Function Follows Form: Modeling Ventricular Assist Device-Mediated Blood Flow

La filosofia è scritta in questo grandissimo libre che continuamente ci sta aperto innanzi a gli occhi (io dico l’universo), ma non si può intendere se prima non s’impara a intender la lingua, e conoscer i caratteri, ne’ quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi è impossibile a intenderne umanamente parola. Philosophy is written in this grand book which stands continually open to our gaze (I speak of the universe). But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it. —Galileo Galilei, The Assayer, 1623 Scientific theories ought to accomplish two tasks: 1) describe and explain that which has been observed in uncontrolled (“natural”) or under experimental circumstances, and 2) make predictions based on an understanding of causative mechanisms—preferably experimentally testable—about that which has not yet been observed in uncontrolled or experimental circumstances (not observed because it has not yet been investigated). Mathematics has been, is, and likely will continue to be, the language, or terminological framework, within which scientific theories are understood. Importantly, natural science and mathematics have co-evolved over time, each influencing the progress of the other. The cardiovascular system in general, and the ventricular assist device (VAD)-supported circulation in particular, exhibit highly complex solid and fluid mechanical behaviors, for which matching complex mechanically accurate and mathematically appropriate models must be developed—and once developed, then employed. In this issue of ASAIO Journal, Grinstein et al.1 from the University of Chicago have applied computational fluid dynamics approaches to the comparative study of model systemic/left-sided circulations supported by three different choices of left ventricular assist devices (LVADs): 1) HeartMate 2, 2) HeartMate 3, and 3) HVAD. The authors specifically examined profiles of wall shear stress, blood flow velocity, and wall shear rate, in the outflow graft, aortic root, ascending thoracic aorta, and aortic arch, under conditions of steady flow versus unsteady (specifically, via LVAD-induced pulsatility; the aortic valve was persistently closed) flow. They found that 1) for matched volumetric flow rates between devices, the HVAD had higher blood flow velocities, shear rates, and shear stresses in the aortic arch and head vessels (this was not the case in the ascending thoracic aorta, with far lower wall shear stresses for all devices analyzed), 2) consistent with this, virtual contrast flushing from the ascending thoracic aorta was greatest with the HVAD, and 3) addition of pulsatility to HeartMate 32 function (but not the Lavare cycle3 for HVAD function) resulted in improved contrast flushing, with this being associated with a relative increase in wall shear stress in the ascending thoracic aorta (again, the absolute values of wall shear stress in the ascending thoracic aorta were 5-10-fold lower than in the aortic arch; see below). The clinical implications of these findings are unclear at present, but the authors provide thoughtful speculations in the Discussion. The construction of the model bears analysis. The Navier-Stokes equations,4,5 the partial differential equations most commonly used to describe the flow of viscous fluids, are employed. This level of complexity with regard to the modeling, although incomplete and potentially importantly flawed (see below), is welcomed, and often absent from most studies of blood flow. The Navier-Stokes equations arise as a consequence of applying the Navier-Stokes constitutive relation to the balance of linear momentum. Blood is modeled as a shear-thinning “non-Newtonian” fluid. A Newtonian fluid exhibits constant viscosity over a range of shear of shear rates (the first derivative of viscosity with respect to shear rate is zero), whereas a shear-thinning fluid exhibits a negative first derivative of viscosity with respect to shear rate. This is somewhat accurate. In reality, blood exhibits thixotropy,6 or time-dependent shear thinning. This is because high shear induces irreversible blood damage, which reduces viscosity at the time of exposure to high shear, but these effects last following the initial damage. In contrast, shear thinning alone suggests time-independence of viscosity based merely on the current value of the shear rate; blood exposed to low → high → low shear would exhibit a baseline viscosity under low shear, reduced viscosity under high shear, which then would return to the level previously observed under low shear. While blood commonly is modeled as shear thinning, thixotropy may be more appropriate in the setting of LVAD-mediated blood flow. More importantly, when one uses a shear-thinning characterization of blood and applies it to the balance of linear momentum, the Navier-Stokes equations are “not” what arises. The equations for shear-thinning fluids and their derivation, are provided below, and compared to the Navier-Stokes equations, which characterize Navier-Stokes (a.k.a. Newtonian) fluids. The consequences of this error for the flows under consideration are unclear, but in essence, modeling blood as shear thinning is incompatible with using the Navier-Stokes equations. This is the balance of linear momentum, in which ρ is density, v is velocity, t is time, T is the Cauchy stress tensor (for which div refers to its divergence, which the authors denote as ∇• in their article), and b denotes a body force (e.g., gravity): Since the fluid is incompressible, the fluid can undergo only isochoric motions, and the balance of mass reduces to This is the Navier-Stokes constitutive relation, in which p is the mechanical pressure (and is also the indeterminate part of the stress due to the constraint of incompressibility, I is the identity matrix, D is the symmetric part of the velocity gradient, and x is position (the superscript T denotes the transpose): When μ is constant (Newtonian fluid), and the body force b is neglected, and using the balance of mass, the “Navier-Stokes equation” is obtained: However, when μ is a function of the shear rate, as it is for a shear-thinning fluid, in which tr denotes its trace: To clarify, μ(D) simply denotes that μ is a function of D, “not” multiplied by D; is termed the generalized viscosity. Now, when the divergence of the Cauchy stress tensor is determined for the balance of linear momentum for a fluid with shear-dependent viscosity: This may be written more simply as: where is a function of D, as stated above. The third term on the right-hand side of the equation is absent in the balance of linear momentum for a Navier-Stokes/Newtonian fluid, but necessarily arises when modeling a fluid with shear-dependent viscosity, as the authors have done for blood. It is unclear how important this term is with specific reference to the problems that the authors have studied, but neglecting the term ought to be done only after a priori determination of the physical soundness of doing so. Last, the properties of the systemic arterial circulation are expressed as “lumped” parameters. This is again a commonly used oversimplification, but in contrast to blood flow being described by a system of nonlinear partial differential equations, is comparatively crude. Lumped parameters are allowed to vary temporally but are held constant spatially, and in the case of the cardiovascular system, are electrical circuit theory-derived variables that are analogously forced onto mechanical systems. In reality, “resistance” or “impedance” is not exactly a property of the systemic arterial circulation. The energy of flowing blood is dissipated across impedances in the circulation, but the “physical basis” for this impedance is a property of blood, namely, viscosity. Fluids without viscosity, also known as inviscid fluids, do not exhibit energy dissipation, across large conduit narrowings, or small conduits. Moreover, even if resistance or impedance were inherent to the circulation, it very obviously is different in different compartments of the circulation, that is, it varies as a function of space. The same spatial variation is true, albeit in a different way, for “elastance” or “capacitance/compliance” (see below). With respect to physiologic conditions, for the sake of simplicity, aortic valve closure is assumed. It seems obvious that the findings that the authors observe would be substantially different with antegrade flow through the aortic root. With persistent closure of a competent aortic valve, wall shear stress and shear rate in the ascending thoracic aorta and aortic root are very low. This is because very little actual blood flow, that is, actual blood “transit through” the ascending thoracic aorta and aortic root, can occur. The only outflow that exists is via coronary arterial blood flow. In contrast, the remainder of the LVAD output (which equals the cardiac output under these conditions) is via all of the branches of the aorta distal to the outflow graft anastomosis. In addition, although the aorta is given a compliance or capacitance, there is no evidence that the 3D structure of the aorta undergoes time-varying displacement in these studies, which would have to occur for capacitance to have real meaning. Fluid-structure interaction approaches, with moving blood-aortic wall boundaries as functions of time, would have to be added to the model. Finally, there may be a very simple explanation for the bulk of the findings observed. The outflow graft of the HVAD is 10 mm in diameter, whereas the outflow graft for the HeartMate 2 and 3 LVADs is 14 mm. If the volumetric flow rate through these grafts is approximately the same, as it was set to be for these studies, then mean flow velocity will be almost exactly twofold higher within the outflow graft. Within the portion of aorta just distal to the outflow graft anastomosis, that is, the aortic arch (and its branches), it would be reasonably hypothesized that wall shear stress and shear rate (which are proportionally related to flow velocity) would also be higher when LVAD outflow grafts are of smaller diameter, although the magnitude of this increase almost certainly is not the same as that of the mean flow velocity. Thus, it is quite possible, and perhaps even probable, that the authors’ findings are predominantly consequences of this very simple difference. Different inflow cannulae, pump mechanisms, outflow grafts, details of anatomic construction, native ventricular properties, valvular function, and arterial circulatory properties, all result in different blood flow dynamics. That is, function follows causally from form. High-quality modeling permits us to understand how particular alterations in one or more of these components influence blood flow dynamics.