Magnetic Resonance in Medicine 51:816 – 827 (2004)
Modeling 1H Exchange: An Estimate of the Error Introduced in MRI by Assuming the Fast Exchange Limit in Bolus Tracking G.R. Moran1,2* and F.S. Prato2 A simulation is presented which calculates the MRI signal expected from a model tissue for a given pulse sequence after a bolus injection of a contrast agent. The calculation assumes two physiologic compartments only, the intravascular and extravascular spaces. The determination of the concentration of contrast in each compartment as a function of time and position has been outlined in a previous publication (Moran and Prato, Magn Reson Med 2001;45:42– 45). These contrast agent concentrations are used here to determine the NMR relaxation times as a function of time and position within the tissue. Knowledge of this simulated tissue ‘map‘ of relaxation times as a function of time provides the information required to determine whether the proton exchange rate is fast or slow on the NMR timescale. Since with a bolus injection the concentration of contrast and hence the relaxation time may vary with position along the capillary, some segments of the capillary are allowed to be in fast exchange with the extravascular space, while others may be in slow exchange. Using this information, and parameters specific to a given tissue, the MRI signal for a given pulse sequence is constructed which correctly accounts for differences in proton exchange across the length of the capillary. It is shown that extravascular contrast agents show less signal dependence on water exchange, and thus may be more appropriate for quantitative imaging when using fast exchange assumptions. It is also shown that nondistributed compartment models can incorrectly estimate the water exchange that is occurring at the capillary level if exchange-minimizing pulse sequences are not used. Magn Reson Med 51:816 – 827, 2004. © 2004 Wiley-Liss, Inc. Key words: exchange; perfusion; compartment modeling; MRI
One approach to the determination of physiologically important parameters such as tissue blood flow and vascular volume has been to inject, as a bolus, a paramagnetic contrast agent through the tissue of interest and to evaluate the concentration of that agent as a function of time. In order to use MRI to measure this concentration, it is most often assumed that the change induced by the paramagnetic contrast agent in the MR relaxation rate is directly proportional to the tissue contrast agent concentration.
1 Medical Physics and Applied Radiation Sciences Unit, McMaster University, Hamilton, Ontario, Canada. 2 The Department of Nuclear Medicine and Diagnostic Radiology, The Lawson Health Research Institute, St. Joseph’s Health Care, London, Ontario, Canada. Grant sponsor: Medical Research Council of Canada; Grant number: MT9467 (to F.S.P.); Grant sponsor: NSERC Industrial Research Fellowship (to G.R.M.). *Correspondence to: Gerald R. Moran, Medical Physics and Applied Radiation Sciences Unit, McMaster University, 1280 Main St. W, Hamilton, Ontario L8S 4K1, Canada. E-mail:
[email protected] Received 17 September 2003; revised 28 October 2003; accepted 31 October 2003. DOI 10.1002/mrm.20002 Published online in Wiley InterScience (www.interscience.wiley.com).
© 2004 Wiley-Liss, Inc.
This assumes, however, that the MRI relaxation of the tissue at all concentrations used can be represented by a single exponential behavior, i.e., the protons contained in the water molecules, for instance, are moving rapidly between different tissue “compartments,” or are in fast exchange. It has been reported (1,2), however, that at the high concentration of contrast medium present during the peak of a bolus injection, the fast exchange limit breaks down for some contrast agents. Several groups have investigated tissue compartmentalization and exchange effects with various models. Donahue and co-workers (3,4) are continuing the exchange minimization techniques introduced in the frequently referenced 1996 work to determine the optimum pulse sequence parameters in order to make MRI contrast-enhanced measurements independent of exchange and inflow effects. These measurement techniques are promising and have been analyzed with the two site exchange model introduced by Hazlewood et al. (5). Larsson et al. (6) used a two-compartment tissue model and the Hazlewood model of exchange to calculate perfusion and extraction fraction in the myocardium. Comparison with experiment indicated that, when using an intravascular contrast agent, exchange significantly affected the perfusion estimation unless the inversion time was kept short. Brix et al. (7) and Tofts et al. (8) also presented similar models. Noseworthy et al. (9) also use a two-compartment tissue model and the Hazlewood exchange model to study the use of contrast agents to identify tumor microvasculature. Judd et al. (2), modifying a model presented by Bauer and Schulten (10), introduced a model where the tissue was broken down into cellular, interstitial, and intravascular components. Water exchange across the cellular–interstitial boundary is assumed to be fast, and thus exchange is only limited at the intra– extravascular boundary. They concluded that water exchange in the myocardium strongly affects firstpass enhancement, but that reasonable estimates of the perfusion differences can be obtained by neglecting exchange. All of these models, however, treat the tissue compartments as single-volume elements, not allowing the concentration to vary with position along the compartment volume. It is expected in bolus tracking experiments that at any given instant in time the intravascular contrast agent concentration will vary along the length of the capillary. As the concentration of contrast will be a factor in determining the rate of the exchange across the intravascular– extravascular boundary, these models do not address the notion that some portions of the intravascular space may be in fast exchange with the extravascular space, while other portions may be in slow or intermediate
