Rationale and Objectives
Perturbations in cerebral blood volume (CBV), blood flow (CBF), and metabolic rate of oxygen (CMRO 2 ) lead to associated changes in tissue concentrations of oxy- and deoxy-hemoglobin (Δ O and Δ D ), which can be measured by near-infrared spectroscopy (NIRS). A novel hemodynamic model has been introduced to relate physiological perturbations and measured quantities. We seek to use this model to determine functional traces of cbv( t ) and cbf( t ) − cmro 2 ( t ) from time-varying NIRS data, and cerebrovascular physiological parameters from oscillatory NIRS data (lowercase letters denote the relative changes in CBV, CBF, and CMRO 2 with respect to baseline). Such a practical implementation of a quantitative hemodynamic model is an important step toward the clinical translation of NIRS.
Materials and Methods
In the time domain, we have simulated O ( t ) and D ( t ) traces induced by cerebral activation. In the frequency domain, we have performed a new analysis of frequency-resolved measurements of cerebral hemodynamic oscillations during a paced breathing paradigm.
Results
We have demonstrated that cbv( t ) and cbf( t ) − cmro 2 ( t ) can be reliably obtained from O ( t ) and D ( t ) using the model, and that the functional NIRS signals are delayed with respect to cbf( t ) − cmro 2 ( t ) as a result of the blood transit time in the microvasculature. In the frequency domain, we have identified physiological parameters (e.g., blood transit time, cutoff frequency of autoregulation) that can be measured by frequency-resolved measurements of hemodynamic oscillations.
Conclusions
The ability to perform noninvasive measurements of cerebrovascular parameters has far-reaching clinical implications. Functional brain studies rely on measurements of CBV, CBF, and CMRO 2 , whereas the diagnosis and assessment of neurovascular disorders, traumatic brain injury, and stroke would benefit from measurements of local cerebral hemodynamics and autoregulation.
Near-infrared spectroscopy (NIRS) can assess noninvasively cerebral hemodynamics and brain function by being sensitive to cerebral concentrations of deoxyhemoglobin ( D ) and oxy-hemoglobin ( O ). Noninvasive measurements of task-related functional activity with NIRS, or fNIRS, have been reported . These hemodynamic changes result from changes in the cerebral blood volume (CBV), cerebral blood flow (CBF), and metabolic rate of oxygen (CMRO 2 ) as a result of brain activation and neurovascular coupling. Understanding the interplay between these physiological/functional/metabolic processes and the measured signals with functional neuroimaging techniques such as fNIRS and functional magnetic resonance imaging is the major objective of hemodynamic models (for a review, see Buxton, 2012 ).
A novel hemodynamic model has been recently introduced to provide an analytical tool for the study of oscillatory (frequency domain) and time varying (time domain) hemodynamics that are measurable with NIRS . The model relates normalized perturbations in CBV, CBF, and CMRO 2 to the dynamics of O and D concentrations in tissue. In particular, this model treats the cerebral microvasculature in terms of three compartments (arterial, capillary, venous) and describes the effects of changes in blood volume in all three compartments (even though the capillary contribution to blood volume changes may be negligible), and the effects of changes in blood flow and metabolic rate of oxygen in the capillary compartment (direct effects) and the venous compartment (indirect effects). This novel model can be applied to measurements in the time domain ( O ( t ), D ( t )), where hemodynamic changes are induced over time, and in the frequency domain (via the phasors O (ω), D (ω)), where induced hemodynamic oscillations are measured as a function of the frequency of oscillation. Hemodynamic oscillations at a specific frequency can be induced by a number of protocols including paced breathing , head-up-tilting , squat-stand maneuvers , and pneumatic thigh-cuff inflation , leading to a technique that we have recently proposed, coherent hemodynamics spectroscopy (CHS) .
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. In the frequency domain, we demonstrate how the model can be used to measure a number of physiologically relevant parameters such as the blood transit time in the microvasculature and the cutoff frequency for cerebral autoregulation. The work presented here demonstrates, in practical terms, that the new hemodynamic model is a workable model for translation of NIRS measurements into functional and physiological parameters. The feasibility of a practical implementation of this mathematical model, in combination with noninvasive NIRS and fNIRS measurements, is a critical element for its translation toward functional and clinical studies.
