A New Functional MRI Approach for Investigating Modulations of Brain Oxygen Metabolism

Abstract

Functional MRI (fMRI) using the blood oxygenation level dependent (BOLD) signal is a common technique in the study of brain function. The BOLD signal is sensitive to the complex interaction of physiological changes including cerebral blood flow (CBF), cerebral blood volume (CBV), and cerebral oxygen metabolism (CMRO2). A primary goal of quantitative fMRI methods is to combine BOLD imaging with other measurements (such as CBF measured with arterial spin labeling) to derive information about CMRO2. This requires an accurate mathematical model to relate the BOLD signal to the physiological and hemodynamic changes; the most commonly used of these is the Davis model. Here, we propose a new nonlinear model that is straightforward and shows heuristic value in clearly relating the BOLD signal to blood flow, blood volume and the blood flow-oxygen metabolism coupling ratio. The model was tested for accuracy against a more detailed model adapted for magnetic fields of 1.5, 3 and 7T. The mathematical form of the heuristic model suggests a new ratio method for comparing combined BOLD and CBF data from two different stimulus responses to determine whether CBF and CMRO2 coupling differs. The method does not require a calibration experiment or knowledge of parameter values as long as the exponential parameter describing the CBF-CBV relationship remains constant between stimuli. The method was found to work well for 1.5 and 3T but is prone to systematic error at 7T. If more specific information regarding changes in CMRO2 is required, then with accuracy similar to that of the Davis model, the heuristic model can be applied to calibrated BOLD data at 1.5T, 3T and 7T. Both models work well over a reasonable range of blood flow and oxygen metabolism changes but are less accurate when applied to a simulated caffeine experiment in which CBF decreases and CMRO2 increases.

Full text

A New Functional MRI Approach for Investigating
Modulations of Brain Oxygen Metabolism
Valerie E. M. Griffeth
1
, Nicholas P. Blockley
2
, Aaron B. Simon
1
, Richard B. Buxton
2,3
*
1Department of Bioengineering and Medical Scientist Training Program, University of California San Diego, La Jolla, California, United States of America, 2Center for
Functional Magnetic Resonance Imaging, Department of Radiology, University of California San Diego, La Jolla, California, United States of America, 3Kavli Institute for
Brain and Mind, University of California San Diego, La Jolla, California, United States of America
Abstract
Functional MRI (fMRI) using the blood oxygenation level dependent (BOLD) signal is a common technique in the study of
brain function. The BOLD signal is sensitive to the complex interaction of physiological changes including cerebral blood
flow (CBF), cerebral blood volume (CBV), and cerebral oxygen metabolism (CMRO
2
). A primary goal of quantitative fMRI
methods is to combine BOLD imaging with other measurements (such as CBF measured with arterial spin labeling) to derive
information about CMRO
2
. This requires an accurate mathematical model to relate the BOLD signal to the physiological and
hemodynamic changes; the most commonly used of these is the Davis model. Here, we propose a new nonlinear model
that is straightforward and shows heuristic value in clearly relating the BOLD signal to blood flow, blood volume and the
blood flow-oxygen metabolism coupling ratio. The model was tested for accuracy against a more detailed model adapted
for magnetic fields of 1.5, 3 and 7T. The mathematical form of the heuristic model suggests a new ratio method for
comparing combined BOLD and CBF data from two different stimulus responses to determine whether CBF and CMRO
2
coupling differs. The method does not require a calibration experiment or knowledge of parameter values as long as the
exponential parameter describing the CBF-CBV relationship remains constant between stimuli. The method was found to
work well for 1.5 and 3T but is prone to systematic error at 7T. If more specific information regarding changes in CMRO
2
is
required, then with accuracy similar to that of the Davis model, the heuristic model can be applied to calibrated BOLD data
at 1.5T, 3T and 7T. Both models work well over a reasonable range of blood flow and oxygen metabolism changes but are
less accurate when applied to a simulated caffeine experiment in which CBF decreases and CMRO
2
increases.
Citation: Griffeth VEM, Blockley NP, Simon AB, Buxton RB (2013) A New Functional MRI Approach for Investigating Modulations of Brain Oxygen Metabolism. PLoS
ONE 8(6): e68122. doi:10.1371/journal.pone.0068122
Editor: Essa Yacoub, University of Minnesota, United States of America
Received August 31, 2012; Accepted May 29, 2013; Published June 27, 2013
Copyright: ß2013 Griffeth et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This research was supported by funding from National Institutes of Health (NIH) grants NS-36722, MH-095298, EB-00790 and EB-009830 (www.nih.gov).
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: rbuxton@ucsd.edu
Introduction
Functional magnetic resonance imaging (fMRI) is commonly
used to map patterns of brain activation based on blood
oxygenation level dependent (BOLD) signal changes [1]. A neural
stimulus rapidly causes a large increase in cerebral blood flow
(CBF) that is not matched in magnitude by the change in the
cerebral metabolic rate of oxygen (CMRO
2
) [2]. This mismatch,
defined as the coupling ratio n(DCBF/DCMRO
2
), leads to an
increase in blood oxygenation that in large part determines the
magnitude of the BOLD response. The coupling ratio is of interest
because it is not constant but rather depends on factors such as
brain region, stimulus type, aging and alterations in the baseline
state due to drugs such as caffeine [3–8]. The current paradigm for
examining variability in nrelies on the Davis model [9] to analyze
combined BOLD and CBF data from two stimulus response
experiments along with data from an additional calibration
experiment. This is a complicated data acquisition, and the
analysis is further complicated by the mathematical form of the
Davis model, which tends to obscure an underlying simplicity in
the relationship between BOLD, CBF and CMRO
2
[10].
Davis and colleagues introduced this model for the BOLD effect
using it as the foundation for the calibrated BOLD method, and
this work remains the basis for calibrated BOLD studies today [9].
In the Davis model the BOLD signal is a nonlinear function of
fractional changes in CBF and CMRO
2
, multiplied by a scaling
parameter M. The factor Mis a lumped parameter, which includes
many variables that could scale the BOLD signal and depends on
both aspects of the physiological baseline state (oxygen extraction
fraction, venous blood volume, and hematocrit) and also on
parameters of the data acquisition (magnetic field strength and the
echo time) [10,11]. The essence of the calibrated BOLD method is
that this scaling parameter, M, is measured in a separate
experiment. In the original Davis method and still the most
commonly used approach [12–22], the calibration experiment to
calculate Mutilizes inhalation of a hypercapnic gas mixture to
elicit BOLD and CBF responses with the assumption that CO
2
alters CBF but not CMRO
2
[23,24].
However, the original derivation of the Davis model neglected
intravascular signal changes and volume exchange effects associ-
ated with changes in cerebral blood volume (CBV), including
changes on the arterial side that are thought to be the dominant
site of CBV changes [25,26]. Recently we developed a detailed
biophysical model of the BOLD signal (DBM) [10], which includes
all of these additional effects while also specifically modeling effects
related to arterial, capillary and venous blood volume changes
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with activation. While this model is too detailed to apply routinely
in the calibrated BOLD experiment because many of the
physiological parameters are unknown, it provides the solid
theoretical framework necessary for relating the underlying
metabolic and hemodynamic changes to the measured signals.
We previously used this DBM to test the accuracy of the Davis
model when applied to the analysis of calibrated BOLD data,
finding that errors in the estimated CMRO
2
change were
surprisingly modest given that important components of the
BOLD effect were neglected in the original derivation [10].
