Modelling
Moyamoya disease

Background: Paired cerebral blood flow (CBF) measurement is usually acquired before and after vasoactive
stimulus to estimate cerebrovascular reserve (CVR). However, CVR may be confounded because of variations intime-to-maximum CBF response (tmax) following acetazolamide injection. With a mathematical model, CVR canbe calculated insensitive to variations in tmax, and a model offers the possibility to calculate additional modelderived parameters. A model that describes the temporal CBF response following a vasodilating acetazolamide
injection is proposed and evaluated. Methods: A bi-exponential model was adopted and fitted to four CBF
measurements acquired using arterial spin labelling before and initialised at 5, 15 and 25 min after acetazol­
amide injection in a total of fifteen patients with Moyamoya disease. Curve fitting was performed using a nonlinear least squares method with a priori constraints based on simulations. Results: Goodness of fit (mean ab­
solute error) varied between 0.30 and 0.62 ml⋅100 g-1⋅min-1. Model-derived CVR was significantly higher
compared to static CVR measures. Maximum CBF increase occurred earlier in healthy- compared to diseased
vascular regions. Conclusions: The proposed mathematical model offers the possibility to calculate CVR insen­
sitive to variations in time to maximum CBF response which gives a more detailed characterisation of CVR
compared to static CVR measures. Although the mathematical model adapts generally well to this dataset ofpatients with MMD it should be considered as experimental; hence, further studies in healthy populations andother patient cohorts are warranted.

1. Introduction
Cerebrovascular reserve (CVR) describes the capacity of the brain to
increase cerebral blood flow (CBF) during changing physiological con­ditions and can be measured as the relative maximum change in CBF
induced by a vasoactive stimulus [1,2]. Impaired CVR is an importantbiomarker for the evaluation of future stroke risk in patients with stenoocclusive disease [3–7] and is compromised in other neurological dis­orders such as cerebral small vessel disease, Alzheimer’s disease andmultiple sclerosis [3,4,8–10]. Acetazolamide (ACZ) is a well-known
carbonic anhydrase inhibitor used to produce vasodilatory stimuli
with a long-lasting vasodilatory response [11]. ACZ has limitations that

affect its ability to be standardised; reported limitations include indi­vidual variability in response to ACZ for a given dose and variations intime-to-maximum CBF (tmax) augmentation [12,13]. Usually, a single
CBF measurement is acquired at an arbitrarily chosen post-ACZ injection
time, generally between 10 and 20 min, and normalized to a preinjection CBF measurement to calculate a static CVR measure [14].
Studies have shown that tmax occurs earlier in healthy controls than inpatients with cerebrovascular disease and in healthy regions than indiseased regions [14–18]. Single CBF measurements following ACZ in­
jection in combination with the reported variability in tmax may there­
fore contribute to an underestimation of static CVR measures because it
is not certain that the single CBF measurement after ACZ injection is* Corresponding author at: Molecular Imaging and Medical Physics, Department of Surgical Sciences, Uppsala University, Uppsala University Hospital, SE 751 85
Uppsala, Sweden.
E-mail addresses: Markus.Fahlstrom@uu.se (M. Fahlström), Joao.Sousa@uu.se (J.M. Sousa), Teodor.Svedung-Wettervik@uu.se (T. Svedung Wettervik), Johan.
Berglund@uu.se (J. Berglund), Per.Enblad@uu.se (P. Enblad), Anders.Lewen@uu.se (A. Lewén), Johan.Wikstrom@uu.se (J. Wikström).
https://doi.org/10.1016/j.mri.2024.03.044
Received 25 February 2024; Received in revised form 27 March 2024; Accepted 31 March 2024
Available online 2 April 2024
0730-725X/© 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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Magnetic Resonance Imaging 110 (2024) 35–42

performed at tmax. Additionally, the normalization of static CVR mea­sures in diseased compared to healthy regions (contralateral or cere­
bellum) may be confounded because of variations in tmax between
regions [19]. Consequently, other strategies are warranted to derive adynamic CVR independent of variations in tmax. In this study, we pro­pose a mathematical model to describe the temporal CBF response
following ACZ injection that provides the possibility to calculate CVR
insensitive to variations in tmax and extract additional model-derived
parameters with unexplored clinical potential. Moreover, the present
study aimed to evaluate this model and the possible clinical potential of
additional model-derived parameters in patients with Moyamoya dis­ease (MMD).
2. Materials and methods
2.1. Theory
The development of the mathematical model was based on a set of
preconditions.

