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Estimating cerebral oxygen metabolism from fMRI with a dynamic

multi-compartment windkessel model

Theodore J. Huppert1,2, Monica S. Allen3, Solomon G. Diamond2,4, and David A. Boas2,5

1 Assistant Professor of Radiology. University of Pittsburgh, Pittsburgh PA 15213, USA

2 Athinoula A. Martinos Center for Biomedical Imaging Massachusetts General Hospital, Charlestown, MA

02129, USA

3 Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX 76019, USA

4 Assistant Professor of Engineering. Thayer School of Engineering, Dartmouth College, Hanover, NH 03755,

USA

5 Associate Professor of Radiology, HMS, MGH and Member of the Affiliated Faculty Member in the Harvard-

MIT Division of Health Sciences and Technology

Abstract

Stimulus evoked changes in cerebral blood flow, volume, and oxygenation arise from vascular

responses to underlying neuronally mediated changes in vascular tone and cerebral oxygen

metabolism. There is increasing evidence that the magnitude and temporal characteristics of these

evoked hemodynamic changes are additionally influenced by the local properties of the vasculature

including the levels of baseline cerebral blood flow, volume, and blood oxygenation. In this work,

we utilize a physiologically motivated vascular model to describe the temporal characteristics of

evoked hemodynamic responses and their expected relationships to the structural and biomechanical

properties of the underlying vasculature. We use this model in a temporal curve-fitting analysis of

the high-temporal resolution functional MRI data to estimate the underlying cerebral vascular and

metabolic responses in the brain. We present evidence for the feasibility of our model-based analysis

to estimate transient changes in the cerebral metabolic rate of oxygen (CMRO2) in the human motor

cortex from combined pulsed arterial spin labeling (ASL) and blood oxygen level dependent (BOLD)

MRI. We examine both the numerical characteristics of this model and present experimental evidence

to support this model by examining concurrently measured ASL, BOLD, and near-infrared

spectroscopy to validate the calculated changes in underlying CMRO2.

Keywords

Cerebral metabolism; functional neuroimaging; vascular modeling

Introduction

Functional neuroimaging techniques, such as magnetic resonance imaging (MRI) (Belliveau

et al 1991; Ogawa et al 1990) can record changes in cerebral blood flow, volume, or oxygen

saturation that are indirectly related to neuronal activity and reflect the consequences of both

neural-metabolic and neural-vascular coupling. While the ability to image these signals has

Correspondence should be addressed to Theodore Huppert, PhD, Assistant Professor of Radiology, University of Pittsburgh, UPMC

Presbyterian, 200 Lothrop Street, Pittsburgh, PA 15213, E-mail: E-mail: huppertt@upmc.edu.

NIH Public Access

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Published in final edited form as:

Hum Brain Mapp. 2009 May ; 30(5): 1548–1567. doi:10.1002/hbm.20628.

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been invaluable in developing spatial maps of the functional organization of the brain, the

dependence of measured hemodynamic signals on both vascular and metabolic function results

in ambiguous and potentially non-linear relationships with the underlying electrophysiological

and metabolic responses. The potential use of cerebral oxygen metabolism (CMRO2) as a more

accurate indicator of neuronal activation has motivated the recent development of model-based

methods to extract these estimates from functional MRI (fMRI) or optical measurements

(reviewed by (Buxton et al. 2004)).

The majority of studies that have been performed to estimate CMRO2 by fMRI methods have

focused on examining the magnitude of evoked signals. In many cases, these studies have used

long duration functional stimulation and steady-state approximations of the vascular and

oxygen transport models. In addition, many of these studies rely on a hypercapnic calibration

of the blood oxygen level dependent (BOLD) signal, as introduced by Davis et al (Davis et al.

1998), to calibrate the magnitude of signal changes; as these depend on the baseline

physiological and biophysical properties of the vasculature. However, such studies have largely

overlooked the potential to utilize the temporal dynamics of the hemodynamic measurements

and the inter-relationships between the flow, volume, and oxygen-saturation changes, which

can reveal information about the underlying biomechanical properties of the vasculature, such

as vascular transit time, vascular compliance, and baseline vascular features. Several recent

experimental studies have demonstrated that baseline blood flow, volume, and venous oxygen

saturation can have significant effects on the relative temporal characteristics of the overall

hemodynamic changes (e.g. (Cohen et al 2002; Liu et al 2004; Sicard and Duong 2005)). In

light of such experimental data, we hypothesized that the utilization of dynamic information

from multimodal measurements will help to characterize flow and oxygen metabolism changes

more accurately by providing additional information about the properties of the vascular

network based on the relationships of both the temporal evolution and magnitude of the evoked

hemodynamic signals.

Several recent studies have described bottom-up methodologies that use model-based curve-

fitting of the hemodynamic response to estimate the underlying vascular and metabolic states

(Deneux and Faugeras 2006; Huppert et al 2007; Riera et al 2005). In this work, we extend

our previous model described in (Huppert et al. 2007) to use high-temporal resolution,

multimodal measurements to estimate parameters depicting the vascular biomechanics. We

find that this approach allows us to estimate dynamic changes in CMRO2 from measurements

of blood flow and BOLD evoked changes by using our vascular model as the basis for a non-

linear inverse problem. We present three examples to demonstrate the advantages and

limitations of the inverse problem by examining i) the local sensitivity and stability of the

inverse problem, ii) the global robustness of the parameter identification procedure, and iii)

the application of our model to experimental data. In the experimental data, we estimate relative

CMRO2 changes from blood flow and BOLD responses measured during the performance of

a brief motor task activity as measured by pulsed ASL imaging. Finally, we compare the results

of this analysis of fMRI signals with the analysis of measured changes in blood flow, volume,

and oxygen saturation obtained using simultaneously measured fMRI and near-infrared

spectroscopy (NIRS) data.