Effectively, the Davis model parameters were providing an
approximate description of the factors that were left out, beyond
the parameters’ original definition in the model, and thus
complicating their interpretation in physiological terms. In
addition, the choice of parameter values had a relatively weak
effect on the accuracy of the estimated CMRO
2
change, provided
the model employed was used consistently to calculate both M
from the hypercapnia experiment and also the CMRO
2
change
from the activation experiment. This observation suggested that
the Davis model may be more complicated than it needs to be
(despite the fact that important effects were missing from its
original derivation). This prompted us to look for a model that
would be both simpler mathematically and that would explicitly
include the effects left out of the Davis model allowing
straightforward parameter interpretation.
Here we present a new, heuristic model of the BOLD response
that is a pure nonlinear function of CBF scaled by a lumped factor,
which includes the CBF/CMRO
2
coupling ratio n. Inspired by the
simple mathematical form of this new model, we present a
straightforward ‘‘ratio method’’ to test whether the blood flow-
oxygen metabolism coupling ratio is the same for two stimuli using
only a comparison of the BOLD and CBF response ratios. This
method is independent of model parameters assuming they remain
consistent across experimental states, and it does not rely on an
additional calibration experiment. The reliability of the new
method was tested using the DBM [10] and as a demonstration the
model was used to analyze data from a recent study of visual
stimulus contrast [27]. Application of this technique will expand
our understanding of why the mismatch between blood flow and
oxygen metabolism occurs by simplifying the approach for
detecting variations in the coupling ratio for different stimuli from
combined BOLD and CBF data.
When quantitative information about the CMRO
2
change is
necessary, the heuristic model can also be used in the same way as
the Davis model to analyze calibrated BOLD data. To examine
the accuracy of the heuristic model in this application, we again
used the DBM to simulate measurements of both stimulus
responses and calibration responses for different combinations of
physiological states. This assessment was complementary to our
previous examination of the Davis model as we again compared
the results against the ‘‘true’’ CMRO
2
change from the DBM
[10]. This analysis demonstrates that the heuristic model has
comparable accuracy to the Davis model.
Methods and Results
Modifications to the Detailed Biophysical Model of the
BOLD Signal
The DBM includes effects of intravascular and extravascular
signal changes, hematocrit (Hct), baseline oxygen extraction
fraction (E
0
), blood volume fractions for different vascular
compartments, changes in these volumes as CBF changes, tissue
signal properties and imaging parameters [10]. In the current
work, an additional feature in which the arteries are split into two
compartments (large arteries - Aand arterioles - a) was added in
order to allow for partial oxygen desaturation of the arterioles. For
simplicity, the second arteriolar compartment was modeled as
equal in size to the fully saturated compartment with their sum
comparable in size to previous modeling for the total arterial
compartment. Desaturation occurring in the arteriolar compart-
ment was modeled as a weighted average of arterial and venous
hemoglobin saturation (Table 1, s= 0–0.2).
To permit modeling of the effect of hyperoxia on the BOLD
signal, we also updated the DBM to calculate compartmental
oxygen saturation from oxygen partial pressures using the
Severinghaus equation [28]. The arterial oxygen concentration
was calculated first followed by the venous oxygen concentration
using the E
0
and Eq (10–13) from Chiarelli et al. [29]. Venous
oxygen saturation was then calculated using linear interpolation of
the Severinghaus equation. Arteriolar saturation was calculated as
noted in the previous paragraph and capillary saturation was
calculated also using a weighted average of arteries and veins [10].
We also expanded the DBM to simulate the BOLD signal at
1.5T and 7T, since the original model was only for 3T. This
required adjusting the DBM to include magnetic field specific echo
time (TE) and baseline extravascular signal decay rate (R
2)
(Table 2) [30]. Intravascular signal decay rates were again
determined using a quadratic model fit to data relating intravas-
cular R
2to oxygen saturation. At 1.5T hematocrit-dependent
values were calculated according to Silvennoinen et al. [31]. At 7T
data from Blockley et al. [32] was used to determine intravascular
R
2dependence on oxygen saturation independent of hematocrit.
Changes in extravascular signal decay rates are linearly dependent
on B
0
, which was already included in the model [1]. Calculations
of oxygenation and blood volume were performed as published
previously [10].
Simple BOLD Signal Models
The new model, as derived in Appendix S1, is:
BOLD(%)~A1{1=fðÞw1{av{1=nðÞð1Þ
Important terms in this model include the scaling parameter (A),
CBF in the active state normalized to its value in the baseline state
(f), the ratio of fractional changes in CBF and CMRO
2
(n), and the
exponent relating the CBF change to the venous CBV change (a
v
).
One additional parameter of importance is r, which is CMRO
2
in
the active state normalized to its value in the baseline state and is
related to nand fthrough n=(f21)/(r21). In the following we
refer to Eq. (1) as the heuristic model, because it clearly shows the
basic physiological factors that affect the BOLD response: it is
driven by the CBF change, but strongly modulated by both the
venous CBV change and the CBF/CMRO
2
coupling ratio. The
parameter a
v
is from the Grubb relationship, which relates the
normalized venous CBV change (v)tofthrough the equation
v~fav. For calculations using the heuristic model, we set
a
v
= 0.2 as determined by Chen and Pike [33].
The two underlying assumptions of the heuristic model
discussed and illustrated in the Appendix are: (1) The fractional
BOLD signal change is directly proportional to the absolute
change in total dHb content in a voxel (simulations for 1.5T, 3T
and 7T are shown in Figures S1–S3); and (2) the fractional change
in tissue concentration of total dHb is equal to the fractional
change of venous dHb (Figure S4). Figure S5 examines the
relationship between changes in CBF and CMRO
2
in comparison
to changes in both the BOLD signal and dHb content. At first
A New fMRI Approach Testing for CMRO2 Modulation
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glance, these assumptions appear to be too restrictive for the full
complexity of the BOLD response, so we used the DBM to explore
the errors in these assumptions and the ultimate effect of using the
heuristic model for estimation of CMRO
2
changes, which is
discussed below.
Of note in the heuristic model is the non-linear dependence of
the BOLD signal on the CBF change, which is reflected in the
term incorporating f. This term reflects the ceiling effect on the
BOLD response: very large increases in CBF will tend to produce
the largest BOLD signals as 1/fand 1/napproach zero (Eq. B2).
Physically this corresponds to a clearance of dHb from the
vasculature. For changes in CBF approaching zero (f= 1), the
BOLD response is a linear reflection of the fractional CMRO
2
change as shown in Appendix S2 (Eq. B7): BOLD(%)~A(1{r).
The second term in the heuristic model relates the BOLD signal
change to nwhile also incorporating the dependence of CBV on
CBF. This term reflects that the largest BOLD signal will result
from a large n, when the saturation of hemoglobin is maximized
through a much lower oxygen metabolism change relative to the
blood flow change (e.g. hypercapnia) [24,34,35]. This term also
reflects that smaller changes in venous CBV relative to CBF
(smaller a
v
) will lead to larger BOLD signal changes. The physical
interpretation of this is that a smaller increase in dHb containing
blood volume leads to a larger BOLD signal, because any increase
in volume will increase the dHb content of a voxel in opposition to
the oxygen extraction fraction decrease, which dominates the
BOLD signal change.
