BIOPHYSICS AND BASICBIOMEDICAL

RESEARCH - Full Papers

Modeling of Look-Locker Estimates of the MagneticResonance Imaging Estimate of Longitudinal RelaxationRate in Tissue After Contrast Administration

Ramesh Paudyal,1 Hassan Bagher-Ebadian,1,2 Tavarekere N. Nagaraja,3

Joseph D. Fenstermacher,3 and James. R. Ewing1,2,4*

This paper models the behavior of the longitudinal relaxationrate of the protons of tissue water R1 (R1 5 1/T1), measuredin a Look-Locker experiment at 7 Tesla after administration ofa paramagnetic contrast agent (CA). It solves the Bloch-McConnell equations for the longitudinal magnetization of theprotons of water in a three-site two-exchange (3S2X) modelwith boundary conditions appropriate to repeated sampling ofmagnetization. The extent to which equilibrium intercompart-mental water exchange kinetics affect monoexponential esti-mates of R1 after administration of a CA in dynamic contrastenhanced experiment is described. The relation between R1

and tissue CA concentration was calculated for CA restrictedto the intravascular, or to the intravascular and extracellularcompartments, by varying model parameters to mimic experi-mental data acquired in a rat model of cerebral tumor. Themodel described a nearly linear relationship between R1 andtissue concentration of CA, but demonstrated that the appa-rent longitudinal relaxivity of CA depends upon tissue type.The practical consequence of this finding is that the extendedPatlak plot linearizes the DR1 data in tissue with leaky micro-vessels, accurately determines the influx rate of the CAacross these microvessels, but underestimates the volume ofintravascular blood water. Magn Reson Med 66:1432–1444,2011. VC 2011 Wiley Periodicals, Inc.

Key words: look locker pulse sequence; relaxation rateconstant; contrast mechanism; Bloch-McConnell’s equations;shutter-speed; water exchange rate

Dynamic contrast enhanced MRI (DCE-MRI) often usesthe assumption that a change in the tissue water protons’longitudinal relaxation rate (DR1, where DR1 ¼ D1/T1)induced by a paramagnetic MR contrast agent (CA) suchas Gd-DTPA is proportional to the tissue concentrationof CA and can therefore serve as a measure of CA tissueconcentration as it varies with time. This concentration-time curve is then used to estimate vascular kinetic pa-rameters in cerebral diseases such as stroke and tumors

using a pharmacokinetic theory that is decades old (1–5).We, along with others, call this model ‘‘the standardmodel’’ (abbreviated as SM). The linear relationshipbetween DR1 and tissue concentration has been thoughtto depend on the rapid exchange (relative to the ensem-ble’s longitudinal relaxation rate) of water protonsbetween the different tissue compartments.

Contrariwise, it has been proposed that estimates ofthe relaxation rate of tissue water protons produced byusing a single exponential as a model for the recovery oflongitudinal magnetization may yield an estimate of tis-sue CA concentration that is significantly biased due tothe finite rates of exchange of water protons between thetissue compartments (6,7). This latter work was con-ducted first in a 2-compartment model (2SX) that studiedproton exchange across parenchymal cell membranesand essentially ignored the vascular contribution (7–11);later this was extended to a 3-compartment model (3SX)that included intravascular (blood), interstitial, and intra-cellular spaces (12–14). Because, in both the 2SX and3SX models, the apparent longitudinal relaxivity of thetissue depends on the rate constants of intercompartmen-tal water exchange, the model has been named by itsauthors the ‘‘shutter-speed model’’ (SSM). In a recentapplication, a 2-compartment (2SX) SSM that ignores thevascular compartment has shown promise in discrimi-nating malignant lesions in breast cancer (13,14).

The SSM has recently been challenged (15). The basisof this challenge was twofold: (1) the SSM as imple-mented was held to be an incomplete representation ofthe actual experimental condition in typical DCE experi-ments and (2) experimental fitting results for the stand-ard model versus the SSM showed no advantage for theSSM. This latter result does not necessarily disprove theSSM; rather, it may point to the complexity of the SSM(5 free variables) in contrast to the standard model (1, 2,or 3 free variables) (5) and the consequent requirementof data with much higher signal-to-noise for the SSM.

The question as to how the change in an estimate of asingle longitudinal rate constant relates to the tissue con-centration of CA retains a great deal of practical signifi-cance in current clinical considerations. This article pro-poses to particularize this question to Look-Locker (LL)studies in a rat model of cerebral tumor at 7 Tesla, andthus examine the operating characteristics of thisapproach to estimates of vascular permeability Ktrans. Weemphasize that it is not the purpose of this paper tocompare models—that is a much more complex task—

1Department of Neurology, Henry Ford Hospital, Detroit, Michigan.2Department of Physics, Oakland University, Rochester, Michigan.3Department of Anesthesiology, Henry Ford Hospital, Detroit, Michigan.4Department of Neurology, Wayne State University, Detroit, Michigan.

Grant sponsor: NIH; Grant number: R01 CA135329-01; Grant sponsor: MRIBiomarkers of Response in Cerebral Tumors; Grant sponsor: MRI Measuresof Blood Brain Barrier Permeability; Grant number: RO1 HL70023-01A1

*Correspondence to: James. R. Ewing, Ph.D., Neurology NMR Facility, E&RB126, Henry Ford Hospital, 2799 W. Grand Blvd, Detroit MI 48202. E-mail:jre@neurnis.neuro.hfh.edu

Received 11 August 2010; revised 7 December 2010; accepted 5 January2011.

DOI 10.1002/mrm.22852Published online 31 May 2011 in Wiley Online Library (wileyonlinelibrary.com).

Magnetic Resonance in Medicine 66:1432–1444 (2011)

VC 2011 Wiley Periodicals, Inc. 1432

but rather to examine in theory the response of tissuemagnetization under a particular experimental condition.This article addresses the theory of equilibrium inter-compartmental water exchange in its complete form (12)with correct boundary conditions and both theory andmodeling of a three-site two-exchange model (3S2Xmodel) as applied to LL data at 7 Tesla, but does notaccount for blood inflow effects (16). The modelingresults are used to examine the relation between a mono-exponential estimate of tissue R1 and tissue CA concen-tration. The findings are that the apparent relaxivity of aCA in a given tissue does not vary appreciably over thecourse of a typical experiment. However, it is demon-strated that the apparent relaxivity of CA does varybetween tissues, and thus lends a note of caution to a na-ıve interpretation of experimental results in a DCE esti-mate of vascular permeability.

THEORY

Three-Site Two-Exchange Model

While there are many biological compartments in tissue,it is usually assumed that the protons associated withtissue water can be characterized as residing in one ofthree compartments: intravascular, extracellular, or intra-cellular with equilibrium water exchange kinetics takingplace between these compartments (7,8,11,12). The waterin red blood cells and in the plasma of the intravascularcompartment is assumed to form a single compartment,since the protons of water exchange rapidly between thetwo moieties (mixing time �10 ms (8), which is muchshorter than the ensemble’s longitudinal relaxationtime). On the other hand, water exchange between theintracellular and interstitial spaces (i.e., across parenchy-mal cellular membranes) and between the intravascularand the interstitial spaces (i.e., across the microvascularendothelium) are at least an order of magnitude slowerthan across the red blood cell membrane. It is generallyassumed that there is little or no direct exchange ofwater between the intravascular space and the intracellu-lar space via the intercellular tight junctions of the endo-thelium. For modeling longitudinal relaxation rate of tis-sue water protons, the intercompartmental equilibriumwater exchange kinetics can thus be described by a lin-ear three-site two-exchange [3S2X] model (12). A sche-matic diagram of a 3S2X model and its characteristic pa-rameters is given in Fig. 1.