Fig. 1. Graphical illustration of model-derived parameters.

2.2. A priori constraints for free parameters• The model defines a rapid increase in CBF after ACZ injection fol­
lowed by a gradual reduction in CBF as described by most previous
studies (see Supplementary Table 4).
• The model has a non-zero value at t = 0 equal to CBF at baseline.
• CBF returns to baseline at a given time t.

Simulation experiments were performed to assess possible a priori
constraints for r1 and r2 given clinically relevant values of CVR and tmax
obtained from current literature, see Supplementary Table 4 and [14].
Of note, A was bound to 0 ≤ A ≤ 110 ml/100 g/min which is considered
to cover normal, hypo-, and hyper-perfusion conditions with added
margin. A 2D grid was constructed based on the vectors r1, r2 =0:0.0001:2 where 0.0001 defines the increment between elements. CVR
and tmax were calculated element-wise using eqs. 2 and 4, respectively.
Clinically relevant limits of CVR and tmax values were determined as[1,2], which equals a relative increase in CBF of 0% to 100% and [3 min,40 min] respectively. Furthermore, twashout was limited to values above
20 min. All elements with CVR and/or tmax outside these limits or
twashout below 20 min were set to NaN and omitted from further analysis.
A priori constraints for r1 and r2 were determined to [0.005, 0.04] and[0, 0.8] by visual assessment of contour plots presented in Supplemen­tary Fig. 1A-B. Combinations of r1 and r2 <0.005 and higher than 0.8,
respectively were omitted since the distribution of CVR and tmax com­binations were heavily skewed towards high CVR and low tmax values.
Corresponding contour plots showing CVR and tmax with constraints
applied are presented in Supplementary Fig. 1C-D. All simulations were
performed using Matlab version 2023b (Mathworks, MA, United States).

We adopted a bi-exponential equation to describe the temporal CBF
response following ACZ injection which fulfils all the above-stated re­quirements.
f (t)CBF = A(2exp− r1 t − exp− r2 t )(1)where A, r1 and r2 are free parameters and t represents time. Further­more, A equals pre-injection CBF at t = 0 given in ml⋅100 g− 1⋅min− 1 andr1 and r2 are rate constants both given in s− 1 which governs the washout
and the augmentative effects of ACZ respectively (more details can befound in Supplementary Methods). Based on the model, we can define a
set of model-derived parameters:
tmax = ln(2r1 /r2 )/(r1 − r2 )(2)where, tmax is the time required to reach maximum CBF augmentation.
twashout = t − tmax : t > 0, fCBF (t) = fCBF (0)(3)

where, twashout is the time between tmax and the complete return to preinjection CBF values.
CVRmodel = 2exp(− r1 tmax ) − exp(− r2 tmax )2.3. Patient data

(4)

Patient data used for the evaluation of the model were retrospec­tively obtained as part of an ongoing larger research project. Patients
eligible for this study were MMD patients with available CBF measure­ments obtained with arterial spin labelling (ASL) MRI and whole-brain
coverage, which included the cerebellum. Patients with previous
revascularisation surgery and non-operated patients were included.
Fifteen patients with confirmed MMD were studied to assess the validityof the model. The stages of a patient’s disease (uni- or bilaterality) were
graded using the Suzuki score system by an experienced neuro­interventionist based on digital subtraction (n = 13) or magnetic reso­
nance angiography (MRA) (n = 2) [20]. All patients were intravenously
injected with ACZ (1000 mg for adults and 10 mg/kg for children). This
retrospective study was performed in accordance with the Declaration of
Helsinki and approved by the Swedish Ethical Review Authority. All
included patients or legal guardians signed an informed consent form.

where, CVRmodel is the model-derived CVR at tmax. Of note, themaximum CVRmodel values is 2.0 (i.e. a 100% increase in CBF following
ACZ injection), which is further elaborated on in the discussion.
k=fCBF (tmax ) − fCBF (0)
tmax

(5)

where, k, given in ml⋅100 g− 1⋅min− 2, is defined as the slope of a line
between the two points defined by (t0, fCBF(0)) and (tmax, fCBF(tmax)) andcan be considered as an estimation of the rate of increase in CBF during
the first phase after ACZ injection.
Model-derived parameters are graphically illustrated in Fig. 1.
Additionally, the area-under-the-curve (AUC45min) was also included,
defined as the area between t = 0 and t = 45 min (chosen to allow
comparison with other studies), and limited by baseline CBF, in units ofml⋅100 g− 1.