2. Methods

2.1 Overview of Model

In our dynamic model, evoked changes in the vascular tone and the cerebral metabolic rate of

oxygen (CMRO2) are estimated by the inversion of a non-linear model that describes the

changes in blood flow, volume, and oxygenation in response to the driving forces as

schematically depicted in Fig. 1A. Our approach is similar to that discussed in (Deneux and

Faugeras 2006;Huppert et al 2007;Riera et al 2005). Our model can be divided into three

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components: i) a dynamic forward model that describes the physiological response of the

vascular system to flow and CMRO2 changes (Section 2.1.1 and 2.1.2); ii) the observation

model that describes the biophysics of the fMRI or optical measurement process (Section

2.1.3); and iii) the inverse model that estimates values of the input states and parameters via a

measurement variance weighted least-squares fit of experimental data (Section 2.1.4). The

details of this forward model have been presented previously in (Huppert 2007;Huppert et al.

2007).

2.1.1. Vascular Forward Model—The vascular component of this model is derived from

basic principles of fluid mechanics and is based on the flow and volume changes within a

system consisting of a dilating arteriole, two compliant vascular windkessel compartments and

a constant volume compartment that models the pial veins as depicted in Fig. 1B and is

summarized in Table 1. Briefly, the blood flow between each vascular compartment (Fn-1,n)

is derived from the gradient of the hydrostatic pressure (Pn) between the compartments and

the vascular resistance of the blood vessels (Rn). The vascular resistance is inversely related

to the diameter of the blood vessels according to Poiseuille's law (Washburn 1921) and thereby

related to vascular volumes. A decrease in arterial resistance drives an increase in blood flow.

The resulting hydrostatic pressure between the vascular compartment and the surrounding

tissue (Psurround) causes an increase in the vascular blood volume (Vn). As the capillary and

venous vascular compartments expand against the surrounding tissue, the extra-vascular space

resists compression which gives rise to a saturation of the volume expansion function. Thus,

the vascular capacitance (Cn) decreases as a function of the pressure gradient between the

compartment and the surrounding tissue (Mandeville et al. 1999). In the aforementioned

variables, n indicates the vascular compartment (arterial, capillary, vein or pial vein).

In our model, we assume that the changes in vascular resistance can be approximated using a

canonical driving function to the vascular system similar to the canonical neurovascular model

described in (Buxton et al.). We define the dilation and contraction of the arteries as the sum

of a pair of independent canonical gamma-variant temporal basis functions with variable timing

parameters (refer Table 1: Arterial Dilation). We use these non-linear canonical functions with

variable onset time (τonset), width (σ), and amplitude, to impose a smooth temporal basis for

the arterial resistance state instead of modeling the entire time-series of arterial dilation. A total

of six unknown model parameters describe the arterial dilation and contraction states (refer

Tables 1 and 3).

2.1.2. Oxygen Transport Forward Model—In the second part of the physiological

forward model, net changes in blood oxygenation are modeled by a set of first-order differential

equations that describe the balance between the oxygen supplied by the vascular response and

the oxygen consumed within the tissue. We calculate the oxygen content (cnO2; designating

the nth compartment) in both the vascular segments and the bulk extra-vascular parenchyma

tissue using the oxygen transport model described in (Herman et al 2006; Huppert et al

2007). The rate of transport across each vascular segment can be described by first-order rate

equations and depends on (i) the steady-state flow and (ii) the relative permeability of the

segment which can be determined from the baseline oxygen saturation.

Increased CMRO2 in the extra-vascular compartment is approximated by a canonical temporal

basis function as described in (Huppert et al. 2007). As with the arterial resistance state,

CMRO2 changes are described by a gamma-variant functional form (refer Table 2). Both the

timing and amplitude of this function are estimated as parameters within the model. Unlike

arterial resistance changes only one basis function is used as a regressor maintaining that

evoked CMRO2 changes are expected to be elevated compared to baseline.

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2.1.3. Hemodynamic Observation Models—The second component of our overall

model is a set of observation equations that represent the relationships between the underlying

physiological variables in the model and the fMRI or optical measurements. Such an unified

model of the physiology allows different multimodal information to be fused into a single

consistent estimate of cerebral activation. This provides a mechanism to combine both

complementary measurements like BOLD and flow, with partially redundant information, such

as the common deoxy-hemoglobin contrast between optical and BOLD signals. Since these

measurement models are modular, this framework allows combinations of the different data

sets, namely fMRI, optical, or multimodal data sets, to be utilized and examined for

consistency.

FMRI Measurement Model: The BOLD signal has a complex origin arising from both intra-

and extra-vascular water signals (Buxton et al 2004; Nair 2005; Obata et al 2004; Ogawa et

al 1990). In our model, we consider the contributions to the BOLD signal of all four vascular

compartments (artery through pial veins) and susceptibility effects on the extra-vascular water

signal. We model the BOLD signal according to the equations for intra- and extra-vascular

contrast described in (Obata et al 2004) based on the theory by Yablonskiy and Haacke

(Yablonskiy and Haacke 1994).

(1)

Extra-vascular signal:

Intra-vascular signal (nth compartment):

At the 3 Tesla field strength used in this study, ro = 100s-1 and vo = 80.6 s-1 (Mildner et al.

2001). SIV and SEV are the fractional baseline contributions of the intra- and extra-vascular

signals respectively. These parameters include a bulk representation of the intrinsic relaxation

rates of water molecules in the intra- and extra-vascular space. Since there is debate over the

exact value of this parameter (Lu and van Zijl 2005; Mildner et al 2001), we include it as an

additional unknown parameter (ɛ ≡ SIV / SEV) estimated by the model. In the above equations,

Vn is the ratio of blood volume of the nth compartment to tissue volume. The term, Vo, represents

the baseline volume fraction and acts as a partial volume correction factor for the BOLD

measurement (i.e.

procedure and is included as a calibration factor for the magnitude of the BOLD signal. Since

this factor has explicit physiological meaning, its value contributes to the calculation of the

baseline physiological values.