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exchange at the same time. This work is designed to address this effect and, in addition, the feasibility of neglecting exchange in cardiac bolus tracking. In order to investigate the effects of exchange during a bolus injection of a contrast medium, it is therefore necessary to model the concentration of contrast within the tissue both as a function of time and position. From this information the induced MR relaxation rate can be ascertained, thus allowing for the exchange limits to be determined. There are currently many different approaches to tracer kinetic modeling of contrast agents within tissue. The modified Kety model (11,12) relates the contrast concentration in the tissue, Ct(t) (in mM), to the concentration in the plasma, Cp(t), and the flow, F, in ml 䡠 min-1 䡠 g-1. This model is not particularly well suited for this work, however, because it only provides the tracer concentration as a function of time for the entire tissue. The concentration of contrast medium as a function of time within each of the physiological compartments would be required in order to quantify the exchange limits within the tissue. As well as the lack of a multicompartment treatment, this model does not allow the concentration to vary with position, necessary to model the flow of a bolus injection through the capillary. Other alternative approaches model the tissue residue function, Q(t) (mM 䡠 g-1), the amount of tracer deposited in the tissue by introducing an impulse residue function of some form, R(t). Then Q(t) can be constructed by a convolution with a measured arterial contrast agent concentration, Ca(t), which is often referred to as the arterial input function (AIF): Q(t) ⫽ FC a(t)RR(t),
[1]
where F is the blood flow, and V represents the convolution operation. These methods are also inappropriate for this work, however, because the tissue is treated as a single compartment, and it does not allow for spatial variation of the contrast concentration. Another model, the multiple indicator multiple path indicator dilution 4 region model, or MMID4 (13), is gaining widespread use. Jerosch-Herold et al. (14) used MMID4 to determine myocardial perfusion reserve in canines. More recently, Schmitt et al. (15) utilized this model to quantify myocardial perfusion and perfusion reserve with MRI after a bolus injection of Gd-DTPA in animals and humans. This detailed model accounts for vascular flow and dispersion, arterial, arteriolar, and capillary volumes, interstitial volume, and capillary permeability. This is a much more sophisticated tracer kinetic model, with many adjustable parameters, driven by the relevant physiology. In a sense, this is limiting because there are so many parameters to fit. A less complicated model was desired so that changes in the output of the model could be attributed to the contribution from various physical phenomena. A model introduced by Johnson and Wilson (16), the Tissue Homogeneity Model, is particularly well suited for this calculation. As mentioned earlier, the literature has suggested that the cellular– extracellular exchange is fast (1–3), and therefore a good first approximation to the myocardial system is a model such as this one, with two
compartments, dividing the tissue into the extravascular space (ev) and the intravascular space (iv). It is assumed (consistent with Ref. 2) that there is rapid mixing in the ev space, implying this concentration is a function of time only, and that it is the exchange across the capillary membrane, the ev–iv boundary, that dominates the exchange calculation. The concentration of the AIF varies with position, and thus the capillary is subdivided into several segments to emphasize that the concentration will vary across its length (17). The rate of diffusion of the contrast agent across the ev–iv boundary is determined by the permeability surface area product, PS. St. Lawrence and Lee (18) introduced this model to study the brain and developed an adiabatic approximation to the solution and rigorously tested the validity of their solution with experiment (19). Their solution, however, provides a tissue residue function of the form of Eq. 1, which is inappropriate for an exchange analysis, as explained above. Recently, we developed an exact solution (17) to this model, solving the appropriate differential equations by taking the Laplace transform. A simulated, or a real experimental, AIF can be input into the calculation to produce the concentration of the tracer within each of the compartments as a function of time and for the intravascular compartment as a function of position as well. It is this solution to the Tissue Homogeneity Model that will be used to generate the tissue contrast agent concentrations, and hence determine the theoretical MRI signal emanating from the tissue, during a simulated bolus injection of contrast medium. In this way the effect of neglecting the distribution of contrast along the capillary can be estimated. Beard and Bassingthwaighte (20) have cautioned against using “computationally simple, less physically realistic models.” Their claim is that the simple two-compartment models incorrectly predict a recruitment of capillary surface volume with increasing flow. The model used in the present work is a two-compartment axially distributed capillary model. It is not as sophisticated as the MMID4 model, for example, by design so that the effects of chemical exchange and the distribution of contrast along the capillary could be isolated. MATERIALS AND METHODS Simulation The parameters used in this MRI tissue model are given, along with a brief description, in Table 1. Once a form for the AIF resulting from a bolus injection of contrast agent was chosen (can be experimentally measured or a theoretical function), the concentration of contrast agent within the tissue as a function of time and position was determined for the two-compartment model, as illustrated in a previous work (17). This determination of contrast concentration over the entire model tissue for the duration of the experiment was required to produce a tissue map of the spin-lattice relaxation time, T1, or equivalently the relaxation rate, R1. This concentration of contrast agent within the tissue determined for both an intravascular and an extravascular agent was considered the starting point for this calculation.