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Hemodynamic model
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Time domain equations
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O(t)=ctHb[S(a)CBV(a)0(1+cbv(a)(t))+<S(c)>Ƒ(c)CBV(c)0+S(v)CBV(v)0(1+cbv(v)(t))]++ctHb[<S(c)>S(v)(<S(c)>−S(v))Ƒ(c)CBV(c)0h(c)RC−LP(t)+(S(a)−S(v))CBV(v)0h(v)G−LP(t)]∗[cbf(t)−cmro2(t)], O
(
t
)
=
ctHb
[
S
(
a
)
CBV
0
(
a
)
(
1
+
cbv
(
a
)
(
t
)
)
+
<
S
(
c
)
Ƒ
(
c
)
CBV
0
(
c
)
+
S
(
v
)
CBV
0
(
v
)
(
1
+
cbv
(
v
)
(
t
)
)
]
+
+
ctHb
[
<
S
(
c
)
S
(
v
)
(
<
S
(
c
)
−
S
(
v
)
)
Ƒ
(
c
)
CBV
0
(
c
)
h
R
C
−
L
P
(
c
)
(
t
)
+
(
S
(
a
)
−
S
(
v
)
)
CBV
0
(
v
)
h
G
−
L
P
(
v
)
(
t
)
]
∗
[
cbf
(
t
)
−
cmro
2
(
t
)
]
,
D(t)=ctHb[(1−S(a))CBV(a)0(1+cbv(a)(t))+(1−<S(c)>)Ƒ(c)CBV(c)0+(1−S(v))CBV(v)0(1+cbv(v)(t))]+−ctHb[<S(c)>S(v)(<S(c)>−S(v))Ƒ(c)CBV(c)0h(c)RC−LP(t)+(S(a)−S(v))CBV(v)0h(v)G−LP(t)]∗[cbf(t)−cmro2(t)], D
(
t
)
=
ctHb
[
(
1
−
S
(
a
)
)
CBV
0
(
a
)
(
1
+
cbv
(
a
)
(
t
)
)
+
(
1
−
<
S
(
c
)
)
Ƒ
(
c
)
CBV
0
(
c
)
+
(
1
−
S
(
v
)
)
CBV
0
(
v
)
(
1
+
cbv
(
v
)
(
t
)
)
]
+
−
ctHb
[
<
S
(
c
)
S
(
v
)
(
<
S
(
c
)
−
S
(
v
)
)
Ƒ
(
c
)
CBV
0
(
c
)
h
R
C
−
L
P
(
c
)
(
t
)
+
(
S
(
a
)
−
S
(
v
)
)
CBV
0
(
v
)
h
G
−
L
P
(
v
)
(
t
)
]
∗
[
cbf
(
t
)
−
cmro
2
(
t
)
]
,
T(t)=ctHb CBV0[1+cbv(t)]. T
(
t
)
=
ctHb CBV
0
[
1
+
cbv
(
t
)
]
.
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O0=ctHb[S(a)CBV(a)0+<S(c)>Ƒ(c)CBV(c)0+S(v)CBV(v)0], O
0
=
ctHb
[
S
(
a
)
CBV
0
(
a
)
+
<
S
(
c
)
Ƒ
(
c
)
CBV
0
(
c
)
+
S
(
v
)
CBV
0
(
v
)
]
,
D0=ctHb[(1−S(a))CBV(a)0+(1−<S(c)>)Ƒ(c)CBV(c)0+(1−S(v))CBV(v)0], D
0
=
ctHb
[
(
1
−
S
(
a
)
)
CBV
0
(
a
)
+
(
1
−
<
S
(
c
)
)
Ƒ
(
c
)
CBV
0
(
c
)
+
(
1
−
S
(
v
)
)
CBV
0
(
v
)
]
,
T0=ctHb CBV0. T
0
=
ctHb CBV
0
.
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h(c)RC−LP(t)=H(t)et(c)e−et/t(c), h
R
C
−
L
P
(
c
)
(
t
)
=
H
(
t
)
e
t
(
c
)
e
−
e
t
/
t
(
c
)
,
h(v)G−LP(t)=10.6(t(c)+t(v))e−π[t−0.5(t(c)+t(v))]2/[0.6(t(c)+t(v))]2, h
G
−
L
P
(
v
)
(
t
)
=
1
0.6
(
t
(
c
)
+
t
(
v
)
)
e
−
π
[
t
−
0.5
(
t
(
c
)
+
t
(
v
)
)
]
2
/
[
0.6
(
t
(
c
)
+
t
(
v
)
)
]
2
,
in which H ( t ) is the Heaviside unit step function— H ( t ) = 0 for t < 0; H ( t ) = 1 for t ≥ 0. We note that both impulse responses are convolved with cbf(t)−cmro2(t) cbf
(
t
)
−
cmro
2
(
t
) in Equations (1) and (2) , as indicated by the convolution operator ∗ ∗ .