). V0 is estimated in the model using the minimization

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The measurement sensitivity weighting factors (wn) in Eq. 1 are assumed to be equal for all

four vascular compartments (wn=1). The measurement sensitivity factors describe the intrinsic

sensitivity of the measurement process and depend on MR physics, the physical vessel

dimensions and vessel orientations. In general, the MR measurement sensitivity is a highly

complex function of the underlying vasculature (e.g. (Boxerman et al. 1995)). Our current

assumption of equal measurement sensitivities implies that the vascular contributions to the

signal directly reflect the relative changes in contrast. For example, the largest contribution to

the BOLD signal arises from the venous compartments because the oxygenation changes are

the largest in these veins due to washout effects. This assumption should be examined more

closely in future work. We use a direct projection of the normalized arterial blood flow changes

to model the ASL signal.

NIRS Measurement Model: The NIRS technique measures spectroscopic changes in the

optical absorption properties of tissue (Obrig et al 1996; Villringer and Chance 1997). These

measurements are related to changes in mean oxy- and deoxy-hemoglobin concentrations by

the modified Beer-Lambert law (Cope et al 1988; Delpy et al 1988). We assume that the

measurements reflect the weighted sum over the vascular compartments i.e.

(2)

In this equation, ε(λ) is the molar extinction coefficient of each hemoglobin species and PPF

(λ) is the effective path-length for wavelength (λ) traveled by photons on the diffuse path

through the brain cortex (Hiraoka et al. 1993). The subscript n indicates the vascular

compartment. In principle, NIRS measurements are related to the absolute changes in

hemoglobin concentrations provided that the effective path-lengths are known (Cooper et al.

1996). However, in practice, such path-length factors are difficult to determine without

additional spatial information (Hiraoka et al 1993; Huppert et al 2006a) or direct measurement

(Duncan et al 1995; Leung et al 2006). In our model, hemoglobin changes are normalized to

the baseline volume and therefore the value of absolute baseline total hemoglobin is needed to

scale between the changes predicted in the model and the true measurements. We include a

scaling value (ΩNIRS) as an additional parameter in the model to compensate for the uncertainty

introduced in the exact quantification of the hemoglobin changes measured with NIRS. The

measurement model for NIRS is given by the equation

(3)

Note that if the optical path-length is known then the scaling factor can be used to calculate

the baseline hemoglobin concentration (Ω=[HbTo]·PPF). The normalized term [ΔHbX /

HbTo] can be calculated directly from within the model. The vascular weights (wn) in Eq. 2

and 3 describe the sensitivity of NIRS to the vascular compartments and are assumed to be

uniform for the arteries through veins (wn=1). The pial venous compartment constitutes larger

blood vessels and contributes to changes measured by both the fMRI and optical imaging.

Sensitivity to vessels decreases with increasing radius for NIRS measurements (Liu et al.

1995) and conversely increases for gradient echo BOLD-fMRI (Boxerman et al. 1995).

Therefore, to account for measurement discrepancies, the weight given to the contribution of

the pial vein in the NIRS measurement is included as an additional model parameter (ωpial<1).

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This allows the NIRS and BOLD measurements to potentially have different weights for the

pial compartment and these measurements can be modeled to sample different effective pial

compartments.

2.1.4. The vascular inverse model—There are a total of 20 unknown parameters in the

full model (see Table 3). The full data model utilizes measurements of blood flow, volume and

oxygen saturation from combined multimodal optical and ASL data. The baseline oxygen

saturation contents of the arterial, capillary, and venous compartments have been included as

three parameters within the model. These oxygen saturation parameters were fixed in the

models using the fMRI only and optical only data. In the fMRI alone and NIRS alone models,

there are a total of 15 unknown model parameters since we fix the values for baseline oxygen

extraction parameters and additionally drop two parameters specific to NIRS/fMRI

respectively.

CMRO2 and arterial resistance changes together with the unknown parameters associated with

the model's differential equations can be determined from the inverse of the model. These

parameters are estimated by non-linear curve fitting of the temporal dynamics of observable

hemodynamic parameters. We used a non-linear Levenberg-Marquardt algorithm (Marquardt

1963) implemented in Matlab (The Mathworks; Natick, MA USA) to calculate the model

parameters that define the CMRO2 and arterial dilation functions (Matlab function

lsqnonlin). In contrast to previous approaches that consider only the magnitude of

hemodynamic changes, we account for the full temporal dynamics of the response and

minimize the model using the entire time course of the hemodynamic changes in a single cost

function. Thus, the model is minimized using the temporally vectorized array of discrete

measurements to simultaneously fit the entire dataset. I.e.

(4)

where R is an estimate of the variance of the measurement error for each measurement type

and superscript T indicates the transpose operation. The estimates of the measurement error

variances were calculated from the error in the estimate of the mean evoked hemodynamic

responses calculated from the linear deconvolution model (described in section 2.2.3) from

each experimental session and normalized across the subjects. Weighting the squared model

error of the different measurement types in the minimization cost function helps to weight the

model fit to more confident observations. This routine provides the minimum variance

unbiased estimator for a general linear model as described in (Huppert et al. 2007) and allows

the fusion of multimodal information using a priori knowledge of the measurement error in

each imaging method. In the model-fitting routine, the error cost metric is minimized by

refining the model parameters. To improve the stability of the inverse problem, we constrain

the parameter set within the range of physiological bounds (Table 3). The fitting routine was

iterated until a predefined convergence criterion (10-6 times the variance of the measurement

error) was met. This took approximately 6-10 hours per fitting procedure (Pentium(R)-4 CPU

3.00 GHz). The error bounds of our model prediction were examined using a Markov Chain

Monte Carlo sampling of the parameter-space as described in (Huppert et al. 2007). Because

of the high computational time involved, it is currently impractical to run this model on a per

voxel basis, but will be explored in future extensions of this work.

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2.1.5. Estimation of baseline physiology from model parameters—In our previous

work (Huppert et al. 2007), we described how values for the baseline physiology could be

extracted from the model estimates of baseline volume, vascular transit time, and baseline

oxygen saturation. For example, the vascular mean transit time, which is defined as the ratio

of baseline blood volume and flow, can be estimated from the temporal delay between the

arterial (i.e. ASL flow) and venous (i.e. BOLD) responses. These biomechanical parameters

provide a link between the dynamics of the evoked response and the baseline state. In addition,

both baseline hemoglobin concentration ([HbT0]) and baseline vascular volume fraction (V0)

are estimated in the model as the scaling factors applied to the NIRS and BOLD observations

respectively (detailed in (Huppert et al. 2007)). These calibration factors allow us to assign a

physiological scale to the normalized changes estimated in the model.