2.4. Magnetic resonance imaging
The patients were scanned on a 3.0 T MRI system (dStream Achieva,
Philips Healthcare, Best, the Netherlands) using a 32-channel head coil.
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Magnetic Resonance Imaging 110 (2024) 35–42

All CBF measurements were performed using a pseudo-continuous ASL(pCASL) labelling scheme with background-suppressed segmented 3D
gradient and spin-echo read-out (GRASE) and the following parameters:
repetition time = 4673 ms, echo-time = 10.7 ms, post-label delay (PLD)= 2500 ms, label duration =1800 ms, turbo factor = 21, EPI factor = 15,
number of segments = 4, SENSE = 2. Scan duration for a pCASL
acquisition was 5 min and 27 s, with a spatial resolution of 3 × 3 × 6mm3 and the number of slices was 22, with four background suppression
pulses and without flow-crushing gradients. An acquired pCASL dataset
consisted of seven control-label pairs and two proton-density weighted
images, used for calibration and acquired with identical parameters as
the control-label pairs, except without background suppression andlabelling. The proton-density weighted images were averaged to derivea mean calibration image used for subsequent processing described
below. With the aid of a phase-contrast MRA survey, the labelling plane
was placed perpendicular to the brain-feeding arteries.
A total of four pCASL datasets were acquired. One before ACZ in­
jection (baseline) and three starting at 5, 15 and 25 min after ACZ in­
jection, respectively. Structural 3D T1-weighted (T1W) images wereacquired with spatial resolution 0.938 × 0.938 × 1 mm3 for tissue
segmentation and registration. Furthermore, a 3D time-of-flight MRA(spatial resolution 0.35 × 0.69 × 1.0 mm3) was included and used for
Suzuki grade assessment in the absence of digital subtraction
angiography.

distributions used in the kinetic model inversion are presented in Sup­
plementary Table 1.
2.5.3. Partial volume correction
ATT artefacts, secondary to slow flow and/or collateral routes ofcirculation, are common in patients with cerebrovascular diseases suchas patients with MMD [30]. Applying partial volume correction (PVC) to
the present pCASL datasets with known ATT artefacts can propagate
erroneous CBF values around these areas [31]. Hence, to reduce partial
volume effects, the regional analysis was restricted to only GM voxels[31]. This was done by transforming the partial volume estimate of GM
from structural to perfusion space using applywarp with super samplingand spline interpolation [31,32]. A “pure” GM mask was defined byapplying a threshold of 70% to the partial volume estimate of GM in
perfusion space, as recommended by Chappell et al. [31,32]. The GM
mask was multiplied with the vascular atlas to remove any contribution
from white matter and cerebrospinal fluid within the vascular regionsof-interests. Regional average CBF values were consecutively extracted
from each vascular region at each time point and used for curve fitting of
the proposed mathematical model and additional CVR calculation.
2.5.4. CVR calculation
Additionally, static CVR measures (CVR5min, CVR15min and
CVR25min) were calculated for all post-injection time points, as the ratio
to baseline CBF.

2.5. Data analysis

2.6. Curve fitting

2.5.1. Structural pre-processing
All steps as described below were performed using analysis tools
within FSL 6.0.5.1 (fsl.fmrib.ox.ac.uk) [21]. Structural T1W images
were segmented using FAST [22] into partial volume estimate images
for grey matter (GM), white matter and cerebrospinal fluid. Registrationfrom perfusion to structural space was performed using FLIRT (BBR)[23]. Structural to standard (MNI) space transformation was achieved
using FLIRT [23] and FNIRT [21]. A deformable vascular atlas defining
four major supra- and infratentorial arterial brain regions in standard
space [24]: anterior (ACA), middle (MCA), posterior cerebral artery(PCA) and vertebrobasilar artery (VBA) was transformed from standard
to perfusion space using applywarp and nearest neighbour interpolation
with the inverse transformation matrix calculated in the previous step.
Based on the Suzuki grading, as described above, all ACA and MCA
vascular regions were classified as diseased, and all PCA and VBA
vascular regions were classified as healthy.

