Simplified Methods for Calculating Cerebral Metabolic Rate of Oxygen Based on 17 O Magnetic Resonance Spectroscopic Imaging Measurement during a Short 17 O 2 Inhalation

Abstract

It has recently been shown that ¹⁷O magnetic resonance (MR) spectroscopic imaging at ultrahigh fields provides a reliable method for measuring CMRO₂ during a short period of ¹⁷O₂ gas inhalation. The mathematical (or complete) model used in the ¹⁷O MR method for calculating CMRO₂ requires simultaneous measurements of multiple parameters including the concentration of H₂¹⁷O produced in the brain tissue from inhaled ¹⁷O₂ gas (C_b), CBF, and the input function for the H₂¹⁷O concentration in the feeding artery (C_a). Both invasive and noninvasive measurements are involved in determining all of these parameters. In this article, two simplified methods are proposed and validated for calculating CMRO₂ based on ¹⁷O MR measurement(s); the first method requires the measurements of C_b and CBF, but not C_a, and the second method only requires a single noninvasive measurement of C_b. The simplified methods were used to calculate CMRO₂ in anesthetized rat brain, and the results were compared with those obtained using the complete model. The results from this work show (1) the validity of the simplified methods for quantifying CMRO₂, and (2) the feasibility for establishing a completely noninvasive ¹⁷O MR approach for imaging CMRO₂ in vivo.

Oxidative metabolism of brain under both normal and pathologic conditions can be evaluated through the measurement of the cerebral metabolic rate of oxygen consumption (CMRO₂) (Siesjo 1978). This can be accomplished by ¹⁵O positron emission tomography (PET), which monitors the concentration change of the metabolic H₂¹⁵O generated by mitochondrial respiration in the brain during an inhalation of ¹⁵O₂ gas (e.g., (Mintun et al., 1984)). However, this is a complex measurement. It requires independent determinations of cerebral H₂¹⁵O content, CBF, and H₂¹⁵O present in the arterial supply to the brain (the arterial input function) (Mintun et al., 1984); in addition, it is necessary to experimentally determine (Mintun et al., 1984) or make assumptions (Ohta et al., 1992) about cerebral blood volume (CBV). The CBV complication arises because the PET method is unable to distinguish the radioactive signals coming from the ¹⁵O₂ bound to hemoglobin or freely dissolved in blood from those emitted by the ¹⁵O atoms in the metabolically generated H₂¹⁵O molecules.

An alternative approach for measuring CMRO₂ is the use of ¹⁷O MR spectroscopy (MRS) or MR spectroscopic imaging (MRSI) techniques to monitor concentration changes of metabolic H₂¹⁷O during an inhalation of the oxygen gas enriched with the nonradioactive ¹⁷O isotope (i.e., ¹⁷O₂) (Arai et al., 1990, 1991; Fiat and Kang, 1992; Fiat et al., 1992; Mateescu et al., 1989; Pekar et al., 1991, 1995; Zhu et al., 2002). This is analogous to the PET approach. However, the ¹⁷O MR method has two major advantages compared with the PET technique. First, the ¹⁷O₂ molecules, either in the form of free gas or bound to the hemoglobin in the blood, are nuclear magnetic resonance (NMR) invisible; consequently, ¹⁷O MR detects the metabolically generated H₂¹⁷O specifically without the effects of confounding signals from other ¹⁷O species (thereby avoiding the CBV complication). Second, the absolute concentration of metabolically generated H₂¹⁷O, and in turn CMRO₂, can be precisely calculated by using the natural abundance H₂¹⁷O signal as an endogenous concentration reference; in the absence of enrichment, the 0.037% of water molecules naturally contain ¹⁷O atoms. Therefore, the natural abundance H₂¹⁷O content per gram tissue or concentration is known in both brain tissue and blood.

The major disadvantage of the NMR method is the low inherent sensitivity of ¹⁷O MR detection due to the small gyromagnetic ratio of the ¹⁷O nucleus. We have, however, recently shown that the sensitivity of ¹⁷O MR detection is improved significantly at ultrahigh magnetic fields, and an approximately fourfold signal-to-noise ratio gain can be obtained at 9.4 T compared with 4.7 T (Zhu et al., 2001). In view of the sensitivity gains available with the use of ultrahigh-field magnets, and the potential advantages compared with the PET technique, ¹⁷O MR approach may provide a viable and more accurate method for measuring CMRO₂. The central remaining problem, however, is the determination of all the necessary parameters for modeling and calculating CMRO₂.

Even though the ¹⁷O MR approach does not have the CBV complication that afflicts the PET technique, it still, in principle, requires three independent determinations of cerebral H₂¹⁷O concentration [C_b(t)], arterial input function for H₂¹⁷O [C_a(t)], and CBF, as in PET. This can be ascertained from a model originally established by Kety and Schmidt (1948a,1948b) (referred to as the “complete model” in this article) and schematically illustrated in Fig. 1. We have recently shown that when all these three parameters were rigorously determined, then the ¹⁷O MR approach indeed yielded an accurate measurement of CMRO₂ (Kety and Schmidt 1948a,b; Zhu et al., 2002). However, the measurements of two of these parameters required for the complete model, namely the arterial H₂¹⁷O input function and CBF, pose technical difficulties (as they do in the PET method), and ultimately limit the utility of the approach.

