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January 29, 2010

The Effects of Body Mass Index on Cerebral Blood Flow Velocity

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Clin Auton Res. Author manuscript; available in PMC 2008 December 10.
Published in final edited form as:
Clin Auton Res. 2008 December; 18(6): 331–338.
Published online 2008 August 22. doi: 10.1007/s10286-008-0490-z.
PMCID: PMC2600600
NIHMSID: NIHMS57514
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The Effects of Body Mass Index on Cerebral Blood Flow Velocity
Magdy Selim, MD, PhD,1 Richard Jones, ScD,2 Peter Novak, MD, PhD,3 Peng
Zhao, PhD,4 and Vera Novak, MD, PhD4
1Beth Israel Deaconess Medical Center, Department of Neurology – Stroke Division, 330 Brookline Avenue, Boston, MA
02215, Tel: 617-632-8913, Fax: 617-632-8920
2Hebrew Senior Life, Institute for Aging Research, 1200 Centre St., Boston, MA 02131, Tel: 617-363-8493, Fax:
617-363-8936
3University of Massachusetts, Dept of Neurology, 55 Lake Avenue N, Worcester, MA 02215. Tel: 508-334-4973, Fax:
508-856-4485
4Beth Israel Deaconess Medical Center, Division of Gerontology, 110 Francis Street, LMOB Suite 1b, Boston MA 02215,
Tel: 617-632-8680, Fax: 617-632-8675, Tel:617-667-0346
Magdy Selim: mselim@bidmc.harvard.edu; Richard Jones: jones@hrca.harvard.edu;
Peter Novak: NovakP@ummhc.org hc.org; Peng Zhao: pzhao1@bidmc.harvard.edu;
Vera Novak: vnovak@bidmc.harvard.edu
Please address correspondence to: Vera Novak MD, PhD, Beth Israel Deaconess Medical Center, Department of
Medicine – Gerontology, 330 Brookline Avenue, Boston, MA 02215, Tel: 617-632-8680, Fax: 617-632-8675, E-mail:
vnovak@bidmc.harvard.edu
The publisher’s final edited version of this article is available at Clin Auton Res.
Top
Abstract
INTRODUCTION
MATERIALS AND
METHODS
RESULTS
DISCUSSION
Abstract
Objective
Obesity is a risk factor for cerebrovascular disease. We aimed to determine the effects of high body
mass index (BMI) on cerebral blood flow regulation in patients with type-2 diabetes mellitus,
hypertension, and stroke.
Methods
We analyzed data from 90 controls, 30 diabetics, 45 hypertensives, and 32 ischemic stroke patients
who underwent transcranial Doppler for evaluation of blood flow velocities (BFV) in the middle
cerebral arteries (MCA) and cerebrovascular resistance (CVR) during supine rest and head-up tilt.
This study was a cross –sectional analysis. We used a structural equation multiple indicators
modeling to determine the effects of BMI and other background variables (age, sex, race, smoking,
alcohol use, and systolic blood pressure) on cerebral BFV.
Results
Higher BMI (p=0.02) and age (p=0.004) were associated with lower mean BFV during baseline,
independent of diagnosis of diabetes mellitus, hypertension or stroke, and after adjusting for all
background variables and vessel diameters. Men, especially those with stroke, had a lower mean
BFV than women (p = 0.01). CVR increased with BMI (p=0.001) at baseline and during head-up tilt
(p=0.02), and was elevated in obese subjects (p=0.004) compared to normal weight subjects across
all groups.
Interpretation
High BMI is associated with a reduction in cerebral BFV and increased CVR. These findings
indicate that obesity can adversely affect cerebral blood flow and resistance in cerebrovascular bed,
independent of diagnosis of type-2 diabetes, hypertension or stroke. Obesity may contribute to
cerebromicrovascular disease, and affect clinical functional outcomes of older population.
