Perfusion Confusion: What is Perfusion and what do we Measure?
Richard B. Buxton
A number of investigational tools are available for exploring the working healthy brain and for diagnosing diseases that alter cerebral perfusion, as described in earlier chapters. This chapter takes a step back from the details of the individual methods to look at the broader question of what we mean by the term perfusion. It is useful to approach this question from two directions. The first is from the direction of basic physiology: What is the most useful quantitative meaning that captures the physiologic significance of perfusion? The second is from the more practical direction: What information on the perfusion state of tissue can our imaging methods provide? Ideally, these two questions have the same answer, so that our imaging methods provide a measurement of the most important physiologic variable. That is certainly our goal in the development of quantitative methods for measuring perfusion. However, it is important to note that a more complex signal, which cannot be reliably decomposed into specific physiologic changes, can nevertheless be very useful. An example is the blood oxygenation level dependent signal used to map patterns of activation in the brain. In physiologic terms, this signal is complex, depending on the change of blood flow, blood volume, and oxygen metabolism with neural activation. For this reason, the blood oxygenation level dependent response is not a good quantitative indication of the magnitude of the change in blood flow because of the unknown changes in the other physiologic variables, but it is nevertheless a very useful tool for identifying areas in which neural activity changed in response to a stimulus.
This chapter’s emphasis will be on a comparison of the arterial spin labeling (ASL) and dynamic susceptibility contrast (DSC) methods, focusing on the basic ideas of these methods in relation to what we mean by perfusion. For this reason many of the details of implementation of these methods will be ignored in this general discussion, and the reader is referred to other chapters for more specific discussions of the techniques. The acronym DSC is here used to mean a technique that works by measuring the dynamics of an injected agent that remains intravascular as it passes through the capillary bed. The context of the discussion is in terms of cerebral blood flow (CBF), but the arguments are generally applicable to any organ in which the agent remains intravascular. Buxton1 provides an expanded discussion of these ideas, and more in-depth discussions of the modeling of ASL and DSC data can be found in Buxton et al.2 and Østergaard et al.3,4 respectively, as well as other chapters in this book. More recently, some studies are combining these approaches.5,6 However, rather than reviewing the many ways these methods can be used, the goal in this chapter is to focus on the underlying ideas. We start with the basic physiology.
What is Perfusion?
Anatomy and Physiology
Figure 31.1 shows a microscopic view of the capillary network in a mouse brain. Clearly this detailed anatomy is a critical component of perfusion. The fundamental role of perfusion is to deliver basic metabolic substrates such as glucose and oxygen to each capillary bed so that these essential molecules are brought close enough to the cells where they are needed to allow diffusion to provide the final mechanism of transport into those cells. Yet these images would not be interpreted as images of perfusion because the anatomy of the vessels and the blood flow through those vessels are distinct physiologic quantities. That is, this image is sensitive to cerebral blood volume (CBV) but not CBF. Specifically, CBV is the fraction of the tissue volume occupied by blood vessels, and a typical value for the human brain is 4% (CBV = 0.04). In principle, there does not need to be any fixed relationship between the volume of a set of pipes and the flow through those pipes (think of standard household plumbing). In the brain, blood flow is dynamically controlled by changing the resistance of the vessels, and this leads to a correlation of CBV and CBF, at least in the healthy brain.7 However, this correlation may break down in disease. For example, CBF has been found to be reduced while CBV is elevated in ischemia, described as a reduced perfusion reserve.8,9
In thinking about the image in Figure 31.1, it is tempting to think that what is missing, in terms of getting information on perfusion, is the velocity of the blood within each vessel. This idea is the basis of laser Doppler flowmetry, a widely used technique in animal models to detect changes in blood flow.10 The method works by shining a laser on an exposed section of brain and measuring the reflected light from the erythrocytes. The frequency of the reflected light is shifted in proportion to the cells’ velocity, and the strength of the reflected light is proportional to the blood volume. This provides a measure of how much blood is moving around within the capillary bed of a tissue element. But is this the most useful definition of perfusion?