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Measuring the Time Course of cbv and the Difference cbf-cmro 2
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cbv(t)=ΔTT0. cbv
(
t
)
=
Δ
T
T
0
.
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cbf˜(ω)−cmro˜2(ω)=ΔO˜(ω)−ΔD˜(ω)T0−(2S(a)−1)CBV(a)0CBV0cbv˜(a)(ω)−(2S(v)−1)CBV(v)0CBV0cbv˜(v)(ω)2[<S(c)>S(v)(<S(c)>−S(v))Ƒ(c)CBV(c)0CBV0H(c)RC−LP(ω)+(S(a)−S(v))CBV(v)0CBV0H(v)G−LP(ω)], cbf
˜
(
ω
)
−
cmro
˜
2
(
ω
)
=
Δ
O
˜
(
ω
)
−
Δ
D
˜
(
ω
)
T
0
−
(
2
S
(
a
)
−
1
)
CBV
0
(
a
)
CBV
0
cbv
˜
(
a
)
(
ω
)
−
(
2
S
(
v
)
−
1
)
CBV
0
(
v
)
CBV
0
cbv
˜
(
v
)
(
ω
)
2
[
<
S
(
c
)
S
(
v
)
(
<
S
(
c
)
−
S
(
v
)
)
Ƒ
(
c
)
CBV
0
(
c
)
CBV
0
H
R
C
−
L
P
(
c
)
(
ω
)
+
(
S
(
a
)
−
S
(
v
)
)
CBV
0
(
v
)
CBV
0
H
G
−
L
P
(
v
)
(
ω
)
]
,
in which the complex transfer functions H(c)RC−LP(ω) H
R
C
−
L
P
(
c
)
(
ω
) and H(v)G−LP(ω) H
G
−
L
P
(
v
)
(
ω
) (which are the Fourier transforms of the corresponding impulse response functions in Eqs. (1) and (2) ) are given by :
H(c)RC−LP(ω)=11+(ωt(c)e)2√e−itan−1(ωt(c)e) H
R
C
−
L
P
(
c
)
(
ω
)
=
1
1
+
(
ω
t
(
c
)
e
)
2
e
−
i
tan
−
1
(
ω
t
(
c
)
e
)
H(v)G−LP(ω)=e−ln22[ω0.281(t(c)+t(v))]2e−iω0.5(t(c)+t(v)). H
G
−
L
P
(
v
)
(
ω
)
=
e
−
ln
2
2
[
ω
0.281
(
t
(
c
)
+
t
(
v
)
)
]
2
e
−
i
ω
0.5
(
t
(
c
)
+
t
(
v
)
)
.
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cbv(t)=CBV(a)0CBV0cbv(a)(t)+CBV(v)0CBV0cbv(v)(t) cbv
(
t
)
=
CBV
0
(
a
)
CBV
0
cbv
(
a
)
(
t
)
+
CBV
0
(
v
)
CBV
0
cbv
(
v
)
(
t
)
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cbv(a)(t)=σCBV0CBV(a)0cbv(t), cbv
(
a
)
(
t
)
=
σ
CBV
0
CBV
0
(
a
)
cbv
(
t
)
,
cbv(v)(t)=(1−σ)CBV0CBV(v)0cbv(t), cbv
(
v
)
(
t
)
=
(
1
−
σ
)
CBV
0
CBV
0
(
v
)
cbv
(
t
)
,
where σ σ is a constant such that 0≤σ≤1 0
≤
σ
≤
1 . If one assumes that cbv(a)(t)=cbv(v)(t) cbv
(
a
)
(
t
)
=
cbv
(
v
)
(
t
) , then σ=CBV(v)0/(CBV(a)0+CBV(v)0) σ
=
CBV
0
(
v
)
/
(
CBV
0
(
a
)
+
CBV
0
(
v
)
) .