For comparison, the Davis model expressed in the same terms
is:
BOLD(%)~M1{fa{bf{1
nz1

b
"#
ð2Þ
The Davis model has two parameters, aand b, and the original
values for these parameters as applied to 1.5T BOLD data were
a= 0.38 and b= 1.5 [9]. In this analysis, we set a= 0.2, consistent
with recent data indicating that most blood volume change occurs
in the arterioles [25,26,33]. As originally derived in the Davis
model, brelates blood oxygenation to transverse relaxivity and is
dependent on magnetic field strength (B
0
). Recent studies based on
previous modeling of this relationship have proposed adjusting b
to reflect this B
0
dependence using the following values: b= 1.5 at
1.5T, b= 1.3 at 3T and b= 1 at 7T [9,36,37]. We refer to the
Davis model with these parameter values as the B
0
-adjusted Davis
models (e.g. the 1.5T-adjusted Davis model).
We have also proposed previously treating aand bas free
parameters in the Davis model, and designate the Davis model
using these parameters as the B
0
-‘‘free parameter’’ Davis models
[10]. This approach attempts to provide the best fit to the surface
of BOLD change as a function of CBF and CMRO
2
change using
the mathematical form of the Davis model, but divorcing the
parameters aand bfrom their original physical definitions. The
process of fitting these parameters involves assuming our best guess
of the physiology (Tables 1 and 2) in order to simulate the BOLD
signal for CBF changes between 240% and 80% and CMRO
2
changes between 220% and 40%. We then normalized both the
Davis model and the simulated data using an idealized hypercap-
nia simulation (DCBF = 60% and DCMRO
2
= 0%). This removes
the scaling parameter, M, from the equation leaving only aand b
Table 1. Input parameters to the detailed model.
Variable Description Best Guess Reasonable Variation
V
0
Total baseline CBV fraction 0.05 0.03–0.07
v
A
Arterial fraction of baseline CBV 0.1 0.05–0.15
v
a
Arteriolar fraction of baseline CBV 0.1 0.05–0.15
v
v
Venous fraction of baseline CBV 0.4 0.2–0.6
a
T
Grubb’s constant relating total CBV to CBF 0.38 0.25–0.55
a
v
Exponent relating venous CBV to CBF 0.2 0.1–0.38
Hct Resting hematocrit 0.44 0.37–0.5
E
0
Resting oxygen extraction fraction 0.4 0.3–0.5
sFraction of arteriolar blood reflecting venous saturation 0.1 0–0.2
PaO
2
Arterial partial pressure of oxygen 104 mmHg n/a
lIntravascular to extravascular spin density ratio 1.15 0.9–1.3
doi:10.1371/journal.pone.0068122.t001
Table 2. Input parameter to the detailed model that are sensitive to B
0
.
Variable Description Best Guess Reasonable Variation
1.5T 3T 7T 1.5T 3T 7T
R
2EResting extravascular rate of signal
decay
11.6 s
21
25.1 s
21
35 s
21
9–14 s
21
20–30 s
21
28–42 s
21
TE Echo time 50 ms 32 ms 25 ms 40–60 ms 20–40 ms 15–35 ms
doi:10.1371/journal.pone.0068122.t002
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to be fitted. We discuss the impact of these parameters in a later
section while listing their values here: a= 0.1 and b= 1 at 1.5T,
a= 0.13 and b= 0.92 at 3T, and a= 0.3 and b= 1.2 at 7T. These
values are perhaps counterintuitive, but when treating aand bas
free-parameters they lose their physiological meaning and instead
simply provide the best fit of the model to the simulated data given
the physiological assumptions. In other words our values for a
should not be used as an indication of the relationship between
CBV and CBF, and our values for bshould not be used to describe
the relationship between the magnetic susceptibility due to
deoxyhemoglobin and R
2. In addition to the Davis model
parameter set noted above, we also examined the impact of fixing
b= 1, which makes the form of the Davis model more analogous
to that of the heuristic model.
The Ratio Method
The form of the heuristic model suggests a new method for
analyzing combined BOLD-CBF data independent of the scaling
parameter in order to determine whether nchanges for responses
to different stimuli from the same baseline state without requiring
a calibration experiment. Because the flow response term is
separate from the coupling ratio term, we can use Eq. (1) to
directly compare whether two stimulus responses have the same
flow-metabolism coupling. Denoting one stimulus as a reference
(‘‘ref’’) and the comparison stimulus as ‘‘x’’, we first create a null
hypothesis that nis the same for the two stimuli (n
x
=n
ref
). Taking
the ratio of Eq. (1) for the two stimulus responses makes a specific
prediction for a nonlinear combination of measured BOLD and
CBF responses that is independent of the model parameter values:
BOLDx=BOLDref ~1{1=fx
ðÞw1{1=fref

ð3Þ
This method assumes that both Aand a
v
remain constant
between the two stimulus responses. Under these conditions the
exact values of Aand a
v
are not needed because this ratio is
independent of the model parameters. Differences in ncan then
easily be detected using a sign rank test or similar statistical
analysis comparing the measured BOLD ratio with the ratio
predicted by the non-linear CBF terms for equal values of nin Eq
(3).
To test the accuracy of this new method, we employed the
DBM to simulate BOLD and CBF responses for a reasonable
range of physiological and imaging parameters (Tables 1 and 2). A
reference data set with n
ref
= 2 was produced and compared to
n
x
= 1.8, n
x
= 2 and n
x
= 2.2 at 1.5T, 3T and 7T (Fig. 1A–C). These
values of nare typical for fMRI activation experiments
[7,9,11,18,27,38–40]. Data sets for each value of ncontained
10,000 simulations. Previously published combined BOLD and
CBF data associated with changes in visual stimulus contrast [27]
were then examined using this method (Fig. 1D). A sign rank test
was used to determine whether the flow ratio was statistically
different than the BOLD ratio with results for p noted.
This approach works well for 1.5T and 3T (Fig. 1A–B) as
stimulus responses with values of n
x
not equal to n
ref
are shown to
have BOLD ratios that diverge from the non-linear CBF ratio.
Additionally when n
x
=n
ref
, the BOLD ratios are shown to be
approximately equal to the non-linear CBF ratios reflected in the
blue dots falling along the dashed line of identity. This is most
apparent on the inset histograms taken from additional simulations
for which the non-linear CBF ratio was fixed to 0.5: at 1.5T there
is a very small tendency to underestimate the BOLD ratio when
n
x
=n
ref
, but there is good separation between the data otherwise
(Fig. 1A). Similarly at 3T the blue dots representing n
x
=n
ref
fall
equally on either side of 0.5 and are separated from the data
representing both n
x
.n
ref
and n
x
,n
ref
(Fig. 1B). In certain cases, this
method can also be used to make inferences about changes in
CMRO
2
: when n
x
.n
ref
but DCBF
x
,DCBF
ref
, then DCMRO
2,x
must also be less than DCMRO
2,ref
. Similarly when n
x
,n
ref
but
DCBF
x
.DCBF
ref
, then DCMRO
2,x
must also be more than
DCMRO
2,ref
. Note that when 1{1=fx
ðÞw1{1=fref

then
DCBF
x
.DCBF
ref
.
This approach is less reliable at 7T where the BOLD signal is
more sensitive to changes in dHb (Fig. 1C). Specifically, the ratio
method fails by predicting a difference in n
x
from n
ref
when no
difference exists as reflected in the blue dots deviating from the line
of identify. In the case that DCBF
x
is less than DCBF
ref
, there is a
tendency for this method to predict an increase in n
x
relative to n
ref
,
and when DCBF
x
is greater than DCBF
ref
, there is a tendency for
this method to predict a decrease in n
x
relative to n
ref
. From the
inset histogram, this deviation of the data from the predicted
BOLD ratio of 0.5 designated by the black bar is clearly apparent
(Fig. 1C).
