Improved accuracy of human cerebral blood perfusion measurements using arterial spin labeling: Accounting for capillary water permeability

FAIR

The FAIR (6) technique uses a 180° nonselective RF pulse to invert the longitudinal magnetization of the protons in the head. Inverted or “labeled” protons in the blood outside the image slice flow into the slice, changing the longitudinal magnetization within the slice. After an inversion time (TI) the longitudinal magnetization of this “labeled” image is measured using EPI. For the control image a selective RF pulse is applied, inverting only the protons in the image slice. This time the blood flowing into the slice is unlabeled. The difference in signal between the control and labeled images will depend on the rate at which blood enters the slice, and hence perfusion. Figure 1a shows the tissue magnetization of the difference image ΔM (i.e., control-labeled) with inversion time along with the longitudinal arterial magnetization in the difference image Δm_a which decays with the T₁ of blood, T.

CASL

CASL (2) uses a spatially localized RF field to continuously invert the longitudinal magnetization of the protons in arterial blood water entering the brain. A continuous RF field is applied for labeling time t_L, along with a field gradient, such that the RF field is on resonance in a plane through the carotid artery, called the inversion plane. Flowing protons move through the field gradient and their resonant frequency increases. As they pass through resonance the magnetization of the spins is inverted, effectively “labeling” the blood. On arrival at the image slice, after arrival time t_A, the magnitude of the longitudinal magnetization of the arterial blood has decreased (i.e., Mz has become less negative) due to T₁ relaxation but is nevertheless different from the fully relaxed magnetization of the tissue. The arterial blood reduces the longitudinal magnetization in the slice. At the end of labeling a short delay is introduced, t_D, and the longitudinal magnetization in the slice is measured using echo-planar imaging (EPI). A second, control image is taken without arterial spin labeling. Figure 1b shows the tissue magnetization of the difference image ΔM (i.e., control-labeled) with time after the start of labeling. There are three parts to this curve: part 1 has no signal since the labeled blood has yet to reach the tissue volume (t ≤ t_A); part 2 shows rising signal as the labeled blood continues to enter the slice (t ≤ t_L + t_A); and part 3 shows decaying signal due to longitudinal relaxation and outflow after the end of incoming arterial signal (t > t_L + t_A). The longitudinal arterial magnetization Δm_a is constant during part 2 due to the continuous labeling pulse.

The subtraction image for both ASL techniques has static tissue signal removed with only the signal from the labeled water protons remaining.

Single Compartment Theory

The imaging voxel is assumed to be a single, well-mixed compartment with intra- and extravascular water in perfect communication. Labeled water enters and leaves with perfusion rate f and relaxes with tissue longitudinal relaxation time T₁ as shown in Fig. 2a. The Bloch equation is modified to include the effects of incoming arterial magnetisation (m_a) and outgoing venous magnetization (m_v) (4). The rate of change of longitudinal magnetization, M, in the tissue voxel can be described as:

(1)

This equation is generally true for all types of labeling. M⁰ is the longitudinal equilibrium magnetization of the tissue. For a full definition of the parameters, see Table 1. It is assumed that the labeled blood water molecules immediately equilibrate with the extravascular water molecules such that water in the blood leaving the voxel contains labeled molecules at the same concentration as water in the tissue voxel, weighted by the increased water concentration of blood to tissue, i.e., m_v(t) = M(t)/λ where λ is the brain–blood partition coefficient for water (i.e., ratio of water contents, λ = (v_ew + v_bw)/v, see Table 1). Considering the magnetization of the difference image, we have from Eq. [1]:

(2)

where Δ represents the change in signal between the two images, i.e., control-labeled. Assuming that the physical constants f, T1, M⁰, and λ do not change between the two scans we have:

(3)

where:

(4)

