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Mathematical Modeling of Ventricular Assist Device Function and Blood Flow Generation: Assembling the Heap of Stones

Rajagopal, Keshava

doi: 10.1097/MAT.0000000000000473
Invited Commentaries

From the Department of Advanced Cardiopulmonary Therapies and Transplantation, University of Texas-Houston McGovern Medical School and Memorial Hermann-Texas Medical Center, Houston, Texas.

Disclosures: The authors have no conflicts of interest to report.

Correspondence: Department of Advanced Cardiopulmonary Therapies and Transplantation, University of Texas-Houston McGovern Medical School and Memorial Hermann-Texas Medical Center, 6400 Fannin, Suite 2350, Houston, TX 77030. Email: keshava.rajagopal@uth.tmc.edu.

Scientists try to understand nature. Engineers try to make things that do not exist in nature

Y.C. Fung

Le savant doit ordonner; on fait la science avec des faits comme une maison avec des pierres; mais une accumulation de faits n’est pas plus une science qu’un tas de pierres n’est une maison.

J.H. Poincaré

Translation: the learned man (scientist) must set in order. One performs (creates) science with facts like a house with stones. But an accumulation of facts is no more science than a heap of stones is not a house.

Mechanical circulatory support technologies have been engineered to meet the clinical needs of patients with end-stage heart failure. This is a specific case of a general notion, namely that technology applies scientific principles to accomplish practical tasks that otherwise could not be accomplished; Dr. Fung’s remark underscores this. Yet throughout history, technology development—the output of engineering—has preceded the scientific understanding that underpins it. Perhaps the most dramatic examples are the ancient Egyptian pyramids of Giza. The largest of these structures, built around 2560 BC, was the tallest man-made structure on Earth until the Lincoln cathedral was built in 1300 AD. Remarkably, “classical” engineering mechanics was developed beginning in the 16th century AD—approximately 4000 years after the Great Pyramid and more than 200 years after the Lincoln cathedral.

This general observation that technology development temporally precedes scientific understanding also holds true in the case of mechanical circulatory support. The seemingly simple idea of using an artificial pump to assist or replace the function of a native ventricle or heart has been applied clinically well in advance of a comprehensive and firmly sound theoretical framework. A sound theoretical framework not only can capture and explain clinically observed phenomena, but has predictive capacity with respect to phenomena that may not yet have been observed (because the conditions that result in the phenomena may not yet have been caused to occur). Clinical work thus motivates scientific understanding, which then impacts on subsequent clinical work. Ultimately, although we like to think that the arrow of medical progress is from “bench to bedside,” far more often it is an arrow of biomedical scientific progress from “bedside to bench,” that has post hoc bench to bedside implications for subsequent clinical work.

Approximately 3 decades of clinical work with pulsatile and continuous-flow ventricular assist device (VAD)/artificial heart technology have been the impetus for numerous scientific investigations. This work has been across a wide array of fields, ranging from studies of how VAD support alters essential cardiomyocyte processes, to theoretical work such as the current study of how VAD outflow graft angulation alters particle circulation within the aortic arch and branch vessels. In this issue of the ASAIO Journal, Aliseda et al.1 report the results of state-of-the art mathematical modeling studies of blood flow regimes and cellular component motion as functions of varying angles of the continuous-flow VAD outflow graft-to-ascending thoracic aorta anastomoses. The authors found that outflow graft angulation with the ascending thoracic aorta had substantial influence on aortic arch flow patterns and platelet circulation and recirculation within the aortic arch and branch vessels. Lower degrees of angulation were associated with less chaotic and recirculatory flow patterns, as well as less platelet deposition, than higher degrees of angulation. The authors suggest these findings may impact on surgical techniques for VAD implantation, which if adopted should reduce the risk of aortic arch and arch vessel thrombus formation and embolization.

The studies of Aliseda and colleagues are distinguished by important concepts that are worth reviewing because not all computational fluid dynamics (CFDs) are created equal. Most importantly, the authors employ both “Lagrangian” and “Eulerian” approaches, not solely Eulerian approaches. Most CFD techniques, particularly those used to study complex phenomena, use an exclusively Eulerian approach. An Eulerian specification of a flow field is characterized by examining fluid motion through a fixed specified location in space, as a function of time. This may be thought of as the view of an external observer of fluid motion. In contrast, a Lagrangian specification of a flow field is characterized by examining a specified quantity (with mass and volume) of fluid, and tracking its motion through space as a function of time. This may be thought of as the view of the fluid itself as it moves. The material derivative relates these two specifications of flow fields.2 Although these two approaches to describing fluid flow can thus be interrelated, there is an important difference, as summarized in an introductory fluid mechanics textbook: “However convenient kinematically, the spatial [Eulerian] description is awkward for questions of principle in mechanics, since in fact what is happening to the body [the moving fluid], not to the region of space occupied by the body, enters the laws of dynamics.”2 The Lagrangian approach, although more mathematically complex, truly provides a more comprehensive description of fluid motion, as it tracks particle or parcel motion through time. This is the dominant approach that Aliseda et al. have employed, and is ideally suited to examining the motion of their particles of interest—platelets.

