Fluid structure interaction analysis of bioprosthetic heart valves: Significance of arterial wall deformation

Transcription

1 Computational Mecanics manuscript No. (will be inserted by te editor Fluid structure interaction analysis of bioprostetic eart valves: Significance of arterial wall deformation Ming-Cen Hsu David Kamensky Yuri Bazilevs Micael S. Sacks Tomas J. R. Huges Te final publication is available at Springer via ttp://dx.doi.org/0.007/s Abstract We propose a framework tat combines variational immersed-boundary and arbitrary Lagrangian Eulerian (ALE metods for fluid structure interaction (FSI simulation of a bioprostetic eart valve implanted in an artery tat is allowed to deform in te model. We find tat te variational immersed-boundary metod for FSI remains robust and effective for eart valve analysis wen te background fluid mes undergoes deformations corresponding to te expansion and contraction of te elastic artery. Furtermore, te computations presented in tis work sow tat te arterial wall deformation contributes significantly to te realism of te simulation results, leading to flow rates and valve motions tat more closely resemble tose observed in practice. Keywords Fluid structure interaction Bioprostetic eart valve Variational immersed-boundary metod Arbitrary Lagrangian Eulerian formulation Isogeometric analysis Arterial wall deformation Introduction Heart valves are passive structures tat ensure te unidirectional blood flow troug te eart by opening and closing in response to emodynamic forcing. Hundreds of tou- M.-C. Hsu ( Department of Mecanical Engineering, Iowa State University, 2025 Black Engineering, Ames, IA 500, USA jmcsu@iastate.edu D. Kamensky M. S. Sacks T. J. R. Huges Institute for Computational Engineering and Sciences, Te University of Texas at Austin, 20 East 24t St, Stop C0200, Austin, TX 7872, USA Y. Bazilevs Department of Structural Engineering, University of California, San Diego, 9500 Gilman Drive, Mail Code 0085, La Jolla, CA 92093, USA sands of diseased valves are replaced by prostetics annually [, 2]. Bioprostetic eart valves (BHV are prostetics composed of tin flexible leaflets tat are fabricated from biological materials and mimic te structure of native eart valves to avoid patological emodynamics [2]. Te principal drawback of tis style of prostetic is its durability, wic is limited to 0 5 years [3]. Accurate computational analysis of tese devices could provide insigts into te mecanical processes tat bot contribute to and follow from teir deterioration, streamlining te design process of new prostetics. Te biomecanical significance of arterial elasticity was first clearly described by Hales [4] in 733, after performing a series of pioneering experiments on animals. Hales found tat arteries expand elastically to store te systolic output of te eart, ten gradually release tis blood during diastole. Tis is now known as te Windkessel effect. Frank [5 7] developed te first matematical model of te Windkessel effect in 899. Frank s model may be intuitively understood troug te electronic ydraulic analogy [8], wic substitutes electrical current for volumetric flow and voltage for pressure. In tis analogy, Frank s model te two-element Windkessel model consists of a capacitor and resistor in parallel, downstream of te aortic valve, wic acts as a pulsatile current source. Te capacitor models te elastic arteries, wic accumulate blood to develop pressure, wile te resistor models viscous ead loss witin te circulatory system by analogy to Om s Law. Tis model allows prediction of te time-dependent aortic pressure based on te istory of flow rate troug te aortic valve. Many refinements to Frank s model ave been proposed since is initial contribution, in- Windkessel translates from German to air camber, and likely refers to Hales original analogy between arterial compliance and te air-filled cavities used to smoot ose output from 8 t -century fire engines.

2 2 Ming-Cen Hsu et al. cluding te tree- [9], and four- [0] element Windkessel models. Suc models are referred to as lumped-parameter models. Lumped-parameter models may be coupled wit detailed computational fluid dynamics (CFD simulations of specific arterial sections of interest. Te voltage of te lumped-parameter model acts as a pressure boundary condition on te outflow of te CFD domain, and te volumetric flow from te CFD domain acts as a current source for te lumped-parameter model []. However, to fully account for te Windkessel effect of arterial elasticity, fluid structure interaction (FSI must be incorporated into te detailed model of te section of interest. In tis paper, we demonstrate tat te elasticity of te section of aorta immediately surrounding an implanted BHV can ave profound effects on te dynamics of bot te valve itself and te surrounding blood flow. For te reasons discussed in our earlier work [2], we simulate te BHV leaflets using a non-boundary-fitted (variational immersed-boundary metod, in wic te structural discretization is free to move independently troug a background fluid mes. Detailed reviews of non-boundary-fitted metods for FSI can be found in Sotiropoulos and Yang [3], Mittal and Iaccarino [4], and Peskin [5]. Tese metods are particularly attractive for applications wit complex moving boundaries, suc as eart valve leaflets [6 2]. However, tey ave te inerent disadvantage of uncontrolled mes quality near te fluid structure interface, and may be unable to resolve important boundary layer features tat may globally affect te flow. More accurate results can be obtained using