Magnetization is a measure of the magnetic dipole moment density and so the difference in magnetization between the two images ΔM(t) can be thought of as a measure of the concentration of labeled molecules in the difference image. Equation [3] is a straightforward first order differential equation that can be solved provided we know Δm_a(t), the difference in arterial magnetization between the label and control image. The form of Δm_a(t) will depend on the type of labeling used. For a CASL (2) experiment, we have Δm_a(t) = 2mα(exp(−t_A/T) for t_A ≤ t ≤ t_A + t_L and Δm_a = 0 for t < t_A and t > t_A + t_L, where m is the equilibrium magnetization of arterial blood, α is the inversion efficiency of the labeling, T is the T₁ of blood, t_A is the arrival time of blood from labeling to image plane, and t_L is the labeling time. The factor 2 comes in because we are considering the difference in magnetization between the control and labeled inverted images which is twice the equilibrium magnetization at t = 0. Using this expression for Δm_a in Eq. [3] and solving for ΔM(t), we get:

(5)

(6)

(7)

This is the same solution as derived when considering the two images separately and subtracting or when considering a bolus approach with a specific tissue residue function r(t) = exp(−t(f/λ + 1/T₁)) (13). This residue function assumes immediate equilibration of the intra- and extravascular water molecules and is equivalent to assuming m_v(t) = M(t)/λ when using the differential equation approach.

Table 1. Definitions of Parameters

f	Blood perfusion	ml blood/min/100ml tissue
PS	Permeability surface area product	ml water/min/100ml tissue
λ	Brain:blood partition coefficient λ = (vew + vbw)/v	ml water(ml tissue)−1/ml water(ml blood)−1
tA	Arrival time (from labelling plane to image slice)	s
tL	Labelling time	s
G	Gain	signal per unit of magnetisation
α	Inversion efficiency of labelling pulse	fraction of blood water protons inverted
M(t)	Tissue magnetisation	magnetic moment (ml tissue)−1
M0	Equilibrium tissue magnetisation	magnetic moment (ml tissue)−1
mb(t)	Capillary blood water magnetisation	magnetic moment (ml water)−1
me(t)	Extravascular water magnetisation	magnetic moment (ml water)−1
m	Equilibrium magnetisation of extravascular water	magnetic moment (ml water)−1
ma(t)	Arterial blood magnetisation	magnetic moment (ml blood)−1
m	Equilibrium magnetisation of arterial blood	magnetic moment (ml blood)−1
mv(t)	Venous blood magnetisation	magnetic moment (ml blood)−1
vbw	Blood water volume vbw = vb · v	ml water/ml tissue
vew	Extravascular water volume vew = ve · v	ml water/ml tissue
vb	Blood volume	ml blood/ml tissue
ve	Extravascular volume	ml extravascular space/ml tissue
v	Blood water content	ml water/ml blood
v	Extravascular water content	ml water/ml extravascular space
T1	Longitudinal relaxation time of water in tissue	s
T1b	Longitudinal relaxation time of water in blood	s
T1e	Longitudinal relaxation time of water in extravascular space	s

Two-Compartment Theory

In reality, the labeled blood water does not exchange immediately with the extravascular water, so using a single well-mixed compartment is inaccurate. As described in several studies (16-18) this leads to two opposing effects when measuring perfusion.

First, if labeled water molecules remain in the blood for some time before entering the extravascular space, they will relax with the longer blood relaxation time, T which is measured as 1.4 s (22) compared to extravascular T₁, T which could be as low as 0.6 s in white matter (23) at 1.5T. The signal decay will be slower and the tissue magnetization larger than predicted by the single compartment model. We call this the T₁ effect. The application of diffusion gradients which aim to crush the intravascular signal have little effect in such small vessels (24) and the signal we measure is from both capillaries and tissue. Second, some of the labeled blood water molecules will pass directly through the microvasculature without ever exchanging. The venous magnetization m_v will be larger and the tissue magnetization lower than predicted by the single compartment model. We call this the outflow effect.