FIG. 1.

(A) Schematic illustration of a general model describing the transportation and metabolism processes of oxygen molecules in the brain. After gas exchange in the lung, the inhaled O₂ molecules quickly bind to hemoglobin (Hb) in the blood, forming O₂-Hb complexes (HbO₂). Through the feeding arteries/arterioles, the HbO₂ complex molecules are sent to the cerebral capillaries, where the O₂ molecules dissociate from the complex molecules, transport in the form of free gas across the blood–brain barrier (BBB), diffuse into the brain tissue (or cellular) space, and finally enter mitochondria where they are converted to metabolic water via oxidative metabolism. The metabolic water molecules can cross the mitochondrial membranes, move out of the brain cells, enter the draining venules/veins, and are washed out by the blood circulation. C_a,HbO2 and C_v,HbO2 are the HbO₂ concentration in the arterioles and venules, respectively, C_a,H2O and C_v,H2O are the metabolized H₂O concentration in the arterioles and venules, respectively. (B) Schematic illustration of a simplified “complete model” describing the events in the brain when the ¹⁷O-labeled oxygen gas molecules are introduced via an inhalation. In this model, only the ¹⁷O-labeled water (H₂¹⁷O) is involved because the ¹⁷O-labeled O₂ is NMR invisible.

The determination of the arterial H₂¹⁷O input function, practically requires an invasive procedure such as withdrawing arterial blood (preferably from the carotid arteries) at regular time intervals from the subject and determining arterial H₂¹⁷O concentration, as in PET studies, or using an implanted ¹⁷O radiofrequency (RF) coil around the carotid artery, and continuously measuring arterial H₂¹⁷O concentration with ¹⁷O NMR (Zhang et al., 2003; Zhu et al., 2002). Both of these approaches are highly invasive; the former can induce physiologic perturbations, especially in small-animal studies (e.g., blood volume and the blood pressure changes), potentially affecting the basal CMRO₂ value. The latter, although it is accurate and avoids the deleterious physiologic consequences of blood depletion, does require a complex surgical procedure and is extremely difficult to perform in humans or in nonterminal animal experiments.

Measurement of CBF can be performed directly by using a number of MRI approaches (Detre et al., 1994; Kim 1995; Kwong et al., 1995; Silva et al., 1995). However, the relatively low detection sensitivity of these approaches, especially in the brain areas with lower CBF values (e.g., the white matter), and the difficulty obtaining absolute CBF values using these approaches may lead to significant errors in CMRO₂ calculation.

It is, therefore, important to examine the feasibility of determining CMRO₂ reliably without having to measure C_a(t) and, if possible, CBF. Using different approximations, attempts have already been made to simplify the measuring procedures and modeling for the calculation of CMRO₂ with the ¹⁷O MR method. However, the validity of the approximations proposed in these simplified models has not been rigorously tested by experiments and is still intensely debated (Fiat and Kang 1992, 1993; Fiat et al., 1992; Pekar et al., 1991, 1995). In this article, two simplified methods are proposed for CMRO₂ calculation with the ¹⁷O MRS imaging approach (Zhu et al., 2002). The first method allows calculation of CMRO₂ without the need to experimentally measure C_a(t). The second method is an even simpler approach that requires the noninvasive measurement of C_b(t) only, without having to determine C_a(t) or CBF. The proposed models were experimentally evaluated by explicitly determining all parameters involved in the same animal so that CMRO₂ can be calculated from the same datasets using either the complete model or the simplified models.

THEORY

General considerations

Figure 1A schematically illustrates the transportation and metabolism processes of oxygen molecules in the brain. After gas exchange in the lung, the inhaled O₂ molecules quickly bind to hemoglobin (Hb) in the blood, forming O₂–hemoglobin complexes (HbO₂). Through the feeding arteries and arterioles, the HbO₂ complex permeates cerebral capillaries, where the O₂ molecules dissociate, cross the blood-brain barrier in the form of free gas, diffuse into the brain tissue (intracellular and extracellular) space, and finally enter mitochondria where they are converted to metabolic water via oxidative metabolism. The metabolic water molecules move out of the mitochondria, traversing the same pathway as O₂ entry but in reverse, and are washed out through venules and veins.

The key factor for the success of the ¹⁷O MR approach is the detection sensitivity as well as the visibility of ¹⁷O NMR. The critical parameters for determining the NMR sensitivity at different field strength are the longitudinal (T₁) and the transverse (T₂) relaxation times, and the magnetic field strength. The relaxation times of the ¹⁷O spin, the only magnetic stable isotope of oxygen, are dominated by quadrapolar interactions (Abragam, 1961). In the case of bulk water (excluding “bound” water in cells), for which the extreme narrowing limit is applicable (i.e., τ_cω ≫ 1, where τ_c is the rotational correlation time and ω is Larmor frequency in radians/s), the values of T₁ and T₂ due to quadrapolar interactions become

$\frac{1}{T_1}=\frac{3}{40}\left(\frac{2I+3}{I^2(2I-1)}\right)\left(1+\frac{{\eta }^2}{3}\right){\left(\frac{\pi e^{2}Qq}{h}\right)}^2{\tau }_c$