boundaryfitted approaces by building a fluid mes tat is tailored to te structure and deforms as te structure moves. In suc computations, te fluid subproblem may be posed using an arbitrary Lagrangian Eulerian (ALE formulation [22 24], or a space time formulation [25 27], bot of wic explicitly account for te motion of te fluid mecanics domain and mes. For te parts of an arterial FSI computation wit no contact between solid surfaces, te problem of mes deformation may be effectively solved using a simple fictitious linear elasticity problem [28 32]. Tis makes vascular FSI an ideal application for boundary-fitted approaces. In te analysis of a BHV implanted in a deforming artery, we are faced wit te confluence of two problems tat suggest different computational metods. We terefore elect to use a ybrid metod tat leverages te advantages of bot ALE and immersed-boundary tecniques for FSI. We discretize te valve leaflets separately, and immerse tem into a deforming boundary-fitted mes of te artery volume. Te proposed tecnique falls under te umbrella of te recently proposed Fluid Solid Interface- Tracking/Interface-Capturing Tecnique (FSITICT [33], a metod tat targets FSI problems in wic interfaces tat are possible to track are tracked, and tose too difficult to track are captured. Te FSITICT was introduced as an FSI version of te Mixed Interface-Tracking/Interface-Capturing Tecnique (MITICT [34]. Te MITICT was successfully tested in 2D computations wit solid circles and free surfaces [35,36] and in 3D computation of sip ydrodynamics [37]. Recently Wick [38, 39] made use of te FSITICT approac, coupling a boundary-fitted and immersed-boundary discretizations in a single computation, to compute several 2D FSI bencmark problems. Our immersed-boundary approac for FSI was first developed in Kamensky et al. [2] using te variational framework of augmented Lagrangian metods. Te augmented Lagrangian approac for FSI was proposed in Bazilevs et al. [40] to andle boundary-fitted computations wit nonmatcing fluid structure interface discretizations. We found in Kamensky et al. [2] tat tis augmented Lagrangian framework can be extended to andle non-boundary-fitted CFD and FSI problems, and its efficacy was demonstrated using several computations including te coupling of a BHV and surrounding blood flow at pysiological pressure levels. In tis work, we take te augmented Lagrangian framework for FSI as te starting point of our ALE/immersedboundary ybrid metodology. A single computation combines a boundary-fitted, deforming-mes treatment of some fluid structure interfaces wit a non-boundary-fitted treatment of oters. Tis approac enables us to simulate te FSI of a BHV implanted in an elastic artery troug te entire cardiac cycle, at full scale, under realistic pysiological conditions. Te paper is organized as follows. In Section 2 we present te details of our ybrid ALE/immersed-boundary metod developed for te FSI simulation of a eart valve implanted in a deformable artery. In Section 3 we provide te simulation details and report te results of te FSI computations of an actual BHV design. In particular, we compare te results from te rigid- and elastic-wall simulations and find tat wall elasticity plays an important role in te overall system response. In Section 4 we draw conclusions. 2 FSI modeling using a ybrid ALE/immersed-boundary approac In tis section, we present te computational framework for FSI analysis of a bioprostetic eart valve implanted in a deformable artery. Te blood flow in a deforming artery is governed by te Navier Stokes equations of incompressible flow posed on a moving domain. Te domain motion is andled using te ALE formulation, wic is a widely used approac for vascular blood flow applications [4 46]. Te coupling of te BHV leaflet dynamics to te artery is andled troug te recently proposed variational immersedboundary metod [2], in wic te structural discretization is free to move independently troug a background fluid

3 Fluid structure interaction analysis of bioprostetic eart valves: Significance of arterial wall deformation 3 mes. Te ybrid ALE/immersed-boundary metod will be presented and applied to te simulation of an aortic BHV coupled to an elastic arterial wall and blood flow over cardiac cycles. 2. Augmented Lagrangian framework for FSI Let (Ω t and (Ω 2 t R d, d = {2, 3} represent te timedependent domains of te fluid and structural mecanics problems, respectively, at time t, wit (Γ t and (Γ 2 t representing teir corresponding boundaries. Let (Γ I t R d represent te interface between te fluid and structural domains. Let u and p denote te fluid velocity and pressure, respectively. Let y denote te displacement of structural material points from teir positions in a reference configuration, and define te structure velocity u 2 as te material time derivative of y. We introduce an additional unknown function λ defined on (Γ I t, wic takes on te interpretation of a Lagrange multiplier. Let S u, S p, S d, and S l be te function spaces for te fluid velocity, fluid pressure, structural velocity, and Lagrange multiplier solutions, respectively, and V u, V p, V d, and V l be te corresponding weigting function spaces. Te variational problem of te augmented Lagrangian formulation is: find u S u, p S p, y S d, and λ S l suc tat for all test functions w V u, q V p, w 2 V d, and δλ V l B ({w, q}, {u, p}; û F ({w, q} + w λ dγ + w β(u u 2 dγ = 0, ( (Γ I t (Γ I t w 2 β(u u 2 dγ = 0, (2 t B 2 (w 2, y F 2 (w 2 w 2 λ dγ (Γ I (Γ I t (Γ I t δλ (u u 2 dγ = 0. (3 In te above, te subscripts and 2 