The estimates of baseline blood volume fraction from the BOLD measurement model (or

equivalently total hemoglobin concentration) and the mean vascular transit time (τ) are

sufficient to estimate the value of baseline blood flow using the steady-state relationship,

(5)

MWHb is the gram molecular weight of hemoglobin (64.5 kDa) and ρtissue is the density of

brain tissue (=1.04 g/mL (DiResta et al. 1991)). When the full set of measurements (ASL,

BOLD, and NIRS) is used to cross-calibrate the estimates of absolute values for the baseline

blood volumes, the ratios of the vascular volume fraction (in volume blood per volume tissue)

and baseline hemoglobin concentration (moles Hb per volume tissue) can be used to calculate

the hemoglobin content of blood (HGB) in gm Hb/volume blood. In the absence of estimates

of both V0 and [HbT0], in equation 5 HGB can be assumed from literature or measured (in adult

humans HGB is typically 12-18 gm/dL (Habler and Messmer 1997)).

At baseline steady state, relative CMRO2 is the product of blood flow and the oxygen extraction

fraction across the compartments. In this model, baseline oxygen saturation is either estimated

(using the full multimodality data set only) or assumed from the analysis of the complete data

set. Thus, baseline CMRO2 is calculated directly from the baseline blood flow and absolute

oxygen extraction (OE; OE =sinO2-soutO2),

(6)

where Hn is the Hufner number (Hn=1.39 mL O2/gm Hb (Habler and Messmer 1997)).

2.2. Experimental methods

The data used in this study previously published in (Huppert et al. 2006b) and recorded

simultaneous pulsed arteriole spin labeling (PASL) and NIRS measurements. Five subjects

were included in this study (4 male: 1 female). Subjects performed a brief 2-second duration

finger walking task using their right hand. All subjects were right-handed. The stimulus was

jittered evenly on a 500ms time-step, which allowed for the response to be estimated at 2Hz

with fMRI. The length of the inter-stimulus interval (ISI) ranged between 4 and 20 seconds

with an average inter-stimulus interval period of 12 seconds. The timing of the stimulus

presentation was synchronized with the MR image acquisitions and generated with a custom

written Matlab script. Each run lasted 6 minutes and consisted of between 27 and 32 stimulus

periods. This was repeated 4-6 times for each subject during the course of one scan session

(approximately 90 minutes in total time for each session including position of the NIRS probe

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and structural scans). The human research committee at the Massachusetts General Hospital

approved all protocols.

2.2.1. NIRS acquisition—The NIRS probe consisted of two rows of four detector fibers and

one row of four source fibers arranged in a rectangular grid pattern and spaced 2.9 cm between

nearest neighbor source-detector pairs as shown in Fig. 2. The probe was secured to the subject's

head over the contralateral primary motor cortex (M1). This was verified using vitamin-E

fiducial markers in the anatomical MRI. NIRS measurements were taken using a multi-channel

continuous-wave optical imager (690nm and 830nm; 18mW and 7mW respectively) (CW4

system, TechEn Inc., Milford, MA USA). These signals were synchronized to the timing of

the fMRI. Monte Carlo simulations were used to determine the mean photon trajectories by

simulating the diffusion through tissue-segmented anatomical volumes for each subject using

the locations of the optodes referenced by the fiducial markers (Huppert et al. 2006a). A mean

partial path-length correction of 4.7±1.5 and 4.8 ±1.5 (830nm and 690nm respectively; N=5)

was calculated from the fraction of the path-length through the brain cortex based on these

simulation results (see Fig. 2B). Methods and results for the collection of this data set have

been previously published in (Huppert et al. 2006b). Optical data was sampled at 100 Hz and

down-sampled with a Nyquist filter to 10Hz for analysis in the model.

2.2.2. Functional MRI acquisition—Pulsed ASL measurements were carried out at 3T on

a Siemens Allegra MR scanner using PICORE (proximal inversion with control for off-

resonance effects) labeling geometry (Wong et al. 1997) with Q2TIPS (second version of

quantitative imaging of perfusion by using a single subtraction with addition of thin-section

periodic saturation after inversion and a time delay sequence) saturation (Luh et al. 1999) to

impose a controlled label duration. A post-label delay of 1400ms and label duration of 700ms

were used, with repetition and echo times of 2s and 20ms, respectively [α=90°]. Echo Planar

Imaging (EPI) was used to image five 6mm slices (1mm spacing) with 3.75mm in-plane spatial

resolution. The PICORE labeling scheme allowed collection of BOLD signals using the

interspersed control images from the acquisition. The fMRI images were motion-corrected

(Cox and Jesmanowicz 1999) and spatially smoothed with a 6mm Gaussian kernel. Blood flow

and BOLD signals were separated by first extracting the even-numbered (ASL label and BOLD

contrast) and odd-numbered (BOLD contrast only) series of images. These two time-series

were independently interpolated using cubic spline functions and then the temporally-aligned

series were subtracted to yield the isolated spin-label contrast. This approach was similar to

the linear “surround subtraction” method described by Lu et al to give the minimal cross-talk

between the ASL and BOLD signals (Lu et al. 2006). Structural scans were performed using

a T1-weighted MPRAGE (magnetization prepared rapid gradient echo) sequence

[1×1×1.33mm resolution, TR/TI/TE/α = 2530ms/1100ms/3.25ms/7°].