A complex interplay of flow rate, field strength (which affects T and T), measurement time, and tissue characteristics will determine which effect is dominant, and hence whether perfusion is over or underestimated when ignoring these effects. To model these effects we consider a two-compartment system. A blood water compartment and an extravascular water compartment, each with corresponding volumes and longitudinal relaxation times, are separated by semipermeable endothelium (Fig. 2b). An extra component must be introduced to the differential equation, which accounts for labeled water crossing between the two compartments through the permeable capillary wall. To do this we use PS, the permeability surface area product. Equation [1] can be extended to include permeability and applied to each compartment. The total magnetization in the difference image will then be the sum of the magnetization from each compartment multiplied by the relative volumes of the compartments:

(8)

where v_ew and v_bw are the respective volumes of extravascular and blood water per unit tissue volume and m_e and m_b are the magnetization of water in the extravascular water and blood water spaces, respectively.

Blood Compartment

The blood compartment includes all of the blood in the voxel which could contain labeled water molecules, including arterioles and venules supplying and draining the capillary beds within the voxel. Equation [9] describes the change in magnetization of the blood compartment, m_b(t) with time:

(9)

It is a modified version of Eq [1]. Magnetizations m_b and m_e are represented by lower-case letters since they refer to the magnetization per unit volume of water within a given compartment. Blood water volume v_bw (i.e., blood water volume per unit volume of tissue) is used so that every term is expressed in units of magnetization per volume of tissue per unit time. v_bw is the product of v_b, the volume of blood per unit volume of tissue and v, the volume of water per unit volume of blood: v_bw = v_bv. M⁰ is replaced by the equilibrium magnetization of blood water, m and T₁ by T. The final term describes the flow of labeled water protons between the two compartments. This is normally expressed as PS multiplied by the difference in tracer concentration between the two compartments. The difference in magnetization m_e(t) − m_b(t) is equivalent since this is proportional to the concentration of labeled water molecules or “tracer” in the water that can move between the two compartments.

Extravascular Compartment

This equation is simpler than that for the blood compartment since there is no inflow or outflow of blood and follows easily by analogy with Eq. [9]. The only entry of labeled molecules is through the permeable capillary wall.

(10)

Here v_ew is the product of v_e, the volume of extravascular space per unit volume of tissue and v, the volume of water per unit volume of extravascular space: v_ew = v_ev. v_e = 1 − v_b and λ can be used to calculate v_ew. T is the T₁ of extravascular water which can be approximated as the T₁ of tissue since the blood contribution is small. These equations are similar to those of Zhou et al. (17) and Schwarzbauer et al. (12) but with PS stated explicitly from the start. Our model is described in terms of the extravascular water and blood water volumes with the blood–brain partition coefficient, λ, used only to link these two quantities. The compartments are defined as water compartments such that m_b(t) and m_e(t) are per unit volume of water. At equilibrium (t = ∞) m_e(t) = m, and Eq. [10] shows m = m, i.e., the labeled water molecules are equally distributed throughout the accessible water.

Note that for the CASL technique the magnetization transfer (MT) effects of the labeling pulse will significantly reduce T. The MT effects will reduce with time for t > t_L after the labeling pulse is switched off, giving a time-varying T.

Assuming that the physical constants v_bw, v_ew, T, T, m, m, f, and PS do not change between the labeled and control scans, we have from Eq. [9] and Eq. [10]:

(11)

(12)

Solutions to the Two-Compartment Model

Equations [11] and [12] can be solved to find m_b and m_e and hence the total magnetization using Eq. [8], provided we know Δm_a and Δm_v. Δm_a can be calculated dependent on the labeling strategy. However, the assumption that Δm_v = ΔM/λ as for the single compartment theory cannot be made since we are allowing for the possibility that equilibrium between blood and extravascular water molecules is not reached.

It may seem natural to assume that the venous blood has the same magnetization as the voxel blood compartment. However, this will overestimate the venous magnetization since at short times after labeling the outflow of labeled molecules will be small, as they have not yet had time to travel through the vasculature. For a precise definition of Δm_v the distribution of labeled molecules with distance along the vessels within the voxel must be considered (16). In truth, a compartmental model can never deal with this situation properly and assumptions must be made. However, in certain conditions reasonable assumptions can be made about Δm_v that will simplify the model solutions.

