Quantification of myocardial perfusion by cardiovascular magnetic resonance
© Jerosch-Herold; licensee BioMed Central Ltd. 2010
Received: 3 September 2010
Accepted: 8 October 2010
Published: 8 October 2010
The potential of contrast-enhanced cardiovascular magnetic resonance (CMR) for a quantitative assessment of myocardial perfusion has been explored for more than a decade now, with encouraging results from comparisons with accepted "gold standards", such as microspheres used in the physiology laboratory. This has generated an increasing interest in the requirements and methodological approaches for the non-invasive quantification of myocardial blood flow by CMR. This review provides a synopsis of the current status of the field, and introduces the reader to the technical aspects of perfusion quantification by CMR. The field has reached a stage, where quantification of myocardial perfusion is no longer a claim exclusive to nuclear imaging techniques. CMR may in fact offer important advantages like the absence of ionizing radiation, high spatial resolution, and an unmatched versatility to combine the interrogation of the perfusion status with a comprehensive tissue characterization. Further progress will depend on successful dissemination of the techniques for perfusion quantification among the CMR community.
The title of this review, "perfusion quantification" refers to the approaches used with cardiovascular magnetic resonance (CMR) to assess or measure myocardial blood flow from the contrast enhancement observed during the first pass of a contrast agent bolus. This technique is often referred to by the name of "first pass" imaging, because the first pass of a contrast agent represents the phase of contrast enhancement most sensitive to changes in blood flow, be it from disease, pharmacological intervention, or exercise. Another approach, not requiring exogenous contrast, and referred to as arterial spin labeling, can also be employed to assess myocardial blood flow. Although a promising technique that is not burdened by concerns about contrast administration in patients with poor renal function, it still remains largely confined to experimental studies, challenged by poor contrast-to-noise, and arguably only practically relevant for cardiac studies at 3 Tesla, or higher field strengths. Another promising technique for assessing myocardial perfusion is based on blood-oxygen-level dependent (BOLD) myocardial imaging[2, 3]. This type of endogeneous contrast mechanism is dependent on a combination of factors such as blood flow, but also oxygen extraction fraction, with the latter normally declining with increasing myocardial blood flow. BOLD imaging does not provide as direct an assessment of myocardial blood flow, as contrast-enhanced perfusion imaging. This review will focus on the quantification of myocardial perfusion by the use of contrast-enhanced techniques, which in the future may include novel contrast agents such as hyper-polarized media, but at the present time is mostly performed by the administration of gadolinium chelates as contrast agent. For the purpose of this review article, we define the quantification of myocardial perfusion as methods or approaches to determine the tissue blood flow in the heart muscle. A recent comprehensive review with considerable detail on the technical aspects of myocardial perfusion imaging can be found in.
The quest to quantify myocardial perfusion has been largely motivated by the desire to obtain quantitative, observer-independent, and reproducible measures of the myocardial perfusion status. Whether a quantitative approach improves the accuracy of myocardial perfusion imaging, such as for the detection of coronary artery disease, remains controversial, with circumstantial evidence pointing to the benefits of a quantitative analysis, versus a qualitative interpretation in cases of multi-vessel disease. But in the research realm, there is by now a long line of evidence proving that the measurement of myocardial perfusion leads to new insights in coronary physiology and the etiology of cardiac diseases.
CMR Acquisition Methods
In the context of perfusion quantification, there are two aspects of myocardial perfusion studies that warrant some discussion about the methods for dynamic image acquisition. First of all, one assumes that the observed contrast enhancement is proportional to the change of contrast concentration in the tissue. For example, if one were to use a pulse sequence giving mixed T1/T2* contrast, with T1 effects prevailing at lower contrast concentrations, and T2* effects dominating at higher concentrations, then this would give rise to signal increases at lower concentrations, which would be confounded by signal loss at higher concentrations. It is preferable to have a sequence technique that makes one contrast mechanism (e.g. T1) prevail. T1-weighted imaging is predominantly used for myocardial perfusions studies, not the least because the vascular volume and also the distribution volume of Gd-chelates is sufficiently large to yield appreciable signal intensity changes in myocardial tissue. The short echo times inherent in weighting the signal towards T1 are advantageous for minimizing the effects of myocardial motion and flow, as one would otherwise have increasing signal attenuation due to motion and flow when the echo time (TE) becomes longer.