We also tested whether the coupling of CBF and CMRO
2
impacts the effectiveness of the ratio method. As suggested by the
form of the heuristic model, the ratio method is most sensitive to
1/n. By testing different values of n, we found that for positive
coupling of CBF and CMRO
2
the ratio method is most effective
when differences in 1/nare greater than 0.05. For n= 2, this
corresponds to n= 1.8 or n= 2.2. For n= 4, this corresponds to
n= 3.3 or n= 5 (Fig. S6). We examined a wide range of both
positive and negative values of n, and included in Figure S6 n=21
corresponding to a decrease in CBF and an increase in CMRO
2
.
A general pattern emerged from simulations across a broad range
of coupling parameter values showing that the ratio method breaks
down close to a coupling of n= 1.3, which is frequently associated
with the null point of the BOLD signal (data not shown).
Specifically at 1.5T and 3T, the ratio method appears to fail for
0.75,n,1.5. The limits at 7T extend somewhat higher such that
the model fails for 0.75,n,2.25. In addition to this limitation on
the range of nthat can be examined, these tests also revealed a
systematic bias in the predicted BOLD signal ratio. For positive n,
any difference between the BOLD ratio and non-linear CBF ratio
less than 0.02 should be viewed with caution and outside the
ability of the ratio method to discriminate. For example, if the
non-linear CBF ratio predicts a BOLD ratio of 0.5, any BOLD
ratio between 0.48 and 0.52 should be considered to have the
same n. For negative n, this difference is 0.04. These biases are
likely due to error inherent to the use of the relatively simple
heuristic model to describe the full complexity of the BOLD signal.
Having confirmed the accuracy of the ratio method for
simulated data at 3T, we applied this approach to a study of 9
subjects comparing different levels of visual stimulus contrast [27].
Consistent with the results from the previous analysis using the
Davis model, we found that the response to 1% contrast has a
lower nthan the response to 100% contrast (p,0.01) (Fig. 1D).
The BOLD ratios at 4% and 9% contrast also fall below the
prediction by the CBF ratios, but the results do not reach statistical
significance. Assuming n
ref
= 2.3 at 100% contrast consistent with a
previous calibrated-BOLD study [22], these ratio differences
translate to nvalues of 1.66, 2.14 and 2.25 (with associated Cohen-
dstatistics of 0.6, 1.04 and 1.71) respectively.
Simulating the Calibrated-BOLD Experiment
Next we simulated a calibrated-BOLD experiment to compare
the heuristic model to the B
0
-adjusted Davis model [9] for
accuracy in determining the CMRO
2
change at 3 magnetic field
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strengths. To examine the effects of various parameters on
calculations of the CMRO
2
change, we used two ranges for n
(n= 2 and n=21). Activation studies typically show increases in
both CBF and CMRO
2
with nabout equal to 2 [18,22,40]. In
contrast, we found that caffeine as a stimulus decreased CBF and
increased CMRO
2
, with nabout equal to 21 [7]. To determine
the effectiveness of the simple models, this comparison required
three steps: (1) using the detailed model to simulate the
hypercapnia response assuming DCBF = 60% and
DCMRO
2
= 0%, (2) using the detailed model to simulate the
stimulus response with n=2 (%DCBF = 50% and
%DCMRO
2
= 25%) or the caffeine response with n=21
(%DCBF = 225% and %DCMRO
2
= 25%), and (3) using the
B
0
-adjusted Davis model and the heuristic model to analyze this
data in order to calculate the CMRO
2
change in response to either
the simulated stimulus or caffeine experiments. Inputs to the
detailed model were varied individually over the ranges specified
in Tables 1 and 2 to determine the effect on DCMRO
2
calculations. Parameters other than the one specified were kept
constant at the best guess values.
Figure 2 presents deviations from the DBM simulated CMRO
2
response when using the simple models at 1.5T, 3T and 7T. These
results demonstrate that %DCMRO
2
calculated using the heuristic
model is consistent with %DCMRO
2
produced by the Davis
model. It shows that even for variation in multiple physiological
inputs to the DBM (Tables 1 and 2), the heuristic model with
a
v
= 0.2 [33] predicts changes in CMRO
2
comparable to
predictions by the B
0
-adjusted Davis model at 1.5T, 3T and 7T.
These simple models are both quite accurate at the typical
coupling ratio of n= 2, and at 3T there is a small underestimation
bias of 26.4% error by the heuristic model compared to 22.1%
for the Davis model (Fig. 2B). Both models are most sensitive to
differences in a
v
, reflecting the impact of DCBF and venous DCBV
coupling. If a
v
is allowed to vary across a range of 0.1–0.3 within
the DBM while the assumptions about a
v
in the heuristic and
Davis models are kept constant, then the 3T-adjusted Davis model
Figure 1. The ratio method for analysis of combined BOLD (dS) and CBF data. The DBM was used to simulate BOLD data from changes in
CBF and set values of n. 10,000 simulations were performed using the ranges for the model inputs noted in Tables 1 and 2. The data was compared to
a reference of n
ref
= 2. Inset histograms show the distribution of dS ratios for a non-linear CBF ratio of 0.5. At 1.5T (A) and 3T (B), the ratio method
separates the data well while predicting n
x
=n
ref
data will fall along the line of identity. At 7T (C), the ratio method does not perform as well,
particularly for n
x
=n
ref
for which the data deviates from the line of identity. (D) Application of the ratio method to data examining the effect of visual
stimulus contrast on the coupling of CBF and CMRO
2
in 9 subjects. 100% contrast flickering checkerboard was used as the reference with results
showing that 1% contrast has a significantly lower n.
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will predict %DCMRO
2
between 22.2% and 29.6% (for a true
value of 25%, with a maximum error of 18.4% of that 25%
change in CMRO
2
) while the heuristic model predicts
%DCMRO
2
between 21.5% and 28.5% (maximum error of
614.0%). These results are consistent with the pattern found
previously using slightly different values for aand bin the Davis
model [10]. At 1.5T the same pattern of underestimating the
CMRO
2
change at n= 2 was found: for the 1.5T-adjusted Davis
model the underestimation bias was 24.5% and for the heuristic
model the bias was 213.3% (Fig. 2A). At 7T the basic models both
overestimate %DCMRO
2
with overall bias percent errors of 7.7%
using the heuristic model and 2.9% using the Davis model
(Fig. 2C). These patterns of bias due to parameter variation are
consistent when values of nup to 6 (%DCBF = 60%) are examined
(not shown).
These basic models are less accurate when used to analyze
changes associated with caffeine consumption (Fig. 2D–F), which
we modeled in the DBM as a 225% CBF decreases and 25%
CMRO
2
increase. This is a slightly extreme test case of CBF/
CMRO
2
coupling changes due to caffeine, since previous findings
estimated a smaller CMRO
2
increase for this level of CBF
decrease [7]. At both 1.5T and 3T, the models systematically
underestimate this simulated change in CMRO
2
. For example at
3T, the B
0
-adjusted Davis model calculates a CMRO
2
increase of
only 17.7% (error of 229.2%) while the heuristic model calculates
18.8% (error of 224.8%) (Fig. 2E). Of note at this value of flow-
metabolism coupling, the Davis and heuristic models at 3T and 7T
are most sensitive to variation in baseline dHb content as
determined by v
v
and E
0
, and less sensitive to changes in blood
flow-blood volume coupling, a
v
. At 1.5T the simple models are
most sensitive to the intravascular/extravascular proton density
ratio (l) followed by tissue R
2while showing less overall sensitivity
to parameter variability. At 7T the heuristic model is more
accurate with an error bias of 21.6% while the Davis model
overestimates %DCMRO
2
with an error of 15.2%. For this
combination of B
0
, CBF and CMRO
2
, the magnitude of error
bars is also much larger suggesting greater sensitivity to changes in
dHb at 7T.