2.2.3. Preprocessing of evoked hemodynamic responses—Both the NIRS and fMRI

hemodynamic responses were separately estimated using a 2-Hz resolution finite impulse

response (FIR) linear model as described in (Huppert et al. 2006b). High temporal resolution

was achieved by designing the stimulation protocol using a jittered inter-stimulus interval and

lowering spatial coverage and resolution for the EPI sequence. This may limit the use of this

method in general studies but were necessary to get the effective sample rates needed for our

model analysis. The FIR approach assumes a linearly additive hemodynamic model to estimate

the high-temporal resolution responses but avoids canonical assumptions that could restrict the

temporal characteristics of the estimates. Second order linear trends were removed from the

estimated responses. The epoch timing for this deconvolution was based on the stimulus

presentation timing rather then the subject's motor response. However, the subject motor

response times were similar to presentation times with a jitter of ∼100 ms as judged by an

optical sensor placed on the fingers which was used to record the tapping event as described

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in (Huppert et al. 2006b). Due to technical limitations, this recording was only available on 3

of the 5 subjects. Region-of-interest averages for both data sets were independently chosen

from measurements showing statistical changes (p<0.05) based on mixed effects analysis using

an activation time window of 2-7 seconds to define the t-statistic as described in (Huppert et

al. 2006b). The fMRI responses were restricted to voxels beneath the optical probe judged

manually based on the fiducial markers on the probe. ASL and BOLD responses were

independently estimated. Significant NIRS channels were defined based on the oxy-

hemoglobin estimate.

The resulting BOLD, ASL, and NIRS time-courses for each subject were variance normalized

and averaged into a single group average. In this work, the high computational load of the

inverse routine required us to predetermine the single-trial hemodynamic response function

using the FIR model rather than run our state-space analysis directly on the full data set (c.f.

(Deneux and Faugeras 2006)). However, the linear assumption of the FIR model can be

justified (e.g. (Cannestra et al. 1998)) for the minimum inter-stimulus interval (>4 seconds)

used in this study. We acknowledge that the temporal resolution is limited by the acquisition

techniques of fMRI and ASL methods and may currently limit the utility of this method in

general studies since the response characterization of our vascular model makes use of high-

temporal resolution response to estimate the vascular parameters.

3. Results

The underlying vascular and metabolic changes are provided by a non-linear inverse solution

to our vascular forward model and estimated by curve-fitting the observed hemodynamic

responses. In order to illustrate the performance of this inverse model, we demonstrate the

characteristics of our model using three examples that examine the local precision, global

accuracy, and physiological consistency of our model when used to analyze fMRI, NIRS, or

multimodal data.

3.1 Example 1. Parameter sensitivity and local stability of the inverse problem

A requirement of a well-poised inverse model is that the number of independent measurements

is equal to or greater than the number of free model parameters. This is informally referred to

as the “counting rule” and can provide a qualitative common-sense check for such a model

(e.g. (Bamber and van Santen)). To begin our investigation of our windkessel model, we will

first discuss this qualitative rule. Our model used for analysis of the fMRI signals has a total

of 15 free parameters. Although the ASL and BOLD responses (recovered at 2Hz) contain a

greater number of sample points, each sample point does not represent an independent

measurement due to the inherent temporal autocorrelation of the evoked response and partial

correlation between measurements. In order to apply the counting rule, we must estimate the

number of independent degrees-of-freedom to these measurements. The BOLD response is

believed to have up to three distinct phases; an initial dip period, a main response, and a post-

stimulus undershoot. In order to correctly model the overall BOLD response, for example in

a general linear model, three independent canonical functions should be used with one for each

of these phases. Since each canonical function requires three degrees-of-freedom for the

amplitude, first order timing (e.g. onset or time-to-peak) and second order timing (e.g. temporal

width), it follows that a total of nine parameters should be included to model all three phases

of the BOLD response. Similar arguments can be applied to determine that oxy- and deoxy-

hemoglobin are expected to contain nine degrees-of-freedom each and flow to contain six (with

an assumption of no “initial dip” period). By this argument, we can expect that the BOLD /

ASL fMRI data should have the minimum number of independent degrees-of-freedom needed

for our model. This argument, however, is at best only a qualitative deduction. In order to more

rigorously investigate the ability to determine all model parameters from fMRI measurements,

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we must turn to more formal mathematical definitions of model sensitivity and parameter

identifiability.

The stability of our inverse model indicates the feasibility of our model to estimate the model

parameters from experimental data and requires that measurements be sensitive to changes in

each of the estimated model parameters (reviewed in (Bamber and van Santen)). Model

sensitivity can be examined using a variety of numerical methods and has been recently

discussed in the context of a similar vascular model by (Deneux and Faugeras 2006). However,

model sensitivity does not guarantee that the estimated values are accurate or unique. Instead,

it allows us to test whether a solution to the inverse problem exists. Because of the non-linear

nature of our model, we focused on local data perturbation methods to examine the theoretical

sensitivity of measurements to changes in the model parameters.

Model sensitivity quantifies the influence of each parameter on the output of the forward model.

Two parameters; namely, the mean vascular transit time (τ) and the Windkessel vascular

compliance factor (β), are primarily involved in modeling the dynamic relationship between

flow and volume. We begin our investigation of model sensitivity by qualitatively examining

whether these two parameters affect the temporal dynamics of the hemodynamic response and

if it is plausible to estimate these values based on relative temporal dynamics. Based on our

model, we predicted that these two parameters would have distinguishable effects on the

relative temporal dynamics of the BOLD and ASL measurements. For example, in Fig. 3A we

see that if vascular transit time is increased (i.e. with constant baseline flow and thus increased

baseline volume), the BOLD response amplitude is reduced and the initial dip component is

removed. In particular, a longer transit time increases the temporal delay between arterial and

venous responses. Likewise, increases in the value of β lead to larger and quicker washout

changes that produce larger magnitude changes in BOLD and deoxy-hemoglobin and reduce

the relative time-to-peak value of these responses. The compliance parameter also has

pronounced effects on the relative magnitude (i.e. ratio) of the oxy-, deoxy-, and total-

hemoglobin changes that are measured by NIRS reflecting differing degrees of washout. We

note that the relationships between multimodal measurements are uniquely related to these

parameters when compared to the individual evoked responses considered independently.

