Response

The Letter to the Editor by Kuikka debates some issues within our recent publication in this journal (1). It is argued that single capillary models are mathematically poor representations of the physical phenomena of interest and that axially distributed multiple capillary models should be used instead. The claim is also made that the published model may sometimes overestimate, perhaps by 2–4 times, the true permeability-surface area product (PS). The model that was used is based on an adiabatic approximation to the tissue homogeneity model appearing in the tracer kinetic modeling (brain) literature recently (2). We attempted to make it clear in the communication that our interest was in proton (water) exchange across the intravascular–extravascular boundary. Further, we stated that (for longer transit times through the myocardium in particular) there will exist a contrast concentration gradient along the capillary length. This implies that the relaxation rate will vary with position along the capillary length. Even though the physical exchange rate of the protons remains the same, this exchange as visualized with MRI will appear to change depending on the local relaxation times (3). Our aim was therefore to study whether the proton exchange rate as perceived by MRI can be in slow exchange at some positions along the capillary, and in fast exchange at others at the same instant in time. Further, the assignment of a single exchange regime for the entire tissue is not possible in this light. It was this single effect that we wished to study, and outline its implications for MRI contrast agent kinetics. Thus, for future modeling the contrast concentration as a function of time and position was desired, and therefore a relatively simple single capillary (distributed) model which allows contrast concentration to vary with capillary position was chosen. This is addressed further in a later manuscript; however, the disputed article (1) is a description of a very necessary first step in the work. The development of a single capillary (distributed) model which allows contrast concentration to vary with position along the capillary was undertaken because we felt that multiple capillary models had too many other confounding factors which would complicate the study of this one effect. As well, models which compute residues by convolution with impulse residue functions, be they multicapillary or not, do not explicitly provide the contrast concentration as a function of both time and position within the capillary—the desired quantity for our enquiry into the exchange problem. Single capillary models as well will represent the worst-case scenario with regard to slow exchange—they are a good approximation, but we have in a sense exaggerated the problem. A bolus entering into a single capillary model will see the maximum concentration gradient across the capillary membrane, as opposed to a multiple capillary model where it would be possible for contrast to have leaked into the interstitial space from another capillary before contrast had reached the capillary under study. We have also chosen a lower estimate of PS = 0.65, again to accentuate the slow exchange problem. We can understand the disappointment that multiple capillary models are not discussed in our brief communication, and we regret this oversight. One might note, however, that we indicate in the later manuscript—now in review—that nondistributed single capillary models may not be appropriate for some cases, particularly when proton exchange becomes more slow, and that a distributed model is necessary. We are aware of Bassingthwaighte and Goresky's work on canines and their modeling—in particular, the development of the MMID4 model (4). We are also aware that the slow exchange rates observed in MRI for water crossing the capillary boundary (5) are at odds with model-independent measures of permeability (6). Personal communication with Bassingthwaighte has been very helpful in directing us to a possible resolution of this problem and we are currently investigating several hypotheses to explain this discrepancy. If it is being suggested, however, that multiple capillary models must be used in all cases, because single capillary models do not capture the complexity of the physical system, then we disagree. First of all, we are not proposing a whole-organ model, and flow heterogeneity within the myocardium has not been ignored. Our laboratory, for instance, has performed studies of the extraction fraction in normal and diseased canine myocardium (7, 8). The second of these articles by Tong et al. (8), and one more recently by Bellamy et al. (9), measure the global extraction fraction within the myocardium as a function of time using reference tracer techniques, which clearly demonstrated the heterogeneity of intramyocardial flow. Local extraction fractions within small regions were then determined by fitting to the modified Kety model, using the local myocardial flow measured with microspheres in that region. The distribution of these extraction fractions were consistent with those observed with tracer techniques. Plots of extraction fraction vs. flow (all within the same animal) were then used to determine the PS product. Although we concur that multiple capillary models are a more accurate representation of what we believe to be physiologically accurate, we disagree that single capillary models have lost their utility. Care can be taken to account for intramyocardial flow and permeability surface area products can be accurately estimated. Perhaps multiple capillary models should have been briefly discussed in our brief communication and reasons given for using the distributed single capillary model. We apologize for the oversight. “All of these models are incomplete, inexact, or just wrong in one way or another. The biology is never so precisely ordered that any model can be correct. Consequently, there is no basis for debates between users of different models where each claims that he has the 'right way.' All models are compromises.” (10).

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