Mathematical model for the estimation of hemodynamic and oxygenation variables by tissue...

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Journal of Theoretical Biology 241 (2006) 262–275

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Mathematical model for the estimation of hemodynamic andoxygenation variables by tissue spectroscopy

Laszlo Kocsis, Peter Herman, Andras Eke�

Institute of Human Physiology and Clinical Experimental Research, Semmelweis University, Faculty of Medicine, P.O. Box 448, Budapest 1446, Hungary

Received 15 August 2005; received in revised form 23 November 2005; accepted 25 November 2005

Available online 18 January 2006

Abstract

This article presents a quasistatic, compartmental model of tissue-level hemodynamics and oxygenation that leads to a set of formulas,

which is suitable to calculate important physiological variables from the mean tissue concentration and saturation of hemoglobin,

measured by tissue spectroscopy. Dimensioned quantities are represented relative to their baseline value in the equations (relative

value ¼ perturbed/baseline). All model parameters are non-dimensional. The model is based and extends on a number of previous works:

previous models of similar aim and scope are consolidated, and every critical assumptions and approximations are treated explicitly;

extensions include for example the incorporation of the Fahraeus-effect and the separate estimation of the volume changes of the arterial

and the venous compartments. The information content of spectroscopic data alone is shown to be valuable, but limited: the relative

venous volume, the oxygen extraction fraction and the relative cellulovascular coupling (defined as the ratio of blood flow and oxygen

consumption) can be calculated from these data, if the alterations in arterial blood volume are negligible. The number of variables

estimated by the derived formulas can be increased if local blood flow is measured simultaneously: in this case, the relative arterial and

venous volume and resistance, the oxygen extraction fraction, and the relative oxygen consumption can be determined. Given that this

model considers arterial blood pressure, saturation and hematocrit as its inputs, when measured, the model becomes applicable in such

conditions as hyper- or hypotension, hypoxic hypoxia, hemodilution and hemorrhage, where these variables do change. The estimation

of the changes in arterial resistance can be applied to estimate the extent of an autoregulatory response.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Mathematical model; Tissue spectroscopy; Hemoglobin; Hemodynamics; Oxygenation

1. Introduction

During the nearly three decades of tissue spectroscopy(Jobsis, 1977), a large number of methods have beendeveloped to determine the tissue concentrations of oxy-and deoxyhemoglobin in organs like the brain, the musclesor the breast (Chance, 1998; Delpy and Cope, 1997; Ferrariet al., 2004). Near-infrared light (about 700–1000 nm)penetrates deeply hence is widely used to gain informationabout deeper tissue layers, typically in a non-invasivemanner (near-infrared spectroscopy, NIRS). Visible light(about 400–700 nm) can be utilized to collect high-

e front matter r 2006 Elsevier Ltd. All rights reserved.

i.2005.11.033

ing author. Tel.: +361 210 0290x3139;

162.

esses: [email protected], [email protected] (A. Eke).

resolution images from the exposed superficial layer ofthe tissue (Dunn et al., 2003; Eke, 1992).Mere oxy- and deoxyhemoglobin concentrations provide

only indirect insight into the equilibrium between oxygensupply to and oxygen demand of the tissue. Mathematicalmodeling of the relationship between measured hemoglo-bin concentrations and the underlying physiological vari-ables is needed to quantitatively investigate hemodynamicand oxygenation processes of the tissue. Previous mathe-matical models and formulas (Boas et al., 2003; Culver et al.,2003; Fantini, 2002; Mayhew et al., 2001; Rostrup et al.,2002; Zheng et al., 2002) emerged from models developedfor the interpretation of blood-oxygen-level-dependentfunctional magnetic resonance imaging (BOLD-fMRI) andpositron emission tomographic (PET) data from the brain(Buxton and Frank, 1997; Buxton et al., 2004; Hoge et al.,1999; Hyder et al., 1998; Mandeville et al., 1999).

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Table 1

Variables of the model. A consistent set of units is given, although

dimensioned quantities will be represented relative to their baseline value,

i.e. in a non-dimensional form. Hemoglobin concentrations refer to

tetramers. Possible subscripts may indicate compartments (T, A, C, V, B:

see Fig. 1.) or baseline state (0). Relative (compared-to-baseline) values

will be denoted by a prime (see text)

L. Kocsis et al. / Journal of Theoretical Biology 241 (2006) 262–275 263

In the present study, by integrating and extending thesemodels, we developed a general mathematical model forthe estimation of hemodynamic and oxygenation variablesby tissue spectroscopy. The main aspect of model buildingwas the widest practical applicability, which was achievedby applying algebraic equations only, with a small set ofparameters, all of which can be accurately estimated.

Variable Unit Description

H(oxy) mmol/mL Concentration of oxyhemoglobin

H(deoxy) mmol/mL Concentration of deoxyhemoglobin

H mmol/mL Total hemoglobin concentration

S — Saturation of hemoglobin

V mL Volume of a compartment

J mmol/s Oxygen flux into or out of a compartment

M mmol/s Metabolic rate of oxygen in tissue compartment

E — Oxygen extraction fraction

Q mL/s Blood flow through compartments (perfusion)

P mmHg Blood pressure

R mmHg � s/mL Vascular resistance

2. Assumptions of the model

The schematic representation of the model and theapplied notations are shown in Fig. 1 and Tables 1 and 2,respectively. Our assumptions are as follows:

(a1) Absolute concentrations: Tissue spectroscopy isassumed to be able to accurately measure absolute molarconcentrations of oxy- and deoxyhemoglobin in theilluminated tissue volume.

(a2) Independent regions: From a hemodynamic andoxygenation point of view, every illuminated tissue regionis assumed to be independent of its neighbors: regions areassumed to be coupled in parallel hemodynamically, withno oxygen diffusion between them, meaning that allcalculations refer to a single region of interest (ROI).

(a3) Compartmental model: Inside the illuminated tissuevolume, arterial, capillary and venous segments of theintravascular space are treated as three lumped compart-ments, coupled in series. The capillary compartment isassumed to be the only site for oxygen transport to tissue(Fig. 1).

(a4) Quasistatic changes: The rate of the volume changeof vascular compartments (mL/s) is assumed to be muchslower than blood flow (mL/s). Transients are assumed tobe slow enough so that outward oxygen diffusion ratethrough the capillary wall (mmol/s) always equals to theconsumption rate of oxygen by the tissue (mmol/s). These

Fig. 1. The scheme of our compartmental model. T: tissue compartment

probed by visible or near-infrared light; A, C, V: arterial, capillary and

venous compartments (B ¼ Aþ C þ V : blood compartment). Downward

and upward arrows indicate incident and backscattered light, respectively.

