2.2.1.

Static cerebral autoregulation

A first review on static CA was written by Lassen.²⁰ The MAP–CBF curve reported in this work showed a constant CBF for MAP values between 60 and 170 mmHg, indicating a highly active static regulatory system. The work of Lassen had a profound effect on the scientific and medical community, and the MAP–CBF curve presented in his paper was considered an important reference for the upper and lower cutoff values of MAP within which CA was effective. Figure 1 shows the CBF plateau over the MAP range for static autoregulation. The MAP–CBF curve in Lassen’s work was obtained by combining the results from seven human studies with 11 different subject groups, where in each group CBF was measured at a single MAP. The results on different subject groups were mixed regardless of whether the subjects were healthy, diseased, or under medication. Therefore, the curve represented intersubject values under different health conditions, rather than an intrasubject MAP–CBF relationship measured on a cohort of subjects in similar health conditions. As previously noted, this way of extrapolating the CA static curve from a limited number of different subjects can lead to misleading results, even if the subjects are all healthy, because of individual variability and unaccounted for effects of other variables.¹¹ It is possible that the static CA curve is more pressure passive than previously described by Lassen, or, in other words, that the CBF–MAP curve in the autoregulation range is not exactly a flat plateau (as in Fig. 1), but has a slightly positive slope. This is in agreement also with some theoretical models of CA based on a feedback loop.²¹ Moreover, it is nowadays known that the static CA curve is affected by other variables, like the concentrations of carbon dioxide (${\mathrm{CO}}_2$) and ${\mathrm{O}}_2$ in blood. For a theoretical model of the influence of blood gas levels on CA, we refer to the work of Payne et al.²²

Fig. 1

Static autoregulation. The classic static autoregulation curve shows a plateau of cerebral blood flow (CBF) versus mean arterial pressure (MAP) in the MAP range from about 50 mmHg to about 170 mmHg.

In a more recent review of static CA,¹⁸ the authors have reanalyzed 49 case studies of healthy subjects, dividing them into two groups according to whether MAP was increased or decreased. Since CBF was measured in different ways in the 49 studies, the authors considered the percentage change of CBF (relative to baseline) as a function of the percentage change of MAP (relative to baseline). The linear regression coefficient between MAP and CBF was positive and significantly greater during decreases than during increases in MAP, suggesting that autoregulation is more effective against positive changes in MAP than against negative changes in MAP. An asymmetric CBF response to an increase versus a decrease of MAP was also found in dynamic studies involving patients with TBI.^23,24

We conclude this section with a comment about studies on static CA. As it is highlighted in the work of Tzeng and Ainslie,³ static CA studies pose some difficulties, especially if one wants to reproduce the entire static CA curve, because of confounding effects of some covariates of MAP changes. One of the covariates is the partial pressure of ${\mathrm{CO}}_2$, which can be altered after changes in MAP, and, therefore, it can affect CBF independently from MAP. Also, the use of drugs is necessary for studying large changes in MAP because of the baroreflex mechanism that limits the range of MAP induced by mechanical maneuvers. Therefore, one should be aware of the potential side effects of drugs on the tonus of both large and small arteries, which can also alter the CA response. Since static studies imply relatively long time scales (in the order of minutes or longer) to reach steady state values of MAP and CBF, it can be more difficult to disentangle the pure CA response to MAP changes from the vascular effects of drugs and from other physiological mechanisms.

2.2.2.

Dynamic cerebral autoregulation

CA operates at relatively short time scales, as already observed in studies dating back to the 1930s, in which the arteriolar dilation was directly observed on a time scale of seconds following changes in arterial blood pressure (ABP).²⁵ In later studies, it was also confirmed that the mechanism of vessel dilation in response to hypotension occurred at a similar temporal scale, as long as the change in MAP was strong enough to elicit a CA response. For example, in some studies, a MAP change of 5 mmHg is used as a threshold for eliciting a CA response.¹⁸

Aaslid et al.¹⁹ proposed a method for studying dynamic CA that still remains one of the most commonly used. It is based on the transient change in MAP following a thigh cuff pressure release, after a sustained cuff inflation above the systolic pressure for about 2 to 3 min. The sudden cuff release causes a decrease in MAP of about 20 mmHg, which develops over 3 to 5 s postrelease, before the recovery phase starts. Usually, the entire MAP transient from the moment of cuff release to the recovery to steady state unfolds in a time range of 15 to 20 s (see Fig. 2). A trend similar to that of MAP is observed in the arterial blood flow velocity (BFv), which is monitored with TCD, usually in the MCA. We observe that, strictly speaking, BFv is a surrogate of CBF and it can be considered as a reliable substitute of CBF only if the vessel’s diameter does not change. This might be the case in several protocols (including the thigh cuff release protocol), but not in others, as questioned in recent literature.²⁶ Also, we note that the BFv and MAP transients usually follow slightly different dynamics, with a faster recovery of BFv to baseline, at least in the case of a properly functioning CA.

Fig. 2

Dynamic autoregulation. Schematic illustration of the mean arterial pressure (MAP) and cerebral blood flow (CBF) transients induced by the fast release of pneumatic thigh cuffs (at time 0) for the assessment of dynamic autoregulation. MAP quickly drops and CBF passively follows this fast change. Then MAP remains at a value lower than at baseline for about 5 s, and during this time autoregulation mechanisms increase CBF toward its precuff-release value. After this initial $\sim 5\mathrm{s}$ period, baroreflex mechanisms increase MAP, and CBF changes reflect both MAP changes and autoregulatory effects.

Aaslid et al.¹⁹ have shown that different CBF responses to MAP changes are measured for different partial pressures of ${\mathrm{CO}}_2$ in arterial blood (${\mathrm{PaCO}}_2$). Basically, three cases were studied: hypocapnia (induced by hyperventilation), normocapnia (under normal breathing conditions), and hypercapnia (induced by breathing a mixture of 5% ${\mathrm{CO}}_2$ in air). The end-expiratory partial pressures of ${\mathrm{CO}}_2$ (${\mathrm{pCO}}_2$, also referred to as end-tidal ${\mathrm{CO}}_2$) in the three cases were: 22 mmHg (hypocapnia), 37 mmHg (normocapnia), and 47 mmHg (hypercapnia). The lower the level of ${\mathrm{CO}}_2$, the faster was the recovery of BFv to baseline after cuff release. This behavior can be expected by considering the known effect of ${\mathrm{CO}}_2$ content in blood on the MAP–CBF curve measured during static CA studies: the curve becomes more parallel to the horizontal MAP axis (indicating better autoregulation) during hypocapnia, whereas it shows a positive slope (i.e., CBF follows more passively MAP changes) during hypercapnia. An intuitive way to explain these results is the following: during hypocapnia the vascular tone is increased with respect to normocapnia (vessels are constricted) and therefore any CA response to hypotensive maneuvers (like during the cuff release) occurs on a faster time scale because it implies a relaxation of the vascular tone (i.e., a dilation of the blood vessels). On the contrary, during hypercapnia, the vascular tone is relaxed with respect to normocapnia (vessels are dilated), and any CA response to hypotensive maneuvers will occur on a slower time scale because it implies a further relaxation of the tonic level (i.e., a further dilatation of the blood vessels). More precisely, the dynamic CA measurements based on the transient changes of MAP and CBF induced by the thigh cuffs release mostly focus on the initial $\sim 5\mathrm{s}$ after cuff release, when MAP remains low before baroreflex mechanisms raise it back toward its baseline value. It is the rate of CBF recovery during this initial period, when MAP is approximately constant (at a value lower than at baseline), that is most closely indicative of CA. The main features of MAP and CBF transient changes following thigh cuffs release are schematically illustrated in Fig. 2.

