Regarding “Aneurysm Rupture Following Treatment with Flow-Diverting Stents: Computational Hemodynamics Analysis of Treatment”

Letters to the Editor Letters to the Editor

Published ahead of print on April 21, 2011
doi: 10.3174/ajnr.A2534

American Journal of Neuroradiology 32:E95-E97, May 2011
© 2011 American Society of Neuroradiology

D. Fiorellaa, C. Sadasivana, H. H. Wooa and B. Liebera
aDepartment of Neurosurgery
Stony Brook University Medical Center
Stony Brook, New York

We1 read with great interest the recent publication by Cebral et al entitled “Aneurysm Rupture Following Treatment with Flow-Diverting Stents: Computational Hemodynamics Analysis of Treatment.”1 The postprocedural rupture of previously unruptured aneurysms after flow diversion (FD) is an uncommon but devastating complication. Correspondingly, any analysis that would allow operators to identify aneurysms at high risk for rupture after FD could improve the safety profile of the treatment strategy.

Cebral et al1 used a computational fluid dynamics (CFD) model to generate results, which suggest that FDs, in selected cases, may induce dramatic elevations of intra-aneurysmal pressure leading to postprocedural rupture. The authors further suggested that their numeric simulations would allow them to prospectively identify those aneurysms at risk for rupture after FD. Theypresent 7 aneurysms, 3 with postprocedural rupture and 4 that were treated successfully. The selected aneurysms and arterial segments, along with assumptions about flow through the segments, constant pressure at the outlet of the segments, and rigid arterial walls, were evaluated within a CFD model. Using this model, their calculations showed that all 3 aneurysms which went on to rupture after FD demonstrated severe increases in intra-aneurysmalpressure (>20 mm Hg) after treatment, while those aneurysms that did not rupture after FD did not exhibit such dramatic pressure rises (<3 mm Hg).

While this hypothesis is very attractive, the modeling upon which these conclusions are based, seems flawed. The article underscores the general need to critically evaluate the results and carefully parse the pictures output by CFD programs, especially when pressures are reported. Given that the concepts are presented within the context of a complex mathematical argument, it may be difficult for readers who are inexperienced in fluid mechanics or CFD to fully appreciate the details of what the authors areproposing.

The major problems with the study include the following: 1) a vast overestimation of the physiologic pressure gradientsoccurring across short, tapered, and/or tortuous vascular segments; and 2) a selective invocation of cerebral autoregulation.

The Baseline Pressure Gradient Values in the Cebral Model Are Incorrect

A computational domain, which is usually a truncated segment of a vascular bed that is used to model hemodynamics, requires the specification of inlet and outlet flow conditions as well as the interaction of the fluid with the confining vasculargeometry. Because the actual flow rates into the selected segments were not measured, they were calculated to generate an average (over the cardiac cycle) wall shear stress at the inlet of 15 dyne/cm2. The authors also elected to set the outlet pressure to zero throughout the cardiac beat. The inlet pressure was then calculated by using the Navier-Stokes equations. Pressure distributions were then obtained by increasing the systolic input pressure to 120 mm Hg, while they also increased the output pressure to a nonzero value such that the pressure drops through the domain remained unchanged. The authors elected not to model the distensibility of the artery (ie, they neglected the important windkessel effect when attempting to predict pressure drops). Previous studies have shown and the authors themselves note that using rigid walls can result in unrealistic pressure gradients across the domain.2,3 Nonetheless, applying this set of assumptions to the modeled vascular segments, the CFD computations calculated pressure drops across these short vascular segments ranging between 20 and 40 mm Hg. These computations are vast overestimates that are in direct conflict with in and ex vivo experimental data as well as conventional fluid mechanics.

Banerjee and Back3 measured the pressure gradient that is required to drive blood through a 5.2-cm segment of the canine femoral artery in vivo (3.8-mm diameter with a slight taper) under physiologic conditions of flow approximating those found in the internal carotid artery of humans (ie, the values of the Reynolds and Womersley numbers are equivalent and similar to those applied by Cebral et al1–3) In these experiments, Banerjee and Back measured a cyclic peak pressure drop of approximately 4.3 mm Hg. This pressure drop is attributable to the combination of the pressure required to overcome viscous dissipation (the cycle average pressure drop, approximately 0.6 mm Hg) and inertia during acceleration to peak systole (approximately 3.7 mm Hg). Thus, these experimentally measured pressure drops are almost an order of magnitude smaller than those calculated by Cebral et al.