Keywords: Body Mass Index, Obesity, Cerebral Blood Flow, Transcranial Doppler, Stroke, Diabetes,
Cerebrovascular resistance, Tilt
Top
Abstract
INTRODUCTION
MATERIALS AND
METHODS
RESULTS
DISCUSSION
INTRODUCTION
Patients with diabetes, hypertension, and previous history of stroke have increased risk for
cerebrovascular diseases, stroke, and cognitive decline. [19;26;32] Body mass index (BMI) is being
increasingly recognized as a risk factor for stroke, cardiovascular disease, and cognitive decline, in
addition to known factors such age, hypertension, smoking, and alcohol.[9;11] Diabetes mellitus,
hypertension and cardiovascular risk factors exert complex effects on cerebral microvasculature
[5;15] which accelerate cerebral blood flow (CBF) decline that occurs with normal aging. Little is
known about the impact of patients’ characteristics, including BMI, on cerebral hemodynamics in
these conditions. Transcranial Doppler (TCD) ultrasound is used as a surrogate for non-invasive
assessment of CBF [22] by measuring blood flow velocities in major arteries of the brain at baseline
and during orthostatic stress [27]. Therefore, we aimed to determine the impact of BMI and
background clinical characteristics on blood flow velocities (BFV) in middle cerebral arteries and
cerebrovascular resistance in patients with diabetes, hypertension, and stroke in comparison with
controls at baseline and during head-up tilt.
Top
Abstract
INTRODUCTION
MATERIALS AND
METHODS
RESULTS
DISCUSSION
MATERIALS AND METHODS
Subjects
Initial recruitment began in the Autonomic Nervous System Laboratory at the Department of
Neurology at The Ohio State University. Recruitment during the latter part of the study was carried
through the Syncope and Falls in the Elderly (SAFE) Laboratory at the Beth Israel Deaconess
Medical Center in Boston under the direction of the same investigators (VN, PN) and using identical
protocols and methodology. All subjects provided a written informed consent to an IRB approved
protocol. Participants were prospectively recruited according to the following inclusion/exclusion
criteria: Age ≥ 50 to ≤ 85 years. Control group – subjects who were normotensive, had normal
hemoglobin A1c (HbA1c) level, had no history of stroke or transient ischemic attacks (TIA), and
were not treated for any systemic disease except hypercholesterolemia. Diabetes group – patients
diagnosed with type-2 diabetes mellitus (average 12.8±11.5 yrs), and had no history of stroke or TIA.
Hypertension group – subjects treated for essential hypertension, who had no history of stroke or
TIA, and had normal HbA1c. Stroke group – included subjects with history of ischemic stroke, who
had a documented infarct on MRI or CT scan affecting < 1/3 of MCA territory with a modified
Rankin Scale score < 4.
Fifty-three percent of patients in the stroke group had a left-sided infarct; 47% had a right-sided
infarct. Forty-one percent of strokes were attributed to large artery disease; 25% to small vessel
(lacunar) disease; and 10% to cardioembolism. Stroke mechanism was undetermined in the
remaining 24% of patients. Approximately 7% of participants in the diabetes group were also
hypertensives, while 19% of stroke patients were also treated for hypertension, and 3 % were
diabetics. Subjects with a history of stroke (except for the stroke group), clinically significant cardiac
disease, arrhythmias, significant nephropathy, kidney or liver transplant, renal or congestive heart
failure, uncontrolled hypertension, known carotid artery stenosis > 50%, neurological or other
systemic disorders, and hemorrhagic stroke were not eligible to participate in this study.
Eligibility and Risk Factors Assessment
We screened potential subjects with detailed medical history and physical and neurological
examinations, electrocardiogram, and routine laboratory tests that included serum glucose and renal
function, HbA1C, lipid panel (including triglycerides, and total, LDL and HDL cholesterol),
complete blood cell and differential count, and urine analysis. We calculated the atherogenic index in
the plasma as log (triglycerides/HDL-cholesterol, mmol/l).[10] We measured the subjects’ weight
and height, and calculated the BMI in kilograms per square meter.
TCD Examinations
All TCD examinations were conducted early in the morning, at least two hours after the last meal,
and performed by the same sonographer (VN) using MultiDop x4 TCD machine (Neuroscan Inc., El
Paso, TX). Antihypertensive medications were tapered over a one-week period and withdrawn on the
day of the examination. Anticholinergics and other cardioactive medications were held before the
study on the same day. Hypoglycemic agents, anticoagulants and other medications were allowed.