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Frequency Domain Equations
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O(ω)=ctHb[S(a)CBV(a)0cbv(a)(ω)+S(v)CBV(v)0cbv(v)(ω)]++ctHb[<S(c)>S(v)(<S(c)>−S(v))Ƒ(c)CBV(c)0H(c)RC−LP(ω)+(S(a)−S(v))CBV(v)0H(v)G−LP(ω)][cbf(ω)−cmro2(ω)], O
(
ω
)
=
ctHb
[
S
(
a
)
CBV
0
(
a
)
cb
v
(
a
)
(
ω
)
+
S
(
v
)
CBV
0
(
v
)
cb
v
(
v
)
(
ω
)
]
+
+
ctHb
[
<
S
(
c
)
S
(
v
)
(
<
S
(
c
)
−
S
(
v
)
)
Ƒ
(
c
)
CBV
0
(
c
)
H
R
C
−
L
P
(
c
)
(
ω
)
+
(
S
(
a
)
−
S
(
v
)
)
CBV
0
(
v
)
H
G
−
L
P
(
v
)
(
ω
)
]
[
cbf
(
ω
)
−
cmr
o
2
(
ω
)
]
,
D(ω)=ctHb[(1−S(a))CBV(a)0cbv(a)(ω)+(1−S(v))CBV(v)0cbv(v)(ω)]+−ctHb[<S(c)>S(v)(<S(c)>−S(v))Ƒ(c)CBV(c)0H(c)RC−LP(ω)+(S(a)−S(v))CBV(v)0H(v)G−LP(ω)][cbf(ω)−cmro2(ω)], D
(
ω
)
=
ctHb
[
(
1
−
S
(
a
)
)
CBV
0
(
a
)
cb
v
(
a
)
(
ω
)
+
(
1
−
S
(
v
)
)
CBV
0
(
v
)
cb
v
(
v
)
(
ω
)
]
+
−
ctHb
[
<
S
(
c
)
S
(
v
)
(
<
S
(
c
)
−
S
(
v
)
)
Ƒ
(
c
)
CBV
0
(
c
)
H
R
C
−
L
P
(
c
)
(
ω
)
+
(
S
(
a
)
−
S
(
v
)
)
CBV
0
(
v
)
H
G
−
L
P
(
v
)
(
ω
)
]
[
cbf
(
ω
)
−
cmr
o
2
(
ω
)
]
,
T(ω)=ctHb[CBV(a)0cbv(a)(ω)+CBV(v)0cbv(v)(ω)], T
(
ω
)
=
ctHb
[
CBV
0
(
a
)
cb
v
(
a
)
(
ω
)
+
CBV
0
(
v
)
cb
v
(
v
)
(
ω
)
]
,
in which H(c)RC−LP(ω) H
R
C
−
L
P
(
c
)
(
ω
) and H(v)G−LP(ω) H
G
−
L
P
(
v
)
(
ω
) are the complex transfer function given in Equations (11) and (12) , and we have set cbv(c)(ω)=0 cb
v
(
c
)
(
ω
)
=
0 because of the negligible dynamic dilation and recruitment of capillaries in brain tissue . The notation in Equations (16)-(18) matches that in Equations (1)-(3) , and we stress that the cbv , cbf , and cmro2 phasors are all dimensionless, with their magnitude indicating the amplitude of oscillations normalized to the average, or baseline, values. Because of the high-pass nature of the cerebral autoregulation process that regulates CBF in response to blood pressure changes , we consider the following relationship between cbf and cbv :
cbf(ω)=kH(AutoReg)RC−HP(ω)cbv(ω)=kH(AutoReg)RC−HP(ω)[CBV(a)0CBV0cbv(a)(ω)+CBV(v)0CBV0cbv(v)(ω)], cbf
(
ω
)
=
k
H
R
C
−
H
P
(
AutoReg
)
(
ω
)
cbv
(
ω
)
=
k
H
R
C
−
H
P
(
AutoReg
)
(
ω
)
[
CBV
0
(
a
)
CBV
0
cb
v
(
a
)
(
ω
)
+
CBV
0
(
v
)
CBV
0
cb
v
(
v
)
(
ω
)
]
,
in which k is the inverse of the modified Grubb exponent, H(AutoReg)RC−HP(ω) H
R
C
−
H
P
(
AutoReg
)
(
ω
) is the RC high-pass transfer function with cutoff frequency ω(AutoReg)c ω
c
(
AutoReg
) that describes the effect of autoregulation, and the second equalities follows from Equation (13) . More precisely, the expression of the RC high-pass transfer function is:
H(AutoReg)RC−HP(ω)=11+(ω(AutoReg)cω)2⎷eitan−1(ω(AutoReg)cω) H
R
C
−
H
P
(
AutoReg
)
(
ω
)
=
1
1
+
(
ω
c
(
AutoReg
)
ω
)
2
e
i
tan
−
1
(
ω
c
(
AutoReg
)
ω
)
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Measuring Physiological Parameters with CHS
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D(ω)O(ω)=|D(ω)||O(ω)|ei{Arg[D(ω)]−Arg[O(ω)]}, D
(
ω
)
O
(
ω
)
=
|
D
(
ω
)
|
|
O
(
ω
)
|
e
i
{
Arg
[
D
(
ω
)
]
−
Arg
[
O
(
ω
)
]
}
,
O(ω)T(ω)=|O(ω)||T(ω)|ei{Arg[O(ω)]−Arg[T(ω)]}. O
(
ω
)
T
(
ω
)
=
|
O
(
ω
)
|
|
T
(
ω
)
|
e
i
{
Arg
[
O
(
ω
)
]
−
Arg
[
T
(
ω
)
]
}
.