[1]

$\frac{1}{T_1}\cong \frac{3}{40}\left(\frac{2I+3}{I^2(2I-1)}\right)\left(1+\frac{{\eta }^2}{3}\right){\left(\frac{\pi e^{2}Qq}{h}\right)}^2{\tau }_c$

[2]

where the term of πe² Qq/h) is the quadrapolar coupling constant, and η is an asymmetry parameter (0≤ η ≤1) (Abragam 1961). This mechanism results in rapid relaxation and very short T₁ and T₂ values (a few milliseconds), as well as broadening of the ¹⁷O water resonance line width to approximately 100 to 200 Hz (Zhu et al., 2001). There does exist a contribution from the ¹⁷O-¹H scalar coupling on the ¹⁷O T₂ relaxation in water, which is sensitive to the proton chemical exchange (Meiboom 1961); however, this effect is relatively small in in vivo tissues.

In contrast to ¹⁷O water, ¹⁷O₂, either as free gas or in a dissolved form in water, is strongly paramagnetic because of its two unpaired electrons, and hence undetectable because of the strong dipolar coupling between the electrons and the nucleus. When bound to hemoglobin, the ¹⁷O₂ resonance is broadened beyond NMR detection owing to the extremely slow rotational motion of the oxyhemoglobin complex with large molecular weight. Saturation transfer electron paramagnetic resonance studies have shown that the τ_c value for rotational motion of the hemoglobin molecule is 8 × 10⁻⁶ seconds when the hemoglobin molecule is encapsulated within the erythrocyte membrane (Cassoly, 1982). This is approximately 10⁶ times slower than the rotational correlation time of bulk water (10⁻¹²–10⁻¹¹ s) (Steinhoff et al., 1993). Such slow rotational motion leads to extremely fast T₂ relaxation and renders the ¹⁷O₂ molecule bound to hemoglobin invisible in ¹⁷O MR data. Therefore, ¹⁷O MR signal becomes detectable only when ¹⁷O₂ is incorporated into water. This unique specificity makes it possible that ¹⁷O NMR detects the metabolically generated H₂¹⁷O without confounding signals from the ¹⁷O₂ molecules bound to hemoglobin or dissolved in tissue space. The general model described in Fig. 1A, therefore, can be significantly modified to a simple model via the elimination of the ¹⁷O₂ components. Figure 1B illustrates the modified “complete model” for ¹⁷O MR studies, where only the MR detectable ¹⁷O-labeled H₂¹⁷O molecules in the three compartments (artery, venule, and the brain tissue including capillaries) are involved.

In the modified “complete model” for ¹⁷O MR measurements (Fig. 1B), the vascular compartment of arteriole is replaced by artery because of the approximately identical H₂¹⁷O contents in arteriole and artery. Another approximation applied in this model is to include the vascular compartment of capillary into the brain tissue. In fact, the contribution of the H₂¹⁷O contents inside the capillaries is negligible because of the small capillary blood volume in the brain (approximately 2%). Henceforth, we will refer to the model in Fig 1B as the “complete model” accounting for all relevant parameters in this modified model.

Complete CMRO₂ model for the ¹⁷O magnetic resonance approach

As illustrated by Fig. 1B, the mass balance relation of the concentration of labeled H₂¹⁷O in the brain tissue during the period of inhalation of ¹⁷O₂ gas is given by:

$\frac{d{C}_b(t)}{dt}=2\alpha f_1(CMR{O}_2)+CBF(f_2[C_a(t)-C_v(t)])$

[3]

where C_a, C_b and C_v are the H₂¹⁷O concentration expressed in excess of the natural abundance H₂¹⁷O concentration in the arterial blood, brain tissue, and venule blood, respectively, as a function of ¹⁷O₂ inhalation time t (unit = minute). One labeled ¹⁷O₂ converts to two H₂¹⁷O molecules through oxidative metabolism; thus, a constant of 2 is used in Eq. 3. The parameter α in Eq. 3 is the ¹⁷O enrichment fraction in the inhaled ¹⁷O₂ gas (no unit). Appropriate use of Eq. 3 requires the units to be consistent among all parameters contained in the equation. Natural abundance H₂¹⁷O concentration is equal to 20.35 μmol per gram brain water for brain tissue and μmol per gram blood water in blood; this is based on the natural abundance H₂¹⁷O enrichment (0.037%) and the molecular weight of H₂¹⁷O (19.0). This concentration can be used to calibrate the absolute concentrations of C_b(t), C_a(t), and C_v(t). Therefore, for convenience, the unit for C_b is chosen as μmol/(g brain water) and the units for C_a and C_v are μmol/(g blood water). The parameter f₁ is a conversion factor that converts the CMRO₂ value to a conventional unit of μmol · (g brain tissue)⁻¹ · min⁻¹ according to:

$\begin{array}{lll}{f}_1=\frac{unit\hspace{0.17em}of\hspace{0.17em}d{C}_b(t)/dt}{unit\hspace{0.17em}of\hspace{0.17em}CMR{O}_2}\hfill & =\hfill & \frac{\frac{\mu mole}{(g\hspace{0.17em}brain\hspace{0.17em}water)\cdot min}}{\frac{\mu mole}{(g\hspace{0.17em}brain\hspace{0.17em}tissue)\cdot min}}\hfill \\ \hfill & \hfill & \frac{1}{{\beta }_{brain}}=1.266\hfill \end{array}$

[4]

where β_brain is defined as (g brain water)/(g brain tissue) and is equal to 0.79 (Pekar et al., 1995; Ter-Pogossian et al., 1970).