A second aspect relates to the nature of contrast enhancement in the blood pool of the ventricular cavity or the proximal aorta. Intuitively, it is obvious that myocardial contrast enhancement is driven by arterial enhancement. In fact, we will see that the myocardial contrast enhancement can be considered as a linear response to the arterial contrast enhancement, which implies that myocardial contrast enhancement can never proceed at a rate faster than for the arterial contrast enhancement. For this reason it is important to measure the contrast enhancement in the blood pool as reference for the analysis of the myocardial contrast enhancement. Ideally, the arterial contrast enhancement would be measured as close as possible to the myocardial region under consideration, but in practice one can only expect to measure enhancement in the blood pool at a location in the left ventricle, or in the proximal aorta. Either way, the pulse sequence should not be overly sensitive to blood flow in the great vessels or the ventricular cavities, nor should it suppress the signal from flowing blood. For example, pulse sequences with long echo-trains after each radio-frequency excitation would suppress signal from blood flow in the ventricular cavities and should be avoided for quantitative perfusion studies.
The contrast enhancement in the blood pool has a non-linear or sub-linear dependence on contrast concentration, with the nonlinearity, or contrast-enhancement saturation, becoming more pronounced as contrast concentration increases [7, 8]. Eventually, the signal intensity would even start to decrease above a certain contrast level, due to T2* effects. Signal saturation effects need to be avoided or corrected, if one wants to accurately measure the arterial input of contrast. Methods for quantifying myocardial perfusion use the arterial input as reference. Any systematic underestimate of arterial contrast enhancement results in an overestimate of myocardial perfusion, i.e. relative to the arterial contrast enhancement, myocardial contrast enhancement appears to be higher than is truly the case, and assuming negligible myocardial signal saturation.
Possible solutions to avoid arterial signal saturation are: a) the use of lower contrast dosages, which avoid the saturation effect, albeit at the price of reduced contrast-to-noise in the myocardium; b) employing dual contrast sequences[8, 9]; c) using a dual contrast bolus protocol, and d) correcting retrospectively for signal saturation, e.g. by using calibration curves[11, 12] (see section below on Signal Intensity and Contrast Concentration). Briefly, the dual-contrast sequences include low-resolution dynamic imaging of the enhancement in the ventricular cavity, in addition to high(er) resolution imaging of myocardial enhancement. With a 2D dual-contrast sequence one images several slices (e.g. in short axis view) during a heart beat to capture the myocardial contrast enhancement with strong T1 weighting, and during the same heart beat also acquires a low resolution image of the contrast enhancement in the ventricular cavity, with lower T1-weighting than for the myocardium, to avoid saturation at higher contrast concentrations. The low resolution blood pool image has a low T1 weighting, because fewer phase-encodings are carried out between magnetization preparation and read-out of central phase-encodings, resulting in a shorter delay after inversion (TI) or saturation. The short delay, and resulting low-T1 weighting, improves the linearity of the signal-intensity vs. R1 relationship at higher contrast concentrations.
The dual bolus approach involves giving a low dosage contrast bolus to characterize the arterial input of contrast, followed by a higher dosage bolus to image the myocardial contrast enhancement. The two bolus dosages are in a pre-determined ratio (e.g. 1:10) that is then used to scale the arterial input function (AIF) from the low-dosage bolus to analyze the myocardial contrast enhancement with the rescaled and time-shifted AIF.
A further important aspect is the administration of contrast: If the contrast is injected slowly, then the observed myocardial contrast enhancement is bottlenecked by the slow arterial enhancement, and myocardial enhancement becomes relatively insensitive to blood flow. To achieve good sensitivity to myocardial blood flow, the contrast agent should be injected as a bolus, at a rate somewhere between 3-5 ml/s, in particular for measurements of the hyperemic response. In other words, as the blood flow increases, the requirement for a bolus injection become more demanding, while for resting flows, injection rates as low as 1 ml/s are acceptable. There is no stringent reason why the injection rates for rest and hyperemic flow measurements need to be the same, although in practice this is generally the case.