A priori constraints were applied to all free parameters based on
simulations as described above. Curve fitting was performed using thenon-linear least squares method in an in-house developed Python-based
script using SciPy curve_fit() function. Initial values were set to A = 50,
r1 = 0.007 and r2 = 0.33. Time points used for curve fitting was 0, 7.5,
17.5 and 27.5 min corresponding to the middle of the CBF measurement
duration in relation to ACZ injection.
2.7. Statistical analysis
The evaluation of the model was performed by analysing goodness of
fit and comparing derived model-derived parameters to those given in
the current literature. Left and right vascular regions were pooled
together; hence the sample size was n = 30 for all analyses performed.
The Shapiro–Wilks test was performed to test for normality. Descriptive
statistics are expressed as mean (SD) or median with interquartile range(IQR). Derived p-values are two-sided and presented as exact values or< 0.001, p < 0.05 being considered significant. GraphPad Prism 10 for
Mac (GraphPad Software, La Jolla, CA, USA) was used for statisticalanalysis and graph design.

2.5.2. CBF quantification
The Bayesian Inference for Arterial Spin Labeling MRI (BASIL)
toolkit [25], distributed as a part of FSL, was used for pCASL preprocessing and kinetic model inversion. The control-label pairs werecorrected for motion and registered to the mean calibration image, using
rigid-body registration with FLIRT [23]. Perfusion-weighted images
were created by pairwise subtraction of pre-processed pCASL data.
The kinetic model used in this work was based on the general kinetic
model proposed by Buxton et al. in 1998 [26] incorporating a Gamma
dispersion kernel to account for the dispersion of labelled blood duringtransit from the labelling plane to the measurement location [27]. T1 oftissue was fixed to 1300 ms, T1 of arterial blood to 1650 ms and labelingefficiency to 0.85. Uncalibrated perfusion and arterial transit time (ATT)
were included as non-spatial priors described by a normal distribution
with mean and standard deviation (SD) of mean 0 (SD 1000) and 1300
ms (SD 316 ms) for perfusion and ATT, respectively. Adaptive spatial
regularization was applied to the estimated perfusion image [28]. The
mean calibration image was corrected for short TR (< 5 s) and protondensity differences between tissue and blood using λ = 0.9 (the brain
average water partition coefficient between blood and water) after
which calibration was performed voxel-wise to obtain CBF measures inabsolute units of ml⋅100 g− 1⋅min− 1 [29]. Parameters and prior

2.7.1. Goodness of fit
The mean absolute error (MAE) was calculated, where a low MAE
value in relation to the range of the CBF scale is considered a marker ofwell-performed curve fitting. MAE did not pass the test for normality. Rsquared generally presents with issues when applied to non-linear
regression, which is why we omitted this measure in the present study.
2.7.2. Analysis of model-derived parameters
Model-derived parameters included in this analysis are CVRmodel,
tmax, twashout, k and AUC45min. To test whether any differences were
present between vascular regions for a given model-derived parameter,
we performed a repeated measure ANOVA test with Sidak’s multiple
comparison on normal distributed parameters or Friedman’s test with
Dunn’s multiple comparisons on non-normal distributed parameters.
Post-hoc analysis was performed on the possible diseased-healthy pairs,
ACA vs PCA, MCA vs PCA, ACA vs VBA and MCA vs VBA. Additionally,
we performed a repeated measure ANOVA test with Sidak’s multiple
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Magnetic Resonance Imaging 110 (2024) 35–42

comparison to test possible differences between CVRmodel and CVR5min,
CVR15min and CVR25min. All CVR measures and AUC45min passed the test
for normality.