The pressure drop across a vascular segment of known dimensions can be calculated by using basic principles of conventional fluid mechanics as well—namely, the Poiseuille calculations for the viscous pressure drop and Newton’s second law for the pressure drop required to overcome inertia. Over a vascular segment with dimensions similar to those described by Banerjee and Back3 and Cebral et al,1 these calculations yield 0.51 and 3.12 mm Hg, respectively, for viscous dissipation and inertial pressure drop, for a total pressure drop of 3.63 mm Hg. Thus, values calculated by using the basic principles of fluid mechanics under ideal flow conditions, while somewhat lower (by 16%) than the measured experimental values mentioned, are nevertheless within a range of reasonable physiologic variance and again are almost an order of magnitude lower than those calculated by Cebral et al.

Differing Mechanisms are Employed to Obtain Uniformity Among the Ruptured Aneurysms after FD

According to the authors, the intra-aneurysmal pressure increases were caused by 1 of 2 different mechanisms:

Mechanism 1

Patients 1 and 3

The authors assume that the prestent arterial configuration proximal to the aneurysm created an enormous resistance to flow (and therefore a marked pressure drop) that was substantially relieved by the placement of the FD, such that the higher pressure at the inlet propagated further downstream and into the aneurysm.

In patient 1, a mild (50%) stenosis proximal to the aneurysm was resolved by placement of the FD. The dilation of this 50%stenosis after FD placement resulted in a calculated (“virtual”) reduction in the trans-segmental pressure drop from 25 to 5mm Hg. The authors suggested that this marked reduction in the trans-segmental pressure drop of 20 mm Hg resulted in higher pressures over the entire parent artery segment and that these higher pressures were transmitted directly to the aneurysm fundus. They state, “As a consequence [of the preaneurysm stenosis dilation], the intra-aneurysmal pressure was increased by 20 mm Hg.”

According to basic hemodynamic principles, the removal of a 50% stenosis reduces segmental resistance only minimally. Based on previous experiments using vessels with dimensions analogous to those described by the authors, removal of a 65% stenosis results in a reduction in the trans-segmental peak pressure drop of half the value reported by the authors.4,5Therefore, the calculations performed by Cebral et al1 result again in an overestimation of the trans-segmental pressure drop by at least a factor of 2 compared with traditional hemodynamic calculations and measurements. The authors opine that “This effect is well-known by endovascular specialists and is readily understandable.” We speculate that they might have difficulty finding an experienced endovascular interventionist who would predict a 20 mm Hg reduction in the trans-segmental pressure drop as a result of the dilation of a mild focal pre-aneurysmal stenosis.

In patient 3, the authors attribute a large baseline pressure drop of 45 mm Hg to parent artery tapering proximal to the aneurysm and a poststent pressure drop of approximately 20–25 mm Hg to a sharp turn distal to the aneurysm. If such a high pressure drop could be attributed to 1 arterial segment, the heart would have to generate super-physiological pressures to overcome all of the sharp arterial turns and vascular taperings that occur between the aortic outlet and through the tortuous cerebrovasculature to reach the capillary outflow. As in patient 1, the authors noted that this pre-existing taper was improved after FD placement. Thus, on the basis of these minimal changes in the vessel diameter after treatment, the authors calculate a profoundly exaggerated effect on the posttreatment pressures.

In evaluating the accuracy of the proposed calculations of Cebral et al,1 we can once again consider previous experimental studies and conventional fluid mechanics on arterial tapers6 and turns7 as a reference point for physiologically relevant pressure drops. For a 33% reduction in the diameter between an inlet and outlet of a 6-cm-long artery, the viscous pressure drop increases by 15% due to the taper compared with the Poiseuille value.6 There will be an additional pressure drop due to the change in momentum of approximately 4 mm Hg (derived from the Bernoulli equation). Then, the pressure drop required to overcome inertia in the vessel segment during acceleration to peak systole needs to be considered. Adding these 3 components, the total pressure drop across a “significantly” tapered artery would be around 5.6 mm Hg, which, once again, is an order of magnitude lower than the 45 mm Hg value proposed by Cebral et al. In calculating the flow through pipes, the energy losses attributable to turns are classified as minor losses and (for a 180° turn) are estimated to be approximately 20% of the kinetic energy of the fluid7; for blood under the proposed conditions, this value would be insignificant.