The signal-intensity-based analysis of myocardial perfusion can be confounded by spatial variation of the coil sensitivity profiles, e.g. with phased-array receive coils, and also by B1 inhomogeneity effects, e.g. due to dielectric effects, in particular at magnetic field strengths of 3 Tesla or higher. Body RF coils, used almost exclusively for RF excitation, are designed to achieve excellent B1 homogeneity, albeit under assumed ideal conditions. B1 inhomogeneity over a region with the dimensions of the heart can be brought about by two conditions: a) the electromagnetic wavelength of the RF excitation can approach the field of view dimensions at higher field strengths, and in tissue the wavelength can be even shorter, because tissue has a comparatively high relative dielectric constant or permittivity (e.g. εr~70 at 100 MHz for muscle tissue); and b) the dielectric properties within the anatomical region being imaged can be highly heterogeneous, thereby giving rise to dielectric resonances, a form of constructive B1 interference effects that can be set up in a dielectric "cavity". B1 inhomogeneity can contribute to signal intensity variation over the heart, but equally important, it also causes unintended changes for the magnetization preparation. For the latter, it is useful to realize that a magnetization preparation with a nominal inversion pulse that deviates from the ideal 180° flip angle can be confounded with a more rapid relaxation recovery, and similar observations apply to saturation pulses. In other words, in the presence of B1 inhomogeneity, the signal intensity variations can mimic perfusion defects. Currently, the most common solution for myocardial perfusion imaging at ≥ 3T, is the use of adiabatic, i.e. B1-insensitive, RF inversion or saturation pulses, or composite pulses. Adiabatic pulses increase the SAR burden, because the B1 amplitude intrinsically has to be considerably higher than normal (to meet the adiabaticity condition), and the pulses also have 5 to 10-fold longer durations, both of which increase the deposited RF energy (proportional to B 1 2 ·Δt, over a time interval Δt of the pulse, during which B1 is approximately constant). Composite pulses represent a practical compromise with lower SAR burden than adiabatic pulses, but they still compensate for B1 inhomogeneity or flip-angle variation.
The spatial variation of receive coil sensitivity, although also a potential source of misinterpretation of contrast enhancement, can be corrected for in practice by mapping out, or estimating the coil sensitivy profile(s). Mapping of the coil sensitivity profiles is a prerequisite for reconstruction of images acquired with parallel imaging acceleration, which intrinsically also takes care of the coil-sensitivity variation in the reconstructed images. Otherwise, coil sensitivity has to be estimated from one of the pre-contrast images, and preferably images acquired with proton-density weighting, and insensitive to in-flow in the ventricular cavity ("bright blood effect"). Recent implementations of myocardial perfusion sequences include such proton-density acquisitions as the first 1-2 images in a dynamic imaging scan, by leaving out the magnetization preparation, and using a small flip-angle for the image read-out. To reduce the impact of noise, the data of different myocardial segments within one slice can be fitted to a sinusoidal function. A sinusoidal variation would be a first-order approximation, assuming a locally linear spatial variation of the signal intensity, and a circular short axis cross-section of the left ventricle (LV). A common way to correct for intrinsic, spatial signal intensity variations is to divide the myocardial signal by its pre-contrast value. This corrects for variations for the signal within the myocardium, due to receive-coil inhomogeneity, but for the blood pool a similar correction is only feasible if pre-contrast-signal is not enhanced by flow, and predominantly proton-density weighted. It is also implicitly assumed that the proton density of myocardium and blood are about the same.
Pulse Sequence Techniques and Perfusion Quantification
Techniques for image acquisition in myocardial perfusion studies have spanned, in approximate order of image acquisition speed, the range from (spoiled) gradient echo imaging with cartesian, or radial k-space trajectories, through gradient echo imaging with steady state free precession[17, 18], to echo-planar[19–22], and spiral techniques. The spoiled gradient-echo technique is the slowest, but least susceptible to artifacts from off-resonance shifts (i.e. field inhomogeneities), T2* and susceptibility effects, and flow and motion. Compared to an acquisition with steady state free precession, the spoiled gradient echo technique suffers from nearly two-times lower signal-to-noise, but steady-state free precession techniques are currently not a viable option for perfusion imaging at 3 Tesla or higher field strengths, because of image artifacts. Echo-planar techniques are mostly used in a hybrid form, where the echo-train length is limited to approximately 3-6 echoes, depending on field strength and T2*. Similarly, the spiral technique is not used in a single-shot read-out mode for contrast-enhanced myocardial perfusion imaging, but instead interleaved spirals are acquired to reduce the effective T2* weighting of the signal. Whether a particular technique is suitable for quantitative myocardial perfusion imaging depends on details of its implementation on any particular scanner platform, field strength, and contrast agent. It is conceivable that a particular technique may work with a standard gadolinium-chelate, but that with a iron-based contrast agent, T2* effects cause too much susceptibility artifacts.
Signal Intensity and Contrast Concentration
where R1n denotes the native T1 relaxation rate constant (i.e. in the absence of any contrast agent) and [Gd], the concentration of Gd contrast in the voxel. The relaxivity of most Gd-chelate agents is unchanged as it transitions from blood into tissue, with albumin-binding agents being one notable exception.
where S 0 is the equilibrium signal, TR the repetition time per phase encoding, R1 the T1 relaxation rate (R1 = 1/T1), TD, the delay between magnetization preparation and start of FLASH read-out, E = exp(-R1·TR), and a = E·cos(α), with α denoting the flip angle. The signal intensity of the central-k-space echoes determines the overall contrast characteristics of an image, although contrast enhancement in smaller structures, is also weighted by the T1-weighting of higher-frequency k-space echoes. The effects of the modulation of the phase-encoding amplitudes by an inversion, or saturation recovery have not been systematically investigated, but could impact negatively on the quantification of blood flow, e.g. in thinned LV walls.