Horizontal arrows show the direction of blood flow through vascular

compartments. Fan-like arrows illustrate oxygen diffusion out of

capillaries.

assumptions entail that algebraic equations can be usedinstead differential equations.(a5) Laminar blood flow and homogeneous volume change

in arteries and veins: We assume that in case of arteries andveins, blood flow is laminar, and all vessel diameters alwayschange approximately by the same factor within a givenvascular compartment (but this factor can be different incase of arteries, capillaries and veins, i.e. their volume isassumed to change independently). These assumptionsimply that the hemodynamic resistance of the arterial andthe venous compartment are approximately inverselyproportional to compartmental volume squared, becausevessel volume is directly proportional to diameter squared(cylinder model), and vessel resistance is inversely propor-tional to diameter to the fourth power (Hagen–Poiseuille-equation) (Nichols and O’Rourke, 1990).(a6) Spatially homogeneous oxygen consumption: We

assume that the oxygen consumption is the same in allportions of the capillary bed, and therefore the totaloxygen content of blood decreases linearly along thecapillary (Mintun et al., 2001; Sharan and Popel, 2002;Herman et al., 2006). Because 97–99% of the oxygencontent is bound to hemoglobin (Fantini, 2002), saturationcan also be assumed to decrease linearly.

3. Equations of the model

3.1. Modeling tissue spectroscopic data

Instead of the concentrations of oxy- and deoxyhemo-globin (H(oxy) and H(deoxy)), total concentration andsaturation of hemoglobin (H and S) will be used. It is wellknown that the latter two can be calculated from theformer ones as

H ¼ H ðoxyÞ þH ðdeoxyÞ, (1)

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Table 2

Parameters of the model. These representative values refer to the brain, but the parameters of other organs probably do not differ considerably

Parameter Description Value Source

h Hemoglobin ratio 0.38 Krolo and Hudetz (2000), Sakai et al. (1985)

vA Baseline volume fractions 0.20 Ferrari et al. (2004), Van Lieshout et al. (2003)

vC 0.05

vV 0.75

rA Baseline resistance fractions 0.70 Nichols and O’Rourke (1990), Gao et al. (1998)

rC 0.15

rV 0.15

Fig. 2. Incorporation of the Fahraeus-effect into the model: discharge (H)

and tube (HA, HC and HV) hemoglobin concentrations in vascular

compartments. The latter three are compartmental averages. SA, SV:

systemic arterial and venous compartments.

L. Kocsis et al. / Journal of Theoretical Biology 241 (2006) 262–275264

S ¼H ðoxyÞ

H. (2)

Tissue spectroscopy determines oxy- and deoxyhemoglobinconcentrations in a tissue compartment as volume-weighted averages of arterial, capillary and venousconcentrations (Culver et al., 2003; Fantini, 2002):

HðoxyÞT ¼

V A

VT

HðoxyÞA þ

V C

VT

HðoxyÞC þ

VV

VT

HðoxyÞV , (3)

HðdeoxyÞT ¼

VA

V T

HðdeoxyÞA þ

V C

VT

HðdeoxyÞC þ

V V

V T

HðdeoxyÞV , (4)

where VT is the tissue volume probed by photons, and VA,VC and VV are the respective volumes of the arterial,capillary and venous compartments.

By applying Eqs. (3)–(4), and the compartment-specificversions of Eqs. (1)–(2), we get

HT ¼V A

VT

HA þV C

V T

HC þV V

V T

HV , (5)

ST ¼V ASAHA þ V CSCHC þ V V SV HV

VAHA þ V CHC þ V V HV

, (6)

where SC is the average saturation in capillaries, which,according to (a6), can be expressed as the arithmeticaverage of arterial and venous saturations (for details, seeAppendix C):

SC ¼SA þ SV

2. (7)

It will prove useful to define here the average hemoglobinconcentration of the intravascular (blood) compartment ofthe ROI:

HB ¼V A

VB

HA þV C

V B

HC þV V

VB

HV , (8)

where VB ¼ VA+VC+VV denotes the intravascular(blood) volume inside the ROI. Eq. (8) will be used toeliminate HA, HC and HV from Eq. (5).

In every vascular compartment, total hemoglobin con-centration is proportional to hematocrit. Owing to theFahraeus-effect, tube and discharge hematocrit should bedistinguished (Pries et al., 1996), therefore tube anddischarge concentrations of hemoglobin need to be defined.

Discharge hematocrit (and discharge hemoglobin concen-tration) is the same in all vascular compartments. Tubehematocrit (and tube hemoglobin concentration) decreaseswith vessel diameter, that is, its value is the largest insystemic vessels, smaller in the arterial and venouscompartment of the ROI, and the smallest in the capillaries(Fig. 2). Light absorption, measured by tissue spectro-scopy, is determined by the instantaneous blood content ofvascular tubes, therefore HA, HC and HV are tubeconcentrations, and the following approximations can beused (Fig. 2):

HV ¼ HA, (9)

HC ¼ hHA, (10)

where 0oho1 will be termed as ‘‘hemoglobin ratio’’.Eqs. (9)–(10) will be used to eliminate HA, HC and HV fromEq. (6).

ARTICLE IN PRESSL. Kocsis et al. / Journal of Theoretical Biology 241 (2006) 262–275 265

3.2. Modeling oxygenation

The oxygen flux from arteries to capillaries:

JA ¼ 4SAHQ. (11)

The oxygen flux from capillaries to veins:

JV ¼ 4SV HQ. (12)

In Eqs. (11)–(12), the factor 4 is the number of oxygenbinding sites of a hemoglobin tetramer, H is the dischargeconcentration of hemoglobin in blood, and Q denotesblood flow. Note that there is no subscript on H, becauseits value is the same in all vascular compartments (seeabove and Fig. 2).

The oxygen flux through the capillary wall:

JC ¼ JA � JV . (13)

According to assumption (a4), the metabolic rate ofoxygen is

M ¼ JC . (14)

By definition, the oxygen extraction fraction is

E ¼JC

JA

. (15)

From Eqs. (11), (14) and (15), we obtain

M ¼ 4SAHQE (16)

and, from Eqs. (7), (11)–(13) and (15):

SC ¼ ð1� E=2ÞSA, (17)

SV ¼ ð1� EÞSA. (18)

The latter two equations will be used to eliminate SC andSV from Eq. (6).

3.3. Modeling hemodynamics

According to assumption (a4), the effect of vascularcompliances can be neglected, and from a hemodynamicpoint of view, arterial, capillary and venous compartmentscan be treated as three vascular resistances, coupled inseries (Nichols and O’Rourke, 1990). That is, the totalvascular resistance,

R ¼DP

Q(19)

is the sum of the arterial, capillary and venous resistances:

R ¼ RA þ RC þ RV . (20)

In Eq. (19), DP ¼ PA � PV is the pressure head, where PA

and PV are the blood pressures at the proximal end of thearterial, and the distal end of the venous compart-ment, respectively. These are the same for all regions,because we treat them as being coupled in parallel, seeassumption (a2).

According to assumption (a5):

RA /1

V 2A

, (21)

RV /1

V2V

. (22)

A third relationship exists, the one between RC and VC, butthis cannot be written as simply as Eqs. (21) and (22), sinceblood flow cannot be regarded laminar in the capillaries(Pries et al., 1996).

3.4. Need for non-dimensional equations

In order to avoid the use of dimensioned constants,which can be difficult to estimate, and to reduce thenumber of model parameters, the equations for theestimation of hemodynamic and oxygenation variablesfrom tissue spectroscopic data must be written in a unitless(non-dimensional) form, by representing dimensionedvariables in relative terms, defined as the ratio of perturbedand baseline values (Culver et al., 2003). Baseline valueswill be indicated by a ‘‘0’’ subscript. Relative values will bedenoted by a prime, and are defined in Appendix A. Ourmodel has seven, non-dimensional parameters: the hemo-globin ratio (h), the baseline volume fractions (vA, vC andvV), and the baseline resistance fractions (rA, rC and rV).The first one is defined by Eq. (10), and the latter six aredefined in Appendix B.