Mechanism 2

Patient 2

The authors assume that the implantation of the FD construct caused a significant increase in the resistance of the treatedvascular segment, resulting in a subsequent reduction in flow. At this point, they selectively invoke “autoregulatory mechanisms” to raise the inlet pressure (ie, the systemic blood pressure) to maintain a constant flow rate through the segment. This “flow diverter–induced systemic hypertension” is then assumed to be transmitted directly to the aneurysm.

These are perhaps the most questionable assumptions in the entire study. The authors report that after the implantation of the FD construct, the pressure gradient required to drive equivalent flow though the analyzed segment increased from 20 to 48 mm Hg! They attributed this marked increase in the trans-segmental resistance to the tortuosity and tapering of the vascular segment being exacerbated by the implantation of the FD construct and to the “removal” of the aneurysm from the vascular circuit.

First, with respect to the effect of the FD on the segmental vascular resistance, such an increase in the trans-segmentalpressure drop over a domain only a few centimeters in length, whether or not it includes an aneurysm, would be an impossibility from either a physiologic or hemodynamic standpoint. From a hemodynamic standpoint, the only way that the pressure would increase so dramatically would be for viscous dissipation and inertial forces to be tremendously increased after FD placement. Even if we assume that 2 FDs were placed concentrically inside each other inside the treated segment causing a luminal loss equal to approximately 200 µm of cross-sectional diameter, the increases in pressure required to overcome the minimal stenotic effects of the FDs on the luminal cross-sectional area would be trivial. They can be calculated by using the Poiseuille and Newton laws to be 0.025 mm Hg for viscous dissipation and 0.030 mm Hg for inertial resistance or a total of 0.055 mm Hg increase in peak pressure.

Second, the authors suggest that the placement of an FD results in the removal of the “low resistance” aneurysm from the vascular circuit, which now requires that all of the blood flow be shunted through the “higher resistance” reconstructed parent artery. They illustrate this “circuit” in Fig 5 that shows 2 parallel unlabeled individual resistors—1 representing the short arterial treated segment and the other representing the aneurysm itself. In analogy to basic electric circuits, hydraulic resistance is defined as pressure drop divided by the through-flow. Because flow has to enter and leave the aneurysm through the neck, it is reasonable to expect that the pressure gradient between the proximal and the distal side of the neck is zero or very close to zero. Therefore, the resistance to flow into and out of the aneurysm is nearly zero. However, even if the aneurysm “shunt” is removed and hypothetically replaced by a healthy blood vessel segment or a stent graft, only the “high resistance” over a short 2-cm segment would remain. The peak pressure drop over such a short segment can be estimated as approximately 40% of the pressure drop measured by Banerjee and Back3 over a longer segment of 5.2 cm2,3; using linear interpolation, the peak pressure drop would then be approximately 1.6 mm Hg. Because the aneurysm is not completely blocked by the FD, the total resistance through the parallel branches of the FD and the aneurysm would actuallybe less than that through the arterial segment alone; therefore, the actual peak pressure drop would likely be even lower than 1.6 mm Hg.

This profound increased resistance and exaggerated pressure drop after flow diversion is the opposite of the effects observed for patients 1 and 3, and runs counter to the proposed theory of elevated intra-aneurysmal pressure leading to rupture. For this case to conform to the “elevated intra-aneurysmal pressure” theory, it was necessary to substantially increase the inlet pressure through a “complex system of autoregulation” to maintain cerebral blood flow. This assumption of “flow diverter-induced systemic hypertension” then makes the aneurysm of Patient 2 conform to the elevated intra-aneurysmal pressure theory. After the authors apply the assumption that the systemic blood pressure increases by 25 mm Hg to compensate for this elevated trans-segmental resistance and reduced flow, they are able to conclude that the intra-aneurysmal pressure increases by a similar amount.

That the implantation of an FD induces an increase in the systemic blood pressure of 25 mm Hg would seem to be easily measurable by the operators, their anesthesiologists, and critical care support staff. The blood pressure tracings from the peri-procedural period could be presented to validate such a claim.