The thigh pressure cuff release method is termed a time domain method, and under the assumption that CA can be approximated to be a linear process (at least for small change of MAP²⁷), one can use typical methods of linear systems to analyze the data (see Sec. 5). Other maneuvers to study dynamic CA have been proposed on the basis of induced MAP oscillations; one such method is the lower body negative pressure protocol.²⁸ These methods, which are based on inducing oscillations at different frequencies, are termed frequency-domain methods, and can also be studied with typical methods of linear systems (see Sec. 5).

We conclude this section by listing a few advantages of dynamic versus static CA studies. First, the fast MAP–CBF temporal dynamics implied in dynamic CA studies make the confounding effects of some covariates of MAP changes less important than in static CA studies. Second, the study of MAP and CBF transients, like those measured in the thigh cuff release protocol, has elicited a new concept of CA, in which different “levels” of dynamic CA can be distinguished, based on the particular features of the temporal trends of CBF (see Sec. 5). In other words, CA is viewed as a process characterized by different levels of effectiveness and not as an all-or-nothing mechanism (as it was conceived in previous static CA studies). Third, dynamic studies are easier to perform, do not require invasive maneuvers (like drug injections), and involve shorter acquisition times, facilitating longitudinal studies. One of the major problems of dynamic studies of CA is that they cannot be used to study the vasoconstriction response to hypertension. The reason is that it is difficult to induce hypertension without using drugs that act on a longer temporal scale.

3.1.1.

Fick principle

A general approach to measuring CBF is based on a mass balance relationship, known as the Fick principle, applied to a physiologically inert substance $x$ (i.e., a substance that is not metabolized in the brain) in the blood stream. In its simplest form, the Fick principle states that the amount of substance $x$ ($\mathrm{d}{Q}_x$; units: ${\mathrm{mol}}_x$) that is accumulated in (or dissipated from) a brain tissue volume $V_T$ over the infinitesimal time interval $\mathrm{d}t$ about time $t$ is given by the difference between the amounts of the substance delivered to (by arterial inflow) and removed from (by venous outflow) the tissue volume between times $t$ and $t+\mathrm{d}t$. We observe that the tissue volume $V_T$ is intended to include the blood that perfuses it, and that the blood volume fraction, or cerebral blood volume (CBV), is assumed to stay constant. With these definitions and under these conditions, the Fick principle is written as follows:

Fig. 3

Three basic approaches to the measurement of cerebral blood flow. (a) The Fick principle, (b) the central volume principle, and (c) the Doppler effect or autocorrelation methods. (a) A global CBF measurement is based on recording time traces of the arterial and venous blood concentrations (${[x]}_a$ and ${[x]}_v$, respectively) of a diffusible and physiologically inert intravascular tracer $x$ over a time $\mathrm{\Delta }t$ that is sufficiently long to achieve equilibrium in the blood–brain tracer diffusion (in the case of the Kety–Schmidt method, the intravascular tracer was nitrous oxide (${\mathrm{N}}_2\mathrm{O}$), breathed continuously by the subject [see Sec. 3.2.1]). (b) A regional CBF measurement in a volume of interest $V_T$ is based on the measurement of the temporal dynamics of the arterial and tissue concentration ($C_a$ and $C_T$, respectively) of an intravascular bolus ($H$ and $A$ represent the peak value and the total area under the curve of the temporal trace of $C_T$ (this approach is the basis for a number of nuclear medicine [Sec. 3.3], X-ray [Sec. 3.4], and MRI techniques [Sec. 3.5]). (c) The Doppler effect applies to ultrasound or optical waves that interrogate the brain tissue at a certain frequency $f$, and results in a frequency shift ($\mathrm{\Delta }f$) and in a decay rate for the normalized intensity autocorrelation function ($g_2$) that are directly related to the speed of blood flow (these methods are employed by Doppler ultrasound [Sec. 3.6.1], laser Doppler flowmetry [Sec. 3.8.1], and diffuse correlation spectroscopy [Sec. 3.8.2]).

3.1.2.

Central volume principle

A case in which a measurement of the tracer concentration in venous blood is not required [unlike the case of Eq. (4)] is when the tracer concentration is directly measured in the brain volume of interest (VOI). If the tracer thoroughly mixes with blood, its concentration in brain tissue ($C_T$, units: $\mathrm{mol}/{\mathrm{g}}_{\text{tissue}}$, where, again, we consider brain tissue as a whole, including the vascular space) at a given time $t$ is given by the integration of its concentration in arterial blood ($C_a$, units: $\mathrm{mol}/{\mathrm{ml}}_{\text{blood}}$) at a previous time τ (ranging from $-\infty$ to $t$) times the probability that the tracer transit time in the VOI is longer than $t-\tau$ [$R(t-\tau )$, dimensionless]:

Now, let us introduce the probability density function, $h(t)$, for the tracer transit time within the tissue VOI ($V_T$). This probability density function leads to the definition of the tracer mean transit time (MTT) within such tissue region of interest

Nondiffusible tracer: In the case of a nondiffusible tracer, i.e., a tracer that remains confined to the vascular space, its MTT transit [given by Eq. (6)] coincides with the blood transit time, and, therefore, we denote it with a subscript “$B$” (${\mathrm{MTT}}_B$). Now, let us consider the blood volume that is contained within the tissue VOI, $V_T$, and let us denote it with $V_B$ (this blood volume has units of ml, and should not be confused with CBV, which has units of ${\mathrm{ml}}_{\text{blood}}/{\mathrm{ml}}_{\text{tissue}}$ or ${\mathrm{ml}}_{\text{blood}}/100{\mathrm{g}}_{\text{tissue}}$ and actually represents a blood volume fraction). Such blood volume $V_B$ can be decomposed into infinitesimal elements [$\mathrm{d}{V}_B(t)$] that are associated with a given blood transit time $t$ or, more accurately, with a transit time within the interval ($t$, $t+\mathrm{d}t$). This infinitesimal element $\mathrm{d}{V}_B(t)$ is given by the product of two factors: (1) the rate at which blood volume elements with transit times between $t$ and $t+\mathrm{d}t$ enter the tissue region of interest (which is the same as the rate at which they exit it because of the assumed stationarity of CBF and CBV), which is given by $F_{V_T}h(t)\mathrm{d}t$ (where $F_{V_T}$ represents the amount of blood flowing through the tissue volume $V_T$, and has units of ${\mathrm{ml}}_{\text{blood}}/\min$); (2) the time spent by such blood volume elements in the tissue VOI, which is t by definition. Therefore, one can write^29,30

Diffusible tracer: Equation (10) may be generalized to the case of a diffusible tracer. In this case, $V_B$ in Eq. (9) is replaced by the equilibrium volume of distribution of the tracer, i.e., the blood volume that contains the same amount of tracer that is in the tissue volume $V_T$ at equilibrium.³¹ We stress again that “tissue” is intended here to include the blood in the vasculature.³² The numerator on the right-hand-side of Eq. (10) then becomes the volume of distribution of the tracer per unit mass of tissue, which is referred to as the tissue-blood partition coefficient, $\lambda$, also defined as the concentration of the diffusible tracer in tissue (at equilibrium) divided by the concentration of the tracer in blood. If the same units for the concentrations in tissue and blood are used, $\lambda$ is dimensionless (or expressed in ${\mathrm{ml}}_{\text{blood}}/{\mathrm{ml}}_{\text{tissue}}$ or $g_{\text{blood}}/g_{\text{tissue}}$), whereas it takes units of $\mathrm{ml}/\mathrm{g}$ (i.e., ${\mathrm{ml}}_{\text{blood}}/g_{\text{tissue}}$) if the tissue and blood concentrations of the indicator are expressed per unit tissue mass and per unit blood volume, respectively. Both conventions are used in the literature, but here we use the latter convention, so that the units of $\lambda$ and CBV are the same. For example, the brain–blood partition coefficient for water is $0.90\mathrm{ml}/\mathrm{g}$,³³ which can be interpreted by saying that 0.90 ml of blood contain the same amount of water as 1 g of brain tissue. On the basis of the above definitions, the central volume principle equation for a diffusible tracer is

3.1.3.