We also used this method of simulating the calibrated BOLD
experiment to examine the efficacy of these simple models over a
larger range of CBF and CMRO
2
combinations while keeping
other physiology constant at our best guess (Tables 1 and 2). We
included in this comparison the Davis model with b= 1 (Fig. 3). As
an example at 3T and for our best guess of the physiology, the
simulated hypercapnia BOLD signal was 4.6% for a 60% CBF
increase producing the following estimates of the scaling param-
eters: B
0
-adjusted, M
HC
= 11.4%; fixed b=1, M
HC
= 14.6%; and
heuristic model A
HC
= 15.3%. For an activation resulting in
%DCBF = 25% and %DCMRO
2
= 10%, the BOLD signal was
1.3%, and in this case the estimates of the CMRO
2
change with
activation were: 10.3% for the B
0
-adjusted Davis model, 9.3% for
the fixed b= 1 Davis model, and 9.7% for the heuristic model. We
also tested the impact of treating aand bas free fitting parameters
within the Davis model to minimize error in CMRO
2
calculations,
and using this model DCMRO
2
was estimated to equal 10.0%.
Most apparent from Figure 3 is the performance similarity of
these models at different field strengths. Although subtle differ-
ences between the models exist, they all appear to function
reasonably well for positive coupling of CBF and CMRO
2
changes, particularly at 3T. Across all field strengths, the B
0
-‘‘free
parameter’’ Davis models perform the most consistently while the
B
0
-adjusted Davis models also perform well. While we had
expected the heuristic model to perform with the most similarity to
the Davis model with b= 1, it in fact shares similarity to both the
B
0
-adjusted and b= 1 Davis models. Notably, most of the models
have difficulty correctly determining a CMRO
2
change when it is
associated with a decrease in CBF, as in changes associated with
caffeine consumption. The exception to this are the free parameter
Davis models and surprisingly the heuristic model at 7T (Fig. 3D,
H, J, L). The drawback to the free parameter Davis model is that it
requires one to discard physiological meaning for the parameters a
and b. Furthermore values for aand bwould need to be updated
as new information affecting the DBM becomes available.
Specifically ano longer corresponds to blood volume changes
alone, so updating the model as new information about the true
venous CBV values becomes available is more complicated. Note
this is also clear from the values of aand bat 3T, which differ
slightly from those published previously due to the inclusion of a
desaturated arteriolar compartment here [10].
Calibrated BOLD Analysis of Experimental Data
Using these two simple models, we examined experimental data
by reanalyzing CMRO
2
changes in response to a visual stimulus
pre- and post-caffeine as well as changes due to caffeine alone
[7,22]. This data set was acquired on a GE Signa Excite 3T whole-
body system using a spiral dual-echo ASL PICORE QUIPSS II
pulse sequence [41]. Responses to 20 s blocks of an 8 Hz flickering
checkerboard were measured pre- and post-caffeine. For complete
details of the experiment see Perthen et al. [22]. Results were
compared to the same data published previously so that in
addition to the heuristic model we examined DCMRO
2
calcula-
tions by the original Davis model (a= 0.38 and b= 1.5), 3T-
adjusted Davis model (a= 0.2 and b= 1.3), and fitted free
parameter Davis model (a= 0.13 and b= 0.92).
Results from this analysis using these models are shown in
Table 3. The estimated values of CMRO
2
were similar for all the
models with slight systematic differences consistent with the
simulations in Figures 2 and 3. The small differences in
%DCMRO
2
predictions reflect the similarity of these models in
calculating changes in CMRO
2
when both blood flow and
metabolism increase. In contrast, the models diverge when
calculating the CMRO
2
response to caffeine alone (n<21). While
the 3T free parameter model calculated a CMRO
2
change of
21.7%, the heuristic model found 17.1%, the 3T-adjusted Davis
model found 15.7%, and the original Davis model calculated
13.3% (Table 3). We anticipate that the free parameter values are
most accurate in this area of CBF-CMRO
2
and that the other
models all underestimate the CMRO
2
change for caffeine. This is
consistent with our previous findings [10].
Scaling Parameters and Limits of the Davis and Heuristic
Models
The above tests comparing the simple models have focused on
the effects of physiological variation in properties such as CBV and
hematocrit, and we can think of estimates of the scaling parameter
(Mor A) as the fitted value that best approximates the BOLD
signal behavior over the defined physiological range. However, it
is useful to also consider the limits implied by these mathematical
expressions because the scaling parameter is often described in
physical terms as the maximum possible BOLD signal produced
when all dHb has been eliminated [9]. By this interpretation, one
could in principle determine the scaling factor by extreme
physiological manipulations to eliminate deoxyhemoglobin. This
raises the basic question of whether these simple models remain
accurate under these extreme physiological conditions. That is, is
the scaling parameter in the model best thought of as an absolute
physiological variable or as a fitting parameter that adjusts the
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mathematical form to be accurate over a normal physiological
range?
To address this question we considered the limiting forms of the
simple models and compared them with the limits calculated from
the DBM. This is a somewhat subtle question because the
elimination of dHb can be accomplished through two basic paths:
a dramatic increase of CBF (perhaps augmented with hyperoxia),
or a reduction of absolute CMRO
2
to zero. We considered both
scenarios with the DBM. First we modeled the elimination of dHb
based on the carbogen-10 experiments by Gauthier et al. [42]
allowing CBF to increase by 200%, slightly more than their
finding of 160% produced using combined visual stimulus with
10% hypercapnia. We then combined this increase in CBF with
an increase in arterial oxygen partial pressure (PaO
2
)upto
600 mmHg consist with about 90% inspired O
2
[29]. We also
included a simulation with DCBF = 100% (f= 2) and
PaO
2
= 390 mmHg to mimic the actual findings from the
carbogen-10 experiments [42]. No literature was found on the
relationship of CBV to CBF as blood flow increases beyond typical
physiological measurements to provide an empirical basis for
modeling such effects within the DBM, so here we kept a
T
and a
v
constant at 0.38 and 0.2 respectively [33,43]. Second, to simulate
oxygen metabolism cessation, the CMRO
2
input to the DBM was
simply decreased to zero without altering CBF or any other input.
Figure 2. Heuristic vs. B
0
-adjusted Davis model applied to the calibrated BOLD experiment: Estimating %DCMRO
2
calculation bias
due to variability in physiological parameters. Eleven input parameters to the detailed model were varied around reasonable values as defined
in Tables 1 and 2 while other parameters were held constant at the best guess of physiology. The activation and ideal hypercapnia experiments were
simulated for each of these physiological states at (A,B) 1.5T, (C,D) 3T and (E,F) 7T. The true CMRO
2
change with activation is shown as a dashed line,
while the bars showing the range of calculated values is shaded from dark to light for increasing values of the associated physiological parameter.