Table 2
Descriptive statistics for model-derived parameters including MAE.
Parameter

3. Results

MAE [ml⋅100
g− 1⋅min− 1]a3.1. Patient characteristics

CVRmodel [unitless]Fifteen patients with MMD (average age, 37 years; range, 13 to 56; 1
male, 14 female) underwent one MRI examinations each. All patientswere diagnosed with bilateral MMD. Indirect revascularisation surgery
had been performed in six patients and direct revascularisation surgery
had been performed in two patients. Patient characteristics are given in
Table 1.

tmax [min]a
AUC45min [ml⋅100 g− 1]
twashout [min]ak [ml⋅100 g− 1⋅min− 2]a

MAE values across all vascular regions were generally low, with only
minor nonsignificant differences between regions. Median MAE (IQR)
values, in ml⋅100 g− 1⋅min− 1, were 0.62 (0.61), 0.58 (0.58), 0.30 (0.61)
and 0.49 (0.60) for ACA, MCA, PCA and VBA, respectively. The highest
MAE found was 4.00 ml⋅100 g− 1⋅min− 1 for PCA, and the highest MAEs
for ACA, MCA and VBA were 3.40, 3.30 and 3.40 ml⋅100 g− 1⋅min− 1,
respectively. Of note, the highest MAE values were found in the same
patient, which is further elaborated on in the discussion. Scatter plotspresenting MAE values for each vascular region can be found in Fig. S3;
median MAE values are also presented in Table 2.

As expected in patients, lower values of CVRmodel were generally
found in diseased vascular regions (see Table 2). Mean CVRmodel (SD)were 1.42 (0.17) in ACA and 1.40 (0.15) in MCA. Corresponding values
for PCA and VBA were 1.68 (0.17) and 1.53 (0.21), respectively.
CVRmodel was significantly lower for all diseased vascular regions
compared to healthy vascular regions (see Supplementary Table 2).
Mean differences varied between − 0.28 to − 0.11, largest difference wasfound between MCA and VBA. Moreover, CVRmodel was significantly
higher compared to CVR5min, CVR15min and CVR25min for both diseased
and healthy vascular regions (see Supplementary Fig. 4 and Supple­
mentary Table 3).
Maximum CBF increase occurred earlier in healthy- compared todiseased vascular regions (see Table 2). Median tmax (IQR) values were
13.6 min (5.0) and 15.4 min (6.3) in ACA and MCA, respectively. In
comparison, corresponding values in PCA and VBA were 12.7 min (8.2)
and 10.9 min (4.6). Median differences reached significance for all

Suzuki grade (L / R)

Operation

31/F
34/F
56/F
13/F
55/F
21/F
53/F
15/F
48/F
37/F
42/F
21/F
48/F
53/F
30/M

V/I
III / IIIIV / IV
III / III
III / IIIIV / IV
IV / IV
III / III
V/V
IV / IV
III / IIIIV / III
III / III
III / III
III / III

MBH R
MBH L/R
MBH L/R
MBH L/R

PCA

VBA

0.58
(0.58)
1.40(0.15)
15.4 (6.3)
324 (139)
35 (24)0.72(0.60)

0.30(0.61)
1.53(0.21)
12.7 (8.2)
373 (160)
43 (29)0.90(0.90)

0.49(0.60)
1.68(0.17)
10.9 (4.6)
445 (176)
49 (42)1.48(0.79)

4. Discussion
4.1. Mathematical model goodness of fit
Herein we present a novel mathematical model to describe thetemporal CBF response following ACZ injection. The model fundamen­tally describes a rapid increase in CBF after ACZ injection followed by agradual reduction in CBF (see example in Fig. 1), which is in agreement
and comparable to the majority of previous publications (see Table S4)[8,11,33–37]. In contrast, Dahl et al. [38] reported that the CBF
response reaches a plateau between 8 and 15 min after injection of ACZ
in healthy controls which persists for up to 30 min after injection. In
addition, Vorstrup et al. [2] reported an increase in CBF that lasted for
up to 20 min after injection of ACZ in healthy controls, but no CBF
measurements were performed past 20 min; hence the maximum CBF
response post-ACZ injection could not be detected.
Individual curve fitting was performed in eight different vascularregions: bilateral ACA, MCA, PCA and VBA for each of the fifteen
included patients, respectively. Generally, the fitting performance wasgood (an example of a good curve fit is given in Fig. 2A) as indicated by
the relatively low average MAE (see Table 2 and Supplementary Fig. 3),
however in some cases the fitting performance was poor. The fivehighest MAE values were found in the same patient (see Supplementary
Fig. 3, indicated by filled circles), the highest derived MAE was 4.0
ml⋅100 g− 1⋅min− 1, an error of 15% in relation to baseline CBF for thatspecific vascular region. The non-optimal fitting is caused by themeasured CBF value at 15 min post-ACZ injection being lower thancorresponding value at 25 min post-ACZ injection (see Fig. 2B), breaking
the general assumption of the response in which the model is based on.
We did not find any apparent methodological issues explaining such