Doppler effect and intensity fluctuations

The interaction of a probing wave, be it an ultrasound pressure wave or an optical electromagnetic wave, with moving red blood cells in the blood stream results in a frequency shift and in intensity fluctuations of the detected wave. The frequency shift is a manifestation of the Doppler effect; the intensity fluctuations can be characterized by the field or intensity autocorrelation functions and their rate of decay. The Doppler effect, which results in spectral line broadening, and the intensity fluctuations are different aspects of the same phenomenon. In fact, the intensity power spectrum and the field autocorrelation function are related by a Fourier transformation, known as the Wiener–Khinchin theorem. BF velocity, and therefore CBF, is directly related to the Doppler shift ($\mathrm{\Delta }f$) and directly related to the rate of decay of the normalized intensity autocorrelation function ($g_2$). This approach to the measurement of BF is schematically illustrated in Fig. 3(c). It is important to observe that Doppler and autocorrelation methods do not really measure CBF, but they rather yield measures of the speed of BF, either in large vessels (Doppler ultrasound, Sec. 3.6.1) or in the microcirculation [laser Doppler flowmetry (LDF), Sec. 3.8.1; diffuse correlation spectroscopy (DCS), Sec. 3.8.2]. Because CBF is intended to represent the rate of inflow of arterial blood into the capillary bed rather than the speed of BF within brain tissue, the data provided by Doppler and autocorrelation methods need to be carefully interpreted and are often taken to provide relative measures of CBF changes.

3.2.1.

Kety–Schmidt arteriovenous difference method

A seminal paper by Kety and Schmidt,³⁴ which set the stage for the development of a variety of CBF measurements in the human brain,³⁵ considered nitrous oxide (${\mathrm{N}}_2\mathrm{O}$) as a freely diffusible intravascular tracer. It was administered by inhalation, and the dynamics of ${[{\mathrm{N}}_2\mathrm{O}]}_a$ and ${[{\mathrm{N}}_2\mathrm{O}]}_v$ were measured in blood drawn by femoral artery puncture and from a needle in the right internal jugular vein, respectively. The dynamic measurements were performed over a time interval $\mathrm{\Delta }t=10\min$ following the beginning of ${\mathrm{N}}_2\mathrm{O}$ inhalation, at which time a steady state was reached, such that ${[{\mathrm{N}}_2\mathrm{O}]}_a={[{\mathrm{N}}_2\mathrm{O}]}_v$. This steady state carries no information about CBF, but the time required to reach it is inversely related to CBF. Specifically, the measurement of CBF is based on the amount of ${\mathrm{N}}_2\mathrm{O}$ delivered to the brain, per unit brain mass, over the entire time $\mathrm{\Delta }t$ ($\mathrm{\Delta }{q}_x$), and the total arteriovenous difference integrated over time $\mathrm{\Delta }t$, as given by Eq. (4), written again here with the generic tracer $x$ replaced by ${\mathrm{N}}_2\mathrm{O}$:^34,36

Following its introduction in 1945, the Kety–Schmidt method quickly became a standard approach for quantitative measurements of the global CBF. However, its application is invasive and cumbersome, requiring inhalation of nitrous oxide, puncture or catheterization of the carotid or femoral artery, and accurate measurements of ${\mathrm{N}}_2\mathrm{O}$ concentrations in blood samples. Furthermore, it relies on some assumptions (jugular venous blood representing brain venous drainage, reaching a steady state of brain saturation with ${\mathrm{N}}_2\mathrm{O}$ within 10 to 15 min, and so on) that may not always be accurate. Finally, the Kety–Schmidt method allows for only global CBF measurements, but regional CBF measurements are highly desirable in a number of research and clinical situations. For these reasons, other techniques have been considered to improve upon the intravascular tracer approach of Kety and Schmidt. For example, the introduction of radioactive tracers (such as ${}^{133}\mathrm{Xe}$ and ${}^{85}\mathrm{Kr}$), in conjunction with measurements of their clearance curve (i.e., their concentration decay versus time), leads to measurements of regional CBF (see Sec. 3.3.1).

3.2.2.

Jugular thermodilution

As an alternative to intravascular tracers that diffuse into the brain, one may inject a cold fluid miscible with blood (typically saline) that introduces a thermal perturbation to perform a direct measurement of BF velocity within a large blood vessel that drains the brain, such as the internal jugular vein. This method allows for continuous or repeated measurements (in the cases of continuous infusion or cold bolus input, respectively) and is referred to as jugular thermodilution.³⁷ While this method does not require arterial puncture or catheterization (but it requires jugular vein catheterization) it provides only a local measurement of jugular venous flow (in units of ${\mathrm{ml}}_{\text{blood}}/\min$). In the case of a continuous infusion, the jugular blood flow (JBF: $\mathrm{ml}/\min$) is expressed as follows in terms of the injection rate of the indicator fluid ($F_I$: $\mathrm{ml}/\min$), the specific heat of blood and indicator [$c_B$, $c_I$: $\mathrm{cal}/(g°\mathrm{C})$], and the densities of blood and indicator (${\rho }_B$, ${\rho }_I$: $\mathrm{g}/\mathrm{ml}$)

While jugular thermodilution allows for continuous measurements of BF, it does require injection of the fluid indicator, so that the ability of the patient to handle the fluid load poses a limitation to the duration of flow monitoring.

3.3.1.

Intra-arterial injection of a radioactive inert gas (¹³³Xe or ⁸⁵Kr)

A common intraoperative method to measure CBF is based on the intra-arterial injection of a radioactive bolus that emits $\gamma$-rays (photons) or $\beta$ particles (electrons) and the detection of the diffusible tracer’s clearance curve in brain tissue regions by scintillation detectors,^31,39 an Anger camera,⁴⁰ or a Geiger–Müller tube.^41,42 Two radioactive isotopes, ${}^{133}\mathrm{Xe}$ and ${}^{85}\mathrm{Kr}$, the former more commonly used than the latter, decay by emitting $\gamma$-rays and $\beta$ particles, respectively, that are detected to yield regional radioisotopes clearance curves. This approach relies on the same concepts as the Kety–Schmidt method (Sec. 3.2.1), but allows for regional measurements of CBF, and measures the clearance curve of the tracer continuously rather than at a number of time points. Three methods have been proposed to analyze the clearance curve to generate CBF measurements:

3.3.2.

Single-photon emission computed tomography

Single-photon emission computed tomography (SPECT) uses the injection of a delivery compound, such as hexamethylpropyleneamine oxime (HMPAO) or ethyl cysteinate dimer (ECD), labeled with the radioisotope technetium-99m (${}^{99\mathrm{m}}\mathrm{Tc}$), the most common of the gamma ray producing radionuclides.^47,48 The compound is injected intravenously, passes the blood brain barrier, and is metabolized and retained intracellularly.^49,50 The SPECT system measures the spatial distribution of the radiotracer in the cerebral tissue and its temporal evolution. Rather than absolute CBF, SPECT measurements yield brain perfusion indices^51,52 that reflect CBF as well as the radiotracer’s kinetics.

3.3.3.

Positron emission tomography

CBF imaging with PET, which achieves a spatial resolution on the order of $1{\mathrm{cm}}^3$, uses ${}^{15}\mathrm{O}$-labeled oxygen (${{}^{15}\mathrm{O}}_2$), carbon dioxide (${\mathrm{C}}^{15}{\mathrm{O}}_2$), or water (${\mathrm{H}}_2^{}{}^{15}\mathrm{O}$) as a radioactive diffusible contrast agent. After intravenous injection or inhalation of the contrast agent, the local instantaneous tissue radiotracer concentration (expressed as $c_T^{({}^{15}\mathrm{O})}(t)$, in units of $\mathrm{cps}/\mathrm{g}$, where cps are counts-per-second) is written as follows by the autoradiographic method, developed from the one-compartment Kety–Schmidt method:^53,54

In addition to CBF, PET can measure CBV, cerebral metabolic rate of oxygen or glucose, and oxygen extraction fraction with rapid sequential scans due to the short half-life of the tracers.^56,57 PET is the most expensive of the tomographic modalities for measuring CBF, making it more favorable as a research tool, to study physiology or validate other perfusion techniques, than as a clinical tool. However, recent advances in detector technology and greater availability of radiopharmaceuticals have improved the desirability of PET for clinical use.⁵⁸

3.4.1.