Under equilibrium exchange, the Bloch–McConnellequations describe the longitudinal magnetization ofwater protons in each pool of a 3S2X system (17):

dMb tð Þdt

¼ M0b �Mbð ÞR1b � kbe Mb þ keb Me;

dMe tð Þdt

¼ M0e�Með Þ R1e�keb Me�kei Meþkbe Mb þ kie Mi;

dMi tð Þdt

¼ M0i �Mið Þ R1i � kie Mi þ kei Me;

½1�where Mb, Me, and Mi are the time-dependent longitudinalmagnetizations of the water protons in the blood, intersti-tial, and parenchymal intracellular spaces, respectively.M0b, M0e, and M0i denote their equilibrium magnetization

in the main magnetic field. R1b, R1e, and R1i are the longi-tudinal relaxation rate constants of the water protons inthe absence of exchange within the blood, interstitium,and intracellular spaces, respectively. The process of equi-librium water exchange from the index site ‘‘m" to ‘‘n" and‘‘n’’ to ‘‘m" is parameterized by the exchange rates Kmn andKnm, (m, n ¼ b, e, i), and the mean pre-exchange lifetime, aninverse of the exchange rate constant in a given compart-ment or pool, is denoted by tm (e.g., m ¼ b, e, i).

Equation 1 can be written in a matrix form (18):

dM

dt¼ AMþ C; ½2�

where M is a column vector,

M ¼Mb

Me

Mi

0@

1A; ½3�

A is the relaxation rate exchange matrix.

A ¼� R1b þ kbeð Þ keb 0

kbe � R1b þ keb þ keið Þ kie0 kei � R1i þ kieð Þ

24

35;½4�

and C is a column vector

C ¼Mob R1b

Moe R1e

Moi R1i

0@

1A: ½5�

FIG. 1. The three-site two-exchange (3S2X) model. R1b, R1e, and

R1i are the longitudinal relaxation rates of the protons of water inthe blood, interstitial, and intracellular spaces, respectively. Mob,

Moe, and M0i are equilibrium magnetizations of the protons inthese three spaces, respectively; their corresponding fractionalwater protons sizes are ub, ue, and ui. The rate of the water pro-

tons exchange from the blood to extracellular space is kbe andfrom the extracellular space back to blood is keb. The rate of

water protons exchange from intra- to extracellular space is kieand from extra- to intracellular space is kei. In general, paramag-netically labeled contrast agents (e.g., Gd-DTPA) can exchange

between plasma and the extracellular space across a leaky endo-thelium but will not enter the intracellular space (ui). The bidirec-

tional exchange of Gd-DTPA between the intravascular space andextracellular is described by the transvascular transfer rate con-stants Ktrans and kb, respectively.

Modeling LL Estimates of R1 After CA Administration 1433

Equation 2 can be solved using standard methods fordifferential equations. The characteristic equation thatsolves the exchange matrix A of an Eq. 4 leads to a cubicequation. Its solution at the particular boundary condi-tions gives three distinct real eigenvalues, which arecharacterized as the short (R1S), intermediate (R1l), andlong (R1L) longitudinal relaxation rate constants, respec-tively. These rate constants can be calculated by the ker-nels detailed in the Appendix. As in Eq. A1, the relaxa-tion rate constants are functions of the compartmentalrelaxation rates, intercompartmental water exchangerates, fractional water content of each pool, and CAconcentration.

A mass balance condition relates the exchange rateconstant to the fractional water content of the compart-ment in a 3S2X model:

kmnM0n ¼ knmM0m: ½6�

In addition, the total equilibrium longitudinal magnet-ization of a 3S2X system is M0 ¼ M0b þM0e þM0i, whichleads to ub þ ue þ ui ¼ 1, where ub, ue, and ui denote thefractional water content in the intravascular, interstial,and parenchymal intracellular spaces, respectively.

The rate of exchange across the plasma membrane, kie,is a function of membrane water permeability, P, and theratio of surface area, A, to the volume, vi, of the intracel-lular space is given by (8):

kie ¼ PA

vi

� �: ½7�

When CA is introduced to the blood, it may or maynot extravasate. If it extravasates, the CA distributes inthe extracellular space but does not cross intact cellmembranes in the parenchyma. The administration ofCA may therefore significantly increase the longitudinalrelaxation rate of the water protons within the intravas-cular space, R1b (R1b ¼ 1/T1b), and the extracellularspace, R1e (R1e ¼ 1/T1e), but not within the intracellularspace, R1i (R1i ¼ 1/T1i).

Under the well-justified assumption that all the waterprotons in blood undergo longitudinal relaxation as one en-semble; the effective longitudinal relaxation rate of theseprotons after the administration of CA is given by (19):

R1b ¼ R10b þ 1�Hctð Þ <p ½CAp� ½8�

where R10b and R1b are pre- and post-contrast relaxationrates [s�1] of water protons in blood, Hct is the microvas-cular hematocrit, [CAp] is the arterial plasma concentra-tion of the CA [mM], and Rp [mM�1s�1] is thelongitudinal relaxivity of the CAp, respectively.

In an in vivo MRI study, if CA diffuses from blood tothe interstitial compartment through leaky endothelium,then the relaxation rate of the protons in interstitialwater, R1e, is assumed to be linearly correlated to tissueconcentration of CA as (19):

R1e ¼ R10e þ <e ½CAe� ½9�

where R10e and R1e are the pre- and post-contrast relaxa-tion rates [s�1], [CAe] is CA concentration [mM] in inter-

stitial water, and Re [mM�1s�1] is the relaxivity of theCAe, respectively.

Paramagnetic chelates rely on short-range (�nm) inter-actions with water protons to increase R1, and thischange that serves as a ‘‘marker’’ or ‘‘surrogate’’ of CAdistribution. Unlike the other two compartments, achange in the R1i (about 80% of total tissue water) afterthe administration of a CA is effected mainly, if notexclusively, via the transfer of water molecules acrossthe cell membrane. Accordingly, both the interstitial CAconcentration and the biophysical properties of paren-chymal cell membranes influence R1i.

Equilibrium intercompartmental water exchangekinetics can be characterized (fast, intermediate, andslow) by comparing the rate of intercompartmental waterexchange with the relaxographic ‘‘shutter speed’’ (i.e.,the absolute difference in relaxation rates between thetissue compartments). In the fast exchange limit (FXL),ensembles of water protons relax with a single rate con-stant that is equal to the population-weighted average ofthe relaxation rates of the system. Such a physicalcondition is defined as kt >> R1b � R1ej j andkc >> R1e � R1ij j. The terms kt ¼ kbe þ keb andkc ¼ kie þ kei refer to the rate of proton (mainly water)exchange across the microvascular endothelium and theparenchymal cellular membranes, respectively, andR1b � R1ej j and R1e � R1ij j denote their correspondingshutter speeds. In the slow exchange limit (SXL), waterprotons’ magnetization relaxes with multiple time con-stants. For the microvascular endothelium and theparenchymal cellular membranes, this condition is set askt << R1b � R1ej j and kc << R1e � R1ij j, respectively. In-termediate to these limits is the fast exchange regime,where distinct multiexponential relaxation rates may ormay not be evident, depending on such variables as com-partment sizes, exchange rates, CA concentration, andMRI procedures. Of note, in DCE-MRI studies, thekinetics of water exchange is invariant; only the shutterspeeds vary with a passage of CA.

The set of differential equations of the longitudinalmagnetization of each pool in Eq. 1 was solved analyti-cally using standard methods. The resulting solutions toEq. 1 under an initial condition: M0 ¼ Mz (t=0) describea triexponential behavior of the magnetization of the pro-tons of water within the blood, interstitium, and intracel-lular space. The general form can be written as (20):

Mb tð Þ ¼ Mob þ c1 Y11 E1 þ c2 Y12 E2 þ c3 Y13 E3

Me tð Þ ¼ Moe þ c1 Y21 E1 þ c2 Y22 E2 þ c3 Y23 E3

Mi tð Þ ¼ Moi þ c1 Y31 E1 þ c2 Y32 E2 þ c3 Y33 E3

½10�

where Ej ¼ e�R1lt; j ¼ 1;2;3; l ¼ S; I;L, Y11–Y33 are eigen-vectors associated with the eigenvalues of A of Eq. 4,and the set of constants {cj} is chosen to satisfy the initialconditions (20). The coefficients {cj} and Y11–Y33 arefunctions of the CA concentration, relaxation rate con-stants, exchange rates, and the fractional size of eachpool. The coefficients {cj} in Eq. 10 can be determined byconsidering the initial values of the magnetization vectorM, which are dependent on initial conditions (20).