denote te fluid and structural mecanics quantities, and β is a penalty parameter, wic we leave unspecified for te moment. B, B 2, F, and F 2 are te semi-linear forms and linear functionals corresponding to te fluid and structural mecanics problems, respectively, and are given by ( u B ({w, q}, {u, p}; û = w ρ t + (u û u dω ˆx (Ω t + ε(w : σ dω + q u dω, (Ω t (Ω t F ({w, q} = (Ω t w ρ f dω + (4 (Γ t w dγ, (5 2 y B 2 (w, y = w ρ 2 t 2 dω + ε(w : σ 2 dω, (6 X (Ω 2 t (Ω 2 t F 2 (w = w ρ 2 f 2 dω + w 2 dγ, (7 (Ω 2 t (Γ 2 t were ρ and ρ 2 are te densities, σ and σ 2 are te Caucy stresses, f and f 2 are te applied body forces, and 2 are te applied surface tractions, (Γ t and (Γ 2 t are te boundaries were te surface tractions are specified, ε( is te symmetric gradient operator given by ε(w = 2 ( w + wt, û is te velocity of te fluid domain (Ω t, ( t is te time ˆx derivative taken wit respect to te fixed spatial coordinate ˆx in te referential domain (wic does not follow te motion of te fluid itself, and ( t is te time derivative olding te material coordinates X fixed. Te gradient is taken X wit respect to te spatial coordinate x of te current configuration. We assume tat te fluid is Newtonian wit dynamic viscosity µ and Caucy stress σ = pi + 2µε(u. Bazilevs et al. [40] demonstrate ow te Lagrange multiplier, λ, may be formally eliminated by substituting an expression for te fluid structure interface traction in terms of te oter unknowns. Tis leads to te following variational formulation for te coupled problem: find u S u, p S p, and y S d suc tat for all w V u, q V p, and w 2 V d B ({w, q}, {u, p}; û F ({w, q} + B 2 (w 2, y F 2 (w 2 (w w 2 σ (u, p n dγ (Γ I t δσ (w, q n (u u 2 dγ (Γ I t + (w w 2 β(u u 2 dγ = 0. (8 (Γ I t In te above, δσ (w, qn = 2µε(wn + qn. Equation (8 may be interpreted as an extension of Nitsce s metod [47], wic is a consistent, stabilized metod for imposing constraints on te boundaries by augmenting te governing equations wit additional constraint equations. Tis augmented Lagrangian approac for FSI was originally proposed by Bazilevs et al. [40] and furter studied in Hsu and Bazilevs [48] to andle boundary-fitted computations wit non-matcing fluid structure interface discretizations. In Kamensky et al. [2], we found tat tis framework can be extended to andle non-boundary-fitted FSI problems and te accuracy and efficiency of te metodology was examined troug several computations. In tis work, we take te augmented Lagrangian framework for FSI as te starting point of our ybrid ALE/immersed-boundary

4 4 Ming-Cen Hsu et al. metod. A single computation can combine a boundaryfitted, deforming-mes treatment of some fluid structure interfaces wit a non-boundary-fitted treatment of oters. Remark In te above developments we assumed tat te trial and test function spaces of te fluid and structural subproblems are independent of eac oter. Tis approac provides one wit te framework tat is capable of andling non-matcing fluid and structural interface discretizations. If te fluid and structural velocities and te test functions are explicitly assumed to be continuous (i.e. u = u 2 and w = w 2 at te interface, te FSI formulation given by Eq. (8 reduces to: find u S u, p S p, and y S d suc tat for all w V u, q V p, and w 2 V d B ({w, q}, {u, p}; û F ({w, q}+ B 2 (w 2, y F 2 (w 2 = 0. Tis form of te FSI problem is suitable for matcing fluid structure interface meses. Altoug somewat limiting, matcing interface discretizations were successfully applied to cardiovascular FSI in many earlier works [32, 44, 49 54]. 2.2 Semi-discrete fluid formulation wit weak boundary conditions Te fluid subproblem may be obtained by setting w 2 = 0 in Eq. (8. Tis approac gives a formulation for weak imposition of Diriclet boundary conditions for te fluid problem, wic was first proposed by Bazilevs and Huges [55] and furter refined in Bazilevs et al. [56, 57] to improve te performance of te fluid mecanics formulation in te presence of underresolved boundary layers. Tis weak imposition of te Diriclet boundary conditions is also te starting point of te variational immersed-boundary approac [2]. In a nonboundary-fitted metod, te elements of te fluid discretization may extend into te interior of an immersed object. Imposing Diriclet boundary conditions is no longer straigtforward given tat te basis functions are non-interpolating at te object boundaries. In order to enforce essential boundary conditions, one can eiter modify te basis functions so tey vanis at te interface [58] or augment te governing equations wit additional constraint equations. In tis work we coose te latter approac. Consider a collection of disjoint elements {Ω e }, e Ω e R d, wit closures covering te fluid domain: Ω e Ω e. Note tat Ω e is not necessarily a subset of Ω. {Ω e }, Ω, and Γ I remain time-dependent, but we drop te subscript t for notational convenience. Te mes defined by {Ω e } deforms wit a velocity field û and te boundary Γ I moves wit velocity u 2. Te semi-discrete fluid problem is given by: find u S u and p S p suc tat for all w V u and q V p B VMS ( {w, q }, {u, p }; û F VMS ( {w, q } (9 w ( p n + 2µε(u n dγ Γ I ( 2µε(w n ( + q n u u 2 dγ Γ I w ρ (( u û ( n u u 2 dγ (Γ I ( + w ( w n n Γ I + Γ I τ B TAN τ B NOR (( u u 2 (( u u 2 n n dγ ( w n (( u u 2 n dγ = 0, (0 were (Γ I is te inflow part of Γ I, on wic (u û n < 0, te constants τtan B and τb NOR correspond to a splitting of te penalty term into te tangential and normal directions, respectively, and Γ I may cut troug element interiors. Te discrete trial function spaces S u for te velocity and S p for te pressure, as well as te corresponding test function spaces Vu and V p are assumed to be equal order, and, in tis work, are comprised of isogeometric [59,60] functions. Te forms B VMS and F VMS are te variational multiscale (VMS discretizations of B and F, respectively, given by B VMS ({w, q}, {u, p}; û ( u = w ρ t + (u û u dω ˆx (Ω t + ε(w : σ dω + q u dω (Ω t (Ω t ( + (u û w + q u dω ρ e Ω e Ω + w ρ p dω and e Ω e Ω e Ω e Ω e Ω e Ω + e Ω e Ω w (u u dω w ρ : ( u u dω ( u w τ (u u dω, ( F VMS ({w, q} = F ({w, q}, (2 were ( ( u u = τ M ρ t + (u û u f σ, (3 ˆx