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Table 1 Simulation Parameters Parameter Pa Pb Ta Tb TI TR ␣ PS
Value 0.1 0.9 1.2 0.823 50 2.3/2.4 9/15
 ka
1.11(ev) 0.015(iv) 4 3
kb
1/3
Units ⫺1
ml 䡠 g ml 䡠 g⫺1 s s ms ms ° ml 䡠 min⫺1 䡠 g⫺1 s⫺1 䡠 mM⫺1 s⫺1 s⫺1
Description Volume of compartment a in the absence of exchange. Volume of compartment b in the absence of exchange. Spin-lattice relaxation time for protons in a, in the absence of exchange. Spin-lattice relaxation time for protons in b, in the absence of exchange. Inversion time for simulated inversion recovery experiment. Repetition time for simulated T1-FARM(24)/saturation recovery turbo FLASH experiment. Tip angle of pulses used in simulated T1-FARM(24)/saturation recovery turbo FLASH experiment. Permeability-surface area product assumed for myocardium.(2,6,12) Relaxivitty assumed for Gd-DTPA in myocardium.(2,6,12) Rate of exchange for protons (water molecules) to leave compartment a. IE) 1/(residence time) (1–4,6) Rate of exchange for protons (water molecules) to leave compartment b. IE) 1/(residence time) (1–4,6)
An AIF was simulated using a gamma variate function of the form: c(t) ⫽
C(t ⫺ t 0) s ⫺ 1e ⫺(t ⫺ t0) , ⌫(s ⫹ 1)
[2]
where s ⫽ 13, C ⫽ 100, and t0 ⫽ –2 are constants which were adjusted to produce an AIF typical of that observed in our canine experiments. The simulation was designed as well to allow the input of an experimentally determined AIF. It is required that the AIF be Laplace transformed. Thus, rather than a numerical method, the AIF was fit to a function of the form of Eq. 2 to represent the main bolus and an exponentially decreasing function of time to represent the clearance of contrast. In this work the recirculation peak is neglected. The goal here was to use the calculated contrast agent concentrations to produce a map of the R1 values over the model tissue. In some cases where contrast agents are used in MRI, the effect on the relaxation rate is modeled by introducing the relaxivity, , in s-1 䡠 mM-1, with the change in relaxation rate, ⌬R1, induced by the contrast being given by: ⌬R 1 ⫽ [C],
[3]
where [C] is the concentration of contrast within the tissue. This is correct if the protons in the tissue (water molecules) are in fast exchange so that a single relaxation time is observed for the entire tissue, which is essentially the assumption being tested by this model. Thus, this expression cannot be adopted globally, i.e., for the tissue as a whole, if we wish to test this assumption. However, it is reasonable to assume that this expression is obeyed on a more local level, for instance, within the small subdivisions of the capillary and within the extravascular space where fast mixing is assumed. Thus, using a reasonable value for the relaxivity (4.0 s-1 䡠 mM-1 at 1.5 T) for GdDTPA in the myocardium (2) (15.0 s-1 䡠 mM-1 at 1.5 T is assumed for Gadomer-17), the contrast agent concentrations over the tissue were used to calculate the relaxation rate at every position within the model tissue as a function
of time during the bolus transit for both an extravascular (PS ⫽ 1.11 ml 䡠 min-1 䡠 g-1) and an intravascular (PS ⫽ 0.015 ml 䡠 min-1 䡠 g-1) agent (21). A normal flow of 1.0 ml 䡠 min-1 䡠 g-1 is also assumed. The concentration of the bolus injection used for all of these simulated experiments corresponds to the AIF defined in a previous work (17) which reaches a peak concentration in the plasma of 2.5 mM. Once the relaxation rates were determined, these were then used to construct the theoretical MRI signal (see below) from the tissue, which is dependent on a number of factors, including the computed contrast agent concentrations, the parameters used in the model (Table 1), the MRI pulse sequence chosen, and the exchange model used to calculate exchange. Exchange For the purposes of this work, exchange will be taken to mean proton exchange, or the physical motion of protons from one physiologic compartment to another. As an example, our tissue of interest, the myocardium, can be divided into four physiological compartments, the extravascular extracellular space, the extravascular cellular space, the intravascular extracellular space, and the intravascular cellular space. Since the literature has suggested that the cellular– extracellular exchange is fast (1–3), a good first approximation to this system is a model with two compartments, one intravascular and one extravascular, and it is the exchange across the capillary membrane which dominates. Consider that protons (within water molecules, for instance) are moving between two compartments at a particular rate, ka, or the exchange between the compartments has a certain lifetime, ⫽ 1/ka. To determine whether this exchange rate is fast or slow, more information regarding the method of measurement is required. If a measurement technique is used such that the water moves between the compartments several times during the measurement, then the exchange is fast. If, however, a more rapid measurement technique is used, so that the water molecules do not have time to change position during the measurement, then the exchange is slow. In MRI measurements, a proton must exchange to the second
Error Introduced by Assuming Fast Exchange
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compartment before its magnetization is destroyed by the local relaxation time, in order for the exchange to be rapid. This scenario is illustrated by the conditions: for slow exchange
m⫽
冏
1 . 1 1 ⫺ Ta Tb
[7]
冏
Then the overall exchange rate would be given by: ⬎⬎
1
冏 T1 ⫺ T1 冏 a
,
[4] K⫽
b
1 1⫹
m
,
[8]
for fast exchange
⬍⬍
冏
1 , 1 1 ⫺ Ta Tb
冏
[5]
where Ta and Tb are NMR relaxation times within the two compartments labeled a and b in the absence of exchange. Whether considering a two-compartment system or even a more general multicompartment system, if the exchange rate is rapid compared to the NMR measurement, then the protons can sample the local environments of all of the compartments, with the result being that a single relaxation rate is observed which is a weighted average of the relaxation rates within all of the compartments. That is, for fast exchange the two-compartment system relaxes with a single exponential relaxation time, Tave, given by: 1 Pa Pb ⫽ ⫹ , T ave Ta Tb
[6]