With pre-contrast T1 measurements, knowledge of the sequence parameters, and numerical simulation, it is possible to correct the signal saturation, by generating calibration curves to replace the measured signal intensity, or percent contrast enhancement, by their corresponding values in the absence of signal saturation[11, 12]. Alternatively, one can measure the signal intensity calibration curves, but this turns into a more tedious approach, and with a change of sequence parameters it may be necessary to regenerate a calibration curve. Details on methods for saturation correction can be found in[11, 12]. If the saturation effect is neglected, then the contrast enhancement in the myocardium will appear to be larger relative to the arterial contrast enhancement, and myocardial perfusion is overestimated. The overestimate of blood flow can be almost proportional to the peak saturation effect, i.e. a 30% reduction of the peak signal intensity of the arterial input due to saturation can cause an overestimate of myocardial blood flow of similar relative magnitude.
Signal Intensity Artifacts
Although it may go without saying that signal intensity curves for myocardial regions of interest should represent only the underlying tissue properties, this may be difficult to achieve for the endocardial layer, because the regional signal intensity average may include an admixture from the ventricular blood pool. Several factors can contribute to this, all too often, subtle artifact: the endocardial border definition may be poor, which can lead to the inadvertent inclusion of some blood pool region, or some voxels may only partially be filled by myocardial tissue over the full slice thickness, which generally exceeds by a factor of 4-5 the in-plane voxel dimensions ("partial volume effect"). These partial-volume effects are also referred to as "spillover" from the blood pool, a term particularly prevalent in nuclear medicine. In nuclear medicine spillover correction was incorporated by adding to any model for the myocardial contrast enhancement, a "spillover" term, which essentially amounts to a scaled (and time-shifted) arterial input function, with the scaling factor (and time shift) as variable parameter(s). Such an approach has also been tested successfully in MRI studies of myocardial perfusion. By virtue of the higher spatial resolution of MRI, it may be preferable in the future to further optimize the image acquisition to avoid spillover, rather than trying to estimate any signal admixture from the blood pool, in addition to the tissue perfusion parameters.
A persistently vexing problem in myocardial perfusion studies has been the appearance of dark rim artifacts at the endocardial border, when a contrast bolus first enters the ventricular cavity[27, 28]. Several potential mechanisms have been proposed to explain this phenomenon, with arguably the two most likely causes being, the effect of Gibbs ringing near a sharp and large signal intensity jump, and magnetic susceptibility effects, which cause intra-voxel dephasing at the endocardial border. At the stage of post-processing, the choices to deal with this artifact are limited. One can avoid the region with the dark rim when the endocardial contour is drawn, by pulling the contour back towards the epicardial border, but then runs the risk of missing a perfusion defect limited to the subendocardial layer, which is the layer most vulnerable to ischemia. Alternatively, one can leave out the data points in the signal intensity curve where the signal intensity drops significantly below the pre-contrast baseline level. This is clearly also an unsatisfactory solution, and generally requires that the signal intensity curve is constrained by a model to estimate blood flow. With a semi-quantitative, model-independent analysis, such as determination of the signal-intensity up-slope, the missing data-points may render it impossible to estimate the semi-quantitative perfusion parameter.
Units of Perfusion Measurements
Measurements of arterial and myocardial contrast enhancement do not require any calibration in terms of absolute units for contrast enhancement. It is sufficient that they are measured on the same linear scale, irrespective of whether that scale has absolute or arbitrary units. The time points for signal intensity readings need to be recorded in absolute units, for example seconds, relative to a common reference for all images, such as the start of the image acquisition. One can then quantify the contrast enhancement as fraction of the arterial contrast enhancement (ml of arterial input per ml of tissue), and per unit of time. While the arterial contrast enhancement is typically measured per unit volume of blood, tissue blood flow is quoted in mL of arterial input per gram of tissue, which requires that the unit volume of myocardial tissue be converted into its mass equivalent, using the specific gravity of myocardial tissue, which averages 1.05 g per ml of tissue. Units of ml/min per g of tissue follow naturally from the procedures for microsphere tissue blood flow measurements, where the deposition of tracer is measured per g of tissue sample. The use of microspheres as reference standard made the units of ml/min per g of tissue also a natural choice for estimates of myocardial blood flow with external detection by an imaging device (MRI, CT, PET, etc.).