The conventional CBF unit is (mL blood) · (g brain tissue)⁻¹ · min⁻¹. Based on one common assumption that water in the brain tissue is in rapid equilibrium with water in venous blood, C_v(t) is proportional to C_b(t) (f₂C_v(t) = C_b(t)/λ) where λ (=0.90) is the brain/blood partition coefficient with the unit of (mL blood)/(g brain tissue) (Herscovitch and Raichle 1985). The constant f₂ (= 1.077) is another conversion factor used for the consistency of unit for the expression of C_b(t)/λ and C_v(t) terms. It is based on

$\begin{array}{l}{f}_2=\frac{unite\hspace{0.17em}of\hspace{0.17em}{C}_b(t)/\lambda }{unit\hspace{0.17em}of\hspace{0.17em}{C}_v(t)}=\frac{(g\hspace{0.17em}brain\hspace{0.17em}tissue)\cdot (g\hspace{0.17em}blood\hspace{0.17em}water)}{(mL\hspace{0.17em}blood)\cdot (g\hspace{0.17em}brain\hspace{0.17em}water)}\hfill \\ \hspace{1em}=\frac{{\rho }_{blood}\cdot {\beta }_{blood}}{{\beta }_{brain}}=\frac{1.05\times 0.81}{0.79}=1.077\hfill \end{array}$

[5]

where ρ_blood = (g blood)/(mL blood) = 1.05 (Altman, 1961), and β_blood = (g blood water)/(g blood) = 0.81 (Ter-Pogossian et al., 1970).

Substituting the C_v(t) term in Eq. 3 using the equality f₂C_v(t) = C_b(t)/λ, as well as introducing two correction factors (m and n) into Eq. 3 leads to:

$\frac{d{C}_b(t)}{dt}=2\alpha f_{1}CMR{O}_2+mCBF\left(f_2{C}_a(t)-n\frac{C_b(t)}{\lambda }\right).$

[6]

The correction factor m is used to account for the fact that water is not completely freely diffusible across the blood–brain barrier, which effectively reduces the rates of H₂¹⁷O perfusion and washout across this barrier. The value of m depends on CBF and is 0.84 for the CBF range of our experimental circumstances (Herscovitch et al., 1987). The other correction factor n accounts for the fact that the washout rate of the metabolic H₂¹⁷O after the cessation of ¹⁷O₂ inhalation was significantly slower than the washout rate of the H₂¹⁷O that permeates brain tissue subsequent to a bolus injection of H₂¹⁷O through the internal carotid artery in the rat (Zhu et al., 2002). The value of n is given by the ratio of the washout rate of the metabolic H²¹⁷O after the cessation of ¹⁷O₂ inhalation versus that of the cerebral H₂¹⁷O after a bolus injection of H₂¹⁷O. This difference may be due to the additional restrictions imposed on the diffusion of metabolic water across the mitochondrial membrane.

Solving the linear differential equation of Eq. 6 gives:

$\begin{array}{lll}{C}_b(t)\hfill & =\hfill & \frac{2\alpha \lambda f_1}{mnCBF}{\text{CMRO}}_2\left[1-{\mathrm{e}}^{-\frac{mnCBF}{\lambda }t}\right]\hfill \\ \hfill & \hfill & +{\text{mf}}_2\text{CBF}{\int }_0^{\mathrm{t}}{\mathrm{C}}_{\mathrm{a}}({\mathrm{t}}^{\prime }){\mathrm{e}}^{-\frac{mnCBF}{\lambda }(t^{\prime }-t)}d{t}^{\prime }.\hfill \end{array}$

[7]

In principle, the CMRO₂ value can be precisely calculated using Eq. 7 and the experimental data on C_b(t), CBF, n, and C_a(t). The primary aim of the two methods as presented in the upcoming sections of this article is to simplify CMRO₂ calculation without requiring C_a(t) measurement or CBF measurement. In the first method (method I), CBF measurement for each image voxel is required. The second method (method II) provides a further simplified approach for CMRO₂ calculation, which does not require CBF and C_a(t) measurements.