The subjects rested in a supine position for 10 minutes and then the table was titled upright to 70
degrees for 10 minutes. The right (MCAR) and left (MCAL) MCA were insonated from the temporal
windows with 2-MHz pulsed Doppler probes. Each probe was positioned to record the maximal flow
velocities and stabilized using a 3-dimensional head frame positioning system. Peak-systolic,
end-diastolic, and mean BFV were measured for each MCA. Systolic (SBP) and diastolic (DBP)
blood pressures during the examination were recorded beat-to-beat from a finger with a Finapres
device (Ohmeda Monitoring Systems, Englewood, CO) and intermittently with BP measurement
tonography.[24] Beat-to-beat BP recordings were averaged over the resting period (5–10 minutes).
Cerebrovascular resistance was calculated as mean BP divided by mean BFV in MCAR and MCAL
and as a Gosling’s pulsatility index (systolic-diastolic BFV/mean BFV).[2]
Magnetic Resonance Imaging
A subset of 79 patients underwent imaging studies at the Magnetic Resonance Imaging Center at the
Beth Israel Deaconess Medical Center at the GE 3 Tesla VHI scanner using a quadrature head coil.
Anatomical images of intracranial vasculature were obtained using 3D-MR angiography (time of
flight, TOF) with the following parameters: TE/TR = 3.9/38 ms, flip angle of 25 degrees, 2 mm slice
thickness, −1 mm skip, 20 cm × 20 cm FOV, 384 × 224 matrix size, pixel size 0.39 × 0.39 mm and
tissue imaging included T1-weighted inversion-recovery prepared fast gradient echo (IR-FGE), 3D
magnetization prepared rapid gradient echo (MP-RAGE) °and fluid-attenuated° inversion° recovery°
(FLAIR)°sequences. The MCA and internal carotid arteries (ICA) radii were measured by the
software “Medical Image Processing, Analysis, and Visualization” (MIPAV), Biomedical Imaging
Research Services Section, NIH, USA. The scale for an image can be defined to achieve accurate
measurements with resolution up to one pixel size (0.39 mm × 0.39 mm). For each vessel, the
diameter was measured at 3 locations and averaged.
Statistical Analysis
We conducted two types of analyses. In the first set of analyses we used a structural equation
modeling procedure called multiple groups, multiple indicators, and multiple cause (MIMIC)
modeling to explore the relationship between diabetes, hypertension and stroke, and clinical and
behavioral characteristics related to CBF [13]. MIMIC model details are provided in the Appendix.
This model does not aim to predict absolute CBF values. The MIMIC model uses CBF as a latent
variable that cannot be directly measured but it is represented by two indicators, in our case mean
BFV in right and left MCAs. The model predicts the effects of multiple covariates (age, sex, race,
BMI, SBP, smoking and alcohol use); and test for differences in the prediction across clinical group
while allowing for heteroskedasticity in CBF across study population. MIMIC model parameters
were interpreted as ANCOVA-type regression parameters. Overall model fit was evaluated by using
chi-square statistic, where degrees of freedom are tied to the number of parameter estimates in the
means and covariance matrix (high P-values implying good fit). We also used the root mean square
error of approximation (RMSEA) and the comparative fit index, where RMSEA provides a measure
of discrepancy per model degree of freedom.[1] The RMSEA approaches 0 as model fit improves.
We accepted values close to 0.06 or less that represent adequately fitting models[6], and comparative
fit index values greater than 0.95 that are generally accepted as describing adequately fitting models.
In the second set of analyses we used the MANOVA and the least square models to evaluate the
effects of BMI on CVR at baseline and during tilt, atherogenic index, Gosling’s pulsatility index,
MCA and ICA diameters, and other laboratory variables. Models were adjusted for age, sex and
group. Statistical significance was set as p ≤ 0.05.
Top
Abstract
INTRODUCTION
MATERIALS AND
METHODS
RESULTS
DISCUSSION
RESULTS
A total of 212 subjects were enrolled into the study. Of these, 15 subjects were excluded because of
poor quality TCD examinations, poor temporal windows, or missing elements of the dataset. Data
from the remaining 197 subjects (90 healthy controls, 30 diabetics, 45 hypertensives, and 32 stroke
patients) were included in the analysis. MRI analysis is based on data from 79 (40 controls, 22
diabetics, 10 hypertensives, and 7 stroke patients). summarizes the characteristics of each of
these 4 groups including demographics, risk factors, laboratory values, pulsatility index, intracranial
vessels diameters and medications. Demographic factors and hematological parameters including
lipids were similar among the groups, except, as expected, for systolic blood pressure (p=0.008) and
glucose (p=0.02). History of smoking, alcohol consumption was not different. MCA and ICA
diameters for both sides were not different among the groups. There were no significant differences
among subjects in the diabetes, hypertension and stroke groups who were treated with angiotensinconverting
enzyme inhibitors (ACE inhibitors), diuretics, β-blockers, statins, or antithrombotics. We
found no significant interaction between antithrombotics, ACE inhibitors, or statins and BFVs.