Testing the Two-Compartment Model

The solutions to Eq. [11] and Eq. [12] are complex and there is always a possibility of error. They were tested in two ways: First, it was verified that as PS reaches infinity the solutions reduce to the single compartment solution. This is to be expected, since if we sum Eq. [11] and Eq. [12] the permeability terms cancel and we are left with Eq. [3] for the single compartment case with:

(13)

i.e., the tissue relaxation rate is a weighted sum of contributions from water in blood and extravascular spaces. To check for mathematical errors the solutions were successfully substituted back into the differential equations.

Simulated Data

Imaging

In order to test the accuracy of the two-compartment model, we collected data from three subjects (female, mean age 28 years) using a CASL (2) pulse sequence (described in the Introduction) with gradient echo EPI collection. The imaging was repeated on one subject in order to test reproducibility. In this implementation labeling is psuedo-continuous for a time t_L and then there is a postlabeling delay t_D before image collection at time t = t_L + t_D. The postlabeling delay time was varied in order to sample phases 2 and 3 of the signal curve (Fig. 1). Multislice CASL imaging was performed on three subjects with a range of 7 postlabeling delay times from 0 to 1.5 s + t_i. t_i is the imaging time, which is different for each slice (34–577 ms) depending on the slice collection order. Scanning was conducted on a General Electric Signa 5.6 1.5T scanner with scan parameters: repetition time, TR_ASL = 4 s (time between the starts of successive labeling pulses), echo time = 34 ms, t_L = 1.7 s, 45 averages, 64 × 64 resolution, 24 cm field of view (giving pixel size 3.75 × 3.75 mm), eight slices with 7 mm slice thickness and 2 mm gap, slice collection order 1,3,5,7,2,4,6,8 with slice 1 being the inferior slice. The labeling plane was positioned 4 cm below the inferior slice; control and labeled image collection were interleaved. This perfusion protocol took 42 min.

A T₁ map was also collected at the same resolution and slice positions in order to segment the tissue into gray and white matter regions. A gradient echo EPI inversion recovery technique was used in order to exactly match the image distortions of the perfusion images. Three image sets were collected with TI = 1 s, 1.6 s, and one without inversion, TR = 7.2 s, which took a total of 3 min. The T₁ map was calculated using the equation:

(14)

where S⁰ is the magnitude signal from the image collected without inversion, β is the degree of inversion, and TI is the inversion time. Maps for β and T1 were calculated by fitting to the equation using least-squares minimization.

Analysis

The total signal measured in the difference image is given by S = GΔM where ΔM is given by Eq. [21] or Eq. [22], depending on the measurement time (or the solution to Eq. [3] for the single compartment model) and G is the scanner gain. G also includes any signal loss due to transverse relaxation (T), which will be the same for both labeled and control images. The signal in the control ASL images is given by:

(15)

where S = GM⁰ = Gλm. Since TR_ASL ≫ T₁, the control ASL images are taken to be approximate maps of Gλm. We can then use a whole brain, mean estimate of λ (taken as 0.9 from the literature (27)) to make an estimate of Gm. Dividing the measured difference signal S(t) by Gm gives estimates of ΔM(t)/m; these are used to fit Eq. [21] or Eq. [22] for f, t_A, and PS/v_bw with fixed parameters T, T and α.

To identify large regions of gray and white matter the difference images were segmented on the basis of T₁ values: 0.5–0.9 s (white matter), 1.0–1.8 s (gray matter), and single pixels in the vicinity of large vessels, which were identified as having high signal in both the T₁ and perfusion difference images. The single compartment solution, Eqs. [6] and [7], and the simplified two-compartment solutions, Eqs. [21,22], were fitted (using least-squares minimization) to region of interest data from each slice for the free parameters f, PS/v_bw, and t_A. Fixed parameters were: T = 1.4 s ± 0.1 s, T = 0.75 s (gray matter) 0.52 s (white matter) ± 0.1 s, α = 0.7 ± 0.1, λ = 0.9 ± 0.1. The uncertainties of these values (found from considering a range of published values (2, 14, 22, 27)) were propagated to produce systematic errors in the fitted parameters. For the gray and white matter regions the slow solution was used and for the large vessel pixels the fast solution was used with the extra parameter v_b = 0.05 (note that an artery of 1 mm diameter gives v_b ≈ 0.05). Note that the T values are chosen to take account of the MT effects of the labeling pulse (14) assuming complete saturation.