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D(ω)O(ω)=(1−S(a))CBV(a)0cbv(a)(ω)CBV(v)0cbv(v)(ω)+(1−S(v))−[<S(c)>S(v)(<S(c)>−S(v))Ƒ(c)CBV(c)0CBV(v)0H(c)RC−LP(ω)+(S(a)−S(v))H(v)G−LP(ω)]kCBV(v)0CBV0H(AutoReg)RC−HP(ω)[CBV(a)0cbv(a)(ω)CBV(v)0cbv(v)(ω)+1]S(a)CBV(a)0cbv(a)(ω)CBV(v)0cbv(v)(ω)+S(v)+[<S(c)>S(v)(<S(c)>−S(v))Ƒ(c)CBV(c)0CBV(v)0H(c)RC−LP(ω)+(S(a)−S(v))H(v)G−LP(ω)]kCBV(v)0CBV0H(AutoReg)RC−HP(ω)[CBV(a)0cbv(a)(ω)CBV(v)0cbv(v)(ω)+1] D
(
ω
)
O
(
ω
)
=
(
1
−
S
(
a
)
)
CBV
0
(
a
)
cb
v
(
a
)
(
ω
)
CBV
0
(
v
)
cb
v
(
v
)
(
ω
)
+
(
1
−
S
(
v
)
)
−
[
<
S
(
c
)
S
(
v
)
(
<
S
(
c
)
−
S
(
v
)
)
Ƒ
(
c
)
CBV
0
(
c
)
CBV
0
(
v
)
H
R
C
−
L
P
(
c
)
(
ω
)
+
(
S
(
a
)
−
S
(
v
)
)
H
G
−
L
P
(
v
)
(
ω
)
]
k
CBV
0
(
v
)
CBV
0
H
R
C
−
H
P
(
AutoReg
)
(
ω
)
[
CBV
0
(
a
)
cb
v
(
a
)
(
ω
)
CBV
0
(
v
)
cb
v
(
v
)
(
ω
)
+
1
]
S
(
a
)
CBV
0
(
a
)
cb
v
(
a
)
(
ω
)
CBV
0
(
v
)
cb
v
(
v
)
(
ω
)
+
S
(
v
)
+
[
<
S
(
c
)
S
(
v
)
(
<
S
(
c
)
−
S
(
v
)
)
Ƒ
(
c
)
CBV
0
(
c
)
CBV
0
(
v
)
H
R
C
−
L
P
(
c
)
(
ω
)
+
(
S
(
a
)
−
S
(
v
)
)
H
G
−
L
P
(
v
)
(
ω
)
]
k
CBV
0
(
v
)
CBV
0
H
R
C
−
H
P
(
AutoReg
)
(
ω
)
[
CBV
0
(
a
)
cb
v
(
a
)
(
ω
)
CBV
0
(
v
)
cb
v
(
v
)
(
ω
)
+
1
]
O(ω)T(ω)=S(a)CBV(a)0cbv(α)(ω)CBV(v)0cbv(v)(ω)+S(v)+[<S(c)>S(v)(<S(c)>−S(v))F(c)CBV(c)0CBV(v)0H(c)RC−LP(ω)+(S(α)−S(v))H(v)G−LP(ω)]kCBV(v)0CBV0H(AutoReg)RC−HP(ω)[CBV(α)0cbv(α)(ω)CBV(v)0cbvv(ω)+1]CBVα0cbv(α)(ω)CBV(v)0cbv(v)(ω)+1. O
(
ω
)
T
(
ω
)
=
S
(
a
)
CBV
0
(
a
)
cbv
(
α
)
(
ω
)
CBV
0
(
v
)
cbv
(
v
)
(
ω
)
+
S
(
v
)
+
[
<
S
(
c
)
S
(
v
)
(
<
S
(
c
)
−
S
(
v
)
)
F
(
c
)
CBV
0
(
c
)
CBV
0
(
v
)
H
R
C
−
L
P
(
c
)
(
ω
)
+
(
S
(
α
)
−
S
(
v
)
)
H
G
−
L
P
(
v
)
(
ω
)
]
k
CBV
0
(
v
)
CBV
0
H
R
C
−
H
P
(
AutoReg
)
(
ω
)
[
CBV
0
(
α
)
cbv
(
α
)
(
ω
)
CBV
0
(
v
)
cbv
v
(
ω
)
+
1
]
CBV
0
α
cbv
(
α
)
(
ω
)
CBV
0
(
v
)
cbv
(
v
)
(
ω
)
+
1
.