Table 1
Characteristics of the study population.
Cerebral Blood Flow Velocities and Vascular Resistance
We modeled the effects of patient characteristics on baseline CBF (as a latent variable) by using
mean BFV in both MCAs. Diabetics had a lower mean baseline BFV compared to controls
(p=0.017), but mean BFV was similar among the other groups. This effect was no longer significant
after adjusting for background variables.
summarizes the results of the final fitted MIMIC model. This model estimated the effects of
background variables (age, BMI, SBP, smoking and current alcohol use) within each group and
across the whole population. The model reveals that older age (p=0.004) and higher body mass index
(p=0.022) are associated with lower mean BFV in all 4 groups; SBP is positively related to mean
BFV among hypertensive subjects (p=0.007); and that men relative to women in the stroke group
have lower mean BFV (p=0.01). In the control group, age (p=0.004) and BMI (p=0.022) were
associated with lower BFV. No significant relationship was found for smoking and alcohol use.
Table 1
Table 2
Higher BMI remained associated with lower BFV after adjusting background variables and vessel
diameters (p=0.017). shows a plot of BMI, and age-adjusted mean MCAR and MCAL
BFV, and shows that mean BFV in MCAR (p=0.017) and MCAL (p=0.0002) were higher
for normal weight (BMI<25 kg/m2) than overweight (BMI 25–30 kg/m2) and obese subjects
(BMI>30 kg/m2) in all study groups.
Table 2
Standardized regression parameters of MIMIC models of BFV on subject background
characteristics.
Figure 1
Panels A and B show the relationship between body mass index (BMI) and
age-adjusted mean blood flow velocities in right and left middle cerebral artery
(□MCAR, ■MCAL) during baseline in all groups. Panels C and D show the average
cerebrovascular (more …)
summarizes the hemodynamic characteristics of each of the 4 groups during baseline and
head-up tilt. Controls had significantly lower CVR in MCAR and MCAL at baseline and during
head-up tilt. After adjusting for age, sex and group, BMI was independently associated with
increased vascular resistance (CVR MCAR p=0.03, CVR MCAL p=0.0002) during baseline and
head up tilt (CVR MCAR p=0.02, CVR MCAL p=0.04). Gosling’s pulsatility index was used as a
second measure of vascular resistance and was also associated with a higher BMI at baseline
(MCAR BMI<25 0.71±0.17; BMI 25–30 0.80±0.2, BMI>30 0.71±0.19, p=0.037) and
MCAL(.BMI<25 0.71±0.13; BMI 25–30 0.80±0.12, BMI>30 0.71±0.15, p=0.01).
Table 3
Hemodynamic parameters during baseline and head up tilt
shows that baseline CVR increases with BMI for normal weight, overweight and obese
subjects (CVR MCAR=p=0.008, CVR MCAL p=0.002) in all groups, and was elevated in obese
subjects (p=0.004) compared to normal weight subjects across all groups. shows that CVR
during head-up tilt (corrected for hydrostatic pressure change) also increased with BMI (CVR
MCAR=p=0.009, CVR MCAL p=0.001).
MCA diameter
MCA diameters were not significantly different between the groups and among normal weight,
owerweight and obese subjects. MCA diameters were not significantly associated with BMI (right
MCA p=0.39, left MCA p=0.16). BFVs in MCA s were not significantly associated with MCA
diameters (MCAR-r2=0.025, p=0.20; MCAL r2=0.034, p=0.1). Higher BMI remained associated
with lower BFV after adjusting background variables and vessel diameters (p=0.017).