5 Fluid structure interaction analysis of bioprostetic eart valves: Significance of arterial wall deformation 5 p = τ C u. (4 Equations ( (4 correspond to te ALE VMS formulation of te Navier Stokes equations of incompressible flows [6 63]. Te additional terms may be interpreted bot as stabilization and as a turbulence model [64 72]. Te stabilization parameters are τ M = ( ( Ct s(x, t t + (u û G(u û + C Iν 2 2 G : G, 2 (5 τ C = (τ M tr G, (6 τ = ( u Gu 2, (7 were t is te time-step size, ν = µ/ρ is te kinematic viscosity, C I is a positive constant derived from an appropriate element-wise inverse estimate [73 76], G is te element metric tensor defined as G = ξ T ξ x x, (8 were ξ/ x is te inverse Jacobian of te element mapping between te parametric and pysical domain, tr G is te trace of G, and te parameter C t is typically taken equal to 4 [66, 70, 77]. Te scalar function s(x, t in Eq. (5 is a dimensionless scaling factor introduced in Kamensky et al. [2] to improve local mass conservation near concentrated loads. Locally increasing s near tin immersed structures can greatly improve te quality of approximate solutions wen te concentrated surface force due to te structure induces a significant pressure discontinuity. In most of te domain, we keep s =, as in te usual VMS formulation, but, in an O( neigborood around tin immersed structures, we increase it to equal te dimensionless constant s sell. Remark 2 On te fluid mecanics domain interior, te mes velocity, û, may be obtained by solving a linear elastostatics problem subject to te displacement boundary conditions coming from te motion of te boundary-fitted fluid solid interface [28 32]. Tis metod is effective for relatively mild deformations, suc as tose of te artery. However, for scenarios tat involve large translational and/or rotational structural motions, suc as eart valve dynamics, te boundary-fitted fluid mes can become severely distorted. Non-boundary-fitted approaces could be an alternative for tese type of problem. Remark 3 Te last term of Eq. ( provides additional residual-based stabilization and originates from Taylor et al. [78]. Te term is consistent and dissipative, and as similarities wit discontinuity-capturing metods suc as te DCDD [68, 79, 80] and YZβ [8 83] tecniques. Te terms from te second to te last line of Eq. (0 are responsible for te weak enforcement of kinematic and traction constraints at te non-matcing or immersed boundaries. It was sown in earlier works [55 57, 84, 85] tat imposing te Diriclet boundary conditions weakly in fluid dynamics allows te flow to slip on te solid surface wen te wall-normal mes size is relatively large. Tis effect mimics te tin boundary layer tat would oterwise need to be resolved wit spatial refinement, allowing more accurate solutions on coarse meses. In te immersed-boundary metod, te fluid mes is arbitrarily cut by te structural boundary, leaving a boundary layer discretization of inferior quality compared to te body-fitted case. Terefore, in addition to imposing te constraints easily in te context of non-boundary-fitted approac, we may obtain more accurate fluid solutions as an added benefit of using te weak boundary condition formulation (0. In Eq. (0, te parameters τtan B and τb NOR must be sufficiently large to stabilize te formulation, but not so large as to degenerate Nitsce s metod into a pure penalty metod. Based on previous studies of weakly-enforced Diriclet boundary conditions in fluid mecanics [55 57], we expect tese parameters to scale as τ B ( = CB I µ (9 were is a measure of te element size at te boundary and CI B is a dimensionless constant. However, in te case of an immersed boundary, neiter te appropriate definition of nor te principle for deriving CI B is straigtforward. As a result, we cose te penalty-parameter values troug numerical experiments. Integrating te fluid formulation ( over elements tat are only partially contained in Ω typically requires special quadrature tecniques, as discussed in Kamensky et al. [2]. In te present work, we do not need tese quadrature tecniques, because fluid elements only overlap spatially wit tin sell structures, wic are modeled geometrically as (d -dimensional surfaces and terefore ave zero Lebesgue measure in R d. To evaluate te surface integrals of Eq. (0 over immersed boundaries, we define a Gaussian quadrature rule wit respect to a parameterization of te immersed surface, ten locate te quadrature points of tis rule in te parameter space of te background mes elements to evaluate traces of te fluid test and trial functions. Unsteady flow computations may sometimes diverge due to flow reversal on outflow boundaries. Tis is known as backflow divergence and is frequently encountered in cardiovascular simulations. In order to preclude backflow divergence, an outflow stabilization metod originally proposed in Bazilevs et al. [50] and furter studied in Esmaily- Mogadam et al. [86] is employed in our fluid mecanics formulation.