 FIGURE 31.1. Detailed vascular anatomy in somatosensory cortex of a mouse brain imaged with two photon microscopy. The intricate complexity of the vascular bed is apparent, with the larger network of surface vessels shown on the left and a closer view, projected over the upper cortical layers, of the finer vessels shown on the right. (Image courtesy of Anna Devor.) |
Cerebral Blood Flow is a Measure of the Delivery of Arterial Blood
In fact, the quantitative definition of perfusion is not related to how blood moves around within a capillary bed, but rather to the amount of arterial blood delivered to the capillary bed. CBF is defined as the volume of arterial blood (milliliters) delivered to 100 g of tissue per minute, and a typical human CBF in gray matter is about 60 mL/100 g/min. In imaging applications, localization is in terms of an imaging voxel, so it is easier to define CBF in terms of a volume of tissue. Because the density of brain is near 1 g/mL, the typical human CBF value is 0.6 mL/mL/min. Note that when referenced to tissue volume, the dimensions of CBF are really just inverse time, the dimensions of a rate constant. That is, a CBF of 0.6 min−1 means that for a tissue element of volume V, a volume of arterial blood equal to 0.6 V is delivered in 1 minute. This highlights the primary role CBF plays in determining the delivery of metabolic substrates and the clearance of metabolic products. For example, consider the cerebral metabolic rate of oxygen (CMRO2). If CA is the arterial concentration of O2, then the rate of delivery of O2 to a capillary bed is CBF·CA, and if a fraction E of that delivered O2 is extracted and metabolized, then CMRO2 = E·CBF·CA. In this sense, CBF provides the essential time-dependent quantity that links a blood concentration to a metabolic rate.
 FIGURE 31.2. Schematic illustration of the vasculature to define cerebral blood flow (CBF) and cerebral blood volume (CBV). The left figure shows arterial, capillary, and venous vessels, and CBV is the net fraction of the total tissue volume occupied by blood vessels. The rate of delivery of arterial blood to the capillary bed is CBF. Interestingly, the combination of CBV with the velocity of blood in the vessels within the tissue element is not sufficient to determine CBF. on the right is a possible scenario in which the CBV and the blood velocity are identical in two tissues but the CBF is twice as large in one tissue compared with the other. Note that the transit time through the capillary bed is only half as large in the one with the higher flow. This illustrates the idea that the fundamental dynamic quantity needed to determine CBF is not blood velocity but rather vascular transit time. |
Returning to Figure 31.1, if we were able to measure the volume of each vessel and the velocity of the blood within each vessel, would this be enough information to determine CBF? Surprisingly, the answer is no, as illustrated in Figure 31.2. In this thought experiment, we
consider two simple capillary beds, constructed so the capillaries in each bed have identical volume and blood velocity. The difference is that the first capillary bed is fed by one arteriole and the second is fed by two arterioles. To keep the total capillary volume the same, the capillaries are twice as long in the first capillary bed. If the flow delivered by each arteriole is the same, then by the definition of CBF the first capillary bed has half the CBF of the second bed, despite having the same capillary CBV and blood velocity.
consider two simple capillary beds, constructed so the capillaries in each bed have identical volume and blood velocity. The difference is that the first capillary bed is fed by one arteriole and the second is fed by two arterioles. To keep the total capillary volume the same, the capillaries are twice as long in the first capillary bed. If the flow delivered by each arteriole is the same, then by the definition of CBF the first capillary bed has half the CBF of the second bed, despite having the same capillary CBV and blood velocity.
In short, the definition of CBF emphasizes the delivery of arterial blood to the tissue element and has no fixed relationship with either the amount of blood present in the tissue volume (CBV) or the motion of blood within the tissue element (blood velocity). In practice there are often good correlations of CBV and blood velocity to CBF, in the sense that these variables tend to increase when CBF increases. For that reason, measurements such as laser Doppler flowmetry can be used as a qualitative reflection of CBF in many applications, although one has to be careful in looking at pathology. For example, as noted above, some studies have found evidence for CBV to be increased although CBF is decreased in stroke, a reversal of the expected healthy correlation.
Because CBF depends just on blood delivery, the most robust approaches to quantifying CBF are based on measuring the delivery of an agent carried in arterial blood. The gold standard for measurement of CBF is injected microspheres, which are too big to fit through the capillaries and so remain lodged in the tissue. After injection of a bolus of microspheres, the number of microspheres in each tissue element is directly proportional to CBF. Microspheres cannot be used in human studies, but it is possible to inject agents such as gadolinium-diethylene triamine pentaacetic acid (Gd-DTPA) and measure the delivery and clearance of the agent with imaging. In ASL, nothing is injected, but the magnetization of arterial blood is modified and the delivery of the labeled blood is measured. Both methods borrow from the field of tracer kinetics,11,12,13,14 and the general problem that has to be overcome is that the kinetics of an injected agent do not depend exclusively on CBF, so other physiologic factors have to be included to describe the kinetics.