Davis model parameters aand bwere adjusted for B
0
as noted in the figure. In the heuristic model, a
v
= 0.2 across all B
0
. (A,C,E) Accuracy of the
models at n=2(%DCBF = 50%) and variable B
0
. (B,D,F) Accuracy of the models at n=21(%DCBF = 225%) and variable B
0
. Note values for aand bin
the Davis model are consistent for each B
0
.
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While not physiologically plausible, this simulation mimics
complete removal of all dHb without altering CBV.
Note that one complexity of extending the DBM to these
extreme physiological cases is that we model the intravascular and
extravascular susceptibility difference as being minimized at a
hemoglobin saturation of 95% rather than 100% (SO
2,off
= 95%)
based on the work of [44]. Specifically, this assumes that the
susceptibility of tissue is equal to the susceptibility of plasma (an
assumption that needs to be tested experimentally). This results in
the maximum BOLD signal occurring at a hemoglobin saturation
less than 100%. Since we are after a calibration factor that reflects
a constant relationship between the BOLD signal and hemoglobin
saturation, we chose to extrapolate to the theoretical maximum
BOLD signal at 100% SvO
2
by projecting from the inflection
point using the inverted slope of the BOLD signal for SvO
2
greater
than 95% (Fig. 4, dashed line). We modeled these mechanisms
using the DBM and plotted BOLD vs. venous hemoglobin
saturation (SvO
2
) in order to determine the most appropriate
definition for the scaling parameters (Fig. 4).
At 1.5T the combined hyperoxia with CBF increase appears
nearly identical to elimination of CMRO
2
as both approach a
limit of 8.9%. At higher magnetic field strengths, these cases
diverge as the increase in CBF leads to the displacement of tissue
volume for blood volume, which has a smaller contribution to the
Figure 3. Absolute error in DCMRO
2-
calculations. Simulated calibrated BOLD calculations were made for the best guess of physiology and
imaging parameters noted in Tables 1 and 2. (A–D) Calculations in the absolute DCMRO
2-
error are shown at 1.5T for the 1.5T-adjusted Davis model,
the heuristic model, the Davis model with a= 0.2 and b= 1, and the free parameter Davis model with aand bfitted as noted. Similar calculations are
shown for 3T (E–H) and 7T (I–L).
doi:10.1371/journal.pone.0068122.g003
Table 3. Comparing DCMRO
2
calculations by different models using the calibrated-BOLD approach applied to pre- and post-
caffeine data.
Heuristic
1
B
0
-adjusted
2
Free parameter
3
Original
4
DBOLD (%) DCBF (%) DCMRO
2
(%) DCMRO
2
(%) DCMRO
2
(%) DCMRO
2
(%)
Pre-caffeine visual
response
1.260.09 52.164.25 25.962.3 26.962.3 27.162.4 23.462.0
Post-caffeine visual
response
1.260.07 57.063.57 36.662.2 37.062.3 36.562.4 32.062.0
Caffeine response 26.361.1 226.963.48 17.167.3 15.767.3 21.768.5 13.366.4
1
Using the heuristic model proposed here with a
v
= 0.2,
2
Using the Davis model with a= 0.2 and b= 1.3;
3
Using the Davis model with a= 0.13 and b= 0.92;
4
Using the Davis model with a= 0.38 and b= 1.5.
doi:10.1371/journal.pone.0068122.t003
A New fMRI Approach Testing for CMRO2 Modulation
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signal at higher B
0
. At 3T, elimination of CMRO
2
results in a
maximum signal of 13.2% while combined hyperoxia-CBF
increase produces a signal of 12.5%. The simulated carbogen-10
experiment at 3T resulted in a signal of 8.0%, which is close to the
actual finding of 7.5%. The difference between dHb elimination
methods is even larger at 7T: decreasing CMRO
2
results in a
maximum signal of 24.7%, but combined hyperoxia-CBF increase
produces a maximum signal of 18.9%. The difference at 7T was
expected, because increased CBF leads to increased CBV
replacing tissue volume without contributing to the BOLD signal
at the higher magnetic field strength [45].
The limits of the Davis and heuristic model can also be
examined for these two conditions with interesting differences
arising. At very large values of CBF the heuristic model predicts a
signal that is less than the maximum BOLD signal: A(1-a
v
)
(Appendix S2, Eq. B2) while the Davis model predicts that the
BOLD signal will simply equal M. At 1.5T and 3T, Figure 4
suggests this decrease is too aggressive since in the DBM
simulations a
v
was assumed to equal 0.2, but at 7T the heuristic
model appears to more accurately reflect behavior at this limit.
When absolute CMRO
2
is reduced to zero (r= 0), the heuristic
model predicts dependence of the BOLD signal on both CBF and
a
v
:BOLD %ðÞ~A1{av:DCBF=CBFðÞ(Appendix S2, Eq. B3).
When the CBF change is small as in Figure 4, this limit becomes
the scaling parameter, A. Under the same circumstances, the Davis
model reduces to Mwith no dependence on a
v
or CBF.
These results show that the maximum BOLD signal therefore is
dependent on how elimination of dHb is achieved, and for both
simple models there are discrepancies between the value of the
scaling parameters and the physical limits of reducing deoxyhe-
moglobin. For this reason, it is better to think of the scaling
parameter as a fitted value that makes the equations accurate over
a normal physiological range, rather than having a more absolute
meaning as the maximum possible BOLD change.
Ethics Statement
The institutional review board at the University of California
San Diego approved the study of human subjects in the previously
published work [22,27], and written informed consent was
obtained from all participants.
Discussion
In this study, we revisited basic modeling of the BOLD signal
and derived a new simplified model that has heuristic value in
clearly showing the physiological factors that affect the BOLD
signal. The heuristic model demonstrates the non-linear depen-
dence of the BOLD signal on cerebral blood flow found in
previous studies [46,47], directly incorporates the flow-metabolism
coupling parameter, n, and also incorporates the dependence of
venous CBV on CBF through a
v
. It was inspired by work with the
much more detailed model [10], which appeared to produce a
very smooth BOLD dependence on CBF and CMRO
2
suggesting
that the parameters aand bof the Davis model may be over-
fitting the data. The form of the heuristic model suggests a new
method comparing BOLD signal ratios to non-linear DCBF ratios
in order to determine whether flow-metabolism coupling varies
with the stimulus. Using the previously developed DBM [10], we
demonstated the effectiveness of the ratio method while also
showing that the accuracy of the heuristic model is comparable to
the Davis model when applied in the calibrated BOLD
experiment.
Ratio Method
The new approach to analyzing combined BOLD and CBF
data is straightforward and relies only on measured data to
determine whether nvaries with the stimulus (i.e. different levels of
visual stimulus contrast, frequencies of finger tapping, or level of
drug administration). As an example, we used the method to
reanalyze a previous study investigating how nvaries with the
contrast of a visual stimulus. In the original analysis the conclusion
that nvaries with stimulus contrast was based on many repeated
tests using the Davis model with different values for M,aand b.
Here using the ratio method, the same conclusion is reached in a
more straightforward manner. It is not apparent from the Davis
model that the comparison of the BOLD signal ratio to the non-
linear CBF response ratio would work, but it is readily apparent
from an examination of the heuristic model, which separates the
CBF response from the coupling parameter term. An additional
application of the ratio method could be in the study of brain
diseases with altered vascular responses. For example in diseases
Figure 4. Simulating the maximum BOLD signal through dHb elimination. The maximum BOLD signal results from complete elimination of
dHb, which can be accomplished by increasing CBF, decreasing CMRO
2
and/or increasing PaO
2
. Here the BOLD signal is shown as a function of SvO
2
at (A) 1.5T, (B) 3T, and (C) 7T. Three mechanisms of dHb reduction are included: hyperoxia combined with CBF increase (blue), CBF increase only
(green) and CMRO
2
cessation (red). Also included is a simulation for DCBF = 100% and PaO
2
= 390 mmHg consistent with findings from Gauthier et al.