Semi-Quantitative Perfusion Measures
One advantage of such a perfusion index is the relative simplicity of its estimation from the signal intensity curves. For each perfusion parameter, and with some form of adjustment for hemodynamic conditions, one can in principle define a perfusion reserve index. While these perfusion reserve indices may be proportional to the coronary flow reserve over some limited range of CFR, or the perfusion reserve obtained from flows measured by the microsphere method, they generally each have specific thresholds to distinguish a hemodynamic significant stenosis from a coronary artery without flow-limiting lesions. The semi-quantitative perfusion indices cannot be compared in magnitude directly to the coronary flow reserve ratio measured in the catheterization laboratory. A further drawback of any ratio is the potentially confounding effect of the quantity in the denominator, which purportedly serves the role of "normalization". An example is the measurement of the perfusion reserve in hypertensive patients, where rest perfusion, which increases in proportion to the rate-pressure product, may be abnormally high. Whether the perfusion reserve in a hypertensive patient is reduced because of elevated resting blood flow, or an impaired hyperemic response, or a combination of both cannot be determined from a ratio, unless the quantities in the numerator and denominator can be assessed independently. The latter is only feasible with absolute myocardial blood flows.
Model-based Quantification of Myocardial Blood Flow
The approaches which can be used for quantifying myocardial blood flow from the observed contrast enhancement can be broadly divided into two categories, which we label here as model-based, and model-independent. For model-based approaches one specifies the functional spaces in myocardial tissue, how tracer moves through these spaces, and how it traverses permeable barriers between spaces. A considerable degree of simplification is necessary to arrive at models that can be used for numerical calculations and simulations. As commonly used MR contrast agents, such as Gd-DTPA, are excluded from the intracellular space one can consider a simplified model comprising only the vascular and interstitial spaces. Such a "two-compartment" or "two-space" model can be used to describe the contrast enhancement, i.e. the change of signal above its pre-contrast-injection baseline level. In other words, the background signal, comprising also a contribution from the intracellular space is subtracted before model-based-analysis of the contrast enhancement. Such a baseline correction can also be appropriate for analyzing the "first pass" enhancement after a previous contrast injection, if the background signal has reached an equilibrium level. Generally, one should wait at least 10 minutes between first pass imaging scans. In our experience, T1 in the blood pool at approximately 10 minutes after contrast injection reaches approximately 60% of its pre-contrast level. This corresponds to approximately 1/10th to 1/20th of the peak contrast-enhancement observed with a 0.03-0.04 mmol/kg bolus of Gd-DTPA, and therefore amounts to approximately 5-10%, or less, of the peak contrast enhancement observed during a second injection, an arguably acceptable level for the background signal increase in the blood pool. In the myocardium, the signal intensity level at 10 minutes after contrast injection is only ~10-20% higher than before contrast injection, which is a small fraction of the peak myocardial contrast enhancement during a first pass. After 10 minutes or longer, the background signal is therefore unlikely to make a significant contribution to signal saturation effects, which is arguably the primary concern related to the contrast residue from a previous injection. Higher contrast dosages may require longer delays before a repeat injection.
For an extracellular contrast agent an important contribution to the total observed or measured tissue contrast concentration comes from the fraction of contrast that has traversed the capillary barrier and leaked into the interstitial space. On signal intensity curves from myocardial perfusion studies the difference becomes clearly visible in the form of elevated signal intensity after the first pass of the contrast: contrast has passed into the interstitial space, and although it will eventually return to the vascular space and be washed out of the tissue region, the overall effect is a delay of the wash-out of contrast from the tissue region. For the extracellular contrast agent one needs to specify the volume of the interstitial space region, and also specify the rate at which contrast can traverse the capillary barrier, namely the permeability-surface area product (PS). Note, that it is not only the permeability of the capillary barrier that counts, but also the total area of this barrier, as a larger capillary surface area per unit tissue volume will result with constant permeability in a higher rate of contrast traversal from the vascular to the interstitial side. This means that capillary recruitment during vasodilation results in an increase of both blood flow and PS . For an extracellular contrast agent one can assume that the capillary barrier passage is driven by the concentration difference between interstitial and extracellular spaces, and the PS rate constant is the same for either direction of transit. If the primary goal of the analysis is the quantification of blood flow, the other model parameters may appear as a nuisance. The focus on blood flow often captures only a very limited view of pathological changes, and effects like limited capillary recruitment, and a blunted vasodilator capacity can have a significant effect on PS and vascular volumes. Unfortunately, it is challenging to quantifying the permeability surface area product from first pass studies. A reliable estimate of PS may only be feasible with two contrast injections, using intravascular and extracellular contrast agents respectively.