In a 3S2X model, the temporal evolution of the totallongitudinal magnetization consists of magnetization of

1434 Paudyal et al.

the water protons within the blood, interstitial, and in-tracellular compartments. The rearranged Eq. 10 leads to:

M tð Þ ¼ M0 þ c1Y1 E1 þ c2Y2 E2 þ c3Y3 E3 ½11�

where Y1–Y3 are the weighting factors associated withthe short, intermediate, and long relaxing components,given by:

Y1 ¼ Y11 þ Y21 þ Y31

Y2 ¼ Y12 þ Y22 þ Y32

Y3 ¼ Y13 þ Y23 þ Y33

½12�

This model assumes that an RF pulse equally perturbsthe magnetization of the water proton populations dis-tributed in the intravascular, interstitial, and intracellu-lar spaces of a 3S2X model. These water protons prior tothe RF inversion pulse are in thermal equilibrium. Afterthe initial inversion of magnetization, the longitudinalmagnetization is set as:

Mm t ! 0ð Þ ¼ �M0m; m ¼ b; e; i ½13�

A general expression for the temporal evolution of thelongitudinal magnetization following an inversion timeof Dt can be derived by simplifying Eqs. 10–13 and withtheir supplementary equations for each pool in a 3S2Xmodel. Then, Eq. 11 can be rewritten as:

M tð Þ ¼ M0 1� 2 p1 E1 þ p2 E2 þ p3 E3ð Þ½ � ½14�

where Ej ¼ e�R1lDt and p1, p2, and p3 denote the frac-tional populations that are associated with the longitudi-nal magnetization of short, intermediate, and long relax-ing components in a 3S2X model, in which p1þp2þp3 ¼1. These quantities depend on the intrinsic relaxationrates of the water protons within the blood, interstitialand intracellular spaces, intercompartmental exchangerates of water protons, fractional size of each of thesepools, and CA concentration.

Let M� and Mþ denote the longitudinal magnetizationjust before and after the nth RF pulse of a tip angle y inthe T one by Multiple Read Out Pulses (TOMROP)sequence (21), TOMROP being an imaging variant of theLL pulse sequence (22). The analytical solutions formu-lated in Eq. 10 can be used to derive the longitudinal mag-netization that evolves from a 3S2X model, particularizedto the LL experiment, with the following initial condition.

Mnþm t ! 0ð Þ ¼ Mn�

m 0ð Þ cosh ½15�

Using the preceding Eqs. 10–15, the solution with thecorrect initial boundary conditions and proper account-ing of intercompartmental water exchange yields the fol-lowing expression for the MRI signal in a TOMROPexperiment:

S tð Þ ¼ ½Mss þ p1 M 0ð Þ �Mss1ð ÞE�1 þ p2 M 0ð Þ �Mss2ð ÞE�

2

þp3 M 0ð Þ �Mss3ð ÞE�3� sin h e�R�

2TE ½16�

where E�j ¼ e�R�

1lt, t ¼ n � s, in which n and s are thenumber of sampling points and the excitation time inter-val between equally spaced sampling pulses of angle y,

respectively. M(0) is the magnetization between the firstinversion RF pulse and next RF inversion pulse (23).

The relationship between R1l*, R1l, y, and t isexpressed as

R�1l ¼ R1l � ln cos hð Þ

t½17�

The steady-state magnetization of a 3S2X model—thatis, the magnetization of tissue water protons as nbecomes large is given by:

Mssj ¼ M0

1� e�R1lt� �

1� cosh e�R1ltð Þ ½18�

where Mss1, Mss2, and Mss3 are the components of theslow, intermediate, and fast steady state magnetization,and the total steady-state magnetization, Mss, is a linear

sum of these components: Mss ¼P3

j¼1 pjMssj.

An analytical MR signal particularized for a TOMROPsequence of Eq. 16 observed in typical DCE studiesexhibits multiexponential behavior in a 3S2X system. Asnoted in the Introduction, in DCE studies, the relaxationrate constant of tissue water protons, R1, is generallyextracted from an acquired signal via the assumption ofa monoexponential recovery of magnetization (5, 23).Likewise, our use of TOMROP (5,24) has been centeredon the assumption that magnetization recovers in anessentially monoexponential manner. We will nowexamine through modeling how the end-result of amonoexponential TOMROP estimate of R1 varies withtissue concentration for the finite rate of water exchangein a 3S2X model.

The following is the monoexponential model for T1

relaxation in a TOMROP experiment (23, 25):

S tð Þ ¼ Mss þ M 0ð Þ �Mssð Þe�R�1t

� �sin hð Þe�R�

2TE ½19�

where

R�1 ¼ R1 � ln coshð Þ

t½20�

With the steady-state condition

Mss ¼ M0

1� e�R1t� �

1� coshe�R1tð Þ ½21�

We will examine via modeling in a 3S2X model how,and to what extent, R1 varies in relation to total tissueconcentration of CA after a bolus injection of CA.

The values for the relaxation rates constant, rates ofexchange, relaxivity, and other model parameters usedin this study are given in Table 1.

Patlak Theoretical Model of Blood-Tissue Contrast AgentExchange

When a CA, e.g., Gd-DTPA, is administered into venousblood, it initially distributes in the plasma water of thecirculating blood and increases R1 there. Using first-order kinetics, the concentration of the CA in interstitial

Modeling LL Estimates of R1 After CA Administration 1435

fluid after intravenous administration is a function oftime, CAe (t), and can be calculated using a model for-mulated by Kety and modified for MRI DCE studies (26).

CAe tð Þ ¼ Ktrans

ue

Z t

0

e�kb t�tð Þ CAp tð Þ dt ½22�

where the Ktrans is the transfer rate constant of CA fromthe plasma to the interstitium, kb is the rate constantfrom tissue back to plasma, which is assigned askb ¼ Ktrans

ueif transport is bidirectionally passive across the

microvascular wall or endothelium, and [CAp] is theplasma concentration of the CA.

The quantity most closely related to the observablechange in the MRI signal, the concentration of the CA inthe tissue, is then simply the sum of the interstitial andvascular plasma components (5):

CAt tð Þ ¼ Ktrans

Z t

0

e�kb t�tð Þ CAp tð Þ dtþ vD CApðtÞ ½23�

where [CAt] is the tissue concentration of the CA, vD isthe fractional volume of the rapidly reversible space. Ifthe tissue CA concentration can be inferred from achange in the MRI signal (e.g., a change in R1), then theparameters of the model can be estimated by a numberof methods (1–5) including the Patlak Graphical Methodand its variations (5,24).

If the increase in R1 (i.e., the difference between thepost-injection and pre-injection R1 or DR1) in a particulartissue is proportional to its CA concentration, Eq. 23 canbe used to form an observation equation:

ð1�HctÞDR1t tð Þ ¼ Ktrans

Z t

0

e�kb t�tð Þ DR1b tð Þ dt

þ vD DR1b tð Þ ½24�

where DR1b and DR1t are the change in longitudinalrelaxation rate of all the protons of water in arterialblood and tissue, respectively. Herein, the terms

ð1�HctÞDR1t tð ÞDR1b tð Þ and

Rt0

e�kb t�tð Þ DR1b tð Þ dt

DR1b tð Þ will be called the Patlak

ordinate and the efflux-corrected ‘stretch time’ (abscissa),

respectively. If DR1 is linearly proportional to CA concen-tration in both blood and tissue, the plot of the Patlak ordi-nate and ‘‘stretch time’’ will yield a straight line with aslope of Ktrans and an intercept of vD. We note that this isnot the original Patlak model, in which it is assumed thatno backflux of CA from the interstitium to the microvascu-lature occurs, but rather the extended Patlak model, identi-cal to the SM, in which backflux is accounted for.