Some signal time dependence could come from large vessels that contribute signal at short delay times but have washed through at longer delay times. To avoid this effect only the later six time points with delay times 0.25–1.5 s were used for gray and white matter regions to avoid signal from very large vessels. To test that this effect was negligible the data were refitted using only the later five points (delay time 0.5–1.5 s) and the fitted values were compared to those produced using six data points. At a delay time of 0.5 s all blood flowing at a velocity greater than 1.4 cm/s will have washed through the voxel. This leaves blood in capillaries and larger vessels (up to approximately 0.1 mm diameter (29)) which could go on to perfuse blood in the voxel and so should be included in the measurement.

To calculate random errors a Monte Carlo bootstrapping technique was used to resample the data based on measured errors in the data. These errors were found by performing a scan-rescan test and measuring the difference in signal. This previous study (1) gave, for similar volumes, gray matter raw signal reproducibility (95% confidence limit) of 8% and white matter 14%. One hundred sets of resampled data from each dataset were used to find the standard deviation in the fitted parameters for each original dataset. Average values for perfusion and PS/v_bw were found by calculating a weighted mean of values from each dataset. Weights were set as the inverse of the variance for each measurement.

Accuracy of the Single Compartment Solution

The form of the solutions m_e and m_b for pulsed labeling are shown in Fig. 3. They have been multiplied by the relative volumar fractions such that they represent magnetization per unit volume of tissue. It can be seen that m_b peaks before m_e, which is to be expected as the bolus moves through the blood compartment into the extravascular compartment. The average timing of these peaks can be related to the time of exchange, T_ex, which was introduced in our initial, crude two-compartment model (30). Figure 4a–d shows the errors of the single compartment model for both pulsed and continuous labeling at a range of field strengths and perfusion rates. At low perfusion rates the single compartment theory overestimates perfusion due to the dominance of the T₁ effect, while at higher perfusion rates the outflow effect begins to dominate and perfusion is underestimated. Comparing Fig. 4c with 4a, and 4d with 4b, it can be seen that in white matter the T₁ effect is larger due to the greater difference between T and T. These results are in broad agreement with previous studies (16-18) (see the Discussion section).

In addition, the current study also shows the effect of field strength and measurement time. Comparing Fig. 4a with 4b and 4c with 4d, it can be seen how the timing of the measurement affects the interplay of the two effects since the FAIR measurement is taken at an earlier time point (t ≈ 1s) than the CASL measurement (t ≈ 3s). For the later CASL measurement, the outflow effect begins to dominate at a lower perfusion rate because the measurement time is closer to the MTT and unexchanged molecules will begin to exit the voxel. The effect of field strength is to change the T₁ values of the blood and tissue, which in turn determine the strength and relative importance of the venous magnetization, Δm_v. At higher fields Δm_v increases and the outflow effect begins to dominate over the T₁ effect at a lower perfusion rate. The distribution of transit times will determine the smoothness of transition between these two effects. For typical human perfusion rates at 1.5T (see Table 2), we find that for CASL the single compartment theory overestimates perfusion by 17% in gray matter and 62% in white matter.