6 6 Ming-Cen Hsu et al. 2.3 Arterial wall modeling In tis section we sow te variational formulation of te boundary-fitted solid problem for te arterial wall modeling. Te fluid solid interface discretization is assumed to be conforming. Let X be te coordinates of te initial or reference configuration and let y be te displacement wit respect to te reference configuration. Te coordinates of te current configuration, x, are given by x = X + y. Te deformation gradient tensor F is defined as F = x X = I + y X, (20 were I is te identity tensor. Let S d and V d be te trial solution and weigting function spaces for te solid problem. Te arterial wall is modeled as a tree-dimensional yperelastic solid and te variational formulation wic represents te balance of linear momentum for te solid is stated as follows: find te displacement y S d, suc tat for all weigting functions w 2 V d B 2 (w 2, y F 2 (w 2 = 0, (2 were 2 y B 2 (w, y = w ρ 2 t 2 dω + X w : P dω, (22 X (Ω 2 t (Ω 2 0 F 2 (w = w ρ 2 f 2 dω + w 2 dγ. (23 (Ω 2 t (Γ 2 t In te above, (Ω 2 0 is te solid domain in te reference configuration, X is te gradient operator on (Ω 2 0, and P = FS is te first Piola Kircoff stress tensor, were S is te second Piola Kircoff stress tensor given by S = µj 2/3 ( I 3 tr C C + 2 κ ( J 2 C. (24 In Eq. (24, µ and κ are interpreted as te blood vessel sear and bulk moduli, respectively, J = det F is te Jacobian determinant, and C = F T F is te Caucy Green deformation tensor. Equation (24 is a generalized neo-hookean model wit dilatational penalty given in Simo and Huges [87]. Its stress-strain beavior was analytically studied on simple cases of uniaxial strain [32] and pure sear [88]. Te model was argued in Bazilevs et al. [44] to be appropriate for arterial wall modeling in FSI simulations. It was sown tat te level of elastic strain in arterial FSI problems is sufficiently large to preclude te use of infinitesimal (linear strains, yet not large enoug to be sensitive to te nonlinearity of te particular material model. However, te current model as te advantage of stable beavior for te regime of strong compression and terefore is selected in tis work for te modeling of te arterial wall. 2.4 Immersed sell structures We model te eart valve as a sell structure immersed into a deforming background mes covering te lumen of te artery. Te exact solution for te pressure around a sell structure may be discontinuous at te structure, wic presents a conceptual difficulty. Te fluid discretization cannot be informed by te structure s position. Tis means tat our fluid approximation space cannot be selected in suc a way tat te pressure basis functions are temselves discontinuous at te immersed boundary. Tis implies an inerent approximation error in te pressure field. Tis error will converge slowly for polynomial bases [89]. Noneteless, we believe tat solutions of sufficient accuracy for engineering purposes can be obtained in tis fasion and we focus on developing a robust metod for obtaining tese solutions Reduction of Nitsce s metod to te penalty metod Consider integrating te boundary terms of Eq. (0 over bot sides of a tin immersed sell structure. If te velocity and pressure approximation spaces are continuous troug te vanising tickness of te sell (and te velocity approximation space is continuously differentiable, ten te dependence of te consistency and adjoint consistency terms on te normal vector will cause contributions from opposing sides to cancel one anoter. Te only remaining terms will be te penalty and te inflow stabilization. In te case of an immersed sell structure, we may view te inflow term as a velocity-dependent penalty. Te Nitsce-type formulation given by Eq. (0 terefore reduces to te following penalty metod: ( {w, q }, {u, p }; û ( F VMS {w, q } B VMS w ρ (( u û ( n u u 2 dγ (Γ I ( + w ( w n n Γ I + Γ I τ B TAN τ B NOR (( u u 2 (( u u 2 n n dγ ( w n (( u u 2 n dγ = 0, (25 wen te approximation spaces V u and V p are sufficiently regular around te sell. To determine te velocity and pressure about an immersed valve in its closed state, a metod must be capable of developing nearly ydrostatic solutions in te presence of large pressure gradients. Penalty forces will only exist if tere are nonzero violations of kinematic constraints. A pure penalty metod rules out te desired ydrostatic solutions: every term tat could resist te pressure gradient to satisfy

7 Fluid structure interaction analysis of bioprostetic eart valves: Significance of arterial wall deformation 7 balance of linear momentum depends on velocity. Increasing β may diminis leakage troug a structure, but it is a well-known disadvantage of penalty metods tat extreme values of penalty parameters will adversely affect te numerical solvability of te resulting problem. Tis motivates us to return to Eqs. ( (3 and develop a metod tat does not formally eliminate te multiplier field Reintroducing te multipliers Since te introduction of constraints tends to make discrete problems more difficult to solve, we will only reintroduce a scalar multiplier field to strengten enforcement of te nopenetration part of te FSI kinematic constraint, rater tan te vector-valued multiplier field of Eqs. ( (3. Te viscous, tangential component of te constraint will continue to be enforced by only te penalty τtan B. Tis may be tougt of as a formal elimination of just te tangential component of te multiplier field, wic also retains te ability to allow te flow to slip at te boundary, wic tends to produce more accurate fluid solutions as discussed in Section 2.2. For clarity, we redefine te FSI boundary terms on te midsurface of te sell structure, Γ t, rater tan considering te full boundary, Γ I. Tis means tat constants in te current formulation may differ from tose of Eqs. ( (3 by factors of two. We arrive, ten, at te formulation B ({w, q}, {u, p}; û F ({w, q} + w (λ n n 2 dγ + w β(u u 2 dγ = 0, (26 Γ t Γ t B 2 (w 2, y F 2 (w 2 w 2 (λ n n 2 dγ Γ t Γ t w 2 β(u u 2 dγ = 0, (27 Γ t δλ n n 2 (u u 2 dγ = 0, (28 were λ n is te new scalar multiplier field and, to empasize te relation to Eqs. ( (3, te penalty force as not been split into normal and tangential components. Te consistency and adjoint consistency terms associated wit eliminating te tangential component of te multiplier ave been omitted under te assumption tat tey will vanis after integrating over bot sides of te tin sell, as discussed in Section We discretize te multiplier field by collocating kinematic constraints at points of te