Xenon-enhanced computed tomography

Stable xenon is a radiodense diffusible contrast agent that freely crosses the blood–brain barrier. The Xe-CT procedure begins with a baseline CT scan. Then, sequential CT scans are performed during inhalation of $\sim 30\%$ xenon. Baseline values are subtracted from each voxel of the xenon enhanced CT images to determine the concentration of xenon in brain tissue and a curve is computed to describe tracer accumulation over time within each voxel. Additionally, the arterial input function (AIF) is estimated by measuring end-tidal xenon concentration. Similar to the PET case of Eq. (18), the fact that the tracer concentration can be measured directly in brain tissue releases the Kety–Schmidt requirement of venous blood measurements, and the equation for Xe concentration in tissue is⁵⁹

Patient motion is a limitation of Xe-CT that can cause artifacts in images. Clinical safety of xenon was assessed in a widespread study, and the authors concluded that Xe-CT has low risk for adverse events.⁶⁰ CBF was quantified in a study of patients with aneurysmal SAH with a bedside Xe-CT scanner supporting the potential utility of mobile CT scanners in neurointensive care units.⁶¹ For further clinical applications of CBF measurements in the neuro critical care unit, see Sec. 6.2.

3.4.2.

Perfusion computed tomography

The bolus tracking methodology for cerebral perfusion computed tomography (PCT) uses the central volume principle (see Sec. 3.1.2), which relates the mean blood transit time (${\mathrm{MTT}}_B$) through a region of interest, the CBV (units: ${\mathrm{ml}}_{\text{blood}}/g_{\text{tissue}}$), and CBF.⁶² The central volume principle is expressed by Eq. (10), which is repeated here

Although ${\mathrm{MTT}}_B$, CBV, and CBF can be obtained quantitatively, some questions remain about the accuracy and reproducibility of PCT. A correct determination of the AIF is also a challenge that makes absolute quantification of PCT parameters difficult.⁶⁴ Motion artifacts are also a concern and may further limit the accuracy of PCT. For these reasons, relative rather than absolute maps of CBF are often used in PCT.

The biggest impact of PCT has been in imaging of stroke.⁶⁵ Ratios of CBF in brain regions may be used as a quantitative method for interpatient CBF comparisons based on PCT measurements.⁶³ Computed tomography angiography may be performed in the same imaging session as PCT for anatomic investigation of vessels.^66–68 See Sec. 6.2 for additional clinical applications of perfusion CT.

3.5.1.

Dynamic susceptibility contrast, or perfusion-weighted MRI

Similar to the case of the Kety–Schmidt method (Sec. 3.2.1), MRI measurements of CBF with dynamic susceptibility contrast (DSC-MRI), also referred to as perfusion-weighted imaging or bolus-tracking MRI, require the injection (intravenous in this case) of a bolus of contrast agent (Gadolinium-based). A key difference with respect to the Kety–Schmidt method, however, is that the MR contrast agents used for DSC-MRI do not diffuse through the blood–brain barrier and remain in the vascular space. The source of contrast for DSC-MRI measurements is the magnetic field gradient between the capillaries and the surrounding tissue, which results from the presence of the intravascular paramagnetic contrast agent. This field gradient leads to a spin dephasing of the water in the bloodstream (i.e., a decrease in the transverse relaxation time $T_2$ and a corresponding drop in the MR signal) that depends on the local CBF.⁶⁹ Because such spin dephasing occurs during the relatively short transit time of the contrast bolus (in the order of several seconds), the fast technique of echo-planar imaging is ideally suited for DSC-MRI.

The fact that DSC-MRI uses nondiffusible tracers means that the tracer bolus stays in the vasculature, and its residence time in a given brain VOI, described by the probability density function $h(t)$ introduced in Sec. 3.1.2, coincides with the blood transit time in that volume. As expressed by Eq. (7), the probability that a tracer introduced in the brain VOI at time 0 is still present in the VOI at a later time $t$ is given by $R(t)=1-\underset{0}{\overset{t}{\int }}h(\tau )\mathrm{d}\tau$. Therefore, the tracer concentration (per unit tissue volume) in the VOI at any time $t$ [$C_T(t)$] is equal to the temporal convolution between $R(t)$ and the rate at which the tracer enters the VOI per unit tissue volume, $\mathrm{CBF}\times C_a(t)$, where $C_a(t)$ is the arterial tracer concentration (also referred to as the AIF). Based on these observations, the CBF in the VOI satisfies the following equation ⁷⁰

Alternatively, instead of an absolute measurement of CBF, one may perform qualitative CBF measurements on the basis of summary parameters related to the negative peak of the MR signal during the bolus transit in the brain VOI (time to peak, bolus arrival time, peak width, peak area, and so on).⁷¹

As of today, perfusion-weighted MRI remains mostly an investigational tool that has yet to achieve widespread clinical use. However, the practical and technical limitations that have limited the clinical use of cerebral perfusion assessment with MRI can be overcome to lead to a potentially powerful clinical tool.^74,75

3.5.2.

Arterial spin labeling

Contrary to DSC-MRI, presented in the previous section, the MRI technique of arterial spin labeling (ASL) does not use an extrinsic contrast agent and is therefore completely noninvasive. ASL is based on magnetically labeling water in major arteries that perfuse the brain. Such labeling is achieved by inverting the blood magnetization with a combination of a radiofrequency pulse and a field gradient in the direction of BF, thus defining an inversion plane at the base of the brain. Therefore, ASL uses a freely diffusible tracer in the form of magnetically labeled water, which alters the brain tissue magnetization by exchanges with tissue water. The basic concept for ASL is illustrated in Fig. 4, which shows the labeling and imaging planes. In its original realization, ASL uses a series of radiofrequency pulses to continuously saturate blood water spins, and this technique is referred to as continuous ASL (CASL).^76–78 Because the change in tissue magnetization due to the magnetically labeled water is dependent upon BF, CBF measurements with continuous ASL are based on the brain tissue magnetization before the application of the inversion pulse ($M_T^0$) and after steady-state conditions are reached ($M_T^{ss}<M_T^0$). The expression for CBF used by continuous ASL is as follows⁷⁶

Fig. 4

Arterial spin labeling (ASL). Schematic representation of the basic approach for blood flow measurement with arterial spin labeling MRI. The water in the arterial inflow to the brain is magnetically tagged in the labeling plane. In the imaging plane, the change in tissue magnetization is directly related to blood flow and yields a measure of CBF.

The use of a single inversion radiofrequency pulse (typically $\sim 10\mathrm{ms}$ in duration) and a labeling region that is closer to the imaging slice result in pulsed ASL (PASL) [for a review, see Ref. 79]. PASL addresses two problems of CASL: magnetization transfer effects that alter the tissue magnetization independent of flow effects (due to the long application of the series of radiofrequency pulses), and the loss of spin labels as blood travels from the labeling plane to the imaging slice.⁷¹ PASL is implemented by a number of different techniques with creative acronyms such as EPISTAR (echo planar imaging and signal targeting with alternating radiofrequency),⁸⁰ PICORE (proximal inversion with a control for off-resonance effects),⁸¹ FAIR (flow-sensitive alternating inversion recovery),⁸² and QUIPSS II (quantitative imaging of perfusion using a single subtraction, version II).⁸³ These various PASL techniques implement various schemes for the acquisition of the tag and control images, but the concept is always the same: measuring local CBF by quantifying the decrease in tissue magnetization caused by the inflow of labeled arterial blood that carries a negative magnetization (i.e., magnetization that is flipped by 180 deg).