Table 1
Patient characteristics.
Age*/Sex

MCA

0.62(0.61)
1.42(0.15)
13.6 (5.0)
353 (172)39 (23)0.90(0.55)

healthy-disease pairs except for ACA compared to PCA (see Supple­
mentary Table 2).
Median estimated rate of increase, k (IQR) in ml⋅100 g− 1⋅min− 2 were0.90 (0.55) and 0.72 (0.60) in ACA and MCA respectively. Corre­
sponding values for PCA and VBA were 0.90 (0.55) and 1.48 (0.55),
respectively. Diseased-healthy pair comparisons reached significance inall comparisons except for ACA compared to PCA (see Supplementary
Table 2).
Median twashout varied between 35 and 49 min; median difference
ranged from − 3 to − 15 min and reached significance for ACA and MCA
compared to VBA (see Table 2 and Supplementary Table 2). Mean
AUC45min varied between 324 and 445 ml⋅100 g− 1; median difference
ranged from − 20 to − 122 and significance was reached in all diseasedhealthy pairs expect for ACA when compared to PCA (see Table 2 andSupplementary Table 2). Scatterplots of all model-derived parameters
for each vascular region can be found in Fig. 3.

3.3. Analysis of model-derived parameters

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

ACA

Abbreviations: ACA, anterior cerebral artery; AUC, area under the curve; CVR,
cerebrovascular reserve; IQR, interquartile range; MAE, mean absolute error;
MCA, middle cerebral artery; PCA, posterior cerebral artery; VBA, vertebrobasilar artery.
aData are expressed as median (IQR).

3.2. Goodness of fit#

Mean (SD) or Median (IQR)

MBH L/R

MBH L
By-pass L
By-pass R

Abbreviations: F, female; L, left; M, male; MBH, multiple burr hole surgery; R,right.
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Magnetic Resonance Imaging 110 (2024) 35–42

Fig. 2. Three patient-specific curve fittings. Of note, time points on the x-axis are 0, 7.5, 17.5 and 27.5 min corresponding to the middle of the CBF measurement
duration in relation to ACZ injection. Adequate curve fit is shown in A with low MAE. Inadequate curve fitting is shown in B with high MAE. The fitting performanceis substandard because of lower CBF measurement at 15 min compared to 25 min post-ACZ injection. In C, a vascular region with suspected cerebrovascular steal(closed circles) is compared to VBA (open circles). The model does not accept local minimums hence the fitting performance is negatively affected. MAE is above
average and tmax and twashout is considerable prolonged. ACZ, acetazolamide; CBF, cerebral blood flow; MAE, mean absolute error.

deviation in this patient’s data, nevertheless, CBF measurements are
inherently subjected to statistical variations due to several factors.
Hence, these deviations are not surprising, especially during thewashout period where the difference between measured CBF values isusually small. A patient with suspected cerebrovascular steal in left MCA
had higher MAE than average (see Fig. 2C). The model is not able to fit
cases of cerebrovascular steal as it is not able to describe a local mini­
mum, where the CBF decreases and returns to baseline. In this case,derived tmax and twashout becomes abnormally high in comparison to therespective average values.