quadrature rule for integrals over Γ t. Tis entails adding a scalar multiplier unknown at eac quadrature point. In discrete evaluations of integrals, tese multiplier unknowns are treated like point values of a function defined on Γ t. Because te spatial resolution of te discrete multiplier representation is not controlled relative to te background fluid mes, we must relax te collocated constraints to ensure stability of te numerical sceme. We accomplis tis troug te time-discrete algoritm given in Section 2.5. Te algoritmic constraint relaxation is interpreted at te time-continuous level by Kamensky et al. [2], troug an analogy to Corin s metod of artificial compressibility [90], in wic te Lagrange multiplier solves an auxiliary differential equation in time Treatment of sell structure mecanics We assume tat te structure is a tin sell, represented matematically by its mid-surface. Furter, we assume tis surface to be piecewise C -continuous and apply te Kircoff Love sell formulation and isogeometric discretization studied by Kiendl et al. [9 93]. Te spatial coordinates of te sell mid-surface in te reference and current configurations are given by X(ξ, ξ 2 and x(ξ, ξ 2, respectively, parameterized by ξ and ξ 2. Assuming te range {, 2} for Greek letter indices, we define te covariant surface basis vectors g α = x ξ α, (29 g 3 = g g 2 g g 2, (30 and G α = X ξ α, (3 G 3 = G G 2 G G 2, (32 in te current and reference configurations, respectively. Using kinematic assumptions and matematical manipulations given in Kiendl [93], we split te in-plane Green Lagrange strain E αβ into membrane and curvature contributions E αβ = ε αβ + ξ 3 κ αβ, (33 were ε αβ = 2 κ αβ = G α ξ β ( gα g β G α G β, (34 G 3 g α ξ β g 3, (35 are te membrane strain and cange of curvature tensors, respectively, at te sell mid-surface. In Eq. (33, ξ 3 [ t /2, t /2] is te troug-tickness coordinate and t is te sell tickness. Te forms B 2 and F 2 appearing in te structure subproblem ten become, in te case of a tin sell structure, 2 y B 2 (w, y = w ρ 2 t t 2 dγ + δe : S dξ 3 dγ, X Γ t Γ 0 t (36

8 8 Ming-Cen Hsu et al. F 2 (w = w ρ 2 t f 2 dγ + Γ t Γ t w net 2 dγ, (37 were S is te second Piola Kircoff stress, δe is te variation of te Green Lagrange strain, Γ 0 and Γ t are te sell mid-surface in te reference and deformed configurations, respectively, net 2 = 2 (ξ 3 = t /2 + 2 (ξ 3 = t /2 sums traction contributions from te two sides of te sell. For te purposes of tis paper, we assume a St. Venant Kircoff material, in wic S is computed from a constant elasticity tensor,, applied to E. For isotropic materials, te constitutive material tensor may be derived from a Young s modulus, E, and Poisson ratio, ν, and te integral over ξ 3 in Eq. (36 can be computed analytically. Te St. Venant Kircoff material model can become unstable wen subjected to strongly compressive stress states [94], but suc states are not encountered in te present application, because transverse normal stress is ignored by te tin-sell formulation and in-plane stresses witin eart valve leaflets are primarily tensile. Isogeometric analysis [59, 60] is employed for modeling te sell structure. We use C -continuous quadratic B-spline functions to represent bot te geometry and displacement solution field. Te details of tis discretization are given in Kiendl et al. [9 93]. A noteworty aspect of tis discretization is te fact tat it requires no rotational degrees of freedom. Te C -continuous approximation space (for a single patc is in H 2, so we may directly apply Galerkin s metod to te forms defined in Eqs. (36 and ( Time integration and FSI solution strategy We complete te discretization of te coupled FSI formulation by using finite differences to approximate te time derivatives appearing terein. In particular, we employ te Generalized-α tecnique [32, 95, 96], wic is a fullyimplicit second-order accurate metod wit control over te dissipation of ig-frequency modes. Tis produces a nonlinear algebraic system of equations relating te unknown coefficients of te fluid, solid structure, mes-movement, sell structure, and multiplier solutions at time level t n+ to te known solutions from time level t n. An attempt to solve tis system wit a monolitic approac (e.g., by Newton s iteration wit a consistent tangent would encounter te following difficulties: Te sparsity pattern of te nonlinear residual s Jacobian matrix would cange as te immersed sell structure moves troug te background mes. 2 Fluid, structure, and mes solvers would become more difficult to intercange. 3 Te potential for drasticallydifferent multiplier and fluid resolutions could lead to instability. To circumvent te tird issue, at eac time step, we compute te solution using te following two-step procedure:. Solve for te fluid, solid structure, mes displacement, and sell structure unknowns, olding λ n fixed. Note tat te fluid and sell structure are still coupled in tis problem, due to te penalty term. 2. Update te multiplier λ n, by adding te normal component of penalty forces present in te solution from Step. Te solution from Step will not satisfy te kinematic constraints exactly at all quadrature points on Γ t. Tis is a deliberate weakening of te constraints to improve stability, as mentioned in Section Te two-step solution procedure may be interpreted as penalization of an implicitlyevaluated time integral of te velocity difference between te fluid and sell structure, as detailed in Kamensky et al. [2], and is conceptually-similar to te metod of artificial compressibility [90] for incompressible flow problems. Note tat te time integral of te velocity difference is a displacement: we effectively implement spring-like sliding contact elements between te fluid and sell structure. Tis prevents te steady creeping flow troug sell structures tat can occur wen only te current velocity difference is penalized, as in te penalty approac coming from Nitsce s metod. To solve te nonlinear coupled problem in Step, we apply a fixed-point iteration based on Newton s metod. Te linear system to be solved witin eac iteration of Newton s metod would ave te form R fl U fl R so U fl R me U fl R s U fl R fl U so R so U so R me U so R s U so R fl U me R so U me R me U me R s U me R fl U s R so U s R me U s R s U s U fl R fl U so R = so, (38 U me R me U s R s were R ( and U ( are te nonlinear residuals and discrete unknowns of te fluid (fl, solid structure (so, mes (me, and sell structure (s. U ( are te corresponding solution increments. To avoid te aforementioned disadvantages of assembling te full consistent tangent, we approximate it wit te block-diagonal matrix R fl U fl R so U fl 0 0 R fl U so 0 0 R so U so R me U me 0 R s U s, (39