MATERIALS AND METHODS

Simulation of Water Exchange Kinetics and CAConcentration in Vascular and Interstitium

We emphasize that it was our intention to simulate theresponse of a monoexponential estimate of tissue R1

under conditions of a real experiment conducted at 7Twith repeated sampling of tissue magnetization. Usingan analytical 3S2X model, simulations were performedto examine the effects of equilibrium intercompartmentalwater exchange kinetics on the relationship between R1

of the protons of tissue water and the tissue CA concen-tration in typical dynamic MRI experiments. In one sim-ulation, CA was restricted to the plasma space and [Gd-DTPA] varied from 0 to 20.0 mM. In the other condition,the interstitial CA concentration [CAe] was allowed tovary to about 5.0 mM as it leaked into the interstitialspace. Then R1 versus [CAt] curves were analyzed for arange of exchange rates of water protons across endothe-lial and cellular membranes. The estimated R1 values arereported as a function of voxel Gd concentration up to1.0 mM. These simulations were also performed usingan experimentally measured plasma concentration-timecurve or arterial input function (AIF) obtained after abolus injection of contrast agent in a rat 9L stroke model(27).

In these simulations, the relaxation rates of water pro-tons in blood (R1b) and extracellular fluid (R1e) wereallowed to vary over ranges typical of dynamic MRIexperiments. The R1b (t) and R1e (t) were also modeledusing the compartmental CA concentrations: [CAp(t)]and [CAe(t)] in a typical experimental condition. Theintrinsic relaxation rate constants of blood and extracel-lular fluid were set to 0.5 s�1, whereas the intracellularrelaxation rate R1i was assumed to be 0.56 s�1 (28). Therate of water proton exchange across the endothelium,kbe, for a normal brain (ub �2%) was set to 2.0 s�1, and

Table 1Simulation Parameters of a 3S2X Model

Parameters description Value Units

Range of relaxation rate for protons’ of water in blood (R1b) 0.5–40 s�1

Range of relaxation rate for protons’ of water in extracellular space (R1e) 0.5–20 s�1

Range of relaxation rate for protons’ of water in intracellular space (R1i) 0.56 s�1

Range of rate of exchange for protons’ of water from blood to extracellular space (kbe) 0.5–10 s�1

Range of rate of exchange for protons’ of water from intra- to extracellular space (kie) 0.5–10 s�1

Blood water protons’ content fraction (ub) 0.02–0.05

Intracellular water protons’ content fraction (ui) 0.8Range of blood to tissue CA transfer constant (Ktrans) 0.00125–0.01 min�1

Sampling interexcitation times (t) 50 msTip angle used in LL experiment (y) 18�

Longitudinal relaxivity of protons’ of water (R) 4.2 mM�1s�1

1436 Paudyal et al.

for brain with highly permeable microvessels (ub �5%)(19) to 5.0 s�1 (29), respectively. Then kbe and kie weremodeled from slow to fast water exchange [0.5–10.0 s�1]to assess their influence on a monoexponential estimateof R1. The rate of water protons’ exchange across paren-chymal cell membranes previously estimated in brain tis-sue was set to kie ¼ 1.81 s�1, and the intracellular watercontent fraction, to ui ¼ 0.8 (28). The CA relaxivity ofGd-DTPA (R) was set to 4.2 [mM�1s�1] (30) and wasassumed to be equal for the water protons of microvascu-lar blood and extracellular fluid. The model parametersvalues chosen for simulation were thus fairly representa-tive of a typical experiment at 7 Tesla.

We used Eq. 16 and associated expressions, withappropriate boundary conditions, to simulate the recov-ery of longitudinal magnetization in tissue after nonselective adiabatic inversion, with readouts performedvia a series of small tip-angle pulses and gradient-echoimaging, and including the effects of equilibrium inter-compartmental water exchange. We then fitted the simu-lated signal train generated with Eq. 16 to produce a sin-gle monoexponential recovery according to Eq. 19 (23).In this study, by varying the relaxation rates in the vas-cular and interstitial compartments, the response of thesystem to the administration of CA, and the relation ofR1 to changes in overall tissue concentration and R1 timecourses with Ktrans, ub, kbe, and kie were simulated. Themodel parameters (timing and tip angles) were those typ-ical of our post-inversion 24-echo experimental tech-nique for estimating vascular permeability in the rat 9Lcerebral tumor model at 7T (5). The tip angles (y) andinter- excitation times (t) were set to 18� and 50 ms,respectively for the LL pulse sequence. The transverseattenuation component was taken to be constant andsmall across the time of the study, e�R�

2 TE � 1. The timeinterval, Dt, after inversion and just before the first RFpulse was set to 12 ms. The other model parameters for

cerebral tissue of rat brain utilized in the simulations arelisted in Table 1.

Simulations were performed using programs written inANSI C implemented in a UNIX system (Solaris 8.0–SunMicrosystems, Santa Clara CA). Pseudo-random Gaussiannoise was generated via the Box-Muller algorithm (31).Noise was added to the simulated TOMROP signals toachieve desired signal to noise ratio, but signal to noiseratio in the model experiment was kept very high—about300:1—because our main focus was on the relationbetween R1 and CA tissue concentration. Monoexponen-tial fits were performed on the simulated inversion re-covery curves to construct the R1 of tissue water protonsusing standard methods (21,22). The central element ofthis investigation was characterizing the variation of amonoexponential fit of R1 over a range of tissue concen-trations of Gd- labeled compound in a 3S2X model.

RESULTS

The curves for the three longitudinal relaxation rate con-stants R1S, I, L, (see Appendix) associated with in Eq. 14were generated using typical values of the intravascularand intracellular water fractions and their rate of transferconstants (ub ¼ 0.02 and kbe ¼ 2.0 s�1, ui ¼ 0.8 and kie¼ 1.81 s�1) and the other model parameters given in Ta-ble 1. In Fig. 2a, the variation of the three components ofthe relaxation rate constants (e.g., R1S, I, L) as a functionof interstitial CA concentration ([CAe]) (mM) is illus-trated. Herein, we assumed CA has equilibrated betweenintravascular and extracellular compartments and notethat R1S, I, L are dependent on the spin-lattice relaxationrate constants, rate of exchange constants, water protondistribution spaces, and CA concentration. The shorterrelaxation rate constant R1S gradually increases initiallyand saturates with increasing [CAe] while the other tworates R1I and R1L keep increasing as [CAe] increases.

FIG. 2. a: Graph of simulated results of the analytical longitudinal relaxation rate constant R1 (e.g., R1S, I, L) as a function of the intersti-tial CA concentration [CAe] in a 3S2X model. Equation A1 given in the Appendix was used to construct these curves. b: Simulated sig-

nal intensity time course curves at 0.95 mM Gd-DTPA in interstitial CA concentration for the LL imaging sequence and with tissueparameters ub ¼ 0.02 and kbe ¼ 2.0 s�1. The lower curve, labeled SSM, demonstrates the recovery of the longest component of longi-

tudinal magnetization, sometimes used in previous studies (10,11) as the only component. The contribution of the components R1I andR1L on a signal that evolved 3S2X model, which were ignored in the SSM, can be seen. Model parameters values are listed in Table 1.

Modeling LL Estimates of R1 After CA Administration 1437

Because the intrinsic relaxation rate constants of the intra-and extracellular space are similar, the mean rate ofexchange across the cellular membranes, kc �10.0 s�1 (ui¼ 0.8 and kie ¼ 1.81 s�1) (28) is much greater than the in-tracellular shutter speed R1e � R1ij j without CA. As CAextravasates, R1e � R1ij j increases, and this can vary theexchange regimes, FXL, fast exchange regime and SXL,from the extreme left end to the right along the abscissa[CAe], because the sum of the rate of exchange kc remainsinvariant. The cellular shutter speed exceeds kc at about2.40 mM of interstitial CA concentration and thus becomesstrongly effective in upper half of [CAe] in Fig. 2a.