3.5. The non-dimensional equations

By substituting Eq. (8) into Eq. (5), and Eqs. (9)–(10)and (17)–(18) into Eq. (6) we get

H 0T ¼ ðvAV 0A þ vCV 0C þ vV V 0V ÞH0B, (23)

ST ¼vAV 0A þ hð1� E=2ÞvCV 0C þ ð1� EÞvV V 0V

vAV 0A þ hvCV 0C þ vV V 0VSA, (24)

where we considered the fact that the probed volume, VT isconstant. The factor 4 drops out of the non-dimensionalform of Eq. (16):

M 0 ¼ S0AH 0Q0E0, (25)

The non-dimensional form of Eq. (19) is

R0 ¼DP0

Q0, (26)

where, according to Eq. (20):

R0 ¼ rAR0A þ rCR0C þ rV R0V (27)

and due to Eqs. (21) and (22):

R0A ¼1

V 02A

, (28)

R0V ¼1

V 02V

. (29)

ARTICLE IN PRESSL. Kocsis et al. / Journal of Theoretical Biology 241 (2006) 262–275266

This system of non-dimensional equations (Eqs.(23)–(29)) can be applied to calculate physiologicallyimportant hemodynamic and oxygenation variables fromthe measured raw data.

4. Possible solutions

4.1. Input variables

The input, i.e. measurable variables in Eqs. (23)–(29) arethe followings:

(i1)
H0T and ST are determined by tissue spectroscopy. (i2) Since the red cell hemoglobin concentration is con-
stant, H 0 practically equals to the relative systemicarterial hematocrit. SA and S0A can be approximatedby systemic arterial saturation and its relative value,respectively. Systemic arterial hematocrit and satura-tion can be measured by arterial blood sampleanalysis.

(i3)
H 0B equals to relative average local hematocrit, whichcan be approximated by H 0 or measured by anappropriate indicator dilution method (Eke, 1988).
(i4)
DP0 can be determined by blood pressure measure-ment.
(i5)
Relative local blood flow, Q0 can also be measuredsimultaneously by special techniques. First (in Section4.3), we will assume that Q0 is unknown, and this casewill only be dealt with later (in Section 4.4).
The number of input variables can be decreased byapproximating SA by 1. The application of relative valueshas an important advantage: if any given input variable isonly moderately perturbed, e.g. by an experimentalintervention, its relative value can be approximated by 1,which further reduces the number of variables to bemeasured.

4.2. Output variables

The output variables (unknowns) in Eqs. (23)–(29) arethe followings:

(o1)
The relative volume of the vascular compartments:V 0A, V 0C and V 0V .
(o2)
The oxygen extraction fraction: E (E0 can becalculated from E, see Appendix A).
(o3)
The relative oxygen consumption: M0. (o4) The relative total vascular resistance: R0, and its
components, the relative compartmental resistances:R0A, R0C and R0V.

(o5)
If it is not measured, the relative local blood flow, Q0
must be treated as an unknown (see Section 4.3),otherwise it is an input variable (see Section 4.4).

The above seven equations (Eqs. (23)–(29)) contain tenunknowns (or nine, if Q0 is measured). This number can be

decreased by two, if the volume change of capillaries isneglected, i.e. by using the approximation V 0C ¼ 1, whichimplies that R0C ¼ 1.

4.3. Solution #1: when local blood flow is unknown

If Q0 is unknown, we have seven equations with eightunknowns. If not only the volume changes of capillaries,but that of the arterial compartment is regarded negligible,the approximation V 0A ¼ 1 can be applied, which entailsthat R0A ¼ 1. The hemodynamic equations (Eqs. (26)–(29))lose their importance at this point (and therefore there is noneed for assumption (a5)), because the main component ofvascular resistance, the resistance of the arteries, is assumedto be constant. The remaining three equations (Eqs.(23)–(25)) have a simple, analytical solution:

V 0V ¼1

vV

H 0TH 0B� vC � vA

� �, (30)

E ¼ 1�ST

SA

� �1þ

vA þ hvC=2

vV V 0V þ hvC=2

� �(31)

and

Q0

M 0 ¼1

S0AH 0E0, (32)

where E0 is calculated from E (see Appendix A). That is, bysuccessive substitution, we get the relative venous volume,the oxygen extraction fraction and the relative value of theratio of blood flow and oxygen consumption (because (Q/M)0 ¼ Q0/M0). The latter ratio (Q/M) may be termed‘‘cellulovascular coupling’’, on the analogy of neurovas-cular coupling (Buxton et al., 2004). The relative value ofcellulovascular coupling may be suitable to evaluate theequilibrium between oxygen supply (p perfusion, Q) andoxygen demand ( ¼ consumption, M) in a perturbed state,relative to the baseline level.The investigation of the above equations allows the

determination of unphysiological value combinations ofthe input variables: these combinations would yieldunphysiological results when substituted, namely valuesfor which the restrictions V 0V X 0 or 0pEp1 do not hold.This is demonstrated in Fig. 3, for the simple case whenH 0B ¼ 1, and SA can be approximated by 1:

(1)
For a given H 0T, there is a smallest ST (line 1), belowwhich the oxygen extraction fraction, E would be largerthan 1 (domain *).
(2)
There is a smallest H 0T (line 2), below which the relativevenous volume, V 0V would be negative (domain **).
The values of V 0V and E, calculated from Eqs. (30) and(31), respectively, are shown in Figs. 4b and c. These figurescan be used as nomograms, but only in case of cerebralmeasurements, because they were calculated by usingparameters typical of cerebral tissue (Table 2). Newnomograms have to be calculated in case of other organs.

ARTICLE IN PRESS

1

0.7

(*)

(**)

(1)

ST

0

(2)

0 1 2

HT′

Fig. 3. Valid and invalid domains in the input variable space, in case of

solution #1 (see text). In this and the following figures, all calculations are

based on parameters typical of cerebral tissue (Table 2), and the dot

represents the baseline state, with ST,0 ¼ 0.7, which is also typical of the

brain (Elwell et al., 1994).

1

0.71

ST

0

10.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

1.2 2.21.4 1.6 1.8 2

0 1 2

1

0.7

ST

00 1 2

1

0.7

ST

00 1 2

HT′

E

VV′

VA′

(a)

(b)

(c)

Fig. 4. The values of (a) V 0A, (b) V 0V and (c) E, given by solution #1: V 0Awas approximated by 1; V 0V and E were calculated from Eqs. (30) and

(31). In this figure and Figs. 6–8, arrows on the right indicate the main

trend of increase in the value of the output variables. No arrow is given if a

variable do not change considerably.