Simplified model: method I

The source of labeled water in the arterial input [i.e., C_a(t)] is the metabolically created water in all the aerobic tissues in the entire animal which exchanges with the capillary water and enters into the venous and subsequently the arterial part through the blood circulation. This is often referred to as “recirculation.” Assuming that there are no nonlinear effects introduced by water transport from tissues to capillaries, the concentration of H₂¹⁷O in blood (venous or arterial) should increase linearly during a brief ¹⁷O₂ inhalation period (e.g., 2 minutes in our experiments). This assumption is justified based on the approximation that the metabolic H₂¹⁷O from all organs can reach equilibrium with the water in venous blood rapidly compared with the rate it is produced by the mitochondria. Therefore, the concentration of H₂¹⁷O in the arterial blood, C_a(t), can be assumed to have a linear dependence on the inhalation time t, i.e.,

$C_a(t)=At$

[8]

where A is a constant. This linear relationship was in fact shown in our previous experiments based on invasive methods that allowed the continuous in vivo measurement of C_a(t) (Zhu et al., 2002). It is also supported by other studies (Arai et al., 1991; Fiat et al., 1992; Pekar et al., 1991). Substituting Eq. 8 into Eq. 7 provides the following solution for Eq. 3:

$\begin{array}{lll}{C}_b(t)\hfill & =\hfill & \left(\frac{2\alpha \lambda f_1}{mnCBF}{\text{CMRO}}_2+\frac{{\text{Af}}_2\lambda \mathrm{t}}{\mathrm{n}}\right)\left(1-{\mathrm{e}}^{-\frac{mnCBF}{\lambda }t}\right)\hfill \\ \hfill & \hfill & +\frac{{\text{Af}}_2{\lambda }^2}{m{n}^{2}CBF}\left(\frac{\text{mnCBF}}{\lambda }{\text{te}}^{-\frac{mnCBF}{\lambda }t}+{\mathrm{e}}^{-\frac{mnCBF}{\lambda }t}-1\right)\hfill \end{array}$

[9]

or,

$\begin{array}{l}{\text{CMRO}}_2(t)=\hfill \\ \frac{\left[\frac{C_b(t)-\frac{{\text{Af}}_2{\lambda }^2}{m{n}^{2}CBF}\left(\frac{\text{mnCBF}}{\lambda }{\text{te}}^{-\frac{mbCBF}{\lambda }t}+e^{-\frac{mnCBF}{\lambda }t}-1\right)-\frac{{\text{Af}}_2\lambda t}{n}}{1-{\mathrm{e}}^{-\frac{mnCBF}{\lambda }t}}\right]}{\frac{2\alpha \lambda f_1}{mnCBF}}\hfill \end{array}$

[10]

According to Eq. 10, besides C_b(t) and other known constants, both constant A and CBF are required to calculate CMRO₂. In our experiments, CBF was measured from the “wash-out” rate of the inert diffusible tracer H₂¹⁷O in the brain tissue after a rapid bolus injection of H₂¹⁷O via a close feeding artery (Zhu et al., 2001). Because metabolic H₂¹⁷O molecules are not generated in the absence of ¹⁷O-labeled oxygen gas inhalation (i.e., the contribution from the first term on the right side of Eq. 3 is negligible), and C_b(t) ≫ C_a(t) subsequent to the passage of the bolus of H₂¹⁷O injected via a close feeding carotid artery, then

$\frac{d{C}_b(t)}{dt}=-mCBF\frac{C_b(t)}{\lambda }.$

[11]

Solving Eq. 11 gives:

${\mathrm{C}}_{\mathrm{b}}(\mathrm{t})=[{\mathrm{C}}_{\mathrm{b}}({\mathrm{t}}_0)-{\mathrm{C}}_{\mathrm{o}}]{\mathrm{e}}^{-\frac{\text{mCBF}}{\lambda }(\mathrm{t}-{\mathrm{t}}_0)}+{\mathrm{C}}_{\mathrm{o}}\hspace{1em}(\text{for}\hspace{0.17em}\mathrm{t}\ge {\mathrm{t}}_0)$

[12]

where C_b(t₀) is the H₂¹⁷O concentration in the brain tissue at a time point t₀ when the bolus has essentially cleared from the arterial side (i.e., C_b(t) ≫ C_a(t) for t > t₀) and C_o is a constant. Fitting the H₂¹⁷O tracer washout curve by an exponential decay function according to Eq. 12 gives the CBF value.

During a steady state of oxidative metabolism, CMRO₂ values calculated by Eq. 10 at different time points of C_b(t) measurement should be a constant shortly after the onset of ¹⁷O gas inhalation when the ¹⁷O content in brain reaches a value that exceed the K_m of cytochrome oxidase. In the rat, measurements according to the complete modeling for the ¹⁷O MR approach revealed that this period was less than 0.5 minutes. Thus taking a derivative of Eq. 10 for time points beyond this initial delay period, yields an equation where the only unknown is A. Alternatively, the experimental data can be fitted to Eq. 10 using different A values until a time independent CMRO₂ value is obtained, as illustrated in the results section. Method I circumvents the invasive measurement of arterial input function that is extremely undesirable. With this simplified method, a partial or complete noninvasive model for the measurement of CMRO₂ through ¹⁷O MR technology can be established.

Simplified model: method II

Measurement of CBF is still required in method I. This requirement increases the complexity of the experiment, even though this measurement can be accomplished noninvasively by the arterial spin tagging MR approaches; furthermore, another parameter of constant n requiring one invasive measurement (i.e., H₂¹⁷O bolus injection) is also needed in method I. To further simplify CMRO₂ measurements and calculation, the second simplified method is proposed. In this method, neither the CBF value nor the parameter of constant n has to be determined; furthermore, the assumption of a linear arterial input function versus the inhalation time is not required.