Figure 2b shows the time course of a simulated signalconstructed for FXL (Eq. 19), full 3S2X model (Eq. 16)and a phenomenological curve, labeled SSM, which con-siders only the R1S component of Eq. 16, a componentthat evolves from the 3S2X model but does not, as thegraph shows, accurately reflect the true signal evolutionin a typical LL experiment conducted in our laboratoryat 7T. These curves were constructed using the LL imag-ing sequence employed here: number of sampling points(n) ¼ 24, interexcitation time (t) ¼ 50 ms, tip angle (y) ¼18� at 0.95 mM of interstitial CA concentration withmodel parameters as in Fig. 2a. The contribution of theR1I and R1L components can be seen on the full exchangemodel in comparison to the phenomenological model,SSM. While Fig. 2b was generated from a 3S2X model; itbears a strong resemblance in signal behavior to that of

Buckley’s Fig. 6 (15), although that figure was generatedusing a 2SX model.

Figure 3 displays the response of R1 to changes in tis-sue concentration under a range of conditions typical ofthose experienced in DCE studies. The temporal evolu-tion of a LL signal was calculated using all the kernels ofthe relaxation rates (short, intermediate, and long) andtheir associated fractional populations in a 3S2X model.As before, the model was particularized to our LL experi-ment (n ¼ 24, t ¼ 50 ms, and y ¼ 18�) with model pa-rameters listed in Table 1.

Figure 3a shows, for an entirely intravascular CA, therelationship between LL estimates of the R1 of tissuewater protons versus [CAt]. In this setting, R1 curveswere constructed using a large volume fraction for bloodwater (ub ¼ 0.05), kbe ¼ 5.0 s�1, and cellular parameters:ui ¼ 0.8 and kie ¼ 1.81 s�1 with other model parametersas given in Table 1. The CA concentration in plasma[CAp] was varied up to 20.0 mM. Initially, R1 increases anearly linear manner with [CAt] but curves concave athigher concentrations. Across the range plotted, themodel data deviate about 15% (CAt ¼ 0 [mM]), 6.0%(CAt ¼ 0.5 [mM]), and 10.0% (CAt ¼ 1.0 [mM]) from alinear relationship. As the R1 curve passes from its mini-mum to maximum, the exchange regimes across the ab-scissa [CAt] move from FXL to SXL. The relaxographictransvascular shutter speed R1b � R1ej jrapidly increasesand exceeds the sum of the equilibrium rates of blood-

FIG. 3. Simulated relaxation rate of tissue water protons (R1) versus tissue concentration of the contrast agent ([CAt]) in a 3S2X model.

The curves were constructed with model parameters as follows: ub ¼ 0.05, kbe ¼ 5.0 s�1 and ui ¼ 0.8, kie ¼ 1.81 s�1. a: Simulated withthe CA distributed only in the plasma compartment. The asterisk corresponds to the peak tissue CA concentration in the experimental

condition. b: Simulated with the CA distributed in the blood and interstitial spaces. c: Families of simulated R1 curves for different valuesof kbe (10, 2, and 0.5 s�1; top to bottom). d: Families of simulated R1 curves for different values of kie (10, 2, and 0.5 s�1; top to bottom).The other parameters used in the modeling are given in Table 1.

1438 Paudyal et al.

tissue exchange, kt, �6.67 s�1 around 3.20 mM of [CAp]and becomes effective in the upper half of [CAt]. It mustbe mentioned here that the range of this plot runs far

beyond the usual experimental conditions. The asteriskon the plot points to the maximum value of CA concen-tration in plasma water that might be expected in mostDCE studies; it should be concluded that the relationshipbetween [CAt] and R1 is linear under most experimentalconditions.

Figure 3b shows the LL estimates of R1 of tissue waterprotons versus [CAt] acquired with model data of Fig. 3awhen interstitial concentration was varied up to 5 mM.The model is constructed as if equilibration of the CAhad take place between the blood and interstitial space.With increasing interstitial CA concentration, cellularexchange regimes vary from FXL to SXL. The differencein the longitudinal relaxation rates, R1e � R1ij j, exceedskc at 2.75 mM of [CAe] and becomes effective in theupper half of the [CAt] range. The best fitted line tomodel data of R1 versus [CAt] yielded deviation about15%, 7%, and 13%, respectively, at the lower (R1e ¼ 0.5s�1), middle (R1e ¼10.0 s�1), and upper (R1e ¼ 20.0 s�1)points. Accordingly, the LL experiment predicted anearly linear response between the R1 of tissue waterprotons and [CAt] in a 3S2X model.

In Fig. 3c and d, the effects of varying the rates ofexchange across the microvascular endothelium, kbe,(3c), and the membranes of parenchymal cells, kie (3d),

FIG. 4. The time-course of plasma CA concentration ([CAp]; opencircles) and total tissue CA concentration ([CAt]; filled circles). The

latter, [CAt], was calculated using Eq. 23 with Ktrans ¼ 2 � 10�3

min�1 and ub ¼ 0.02. The other model parameters are listed inthe Table 1. This [CAp] concentration-time curve was used to gen-

erate Figs. 5 and 6.

FIG. 5. a: Concentration-time plot of the change in the relaxation rate of tissue water protons (DR1; open circles) and tissue CA concen-tration (filled circles) for the case of all of the CA in blood. ub ¼ 0.02. b: Concentration-time plot of DR1 (open circles) and tissue CA

concentration (filled circles) for an entirely extracellular CA; this was modeled with Eq. 22, Ktrans ¼ 4.0 � 10�3 min�1 and ub ¼ 0.02. c:Concentration-time plot of DR1 (open circles) and tissue CA concentration (filled circles) for the combined plots of 3A and 3B. The othermodel parameters values are listed in the Table 1. In 5C, the effect of the different apparent relaxivities of CA between blood and inter-

stitium should be noted. While the component DR1’s track very closely with tissue concentration in their respective compartments, thecombination does not.

Modeling LL Estimates of R1 After CA Administration 1439

on R1 are illustrated as a function of total tissue concen-tration. The rates of water exchange for kbe and kie (e.g.,10, 2.0, and 0.5 s�1: top to bottom) span the range formost tissue types. Fig. 3c (ub ¼ 0.05, ui ¼ 0.8, and kie ¼1.81 s�1) and D (ub ¼ 0.05, kbe¼ 5.0 s�1, and ui ¼ 0.8)were constructed with the model parameters used in Fig.3a and b. The R1 curves under varying rates of watertransfer become distinct in the upper region compared tothe lower region of [CAt]. Across this rather wide rangeof tissue concentrations, the variation of kbe, again acrossa wide range of values (kbe ¼ 2.0 s�1 (19) and kie ¼ 1.81s�1 (28) for normal brains) changes the slopes of line by13%, and that of kie by 16%, respectively. This degree ofvariation would probably not be detectable under mostexperimental conditions.

Continuing our focus on the relation between tissue con-centration and R1, we constructed a model of CA concen-tration in tissue in a moderately leaky vascular bed. Figure4 shows the CA plasma concentration time-course of anexperimentally measured [CAp(t)] (left ordinates) and thetotal CA tissue concentration [CAt(t)] (right ordinates)curves. The AIF is that of an intravenous (i.v.) bolus injec-tion standard dose of 0.08 mmol/kg (body weight) of Gd-DTPA (27). The [CAt] time course was generated using the

AIF and a CA transfer rate constant Ktrans ¼ 0.002 min�1

and Eq. 23 and associated expressions for ub ¼ 0.02. Thesemodel concentrations were then used to investigate therelationship between R1 and tissue [CAt] with LL measure-ments of R1 as estimates of the latter.