Volume of Distribution of an Agent
The dynamic tissue concentration curve of an agent delivered by blood flow depends primarily on two quantities: CBF, defined previously, and the volume of distribution of the agent (λ). Also known as the partition coefficient, λ depends on whether the agent crosses the capillary wall or remains in the intravascular space. Specifically, λ is defined as the ratio of the total concentration of the agent in a tissue element to the blood concentration if the agent is allowed to come into equilibrium throughout its volume of distribution (for this reason, λ is also called the tissue–blood partition coefficient of the agent). Two extremes are an intravascular agent, for which λ = CBV, and a freely diffusible agent that fills the tissue space, for which λ = 1. Note that λ is defined as a dimensionless ratio, consistent with our definition of CBF in terms of blood delivery to a voxel volume instead of a mass of tissue. If CBF is expressed in units of milliliters per minute per gram of tissue, λ should be expressed as milliliters per gram of tissue.
Gadolinium contrast agents are intravascular agents in the healthy brain, but they cross into the extravascular space when the blood–brain barrier is damaged. Labeled water is usually considered a freely diffusible tracer, although the blood–brain barrier does slow the equilibration of water with the extravascular space. For this reason, some of the delivered labeled water in an ASL experiment may not be extracted from the blood, a factor that may require correction in some applications.15
How Does Perfusion affect the Dynamics of a Delivered Agent?
Tissue Dynamics of an Agent Delivered by Blood Flow
After injection of an agent, the arterial concentration feeding a tissue element is CA(t) and the total tissue concentration of the agent is CT(t). One can think of CA(t) as the driving function of the system, the arterial input function (sometimes called AIF(t)) and CT(t) as the output. The CBF for the tissue element is denoted as f, and the volume of distribution is λ. The key question is: If we measure CA(t) and CT(t), can we calculate f and λ? What if we only measure CT(t)? The answers depend on how the arterial curve and the values of f and λ affect the tissue curve CT(t).
We can construct a general expression for the tissue curve at time t by adding up the amount of agent delivered up to time t weighted by the probability that the agent is still in the tissue voxel at t. To do this, we define a residue function R(t − t’), the probability that a molecule of the agent that entered the tissue voxel at time t’ is still there at time t. We assume that the underlying physiology is in a steady state (e.g., constant flow throughout the experiment) so that R only depends on the interval t − t’ and not the absolute values of t or t’. Then the number of molecules of the agent delivered during a short interval between t’ and t’ + dt’ is fCA(t’) dt’, and the probability that they are still in the tissue element at time t is R(t − t’). The net tissue concentration at time t is then:
where * indicates convolution as defined by Equation 1. In short, the tissue concentration curve is simply the arterial concentration curve convolved with a local impulse
response function h(t) = f·R(t) (Fig. 31.3). If the arterial concentration is a very sharp bolus, the tissue curve will be simply proportional to h(t).
response function h(t) = f·R(t) (Fig. 31.3). If the arterial concentration is a very sharp bolus, the tissue curve will be simply proportional to h(t).
 FIGURE 31.3. Basic dynamics of the tissue concentration curve. Injection of an agent leads to a dynamic curve of arterial concentration CA(t), and this in turn drives a dynamic tissue concentration curve CT (t). Upper row: The tissue curve is the convolution of the arterial curve with an impulse response function h (t) = f R (t), where f is the local CBF and R (t) is the probability that a molecule of the agent that entered the tissue element at t = 0 is still there at time t. Lower row: The properties of h (t) are: the initial value is f, the area under the curve is λ (the volume of distribution of the agent), and the characteristic temporal width is the mean transit time τ, which is equal to λ/f. Possible shapes for h (t) for an intravascular agent are illustrated with an exponential, appropriate for a single well-mixed compartment, and a single transit time, appropriate for pure plug flow through identical parallel vessels. |
Mean Transit Time and the Central Volume Principle
The full shape of the residue function R(t) is generally not known, but it must satisfy three conditions: (a) R(0) = 1, because the agent must still be there if there has been no time for it to leave; (b) R(t) must monotonically decrease, because the probability that the agent is still there cannot be higher at a later time than an earlier time; and (c) the area under R(t) is the mean transit time τ of a molecule of the agent through the tissue element:
This last relationship is not obvious, but follows from noting that the difference between R(t + dt) and R(t) is the fraction of molecules that are cleared during that interval, so this is the fraction of the delivered agent that has a transit time of t. Mathematically, this means that the distribution of transit times is dR/dt, and Equation 2 results from integrating t times dR/dt to calculate the mean transit time. These three features follow just from the definition of R(t) as the probability that a molecule of the agent that entered the tissue element at t = 0 will still be there at time t.
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