where Pa and Pb are the fractional volumes of the two compartments a and b. If there is slow or no exchange, then the two compartments will not “communicate” via exchange. Each compartment will therefore possess a separate relaxation time, Ta or Tb, and the entire system will display a biexponential relaxation, or a multiexponential relaxation if there are many compartments. For the purposes of this work, two models of proton exchange will be considered. First is the formulation for a two-site exchange model derived from the Bloch equations presented by Hazlewood et al. (5). Since the Larmor frequencies of the two sites are the same, the results are the same as the calculations of Zimmerman and Brittin (22). This model predicts that both the apparent population fractions (or volumes) as well as the apparent relaxation rates will depend on exchange between the compartments. A more simplistic, yet more intuitive, model will also be used, which will enable the visualization of the exchange rate between the compartments. It is assumed as above that there are only two physiological compartments of interest, each with population fractions Pa and Pb, and relaxation times Ta and Tb in the absence of exchange. Further, the overall exchange rate between the compartments is defined to be K, where K ⫽ 0 is the slow exchange limit, and K ⫽ 1 is the fast exchange limit. For example, the exchange might be known to be occurring at a particular rate, 1/, and the conditions outlined in Eqs. 4 and 5 can be used to generate a characteristic time, m, defined by the relaxation rates:
which becomes unity in the limit that 1/ is very large compared to 1/m (fast limit), and becomes zero when 1/ is very small compared to 1/m (slow limit). This exchange rate introduces an additional contribution to the relaxation rate in compartment a from the protons exchanging from compartment b. The apparent relaxation rates would be given by the expressions:
冉
冊
[9a]
冉
冊
[9b]
1 1 1 1 ⫽ ⫹ KP b ⫺ T⬘a Ta Tb Ta
1 1 1 1 ⫽ ⫹ KP a ⫺ , T⬘b Tb Ta Tb
so that in the limit of no exchange, K ⫽ 0, the relaxation rates are those defined in the absence of exchange, and in the limit of fast exchange, K ⫽ 1, both relaxation rates revert to the fast exchange average rate defined in Eq. 6 (recall that Pa ⫹ Pb ⫽ 1). This model has been designed to produce the correct relaxation rates in the limits of fast and slow exchange; however, it assumes that the population fractions do not change with exchange. This model therefore serves two purposes: 1) It provides a simple parameter to rank and plot the exchange at a given location on a scale from 0 to 1 (slow to fast); and 2) It provides a comparison to the Hazlewood model in which there is an effect of the exchange on the population fractions. The presence of a paramagnetic contrast agent in the tissue increases the relaxation rate at that location. During a bolus injection of contrast agent, the contrast agent concentrations may be large enough such that the local relaxation rate increases to the point that Eq. 4 is satisfied and the fast exchange approximation is no longer valid. This is even further complicated because during a bolus injection of contrast medium the concentration within the capillary will vary with both time and position. The presence of paramagnetic ions in various amounts across the capillary will increase the MRI relaxation rate by differing degrees along its length at any instant in time. Since the difference in relaxation rate between the intravascular and extravascular compartments determines whether the exchange is rapid or slow relative to the MRI timescale, then at a given instant in time some positions along the capillary may be in fast exchange while others may be in slow or intermediate exchange with the extravascular space. MRI Signal Construction and Fitting The MRI signals were simulated using both our simple exchange model, which assumes that the compartmental
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volumes are independent of exchange, and the Hazlewood exchange model, which predicts that both the compartment volumes and the relaxation times vary with exchange. The signals were constructed assuming 1) fast exchange, 2) slow exchange, and 3) an exchange rate supported by the literature of ka ⫽ 3 s-1 (1– 4,6) between the intravascular and extravascular spaces. This MRI signal represents the expected result from an MRI experiment with the given pulse sequence and these three exchange scenarios. Often it is assumed that proton exchange within the tissue is in fact rapid, and thus that there exists a single relaxation time for the entire tissue regardless of the true physical circumstances. To model an experiment where this assumption has been made, the constructed MRI signals, assuming fast, slow, and intermediate exchange, were fit to a single exponential relaxation process (see below). This fitting procedure was repeated to measure the relaxation time at several steps during the bolus passage. Then, using Eq. 3, the apparent “measured” contrast concentration which resulted from this procedure was evaluated and compared to the “input” concentration which was determined by summing the contrast concentration over the tissue at the time of each (simulated) MRI measurement. By fitting the data to a single exponential relaxation time, fast exchange is implied. This means that if the MRI signal is simulated assuming fast exchange, then fitting in this manner should accurately predict the true concentration. However, the comparison of the MRI signal where slower exchange rates have been assumed to the fast exchange case provides a direct illustration of the error introduced by fitting the MRI signal representing the transit of a bolus injection of contrast through a tissue assuming fast exchange. Once a relaxation rate, or equivalently, a relaxation time, map is determined the MRI signal can be constructed. This signal will depend on exchange and the choice of MRI pulse sequence as discussed above. For our simple exchange model, the NMR relaxation times will be given by Eq. 9, in which the compartment volumes are assumed constant, whereas for the Hazlewood model (5) the relaxation times and compartment volumes will both vary with exchange. The MRI signal generated via the exchange equations will also be further influenced by the pulse sequence used. For instance, it was mentioned earlier that the signal would be insensitive to exchange for small times. This implies that for short experimental times, as in a saturation recovery experiment, for example, the result would be the same as if the system were in the slow exchange limit. For a saturation recovery Turbo-FLASH (srTFL) experiment the MRI signal after the nth phase encode step, Sn(t), was generated by an equation of the following form (14):
S n(t) ⫽
冘冋
Pa(t)1 ⫺ e⫺T1/Ta(t)aan⫺1 ⫹ (1 ⫺ Ea)
space
冉
⫹ Pb(t) 1 ⫺ e⫺T1/Tb(t)abn⫺1 ⫹ (1 ⫺ Eb)
冊
1 ⫺ aan⫺1 1 ⫺ aa