Accuracy of the Simplified Two-Compartment Solutions

Figure 5a–d shows the accuracy of the simplified fast and slow two-compartment solutions at 1.5T. It can be seen that the slow solution is very accurate for the FAIR technique at typical human perfusion rates in both gray and white matter, being almost indistinguishable from the distributed solution. With continuous imaging, the slow solution becomes inaccurate at lower perfusion rates because of the later measurement times that approach MTT. When the MTT is reached, the outflow effect, which is assumed negligible in the slow solution, becomes important. The fast solution is only accurate for higher perfusion rates with the outflow effect being overestimated at slower rates. Using the impermeable solution, perfusion is underestimated by 3% for gray matter and 7% for white matter with FAIR, and 9% for gray matter and 20% for white matter with CASL at 1.5T.

Errors Due to Fixed Parameters

Figure 6 shows the propagated error in the perfusion estimate for each unknown parameter for gray matter at 1.5T using the parameters in Table 2. While PS/v_bw need not be known very accurately, the parameters T, m, and α must be more precise since they have a larger effect on the perfusion measurement. For less than 5% error in perfusion measurements in gray matter (an acceptable limit), the parameters must be known to the following accuracy: α: 5%; m: 5%; T: 7%; t_A: 21%; T: 27%, and PS/v_bw: 100%. These values represent the accuracy we must strive for. While the accuracy of the T₁ measurement is not too important, it is clear that accuracy of blood parameters T and m are very important—an issue which has been somewhat overlooked. An accurate measurement of T is difficult due to the movement of the blood. Note that the use of T_1app rather than T₁ in current single compartment theory (see Eq. [4]) is insignificant in terms of the accuracy of the perfusion estimate. T_1app is on the order of 1% smaller than T₁, which produces only a 0.2% error in perfusion, which is negligible when compared to other errors.

Imaging

Figure 7a–d shows the typical performance of the single and two-compartment models when fitting for perfusion, arrival time, and PS/v_bw. They show that the two-compartment model fits well to both extremes of data, i.e., white matter and tissue containing a large vessel. It can be seen that the single compartment model decays too quickly to fit the white matter data decay (Fig. 7b), presumably because the labeled water remains in the blood compartment with slower longitudinal relaxation. In some cases the single compartment model compensates for the inability to fit by shifting the curve and overestimating the arrival time, as shown in Fig. 7c. For the single pixels containing large vessels (Fig. 7d) the single compartment model decays more slowly than the data, presumably because of outflow of nonexchanging molecules from the blood compartment.

Table 3 shows the results of the fitting for perfusion and PS/v_bw averaged over all datasets (as described in the Methods section) for three subjects. PS/v_bw is found to be significantly different in different tissue types, being larger in gray matter. There is a wide spread in both perfusion and PS/v_bw measurements. The repeat scan on Subject 1 shows high reproducibility of perfusion measurements (6% change in whole brain value) but poor reproducibility of PS/v_bw (59% change in whole brain value). This perfusion change is within intrasubject biological variability found by repeat scanning at a single delay time throughout the day (21). The errors on the PS/v_bw measurements are large, reflected in the poor reproducibility of this measurement. The results for fitting only the later five data points are not significantly different than with six data points, suggesting that the effect of large vessel through-flow is negligible.

Figure 8 shows the arrival time results averaged over all subjects. We find arrival time to be approximately linear with distance from the labeling plane with a slope in agreement with published values (31). The offset with the origin could be due to the blood traveling faster in the larger arteries closer to the labeling plane. The value at 4 cm agrees with estimates of arrival time (22) of approximately 200 ms based on artery blood velocity of 20 cm s⁻¹. The white matter arrival time is found to be longer than the gray matter arrival time, which could be due to the lower white matter perfusion reflecting a lower blood velocity in feeding vessels (14, 31).