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Methods
Time Domain
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y(t)=A((t−t0)δ−1βδe−β(t−t0)Γ(δ)), y
(
t
)
=
A
(
(
t
−
t
0
)
δ
−
1
β
δ
e
−
β
(
t
−
t
0
)
Γ
(
δ
)
)
,
in which y ( t ) stands for either ΔO ( t ) or ΔD ( t ), t is time, and t 0 is the time at which brain activation starts. Γ represents the gamma function, which acts as a normalizing parameter, and δ and β are constants for which we set values of δ = 8 and β = 0.6 seconds −1 . The amplitude A was set to 3 μMs for ΔO ( t ) and −1 μMs for ΔD ( t ). These parameters were chosen to best represent typical hemodynamic signals during activation , as seen in Figure 1 a, where the ΔO ( t ) and ΔD ( t ) traces peak simultaneously at t − t 0 = 11.7s. We set the baseline total hemoglobin concentration T0 T
0 = 55 μM and the baseline tissue saturation S0 S
0 = 65%. These values fall within typical values reported in the literature for the human brain, which range between 42 and 79 μM for T0 T
0 , and between 55 and 75% for S0 S
0 . Because the hemodynamic model depends on several parameters (as described previously), we have studied the sensitivity of cbf(t)−cmro2(t) cbf
(
t
)
−
cmro
2
(
t
) on the values assigned to these parameters. We further compared the model output with a steady-state approach that has been used extensively in the literature . In comparison to the dynamic model , a steady-state model does not consider any temporal shifts between cbf( t )—or cmro 2 ( t )—and ΔO ( t ) or ΔD ( t ). Mayhew et al. expressed the venous contributions to the tissue concentrations of D and T ( D(v) D
(
v
) , T(v) T
(
v
) ) in terms of the measurable overall tissue concentrations of D and T as follows :
ΔD(v)D(v)∣∣0−ΔT(v)T(v)∣∣0=γrΔDD0−γtΔTT0=−(cbf(t)−cmro2(t)). Δ
D
(
v
)
D
(
v
)
|
0
−
Δ
T
(
v
)
T
(
v
)
|
0
=
γ
r
Δ
D
D
0
−
γ
t
Δ
T
T
0
=
−
(
cbf
(
t
)
−
cmro
2
(
t
)
)
.