3.6.1.

Transcranial Doppler ultrasound

TCD ultrasound provides noninvasive measurements of the BF velocity (units: cm/s) in the basal brain arteries (MCA, proximal anterior cerebral arteries and posterior cerebral arteries),⁸⁴ most commonly in the left or right MCAs. The ultrasound transducer is placed in a subject-specific ultrasonic window in the temporal region above the zygomatic arch, also called the transtemporal window, which is shown in Fig. 5(a).⁸⁵ In Fig. 5(b), it is shown how a TCD ultrasound probe placed at this transtemporal window can measure flow velocity in the MCA.⁸⁵ The basic principle is that an ultrasound wave at frequency $f$ experiences a Doppler frequency shift ($\mathrm{\Delta }f$) as it gets reflected by the red blood cells that are moving at speed $v_B$. The relationship between $v_B$ and $\mathrm{\Delta }f$ is as follows:⁸⁶

Fig. 5

Transcranial Doppler ultrasound. (a) The location of the transtemporal window on the left side of a human subject. (b) A transcranial Doppler (TCD) probe placed against the left transtemporal window has access to the left MCA for measuring blood flow velocity. Reproduced from Ref. 85 with permission.

A further refinement of TCD is transcranial color Doppler sonography, which combines 2D (B-mode) images of the brain (through the intact temporal plate) with a color-coded representation of BF with Doppler ultrasound.⁸⁷

3.6.2.

Transit-time ultrasonic flowmetry

An ultrasound technique to measure flow in an isolated blood vessel consists of placing it within an ultrasonic perivascular flow probe. This technique is applicable only invasively in a surgical setting. The perivascular flow probe features two ultrasound transducers, placed longitudinally on one side of the blood vessel, and an acoustic reflector on the opposite side of the blood vessel. The probe measures the transit times of ultrasound from one transducer to the other when traveling upstream ($t_{12}$) and downstream ($t_{21}$) to the BF. The difference between the upstream and downstream transit times $\mathrm{\Delta }t=t_{12}-t_{12}>0$ is directly proportional to the volumetric BF in the blood vessel (VBF, units: $\mathrm{ml}/\mathrm{s}$), as opposed to the BF velocity ($\mathrm{cm}/\mathrm{s}$) measured by Doppler ultrasound⁸⁸

3.8.1.

Laser Doppler flowmetry

LDF is a technique for continuous monitoring of relative changes in CBF.^97,98 LDF samples a superficial volume of $1–2{\mathrm{mm}}^3$ in tissue⁹¹ so that, while it is conceptually noninvasive, it requires removal of scalp and skull for access to the human brain, thus rendering it an invasive technique for human studies. The LDF instrument consists of a fiber-optic probe and uses monochromatic light in the range of 670 to 810 nm. The movement of red blood cells induces a Doppler shift in the light that is proportional to the velocity and number of red blood cells.⁹⁹ A photodetector measures the scattered light that has been Doppler shifted and these measurements are translated to a time-varying voltage that is proportional to changes in CBF. LDF is the Fourier counterpart to DCS, covered in the next section, although these two techniques do not tend to quantitatively agree due to differences in modeling assumptions.¹⁰⁰

LDF may suffer from artifacts from patient motion and probe displacement.⁹⁰ Clinically, LDF has been used for assessing autoregulation, ${\mathrm{CO}}_2$ reactivity, and responses to therapeutic interventions.^91,101 LDF has also been used in studies in changes of CBF such as for research on ischemic stroke, cortical spreading depression, and neurovascular coupling.¹⁰²

3.8.2.

Diffuse correlation spectroscopy

DCS measures relative changes in BF over much greater tissue volumes than LDF, allowing for noninvasive measurements of the microvasculature of superficial brain cortical regions.¹⁰³ It is a safe, noninvasive technique that can be used continuously at the bedside. Because it uses near-infrared (NIR) light, it has similar resolution and penetration depth as near-infrared spectroscopy (NIRS) methods covered in Secs. 3.8.3 and 3.8.4.

A typical DCS system consists of a single-wavelength, NIR, continuous-wave light source with a long coherence length ($\sim 10\mathrm{m}$), multimode source fibers with diameters of $\sim 1\mathrm{mm}$, single-mode optical fibers for light detection, single-photon counting avalanche photodiodes, and an autocorrelator. The DCS probe is placed against the surface of the subject’s head, and NIR light propagates into the brain. The light is scattered by static scatterers as well as by moving red blood cells. The light then reaches the tissue surface and is measured by the optical detectors. The autocorrelator measures the intensity autocorrelation function of the detected photons. A schematic of a typical DCS system is shown in Fig. 7(a).¹⁰⁴ The normalized intensity auto-correlation function ($g_2$) is related to the normalized electric field correlation function ($g_1$) by the Siegert relation. With this relation, the measurements from the autocorrelator can be fit to solutions of the correlation diffusion equation¹⁰⁵ in order to determine values for $D_B$, the effective Brownian diffusion coefficient of the moving scatterers, and $\beta$, a parameter determined by the optical set up of the experiment. From these fitted values, a blood flow index (BFI [units: ${\mathrm{cm}}^2/\mathrm{s}$]) can be determined as

Fig. 7

Diffuse correlation spectroscopy (DCS). (a) Schematic of typical DCS instrumentation that consists of a long-coherence length source coupled to a multimode fiber for light delivery to the tissue, photon-counting detector(s), and an auto-correlator board that computes the intensity of the autocorrelation function, $g_2(\tau )$, based on photon arrival times (b), (c) Sample $g_2(\tau )$ curves obtained over the frontal cortex in a subject under baseline conditions (black) and under hypercapnia (3% inspired carbon dioxide, gray). The increased decay rate of $g_2(\tau )$ during hypercapnia reflects the increase in CBF by vasodilation. Reproduced from Ref. 104 with permission of the Society of Photo Optical Instrumentation Engineers (SPIE).

The DCS system probes the skin, skull, and CSF, in addition to the tissue region of actual interest, the brain. Similar to the NIRS methods, the effects of superficial, extracerebral layers can be suppressed with special techniques.¹⁰⁶ The DCS measurements of relative changes in BF have been convincingly validated, but quantification of absolute CBF remains a challenge.¹⁰⁴

3.8.3.

Near-infrared spectroscopy

NIRS is a noninvasive technique whose optical probes are placed on the subject’s scalp, through hair if present, for measurements on the cerebral cortex. NIRS has been applied to functional brain studies^107–109 as well as a variety of pilot clinical studies¹¹⁰ aimed at brain monitoring during anesthesia,¹¹¹ after brain injury,^112,113 or during electroconvulsive therapy.¹¹⁴ The spatial resolution is on the order of 1 cm, and the maximum tissue depth that can be probed is 2-3 cm. NIRS is applicable at the bedside and can provide continuous monitoring. It is a diffuse optical method that is sensitive to the tissue concentrations of oxyhemoglobin, [${\mathrm{HbO}}_2$], and deoxyhemoglobin, [Hb], due to the differential absorption of NIR light of the oxygenated and deoxygenated heme group. The sum of [${\mathrm{HbO}}_2$] and [Hb] yields the total hemoglobin concentration, [HbT], and the ratio $[{\mathrm{HbO}}_2]/[\mathrm{HbT}]$ yields the oxygen saturation of hemoglobin in brain tissue, ${\mathrm{StO}}_2$, also called ${\mathrm{ScO}}_2$ to specify cerebral tissue. Because hemoglobin is confined to the red blood cells and the vast majority of oxygen in blood is bound to hemoglobin, [HbT] is proportional to CBV, whereas ${\mathrm{ScO}}_2$ and the concentration difference $[\mathrm{HbD}]=[{\mathrm{HbO}}_2]-[\mathrm{Hb}]$ are indicators of the balance between oxygen supply and oxygen demand in tissue. Such a balance is determined by a number of factors, including BF, blood volume, metabolic rate of oxygen, capillary density, and hematocrit. Even though [HbD] and ${\mathrm{ScO}}_2$, which provide a measure of intravascular oxygenation, are not solely determined by CBF, they are nevertheless linked to CBF. In particular, [HbD] was found to correlate with CBF in piglets,^115,116 in infants undergoing surgery,¹¹⁷ and in critically ill patients.¹¹⁸ Of course, [HbD] and ${\mathrm{ScO}}_2$ can provide only relative measurements of CBF, and they rely on the assumption that other physiological factors, besides CBF, do not contribute significantly to their values. Furthermore, the details of the [HbD] and ${\mathrm{ScO}}_2$ temporal dynamics may not accurately represent the CBF dynamics.