The sensitivity of our model each parameter in the model can be numerically quantified by

examining the Jacobian of the model with respect to the model parameters. We examined the

rank and condition of the Jacobian throughout the bounds of the parameter space to determine

if the set of measurements was sensitive to each model parameter. We examined these

properties for the full multimodal, BOLD and ASL, and NIRS observation models. We found

that the Jacobian was full rank for the parameters estimated in each of these models, which

implies that an inverse of the model around a localized point is theoretically possible. Singular

value decomposition of the Jacobian provided similar conclusions and showed that the number

of independent components was equal to the number of model unknowns (for further discussion

refer to (Huppert 2007)). In comparison, when we examined the model with BOLD only

measurements, we found that it was not possible to uniquely estimate many of the parameters

of the model, including the separation of flow and consumption changes (i.e. the Jacobian was

no longer full rank). This finding was consistent with the previous results from Deneux and

Faugeras (Deneux and Faugeras 2006), which attempted similar analysis of BOLD signals

alone and found that many but not all of their model parameters could be independently

determined. Here, we find that if we use both ASL and BOLD measurements, the model is

better defined because information about the model parameters can be additionally determined

from the interrelationships of these signals. To further explore the limits of the model, we

examined the Jacobian of each model as a function of the simulated sample frequency and

determined that the minimum acquisition speed needed was approximately half of the mean

transit time. This implies that the estimation of this model will require a minimum sampling

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rate of above 1 Hz for both flow and BOLD signals, which is unfortunately faster than is

possible in most current fMRI studies and was only possible in this study using jittered

stimulation presentation.

A further metric of model sensitivity is the variance inflation factor (Γ), which quantifies the

minimum uncertainty that can be theoretically expected in the estimate of each model

parameter based on an analysis of the slope (Jacobian) of the minimization function around a

given local model solution. This analysis allows us to examine how distinguishable the effects

of one parameter are from the other parameters. Using the notation discussed in (Deneux and

Faugeras 2006), for a small change in one parameter (θi), the perturbation of the observation

varies by Jidθi, where Ji denotes the Jacobian of the forward model with respect to the ith

parameter. If the Jacobian associated with the parameter under consideration is not orthogonal

to the Jacobian matrix with respect to the remaining parameters (denoted Jφ), then the cross

talk between parameters results in non-uniqueness of the solution because an error in one

parameter can be compensated by errors in other parameters. The maximum precision of each

parameter can be estimated for a given level of noise in the measurements from the variance

inflation factors as described in (Deneux and Faugeras 2006) using the following equation.

(7)

where represents the transpose of the Jacobian matrix. The smaller the corresponding

diagonal element of the variance inflation matrix (Γi), the more precisely the θi parameter can

be determined for a given variance in the observation (Y) and the more sensitive the data set

is to changes in the parameter under consideration.

In Fig. 3C, we examine the sensitivity of the measurements to changes in model parameters

used in each fitting procedure. Note that the full model (ASL, BOLD, and NIRS) has more

degrees-of-freedom than fMRI alone, NIRS alone, or NIRS and BOLD models since the value

of the baseline blood oxygen saturation is fixed in these models (refer Table 3). The analysis

was conducted for a linearization about the parameter set determined from the model fits to

the empirical data (Table 4). The variance inflation matrix was inspected for other linearization

points (not shown) and was found to be in quantitative agreement to these results.

3.2 Example 2. Simulation results and global identifiability

The analysis of the Jacobian matrix described in our previous example indicated that the

observation sets were theoretically sensitive to the parameters being estimated and thus the

model was expected to be locally invertible. However, the Jacobian can only be defined about

a local linearization point in non-linear systems and such sensitivity analysis can be used to

probe only the local precision of the model but not its global identifiability. Using this analysis

we know that a local solution can be determined but we do not know if this solution is unique.

To investigate the uniqueness of an estimated solution, it is necessary to examine the ability

of the model to correctly determine the parameter set from a global search of the parameter

space. We ran forward simulations of the model using a sampling of the physiological states

and parameters from within the expected physiological ranges (refer to Table 3). To emulate

the fMRI and NIRS experimental data, the measurements were simulated at a 2Hz sample

frequency with a random additive instrumental noise term (10:1 contrast-to-noise ratio to

approximately match the data). We reconstructed system parameters for each simulation using

the fMRI data alone, the NIRS data alone, and the full multimodality data set to explore the

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accuracy of the model to infer the physiological states. For all reconstructions, the Levenberg-

Marquardt minimization algorithm was initialized at a random position within the

physiological range of the parameters. The same initialization was used for each modality set.

Reconstructions were iterated until a set convergence criterion was met (approximately 106

iterations that took around six hours per model estimation). A total of 350 simulations were

run for each of the three data sets.

In Fig. 4A, we show parametric plots of the simulated (truth) values for the parameters used

in the forward simulation against the reconstructed values obtained from each of the three data

sets (NIRS only, BOLD and ASL, or NIRS, BOLD and ASL data). The ability of a model to

predict one subset of a data set (in this case the NIRS measurements) from analysis of a different

subset (the fMRI data) is often used as a strategy to asses if a model has been over-

parameterized. We found that relative CMRO2 and arterial resistance changes (ΔRA) can be

estimated accurately with both single and multi- modality measurement sets, consistent with

the tests of local sensitivity. We examined the accuracy of the estimation of key static and

response-related parameters in the model. We found that the estimates of the Windkessel

vascular reserve parameter (β) had the most variance but could still be estimated fairly

accurately in spite of a slight systematic model under-estimation bias at high values of β with

the optical or fMRI only data sets. In Fig. 4B, the covariance in the error of the estimates for

the parameters is plotted. The lack of high correlation indicated a low interdependence between

variables and minimal cross-talk between these parameters in the model. We noted the largest

cross-talk (R2=0.07) between the vascular transit time through the pial-venous compartment

and the relative change in CMRO2. We also observed negative correlation between CMRO2

and the blood oxygen saturation parameters in the BOLD/ASL/NIRS model as also noted in

(Huppert et al. 2007) (not shown). Recall that baseline oxygen saturation was not varied in the

NIRS or fMRI alone models since we found these parameters were not identifiable in the

reduced data sets.