1 ⫺ ab 1 ⫺ ab
冊册
n⫺1
,
[10]
where 64 phase encode steps where chosen and therefore n ⫽ 32 steps are required to reach the center of k-space, and Ea ⫽ e-TR/Ta(t), aa ⫽ Eacos␣ for compartment a, and similarly for compartment b. A similar expression was used for the inversion recovery Turbo-FLASH (ir-TFL) sequence. Typical values for the inversion time (for the irTFL, or for the srTFL—the time between the 90° pulse and the start of the FLASH acquisition), TI ⫽ 20 ms, the repetition time, TR ⫽ 2.4 ms, and the flip angle, ␣ ⫽ 15°, were chosen. The relaxation times, Ta(t) and Tb(t), and the compartmental volumes, Pa(t) and Pb(t), have been written as functions of time because they are changing as the bolus passes through the tissue. The relaxation times change with contrast and with exchange according to both exchange models. The compartment volumes also change with exchange according to the Hazlewood model, but not for our simple model. The signal is computed by summing over space, over the model tissue as indicated. These equations represent the signal from a saturation recovery turboFLASH (srTFL) experiment using equations described by Jerosch-Herold et al. (14), for example. In addition, we assessed a considerably different MRI pulse sequence which has recently been used to track the flow of a bolus through a tissue: T1 fast acquisition relaxation mapping, or T1-FARM (23,24). This is a Look-Locker type pulse sequence and the MRI signal for this pulse sequence can be represented by (25): S(t) ⫽
冘
兵Pb(t)Yb(t)关1 ⫺ 2e⫺t共1/Tb(t)⫺log(cos␣)/TR兲兴
space
⫹ Pa(t)Ya(t)关1 ⫺ 2e⫺t共1/Ta(t)⫺log(cos␣)/TR兲兴其,
[11]
where TR(⫽2.3ms) is the repetition time and ␣(⫽9°) is the angle of the small angle pulses, and the quantities Yi(t) are given by: Y i(t) ⫽
1 ⫺ e ⫺t/Ti(t) . (1 ⫺ e ⫺t/Ti(t)cos␣)
[12]
The irTFL, srTFL, and the T1-FARM equations (Eqs. 10 and 11, respectively) have been used to generate signals where the slow exchange limit, intermediate exchange, and the fast exchange limit have been assumed between the intravascular/extravascular compartments. The procedure now emulates the situation of a real experiment, where most often the assumption is that these compartments are in fast exchange. In other words, the relaxation time was extracted from the signal by assuming a single exponential recovery. For the tissue, this relaxation time and the noncontrast relaxation time were used via Eq. 3 to estimate the concentration of contrast in the region of tissue. For the case of the simulated data from the srTFL and irTFL sequences, there is no “magnetization recovery curve” to fit, so the equilibrium magnetization was extracted from the simulation and the tissue relaxation time was fit at a single time point. For the T1-FARM pulse sequence the magnetization recovery described in Eq. 11 was fit to extract the relaxation time for the tissue. RESULTS AND DISCUSSION The arterial input function, the tissue contrast agent concentration for an intravascular agent (no contrast in the
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extravascular space, PS ⫽ 0), and the contrast agent concentration for an extravascular agent were used as the starting points for these simulations (17). The extravascular agent is taken to be Gd-DTPA and a typical PS product of 1.11ml 䡠 min-1 䡠 g-1 (2,6,12) is assumed based on average literature values for myocardial tissue, as indicated in Table 1. For the blood-pool agent of Gadomer-17, a typical PS product of 0.015ml 䡠 min-1 䡠 g-1 (21) is assumed, as indicated in Table 1. It was noted in Materials and Methods that the more simple exchange model is useful because it provides an illustration of the exchange. In Fig. 1, the quantity K is plotted as a function of time and position along the capillary for an intravascular agent (Fig. 1a), and an extravascular agent (Fig. 1b) assuming an exchange rate of ka ⫽ 3 s-1. Note that for both agents the exchange appears to be more slow (K ⫽ 0) as the peak of the bolus passes. For the extravascular agent, because of mixing across the capillary membrane, the exchange differences lessen slightly (K closer to 1) at the “back” end of the capillary. The exchange therefore deviates slightly more from the fast limit when using an intravascular contrast agent with the same initial parameters as an extravascular agent.
FIG. 2. Summary of relaxation times (T1) which were fit using a single exponential relaxation to the simulated MRI data (an srTFL sequence with TR/TE ⫽ 2.4/1.2ms, ␣ ⫽ 15°, and flow ⫽ 1 ml 䡠 min-1 䡠 g-1) where fast exchange, slow exchange, and an exchange rate of ka ⫽ 3 s-1 was assumed in the generation of the data for (a) an intravascular contrast agent and (b) an extravascular contrast agent.
FIG. 1. The exchange parameter, K, displayed as a function of time and position along the capillary (in arbitrary units) for (a) an intravascular contrast agent, and (b) an extravascular contrast agent (flow ⫽ 1 ml 䡠 min-1 䡠 g-1 for both).
The results for the simulation of a srTFL experiment, a typical pulse sequence used to track a bolus of contrast, are shown in Fig. 2, where we used 32 phase-encode steps to reach the center of k-space, TI ⫽ 20 ms, TR ⫽ 2.4 ms, and ␣ ⫽ 15°. For this simulation, each signal data point is fit to reconstruct a single relaxation time, T1, for the tissue. It is these fitted relaxation times which are shown in the figure for simulated data where fast exchange, slow exchange, and an exchange rate of ka ⫽ 3 s-1 (“intermediate exchange”) were separately assumed when generating the data. The relaxation times are shown for an intravascular contrast agent (Fig. 2a) and an extravascular contrast agent (Fig. 2b). The relaxation curves are grouped together much more closely for the extravascular agent than the intravascular one. Recall that all of the data are fit assuming a single exponential recovery, or fast exchange. Since the data were generated assuming fast exchange, then the “fast exchange” curve represents fitting assuming fast exchange when it is reasonable to do so. The slow and intermediate exchange data are not expected to be fit accurately by
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FIG. 3. Comparison of the “input” contrast agent concentration to that predicted by the data presented in Fig. 2. The results presented arise from (a) an intravascular contrast agent and the Hazlewood model of exchange, (b) an intravascular contrast agent and the model of exchange given in Eqs. 7–9, (c) an extravascular contrast agent and the Hazlewood model of exchange, and (d) an extravascular contrast agent and the model of exchange given in Eqs. 7–9.