Interestingly, their initial analysis demonstrated only a 2 mm Hg increase in intra-aneurysmal pressure following FD. Thisnegligible rise in intra-aneurysmal pressure following FD required the invocation of an “autoregulatory” mechanism to result in an elevated post-treatment intra-aneurysmal pressure. Contrarily, when the calculations in Patients 1 and 3 resulted in the opposite problem (a reduced pressure drop across the reconstructed segment due to the removal of a stenosis or tapering), systemic autoregulation as a mechanism for maintaining pre-FD perfusion was not used.

Summary

The conclusions presented by Cebral et al1 are based upon results that are not only largely inconsistent with both existing experimental data and basic fluid mechanics but are selectively applied to achieve a consistent conclusion for a group of aneurysms with a known outcome (postprocedural rupture). It is important to recognize that the results or conclusions presented in the manuscript have not been validated by either dynamic angiographic data or direct physiologic measurements. These results are solely the product of mathematic simulations that are only as valid as the assumptions on which they are based. As such, we would urge the readership to exercise extreme caution before incorporating any of the concepts proposed by Cebral et al, into clinical decision-making (eg, in the selection of patients to be treated with flow-diverting devices) or into the design of clinical research studies.

 

Footnotes

Disclosure: Drs Fiorella and Woo are unpaid consultants and proctors for Chestnut Medical/eV3. Dr Lieber is a stockholder in Surpass Medical Inc.

References

  1. Cebral JR, Mut F, Raschi M, et al.Aneurysm rupture following treatment with flow-diverting stents: computational hemodynamics analysis of treatment. AJNR Am J Neuroradiol 2010;32: 27–33.Epub 2010 Nov 11[Medline]
  2. Roy AS, Back LH, Banerjee RK.Evaluation of compliance of arterial vessel using coupled fluid structure interaction analysis. MCB Molecular and Cellular Biomechanics 2008;5: 229–46
  3. Banerjee RK, Back LH.Computed and measured hemodynamics in a compliant tapered femoral artery.In:Proceedings of the 2003 Summer Bioengineering Conference,Key Biscayne, Florida. June 25–29, 2003:261–62
  4. Young DF.Fluid mechanics of arterial stenoses. J Biomech Eng 1979;101: 157–75[CrossRef]
  5. Young DF, Cholvin NR, Roth AC.Pressure drop across artificially induced stenoses in the femoral arteries of dogs. Circ Res 1975;36: 735–43[Abstract/Free Full Text]
  6. How TV, Black RA, Annis D.Comparison of pressure losses in steady non-Newtonian flow through experimental tapered and cylindrical arterial prostheses. J Biomed Eng 1988;10: 225–30[CrossRef][Medline]
  7. Chandran KB, Yoganathan AP, Rittgers SE.Unsteady flow and nonuniform geometric models. In:Chandran KB, Yoganathan AI, Ritters ER eds. Biofluid Mechanics: The Human Circulation.Boca Raton, Florida: Taylor & Francis; 2006

 

Reply

Published ahead of print on April 21, 2011
doi: 10.3174/ajnr.A2560

American Journal of Neuroradiology 32:E98-E100, May 2011
© 2011 American Society of Neuroradiology

C.M. Putmana
aInterventional Neuroradiology
Inova Fairfax Hospital
Fairfax, Virginia

P. Lylykb
bInterventional Neuroradiology
Clinica Sagrada Familia
Buenos Aires, Argentina

J. Cebralc
3Center for Computational Fluid Dynamics
George Mason University
Fairfax, Virginia

We appreciate the opportunity to respond to criticisms of our article in the letter to the editor by Fiorella et al. The comments by Fiorella et al are emphatically presented, full of misrepresentations, and dismissive of our work. We find them insulting and offensive in their tone and attack on our scientific integrity. However, there are several interesting points raised by Fiorella et al that would benefit from discussion.

Fiorella et al are dismissive of computational fluid dynamics (CFD) approaches as complex mathematic constructs that are fantasies without connection to the real world. This opinion may be based on a misunderstanding of how our group used the method and our motivation in using the tool.