[42].
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resulting in reduced blood vessel compliance, increasing stimulus
intensity may not result in the same increase in the CBF response
seen in normal subjects resulting in a constant or decreasing n
rather than increasing n.
There are two limitations of this method: both the scaling
parameter and the relationship between CBF and CBV changes
(a
v
) must remain constant across the comparison. The require-
ment on the scaling parameter to remain constant essentially limits
the technique to a common region of interest and baseline state.
For example, it is not possible using this method to compare flow-
metabolism coupling in the visual cortex to that in the motor
cortex.
An additional limitation emerges at 7T. Simulations using the
DBM confirm that the approach works well for 1.5T and 3T with
the non-linear CBF ratio accurately predicting the BOLD signal
ratio in all cases (Fig. 1A–B), but at 7T the relationship between
the ratios is not a reliable prediction of changes in n. Specifically if
the n-values are in fact the same for two stimuli, the ratio method
at 7T would incorrectly show that the stimulus with the stronger
CBF response had a smaller value of n. Thus although the heuristic
model works reasonably well when calculating %DCMRO
2
from
calibrated BOLD data at all magnetic field strengths (Fig. 2 and 3),
the ratio method is unreliable at 7T. A much smaller bias is
evident at 1.5T, but this deviation from identity is small as
demonstrated from the inset histogram (Fig. 1A).
Calibrated-BOLD using the Heuristic Model
As performed previously for just the Davis model, it is possible
to calculate stimulus associated changes in CMRO
2
using the
heuristic model when BOLD and CBF measurements are
combined with a hypercapnia calibration. To test this we
simulated both an ideal hypercapnia calibration as well as an
activation experiment using the DBM demonstrating that both the
heuristic model and the B
0
-adjusted Davis model produce
reasonable estimates of DCMRO
2
(less than 15% error) for
positive changes in CBF and CMRO
2
(Fig. 2A–C). At 1.5T and
3T both models slightly underestimate changes while they
overestimate changes in CMRO
2
at 7T. As previously reported
[10], the parameter having the largest impact on CMRO
2
calculations by both models and across B
0
is a
v
, emphasizing the
importance of accurately determining the venous CBV-CBF
relationship for future calibrated BOLD studies. In terms of the
effect of other physiological parameters, an interesting standout at
1.5T is l, which is the intravascular/extravascular proton density
ratio. This alters the intravascular to extravascular signal intensity,
a factor that is more important at lower B
0
where the intravascular
signal due to lower intravascular signal decay rates has a relatively
greater impact than at 3T and 7T. At 7T another interesting
standout is a
T
, which emphasizes the importance of total CBV
changes at the higher magnetic field when deoxygenated blood
generates a relatively weak signal so that increases in blood volume
displace tissue without contributing to the BOLD signal leading to
an overall signal decrease. Not shown here, the same pattern of
error is found when using the B
0
-‘‘free parameters’’ in the Davis
model, which is consistent with previous findings [10].
While both models are reasonably accurate for cases in which
both CBF and CMRO
2
increase, the models are less accurate
when CBF and CMRO
2
changes are in opposition (Fig. 2D–F).
Specifically for the Davis model, the B
0
-adjusted values of b
underestimate DCMRO
2
at 1.5T and 3T while overestimating it
at 7T. The heuristic model also does not perform well at 1.5T and
3T in this region of CBF-CMRO
2
coupling, but interestingly it is
much more accurate at 7T. This is consistent with findings in
Figure 3 examining a broad range of CBF-CMRO
2
coupling for
our best guess of physiology. Application of these models to
experimental data showed a similar pattern of CMRO
2
changes
estimated for a visual stimulus response with the simple models in
agreement with the exception of the original Davis model (a= 0.38
and b= 1.5), which estimated a smaller CMRO
2
response
(Table 3). Also consistent with the simulations, the caffeine
CMRO
2
responses calculated by the basic models were more
dissimilar: the original Davis model produced the lowest estimate,
the B
0
-adjusted Davis model and the heuristic model produced
slightly higher estimates, and the free parameter Davis model
produced the highest estimate (Table 3). The inaccuracy of the
Davis model for this region of CBF and CMRO
2
coupling has
been noted previously and can be overcome by treating aand b
both as free parameters in the Davis model then fitting to DBM
simulations [10]. The drawback to this approach is that the
parameters lose their physiological meaning and must be refitted
when new information becomes available.
Examining a full complement of CBF and CMRO
2
changes,
Figure 3 also shows that fixing b= 1 decreased accuracy of the
Davis model for the most common region of CBF-CMRO
2
coupling while there was also unexpected improvement in the
region of CBF decrease and CMRO
2
increase. Although b=1
simplifies the Davis model in line with the simplicity of the
heuristic model, it is still not obvious that the ratio method would
work due to the interaction of the CBF and CMRO
2
terms.
The Scaling Parameter and Additional Comparison of the
Simple Models
When simple models of the BOLD effect are used, the physical
meaning of the scaling parameter (i.e., its relationship to
underlying physiological variables) can become blurred. Here we
considered the question of whether the scaling parameter is
literally the maximum BOLD signal change that would occur if all
of the deoxyhemoglobin was removed, or whether it functions as a
fitting parameter that differs based on the mathematical form of
the particular simple model, adjusting each to fit the data over the
normal physiological range. From Figure 4, it is apparent that the
maximum BOLD signal depends on whether dHb is eliminated by
increasing CBF or decreasing CMRO
2
. Additionally while the
limit of the Davis model in both cases is M, the limit of the
heuristic model is either Awhen CMRO
2
goes to zero or A(1-a
v
)
when CBF approaches infinity. Finally both simulations and
experimental data show that hypercapnia determined values of the
scaling parameter depend not only on the simple model used but
also on the values of the parameters a,band a
v
[7,10]. Therefore
to maximize accuracy of the simple models and avoid ambiguity
introduced by making the scaling parameter equivalent to the
maximum BOLD signal, it is better to determine the scaling
parameter from a calibration experiment, thereby providing a
good fit of the simple models to the physiologically reasonable
range of CBF and CMRO
2
changes.
Our simulations provide further evidence of this: although the
B
0
-adjusted Davis M
HC
is smaller than the heuristic model A
HC
at
3T, both simple models estimate CMRO
2
changes well (Fig. 2 and
3). Furthermore the difference in M
HC
between the original and
free parameter Davis models published previously [10] suggests
strong covariance between the scaling parameter (M), a, and bin
the Davis model. It is through the calibration process that the
simple models become self-correcting, emphasizing that the value
of the scaling parameter depends on the model used to calculate it
rather than on the maximum BOLD response.
A New fMRI Approach Testing for CMRO2 Modulation
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Future Applications
A potentially useful feature of the heuristic model is that if the
variation of nover time during an activation experiment is
relatively small, the BOLD response becomes simply a scaled
version of a pure non-linear function in CBF. In models relating
the BOLD response to underlying physiology (e.g., as a
component of dynamic causal modeling [46]), the ambiguities
due to the baseline state and the CBF/CMRO
2
coupling ratio are
combined into a single scaling parameter, simplifying the
treatment of the forward model from neural responses to
measured BOLD responses.