assuming fast exchange. The difference between the fast exchange case is a representation of the error introduced by assuming fast exchange when it is not the case. The slow and intermediate (ka ⫽ 3 s-1) exchange curves both fall short of predicting the fast exchange T1 minimum by 2.5% in the intravascular case and by 0.5% in the extravascular case. The explanation for this difference can best be seen by considering whether the fitted relaxation times can predict the “input” concentration by calculating the expected “measured” concentration via Eq. 3 and summing over the tissue, i.e., over the length of the capillary. The result of this procedure for the data illustrated in Fig. 2 is shown in Fig. 3 for an intravascular (Fig. 3a,b) and an extravascular contrast agent (Fig. 3c,d). In this figure the “input” concentration is shown as well as data representing a simulated measurement where fast, slow, and intermediate (ka ⫽ 3 s-1) exchange was assumed in the generation of the data. Figure 3b,d results from the “simple” exchange model, and Fig. 3a,c results from the Hazlewood model. Thus, the difference between Fig. 3a,b is that in Fig. 3a the compartment volumes are allowed to vary with exchange.
Figure 3a,b are nearly indistinguishable, similarly for Fig. 3c,d. This implies that this simple exchange model can be used as a good approximation to the Hazlewood model and reasonably depicts the scale of exchange. Consider first the intravascular contrast agent. As expected, the data generated by assuming fast exchange accurately predicts the input concentration. This is reasonable since fast exchange is assumed while generating the data, as well as while fitting the data. In other words, if it is “correct” to fit the data to a single exponential, then the true or input concentration should be recovered—as is the case for the fast exchange data. The simulated data generated by assuming slow exchange falls short of predicting the true tissue contrast agent concentration. This may be represented as the percent error in predicting the peak concentration, in this case 8.8% in Fig. 3a and 8.1% in Fig. 3b. This misfit is also not unexpected, since the MRI data was first generated by assuming a biexponential behavior, and then was fit by assuming a single exponential (fast exchange approximation). The data generated with the “intermediate” exchange rate (ka ⫽ 3 s-1) agrees more closely with the slow exchange data. This is clearly be-
Error Introduced by Assuming Fast Exchange
cause the srTFL (TR ⫽ 2.4 ms) experimental duration is short enough such that exchange minimization effects are beginning to manifest, as mentioned in Materials and Methods. This effect is commensurate with findings from other groups (3,4,6,26). For the case of the extravascular contrast agent in Fig. 3c,d, the slow and intermediate exchange data do not quite agree with the fast exchange data near the peak of the bolus. This misfit, a 0.7% error in predicting the peak concentration in c and 0.4% in d, is less than that observed in the intravascular case. This is a good indication that the extravascular contrast agent is less sensitive to exchange. For the case of an intravascular agent, the result of an inversion recovery experiment with an inversion time greater than, say, 100 ms would be that the experimental duration would be less rapid compared to the exchange and the exchange rate would appear to be more rapid. In Fig. 4 an inversion recovery experiment is illustrated for an intravascular agent with an inversion time of 50 ms. Even at an inversion time of 50 ms, the data generated with the “intermediate” exchange rate (ka ⫽ 3 s-1) is consistent with the data assuming the slow exchange limit. As discussed above, it is expected that the fast exchange data would predict the input concentration; however, for the slow and intermediate exchange cases the percent error in predicting the peak concentration is 14%. Of course, when using MRI to track a bolus injection of a contrast agent, fast pulse sequences with short timescales are typically used. The contrast agent concentrations predicted by using the T1-FARM pulse sequence (TR ⫽ 2.3 ms) are shown in Fig. 5 for (5a) an intravascular and (5b) an extravascular contrast agent. These results are qualitatively similar to those in Fig. 3. However, since it was assumed that several points were required to specify T1 (64 points) and a TR of 2.3 ms was assumed, then the time over which T1 is determined is becoming long enough that the conditions required to minimize exchange are no longer fully met. Therefore, the “intermediate” ka ⫽ 3 s-1
FIG. 4. Comparison of the “input” contrast agent concentration to that predicted by a single exponential fit to the simulated MRI data (irTFL with TI ⫽ 50 ms, TR ⫽ 2.4 ms) where the data was generated by assuming fast exchange, slow exchange, and an exchange rate of ka ⫽ 3 s-1 for an intravascular contrast agent.
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FIG. 5. Comparison of the “input” contrast agent concentration to that predicted by a single exponential fit to the simulated MRI data where the data was generated by assuming fast exchange, slow exchange, and an exchange rate of ka ⫽ 3 s-1, for (a) an intravascular contrast agent and (b) an extravascular contrast agent. This simulated data was generated assuming a T1-FARM pulse sequence with TR ⫽ 2.3 ms and ␣ ⫽ 9°.
exchange curve is becoming more separated from the slow exchange limit. The intermediate (slow) exchange curve has an error of 25% (34%) for the intravascular case and 7.7% (8.8%) for the extravascular case in predicting the input peak concentration. It is clear in this instance that exchange effects are less important with the extravascular agent, and thus the assumption of fast exchange is more correct in this case. It is expected that as the flow rate increases, the extraction of contrast into the extravascular space will be lessened, and thus at higher flows an extravascular agent will begin to resemble an intravascular one. For this reason the calculation summarized by Fig. 3a,c is repeated for a flow of 5 ml 䡠 min-1 䡠 g-1 and presented in Fig. 6. At this value of flow and contrast concentration, the measured concentration predicted for an intravascular agent (Fig. 6a) assuming slow or intermediate exchange deviates from the input concentration, as before. However, it is expected and shown in Fig. 6b that the calculation assuming an inter-
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FIG. 6. The calculation depicted in Fig. 3a,c are repeated here for a flow rate of 5 ml 䡠 min-1 䡠 g-1. Depicted here is (a) the result for an intravascular agent and for (b) an extravascular agent.