Identifiability of a model parameter refers to the ability to measure or detect changes of the model parameter. For example, it turns out that with an extracellular contrast agent, the signal intensity curves measured with a CMR perfusion scan are relatively insensitive to the changes in vascular volume, because of the leakage of contrast into the interstitial space - the detected contrast enhancement corresponds to the effect of contrast in both the vascular and interstitial spaces. This means that signal intensity after the first pass primarily reflects the sum of the vascular and interstitial volume fractions, rather than just the vascular volume fraction. In the context of a "first pass" perfusion study, blood flow is a parameter that has a readily identifiable effect on the signal intensity curves, and this explains in part the reasons why "first pass" perfusion imaging, independent of modality, has mostly focused on the quantification of blood flow. For a model-based analysis of myocardial perfusion with an extracellular contrast agent this means that the vascular and interstitial volume parameters generally have to be kept at fixed assumed values, possibly with the constraint that their sum matches the effective distribution volume.
Central Volume Principle
Model-independent analysis means that one foregoes specifying a functional model of the tissue structure. Model-independent analysis is based on the central volume principle introduced by Kenneth L Zierler in the the 1960's. A 2002 review by Zierler offers an interesting historical retrospective on its conceptual development and experimental validation . For its derivation one can start with an observation due to Eugen Fick, that the rate at which a substance accumulates in a tissue region of interest is given by the difference of concentrations of tracer substance flowing into and leaving the region, multiplied by the flow rate (F): F·(c out - c in ) = dq(t)/dt, where c in, out denote the concentrations at (arterial) inlet and (venous) outlet, respectively, q(t) is the mass of tracer in the region, and dq(t)/dt its rate of change with time (t). Fick's principle is simply a statement of mass balance: tracer that has entered the region and not yet exited remains in the region of interest.
The function R F (t) represents the q(t) response if an impulse input of tracer is applied at the region input - this follows from the integral equation 3, if one replaces c in with a "Dirac-delta" input function. We also note, that with such an impulse input (c in (t) = δ(t)) at time t = 0, there can be no tracer at the output (c out (t = 0) = 0), as this would otherwise require that the tracer or contrast to traverse region instantaneously, i.e. F → ∞. It can be shown then, that R F (t = 0) = F, meaning the initial amplitude of the impulse response is equal to the blood flow through the region. This statement can be generalized to any type of arterial input and it represents the essence of the Central Volume Principle.
The meaning of the impulse response can be further elucidated if interpreted as a probability. The value of R F (t) is normalized for this purpose by its value R F (t = 0). The normalized impulse response, R(t) = R F (t)/R F (t = 0), gives for any time t the probability that a tracer molecule still remains in the region of interest at that time, assuming that it entered at t = 0. In other words, the normalized impulse response gives the probability that the tracer residence time is greater than t. The complement of this statement is that 1-R F (t), is the probability that the transit time of the tracer is < t. Such cumulative distribution functions for the transit times or residence times have to be monotonic functions, which can be a useful constraint for the estimation of an impulse response. From the cumulative distribution of transit times, one can obtain the probability density function (PDF) for transit times, h(t), by taking the derivative of 1-R F (t).
where MTT represents the MTT of tissue, after subtracting the MTT of the first pass of the AIF (i.e. by excluding the recirculation component). This corollary of the central volume principle for the special case of an intravascular tracer is frequently found in the literature on brain perfusion, because Gd-chelates are confined in the brain to the vascular bed by the blood brain barrier. For myocardial perfusion studies, equation 4 only applies when an intravascular contrast agent is used.
The Fourier convolution theorem states that convolution in the time domain is equivalent to point-wise multiplication of the Fourier transforms of the two quantities in the convolution integral. One could therefore think of deconvolution in the context of equation 3 as akin to point-wise division of , the Fourier transform of q(t), by , the Fourier transform of the arterial input, to obtain the impulse response in the frequency domain. If is a relatively slowly varying function of time, it will have many locations at higher frequencies with close to zero, or zero amplitude. This would mean that point-wise division is a mathematically unstable approach for calculating the impulse response from the measured arterial and myocardial contrast enhancement. Although the described approach is useless in practice for deconvolution, the alluded to instability is symptomatic of the difficulties in performing a deconvolution.