For the purposes of a mathematically and mechanically simpler initial study, the authors assume persistent aortic valve closure and a “fully unloaded patient for whom cardiac output solely originates from VAD support.” They also model the systemic arterial circulation as a two-element Windkessel system, with both resistance and capacitance elements. However, this model of the systemic arterial circulation is somewhat at odds with fully unloaded conditions. The cardiac output solely originating from VAD support is a very specific subset of the state of persistent aortic valve closure. Persistent aortic valve closure simply means that the interplay of left ventricular (LV) volume, LV contractility, and the specific afterload to ejection across the aortic valve (because the LVAD is a second pathway for blood flow in parallel to transaortic valvular ejection, it increases the afterload to LV ejection across the aortic valve, yet decreases overall LV afterload because impeller action along the second parallel pathway generates a large transpump pressure gradient with low inflow pressure and high outflow pressure) is such that the aortic valve is always closed. The LV yet may be volume-loaded to an extent that there is phasic LV ejection as a booster pump into the LVAD, with resultant pulsatility. Definitionally under such circumstances, the cardiac output does not solely originate from VAD support despite the aortic valve being persistently closed. The entirety of the cardiac output being VAD function-derived requires either minimal LV preload (i.e., full unloading) or sufficiently poor LV contractility that even with some level of LV preloading, essentially no contribution to the cardiac output is present. In summary, the entirety of cardiac output being dependent on VAD function requires 1) either complete LV decompression or extraordinarily poor LV contractility, and 2) persistent aortic valve closure. However, truly complete LV unloading will result in constant flow at a fixed pump speed; in this state, the second material parameter of the Windkessel model—the capacitance of the systemic arterial circulation—would be rendered functionally irrelevant. Thus, it is unclear whether Aliseda et al. intended to study conditions in which although the aortic valve is closed, LV contractility and some level of preloading result in pulsatile flow through the LVAD and outflow graft or, on the other hand, whether they intended to study constant-flow conditions, in which case capacitance is unimportant. In the former case, we would have to know what the LV stresses and strains (which can be extrapolated to global variables such as pressure and volume) are as functions of time. It appears that the second case, that is, complete LV unloading, was the one studied, based on a specified “continuous” (unclear if constant) flow rate of 5 L/min through the outflow graft. However, the governing equations used in these studies are the Navier–Stokes equations formulated specifically to study unsteady flow, that is, the time derivatives of the stresses and velocities are non-zero; pulsatile flow is a type of unsteady flow. Further work from this group and others may clarify these matters by examining a wide range of LV and LVAD loading and blood flow conditions, perhaps even incorporating aortic valve opening and dual flow paths into the proximal aorta.

Then there is the matter of the fluid medium itself, that is, blood. Blood is treated as a homogeneous, incompressible Newtonian fluid. Macroscopically, this continuum model is a sound and well-established characterization of blood. Platelets are modeled as neutrally buoyant particles within this medium, to undertake the Lagrangian particle tracking that renders these studies unique. Although this is highly innovative, an important oversimplification is present. Erythrocytes are overwhelmingly the highest percentage contributors to the cellular fraction of blood, and account for a substantial amount of blood viscosity. Yet, in these studies, erythrocytes (and leukocytes) are not modeled as particles, but rather as contributors to the continuous phase. This would be reasonable if platelets were substantially larger than erythrocytes, which however is not true. Yet, it is uncertain whether separately incorporating erythrocytes as particles with different properties than platelets, while rigorous, would actually alter the data substantially; the ratio of platelets to erythrocytes is so low that it may be reasonable to treat the continuous phase as essentially a homogeneous erythrocyte/plasma medium. Finally, to study a problem such as thrombus formation, the sizes of the regions studied become sufficiently small such that blood cannot be reasonably modeled as a continuum of any kind; that is, plasma and the cellular fraction are de facto separate. This becomes important for studying problems such as the ones that the authors ultimately look towards—the effects of blood flow patterns on thrombus formation.3,4

These three criticisms notwithstanding, the mathematical and mechanical inventiveness and rigor of the studies of Aliseda et al. is to be admired. Mathematical/physical models are almost invariably approximations of the truth. This is because approximation is necessary to computationally solve complex problems. If the mathematical equations employed and physical characteristics specified are completely accurate, the computations required often are insurmountable. Typically, something must be sacrificed. However, if the mathematics is flawed, then all further studies are tainted by such flaws. In contrast, if some of the physical characteristics (e.g., geometric features, characterization of blood properties, etc.) are oversimplified, additional studies may add these features in an iterative fashion. Accordingly, because Aliseda et al. chose a few oversimplifications of physical characteristics, further studies yet may build on this first effort.

Experience with continuous-flow VADs has led to both clinically important and fundamental scientific questions that were not appreciated before the development of VAD technology. Further scientific and clinical advances require theoretically and experimentally rigorous studies, providing results from which coherent conclusions can be drawn. Poincaré’s comment exemplifies this idea that we must not compile empirical data mindlessly and base practices on an unanalyzed corpus of “evidence” as clinicians. Evidence is necessary, but hardly is sufficient. Rather, we must use the body of existing data to formulate rational hypotheses, and theoretical and experimental studies to test these hypotheses, and ultimately assemble comprehensive conceptual frameworks—theories—with predictive capacities to guide subsequent studies. With innovative work such as that of Aliseda et al., now we are assembling the heap of stones into the beginnings of a house.

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References

1. Aliseda A, Chivukula VK, McGah P, et al: LVAD outflow graft angle and thrombosis risk. ASAIO J 2017.63: 14–23.
2. Truesdell C, Rajagopal KR: An Introduction to the Mechanics of Fluids. Boston, MA, Birkhauser, 2000, pp. 3–5.
3. Anand M, Rajagopal K, Rajagopal KR: A model for the formation and lysis of blood clots. Pathophysiol Haemost Thromb 2006.34: 109–120.
4. Fogelson AL, Neeves KB: Fluid mechanics of blood clot formation. Annu Rev Fluid Mech 2015.47: 377–403.
Copyright © 2017 by the American Society for Artificial Internal Organs