Absolute measurements of CBF have been performed with tracer tracking techniques. The two main tracers have been oxyhemoglobin (endogenous) and indocyanine green (ICG) (exogenous). In the oxyhemoglobin method, the inspired oxygen fraction is first lowered and then sharply increased so that the resulting sudden burst in oxyhemoglobin concentration in blood acts as a bolus.^119,120 For the first few seconds of measurement with NIRS, the venous outflow of the oxygen bolus is assumed to be zero. In other words, NIRS measurements are performed at times ($t$) that are shorter than the bolus transit time in the measured volume. Under these conditions, the Fick principle yields Eq. (5), or Eq. (22), with a tracer resident probability $R(t)=1$, leading to the following expression for CBF in terms of the change in [${\mathrm{HbO}}_2$], the intravascular tracer, that is observed in a time interval $\mathrm{\Delta }t$ that is shorter than MTT¹¹⁹

The ICG method requires an intravenous bolus injection of the contrast agent.¹²¹ Under physiological conditions, ICG binds to plasma proteins (mostly albumin) and does not cross the blood brain barrier, so that it remains confined to the vascular space. Therefore, NIRS measurements can be analyzed with a similar approach to the oxyhemoglobin method.¹²² In these early approaches, the arterial concentration of ICG was measured invasively by arterial catheterization. Another approach is based on the central volume principle, or on the general form of Eq. (5) without requiring that NIRS measurements be performed before any venous ICG outflow takes place.¹²³ The tissue concentration of ICG is measured with NIRS and the ICG arterial concentration is measured with a noninvasive, dye-sensitive oximeter. Deconvolution between arterial and tissue ICG concentrations is performed in order to recover the resident probability function, $R(t)$ [see Eq. (5)], from which ${\mathrm{MTT}}_B$, CBF, and CBV can be computed.¹²³ Instead of doing multiple injections, the NIRS technique can be used to compute a baseline value for CBF in absolute units from a single bolus injection. This baseline value can then be combined with a relative and continuous measure of CBF, such as that provided by [HbD].¹²⁴ In another approach, DCS (see Sec. 3.8.2) was used to obtain quantitative relative CBF changes in combination with ICG and time-resolved NIRS measurements.¹²⁵

One of the challenges with NIRS for measuring CBF is that measurements are contaminated by extracerebral BF.¹²⁶ In fact, with typical methods to measure changes in oxyhemoglobin concentration (modified Beer–Lambert law), $\mathrm{\Delta }[{\mathrm{HbO}}_2]$ in Eq. (27) represents a weighted average of the concentration changes occurring in extracerebral and cerebral tissue. Two-layer diffusion models, short and long source–detector separations, and other approaches for separating extracerebral BF contributions are an active area of research in the field.

3.8.4.

Coherent hemodynamics spectroscopy

Coherent hemodynamics spectroscopy (CHS) is a technique that we have recently developed to translate hemodynamic cerebral measurements into physiological quantities, including CBF.¹²⁷ This technique can be applied to NIRS data, in which case it has the same spatial resolution and reaches the same tissue depth as described in the NIRS section. CHS has the potential to provide noninvasive and continuous bedside monitoring of absolute CBF.¹²⁸

CHS features two alternative methods for measuring physiological parameters. In the first method, operating in the frequency-domain, techniques such as paced breathing or cyclic thigh cuff inflations induce oscillations in MAP, which in turn result in oscillations of cerebral oxy- and deoxyhemoglobin concentrations (indicated with $O$ and $D$, respectively, in CHS) that are highly coherent with MAP. The phase and amplitude of ratios of $O$ and $D$ oscillations are combined to generate CHS spectra that are fit with a hemodynamic model,¹²⁷ whose fitting parameters relate to microvascular flow and volume.¹²⁹ In the second method, operating in the time domain, thigh cuffs are rapidly released after a 2 min arterial occlusion, thereby inducing a transient decrease in MAP and corresponding transient changes in cerebral oxy- and deoxyhemoglobin concentrations, which are fitted with the time-domain version of the hemodynamic model.¹³⁰ Both methods yield several parameters, two of which can be used to compute an absolute value of baseline CBF. These two parameters are the mean blood transit time in the capillaries,$t^{(c)}$, and the capillary blood volume fraction, ${\mathcal{F}}^{(c)}{\mathrm{CBV}}_0^{(c)}/{\mathrm{CBV}}_0$, where ${\mathcal{F}}^{(c)}$ is the Fåhraeus factor that takes into account the hematocrit reduction in the capillaries, ${\mathrm{CBV}}_0$ is the baseline CBV (expressed here in ${\mathrm{ml}}_{\text{blood}}/{\mathrm{ml}}_{\text{tissue}}$), and ${\mathrm{CBV}}_0^{(c)}$ is the capillary blood volume (ml of capillary blood per ml of tissue). From the central volume principle [Eq. (10)] applied to the capillary compartment, the baseline cerebral blood flow, ${\mathrm{CBF}}_0$, can be written as follows:¹²⁸

In addition to absolute measurements of ${\mathrm{CBF}}_0$, CHS can also measure time traces of relative CBF changes, indicated with lower-case notation, $\mathrm{cbf}(t)$ (i.e., $\mathrm{cbf}(t)=[\mathrm{CBF}(t)-{\mathrm{CBF}}_0]/{\mathrm{CBF}}_0$).¹³¹ Because perturbations of CBF and metabolic rate of oxygen cannot be distinguished in NIRS measurements,¹³² we assume a constant metabolic rate of oxygen. Under this condition, the frequency domain version of our hemodynamic model yields the following expression for the Fourier transform of cbf, which can be obtained experimentally from the Fourier transform of the $O$ and $D$ time traces:¹³¹

The first clinical application of CHS revealed a significantly longer cerebral capillary transit time in hemodialysis patients ($1.1\pm 0.2\mathrm{s}$) with respect to healthy subjects ($0.5\pm 0.1\mathrm{s}$),¹³³ and this longer capillary transit time corresponds to a lower baseline ${\mathrm{CBF}}_0$.¹²⁸

5.2.1.

Two representative model-based methods for cerebral autoregulation

The Ursino model²¹ was developed to describe the dynamics of intracranial vascular beds, the production and absorption of CSF and the dynamics of ICP. In this complex mathematical model, the intracranial dynamics are described by electrical analogs. More specifically, the system is divided into three parts, where the first two represent the precapillary vascular bed (proximal pial arteries and small arterioles) and the last one represents the capillary and postcapillary parts of the vascular bed. Each part is defined by its own specific conductance, compliance, ICP, and so on. The autoregulation mechanism is introduced in the two arterial compartments by means of two formally identical equations describing the tension of the vascular walls (due to muscular action) that depend, among other parameters, on one activation factor. The activation factor can be positive (when describing vasoconstriction) or negative (when describing vasodilation). The temporal dynamics of the activation factors in the two arterial compartments are introduced by two differential equations that depend on two key parameters of the CA mechanism: a time constant and a gain. The original model featured 21 equations and 40 free parameters. In the original study,²¹ the model was applied to simulate the ICP pulse wave. In a later study the model was also applied to simulate the dynamics of CBF, of the radius of proximal arteries and arterioles and also of ICP, to a step decrease in MAP.¹⁷⁹ For these simulated data, the time constants describing the CA mechanism were fixed at 20 s (proximal arteries) and 10 s (arterioles). The model could reproduce typical observed behavior of CBF during the step decrease in MAP. The model has been used also for fitting experimental data, in particular the dynamic of ICP after saline injection in the craniospinal space, which is a typical test applied to very sick subjects.¹⁷¹ Of all the model parameters, only four were varied for reproducing the experimental data, including the gain and time constant of the autoregulation mechanism in the distal arterioles. The time constant retrieved by the fitting in eighteen subjects analyzed showed a wide individual variance and covered the range 2-380 s (average value 53 s and standard deviation 108 s).