The parameter t represents time, and the parameters μ, k, and A, do not have a physiological interpretation, and should be viewed simply as "shape" parameters. For example, μ defines the width of the initial plateau, before the function decays at a rate set by the parameter k. Only the amplitude of R F (t) for t = 0 has a physiological meaning: it corresponds to the blood flow, according to the Central Volume Theorem. The "shape" parameters of the Fermi function can be determined with a non-linear least squares fitting algorithm, such as the Marquardt Levenberg algorithm. The fitting function is in this case given by the convolution of the Fermi-function with the arterial input. In an environment such as Matlab (The Mathworks, Natick, MA) it is straightforward to implement this method with a few lines of code.
The representation of the impulse response shape can be generalized to properly reproduce also the features of the impulse response related to the delayed wash-out of contrast that has crossed the capillary barrier into the interstitial space. To that end, one can for example use a representation of the impulse response as a sum of B-spline components. Nevertheless the algorithmic procedure to calculate the impulse response with such a B-spline basis becomes more complicated, and similar results for flow estimates are still obtained with the Fermi-function representation, by the sleight of hand described above.
It lies in the nature of the deconvolution problem that there is no unique solution for the impulse response, which is partially why it is considered an "ill-posed" problem. In fact, the brute force approach of calculating the impulse response from the Fourier transforms, or other naive forms of numerical inversion of the deconvolution operation yield mathematically admissible solutions of the convolution equation, which nevertheless have to be dismissed as unstable, and physiologically unrealistic. Under most circumstances we are not much concerned about the exact details of the impulse response, but are primarily interested in estimating the initial impulse response amplitude, which, by virtue of Zierler's central volume theorem, provides an estimate of the blood flow. But there are some requirements one can impose on the impulse response, e.g. that it has to be a monotonically decaying function - contrast can only leave the region of interest after the initial arterial impulse input and is not replenished. Furthermore one can impose some smoothness constraints, as impulse responses don't have any sudden jumps in the decay from their initial amplitude. Such requirements can help to stabilize the deconvolution operation.
The fact that with the deconvolution analysis one does not need to specify the internal structure of the blood tissue exchange unit also identifies its main limitation. At least with an extracellular contrast agent, one can with the deconvolution analysis determine only the blood flow, but not other perfusion parameters such as the vascular volume or the capillary permeability surface area product. For the latter one has to have some model to identify these parameters, both in the sense of giving them a functional meaning within a model of the blood tissue exchange unit, and also in the sense of being able to determine stable values from measurements of myocardial contrast enhancement.
represents the relaxation rate constant of blood without contrast, r1 is the relaxivity of the contrast agent, and [Gd] its concentration in plasma. With fast water exchange the concentration of [Gd] in plasma is effectively diluted by the (1-Hct) factor.
Within myocardial tissue spaces water exchange (e.g. across the transcytolemmal barrier) often occurs at a rate that falls neither into the fast or slow exchange regimes, and this makes the analysis more tedious, and a description of the formalism is beyond the scope of this review. Suffice it to say that the water exchange necessitates an expansion of the tissue model from a two to a three space model, to include the intra-cellular space, even though the contrast agent itself is excluded from the latter.
The conditions of fast or slow exchange are also determined by effective relaxation rate, which is a function of the pulse sequence applied during read-out of the signal. Assuming that a spoiled gradient echo sequence is used for read-out with higher flip angle radio-frequency pulses, then the effect of water exchange is less noticeable because the radio-frequency pulses, if applied with short repetition times (TR), rapidly erase any "memory" of water exchange. A more formal proof and experimental validation of this can be found in the work of Donahue et al.. More recently Li et al showed how the combination of short TR and higher flip angle renders the observed contrast enhancement relatively insensitive to the effects of water exchange in myocardial tissue. For a relaxation recovery not disturbed by radio-frequency pulses one has to use the full water exchange formalism, as the rate of water exchange falls into a range intermediate between the "no-exchange" and fast-exchange limits.
Practical Recommendations to Facilitate Quantification
The pulse sequence needs to be chosen such that it allows the recovery of an arterial input function. For example, any technique where the signal in the blood pool is suppressed or attenuated by intraventricular flow (echo-planar technique with spin-echo) would not be suited for the goal of blood flow quantification. Another example is the interleaved notched saturation preparation, which is slice selective, and therefore produces T1-weighting in the blood pool that is flow and slice-position dependent. Other advantages of the notched-saturation technique have led to it being offered as standard protocol for myocardial perfusion imaging on one vendor platform.
The contrast dosage should be sufficiently low to allow recovery of contrast concentration, either by using a special acquisition technique, such as the dual-echo technique, or through post-processing, using calibration curves. For the standard gadopentetate dimeglumine agent, this generally means a contrast dosage well below 0.1 mmol/kg. If the non-linearity of contrast-enhancement versus contrast concentration in the blood pool should not exceed more than 10-15%, then the contrast dosage may need to be < 0.03 mmol or less at 1.5 and 3T.