According to Taylor's theorem, C_b(t) can be written as

${\mathrm{C}}_{\mathrm{b}}(\mathrm{t})={\mathrm{a}}_0+{\mathrm{a}}_1\mathrm{t}+{\mathrm{a}}_2{\mathrm{t}}^2+{\mathrm{a}}_3{\mathrm{t}}^3+\cdots .$

[13]

Because all the H₂¹⁷O concentrations are expressed in excess of the natural abundance H₂¹⁷O concentration, C_b is zero at the beginning of the inhalation; thus, a₀ = 0. Similarly, CBFF₂[C_a(t) − C_v(t)] can also be written as:

${\text{CBFf}}_2({\mathrm{C}}_{\mathrm{a}}(\mathrm{t})-{\mathrm{C}}_{\mathrm{v}}(\mathrm{t}))={\mathrm{a}}_1{\hspace{0.17em}}^{\prime }\mathrm{t}+{\mathrm{a}}_2{\hspace{0.17em}}^{\prime }{\mathrm{t}}^2+{\mathrm{a}}_3{\hspace{0.17em}}^{\prime }{\mathrm{t}}^3+\cdots$

[14]

with the boundary condition that C_v and C_a are both zero at the beginning of the inhalation; i.e., C_v(0) = C_a(0) = 0. Using these expansions and the boundary conditions, Eq. 3 becomes:

$\begin{array}{lll}{\mathrm{a}}_1+2{\mathrm{a}}_2\mathrm{t}+3{\mathrm{a}}_3{\mathrm{t}}^2+\cdots \hfill & =\hfill & 2\alpha {\mathrm{f}}_1{\text{CMRO}}_2+{\mathrm{a}}_1{\hspace{0.17em}}^{\prime }\mathrm{t}+{\mathrm{a}}_2{\hspace{0.17em}}^{\prime }{\mathrm{t}}^2\hfill \\ \hfill & \hfill & +{\mathrm{a}}_3{\hspace{0.17em}}^{\prime }{\mathrm{t}}^3+\cdots .\hfill \end{array}$

[15]

Because Eq. 15 holds for any t value, applying the polynomial theorem to Eq. 15 at t = 0 leads to:

${\mathrm{a}}_1=2\alpha {\mathrm{f}}_1{\text{CMRO}}_2$

[16]

${\text{CMRO}}_2=\frac{{\mathrm{a}}_1}{2\alpha {\mathrm{f}}_1}$

[17]

Eq. 17 and all the previous derivations lay the mathematical foundation of method II. If the C_b(t) curve is fitted by a polynomial in the form of Eq. 13 with a₀ = 0, the coefficient of the first linear term (i.e., a₁) can be used to calculate CMRO₂ according to Eq. 17.

Fitting a polynomial equation with infinite degree is obviously impossible for real data. In principle, the higher the order of the polynomial is used, the better the fit. However, this notion is only valid when all the data are noise-free. When noise is taken into account, larger variances in the fitted parameters will be potentially induced by fitting the data to the polynomials of larger degrees as the statistical power of fitting decreases; thus, the fitted results will be less reliable. Practically, only linear and quadratic terms were used for fitting our experimental data on C_b(t) including the zero point of C_b(t = 0) = 0 for each voxel to increase the fitting reliability.

Verification of the validity of the two models

The most straightforward way to verify the validity of the simplified methods is to compare the results obtained by using these simplified methods with the results from the complete model which requires determination of all parameters C_a(t), CBF, and n, as well as other known constants such as m.

MATERIALS AND METHODS

The details of the experimental measurements used in this study are the same as those previously reported in a study we conducted to explore the feasibility of the ¹⁷O MR approach for imaging CMRO₂ in a small-animal model during a brief inhalation of ¹⁷O₂ gas at 9.4 T (Zhu et al., 2002). We briefly describe these measurements here for convenience.

NMR measurements

All NMR experiments were performed on a 9.4-T, 31-cm bore magnet (Magnex Scientific, Abingdon, U.K.) interfaced to a Unity INOVA console (Varian, CA, U.S.A.). The spatial localization of ¹⁷O MRS was achieved by using the three-dimensional Fourier Series Window MRS imaging technique (Zhu et al., 2002). The acquisition parameters for acquiring each three-dimensional ¹⁷O-MRS image were as follows: repetition time for NMR data acquisition, 12 milliseconds; echo time, 0.4 milliseconds; total scan average, 948; spectral width, 30 kHz; 9×9×5 phase encodes; and 0.1 mL voxel size.

Animal preparation

Male Sprague-Dawley rats were anesthetized with approximately 2% (v/v) isoflurane in a mixture of O₂ and N₂O gases (2:3) during surgery. After oral intubations, femoral artery and vein were catheterized for physiologic monitoring, blood sampling, and chemical administration. After surgery, anesthesia was switched to α-chloralose using continuous infusion of 0.4 mL/h of 15-mg/mL α-chloralose solution during the entire NMR measurement period (Zhu et al., 2002). The rectal temperature of the rats was maintained at 37°±1°C by using a heated water blanket.