Given the concentration-time behavior of Figure 4, whatcan be expected of changes in R1? The R1b(t) and R1e(t)(e.g., Eqs 8 and 9) time-course curves were generated usingan AIF and Eq. 22, respectively, with R ¼ 4.2 mM–1s–1.The arterial hematocrit was set to 0.5. Under this condi-tion, the MR signal of the LL experiment (e.g., ¼ 24, t ¼50 ms, and y ¼ 18�) was generated from the componentsof the longitudinal relaxation rate constants R1S, I, L, andthe fractional populations in a 3S2X model. Then R1 wasestimated as described in the preceding paragraphs accord-ing to Eq. 19 from a signal obtained from Eq. 16. When thedata is normalized so that the peak concentrations areplotted at the same ordinate distance, there clearly is a vis-ual difference between tissue concentration and estimatesof tissue concentration by DR1.

In Figure 5a, the normalized values of DR1 (opencircles) predicted by the LL experiment conducted at 7Tand the voxel CA concentration (filled circles) as a func-tion of time are plotted for the case of an entirely

FIG. 6. Families of the plots obtained with the multi-time, graphical method of Patlak in a 3S2X model with the ordinate equal to

1�Hctð ÞDR1tiss

DR1b tð Þ and the abscissa equal to the efflux-corrected term

R t

0e�kb t�tð ÞDR1b tð Þdt

DR1b tð Þ , which is often referred to as ‘‘stretch time’’ when the CA

is injected as a bolus. a: With Ktrans ¼ 0.01, 0.005, and 0.00125 min�1; top to bottom) and ub ¼ 0.02. b: With blood water fractions, ub, setas 0.06, 0.04, and 0.01 (top to bottom) and Ktrans ¼ 4 � 10�3 min�1. c: With the transendothelial rate of influx, kbe, varied from 10 to 2 to 0.5s�1 (top to bottom), Ktrans ¼ 5 � 10�3 min�1, and ub ¼ 0.05. d: Rate of exchange across parenchymal cell membrane set at either 10, 2, or

0.5 s�1 (top to bottom), Ktrans ¼ 3 � 10�3 min�1, and ub ¼ 0.03. The other parameters of this modeling are listed in Table 1.

1440 Paudyal et al.

intravascular CA in a 3S2X model. The two traces werenormalized by setting the 20 min time point on the ordi-nate at the same graphical distance from zero for both.N.B.: the time scales of the two curves are offset, as indi-cated on the top and bottom abscissa labeling. It should beclear that the peak amplitude of the MRI estimate of tissueconcentration when there is no CA leakage is smaller thanthe true peak, thus demonstrating the effect of restrictedwater exchange at the highest concentrations of CA.

In Fig. 5b, the left and right ordinates demonstrate thechange in the relaxation rate DR1 of the protons of tissuewater (open circles) and tissue concentration [CAt] (filledcircles), constructed from Eq. 22, with Ktrans ¼ 4.0 �10�3 min�1 in a 2SX model. Here, it can be seen that thetwo estimates of concentration track each other verywell. Certainly, any differences would be lost if MRI ex-perimental signal to noise ratio were in the range (e.g.,25:1) of what is usually achievable. Thus, despite Figure4, there appears to be a fairly good linearity in signalresponse when the blood and tissue components are con-sidered separately.

Figure 5c plots DR1 (open circles, left ordinates) and[CAt] (filled circles, right ordinates), demonstrates the timebehavior in a model of CA with leaky microvessels andincludes both the blood and extracellular compartmentsgathered from Eq. 23, when Ktrans ¼ 4.0 � 10�3 min–1.Other model parameters were: ub ¼ 0.02, kbe ¼ 2.0 s�1, ui

¼ 0.8, and kie ¼ 1.81 s�1. In general, R1 as a function oftime, when it is considered that CA resides only in theblood compartment or is equilibrated between the latterand the interstitial compartments, is well correlated withtissue concentration except at very high blood concentra-tions. Of note, the apparent relaxivity (the proportionalityconstant), however, differs among the compartments.Additionally, the apparent relaxivity for a given compart-ment will vary with water exchange rates and compart-ment size. This point has been repeatedly underscored inthe development of the SSM (7,9,11–14), but the underly-ing linearity of the R1 response has not been as well repre-sented using the appropriate boundary conditions for MRpulse sequences.

What then is the consequence of both the underlyinglinearity in concentration and the differences in apparentrelaxivities? One of the better ways to examine this ques-tion is to use a method that is inherently linear, since itallows the examination of the model-generated responsefor nonlinearities. We chose the extended Patlak graphi-cal method (2,5), which linearizes the concentration-timecurve when there is appreciable brain-to-blood backfluxof the CA (nonzero kb). Figure 6 shows a number of plotsof concentration-time curves in tissue using the extendedPatlak Graphical method [model 3 of Ewing et al (5)]constructed from Eq. 24. Using a 3S2X model, the influ-ences of the vascular parameters Ktrans, ub, and the ratesof water proton exchange kbe and kie on the Patlak plotsare illustrated. As in the extended Patlak method, the

quantity 1�Hctð ÞDR1tiss

DR1b tð Þ is the Patlak ordinate and efflux-cor-

rected arterial time integral

R t

0e�kb t�tð ÞDR1b tð Þdt

DR1b tð Þ forms the ab-

scissa. The quantity Ktrans is estimated from the slope ofthe plot, while fractional vascular volume is estimatedfrom its intercept at x ¼ 0.In Fig. 6a, the extended Patlak

plot is produced for a range of blood-to-tissue transferrate constants Ktrans (e.g., 0.01, 0.005, and 0.00125 min�1:top to bottom), to examine the systematic errors in esti-mates of Ktrans in a 3S2X model. In this setting, ub ¼0.02 and kbe ¼ 2.0 s�1, ui ¼ 0.8, and kie ¼ 1.81 s�1, andother model parameters given in Table 1 were used. Therange of Ktrans spans the range for Gd-DTPA in most tis-sue types. For this range of Ktrans, the slopes vary by87%, high to low, whereas the estimates of Ktrans varyabout 4% from the true values; systematic errors in Ktrans

are, thus, negligible. Contrariwise, the volume of intra-vascular water (i.e., the blood water space) is underesti-mated by as much as 56% compared to the true value.

Figure 6b shows how the Patlak plot behaves under avariation in the blood water fractions ub (e.g., 0.06, 0.04,and 0.01: top to bottom) in a 3S2X model. In this setting,the influx rate of protons linked to water (kbe ¼ 2.0 s�1),intracellular parameters (ui ¼ 0.8 and kie ¼ 1.81 s�1),and model parameter listed in Table 1 were used. Thisrange of ub is valid for many tissue types. To illustratethe effects of leakage, Ktrans was set to 4 � 10�3 min�1.The curves are distinct, differing in ub and with virtuallyidentical slopes that vary by less than 3%. The volumeof intravascular water is again underestimated by asmuch as 60% compared to the true value.

Figure 6c shows the influence of different rates of thetransendothelial influx of water protons on the extendedPatlak plot. The values of kbe (e.g., 10, 2, and 0.5 s�1: topto bottom) span values that model a highly permeablevasculature to that of a moderately restricted one. In thissetting, Ktrans ¼ 5.0 � 10�3 min�1, ub ¼ 0.05, intracellu-lar parameters (ui ¼ 0.8 and kie ¼ 1.81 s�1), and themodel parameter listed in Table 1 were used. It can beseen that the curves are indistinguishable and slopes ofthe curves are nearly identical, varying by less than 1%.The estimated Ktrans differs by about 2% from the truevalue while ub varies by 58%. This result suggests thatin the LL experiment the effect of intermediate transen-dothelial water exchange will have a negligible effect onthe estimate of Ktrans and possibly a strong effect on vb,depending upon the tissue type.

Figure 6d illustrates how the rates of water protonexchange across parenchymal cell membranes, with kievarying from highly permeable to slightly restricted, inthe 3S2X model influence Patlak plots of the data. Inthis setting, ub ¼ 0.03 and kbe ¼ 2.0 s�1, intracellular pa-rameters (ui ¼ 0.8 and kie ¼ 1.81 s�1), and the model pa-rameter listed in Table 1 and Ktrans ¼ 3.0 � 10�3 min�1

were used. Under these conditions, the slopes of thecurves and the estimates of Ktrans varied less than 1%.This result demonstrates the very small effect of a varia-tion in water exchange cellular membranes has on thePatlak plot estimation of Ktrans.