As shown recently, the heuristic model is also useful for
improving the precision of the CBF response when simultaneous
measurements of BOLD and CBF are acquired [48]. By isolating
dependence of the BOLD signal on the non-linear CBF response,
the unknown parameters A,a
v
and nthat modulate the BOLD
signal can be combined into a single factor. This property relating
the underlying CBF fluctuations simply to the BOLD signal is used
by the BOLD-Constrained Perfusion (BCP) method to dramati-
cally improve the estimate of CBF fluctuations. Specifically, the
heuristic model is used as a constraint in the minimization of the
cost function, which incorporates the measured BOLD signal, the
measured CBF signal, the true underlying BOLD and CBF
signals, and noise.
Conclusions
The heuristic model was inspired by work with the detailed
BOLD model and a desire to develop a simple analysis for
detecting changes in flow-metabolism coupling from combined
BOLD and blood flow data. The heuristic model is advantageous
over previous models, because it simplifies the dependence of the
BOLD signal on blood flow and flow-metabolism coupling and in
doing so suggests the ratio method for analysis of combined BOLD
and CBF data. This approach works very well at 1.5T and 3T, but
does not appear to work at 7T when it predicts a change in nwhen
no change is present. It is remarkable to note that when applied to
calibrated BOLD data the heuristic model with only one fixed
parameter has accuracy similar to the Davis model with
parameters adjusted for the magnetic field strength. At 1.5T, 3T
and 7T, the heuristic model produces consistent results for
DCMRO
2
at n= 2, although they are slightly less accurate than
the B
0
-adjusted Davis model. This small difference is balanced by
greater accuracy of the heuristic model when applied to a
simulated analysis of the response to caffeine particularly at 7T,
which is a somewhat surprising result given the simplicity of the
heuristic model.
Supporting Information
Figure S1 Relationship between the BOLD signal
change and the total change in dHb content (DSdHbT)
at 3T. Scatter plots were produced by independently varying
DCBF (250% to 80%) and DCMRO
2
(230% to 50%) within the
specified ranges. Purple curves are identical in all subplots with the
exception of (D) and represent the best guess physiological case
(Tables 1 and 2). (A) For the best guess of physiological
parameters, the relationship between the BOLD signal and
DSdHbTis linear, but there is a finite width to the curve. In
this case, SdHbT0~0.11 mmol of dHb per liter of tissue. For
DBOLD between 23% and 3%, a fit to this line gives
DBOLD(%) = 2138*DSdHbT. Inset is a histogram of DBOLD
probability distribution around DSdHbT~060.025 mg/mL (i.e.,
variation in the BOLD signal that could result when there is no
change in net tissue dHb). (D) Allowing a wider and still
reasonable distribution of physiology (Tables 1 and 2, Reasonable
Variation) produced more scatter in the relationship between
DBOLD and DSdHbT. For DBOLD between 23% and 3%, a fit
to this line gives DBOLD = 2133*DSdHbT. Inset is a histogram of
DBOLD probability distribution around
DSdHbT~060.025 mg/mL. The remaining panels show how
the curve changes when one of the physiological variables is
altered: (B) varying baseline CBV fraction; (C) varying baseline
venous and capillary CBV fractions; (E) varying the exponent
relating CBF and venous CBV; (F) altering TE.
(TIF)
Figure S2 Relationship between the BOLD signal
change and the total DSdHbTat 1.5T. Scatter plots were
produced by independently varying DCBF and DCMRO
2
as in
Figure S1. (A) For the best guess of physiology, the relationship
between the BOLD signal and DSdHbTis linear, but again there
is a finite width to the curve. For DBOLD between 23% and 3%,
a fit to this line gives DBOLD(%) = 296*DSdHbT. The inset is a
histogram of DBOLD probability distribution around
DSdHbT~060.025 mg/mL is similar to that at 3T As expected,
the BOLD signal shows weaker dependence on the change in dHb
content than at 3T (B–F).
(TIF)
Figure S3 Relationship between the BOLD signal
change and the total DSdHbTat 7T. Scatter plots were
produced by independently varying DCBF and DCMRO
2
as in
Figure S1. (A) For the best guess of physiology, the relationship
between the BOLD signal and DSdHbTis linear with a tighter
distribution than at 3T or 7T. For DBOLD between 23% and
3%, a fit to this data gives DBOLD(%) = 2207*DSdHbT.As
expected, the BOLD signal shows stronger dependence on the
change in dHb content than at 3T (B–F).
(TIF)
Figure S4 Relationship between the normalized venous
change and normalized total change in dHb contents.
Scatter plots were produced as in Figure S1 by independently
varying DCBF (250% to 80%) and DCMRO
2
(230% to 50%)
within the specified ranges. Purple curves are identical in all
subplots with the exception of (D) and represent the best guess
physiological case (Tables 1 and 2). (D) Combined variation of the
parameters within the reasonable ranges (Tables 1 and 2). The
only physiological variable that created a slight deviation from the
identity line is the venous flow-volume relationship expressed as a
v
(E).
(TIF)
Figure S5 Comparing zero BOLD response to zero
change in total dHb content. This plot of the BOLD response
as a function of changes in CBF and CMRO
2
was generated using
our best guess of the physiological inputs to the DBM model at 3T
(Tables 1 and 2). The color scale represents the BOLD signal as a
percent change. The dot-dash line represents DSdHbT~0while
the solid orange line represents DBOLD = 0%. For positive
changes in CBF and CMRO
2
,DSdHbT~0is shown to be
associated with a small positive BOLD signal. This is due to the
intravascular effects of dHb: although the increase in CBV and
decrease in dHb concentration combine to produce no change in
total dHb content and no change in the extravascular signal, the
intravascular signal decay rate decreases due to the decrease in
dHb concentration.
(TIF)
Figure S6 The ratio method for analysis of combined
BOLD (dS) and CBF data: effects of different n.The DBM
A New fMRI Approach Testing for CMRO2 Modulation
PLOS ONE | www.plosone.org 11 June 2013 | Volume 8 | Issue 6 | e68122

was used to simulate BOLD data from changes in CBF and set
values of n. 10,000 simulations were performed using the ranges
for the model inputs noted in Tables 1 and 2. The data was
compared to a reference of n
ref
=21orn
ref
= 4 at B
0
= 1.5T, 3T
and 7T. Inset histograms show the distribution of dS ratios for a
CBF ratio of 0.5. (A,C,E) For n
ref
= 4 at 1.5T (A) and 3T (C), the
ratio method appears to work well, although the data is slightly
more difficult to distinguish, which is expected due to the
decreased sensitivity of the BOLD signal to nat higher values of
n. At 7T (E), the approach is again biased when n
x
=n
ref
. At all
three field strengths, the ratio method separates the data well for
n
ref
=21, although there is bias in the n
x
=n
ref
data. (B,D,F).
(TIF)
Appendix S1 Derivation of the new model.
(DOC)
Appendix S2 Limits of the simple models.
(DOC)
Acknowledgments
The authors are indebted to Christine L. Liang and Joanna E. Perthen for
their work collecting and analyzing data, and to Peter Jezzard and Daniel
P. Bulte for their suggestions and assistance performing cross-field
modeling of the BOLD signal.
Author Contributions
Conceived and designed the experiments: VEMG RBB. Performed the
experiments: VEMG. Analyzed the data: VEG NPB ABS RBB.
Contributed reagents/materials/analysis tools: VEMG NPB. Wrote the
paper: VEMG RBB.
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