Simulations

This work shows that a single compartment model can significantly underestimate or overestimate perfusion, depending on the perfusion rate of the system and the field strength. In the human regime at 1.5T the single compartment theory overestimates perfusion by 62% (white matter) and 17% (gray matter). This error is strongly dependent on the blood T₁, T. Recent work (32) has measured blood T₁ as 1.4–1.7 s at 1.5T, depending on hematocrit, which will increase the errors of the single compartment model. The work by St. Lawrence et al. (16) shows almost identical results for their single pass approximation model, without the assumption of a well-mixed blood compartment. This helps to verify our handling of the venous outflow (Δm_v = 0) to correct for the assumption of a well-mixed blood compartment. Interestingly, the inclusion of a venous compartment in the St. Lawrence model significantly changes the simulation results, reducing the outflow effect. It is therefore important to establish if this venous compartment is required. The work by Zhou et al. (17) does not show an overestimation by the single compartment model at low perfusion rates. There are two possible reasons for this discrepancy. First, Zhou et al. assumed a smaller value of T, which will reduce the T₁ effect. Second, the model used did not take account of the overestimation of venous outflow that results from assuming a well-mixed blood compartment. This will overemphasize the outflow effect, and therefore mask the T₁ effect. Ewing et al. (18) predict the crossover between the two effects occurs around 150 ml blood min⁻¹ (100 ml tissue)⁻¹ at 7T. From Fig. 4a this study gives a similar value of 130 ml blood min⁻¹ (100 ml tissue)⁻¹.

In addition, we find that the balance of the T₁ effect and the outflow effect depends not only on perfusion and relaxation parameters but also on the measurement technique and the measurement time. FAIR is more sensitive to the T₁ effect because measurements are made about 1 s after labeling, when there is negligible outflow of labeled protons. The T₁ effect dominates up to flow rates in excess of 200 ml blood min⁻¹ (100 ml tissue)⁻¹ at 1.5T. CASL imaging is typically around 3 s after the start of labeling, by which time there is some outflow of labeled protons. The outflow effect dominates for flow rates in excess of 100 ml blood min⁻¹ (100 ml tissue)⁻¹ at 1.5T. In either case, for human scanning at 1.5T it is the T₁ effect that dominates and the single compartment model will overestimate perfusion. During human neuronal activation it is likely that the increased perfusion will change the balance of these two effects. It is important to consider what effect this will have on the analysis of functional imaging studies. Note that the simulations assumed zero arrival time. An increased arrival time would tend to delay the outflow effects, causing the T₁ effect to dominate for a wider range of perfusion rates.

The impermeable solution simplifies the model even further to a single blood compartment. For FAIR, errors are found to be only 3% in gray matter, much smaller than assuming the original well-mixed single compartment model. For CASL, at the later measurement time, errors are larger but still an improvement on those of the original single compartment model. The advantage of the impermeable model is that the tissue T₁ map is no longer required for quantification. This is particularly attractive since it will reduce the scan time and the need for registration of the perfusion images to a T₁ map. The error analysis shown in Fig. 6 supports this finding, showing that the accuracy of T is not very important for an accurate gray matter perfusion measurement. The error propagation ratio will, however, be larger for white matter. This model is clearly a good option for FAIR modeling. It may be advantageous to reduce the CASL measurement time in order to improve the accuracy of the impermeable model. Conversely, the error analysis highlights the importance of an accurate measurement of T and m. This has been somewhat overlooked in the current modeling of ASL data.

In Vivo Modeling

The two-compartment model provides a better fit to CASL decay-time data than the single compartment model over all extremes of data. It is important to note that the model can fit to all types of blood flow, from true capillary perfusion to blood flow in a single large vessel, and will simply measure “perfusion” as the total volume of blood flowing through a tissue volume per unit time. The insensitivity of perfusion measurements to increasing delay time after 250 ms (shown in the comparison of fitting five and six data points, Table 3) suggests that after this time all of the blood present in the voxel will go on to perfuse that voxel. It is important to remember that any blood which will eventually perfuse the measurement voxel should be included, even if it is in larger arterioles at the time of image collection.

Our assumption that labeled blood water enters the voxel in exchanging vessels is therefore probably incorrect. The inclusion of an arterial compartment may improve the accuracy of the model, but would also increase the number of free parameters in the model. In terms of modeling in vivo data, the term PS/v_bw could encompass a number of other factors that would have a similar effect on the signal curve. It is instructive to think of PS as an average PS for all of the vessels in the imaging voxel that contain labeled blood water.