CFD is a method of designing scientific experiments in a controlled, safe, efficient, and cost-effective manner. Our study of the current group is not an attempt to perfectly reproduce the exact conditions of each of the patients presented but, in a scientific controlled manner, to explore the relationships among physical parameters. We are interested in the physiologic effects of placement of a flow diverter (FD) in these systems. Our approach is to use CFD to identify hypotheses that can be tested and confirmed with clinical data. If we have confused Fiorella et al or the reader on this, we are concerned and wish to make it perfectly clear that we have consistently advised care in generalizing our conclusions to patient care without further investigation. If the goal of Fiorella et al is to reinforce this notion, we completely agree.

In this long rambling attack, Fiorella et al caution the reader that our calculations must not be correct because the results do not conform to their experience. We firmly believe the calculations are correct for the conditions imposed in this series of experiments. The solver used for this work has been extensively validated in studies comparing with physiologic and imaging data and against commercially available solvers in both medical and nonmedical applications.1–8 The solver and our CFD method have been specifically validated against in vitro data for pressure drops in arterial stenoses.9 We are confident that the mathematic calculations are correct (we can provide the geometries necessary if others wish to perform their own simulations). They have been rechecked and tested in multiple simulations on these specific geometries. In addition, we have done extensive sensitivity analysis to evaluate the impact of changes of input assumptions on our method.10–14

However stated, Fiorella et al are correct in pointing out that the actual values presented are not accurate “measurements”of the values that could have been measured in these individual patients (if that was or could have been done). As stated above, Fiorella et al misunderstand the purpose of this series of experiments. To obtain accurate values, we would need the actual flow conditions in each patient. In that situation, we would not be doing a controlled experiment for this study group for the input and output flow conditions. In this study, we were controlling for everything other than geometry and the placement of an FD. For those conditions, we stand by the relationships we have identified and have proposed a possible mechanism for delayed rupture. Whether this is proved or not is up to further work by anyone interested.

As we stated in our article, there are always limitations to CFD modeling due to incomplete data on which assumptions aremade. We have no direct physiologic measurements, so we used estimations that we think are representative of the physiologic range. Different input and output assumptions will yield significant changes in the magnitude of pressure drops but will not change the relationships between them. Our conclusions are based on the potential increase in intra-aneurysmal pressure in these specific situations, but the absolute values are very dependent on the exact flow rates in the treated arteries. Our estimated flow rates are based on data provided in the literature referenced in our article and yield pressure increases that we found only in the aneurysms that had delayed rupture. This observation leads us to recommend further study of this potential mechanism. We now advise the reader not to dismiss our work due to the criticisms of Fiorella et al.

However, what would a fair review of our data say about the assumptions made in these experiments? Fiorella et al asserted that our baseline pressures are “incorrect” and point out a conflict with “in/ex vivo” experimental data. Careful analysis of our data will show that the pressures are only in conflict in the analyses done on those patients with delayed rupture. The 4 cases successfully treated have pressure drops very similar to the 4–5 mm Hg drops championed by Fiorella et al. The calculations, geometric methods, assumptions, input, and outflow assumptions were appropriately held constant through the study group. If we had available patient-specific physiologic data, we agree that this could be applied to the simulations to improve the accuracy of the absolute values of the pressure changes. We had neither these data nor, for that matter, “dynamic angiographic data,” as implied by Fiorella et al, to refine our assumptions and/or validate our predictions. As we clearly stated, we chose to hold flow conditions constant.

So what is different between the 2 groups and the most important determinant of blood flow in these studies? It is the patient-specific geometry. The studies referenced by Fiorella et al are of arteries with simpler geometries (less irregularity, no aneurysms, stenoses, or extreme tortuosity, and so forth), so they are not readily comparable with the cases we studied. Furthermore, as should be well known to Fiorella et al, Poiseuille’s Law can cause significant inaccuracies in complex anatomies and flow conditions, which is why the Navier-Stokes equations must be solved for situations as complex as we are studying. This is precisely what we are reporting, and we have clearly defined the conditions we used.