Table 3

Results of sensitivity analysis, in case of solution #1. Zero sensitivity

values are indicated by dashes

%/% V0V E0 E (Q/M)0

H0T +1.33 — �0.28 +0.28

H0B �1.33 — +0.28 �0.28

ST,0 — �2.50 — �2.50

ST — — �2.50 +2.50

SA,0 — +2.59 — +3.59

SA — — +2.59 �3.59

H0 — — — �1.00

h — +0.01 +0.01 —

vA �0.27 +0.21 +0.26 �0.06

vC �0.07 +0.01 +0.02 �0.01

vV �1.00 �0.22 — �0.22


Although here the approximation V 0A ¼ 1 was applied, V 0Ais also shown in Fig. 4a, for the sake of comparison withFig. 6. No plot of the relative cellulovascular coupling is

shown, because it can be calculated from Eq. (32) with noadditional constraint.The relative error in the calculated variables, generated

by 1% error in one of the measured variables or modelparameters is given in Table 3 (in %). Calculations werebased on the parameters in Table 2. All numbers refer to asmall vicinity of the baseline, because the differencebetween the baseline and the perturbed state was assumedto be negligible during the calculations. The results showthat Eqs. (30)–(32) are relatively error-tolerant: in mostcases, the relative error is less than 1%. However, largererrors in the measurement of arterial and/or tissuesaturation should be avoided.

4.4. Solution #2: when local blood flow is measured

Sometimes it is necessary to determine the extent ofarterial constriction or dilation, and/or the oxygenconsumption, but, as shown above, tissue spectroscopyalone proves insufficient to determine these variablesdirectly. This limitation can be overcome by the simulta-neous, independent measurement of local blood flow.Combinations of tissue spectroscopy and different bloodflow measuring techniques, like laser-Doppler flowmetry(Jones et al., 2001), laser speckle flowmetry (Dunn et al.,2003) and diffuse light correlation flowmetry (Culver et al.,2003; Durduran et al., 2004) have been already applied. Insuch cases the number of equations and unknownsbecomes the same (seven), hence the calculation of R0,V 0A, V 0V , E, M0, R0A and R0V will be possible. First, R0 has tobe calculated from DP0 and Q0 by Eq. (26). Then, a fourth-order algebraic equation (derived from Eq. (23) and Eqs.(27)–(29)), has to be solved to calculate V 0A:

a4V04A þ a3V

03A þ a2V 0

2A þ a1V 0A þ a0 ¼ 0, (33)

where the coefficients are the following:

a4 ¼ ~rCv2A,

a3 ¼ �2~rCvA ~vC ,

a2 ¼ ~rC ~v2C � rAv2A � rV v2V ,

ARTICLE IN PRESS

1

0.7

1

ST

0

10.6 0.8

0.2

0.4

0.6

0.8

1.2

1.2

2.21.4 1.6 1.8 2

0 1 2

1

0.7

ST

00 1 2

1

0.7

ST

00 1 2

HT′

E

VV′

VA′

(a)

(b)

(c)

R ′ = 1@

Fig. 6. The values of (a) V 0A, (b) V 0V and (c) E, calculated from the0


a1 ¼ 2rAvA ~vC ,

a0 ¼ �rA ~v2C ,

where

~rC ¼DP0

Q0� rC ¼ R0 � rC ,

~vC ¼H 0TH 0B� vC .

The numerical solution (with V0A ¼ 1 as the initialguess), is easier than the analytical one. Subsequently, V0Vand E can be calculated by successive substitution into thefollowing formulas (derived from Eqs. (23)–(24)):

V 0V ¼1

vV

H 0TH 0B� vC � vAV 0A

� �, (34)

E ¼ 1�ST

SA

� �1þ

vAV 0A þ hvC=2

vV V 0V þ hvC=2

� �. (35)

Note that the only difference between Eqs. (30)–(31) andEqs. (34)–(35) is the presence of V 0A in the latter two. Theremaining three unknowns, M0, R0A and R0V are determinedfrom Eqs. (25), (28) and (29), respectively (in Eq. (25), E0 iscalculated from E).

The investigation of the above equations allows thedetermination of unphysiological value combinations ofthe input variables: these combinations would yieldunphysiological results when substituted, namely valuesfor which the restrictions V 0AX0, V 0VX0 or 0pEp1 donot hold. This is demonstrated in Fig. 5, for the simple casewhen H 0B ¼ 1, and SA can be approximated by 1:

equations of solution #2 (Eqs. (33)–(35)), for the special case when R ¼ 1

(section #1).

(1)

ST

Fig.

solu

For a given (H 0T ;R0) pair, there is a smallest ST (surface

1), below which the oxygen extraction fraction, E

would be larger than 1 (domain *).
(2) For a given H0T (4vC), there is a smallest R0 (surface 2),
below which Eq. (33) has no positive, real numbersolution, independently of ST (domain **).

1

0.7

(*)

(**)

(1)

0

(2)

0

0

11

2 2

H T′

R ′

5. Valid and invalid domains in the input variable space, in case of

tion #2 (see text).

The values of V 0A, V 0V and E, calculated from Eqs. (33) to(35), respectively, are shown in Figs. 6–8, for special caseswhen one of the three measured variables do not change.That is, only three orthogonal sections of the input variablespace are plotted: the three which pass through the pointrepresenting the baseline state. We have to emphasize thatH 0T and R0 are not independent but usually change together(because compartmental blood volumes influence both),therefore, in contrast to Fig. 4, Figs. 6–8 cannot be used asnomograms. However, the comparison of these contourplots reveals that the calculated value of V 0A, V 0V and E ismainly determined by only one of the measured variables,namely R0, H 0T and ST, respectively (see the arrows on theright of the plots, which indicate the main trend ofincrease). This can be explained by three, well-known,physiological facts:

(1)
The vascular resistance is mostly determined by thearteries (rA is much larger than rC and rV, cf. Table 2).
(2)
The hemoglobin (i.e. the blood) resides mainly in theveins (vV is much larger than vC and vA, cf. Table 2).

ARTICLE IN PRESS

1

0.7

1

ST

0

10.8

0.2

0.4

0.6

0.8

1.4 1.2 0.8

0 1 2

1

0.7

ST

00 1 2

1

0.7

ST

00 1 2

R ′

E

VV′

VA′

(a)

(b)

(c)

HT′ = 1@

Fig. 7. The values of (a) V 0A, (b) V 0V and (c) E, calculated from the

equations of solution #2 (Eqs. (33)–(35)), for the special case when H 0T ¼ 1

(section #2).

1

0

1

2

10.6 0.8

0.8

1.2

1.2

2.21.4

1.4

1.6 1.8 2

0 1 2

00

1 2

2

2

1 VV′

VA′

(a)

(b)

R′

R′

ST = 0.7@


(3)
The saturation of hemoglobin is essentially deter-mined by the oxygen extraction of the tissue (cf. Eqs.(17)–(18)).
0.4

1

00 1 2

HT′

E

(c)

R′

Fig. 8. The values of (a) V 0A, (b) V 0V and (c) E, calculated from the

equations of solution #2 (Eqs. (33)–(35)), for the special case when

Note that the contour plots in Fig. 6 are very similar tothose of Fig. 4 (except that they are clipped at a higherH 0T ), which supports that the V 0A ¼ 1 approximation ofsolution #1 does not cause a serious error. No plot of M0,R0A and R0V is shown, because no additional constraint isimposed on their calculation from Eqs. (25), (28) and (29).

As in the case of solution #1, the relative error in thecalculated variables, generated by 1% error in one of themeasured variables or model parameters is given in Table 4(in %). The results are very similar: Eqs. (25), (26), (28),(29) and (33)–(35) proved robust, but larger errors in themeasurement of saturations should be avoided.