For temporal resolution a repetition time corresponding to 1-2 R-to-R intervals at rest, and 1 R-to-R interval during stress is strongly recommended. Poor temporal resolution will lead to unacceptably low accuracy for MBF quantification. Spatial resolution affects the accuracy of MBF quantification through partial volume effects, but MBF quantification is more forgiving of lower spatial resolution than temporal resolution, if partial volume effects are avoided. For adults, an in-plane spatial resolution < 2.5 mm should be aimed at. Parallel imaging (SENSE, SMASH, GRAPPA) has become a nearly standard component of myocardial perfusion imaging, using moderate acceleration factors around 2. With higher acceleration, the reduction of signal-to-noise may rapidly become the limiting factor, unless one uses a sequence technique such as steady-state free precession. Practical experience with SSFP perfusion imaging has shown that its use should be limited to 1.5 T or lower field strengths, and probably only with low-dosage contrast bolus.
New acceleration techniques, such as k-t BLAST and k-t SENSE[47, 48], allow perfusion imaging with high spatial resolution, but the algorithms applied for image reconstruction give rise to some degree of low-pass temporal filtering if high acceleration factors (~≥8) are used. Such low-pass temporal filtering, will lead to a bias to underestimate flow. The resolution of this problem will have to await further advances in these combined spatial-temporal acceleration techniques.
CMR at 3 Tesla is steadily gaining ground. For myocardial perfusion imaging with gradient echo techniques (without steady-state free precession) one can make an unreserved recommendation for the higher field strength, because of the concomitant increase of signal-to-noise compared to 1.5 Tesla. Although SSFP techniques yield excellent signal-to-noise ratio at 1.5 Tesla[17, 18], there are well-founded concerns that contrast enhancement with SSFP can be modulated by confounding factors such as frequency shifts during bolus passage.
Arguably one of the most serious limitations of quantitative perfusion analysis is the inability to measure the arterial input accurately and relatively closer to the myocardial region of interest. This has a direct impact on the accuracy of blood flow estimates. Instead the enhancement from the contrast bolus is typically measured in the LV blood pool or the proximal aorta, but transit through the epicardial vessels is bound to cause some dispersion of the contrast bolus, which may be exacerbated by an epicardial stenosis. One is forced to either neglect this dispersion of the contrast bolus during transit through the epicardial vessels, or assume a fixed relative dispersion. A neglect of the dispersion results in an underestimation of blood flow.
Cardiac magnetic resonance can be used to quantify absolute myocardial blood flow with high spatial resolution, thereby avoiding spill-over of signal from the LV cavity, and providing an assessment of the transmural perfusion gradient. The quantification of blood flow is not significantly more time consuming that a semi-quantitative analysis, in particular if a deconvolution method is used for the analysis, requiring minimal user input. Instead the most time-demanding step continues to be the segmentation of the images along endo and epicardial borders. Standardization of the quantitative analysis still appears to be some time off, as this remains a field of active investigation, where further optimization and innovations are still forthcoming at a rate that precludes a broad consensus and standardization. Despite its evolving status, quantitative CMR perfusion imaging has achieved recognized role in studies of coronary physiology and cardiac diseases. Whether it will be adopted for the clinical workflow remains uncertain at this time.
Example of a two compartment model for analysis of myocardial perfusion
It can be shown that the solution of the equations (1) and (2) to an impulse input is given by the sum of two exponential functions with coefficients that are proportional to the volumetric flow rate per unit volume. Obtaining the solution to this set of equations involves some tedious algebra which we spare the reader. Instead we included a Mathemetica notebook (Additional file 1), which can be opened with the Mathematica software package, or with the free Mathematica player software, which can be downloaded from http://www.wolfram.com.
This two-compartment model is used to illustrate the effects of changes in flow, PS, and compartmental volumes on the impulse response, and on the tissue residue curves. The tissue residue curves correspond to the signal intensity curves that are measured for myocardial regions of interest in a CMR perfusion study. We also verify that the amplitude of the impulse response for this lumped two-compartment model corresponds to the tissue blood flow, as predicted by Zierler's central volume theorem.
The author is indebted to the following individuals for long-standing, and productive scientific interactions that substantially benefited his own work in quantitative myocardial perfusion imaging: Norbert Wilke, MD, Arthur Stillman, MD, PhD, Leon Axel, MD, PhD, Ravi Teja Seethamraju, PhD, Keith Kroll, PhD, and James Bassingthwaighte, MD, PhD.
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