A similar (yet more simplified) model, based on a flow dependent feedback mechanism was proposed by Payne and Tarassenko.¹⁷⁵ The arterial compartment is divided into two sections, one characterized by a fixed resistance (reflecting the behavior of larger arteries) and the other characterized by a variable resistance and capacitance (reflecting the behavior of smaller arteries and arterioles). The capillary compartment is represented by a fixed resistance and capacitance, while the venous compartment is represented by two resistances, one lumped together with the capillary resistance, reflecting the behavior of smaller veins, and the other reflecting larger veins. The venous compartment is also characterized by a variable capacitance. In the model, only one (constant) value of ICP is assumed. The model uses differential equations for the dynamics of arterial and venous blood volumes, and the autoregulation mechanism is introduced by a state variable that has a role similar to the activation factor in the Ursino model. The state variable is used to adjust the arterial compliance and its dynamic behavior is specified by a temporal differential equation that (similarly to the Ursino model) depends on two parameters: the gain and the time constant of the autoregulation mechanism. In order to derive the impulse response functions (IRFs) of CBF and CBFv due to a change in MAP, the model equations are linearized around the baseline values. Strictly speaking, the model is used to derive the transfer functions (TFs) of the system, which are expressed in terms of the Laplace transforms of the IRFs. In particular the TF for CBFv depends on five parameters, including the time constant of the autoregulatory mechanism and another parameter that depends on the gain. The TF was fitted to experimental data by using only four (unknown) parameters and fixing the fifth parameter sequentially in several ranges, in order to study the sensitivity of the fitting procedure on that parameter.

5.2.2.

Cerebral autoregulation assessment with coherent hemodynamics spectroscopy

As previously mentioned in Sec. 3.8.4, CHS is based on cerebral hemodynamic oscillations that are highly coherent with MAP oscillations induced by a forcing mechanism (e.g., paced breathing, cyclic thigh cuff inflation, etc.). When cerebral hemodynamic oscillations are properly induced, the cerebral concentrations of oxy- and deoxyhemoglobin become highly coherent and feature a stable phase and amplitude, which depend on the frequency of oscillation. The mathematical model¹²⁷ provides a rather straightforward analytical relationship between CBF changes and the changes in oxy- and deoxyhemoglobin concentrations at different frequencies of the forcing mechanism (this is the frequency-domain version of the method). It also provides a relationship between CBF and oxy- and deoxyhemoglobin concentrations during transients and arbitrary temporal evolution of CBF (this is the time domain version of the method). The idea behind this relatively simple mathematical model is to treat the entire capillary and venous vascular compartments as low pass filters in a circulation system in which the input variables are the changes in blood volume, BF, and oxygen consumption, and the output variables are the changes in oxy-, deoxy-, and total hemoglobin concentrations. By curve fitting the temporal trends of oxy- and deoxyhemoglobin changes (in either the time or frequency domain) we derive six physiological parameters.^129,131 Among these parameters, we have already mentioned the capillary transit time and the capillary blood volume, which can be used to derive an absolute measure of CBF (see Sec. 3.8.4). Another important parameter derived with CHS is the autoregulation cutoff frequency, $f_c^{(\mathit{A}\mathit{R})}$, which is a measure of the upper limit for the frequency of MAP oscillations that can be regulated by CA. In fact, in the literature, the CA process has been often modeled as a high pass filter having MAP as input and CBF as output. In our mathematical model, the CA process is described by a resistor-capacitor (RC) high pass filter, with normalized CBV changes as input and normalized CBF changes as output. The filter is characterized by a cutoff frequency, which is usually in the order 0.01-0.10 Hz. The larger this parameter, the more efficient the CA process is.

A validation test performed on healthy human subjects during hyperventilation-induced hypocapnia has shown that the autoregulation cutoff frequency is significantly greater during hypocapnia ($0.034\pm 0.005\mathrm{Hz}$) with respect to normocapnia ($0.017\pm 0.002\mathrm{Hz}$).¹⁶⁶ This result of enhanced autoregulation during hypocapnia is in agreement with studies based on different methods of grading CA.¹⁹ We place this study in Table 3 although we note that this study was model-based, while the other NIRS studies in the table were based on direct methods and data analysis, as described in Secs. 5.1 and 5.3.

Other hemodynamic models based on sets of unknown physiological parameters have been proposed for the interpretation of NIRS data. However, these models do not address directly the problem of estimating a metric for CA, and most of them are based on complex systems of partial differential equations that make them computationally intensive (for a list of these models we refer to Ref. 180).

5.3.1.

Methods for static cerebral autoregulation

For data analysis of static CA, essentially three methods (with some variations) have been adopted:

The last method has been used mainly in those studies in which MAP (and therefore CBF) was changed only one time with respect to its baseline values. These methods are rather straightforward and they are usable to define CA either in a binary way (i.e., intact or impaired), or to define different levels of CA.

In method (a), a threshold slope was defined, usually in the range $0.5\text{-}3\text{-}4\%\mathrm{CBF}/\mathrm{mmHg}$, for the binary classification, so that any slope greater than the threshold slope was considered indicative of impaired CA.¹¹ In other studies, a multiple linear regression was applied with CBF as dependent variable and MAP, partial pressure of oxygen (${\mathrm{pO}}_2$) and partial pressure of carbon dioxide (${\mathrm{pCO}}_2$) as regressors. In fact, it is well known that CBF is affected independently by other covariates of MAP. In static methods of CA assessment that need longer measurement times, the effect of these regressors on CBF changes is more prominent than for dynamic CA methods (for a list of these studies, we refer to¹¹).

According to method (b), a Pearson correlation coefficient ($r$) can similarly be used to discriminate between intact (low correlation) and impaired (high correlation) CA. The reasoning is that if CA is impaired CBF changes are more passive to MAP changes, and a high value of the correlation coefficient is expected. In some studies $r=0.5$ was chosen as a threshold to discriminate between the two CA conditions.¹⁸¹

In method (c), only two values of MAP were used, and an index of static CA could be defined based on the slope of changes in CVR. This index has been variably named SARI (static autoregulation index)¹¹ or SCA (static CA) index.¹⁸² Despite different names, the two indices share the same definition: $\mathrm{SCA}=\mathrm{SARI}=\%\mathrm{CVR}/\%\mathrm{MAP}$, where %CVR is the change in CVR (in percent of its baseline value) and %MAP is the change in MAP (expressed also in percent of its baseline value). We remind that if $\mathrm{CPP}\cong \mathrm{MAP}$, CVR is defined by

5.3.2.

Methods for dynamic cerebral autoregulation

The methods for assessment of dynamic CA have been devised mainly to take advantage of the continuous monitoring of MAP and CBF (as, for example, with TCD). With these methods, the definition of different levels of CA has become more straightforward than for static methods. Most of the methods assume that the input-output system MAP–CBF (where MAP is the input and CBF is the output) is linear and time invariant. This can be a reasonable assumption for small changes of MAP and CBF, as for example during baseline data, but it should be tested for maneuvers in which MAP changes by relatively large amounts (e.g., thigh cuff release method). These methods have been developed both in the time domain and frequency domain.