mediate exchange rate of ka ⫽ 3 s-1 and slow exchange will begin to deviate from the input and “fast” concentration curves at higher flows or higher contrast concentrations (resembling the intravascular agent in Fig. 6a). The errors associated with the slow and intermediate exchange cases in predicting the peak concentrations are now 16.4% for the intravascular case and 8.1% for the extravascular case. Notice that as the flow increases the extravascular curves resemble more the intravascular ones, as expected. Thus far, it has been shown that the extravascular agents are less sensitive to exchange effects than intravascular agents. However, part of the design of this project was to show that the tissue could not simply be divided into two compartments without allowing for some spatial variation across the length of the capillary. This was in order to properly account for the variation of the limits of exchange along the capillary length. This is illustrated further in Fig. 7, assuming an srTFL sequence and a T1-FARM sequence. Since these methods have shown an extravascular agent to be less sensitive to exchange, only an intravascular contrast agent is studied in three cases. In this figure the curve labeled “Intermediate Exchange,” with the open circles, represents the predicted measurement of contrast agent
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concentration from the model, as before, which calculates the contrast concentration based on a distributed model. The curve labeled “Non-Distributed Model,” with the plus signs, represents the same measured contrast agent concentration predicted, but from a two-compartment model in which the concentration within the capillary at any given instant in time is the same across its entire length. Simulated data is shown for the case of (7a) the same srTFL parameters used for Fig. 3, (7b) the same calculation but with TR ⫽ 5 ms and TI ⫽ 20 ms, in order to enhance the separation between the slow and fast exchange curves, and (7c) the T1-FARM sequence parameters used for Fig. 5a. The nondistributed is consistently elevated with respect to the distributed case. The concentration of contrast within the tissue is the same for each curve at each of the time points. It is only the distribution of contrast which differs between these two curves. This difference will therefore distort the contribution to the signal from exchange, and thus the nondistributed model will estimate incorrectly (perhaps in some cases only slightly) the degree to which exchange influences the MRI signal. Although the method used to produce the “intermediate exchange” curve (distributed model) is still an approximate calculation, it is closer to the reality of what is occurring at the level of the capillary. Examining the intermediate exchange curves, for Fig. 7a, the errors in predicting the peak concentration are found to be 6.7% for the nondistributed model and 10.6% for the distributed case, and for 7b, 17.2% for the nondistributed model and 21.2% for the distributed case. These latter errors are the same for T1-FARM in Fig. 7c. This does not imply that the nondistributed model better predicts the input concentration. Assuming that the contrast is not distributed in the capillary artificially inflates the estimate of contrast concentration. If an srTFL experiment can make measurements rapidly enough, then the exchange rate would appear slow, whereas the differences between the fast and slow exchange cases can be made to vary when pulse sequences which do not minimize exchange are used. For example, the srTFL simulation in Fig. 7b, and the T1-FARM simulation in 7c, are qualitatively similar. The overestimate of concentration resulting from using a nondistributed model appears to be on the order of 4%. These results imply that, regardless of pulse sequence, extravascular agents are less sensitive to exchange, and that for nonexchange-minimizing sequences like T1FARM, an error of ⬃21% in predicting the peak of the bolus of an intravascular agent may be incurred by fitting the data assuming fast exchange. For the exchange-minimizing pulse sequence (srTFL) shown in Fig. 7a, this error is reduced to roughly 11%. To the extent that contrast concentration varies with length along the capillaries, the use of a distributed model over a nondistributed one could lead to an improvement in predicting the peak concentration on the order of 4 –5%, at least based on the parameters chosen to generate Fig. 7. While assuming an intermediate value for water exchange across the capillary membrane, a value of ka ⫽ 3 s-1 has been assumed in accordance with other MRI studies. However, Bassingthwaighte and Goresky (13) in the development of the MMID4 model used values of permeability
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FIG. 7. a: The data presented in Fig. 3a is compared to the apparent contrast agent concentration resulting from a single exponential fit to MRI data simulated by assuming an exchange rate of ka ⫽ 3 s-1 and a nondistributed model for the tissue. The simulation was performed for an intravascular contrast agent and the same srTFL experiment used for Fig. 3 is assumed. b: The calculation is repeated with the modification TR ⫽ 5 ms, TI ⫽ 20 ms. Note the enhanced separation between fast and slow exchange. c: A similar calculation is presented based on the T1-FARM data in Fig. 5a.