So, Fiorella et al are dismissing our calculations because they fall outside of their experience. Perhaps, no one should besurprised. We are presenting work that is outside of everyone’s common experience. Coupling a cerebral aneurysm to vascular stenoses, complex tapering, and tortuous arteries has not been reported for these types of situations. In addition, cases that have a history of delayed hemorrhage following FD treatment are extremely uncommon. We are the first to report such an analysis. Perhaps, we should be surprised! Using “common sense” when considering possible failures related to an FD device, one would naturally consider abnormal and potentially negative changes to the hemodynamic environment that could be caused by affecting the flow in these patients with aneurysms. Logically, an unplanned elevation of the intra-aneurysmal pressure by the treatment came to mind. Somewhat surprising, to our knowledge, no data or discussions on this issue from those promoting this technology have been reported. Fiorella et al and other participants in the FD technology have not produced any experimental or computational data to alleviate this concern. We look forward to the release of this information for an appropriate peer review.

As correctly pointed out by Fiorella et al and discussed in the article, we did not model the distensibility of the artery. We do not agree (and for that matter never stated) “that using rigid walls can result in unrealistic pressure gradients across the domain.” In fact, we only noted that there is an influence on pressure gradients, and we do not agree with the characterization of the production of an unrealistic gradient. Fiorella et al did not calculate the expected impact of this assumption, roughly an overestimation of approximately 10% (estimated by considering the geometric change due to distensibility in a Poiseuille flow). They also did not comment on the implications of implanting an FD on the parent artery and aneurysmal compliance as well as the possibility that these diseased arteries may be stiffer. Common sense would suggest that these effects could make the rigid wall approximation even more realistic. This issue probably disserves further study. Again, in our opinion, neither of these will be large enough to alter our conclusions.

Fiorella et al have made a ridiculous but serious charge that we have twisted the data to “conform” to our preconceived assumption. We are quite insulted by this accusation and strongly confirm that this is absolutely false. We have no vested interest in promoting or attacking this technology. Our interest has been entirely related to advancing the understanding of the pathophysiology of cerebral aneurysm disease by the medical communities. Our interests are entirely misrepresented by the statements of Fiorella et al. The misguided comment of Fiorella et al may relate to an incomplete understanding of the handling of patient 2. Their charge that we have selectively applied certain assumptions to the treatment failure cases is objectively false. As stated above, assumptions have been held constant for all cases, and these data are reported. We did explore changing the hemodynamic conditions (and disclosed the reasoning for this) in patient 2. Both results were reported.

Perhaps one could erroneously come to the conclusion that we purposefully created an unrealistic condition to invoke a pressure increase if one fails to faithfully read and understand our article. If our reasoning is not clear, we again apologize.We reported the results with and without the assumption of “autoregulation” so that the reader can understand the context in which we see a possible clinical concern. We observed an increase in vascular resistance through the parent artery following placement of the FD, implying a reduction in overall parent artery blood flow. As we have reported, no increase in the intra-aneurysmal pressure was observed in our result. We agree that this aneurysm does not share the features we identified in the other 2 patients. From our experience with collateral circulation provided by the circle of Willis, this is the most likely result because distal demand could be met by compensatory flow from collateral circulation and a restoration to pretreatment flow rates would not be required.

However, we also considered the clinical situation in which no compensatory flow is possible (ie, isolated vascular territorywith an absence or inadequate collateral circulation). Our argument is that in that situation, the flow rates would have to be maintained to meet the demand of the brain relying on this artery. We chose to model a range of flow rates up to the pretreatment rates to understand the potential effects. These we have reported, and we have tried to explain our hypothesis. We have not claimed that this has been clinically observed or was active in this particular case. We do not have a sufficient evaluation of the circle of Willis to make this determination. However, as flow rates increased in this particular patient, our calculations show that the intra-aneurysmal pressure increased. Pointing out this hypothesis could provide the incentive to measure systemic blood pressures or intra-arterial pressures so that determination of the clinical importance of this potential effect could be studied in the right clinical context.

Fiorella et al are skeptical of the resistance increase imposed by the placement of the FD in patient 2 and argue that our calculations must be wrong because of the values obtained. The absolute values of the resistance are related to the flow conditions and pressure gradients imposed. We are certain that Fiorella et al have faithfully provided numbers as accurate as possible, but they have not made it clear that they are not repeating our calculations for our specific cases and flow conditions. They have simply applied their own physiologic “guesses” (to use their words) to achieve numbers that they believe. We are most concerned about the relationships that are involved. It is not difficult to understand why the FD is actually a higher resistance system compared with the pretreatment state in patient 2. Simply the cross-sectional area of the arterial system is dramatically reduced (ie, the aneurysm is no longer used as a flow path). Basic hemodynamic principles seem to quite clearly indicate that this would cause an increase in resistance. The actual values obtained are dependent on the flow conditions imposed.