Atherogenic Index, Lipids, Hematocrit
Atherogenic index was not different among the groups, but was lower in women compared to men
(0.26 ± 0.41 vs. 0.64 ± 0.53 mmol/L, p=0.004). Expectedly, the atherogenic index was positively
associated with BMI (p = 0.0006, r=0.39) and male sex (p = 0.02; r = 0.39). Higher BMI (p=0.01)
and male sex (p<0.0001, r = 0.57) were associated with lower HDL levels, and higher LDL levels
Figure 1A
Figure 1B
Table 3
Figure 1C
Figure 1D
(p=0.04, r=0.37) and triglycerides (p=0.0075, r=0.45). Women in our study had lower hemoglobin
and hematocrit (39.3±2.8 vs .43.0±2.3%), and athrogenic index (0.26±0.43 vs. 0.64±0.54 mmol/L,
p=0.004 than men, and lower hematocrit was associated with higher BFV (r=0.42, p=0.01).
Hematocrit was not different in people with higher BMI. There was relative heterogeneity of stroke
group in terms of stroke etiology. Stroke side, etiology and type of antihypertensive medications,
however were not significant factors in our analyses.
Top
Abstract
INTRODUCTION
MATERIALS AND
METHODS
RESULTS
DISCUSSION
DISCUSSION
Our results show that cerebral flow velocities decrease with increasing body mass and age in all
groups, and that male sex is associated with lower BFV especially among stroke patients. Higher
BMI is also associated with increased CVR during supine rest and orthostatic stress. The effects of
BMI on BFV and CVR are independent of those for age and sex and vessel diameter. These findings
indicate that obesity may adversely affect flow velocity and resistance in cerebrovascular bed,
independent of the diagnosis of type-2 diabetes, hypertension or stroke.
Our findings that increased BMI, regardless of age or sex is associated with reduced cerebral BFV
and increased CVR are novel and intriguing. Body mass has been recently recognized as a risk factor
for cerebrovascular disease and cognitive decline in addition to age and other cardiovascular factors.
[9;11] Obesity is associated with increased intima-media thickness that may affect pulsatility large
arteries, and might be the consequence of metabolic dysregulation, associated dyslipidemia,
inflammation, or other mechanisms [12;25]. In multivariate analysis, excess body weight and male
sex were linked to progressive arterial dysfunction and impaired both endothelium mediated and
independent vasodilatation [4],[14] with subsequent decrease in arterial blood flow.[8] In addition,
obesity is also associated with abnormalities in microvascular patterns, reduced small vessel density,
inflammation and impaired endothelial function and vascular reactivity [29;30] in peripheral and
possibly even in central vascular beds.
Our observation of increased CVR suggests that obesity may also affect the cerebral
microvasculature and vasoreactivity during orthostatic stress. Few studies reported on the
relationship between BMI and blood flow regulation and found positive relationship between obesity
and arterial stiffness [33], reduced large and small vessel arterial compliance [3] and reduced
distensibility including carotid arteries. Simillarly, in our study, we found greater resistance in the
larger intracerebral arteries in obese and overweight subjects. Cerebral blood flow during head-up tilt
is maintained by vasodilatation and decreased resistance of arterioles that compensate for reduced
systolic blood pressure and intracranial pressure.[21] Our findings of higher CVR in obese subjects
support the notion that obesity also affects vasoreactivity of cerebral microvasculature during
orthostatic stress. We did not find a relationship between BMI and the diameter of MCAs on 3D
MR-angiography in a subset of 79 patients in this population, thus reputing the notion that reduced
BFV would be due to an increased MCA diameter.
Obesity, as manifested by increased BMI, emerged as the only modifiable characteristic associated
with decreased cerebral BFV in our study. This can have important therapeutic implications. Meyer
et al. [18] reported increased carotid intima-media thickness (IMT), impaired endothelial function,
and flow-mediated vasodilatation in 76 young obese subjects; and showed that regular exercise for 6
months restored endothelial function, decreased IMT by 6%, and increased FMD by 127%,
suggesting that the effects of obesity on blood vessels and flow can be reversed by exercise and
weight reduction.
We found that men had lower BFV, especially among stroke patients, suggesting that changes in
cerebral hemodynamics may be partly sex-dependent. Several pathophysiological, biochemical, and
anatomical factors could potentially account for hemispheric differences in MCA BFV between
sexes.[17] The observed gender differences in BFV in our study could be attributed to differences in
blood rheology and atherogenic burden between the sexes. Women in our study had lower
hematocrit, and athrogenic index than men, and lower hematocrit was associated with higher BFV.
The preferential impact of sex on BFV in stroke patients needs further investigation because of a
small sample size.