CBF measurements

For CBF measurements, one external carotid artery was catheterized for gaining access to the internal carotid artery without interrupting blood circulation into brain (Zhu et al., 2001). A small amount (0.05 to 0.1 mL) of 40%- to 50%-enriched H₂¹⁷O was rapidly injected into the brain via the internal carotid artery, and three-dimensional ¹⁷O MRS imaging was used for monitoring the H₂¹⁷O washout processes and for determining CBF values (Zhu et al., 2001). Fitting the signal decay curve of the H₂¹⁷O tracer in the brain with an exponential decay according to Eq. 12 gave the CBF value in each ¹⁷O image voxel.

C_a(t) measurements

An implanted vascular RF coil was designed and constructed for continuously detecting the ¹⁷O MR signal changes of H₂¹⁷O in the rat carotid artery at 9.4 T. The coil was based on a modified solenoid coil design combined with an RF shielding (Zhang et al., 2003). The entire coil had a compact physical size and was readily implanted into the rat body and wrapped around the rat carotid artery. The RF shielding ensured that the ¹⁷O NMR signal detected by the implanted coil originated only from the artery blood (approximately 7 μL blood volume) covered by the coil, without contaminations from surrounding tissues—therefore, additional spatial localization was not necessary for determining C_a(t). In addition, the RF shielding minimized the electromagnetic coupling between the implanted ¹⁷O coil and the head ¹⁷O surface coil tuned at the same operating frequency (two-coil configuration), which allowed simultaneous measurements of both C_a(t) and C_b(t) with the same temporal resolution (11 seconds).

¹⁷O₂ inhalation experiments

For ¹⁷O₂ inhalation studies, the ¹⁷O-labeled O₂ gas with 72% enrichment (Isotec Inc. OH, U.S.A.) was mixed with N₂O gas (approximately 2:3 volume ratio) and then stored in a cylindrical gas reservoir. The three-dimensional ¹⁷O MRS imaging acquisitions started with the normal gas mixture containing nonlabeled O₂ gas for 2 minutes; these natural abundance three-dimensional ¹⁷O MRS images were used as references for the calibration of metabolic H₂¹⁷O concentrations in the rat brain during and after the ¹⁷O₂ inhalation. Subsequently, the respiration gas was quickly switched to the ¹⁷O₂ labeled gas mixture. After 2 minutes of ¹⁷O₂ inhalation, the gas line was switched back to the normal gas mixture and the three-dimensional ¹⁷O MRS imaging acquisitions were continued throughout the measurement, and a total of 100 images were collected in approximately 19 minutes (Zhu et al., 2002).

Determination of constant n

The ratio between the washout rate of the metabolic H₂¹⁷O after the cessation of 17O2 inhalation and the washout rate of the H₂¹⁷O trace from the H₂¹⁷O bolus injection through the rat internal carotid artery gave the value of constant n for each ¹⁷O image voxel (see Theory).

RESULTS

CMRO₂ calculation results using simplified modeling (method I)

Figure 2A illustrates the simulation results showing the dependence of calculated CMRO₂ values on different values of the parameter A in the simplified model of method I (Eq. 10). In this simulation, the values of CMRO₂ = 2.2 μmol/g/min, A = 2.0 μmol/(g blood water)/min and CBF = 0.6 (mL blood)/(g brain tissue)/min were assumed. As the A value increases, the CMRO₂ values calculated for different time points change from time-dependent with CMRO₂ increasing with time, to time-independent and again to time-dependent with CMRO₂ now decreasing with time. Only the value of A = 2.0 μmol/(g blood water)/min provided a constant value of CMRO₂ = 2.2 μmol · g⁻¹ · min⁻¹ as expected.

FIG. 2.

(A) Simulated data illustrating the tendency of the calculated CMRO₂ values as A is varied. True CMRO₂ value is obtained when the CMRO₂ values calculated for all the timing points converge a constant. (B) CMRO₂ calculation using experimental data and method I. The A value is varied until CMRO₂ values calculated are the least correlated to the inhalation time. In this simulation, the true values of CMRO₂ = 2.2 μmol · g⁻¹ · min⁻¹, A = 2.0 μmol · (g blood water)⁻¹ · min⁻¹ and CBF = 0.6 (mL blood) · (g brain tissue)⁻¹ · min⁻¹ are assumed.

Simplified method I was applied to the experimental data obtained in the form of images of C_b(t) and CBF measured from seven animals, which were previously published (Zhu et al., 2002). Figure 2B demonstrates one example of simplified method I applied to data from a single voxel from a representative rat. In this calculation, the A value varied from 0 to 10 with the step size 0.05 (μmol · (g blood water)⁻¹ · min⁻¹). As discussed in the Theory section, the correct A value should yield a time independent CMRO₂ for this steady state. The first two points (represented by open circles) in this experimental data were not included in this calculation because their signal-to-noise ratio is low due to insufficient H₂¹⁷O buildup and that a delay is expected after the onset of inhalation of the ¹⁷O₂ labeled gas and before the ¹⁷O₂ content in brain reaches a value that exceed the K_m of cytochrome oxidase; only when the ¹⁷O₂ concentration exceeds the K_m of cytochrome oxidase for O₂ should the value calculated for CMRO₂ from the ¹⁷O data become time independent. Linear least square fitting was used to fit the CMRO₂ values calculated according to Eq. 10 based on the experimental C_b(t) data as a function of measurement time as well as the CBF and n values. The linear correlation coefficient (R) of the fitting was calculated (Fig. 2B). Among all the tried A values, the one under which CMRO₂ was the least correlated to the measurement time t (the smallest R square value) was considered to be the true value of A. The average of the calculated CMRO₂ values for all the time points (first two points excluded) under the true A value was used as the true CMRO₂ value. The final value of CMRO₂ using method I for the comparison later was averaged from seven animal measurements.