Similarly, Fiorella et al have characterized the posttreatment changes in pressure gradient in patient 1 as defying basic hemodynamic principles. This appears to be a bit of an overstatement because basic hemodynamic principles predict a pressure gradient across a stenosis. So, Fiorella et al appear to mean not this but that they cannot agree that the magnitudes of changes we report are understandable for what they define as a “mild stenosis.” With the limited images we provided, we doubt they are in a position to accurately assess the geometry of the stenosis. The treating physicians of this patient (Drs Lylyk and Ceratto), independent of this study, reported this value that we relay as an approximation and do not put it forth as a scientific analysis of the actual geometry. Regardless, the geometry we used is obtained from the 3D rotational angiography data obtained by Dr Lylyk and his team at the time of treatment. The calculations we made are faithful to the input geometry we presented. Navier-Stokes equations yielded the result presented.

Fiorella et al again approximate the pressure gradient on the basis of simplified geometry in 3 independent components toachieve a result significantly lower than that obtained by the more appropriate Navier-Stokes equations and the accurate anatomy. This again neglects the complex change in flow and wall shear stress and their effects on inertia and pressure differentials. They cite a 15% drop in viscous pressure in a tapered artery of >6 cm compared with the Poiseuille value. The tapering evident in patient 3 is well in excess of 33% and is a nonsymmetric change in diameter. Because flow patterns are not laminar, making any comparison with the case in patient 3 is problematic. Certainly, a better understanding of why the Navier-Stokes equations gave the reported result will be important in solving this discrepancy.

Rather than consider other possible explanations, Fiorella et al have dismissed the results as impossible and concluded that for this reason, the calculations are erroneous. They seem not to consider the complexity of the situation and all the changes that treatment has imposed on the system. In addition to opening the stenosis, placement of the FD made a significant change in the aneurysmal and downstream environment. The flow into the aneurysm is dramatically reduced; this reduction has the effect of reducing the wall shear stress and the dissipation of the kinetic energy of the flow in the aneurysm. With further analysis, it should be possible to understand why Navier-Stokes would predict this change. Scientifically, we believe it is important to understand this result but cannot support the dismissive attitude put forth by Fiorella et al. As stated earlier, this issue has not been studied and conventional “experience” likely is misleading.

Within this discussion, Fiorella et al go on to misrepresent our statements and then sarcastically criticize. We did not state that the endovascular specialist would predict a 20-mm Hg pressure change by opening the stenosis. We simply said that the basic hemodynamic principle of a pressure gradient being formed at a stenosis is commonly understood by those familiar with endovascular treatment. As Fiorella et al should well know, opening of stenoses has been cautioned in a variety of clinical situations in the cerebrovasculature, including ischemic disease and aneurysms because of the propagation of increased pressure into the distal pathology by the amelioration of the stenosis. This is hardly new to an experienced endovascular specialist. We do not believe the dismissive and derogatory statements of Fiorella et al are justified.

We have suggested a potential adverse mechanism that could lead to posttreatment ruptures. Fiorella et al suggest ignoring this possibility, while our suggestion has been and continues to be to further study this possibility to determine if it actually takes place in some aneurysms. If so, clinicians could be in a position to formulate possible preventive measures to save patients from these devastating complications. Fiorella et al have characterized CFD as “mathematic calculations that are at best physiologic guesses.” They are entitled to their opinion, but we do not find their arguments to be particularly compelling on a theoretic basis, unsubstantiated by any appropriate scientific analysis of these specific cases or any independent experimental work, dismissive without a balanced assessment of the complex interrelations in these systems, and offensive in erroneously assigning unethical and unscientific motives to our work. We do agree and have attempted to clearly relay our caution in applying these results to clinical treatments. We are not in agreement that these issues should be dismissed and believe that our data are a reasonable justification for studying these mechanisms. Because we have no vested interest in the results, we welcome ultimate settlement of this dispute with sufficient scientific methods regardless of what may be the ultimate result.

References

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Regarding “Aneurysm Rupture Following Treatment with Flow-Diverting Stents: Computational Hemodynamics Analysis of Treatment”