In the frequency domain, one test of linearity between two signals is given by the coherence function. High values of the coherence function indicate a linear relationship between two signals. On the contrary, a low coherence value can be interpreted in several ways as being indicative of: (1) a nonlinear relationship, (2) a poor signal-to-noise ratio (SNR), (3) a lack of relationship between the two signals, or (4) a multiple inputs-single output relationship.¹⁸³ Frequency domain methods have been applied to describe phase and amplitude relationships between MAP and CBF either during baseline data or during induced MAP oscillations at a single or multiple frequencies. In the time domain, most of the methods are used to describe the transient of MAP and CBF during a thigh cuff release maneuver.

One simple method for differentiating different levels of CA was introduced by Aaslid et al.¹⁹ during a thigh cuff release maneuver. The method takes advantage of the continuous monitoring of MAP and CBF by defining a time-dependent CVR. The authors demonstrated that in the first few seconds after the cuff release, CVR showed a linear trend as a function of time, in which the slope was sensitive to the level of CA. We note that an ideally working CA mechanism would instantly compensate for the change of MAP, and CVR as a function of time would be a vertical line (infinite slope). However, since the negative feedback loop that is rebalancing CBF through the adjustment of CVR (at least according to the metabolic theory of CA), needs a finite operation time, the slope is also finite. Similarly to the static SARI or SCA indices, the authors defined a rate of autoregulation $\mathrm{RoR}=\%\mathrm{CVR}/\%\mathrm{MAP}/\mathrm{\Delta }t$, where $\mathrm{\Delta }t$ is the time interval where the linear trend of CVR can be assumed. The authors found different RoRs according to the level of ${\mathrm{CO}}_2$ present in the blood, with the steepest slope during hypocapnia, and the shallower slope during hypercapnia.

The same group proposed a more analytical model of CA based on a state space model of the ordinary differential equation (ODE) governing the temporal relationship between MAP and CBF during an ideal step decrease of MAP.¹⁸² Usually, the state variables in a state space model depend on the order of ODE, and for the case of the method proposed by Tiecks et al., the authors chose two state variables.¹⁸² The model depends also on three parameters: a damping factor ($D$), an autoregulation gain ($K$) and a time constant ($T$). Arbitrarily, the authors chose ten different triplets for the three parameters and associated one single index (the autoregulation index, ARI) to describe the level of autoregulation. For each ARI, the state space model is solved and a CBF curve as a function of time is calculated for the transient after a thigh cuff release. An $\mathrm{ARI}=0$ corresponds to no CBF response (which follows passively MAP), and an $\mathrm{ARI}=9$ is characterized by the fastest recovery time for CBF to its baseline value. This “library” of solutions for CBF after thigh cuff release is used for defining the ARI of a measured CBF curve as a function of time by least square fitting.

Instead of studying transients of MAP and CBF during a sudden change of ABP, like in the thigh cuff maneuver, other researchers have induced hemodynamic oscillations at a single^161,184,185 or multiple frequencies¹⁸⁶ and studied the phase, amplitude, and coherence relationships between MAP and CBF. Typical maneuvers adopted to induce oscillations are paced breathing,^161,184 squatting,¹⁸⁵ or lower body negative pressure.²⁸ The systematic study of phase, gain and coherence between MAP (input) and CBF (output) at multiple frequencies (ideally a continuous set of frequencies) is called TFA. Here we note that time and frequency domain analysis (or TFA) of CA are related by a Fourier transformation, and while the former aims at extracting the IRF, the latter aims at extracting the TF between MAP (input) and CBF (output). Pioneering studies using TFA date back to the early 1990s,¹⁸⁷ but here we summarize the results of the study carried out by Zhang et al.,¹⁸³ which have been reproduced in multiple independent studies.

The interesting aspect of the study by Zhang et al. is that TFA was applied to 6 min of rest data taken on healthy subjects, and TFA was applied to estimate the TF of the system MAP (input) – CBF (output).¹⁸³ The TF was estimated by calculating the ratio of the cross spectrum between MAP and CBF, and the auto spectrum of MAP. The coherence function was estimated by the ratio of the absolute value of the cross spectrum between MAP and CBF squared and the product of the autospectrum of MAP and CBF. The group average carried out on ten subjects showed that the amplitude of TF (also called gain) increased in the range of 0.07 to 0.2 Hz, and that the coherence was larger than 0.5 for frequencies larger than 0.3 Hz. The phase of the TF decreased in the range of 0.07 to 0.5 Hz from about 1 radian to 0. These results are typical of a high-pass filter relationship between MAP and CBF, at least in the range where coherence was significant. The authors also discussed the possible interpretations of low coherence values. The IRF, calculated by an inverse Fourier transform of the TF, agreed well with the IRF measured during a thigh cuff release. Strictly speaking, with this maneuver one can measure the step response function; however, this function is related to the IRF by a simple time derivative. Measurements of IRF and TF from baseline data would be ideal for CA studies, since the influence of covariates of MAP changes on CBF changes are minimal. However, the main challenge of using spontaneous oscillations of MAP and CBF is to achieve a significant SNR, or, in other words, to select those spontaneous fluctuations of CBF that are not due to random noise but are caused by similar fluctuations of MAP. For this reason, Panerai et al. proposed the method of coherent averaging^188,189 before applying TFA analysis. This method consists of finding a certain number of MAP fluctuations (about 30 in their original studies) where the peak-to-peak change of MAP is larger than 2% of the baseline value. The coherent averaging of MAP fluctuations and of the corresponding CBF oscillations is done by defining a point of synchronism, which is the point of maximum derivative in the rising part of each MAP oscillation.

Other models of data analysis that assume a linear relationship between MAP and CBF are the autoregressive moving average processes (ARMA). An ARMA model is a process in which a system is described by a single input-single output relationship (as for MAP and CBF in CA) and the output at a certain time is obtained as a linear combination of output values at previous times and of input values at previous and current times. We can say that an ARMA model is applied to stochastic data in an attempt to find the law governing the relationship between input and output. The mathematical form of an ARMA model resembles a discretized differential equation, with the difference that in this case the coefficients must be determined from the data and are not given a priori. Usually, one aims to find the ARMA model with the least number of coefficients that can reliably reproduce the data. A comparison of different ARMA models was performed to study the relationship between MAP and CBF in neonates.¹⁹⁰

Nonlinear models have also been used to describe CA. An interesting comparison between linear and nonlinear models is reported in Ref. 191. Measurements of MAP and CBF were recorded on 47 healthy subjects during two periods of rest (5 min each) and during a sudden thigh cuff deflation protocol. Different methods of predicting the IRF of the CBF response were used: (a) TFA, (b) Tiecks model with ARI, and (c) the method of Volterra–Wiener kernels by considering either the linear term, or both the linear term and the quadratic term. The IRFs were obtained by the different methods during one of the two rest periods (training dataset) and were then tested on the other rest period and during the thigh cuff release maneuver (test datasets) to reproduce the CBF temporal trends. This was done by using the convolution of the IRFs with MAP temporal trends. During the training data set, the correlation between the measured and predicted CBF temporal trends was significant for all the models, but the Volterra kernel method with the quadratic term yielded the highest correlation. However, when the same IRFs were applied to the other rest period and to the thigh cuff release maneuver (test data set) to reproduce the CBF temporal trend, the linear models yielded similar performances that were comparable to those obtained for the training dataset, while the nonlinear model yielded the poorest correlation. The authors explained this result by the lack of randomness in the input data (MAP spontaneous oscillations), which is required by the Volterra–Wiener method and that makes the method too sensitive to the training dataset.

Finally, other methods have released the assumption of linearity and/or stationarity between MAP and CBF (implicit in the TFA method), and require more complex mathematical constructs like Volterra kernels,^191,192 continuous complex wavelet,¹⁹³ the Hilbert transform, and empirical mode decomposition.¹⁹⁴