CMRO₂ calculation results using simplified modeling (method II)

Linear and quadratic polynomials were used to fit the observed C_b(t) curves from individual image voxels as a function of time for CMRO₂ calculation using method II as described in Theory. The experimental data and the fits obtained using only a linear or quadratic dependence on time in a representative rat are shown in Figs. 3A and 3B (single voxel results). The averaged CMRO₂ values over all voxels in the brain for each individual animal (n = 7) using method II (both linear and quadratic fitting results) are summarized in Fig. 4. In comparison with the gray matter, the fraction of the white matter volume in the rat brain is much smaller than that of other animals (e.g., cats and nonhuman primates) and humans. Therefore, the averaged CMRO₂ values in Fig. 4 should approximately reflect the averaged oxygen utilization rates in the gray matter in the rat brain.

FIG. 3.

CMRO₂ calculation using the experimental data of one single voxel from a representative rat and (A) the linear fitting, or (B) the quadratic fitting approach, respectively, as proposed in method II. CMRO₂ was calculated according to Eq. 17.

FIG. 4.

Comparison of calculated CMRO₂ results from seven measurements based on the simplified methods (method I; method II using the linear fitting or quadratic fitting approach) and the complete model. The last (right) column represents the averaged CMRO₂ value of seven animal measurements. All error bars represent the standard deviation errors.

Validation of two simplified methods

As displayed in Fig. 4, the average CMRO₂ values calculated with both the simplified methods were in good agreement with that calculated using the complete model for the ¹⁷O MR-based CMRO₂ measurements from the same ¹⁷O data (Zhu et al., 2002). Paired t-tests were used to compare the CMRO₂ values calculated with different models, and only the linear fitting approach (method II) was found to be statistically significantly different from the complete model (P = 0.03). However, the difference between the calculated CMRO₂ values using the complete model and the linear fitting approach was small (<10%).

DISCUSSION

Method I, in principle, should provide a more accurate CMRO₂ value as long as the assumptions made in this method hold. However, because the criterion for the evaluation of constant CMRO₂ based on the linear correlation coefficient is fairly sensitive to the noise fluctuations of the ¹⁷O signal of cerebral H₂¹⁷O, variance of the CMRO₂ value from a single voxel calculated by method I is relatively large compared with method II. In contrast, method II, though it requires more approximations, is more tolerant to the noise variance and hence may be more reliable and robust than method I. In addition, the parameter of CBF is not necessary in method II. This can be also attributable to less variance in the fitting using method II.

Eq. 3 indicates that the right side in the equation composes two terms: a constant term 2αF₁CMRO₂ and a time variable term of CBFF₂[C_a(t) − C_v(t)]. If C_a(t) and C_v(t) happen to cancel each other, then Eq. 3 can be readily solved with the solution of C_b(t) = 2αF₁CMRO₂t. Under our experimental conditions, this condition is likely to be satisfied because of the observation of good approximation for the linear fitting of the experimental C_b(t) curves. In general, however, this condition cannot be guaranteed under all circumstances. When C_a(t) and C_v(t) deviate from each other, the C_b(t) curve as a function of t can deviate from a straight line. In this case, the use of quadratic or higher-order polynomial fitting should provide a better accuracy for CMRO₂ calculation.

The fact that the linear fitting approach in method II produces a good approximation for providing CMRO₂ values, which are close to the true CMRO₂ values determined by the complete model under our experimental conditions, implies that the difference between F₂C_a(t) and nC_b(t)/λ terms in Eq. 6, where C_b(t) is usually larger than C_a(t), is small and its influence on the CMRO₂ calculation is negligible. One crucial observation supporting this notion is the limited water permeability across the mitochondria membranes leading to a small constant n (< 1) (Zhu et al., 2002); consequently, a small difference between the two terms. It is possible that this approximation may hold under other physiologic conditions; for example, under a stimulus specific activation in the brain when regional CBF is expected to increase significantly but the CMRO₂ increase is relatively less, so that the rate of metabolically produced H₂¹⁷O in the whole body will not be altered significantly. However, this possibility needs further validations in different species at different physiologic conditions. The success of these validations is important for potential human applications, where any elimination of invasive procedures is highly desirable if not absolutely necessary.

CONCLUSION

The results from this work validate the simplified modeling methods described in this article for calculating CMRO₂. They suggest that the ¹⁷O MR approach with the simplified modeling is capable of determining and imaging CMRO₂ even in the absence of the arterial input function or CBF measurements. The combination of the simplified modeling and ultrahigh-field ¹⁷O NMR may potentially provide a robust, reliable, and completely noninvasive tool for studying the central role of oxidative metabolism in brain function and neurologic diseases in vivo.