5. Discussion

In this paper, we present a simple, quasistatic, compart-mental model of tissue-level hemodynamics and oxygena-

ST ¼ 0.7 (section #3).

ARTICLE IN PRESS

Table 4

Results of sensitivity analysis, in case of solution #2. Zero sensitivity values are indicated by dashes

%/% R0 V0A V0V E0 E R0A R0V M0

DP0 +1.00 �0.76 +0.20 — �0.20 +1.52 �0.40 �0.20

Q0 �1.00 +0.75 �0.20 — +0.20 �1.51 +0.40 +1.20

H0T — �0.30 +1.41 — �0.36 +0.61 �2.83 �0.36

H0B — +0.29 �1.40 — +0.36 �0.58 +2.81 +0.36

ST,0 — — — �2.50 — — — +2.50

ST — — — — �2.50 — — �2.50

SA,0 — — — +2.59 — — — �3.59

SA — — — — +2.59 — — +3.59

H0 — — — — — — — +1.00

h — — — +0.01 +0.01 — — —

vA — +0.06 �0.28 +0.21 +0.28 �0.12 +0.57 +0.07

vC — +0.02 �0.07 +0.01 +0.03 �0.03 +0.14 +0.02

vV — +0.23 �1.06 �0.22 +0.06 �0.46 +2.12 +0.28

rA — +0.53 �0.14 — +0.14 �1.06 +0.29 +0.14

rC — +0.11 �0.03 — +0.03 �0.23 +0.06 +0.03

rV — +0.12 �0.03 — +0.03 �0.24 +0.07 +0.03


tion, which is suitable to calculate important physiologicalvariables from raw tissue spectroscopic data. Modelequations were introduced in a detailed manner, in orderto facilitate their straightforward application for theevaluation of experimental data. The information contentof spectroscopic data alone was shown to be valuable butlimited (solution #1): the relative venous volume, theoxygen extraction fraction and the relative cellulovascularcoupling can be calculated from these data. The number ofvariables estimated by the model can be increased if localblood flow is measured simultaneously (solution #2): in thiscase, the relative arterial and venous volume and resis-tance, the oxygen extraction fraction, and the relativeoxygen consumption can be determined.

5.1. Comparison with other mathematical models and

formulas

Modeling tissue spectroscopic data. Fantini (2002)derived the most complete equations that relate the totalconcentration and saturation of hemoglobin, measured bytissue spectroscopy, to the underlying physiological vari-ables. His Eqs. (12)–(15) are practically equivalent with ourEqs. (3)–(6) because his expression for the average capillarysaturation can be substituted by our Eq. (7) (for details, seeAppendix C). The main difference is that we incorporatedthe Fahraeus-effect into our equations, according to thesuggestion of the same author: by applying different totalhemoglobin concentrations in case of the arterial, thecapillary and the venous compartments.

Estimation of blood volume. Blood volume is usuallyassumed to be proportional to the total concentration ofhemoglobin (Jones et al., 2001; Rostrup et al., 2002). Thismeans that

H 0T ¼ V 0B ¼ vAV 0A þ vCV 0C þ vV V 0V , (36)

which does not hold strictly when H 0B, i.e. average localhematocrit changes, e.g. after hemorrhage (Eke, 1988), cf.Eq. (23). No previous work tried to separately estimate therelative volume of the arterial and the venous compart-ment.

Estimation of oxygen extraction fraction and oxygen

consumption. Oxygen consumption is usually calculatedfrom the equation

M 0 ¼ Q0E0, (37)

which implicitly assumes that arterial saturation andsystemic hematocrit is constant (Boas et al., 2003; Culveret al., 2003; Dunn et al., 2003; Durduran et al., 2004; Joneset al., 2001; Mayhew et al., 2001; Zheng et al., 2002), cf.Eq. (25). To apply this formula, Q0 has to be measured andE0 has to be calculated from tissue spectroscopic data.Three main methods have been developed to calculate E0:(1) The ratio method (Hoge et al., 1999; Mayhew et al.,

2001) assumes that SA is approximately 1, starts out fromthat E ¼ 1� SV ¼ 1�H

ðoxyÞV =HV ¼ H

ðdeoxyÞV =HV , that is,

E0 ¼ H0V(deoxy)/H0V (cf. Eqs. (1)–(2) and (18)), and approx-

imates this ratio by H0T(deoxy)/H0T, which can be calculated

from tissue spectroscopic data:

E0 ¼H 0ðdeoxyÞT

H 0T. (38)

This is the original version of this approach (Hoge et al.,1999). Two tuning constants were introduced later(Mayhew et al., 2001), and either E0 and M0 were calculatedover a range of different values of these constants (Joneset al., 2001; Mayhew et al., 2001), or their value was simplyassumed to be 1, and Eq. (38) was directly applied (Dunnet al., 2003).


If SA is approximated by 1, from our Eqs. (1)–(2) and(24) we get

E0 ¼1þ ðvAV 0A þ hvCV 0C=2ÞðvV V 0V þ hvCV 0C=2Þ

1þ ðvA þ hvC=2Þ=ðvV þ hvC=2Þ

H0ðdeoxyÞT

H 0T

(39)

Eq. (38) can be used as an approximation of Eq. (39) ifthe fraction on the left does not differ considerably from 1,that is, in case of relatively small volume changes (cf.Figs. 4 and 6). For example, if VA and VC are constant, andVV decreases or increases by 25%, the relative errorbecomes �6.6% and +4.5%, respectively. The equationsof Culver et al. (2003) are analogous to Eq. (38). With ournotations, their expression for E can be written as

E ¼ 1�ST

SA

� �1þ

vA þ vC=2

vV þ vC=2

� �¼ 1�

ST

SA

� �1

vV þ vC=2,

(40)

where the denominator of the fraction on the right wasdenoted by g in the original article. This formula isequivalent with Eq. (38), if SA is approximated by 1,because HT

(deoxy)/HT ¼ 1�ST, and g drops out. Note thatEq. (40) is similar to our Eqs. (31) and (35), apart from thatthe alterations of blood volumes and the Fahraeus-effect(h) is neglected.

(2) The saturation method (Mayhew et al., 2001) is basedon a mathematical model of the oxygen transport throughthe capillary wall, which has been developed by Buxtonand Frank (1997) and later refined by Hyder et al. (1998).In this model (which is also utilized by Fantini, 2002),tissue oxygen tension is neglected, which implies that theoxygen consumption and the oxygen extraction fraction istightly coupled to blood flow:

E0 ¼1� ð1� E0Þ

D0=Q0

E0, (41)

where D0 denotes the relative value of capillary ‘‘diffusiv-ity’’. In the saturation method, however, the followingempirical formula was chosen to calculate E0, on the basisof computer simulations of this model:

E0 ¼1� ð1� E0Þ

lnðST Þ= lnðST ;0Þ

E0, (42)

E0 and M0 were calculated over a range of different valuesof E0. In our model, this approach could not possibly beapplied given that tissue oxygen tension is significantlylarger than zero (Vovenko, 1999) allowing the tissue toadjust its oxygen consumption to its needs, independentlyof perfusion (Zheng et al., 2002; Valabregue et al., 2003).Moreover, this method neglects the alterations in thevolume of vascular compartments. Note that in our model,E0 can be directly calculated from ST,0 (and SA,0), fromEqs. (31) or (35), by substituting V 0V ¼ 1 and V 0A ¼ 1

(because by definition, relative values equal to 1 atbaseline).(3) The capillary model method (Mayhew et al., 2001;

Zheng et al., 2002) is based on an improved version of theabove mentioned oxygen transport model: the tissueoxygen content is taken into account, and blood volumechanges are assumed to occur (but only in the venouscompartment, cf. our solution #1). Another similardynamic model based method has been developed by Boaset al. (2003). However, these methods seem to be applicablemainly for simulation purposes.