3.3. Example 3. Estimating relative CMRO2 changes from empirical measurements

As a final example of the inverse procedure for our vascular model, the model was used to

analyze empirical measurements of hemodynamic changes in the human motor cortex

following a 2-second finger-tapping task. In order to examine the estimate of the CMRO2

changes and other model parameters given in Table 3, the model was first fit using the combined

NIRS, ASL, and BOLD data set. The values of baseline blood oxygen saturation from the

model fit to the full data set was then fixed and the model was applied to the ASL and BOLD

data and finally to the NIRS only data.

The fits of the fMRI only and NIRS only data sets closely predicted the NIRS and fMRI data

respectively as shown in Fig 5. The 75% confidence bounds for the model fit to each

measurement and the model predictions of the opposing measurement set are shown as dotted

lines in Fig. 5. To verify that the results of the model fit were independent of the initial position

of the Levenberg-Marquardt algorithm, we repeated the minimization for different initial seed

positions (see Fig. 6). The model fit using the group-averaged response curves was consistent

with the mean of the fits to the five individual subjects (Fig. 6). Inter-subject variations in the

parameter estimates were observed, but remained consistent whether the reduced or full data

sets were used in the fitting. The values of the estimated parameters and uncertainties are given

in Table 4. In Fig. 6, we graphically present the estimates for the key model parameters and

uncertainties for both the group and individual subject analysis. The estimated dynamic

changes in arterial diameter and CMRO2 are shown in Fig. 7.

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3.4. Calculated baseline physiological properties

Based on the values of the estimated biomechanical parameters presented in Table 4, several

additional physiological quantities can be determined and are presented in Table 5. Importantly,

the vascular transit time and the vascular volume fraction, needed to scale the BOLD signal

were both precisely determined. Vascular volume fraction, in units of milliliters of blood per

milliliter of tissue, is easily converted to blood volume, in units of milliliters of blood per 100

grams of tissue, given the approximate density of brain tissue (1.04 g/ml (DiResta et al.

1991)). This baseline blood volume divided by the vascular transit time provides an estimate

of baseline blood flow. Likewise, baseline CMRO2 can be calculated from the baseline blood

flow and oxygen saturation within the venous compartment. We were able to precisely estimate

the venous oxygen saturation only in the model using the combined fMRI and NIRS data and

we found that this estimation was not as accurate using only the fMRI only data. Thus, we

fixed the value of baseline vascular oxygen saturations in the model estimates using the fMRI

only or NIRS only data. Prior studies have noted that the oxygen extraction fraction varies only

a few percent between different regions of the brain and this justifies constraining the range of

expected cerebral venous oxygen saturation to the range of 60±9% based on previous

measurements in the healthy adult brain (He and Yablonskiy 2007; Raichle et al 2001). The

estimated baseline blood volume, blood flow, and oxygen consumption using different

combinations of fMRI and NIRS data are shown in Table 5. Although we assumed a value for

the resting oxygen extraction in the fMRI-only and NIRS-only models, we found that the

estimate of baseline blood flow and oxygen consumption based on the estimated scaling

parameters and vascular transit time from the fMRI data alone was consistent with the estimate

from the NIRS alone fit and also with the combined fMRI and NIRS fit. The values of the

estimated baseline physiology for the group data and the mean of the results from the five

individual subjects are presented in Table 5.

4. Discussion

Measured hemodynamic responses are the net result of the competing effects of an increased

metabolic demand for oxygen and an increased supply of oxygen transported by hemoglobin

through the flow-inducing response. However, the role of variable baseline physiology and

vascular biomechanical properties in the interpretation and reproducibility in the measurements

of these responses has been largely under-appreciated. In this work, we have introduced an

inverse model for the analysis of multimodal hemodynamic data which depicts how baseline

physiology and vascular structure affect the relative magnitudes and temporal dynamics of

evoked signals. Although our model is similar to previously published models of the vascular

system (reviewed in (Buxton et al. 2004)), our approach focuses on the dynamics and relative

timing differences observed between multimodal measurements rather than the absolute

magnitude of such signals. This allows us to characterize of the vascular properties more

accurately by considering the interrelated dynamics of multimodal measurements in addition

to the changes in absolute magnitude. The vascular and oxygen transport phenomena are

considered simultaneously in this work, which creates a unified multimodal model that enables

incorporation of both measurements of blood flow and oxygenation changes into a single and

self-consistent description of the underlying physiology. In this way, we demonstrate that

BOLD signals can indirectly add information about both the flow-volume relationship and

CMRO2 changes. We suggest that it is possible to infer blood volume changes from

measurements of blood flow (ASL) and BOLD alone but note that these estimates are subject

to the assumption of baseline values for oxygen extraction from the vascular compartments.

Similarly, we report that blood flow changes can be inferred using measurements of blood

volume and oxygen saturation changes alone (i.e. NIRS measurements). The calculated

estimates of CMRO2 and arterial resistance changes were largely consistent with those obtained

when we used the full complementary set of fMRI and NIRS information.

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4.1 Physiological relevance of estimated model parameters

Since the estimation of the model parameters was restricted to physiological ranges based on

previous literature, it is expected that the parameter values estimated from the data are also

consistent with this literature. The estimate of the Windkessel vascular reserve parameter (β)

is consistent with values estimated from evoked responses found in (Jones et al 2002;

Mandeville et al 1999b; Wu et al 2002; Zheng et al 2005) and in approximate agreement with

the steady-state flow-volume relationship (Grubb et al.). Our estimated total mean vascular

transit time (including pial vein) (τ = 3.3-3.8sec) is consistent with cortical gray matter values

measured with dynamic susceptibility contrast-enhanced MRI methods (DSC-MRI) (τ =

1.2-6.9sec as compiled in (Ibaraki et al. 2006)) and previously applied to modeling the BOLD

signal (Buxton et al. 2004).