DISCUSSION

An analytical equation describing the evolution of mag-netization for tissue water protons was formulated forthe LL experiment via the Bloch�McConnell formalismusing a 3S2X model, incorporating all three componentsof the longitudinal relaxation rates (e.g., short, intermedi-ate, and long) and their associated fractional populations.

Modeling LL Estimates of R1 After CA Administration 1441

A wide range of plasma and extracellular fluid concentra-tions of CA was considered, and signal associated with aLL experiment (n ¼ 24 points, t ¼ 50 ms, y ¼ 18�) wasgenerated. We were most interested in the relationshipbetween the longitudinal relaxation rate of tissue waterprotons and tissue CA concentration, especially how amonoexponential summary of longitudinal signal recovery,R1, would be affected by the finite rate of equilibriumintercompartmental water exchange under the conditionsof a typical DCE study. Thus, we have not considered theblood inflow effects in this study (16).

The results of Fig. 3 illustrate a very important point:under a wide range of typical experimental conditions,a monoexponential estimate of R1 scales nearly linearlywith [CAt]. It should be understood, however, that R1 asa measure of the CA concentration in the microvascula-ture is less linear (Fig. 3a) than as a measure of the in-terstitial CA concentration (Fig. 3b), although notstrongly so. This is because a standard dose of 0.08mmol/kg Gd-DTPA will result in a peak plasma concen-tration of about 3.7 mM (Fig. 4: top), and the corre-sponding peak R1 can reach about 16.0 s�1. However, atthe peak value of 0.77 mM of [CAe], the correspondingR1 is about 4.0 [s�1] when Ktrans ¼ 0.02 min�1. At veryhigh concentrations of CA, the slope of the relationshipbetween R1 and [CAt], i.e., the CA’s longitudinal relax-ivity, varies with the rates of exchange across the endo-thelium and cellular membranes over the range ofhighly to moderately restricted microvascular perme-ability (e.g., top to bottom) in Fig. 3c and d, respec-tively. Qualifying this, the deviations are relativelysmall even at very high concentrations and might easilybe missed under typical experimental conditions withlower signal-to-noise.

The plots of data using the extended Patlak Graphicalmethod are most revealing of the systematic behavior ofthe recovery of the longitudinal magnetization of tissuewater protons when CA is limited to one or two com-partments. Under typical experimental conditions, thecellular exchange regime moves from FXL to fastexchange regime to SXL. If the boundary conditions areproperly modeled and all three components of recoveryare taken into account, it is clear that water exchangeacross parenchymal cell membranes does not stronglyinfluence the estimation of the influx transfer constant ofthe CA. However, if water exchange across the microvas-cular wall is moderately restricted as in normal brain,the blood water volume will be underestimated.

To investigate the effects of equilibrium intercompart-mental water exchange kinetics on R1, this study mainlyfocused on the LL experiment. Water exchange times atthe boundaries of the blood and intracellular compart-ments are about 500 ms and 550 ms, respectively,whereas the LL inter-echo sampling time scale, t, ismuch smaller. However, the total acquisition time (e.g.,1200 ms) is much longer than the residence time ofwater molecules in either blood or intercellular fluid. Itappears that, if the selected experimental time is longenough in comparison to the pre-exchange lifetime, amultipoint technique with an extended and repeatedsampling may average the intercompartmental exchangeof water protons under many conditions.

In all normal cerebral tissue as well as brain tumors,contrast materials are confined to the blood and extracel-lular space, whereas about 75% of the tissue water is in-tracellular. In the cerebral tumors, the change in the tu-mor volume affects on the water diffusion (32). Often thetissue studied by MRI and used for comparison in thisstudy is a cerebral tumor, which has a highly permeablemicrovascular wall in contrast to normal brain. Itappears highly probable that the exchange of wateracross tumor microvessels is more rapid than that acrossthe surrounding normal vasculature. If this is the case,one can expect a general underestimation of blood watervolume in normal brain parenchyma, and possibly in the‘‘normalized" vasculature of the tumor but a correct esti-mate of blood volume in the part of the tumor withhighly permeable microvessels.

This study is not without limitations because severalassumptions were made in the simulation. Probably themost restrictive set of assumptions involve the particulariza-tion to a 24-echo LL inversion-recovery experiment with 50ms echo spacing and 18� tip-angle. However, we have inves-tigated other experimental conditions, namely a 100 msecho spacing, and/or a doubling of number of points. Gener-ally, these variations in experimental condition do notchange the results reported here by more than 1% in esti-mates of the transfer constant where as intercept vary within5%, when Ktrans ¼ 5.0� 10�3 min�1 and ub ¼ 0.03. Anotherrestriction is the specification of 7 Tesla with its longer tis-sue T1’s, and lower CA relaxivities. Blood inflow effectswere not considered, the pre-contrast blood and extracellu-lar R1’s were set equal, and the CA relaxivities in the twocompartments were set equal and assumed constant,although CA relaxivity may vary with macromolecular con-tent (30) by as much as 30%. However, this latter variationsimply changes the apparent relaxivity of the CA, and sincethe modeling covers a wide range of transcellular waterexchange rates, which also changes the apparent relaxivityof the CA, variations in CA relaxivity with macromolecularcontent are probably covered by the modeling. We assumedthat T2* effects were negligible at these echo times. Notwith-standing these limitations, a clear picture evolves from thismodeling: the systematic errors of the SM are likely toappear first in the underestimation of blood water volume.Only in tissue with extremely leaky microvessels and conse-quent high interstitial concentrations of CA will a seriousunderestimation of Ktrans begin to appear. Since the micro-vessels in this case will need to be extremely leaky, the prac-tical consequences of an underestimation of Ktrans may notbe important.

This kind of modeling with a full 3S2X treatment andcorrect boundary conditions needs to be extended toother, more clinically relevant field strengths and to themore widely employed short-TR gradient-echo sequencescommonly used in clinical DCE determinations. An im-portant first step toward this goal has been made andrecently published by those investigators who have pio-neered the SSM (33).

CONCLUSION

We conclude that the analytic results of this study in a3S2X model establish a nearly linear relationship

1442 Paudyal et al.

between tissue CA concentration for a monoexponentialestimate of R1 measured by a LL technique. Therefore,under many conditions, LL estimates of Ktrans will yieldapproximately unbiased estimates (within 4%) of thatparameter in most tissues. However, it is likely that thevolume of blood water in the field of observation can besignificantly underestimated (as much as 60%), espe-cially in normal brain tissue.

APPENDIX

The following are the short, intermediate, and long lon-gitudinal relaxation rate constants of a 3S2X modeldefined in the text:

R1S ¼ � a

3� 2

ffiffiffiffiQ

pcos

d

3

� �

R1I ¼ � a

3� 2

ffiffiffiffiffiffiQ

pcos

d� 2p

3

� �

R1L ¼ � a

3� 2

ffiffiffiffiQ

pcos

dþ 2p

3

� � ½A1�

The relaxation rates are computed through the follow-ing relation:

d ¼ arc cosRffiffiffiffiffiffiQ3

p !

½A2�

The following condition: R2 < Q3, is for the three dif-ferent real roots of eigenvalues.where

Q ¼ a2 � 3b

9

R ¼ 2a3 � 9abþ 27c

54

½A3�

a ¼ � pþ qþ rð Þb ¼ pqþ q r þ r p� kie kei � kbe kebð Þc ¼ pkiekei þ rkbekeb � pq rð Þ

½A4�

with

p ¼ R1b þ kbe

q ¼ R1e þ keb þ kei

r ¼ R1i þ kie

½A5�

REFERENCES

1. Crone C. The permeability of capillares in various organs as deter-

mined by use of the ‘indicator diffusion’ method. Acta physiol Scand

1963;58:292–305.

2. Patlak CS, Blasberg RG. Graphical evaluation of blood-to-brain trans-

fer constants from multiple-time uptake data. Generalizations. J Cere-

bral Blood Flow Metab 1985;5(4):584–590.