4.2. Comparison of model-derived parameters
To further validate the model, we compared our calculated modelderived parameters with previously reported values in the literature.
However, direct comparisons are hampered by methodological differ­ences such as imaging modalities, number of post-injections timepoints
and ACZ dosage, and should therefore be interpreted with caution. A
comparative literature review is presented in Supplementary Table 4.
We found values of tmax in our patient data (10.9 min to 15.4 min, see
Table 2) comparable to other reports (10 min to 17 min) which in thelatter case included values from both healthy controls and patient

Fig. 3. Scatter plots for model-derived parameters for the diseased regions ACA and MCA compared to the healthy regions PCA and VBA. Bars represent mean andstandard deviation for CVRmodel and AUC, and median and interquartile range for tmax, k and twashout. ACA, anterior cerebral artery; AUC, area under the curve; CVR,
cerebrovascular reserve; MCA, middle cerebral artery; PCA, posterior cerebral artery; VBA, vertebra-basilar artery.
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Magnetic Resonance Imaging 110 (2024) 35–42

cohorts [15–18,34,35,39]. We also found significant differences be­
tween diseased vascular regions and healthy vascular regions (seeFig. 3), supporting the hypothesis that maximum CBF response isdelayed in diseased regions, compared to healthy regions and/or healthy
controls [16–19]. The implication of this will be discussed further
below. In contrast, Hauge et al. [33] found that tmax occurred 25 minafter ACZ injection. Of note, the ACZ dose used was lower (500 mg) thanthe more commonly dose of 1000 mg. Additionally, Dogra et al. [15]reported no differences in tmax between ipsi- and contralateral hemi­
sphere using continuous blood oxygenation level-dependent (BOLD)
measurement in patients with unilateral steno-occlusive disease. How­ever, due to methodological differences, comparisons to CVRs derived
from BOLD MRI are difficult.
There are few reports presenting values for k, AUC and twashout,considerable differences to the current study are present making com­
parisons difficult [8,18,33,39]. Hence, we did not include a detailed
comparison between other reports and the current study regarding theabove-mentioned model-derived parameters. However, a brief summaryof the literature review is presented in Supplementary Table 4.

The current study adds support to two hypotheses with major im­
plications for CVR measurements in patients. First, tmax varies between
diseased and healthy vascular regions, thus tmax should be regarded as apossible confounder in CVR measurements. Second, static CVR mea­
surements are inherently based on the suggestive monotonicity of CBF
response to ACZ, but in its absence given the variations in tmax, static
CVR measures are underestimating the true CVR. Implications that fol­lows are that static CVR measures may exaggerate CVR deficits, which
may falsely indicate surgical intervention in patients with cerebrovas­
cular disorders and that errors may be induced when normalization of
CBF measurements are performed to contralateral regions or cerebellum
if not variations in tmax are considered. The discussed variation in tmax
may thus explain some of the inter- and intra-patient variabilities which
often is put forward as a limitation using ACZ as vasodilatory stimulus[12,13]. The key potential of the proposed model is to estimate CVR
independently of variations in tmax. Furthermore, the model can beapplied to other diagnostic imaging modalities but several or continuous
measurement post-ACZ injections are necessary. Additionally, othermodel-derived parameters can be calculated from the model, which may
have diagnostic and prognostic value as shown in the current study.

voxels as described above. We found abnormally high CBF values after
PVC using pvcorr in BASIL (data not shown) compared to uncorrected
data during initial analysis. Both PVC and non-PVC CBF images arederived from BASIL, hence we decided to use the uncorrected dataset.
Further analysis of PVC and non-PVC data was beyond the current scopeof the current paper, hence not performed. Our decision to not use PVC
data is partly based on the discussion on the topic by Chappell et al. [31].
Each CBF measure is averaged over a measurement period of approxi­
mately 5.5 min. This is commonly done to increase the signal-to-noise
ratio but may introduce errors in the quantification if CBF is altered
non-linearly during the duration of measurement. Although, there areseveral technical limitations with single-delay ASL, there are also several
benefits for temporal measurements. Single-delay ASL is highly repeat­able, is non-invasive, does not expose the patient to ionizing radiationand is readily available [44], and measures CBF directly in comparison
to BOLD MRI, where the signal reflects a complex relationship between
CBF, cerebral blood volume, oxygen extraction fraction, cerebral
metabolic rate of oxygen and haematocrit [45,46]. However, BOLD MRI
remains a promising method to measure CVR sharing several benefits
with ASL, using either ACZ or carbon dioxide as vasodilatory agents[15,45]. Although, applying the proposed model to other imaging
methods is important in the future, any detailed comparisons were notwithin the scope of the current study.
The included patients were mainly female, who had or had not un­dergone revascularisation surgery. Thus, the patients represent aheterogenous population, and any comparisons to other study pop­
ulations should be done with caution. The maximum CVR value which
the model for a given fit can assume is 2.0 which equals an increase in
CBF after ACZ injection of 100%. Although this is in theory unlikely
given the cited normal CVR range of 30–60% [1] and is not seen in ourpatient data, but may still be present in healthy controls or patients inregions considered healthy such as cerebellum (highest derived
CVRmodel in an individual patient in our material was 1.90 in VBA).
Lastly, the number of free parameters in the model is three, and inrespect to the number of data points used for fitting (four CBF mea­
surements) should be considered a limitation. Given the limitations
together with cases of inadequate fitting discussed and presented above
the model should be considered experimental. Thus, additional studies
in healthy controls and other patient cohorts are warranted, preferablywith additional CBF measurements following ACZ injection utilizing
multi-delay ASL techniques to reduce potential transit time dependent
artefacts.