We found that secondary calculations of physiological values based on the estimated

parameters were also consistent with previous literature. This is further validation of our model

since our analysis did not explicitly impose this consistency. For instance, while there are

physiological bounds on the magnitude of both flow and consumption changes, there is a much

weaker constraint on the ratio of flow to consumption changes. Our estimated ratios of peak

blood flow to peak blood volume changes of 3.0 to 3.4 (Table 3) were consistent with previous

literature values of 2-4 (Huppert et al 2007; Jones et al 2002; Mandeville et al 1999a; Martin

et al 2006; Wu et al 2002). The determined values of the flow-to-oxygen consumption ratio

between 1.5:1 to 1.8:1 is also consistent with literature measuring these changes in comparable

experiments where values of 1.5:1 to 3:1 have been reported in studies using fMRI (Boas et

al 2003; Dunn et al 2005; Hoge et al 1999; Kastrup et al 2002) and higher ratios of 5-6 have

been reported in PET studies (Fox and Raichle 1986). Although our estimated flow:

consumption ratio is in the range of previous literature values, our estimates that utilize the

ASL data were somewhat lower than the average value from the fMRI literature (∼2:1). We

hypothesize that this may be a result of an unaccounted partial-volume error in the ASL

measurements arising from our defined region-of-interest and require further investigation.

This underestimation of flow is further supported by the higher predicted flow changes from

the model fit of the NIRS data set (Fig. 3A). In the ASL model, the ASL measurements were

considered direct measurements of the relative flow changes and no additional scaling

parameters were applied unlike the BOLD and NIRS measurements where calibration factors

were estimated. We note, however, that fitting a scaling factor on both the ASL and BOLD

signals may give rise to a non-unique estimation of CMRO2 changes and baseline CMRO2

from the ASL and BOLD only data set. The same does not apply to parameter estimation with

the NIRS only data set because only one partial volume term was applied to scale oxy-, deoxy-

and total-hemoglobin measurements. The value of the NIRS scaling factor does not change the

ratio of hemoglobin changes (i.e. the peak oxy-/deoxy- hemoglobin changes normalized to the

peak total-hemoglobin change) which is the relevant measure giving sensitivity to β and τ (see

Fig. 2).

4.2 Estimates of baseline physiology

The estimates of baseline volume, flow, and CMRO2, were consistent between the three dataset

combinations used in the model fitting procedure (Table 5). The estimate of baseline blood

flow from the three data sets was 86 ml/100g/min (mean) [range 82-93 ml/100g/min].

Previously reported values of baseline blood flow in human cortex range from 80-100 ml/100g/

min in gray matter and ∼20 ml/100g/min in white matter as measured using positron emission

tomography (PET) (reviewed in (Coles 2006; Ito et al 2005)). Our estimates derived from

evoked changes represent gray matter estimates of flow, volume, and CMRO2. This explains

why our estimates of blood flow are higher than the reported values that average gray and white

matter (e.g. ∼ 44 ml/100g/min from data complied in (Ito et al. 2005)).

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Our estimate of baseline oxygen extraction determined from the combined BOLD, ASL, and

NIRS data set was 36.6±0.2 %. This value is consistent with values reported in the literature

of 35-43% from PET (Diringer et al 2000; Raichle et al 2001) and quantitative BOLD-fMRI

38±5% (He and Yablonskiy 2007). In the fMRI only and NIRS only models, the value of

oxygen extraction was assumed based on the value obtained from the full model fit. The

additional estimation of this parameter introduced considerably more uncertainty in the

recovered value of CMRO2 from these models. We consider the assumption of oxygen

saturation in these models reasonable given that previous studies have demonstrated fairly

small ranges for these values in normal, healthy populations. However, this assumption poses

a potential source of systematic error for the application of our model to atypical subject

populations.

Lastly, we estimated baseline cerebral oxygen consumption (CMRO2). Our estimate of 5.2 ml

O2/100g/min (mean)[range 5.2-5.4 ml O2/100g/min] is higher than reported values of 3.3 ±

0.5 ml O2/100g/min (Ito et al. 2005). However, the values reported in these PET studies are

based on flow measurements that represent the average of both gray and white matter values.

Baseline CMRO2 in gray matter is estimated to be ∼ 3.5-8 ml O2/100g/min using the literature

ranges for gray matter blood flow, oxygen extraction, and hemoglobin content referred above.

4.3. Advantages of the bottom-up model

Our model used a curve-fitting approach to estimate the underlying changes of CMRO2 and

arterial resistance. The use of such an inverse model allows us to incorporate our multimodal

data into a unified estimate of the underlying physiology. In contrast to our approach, most

previous formulations of vascular models have been built around a top-down (or deductive)

approach where CMRO2 changes are estimated from the set of hemodynamic measurements.

Since deductive models calculate CMRO2 changes directly from measurements, the results

directly reflect the quality of the data used. In such models, CMRO2 changes can be recovered

only at the lowest spatial and temporal resolutions of the data used. In addition, the error in the

estimate of CMRO2 is compounded by the errors in multiple measurements and the resulting

estimates generally have errors greater than the noisiest measurements used.

In an inductive framework, measurements are forward-modeled from changes in the underlying

states (i.e. arterial resistance and CMRO2). The estimate of the underlying states requires the

solution of an inverse problem, which is more computationally intensive than the equations

used in the deductive approach. However, the distinct advantage of the inductive approach is

that model inversion may be possible with limited sets of observations (as in the fMRI alone

analysis) and redundant information can be fused into estimates that maximize the joint

probability of all observations (as in the full multimodal analysis). The inductive framework

predicts all possible observations, not all of which are necessarily measured thus allowing the

use of a smaller subset of possible observations to estimate the parameters in the model.

Additionally, the inductive model may be used to take advantage of advanced state-space

estimation techniques, which are particularly useful for hidden-variable or under-determined

problems. Several methods have been described to solve problems in this framework, such as

the Kalman filter for dynamically variant systems. In this model, we use a time-invariant

approach, but recognize that advanced techniques could provide additional advantages in future

work. Lastly, the model setup described has the advantage of providing a mechanism by which

multimodality information can be fused within a Bayesian framework by incorporating

statistical error of each measurement and the corresponding observation model for each

modality. Recently several similar inductive models have been applied to interpretation of

fMRI data (Deneux and Faugeras 2006; Friston 2002) and the fusion of fMRI and EEG data

(Riera et al. 2005).

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