There are some factors that may limit the impact of this study. This analysis was cross-sectional and
was focused on long-term relationship between BFV and background variables, rather than dynamics
of autoregulation using beat-to-beat BFV-BP variablity or CO
2
reactivity, to assess long-term
adaptation of cerebral vasculature at baseline and during orthostasis. Previous studies have shown
that BFVs in the MCA correlate with invasive measurements of blood flow with xenon clearance
[23], laser Doppler flux [16] and PET. [31] MCA diameter was not different between the groups and
remains relatively constant under physiological conditions.[28] Further studies, however are merited
to assess complex effects of obesity on blood flow and tissue perfusion.
Conclusions
Our study provides evidence that obesity further affects cerebral blood flow velocities and vascular
resistance in older adults in addition to already known effects of diabetes, hypertension and stroke.
These findings may have important therapeutic implications. High BMI is a modifiable risk factor for
stroke and cardiovascular disease, that can be linked to brain atrophy and cognitive decline. [9;11]
This warrants future prospective studies to assess whether the effects of high body mass on cerebral
blood flow and vascular resistance can be reversed by weight reduction.
ACKNOWLEDGMENT
This study was supported by American Diabetes Association Grants (1-03-CR-23 and 1-06-CR-25)
to V. Novak, an NIH Older American Independence Center Grant (2P60 AG08812), an NIH-NIA
Program project (AG004390), an NIH-NINDS Grant (1R01-NS045745-01A2), National American
Heart Association Scientist Development Award (9930119NH) and a General Clinical Research
Center Grant (MO1-RR01032).
Abbreviations
BFV blood flow velocities
CBF cerebral blood flow
BMI body mass index
CVR cerebrovascular resistance
APPENDINX
We used a structural equation modeling approach known as the MIMIC or multiple indicators,
multiple cause model [6]. The MIMIC model is a structural equation modeling approach that
characterizes the relationship between background exogenous variables (causes) and unobserved
(latent) variables, which are indicated by imperfect representations of the underlying latent variable.
In our case, the latent variable (CBF) was modeled as indicated by two observed variables, MCAR
and MCAL BFVs.
We did not aim to model absolute values of CBF. We used the measurement equation (y = ν + Λη +
ε̣), where η is 1×1 and contains the latent BFV variable; y contains MCAR and MCAL BFVs; ν
(vector) captures the intercepts in the measurement relations; Λ the loadings of the latent BFV
variable in the observed variables; and ε the residuals in the measurement relations. To identify the
model, we assumed that λ
11

12
=1 and freely estimated the variance of the latent trait (VAR(η) =
Ψ). The MIMIC model also includes a structural model, which relates the latent variable (η) to
exogenous and so-called ‘causal’ variables. This model can be represented with η = α + Γx + ζ,
where Γ contains regressions parameters expressing the increase in η per unit increase in the
predictor variables in x. Vector ν contains the intercept of η, blood flow velocity, and ζ residuals in
the structural model. We considered the clinical group (controls, diabetics, hypertensives, or stroke
patients) and participant’s background variables (age, sex, and race), physiologic characteristics
(BMI and SBP) and health behaviors (history of smoking, and current alcohol use) in x. Associated
regression parameters in Γ were interpreted as ANCOVA-type regression parameters when the
background variables were discrete or linear regression parameters when the background variables
were more-or-less continuously distributed, such as age, BMI, and SBP. We extended these MIMIC
models to consider multiple groups, where clinical groups were used to separate participants and
MIMIC models estimated simultaneously but separately within group. Initially we assumed the
relationships of background variables and BFV were equal across group, but examined indices of
model misspecification (model modification indices) and individually and iteratively relaxed equality
constraints in Γ that would significantly improve model fit (P<.05). We evaluated the overall model
fit by using chi-square statistic and associated P-value, where degrees of freedom are tied to the
number of parameter estimates and elements in the means and covariance matrix (high P-values
implying good fit). We also used the root mean square error of approximation (RMSEA) and the
comparative fit index (CFI), where RMSEA provides a measure of discrepancy per model degree of
freedom [7]. The RMSEA approaches 0 as model fit improves and values close to 0.06 or less
represent adequately fitting models[6], and CFI values greater than 0.95 are generally accepted as
describing adequately fitting models [20].
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