The simplified slow solution allowed the fitting of arrival time and PS/v_bw as well as perfusion. However, this method is not suggested as a good way to measure PS/v_bw since the reproducibility is poor. We simply show that a realistic, noninfinite, value of PS/v_bw is required to correctly model the perfusion signal. The measurements show a tissue-type dependence of PS/v_bw. This was initially unexpected since the parameter reduces to 2P/rv, with all components being tissue-independent. However, this contradiction could be explained if we consider the location of the blood in the different tissue types when the measurement is taken. For white matter it is thought (14, 31) that the blood takes longer to reach the capillary bed than in gray matter, so we could be measuring PS/v_bw of the arterioles rather than the capillaries, which, as we find, would be lower. This is supported by the trend towards higher PS/v_bw values when only the later five data points are fitted, since at later times more of the labeled molecules will have reached the capillaries. It would be interesting to see if the addition of mild diffusion gradients, which would remove some of the arteriole signal, would increase the estimated PS/v_bw in white matter. The values for PS/v_bw measured from the later five time points are likely to be more accurate estimates of true capillary permeability. The gray matter values of around 25 min⁻¹ agree reasonably well with published values. For example, Herscovitch et al. (10) found PS of 1.5 min⁻¹ (gray matter), giving PS/v_bw values of approximately 30 min⁻¹ using typical blood volume of 5% (33).

There are a number of effects besides restricted vessel permeability that could also produce changes to the difference signal with delay time. These include MT and T effects. The MT effects of the labeling pulse will shorten T and this effect will diminish with increasing delay time. In our simulations we used a fixed value for T taken from a measurement at zero delay time. At maximum delay time T could be up to 50% larger (the value without MT effects). If included in the model this time-varying T would produce a more slowly decaying signal curve, which could perhaps be imitated by a lower value for PS/v_bw. As a result, measurements of PS/v_bw could be underestimated. Simulated gray matter data with varying T (0.75 s to 1.0 s, i.e., saturated to unsaturated) show that assuming a fixed T of 0.75 s estimates perfusion accurately but underestimates PS/v_bw by 55%.

Like T₁, the T values for blood and extravascular water are also different. The difference signal will change as water moves between the two compartments. Different T₂ values can be incorporated into the model using the equation:

(16)

where T is the T of extravascular space and T is the T of blood. In this work we have used S = GΔM instead, which assumes that T = T = T, the transverse relaxation time of tissue. T is very dependent on blood oxygenation, with published values ranging from 130–150 ms for fully oxygenated blood and from 40–80 ms for whole blood ((32) and references therein). Bandettini et al. (34) measured whole brain T (we assume this to be equivalent to T) in four subjects, giving values between 37 and 61 ms. For our echo time of 34 ms, a difference of 20% between T and T gives an error of 7% in perfusion at a delay time of 1 s, using typical gray matter parameters of Table 2. If T > T the signal is weighted towards the water in the extravascular compartment and perfusion is underestimated. In principle, this error could be corrected by using a separate measurement of T and T. Alternatively, the error could be reduced by shortening the echo time, perhaps by using spiral EPI.

There have been a number of studies comparing ASL perfusion estimates using a single compartment model to those using other techniques. The study by Zhou et al. (17) in cats at 4.7T made comparisons with microspheres, finding ASL underestimated perfusion with increasing deviation at higher flow rates. This is in agreement with our predictions (Fig 4a). A study by Ye et al. (35) in humans at 1.5T made comparisons with PET and found good agreement in gray matter but an underestimation of white matter perfusion with ASL, contrary to our model predictions (Fig. 4d). As the authors suggest, this may have been due to an underestimation of white matter arrival time. These results, and other comparison studies, do not all agree with our model predictions, but they are also all fairly inconsistent. This highlights the problem of comparing different perfusion techniques, in that none of them can be assumed to be the “gold standard.” With each technique there are various factors which could cause an error to perfusion quantification. Our approach of testing perfusion models internally on data from a single technique offers an alternative method for model validation.