5.2. Validity of assumptions

(a1) Absolute concentrations: Only the more sophisti-cated spectroscopic techniques are suitable to directlydetermine absolute concentrations, e.g. time-resolvedspectroscopy, intensity-modulated spectroscopy (Delpyand Cope, 1997; Ferrari et al., 2004) and diffuse opticaltomography (Culver et al., 2003; Durduran et al., 2004). Inother cases, when only changes in concentrations can bedetermined, e.g. by conventional continuous-wave NIRS(Ferrari et al., 2004), baseline values have to be estimatedor measured, but, unfortunately, the former is alwaysmore-or-less inaccurate, and the latter requires physiologi-cal perturbations to be made, which cannot always beperformed (Delpy and Cope, 1997). Other technicalproblems of tissue spectroscopy, such as the partial volumeeffect and the crosstalk between oxy- and deoxyhemoglo-bin signals are discussed elsewhere (Strangman et al.,2003). In muscles, myoglobin interferes with the absorptionof hemoglobin, but this effect cannot cause a substantialerror in the estimation of hemoglobin concentrations(Ferrari et al., 2004).(a2) Independent regions: The borders of the ROI are

determined by the propagation of photons in tissue (Wanget al., 1995). Photons do not respect the borders betweenvascular territories, which can cause some error, becauseour model treats regional vascular systems as beingcoupled in parallel. Otherwise, parallel coupling is anacceptable approximation, because pressure drop mainlyoccurs at the level of small vessels, especially arterioles(Nichols and O’Rourke, 1990). Oxygen diffusion from oneregion to another cannot be excluded, which may alsocontribute to model error.(a3) Compartmental model: Focal changes inside the ROI

cannot be resolved (cf. Strangman et al., 2003). Thedistributions of vascular transit times are neglected (cf.Buxton and Frank, 1997). Although gas exchange cannotonly occur through the capillary wall (Vovenko, 1999), SA

and SV can be regarded constant along the arterial and thevenous compartments, respectively (Sharan and Popel,2002), and SA can be accurately approximated by thesystemic arterial saturation.(a4) Quasistatic changes: Fast processes, like temporary

functional activation, are often investigated. Transients ofBOLD-fMRI signals after cerebral activation, like the


initial overshoot and the post-stimulus undershoot, hasbeen explained by an ‘‘elastic balloon’’ model of the venouscompartment (Buxton et al., 2004). These transients can beobserved by NIRS, too (Heekeren et al., 1997). Theaccuracy of our equations have to be assessed before theyare applied to the analysis of these dynamics, becausevascular compliances (Boas et al., 2003; Zheng et al., 2002)are not included in our model. The effect of the oxygenstoring capacity of the interstitial and intracellularcompartments (Goldstick, 1973; Valabregue et al., 2003),and the radial and longitudinal oxygen diffusion along thecapillaries (Secomb et al., 2000) also will have to beevaluated.

(a5) Laminar blood flow and homogeneous volume change

in arteries and veins: Blood flow can be regarded as laminarin smaller arteries and veins, but the error of thisapproximation increases as vessel diameter decreases(Fahraeus–Lindquist-effect; Pries and Secomb, 2003; Prieset al., 1996). This fact has been taken into considerationwhen we assumed laminar flow in the arterial and thevenous compartments only, but not in capillaries. Changeof vessel diameters within these compartments cannot beperfectly homogeneous.

(a6) Spatially homogeneous oxygen consumption: Undersevere but incomplete ischemia, oxygen consumptionbecomes inhomogeneous, because the oxygen efflux occursmostly near the arterial end of capillaries, and theaverage capillary saturation can be expected to approachvenous saturation (for details, see Appendix C). Undercomplete ischemia (no perfusion), the model cannot beapplied.

Inaccuracies from all these assumptions certainly in-crease model error. A more sophisticated, distributed-parameter model could be helpful to assess their extentquantitatively.

5.3. Validity of approximations

We applied two approximations about blood volumealterations, namely V0C ¼ 1 (in solutions #1 and #2) andV 0A ¼ 1 (in solution #1 only), because it is generallyaccepted that tissue spectroscopic signals are determinedmainly by the venous compartment (Ferrari et al., 2004).This can be explained by the differences between the weightfactors of V 0A, V0C and V 0V in Eqs. (23)–(24): the influenceof these relative volumes on H 0T and ST can becharacterized by the following order: {influence ofV 0C}o{influence of V 0A} 5 {influence of V 0V} (cf. Table2). It would be possible to completely omit all membersfrom these equations which contain V 0C or V 0A, but doingso would generate a somewhat larger error than thesubstitution of V 0C ¼ 1 and V 0A ¼ 1. Note that in contrastto other organs (Fantini, 2002), no capillary recruitmentoccurs in the brain (Buxton and Frank, 1997), therefore inthis case the approximation V 0C ¼ 1 can be more safelyapplied.

5.4. Experimental validation of model predictions

It would be a good sign, if the measurement points,defined by the measured (H0T, ST) pairs or (H0T, R0, ST)triples, never fell into the invalid domains of the inputvariable space. However, direct validation of modelpredictions would require simultaneous measurement bypositron emission tomography (PET) and tissue spectro-scopy. There is a series of technical difficulties with such avalidation protocol, like the spatial matching (coregistra-tion) of corresponding tissue regions. Rostrup et al. (2002)in fact has carried out a combined PET/NIRS study tocompare cerebral blood volume changes (DCBV) measuredby PET and the one estimated by a simple formula fromtotal tissue hemoglobin concentration changes, measuredby NIRS. A significant correlation was found, but,unfortunately, the partial volume effect (Strangman etal., 2003) has not been taken into account, which led onaverage to a 6.5 times underestimation in DCBV by NIRSas compared to by PET, where this effect is not present.Note that the partial volume effect is not needed in the caseof our model, because it can be represented by a constantfactor (see, e.g. Durduran et al., 2004), which drops out ofthe expression for H0T and ST.