surface area products for water across membranes on the order of 1000 ml 䡠 min-1 䡠 g-1. This translates roughly to a time constant for crossing the vascular– extravascular boundary of about 0.01 s, or a rate of 100 s-1. This exchange rate is two orders of magnitude larger than values quoted for intravascular– extravascular proton exchange rates within the MRI literature (1– 4,6). Model-independent measurements of water permeability of endothelial cell membranes (27) estimate a permeability of ⬃50 ⫻ 10-4 cm 䡠 s-1. For rabbit heart, assuming an exchange surface area of 560 cm2 䡠 g-1, an estimated PS is 140 ml 䡠 min-1 䡠 g-1. This estimate still predicts a time constant for water to exchange across the capillary on the order of 20 s-1. Saab et al. (28), for instance, determined (assuming compartmentalization) the limits of vascular–intravascular exchange in skeletal muscle to lie between 1 and 3 s-1. Alternatively, Beard and Bassingthwaighte (20) suggested that perhaps this discrepancy is due to the use of more computationally simple two-compartment models. Investigations are currently under way to identify this apparent discrepancy, including identifying phenomena specific to MRI other than proton exchange, which may contribute to this effect, as well as comparing experimentally the various models outlined here, in particular, how well the various techniques model bolus flow in an animal experiment. The comparison of these simulations to con-
trast-enhanced MRI measurements in a canine model is the topic of a future article (in prep.). CONCLUSIONS These simulations have shown that extravascular contrast agents are less sensitive to exchange than intravascular agents at normal flows. This is because the concentration of contrast is allowed to mix well across the capillary boundary, and therefore the concentration gradient across the boundary is minimized, thus making the NMR relaxation times on either side of the capillary membrane comparable. This tends to minimize the effects of water exchange across the membrane. An intravascular contrast agent, however, by definition remains (mostly) within the capillary. Thus, for an intravascular agent the relaxation rates in the intravascular space can be quite different than those in the extravascular space. This tends to increase the effects of water exchange across the membrane. It has also been shown that nondistributed compartment models which are used to track the flow of a bolus injection through a tissue can incorrectly estimate the effect of water exchange that is taking place at the capillary level. This implies a 4 –5% error in predicting the peak concentration of the bolus when using a nondistributed model rather than a distributed one for the parameters used in
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constructing Fig. 7. The ideal solution to this effect would be to utilize pulse sequences which minimize exchange by making MRI measurements rapidly enough for the system to be considered in the no-exchange limit. With rapid pulse sequences, and even an inversion recovery sequence with an inversion time of 50 ms, times are short enough that the exchange minimization predicted by Hazlewood et al. (5) are observed. That is, for these short times the MRI signal is the same as if the system were in the slow exchange limit. For nonexchange-minimizing sequences like T1-FARM, an error of ⬃20% in predicting the peak of the bolus may be incurred by fitting the intravascular data assuming fast exchange. Based on these findings, it is recommended that exchange minimization sequences be used when studying intravascular agents. Although simulations indicate that errors are incurred for both intra- and extravascular agents if one assumes fast exchange, these errors are less for extravascular agents. This model is a preliminary attempt to determine the effects of exchange, and in particular, the exchange variation across the capillary, which will occur with a bolus injection of contrast. As yet, the model still has several limitations. It assumes a homogeneously rapidly mixed extravascular extracellular space, it ignores the water diffusing in the extravascular and the intravascular spaces, it is only a model of a single capillary not a multicapillary model, and it does not attempt to break down the vasculature into an arterial, a capillary, and a venous segment. As these limitations are addressed in the model, the misrepresentation of exchange by utilizing nondistributed models may be further enhanced. Future plans for this model include using as an input an experimentally determined arterial input function and comparison of the theoretical and experimental results from the bolus-tracking MRI experiment in cardiac disease models. As well, in order to utilize this model in simulating experiments identifying tissue viability, this model allows the tissue to be damaged. In other words, more of the extravascular volume will be accessible to the contrast agent. Thus, the model can be compared to results in animal models of ischemic myocardium and test, for example, the flow rate dependence of the signal. This model will also be used to test the theoretical limits of T1-FARM, and the comparison of the behavior of T1-FARM to srTFL, in particular at higher bolus concentrations. Future tests also need to investigate, in particular, the rapid mixing assumption in the extravascular space. Currently, it is assumed that if some of the contrast agent enters the extravascular space at one end of the capillary, it is immediately available to enter the capillary at the opposite end. However, there must be some finite time allowed for mixing and for the diffusion to the other location of the capillary. This model, the calculation of tissue compartment concentrations, and the entire simulation was written in Matlab script, a mathematical and analytical tool developed by The MathWorks (Natick, MA). The code for this work will be made freely available on their website (http://www.mathworks.com). The simulation is composed of two main routines, one simple text file within which the parameters are specified, and a main function which is called from within Matlab. In this way parameters specific to
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other tissues and other situations may be input into the model. The use of a simulated or a measured arterial input function is also an option. By making this simulation freely available, we hope to make it easier for other researchers currently developing similar models to incorporate some or all of these ideas.
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Error Introduced by Assuming Fast Exchange 20. Beard DA, Bassingthwaighte JB. Advection and diffusion of substances in biological tissues with complex vascular networks. Ann Biomed Eng 2000;28:253–268. 21. Henderson E, Sykes J, Drost D, Weinmann HJ, Rutt BK, Lee TY. Simultaneous measurements of blood flow, blood volume, and capillary permeability in mammary tumors using two different contrast agents. J Magn Reson Imag 2000;12:991–1003. 22. Zimmerman JR, Brittin WE. Nuclear magnetic resonance studies in multiple phase systems: Lifetime of a water molecule in an adsorbing phase on silica gel. J Phys Chem 1957;61:1328 –1333. 23. Prato FS, McKenzie CA, Thornhill RE, Moran GR. Functional Imaging of tissues by kinetic modeling of contrast agents in MRI. In: Stergiopoulos S, editor. Advanced signal processing: theory and implementation for radar, sonar, and medical imaging systems. Boca Raton, FL: CRC Press; 2001.
827 24. Bellamy DD, Pereira RS, McKenzie CA, Prato FS, Drost DJ, Sykes J, Wisenberg G. Gd-DTPA bolus tracking in the myocardium using T1FARM. Magn Reson Med 2001;46:555–564. 25. Tong CY, Prato FS. A novel fast T1-mapping method. J Magn Reson Imag 1994;4:701–708. 26. Lombardi M, Jones RA, Westby J, Torheim G, Southon TE, Haraldseth O, Michelassi C, Kvaerness J, Rinck PA, L’Abbate A. Use of the mean transit time of an intravascular contrast agent as an exchange-insensitive index of myocardial perfusion. J Magn Reson Imag 1999;9:402– 408. 27. Michel CC, Curry FE. Microvascular permeability. Physiol Rev 1999; 79:703–761. 28. Saab G, Thompson RT, Marsh GD, Picot PA, Moran GR. Two-dimensional time correlation relaxometry of in vivo skeletal muscle at 3 Tesla. Magn Reson Med 2001;46:1093–1098.