4.4. Limitations

5. Conclusion

The fitting performance and subsequently the precision of the modelderived parameters is dependent on adequate CBF measurements. In
that sense, there are several technical limitations which may affect the
CBF measurements acquired with single-delay ASL. This patient mate­rial was acquired using a PLD of 2500 ms, which is a trade-off between
reducing the impact of ATT artefact common in the patients studied anddecreased signal due to T1 relaxation [30]. Of note, ATT artefacts can befurther reduced using multi-delay ASL techniques [40], however, thistechnique was not available to the authors during the time this patientmaterial was collected. BASIL uses Bayesian interference methods forrobust precision quantification of CBF and includes several imaging
processing steps to further improve quantification such as motion
correction, modelling of dispersion and spatial regularization[25,27,41]. We applied the default ATT prior distribution (mean 1300
ms and SD 316 ms) in the kinetic model inversion for all sequential CBF
measurements to keep the consistency between measurements. As such,
we did not adjust the ATT prior for known increased ATT values atbaseline in patients with MMD and decreased ATT values after ACZ
injection [42,43]. We acknowledge that this will partly affect themeasured CBF at each sampled time point. Furthermore, we did not
perform PVC, instead the regional analysis was restricted to “pure” GM

The proposed mathematical model offers the possibility to calculate
CVR insensitive to variations in time to maximum CBF response which
gives a more detailed characterisation of CVR compared to static CVR
measures. Although the mathematical model adapts generally well tothis dataset of patients with MMD it should be considered as experi­mental; hence, further studies in healthy populations and other patient
cohorts are warranted.

4.3. Potential

Funding statement
This work was supported by the Erik, Karin and Gösta Selanders
Foundation, the Swedish Stroke Association and the Swedish Brain
Foundation. The supporters had no role in conceptualization, design,
data collection, analysis, decision to publish, or preparation of themanuscript.
Ethical approval and consent to participate
This retrospective study was performed in accordance with the
Declaration of Helsinki and approved by the Swedish Ethical Review
Authority. All included patients or legal guardians signed an informed40

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Magnetic Resonance Imaging 110 (2024) 35–42

consent form.

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Consent for publication
All included patients or legal guardians signed an informed consent
form.
CRediT authorship contribution statement
Markus Fahlström: Conceptualization, Methodology, Software,
Validation, Formal analysis, Investigation, Resources, Data curation,
Writing – original draft, Visualization, Supervision, Funding acquisition.
Joao M. Sousa: Methodology, Software, Validation, Resources, Data
curation, Writing – review & editing. Teodor Svedung Wettervik:
Conceptualization, Validation, Writing – review & editing. Johan Ber­glund: Methodology, Software, Writing – review & editing. Per Enblad:
Conceptualization, Supervision, Writing – review & editing. Anders
Lewén: Conceptualization, Supervision, Writing – review & editing,Funding acquisition. Johan Wikström: Conceptualization, Validation,
Supervision, Writing – review & editing.
Appendix A. Supplementary data
Supplementary data to this article can be found online at https://doi.
org/10.1016/j.mri.2024.03.044.
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