5.5. Possible applications of the model

Our equations contain relative-to-baseline values, i.e.measurements have to be performed in a baseline state andthen in one or more states with perturbations, such asfunctional activation, hyper- or hypotension, hypoxichypoxia, hemodilution or blood loss. Note that whenautoregulation is activated (e.g. during transient hypoten-sion), the reduced model (solution #1) becomes inaccurate,and should only be used with caution, because of thesignificant changes in the volume of the arterial compart-ment. In such cases, the simultaneous measurement ofblood flow is called for (solution #2). In fact, thecalculation of the changes in arterial resistance can beapplied to estimate the extent of the autoregulatoryresponse. Human and animal applications are equallypossible; the main limiting factor, the need for accurateabsolute concentrations, is technical (see above), thereforeindependent of the model. As long as monitored tissueregions can be regarded independent (a2), our model canbe applied to analyse imaging data, too (e.g. in case of theskin and the cerebral cortex). Calculations for cases withrapid dynamics should be performed with care. It ispossible that only a more sophisticated model, whichincorporates vascular compliances and uses differentialequations, can characterize dynamic processes accurately.On the other hand, currently available dynamic models(Boas et al., 2003; Zheng et al., 2002) will have to bemodified in order to make them better suit practicalrequirements.In conclusion, an integrated and extended version of

existing models was shown to be applicable to calculate


hemodynamic and oxygenation variables from the rawdata of tissue spectroscopy. Mathematical models such asours can help to explore the physiology behind the changesin raw optical signals, which seems to bear importancegiven the rapid advance of in vivo optical monitoring oftissue functions.

Acknowledgments

This study was supported by OTKA Grant T34122 andEuroBloodSubstitutes (EBS) Consortium LSHB-CT-2004-503023.

Appendix A. Definitions of the relative variables

In Eqs. (5)–(22), the non-dimensional variables are ST,SA, SC, SV, and E, and the variables with dimension areHT, HA, HC, HV, HB, H, VT, VA, VC, VV, VB, JA, JC, JV,M, Q, DP, RA, RC and RV (Table 1). Owing tocancellations, only few members of the latter group arepresent in the derived equations (Eqs. (23)–(29)), and onlyin a non-dimensional, relative (compared-to-baseline)form. Their definitions are listed below.

Relative hemoglobin concentrations:

H 0T ¼HT

HT ;0; H 0B ¼

HB

HB;0; H 0 ¼

H

H0. (A.1)2(A.3)

Relative vascular volumes:

V 0A ¼V A

V A;0; V 0C ¼

V C

V C;0; V 0V ¼

V V

VV ;0. (A.4)2(A.6)

Relative oxygen consumption:

M 0 ¼M

M0. (A.7)

Relative blood flow:

Q0 ¼Q

Q0

. (A.8)

Relative pressure head:

DP0 ¼DP

DP0. (A.9)

Relative vascular resistances:

R0A ¼RA

RA;0; R0C ¼

RC

RC;0; R0V ¼

RV

RV ;0. (A.10)

Although SA and E are non-dimensional, their relativevalue is also need to be defined:

S0A ¼SA

SA;0, (A.11)

E0 ¼E

E0. (A.12)

These ‘‘doubly non-dimensional’’ variables are present inthe same equation (Eq. (25)).

Appendix B. Definition of baseline volume and resistance

fractions

In addition to the hemoglobin ratio (h), we found usefulto define six other non-dimensional parameters, whichrepresent the baseline distribution of blood volume andvascular resistance between arterial, capillary and venouscompartments (Table 2):

Baseline volume fractions:

vA ¼V A;0

V B;0; vC ¼

VC;0

V B;0; vV ¼

VV ;0

V B;0. (B.1)2(B.3)

Note that vA+vC+vV ¼ 1.Baseline resistance fractions:

rA ¼RA;0

R0; rC ¼

RC;0

R0; rV ¼

RV ;0

R0. (B.4)2(B.6)

Note that rA+rC+rV ¼ 1.

Appendix C. The average capillary saturation

The saturation of hemoglobin continuously decreasesalong the capillaries, from the arterial to the venous value.Fantini (2002) derived an exponential relationship todescribe this phenomenon. With our notations:

SCðlÞ ¼ SA expð�klÞ, (C.1)

where 0plpL is the position along a capillary of length L,and k is a constant. The venous saturation is equal to thecapillary saturation at l ¼ L:

SV ¼ SA expð�kLÞ. (C.2)

The average capillary saturation:

SC ¼1

L

Z L

0

SCðlÞ dl ¼ SA

1� expð�kLÞ

kL¼

SA � SV

lnðSA=SV Þ.

(C.3)

By substituting the approximation (Bronshtein et al., 2003)

ln x ¼ 2x� 1

xþ 1(C.4)

into Eq. (C.3), we get

SC ¼SA þ SV

2, (C.5)

which corresponds with Eq. (7). According to Eq. (18), therelative difference between Eqs. (C.5) and (C.3) is about2% if E ¼ 0.4 (which is typical of the cerebral cortex;Mintun et al., 2001), and about 12% if E is increased to 0.7.If E increases over about 0.7 (e.g. under ischemicconditions), (a6) presumably becomes invalid, because theoxygen consumption at the arterial end of capillaries mayexceed that at the venous end. Independently of thederivation of Eq. (C.1), the decrease in saturation maybetter be described by this exponential function than by alinear one, that is, Eq. (7) can probably be replaced by


Eq. (C.3), and therefore Eq. (17) becomes

SC ¼E

lnð1=ð1� EÞÞSA. (C.6)

In such a case, the derivation of an explicit formula forE, like Eqs. (31) and (35), is not possible, and E must becalculated by an appropriate numerical method. However,coming to a decision about whether the linear or theexponential approach is better, is beyond the scope of thisarticle.

References

Boas, D.A., Strangman, G., Culver, J.P., Hoge, R.D., Jasdzewski, G.,

Poldrack, R.A., Rosen, B.R., Mandeville, J.B., 2003. Can the cerebral

metabolic rate of oxygen be estimated with near-infrared spectro-

scopy? Phys. Med. Biol. 48, 2405–2418.

Bronshtein, I.N., Semendyayev, K.A., Musiol, G., Muehlig, H., 2003.

Handbook of Mathematics, 4th ed. Springer, New York.

Buxton, R.B., Frank, L.R., 1997. A model for the coupling between

cerebral blood flow and oxygen metabolism during neural stimulation.

J. Cereb. Blood Flow Metab. 17, 64–72.

Buxton, R.B., Uludag, K., Dubowitz, D.J., Liu, T.T., 2004. Modeling the

hemodynamic response to brain activation. Neuroimage 23,

S220–S233.

Chance, B., 1998. Near-infrared images using continuous, phase-

modulated, and pulsed light with quantitation of blood and blood

oxygenation. Ann. N. Y. Acad. Sci. 838, 29–45.

Culver, J.P., Durduran, T., Furuya, D., Cheung, C., Greenberg, J.H.,

Yodh, A.G., 2003. Diffuse optical tomography of cerebral blood flow,

oxygenation, and metabolism in rat during focal ischemia. J. Cereb.

Blood Flow Metab. 23, 911–924.

Delpy, D.T., Cope, M., 1997. Quantification in tissue near-infrared

spectroscopy. Philos. Trans. R. Soc. London B Biol. Sci. 352,

649–659.

Dunn, A.K., Devor, A., Bolay, H., Andermann, M.L., Moskowitz, M.A.,

Dale, A.M., Boas, D.A., 2003. Simultaneous imaging of total cerebral

hemoglobin concentration, oxygenation, and blood flow during

functional activation. Opt. Lett. 28, 28–30.

Durduran, T., Yu, G., Burnett, M.G., Detre, J.A., Greenberg, J.H.,

Wang, J., Zhou, C., Yodh, A.G., 2004. Diffuse optical measurement of

blood flow, blood oxygenation, and metabolism in a human brain

during sensorimotor cortex activation. Opt. Lett. 29, 1766–1768.