in which γr γ
r and γt γ
t have been assumed to be constants within the range 0.2 to 5 and D ( v ) 0 and T ( v ) 0 are the baseline deoxy hemoglobin concentration and total hemoglobin concentration in the venous compartment, respectively. Equation (26) fits in the definition of steady-state models because it introduces no temporal shift between the changes in the D and T hemoglobin concentrations ( D ( v ) , T ( v ) ), and the blood flow and oxygen consumption perturbations (cbf( t ), cmro 2 ( t )) that cause them. Recently, Fantini derived the following explicit expressions for the coefficients γr γ
r and γt γ
t , under the approximation (1−S(a))≅0 (
1
−
S
(
a
)
)
≅
0 :
γr=(1−<S(c)>)Ƒ(c)CBV(c)0CBV(v)0+(1−S(v))<S(c)>S(v)(<S(c)>−S(v))Ƒ(c)CBV(c)0CBV(v)0+(S(a)−S(v)), γ
r
=
(
1
−
<
S
(
c
)
)
Ƒ
(
c
)
CBV
0
(
c
)
CBV
0
(
v
)
+
(
1
−
S
(
v
)
)
<
S
(
c
)
S
(
v
)
(
<
S
(
c
)
−
S
(
v
)
)
Ƒ
(
c
)
CBV
0
(
c
)
CBV
0
(
v
)
+
(
S
(
a
)
−
S
(
v
)
)
,
γt=1−S(v)<S(c)>S(v)(<S(c)>−S(v))Ƒ(c)CBV(c)0CBV0+(S(a)−S(v))CBV(v)0CBV0ΔCBV(v)ΔCBV. γ
t
=
1
−
S
(
v
)
<
S
(
c
)
S
(
v
)
(
<
S
(
c
)
−
S
(
v
)
)
Ƒ
(
c
)
CBV
0
(
c
)
CBV
0
+
(
S
(
a
)
−
S
(
v
)
)
CBV
0
(
v
)
CBV
0
Δ
CBV
(
v
)
Δ
CBV
.
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Frequency Domain
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Table 1
Upper and Lower Limits for the Six Fitting Parameters of the Model
t ( c ) (s)t ( v ) (s)Ƒ(c)CBV(c)0CBV(v)0 Ƒ
(
c
)
CBV
0
(
c
)
CBV
0
(
v
) (CBV(a)0cbv(a))(CBV(v)0cbv(v)) (
CBV
0
(
a
)
cbv
(
a
)
)
(
CBV
0
(
v
)
cbv
(
v
)
) ω(AutoReg)c2π(Hz) ω
c
(
AutoReg
)
2
π
(
Hz
) kCBV(v)0CBV0 k
CBV
0
(
v
)
CBV
0 Lower limit 0.4 1 0.8 0.2 0 0.4 Upper limit 1.4 3 2.4 5 0.15 1.6
( a ) , contributions from arterial compartments; ( c ) , contributions from capillary compartments; CBV, cerebral blood volume; cbv, relative change in CBV with respect to baseline; k , inverse of the modified Grubb exponent; t ( c ) , capillary blood transit time in seconds (s); t , time; ( v ) , contributions from venous compartments; ω c , cutoff frequency of autoregulation.
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Results
Time Domain Results for cbv(t) and cbf (t) − cmro 2 (t)
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S0=S(a)CBV(a)0+S(a)1−exp(−αt(c))αt(c)Ƒ(c)CBV(c)0+S(a)exp(−αt(c))CBV(v)0CBV0. S
0
=
S
(
a
)
CBV
0
(
a
)
+
S
(
a
)
1
−
exp
(
−
α
t
(
c
)
)
α
t
(
c
)
Ƒ
(
c
)
CBV
0
(
c
)
+
S
(
a
)
exp
(
−
α
t
(
c
)
)
CBV
0
(
v
)
CBV
0
.
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Frequency Domain Results
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|  |
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Table 2
Results of the Fitting Procedure for the Six Parameters of the Model, Reported in Terms of Their Mean Value and Standard Deviation
t ( c ) (s)t ( v ) (s)Ƒ(c)CBV(c)0CBV(v)0 Ƒ
(
c
)
CBV
0
(
c
)
CBV
0
(
v
) (CBV(a)0cbv(a))(CBV(v)0cbv(v)) (
CBV
0
(
a
)
cbv
(
a
)
)
(
CBV
0
(
v
)
cbv
(
v
)
) ω(AutoReg)c2π(Hz) ω
c
(
AutoReg
)
2
π
(
Hz
) kCBV(v)0CBV0 k
CBV
0
(
v
)
CBV
0 Mean ± SD 0.92 ± 0.18 1.29 ± 0.26 1.08 ± 0.27 2.95 ± 0.85 0.035 ± 0.002 0.59 ± 0.10
( a ) , contributions from arterial compartments; ( c ) , contributions from capillary compartments; CBV, cerebral blood volume; cbv, relative change in CBV with respect to baseline; k , inverse of the modified Grubb exponent; t ( c ) , capillary blood transit time in seconds (s); t , time; ( v ) , contributions from venous compartments; ω c , cutoff frequency of autoregulation.
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Discussion
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Conclusions
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