9 Fluid structure interaction analysis of bioprostetic eart valves: Significance of arterial wall deformation 9 aortic sinus artery wall left ventricle ascending aorta aortic valve leaflets Fig. : A scematic drawing illustrating te position of te aortic valve relative to te left ventricle of te eart and te ascending aorta. n S x d x 2 n 2 Fig. 3: Illustration of contact notation. S 2 3 Bioprostetic eart valve simulations In tis section, we use te proposed ybrid ALE/immersedboundary metod to simulate te FSI of an aortic BHV implanted in an elastic artery over cardiac cycles. Te aortic valve regulates flow between te left ventricle of te eart and te ascending aorta. Figure provides a scematic depiction of its position in relation to te surrounding anatomy. Te valve leaflets are discretized separately and immersed into a deforming boundary-fitted background mes of te artery lumen. 3. Heart valve model Fig. 2: B-spline eart valve mes comprised of,404 quadratic elements. Te pinned boundary condition is applied to te leaflet attacment edge. ten assemble and solve eac block of equations in sequence (from top to bottom. We use a number of furter approximations witin eac of te left-and side blocks, but maintain te original nonlinear residuals, R (, of te fullycoupled problem. Converging tese residuals to zero solves te original problem, regardless of any approximations used in te tangent matrix. Te procedure tat we apply at eac step of te fixed-point iteration is not equivalent to a linear solve wit matrix (39. To accelerate convergence, we use te updated solutions from previous blocks to assemble te equations for subsequent ones. We repeat tis fixed-point iteration to converge R ( toward zero and obtain a fullycoupled solution of te fluid-solid-mes-sell system. In practice, we use a fixed number of iterations, cosen to yield typically-satisfactory convergence at te selected time step size. Tis algoritm combines te quasi-direct and blockiterative FSI coupling approaces outlined in Tezduyar et al. [97 99] and Bazilevs et al. [00]. Te BHV leaflet geometry used in tis study is based on a 23-mm design by Edwards Lifesciences. We model eac leaflet using a C -continuous B-spline patc, wic comprises 468 quadratic B-spline elements. Te pinned boundary condition is applied to te leaflet attacment edge as sown in Figure 2. An isotropic St. Venant Kircoff material wit E = 0 7 dyn/cm 2 and ν = 0.45 is applied to te BHV. Te tickness and density of te leaflets are cm and.0 g/cm 3, respectively. Tere is no damping applied to te valve dynamics in tis study. 3.2 Leaflet leaflet contact Contact between leaflets is an essential feature of a functioning eart valve. BHV leaflets contact one anoter during te opening, and especially during te closing to block flow. An advantage of immersed-boundary metods for FSI is tat pre-existing contact algoritms from structural analysis [0 05] may be incorporated directly into te structural subproblem witout affecting te fluid subproblem. We adopt a penalty-based approac for sliding contact and employ contact elements associated wit te quadrature points of te sell structure.

10 0 Ming-Cen Hsu et al. Fig. 4 A view of te arterial wall and lumen into wic te valve is immersed. As detailed in Kamensky et al. [2], a contact element activates wen its associated quadrature point, located on a particular BHV leaflet designated S, is found to penetrate troug anoter leaflet, designated S 2. Penalties are computed using a signed distance, d, from S 2 to te quadrature point on S, and teir activation is controlled by several geometrical conditions omitted from te current paper for brevity. Opposing concentrated loads are applied at te quadrature points on S and teir closest points on S 2. Tis notation is illustrated for a pair of contacting points in Figure 3. Te designation of one leaflet as S and anoter as S 2 is arbitrary, and to preserve geometrical symmetries, we sum te forces resulting from bot coices. 3.3 Artery model Te BHV model mentioned earlier is immersed into a pressure-driven incompressible flow troug a deformable artery. Te fluid density and viscosity are ρ =.0 g/cm 3 and µ = g/(cm s, respectively, wic model te pysical properties of uman blood. Te artery is modeled as a 6 cm long elastic cylindrical tube wit a tree-lobed dilation near te BHV, as sown in Figure 4. Tis dilation represents te aortic sinus, wic is known to play an important role in eart valve dynamics [06]. Te cylindrical portion of te artery as an inside diameter of 2.3 cm and a tickness of 0.5 cm. It is comprised of quadratic NURBS patces, allowing us to represent te circular portions exactly. Te sinus is generated by displacing control points radially from an initial cylindrical configuration, so te normal tickness of te sinus varies. We use a multi-patc design to avoid including a singularity at te center of te cylindrical sections. Cross-sections of tis multi-patc design are sown in Figure 5. Te mes of tis artery, wic includes te fluid-filled interior and solid arterial wall, consists of 69,696 quadratic B-spline elements. For analysis purposes, basis functions are made C 0 - continuous at te fluid solid interface and te discretization is conforming. Mes refinement is focused near te valve and sinus, as sown in Figure 4. Figure 5 sows tat te mes is clustered toward te wall to better capture te boundary-layer solution. As sown in Figure 6, we extend te pinned edges of Fig. 5: Cross-sections of te fluid and solid meses, taken from te cylindrical portion and from te sinus. te valve leaflets wit a rigid stent. Te stent extends outside of te fluid domain and intersects wit te solid region, to properly seal te gap between te pinned edge of te valve and te arterial wall. Te arterial wall is modeled as a yperelastic material a neo-hookean model wit dilatational penalty (see Simo and Huges [87] and Section 2.3 of te present paper wit Young s modulus and Poisson s ratio set to 0 7 dyn/cm 2 and 0.45, respectively. Te density of te arterial wall is.0 g/cm 3. Mass-proportional damping is added to model te interaction of te artery wit surrounding tissues and interstitial fluids. In tis case te inertial term in Eq. (22 is replaced as follows: 2 y ρ 2 t ρ 2 y 2 2 t + aρ y 2 2 t, (40 and te damping coefficient, a, is set to s.