Eke, A., 1988. Reflectometric imaging of local tissue hematocrit in the rat

brain cortex. In: Tomita, M., Sawada, T., Naritomi, H., Heiss, W.-D.

(Eds.), Cerebral Hyperemia and Ischemia: From the Standpoint of

Cerebral Blood Volume. Elsevier Science Publishers BV, Osaka, pp.

247–257.

Eke, A., 1992. Multiparametric mapping of cerebrocortical microcircula-

tion by television reflectometry. In: Frank, K., Kessler, M. (Eds.),

Quantitative Spectroscopy in Tissue. pmi Verlagsgruppe, Frankfurt

am Main, pp. 105–118.

Elwell, C.E., Cope, M., Edwards, A.D., Wyatt, J.S., Delpy, D.T.,

Reynolds, E.O., 1994. Quantification of adult cerebral hemo-

dynamics by near-infrared spectroscopy. J. Appl. Physiol. 77,

2753–2760.

Fantini, S., 2002. A haemodynamic model for the physiological

interpretation of in vivo measurements of the concentration and

oxygen saturation of haemoglobin. Phys. Med. Biol. 47, N249–N257.

Ferrari, M., Mottola, L., Quaresima, V., 2004. Principles, techniques, and

limitations of near infrared spectroscopy. Can. J. Appl. Physiol. 29,

463–487.

Gao, E., Young, W.L., Pile-Spellman, J., Ornstein, E., Ma, Q., 1998.

Mathematical considerations for modeling cerebral blood flow

autoregulation to systemic arterial pressure. Am. J. Physiol. 274,

H1023–H1031.

Goldstick, T.K., 1973. Oxygen transport. In: Brown, J.H.U., Gann, D.S.

(Eds.), Engineering Principles in Physiology. Academic Press, New

York, pp. 257–282.

Heekeren, H.R., Obrig, H., Wenzel, R., Eberle, K., Ruben, J., Villringer,

K., Kurth, R., Villringer, A., 1997. Cerebral haemoglobin oxygenation

during sustained visual stimulation—a near-infrared spectroscopy

study. Philos. Trans. R. Soc. London B. Biol. Sci. 352, 743–750.

Herman, P., Trubel, H.K., Hyder, F., 2006. A multiparametric assessment

of oxygen efflux from the brain. J. Cereb. Blood Flow Metab. 26,

79–91.

Hoge, R.D., Atkinson, J., Gill, B., Crelier, G.R., Marrett, S., Pike, G.B.,

1999. Investigation of BOLD signal dependence on cerebral blood flow

and oxygen consumption: the deoxyhemoglobin dilution model.

Magn. Reson. Med. 42, 849–863.

Hyder, F., Shulman, R.G., Rothman, D.L., 1998. A model for

the regulation of cerebral oxygen delivery. J. Appl. Physiol. 85,

554–564.

Jobsis, F.F., 1977. Noninvasive, infrared monitoring of cerebral and

myocardial oxygen sufficiency and circulatory parameters. Science 198,

1264–1267.

Jones, M., Berwick, J., Johnston, D., Mayhew, J., 2001. Concurrent

optical imaging spectroscopy and laser-Doppler flowmetry: the

relationship between blood flow, oxygenation, and volume in rodent

barrel cortex. Neuroimage 13, 1002–1015.

Krolo, I., Hudetz, A.G., 2000. Hypoxemia alters erythrocyte perfusion

pattern in the cerebral capillary network. Microvasc. Res. 59,

72–79.

Mandeville, J.B., Marota, J.J., Ayata, C., Zaharchuk, G., Moskowitz,

M.A., Rosen, B.R., Weisskoff, R.M., 1999. Evidence of a cerebrovas-

cular postarteriole windkessel with delayed compliance. J. Cereb.

Blood Flow Metab. 19, 679–689.

Mayhew, J., Johnston, D., Martindale, J., Jones, M., Berwick, J., Zheng,

Y., 2001. Increased oxygen consumption following activation of brain:

theoretical footnotes using spectroscopic data from barrel cortex.

Neuroimage 13, 975–987.

Mintun, M.A., Lundstrom, B.N., Snyder, A.Z., Vlassenko, A.G., Shul-

man, G.L., Raichle, M.E., 2001. Blood flow and oxygen delivery to

human brain during functional activity: theoretical modeling and

experimental data. Proc. Natl Acad. Sci. USA 98, 6859–6864.

Nichols, W.W., O’Rourke, M.F., 1990. McDonald’s Blood Flow in

Arteries—Theoretic, Experimental and Clinical Principles. 3rd ed.

Edward Arnold, London.

Pries, A.R., Secomb, T.W., 2003. Rheology of the microcirculation. Clin.

Hemorheol. Microcirc. 29, 143–148.

Pries, A.R., Secomb, T.W., Gaehtgens, P., 1996. Biophysical aspects of

blood flow in the microvasculature. Cardiovasc. Res. 32, 654–667.

Rostrup, E., Law, I., Pott, F., Ide, K., Knudsen, G.M., 2002. Cerebral

hemodynamics measured with simultaneous PET and near-infrared

spectroscopy in humans. Brain Res. 954, 183–193.

Sakai, F., Nakazawa, K., Tazaki, Y., Ishii, K., Hino, H., Igarashi, H.,

Kanda, T., 1985. Regional cerebral blood volume and hematocrit

measured in normal human volunteers by single-photon emission

computed tomography. J. Cereb. Blood Flow Metab. 5, 207–213.

Secomb, T.W., Hsu, R., Beamer, N.B., Coull, B.M., 2000. Theoretical

simulation of oxygen transport to brain by networks of microvessels:

effects of oxygen supply and demand on tissue hypoxia. Microcircula-

tion 7, 237–247.

Sharan, M., Popel, A.S., 2002. A compartmental model for oxygen

transport in brain microcirculation in the presence of blood

substitutes. J. Theor. Biol. 216, 479–500.

Strangman, G., Franceschini, M.A., Boas, D.A., 2003. Factors affecting

the accuracy of near-infrared spectroscopy concentration calcula-

tions for focal changes in oxygenation parameters. Neuroimage 18,

865–879.

Valabregue, R., Aubert, A., Burger, J., Bittoun, J., Costalat, R., 2003.

Relation between cerebral blood flow and metabolism explained

by a model of oxygen exchange. J. Cereb. Blood Flow Metab. 23,

536–545.


Van Lieshout, J.J., Wieling, W., Karemaker, J.M., Secher, N.H., 2003.

Syncope, cerebral perfusion, and oxygenation. J. Appl. Physiol. 94,

833–848.

Vovenko, E., 1999. Distribution of oxygen tension on the surface of

arterioles, capillaries and venules of brain cortex and in tissue in

normoxia: an experimental study on rats. Pflugers. Arch. 437, 617–623.

Wang, L., Jacques, S.L., Zheng, L., 1995. MCML—Monte Carlo

modeling of light transport in multi-layered tissues. Comput. Methods

Programs Biomed. 47, 131–146.

Zheng, Y., Martindale, J., Johnston, D., Jones, M., Berwick, J., Mayhew,

J., 2002. A model of the hemodynamic response and oxygen delivery to

brain. Neuroimage 16, 617–637.

Mathematical model for the estimation of hemodynamic and oxygenation variables by tissue...

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