3. Patlak CS, Blasberg RG, Fenstermacher JD. Graphical evaluation of

blood-to-brain transfer constants from multiple-time uptake data. J

Cerebral Blood Flow Metab 1983;3:1–7.

4. Tofts P, Kermode A. Measurement of the blood–brain barrier perme-

ability and leakage space using dynamic MR imaging. 1. Fundamen-

tal concepts. Magn Reson Med 1991;17:357–367.

5. Ewing JR, Brown SL, Lu M, Panda S, Ding G, Knight RA, Cao Y,

Jiang Q, Nagaraja TN, Churchman JL, Fenstermacher JD. Model selec-

tion in magnetic resonance imaging measurements of vascular per-

meability: Gadomer in a 9L model of rat cerebral tumor. J Cerebral

Blood Flow Metab 2006;26:310–320.

6. Donahue KM, Burstein D, Manning WJ, Gray ML. Studies of Gd-

DTPA relaxivity and proton exchange rates in tissue. Magn Reson

Med 1994;32:66–76.

7. Landis C, Li X, Telang F, Molina P, Palyka I, Vetek G, Springer C, Jr.

Equilibrium transcytolemmal water-exchange kinetics in skeletal

muscle in vivo. Magn Reson Med 1999;42:467–478.

8. Labadie C, Lee JH, Vetek G, Springer CS, Jr. Relaxographic imaging. J

Magn Reson B 1994;105:99–112.

9. Landis CS, Li X, Telang FW, Coderre JA, Micca PL, Rooney WD,

Latour LL, Vetek G, Palyka I, Springer CS, Jr. Determination of the

MRI contrast agent concentration time course in vivo following bolus

injection: effect of equilibrium transcytolemmal water exchange.

Magn Reson Med 2000;44:563–574.

10. Yankeelov TE, Rooney WD, Huang W, Dyke JP, Li X, Tudorica A,

Lee JH, Koutcher JA, Springer CS, Jr. Evidence for shutter-speed vari-

ation in CR bolus-tracking studies of human pathology. NMR in Bio-

medicine 2005;18:173–185.

11. Yankeelov TE, Rooney WD, Li X, Springer CS, Jr. Variation of the

relaxographic ‘‘shutter-speed’’ for transcytolemmal water exchange

affects the CR bolus-tracking curve shape. Magn Reson Med 2003;50:

1151–1169.

12. Li X, Rooney WD, Springer CS, Jr. A unified magnetic resonance

imaging pharmacokinetic theory: intravascular and extracellular con-

trast reagents. Magn Reson Med 2006;54:1351–1359.

13. Li X, Huang W, Morris EA, Tudorica LA, Venkatraman ES, Rooney

WD, Tagge I, Wang Y, Xu J, Springer CS, Jr. Dynamic NMR effects in

breast cancer dynamic-contrast-enhanced MRI. Proc Natl Acad Sci

USA 2008;105:17937–17942.

14. Huang W, Li X, Morris EA, Tudorica LA, Venkatraman ES,

Rooney WD, Tagge I, Wang Y, Xu J, Springer CS, Jr. The MR shut-

ter-speed discriminates vascular properties of malignant and be-

nign breast tumors. Proc Natl Acad Sci USA 2008;105:

17943–17948.

15. Buckely DL, Kershaw LE, Stanisz GJ. Cellular-interstitial water

exchange and its effect on the determination of contrast agent con-

centration in vivo: Dynamic contrast-enhanced MRI of human inter-

nal obturator muscle. Magn Reson Med 2008;60:1011–1019.

16. Barbier EL, St Lawrence KS, Grillon E, Koretsky AP, Decorps M. A

model of blood-brain barrier permeability to water: accounting for

blood inflow and longitudinal relaxation effects. Magn Reson Med

2002;47:1100–1119.

17. McConnell HM. Relaxation rates by nuclear magnetic resonance. J

Chem Phy 1958;28:430–431.

18. Spencer RGS, Fishbein KW. Measurement of spin–lattice relaxation

times and concentrations in systems with chemical exchange using

the one-pulse sequence: breakdown of the Ernst model for partial sat-

uration in nuclear magnetic resonance spectroscopy. J Magn Reson

2000;142:120–135

19. Cao Y, Brown SL, Knight RA, Fenstermacher JD, Ewing JR. Effect of

intravascular-to-extravascular water exchange on the determination

of blood-to-tissue transfer constant by magnetic resonance imaging.

Magn Reson Med 2005;53:282–293.

20. Herbest M, Goldstein JH. Cell water transport measurement by NMR.

A three-compartment model which includes cell aggregation. J Magn

Reson 1984;60:299–306.

21. Brix G, Schad LR, Deimling M, Lorenz WJ. Fast and precise T1 imag-

ing using a TOMROP sequence. Magn Reson Imaging 1990;8:

351–356.

22. Look DC, Locker DR. Time saving in measurement of NMR and EPR

relaxation times. Rev Sci Instrum 1970;41:250–251.

23. Gelman N, Ewing JR, Gorell JM, Spickler EM, Solomon EG. Interre-

gional variation of longitudinal relaxation rates in human brain at

3.0 T: relation to estimated iron and water contents. Magn Reson

Med 2001;45:71–79.

24. Ewing J, Knight R, Nagaraja T, Yee J, Nagesh V, Whitton P, Li L, Fen-

stermacher J. Patlak plots of Gd-DTPA MRI data yield blood-brain

transfer constants concordant with those of 14C-sucrose in areas of

blood-brain opening. Magn Reson Med 2003;50:283–292.

25. Kaptein R, Dijkstra K, Tarr CE. A single-scan fourier transform

method for measuring spin-lattice relaxation times. . J Magn Reson

1976;24:295–300.

26. Kety SS. The theory and applications of the exchange of inert gas at

the lungs and tissues. Pharmacol Rev 1951;3:1–41.

Modeling LL Estimates of R1 After CA Administration 1443

27. Nagaraja TN, Karki K, Ewing JR, Divine G, Fenstermacher JD, Patlak

CS, Knight RA. The MRI-measured arterial input function resulting

from a bolus injection of Gd-DTPA in a rat model of stroke slightly

underestimates that of Gd-[14C]DTPA and marginally overestimates

the blood-to-brain influx rate constant determined by Patlak plots.

Magn Reson Med 2010;63:1502–1509.

28. Quirk JD, Bretthorst GL, Duong TQ, Snyder AZ, Springer CS, Jr.,

Ackerman JJ, Neil JJ. Equilibrium water exchange between the intra-

and extracellular spaces of mammalian brain. Magn Reson Med

2003;50:493–499.

29. Carreira GC, Gemeinhardt O, Beyersdorff D, Jorg Schnorr J, Taupitz

M, Ludemann L. Effects of water exchange on MRI-based determina-

tion of relative blood volume using an inversion-prepared gradient

echo sequence and a blood pool contrast medium. Magn Reson Imag-

ing 2009;27:360–369.

30. Bagher-Ebadian H, Paudyal R, Nagaraja TN, Croxen RL, Fenster-

macher JD, Ewing JR. MRI estimation of gadolinium and albumin

effects on water proton. Neuroimage 2011;54:s176–s179.

31. Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical

recipe in C. Cambridge University Press; New York, N.Y., 1987.

32. Kwee TC, Galban CJ, Tsien C, Junck L, Sundgren PC, Ivancevic MK,

Johnson TD, Meyer CR, Rehemtulla A, Ross BD, Chenevert TL. Intra-

voxel water diffusion heterogeneity imaging of human high-grade

gliomas. NMR Biomed 2010;23:179–187.

33. Li X, Rooney WD, Varallyay CG, Gahramanov S, Muldoon LL, Good-

man JA, Tagge IJ, Selzera AH, Pike MM, Neuwelt EA, Springer CS,

Jr. Dynamic-contrast-enhanced-MRI with extravasating contrast rea-

gent: Rat cerebral glioma blood volume determination. J Magn Reson

2010;206:190–199.

1444 Paudyal et al.