11 Fluid structure interaction analysis of bioprostetic eart valves: Significance of arterial wall deformation LV pressure (mmhg LV pressure (kpa Time (s Fig. 6: Te sinus, magnified and sown in relation to te valve leaflets (pink and rigid stent (blue. Fig. 7: Pysiological left ventricular (LV pressure profile applied at te inlet of te fluid domain. Te duration of a single cardiac cycle is 0.86 s. Te data is obtained from Yap et al. [08] 3.4 Boundary conditions and parameters of te numerical sceme Te solid wall is subjected to zero traction boundary conditions at te outer surface. Te inlet and outlet brances are allowed to slide in teir cut planes as well as deform radially in response to te variations in te blood flow forces (see Bazilevs et al. [44] for details. Tis gives more realistic arterial wall displacement patterns tan fixed inlet and outlet cross-sections. Because te BHV stent is assumed to contain an effectively-rigid metal frame [07], te dynamics of te artery and BHV leaflets are coupled primarily troug te fluid rater tan te sutures connecting te stent to te artery. We terefore constrain te stent to be stationary, and likewise fix te displacement unknowns of any control point of te solid portion of te artery mes wose corresponding basis function s support intersects te stent. Te nominal outflow boundary is cm downstream of te valve, located at te rigt end of te cannel, based on te orientation of Figure 4. Te nominal inflow is located 5 cm upstream at te left end of te cannel. Te designations of inflow and outflow are based on te prevailing flow direction during systole, wen te valve is open and te majority of flow occurs. In general, fluid may move in bot directions and tere is typically some regurgitation during diastole. A pysiologically-realistic left ventricular pressure profile obtained from Yap et al. [08] and sown in Figure 7 is applied as a traction boundary condition at te inflow. Te duration of a single cardiac cycle is 0.86 s. Te traction (p 0 + RQn is applied at te outflow, were p 0 is a constant pysiological pressure level, Q is te volumetric flow rate troug te outflow (wit te conven- tion tat Q > 0 indicates flow leaving te domain, R > 0 is a resistance constant, and n is te outward facing normal of te fluid domain. Tis resistance boundary condition and its implementation are discussed in Bazilevs et al. [50]. In te present computation, we use p 0 = 80 mmhg and R = 70 (dyn s/cm 5. Tese values ensure a realistic transvalvular pressure difference of 80 mmhg in te diastolic steady state (were Q is nearly zero wile permitting a reasonable flow rate during systole. At bot inflow and outflow boundaries we apply backflow stabilization wit γ = 0.5 (see Esmaily- Mogadam et al. [86] for details. Te time-step size is set to t = s and te τ M scaling factor is s sell = 0 6. For te immersed eart valve, we find tat results are relatively insensitive to te tangential-velocity penalty-parameter values, wile conditioning and nonlinear convergence improve wen te values are lower. We terefore set a lower value for τtan B = g/(cm 2 s and a iger value for τnor B = g/(cm 2 s, also because te no-penetration condition is more critical for accuracy. 3.5 Results and discussion We compute bot te rigid- and elastic-wall cases to study te importance of including arterial wall elasticity in te eart-valve FSI simulations. Starting from omogeneous initial conditions, we compute several cardiac cycles until a time-periodic solution is acieved. Figure 8 sows te volumetric flow rate troug te top of te tube trougout te cardiac cycle. Te flow rates computed using rigid and elastic arteries differ primarily in te period immediately following valve closure. Te rigid-wall results sow large oscilla-

12 2 Ming-Cen Hsu et al. 500 Rigid wall Elastic wall Flow rate (ml/s t = s t = 0.36 s t = s. t = s Time (s Fig. 8: Computed volumetric flow rate troug te top of te fluid domain, during a full cardiac cycle of 0.86 s, for te rigid and elastic arterial wall cases. tion in te flow rate, as well as in te valve movement (see Figure 9. Te oscillation is muc smaller wen arterial wall elasticity is included. In te rigid-wall case, te energy of te fluid ammer striking te closed valve is initially converted to elastic potential in te leaflets, transferred back to kinetic energy as te valve rebounds, converted into potential as te fluid moves troug an adverse pressure gradient, ten converted once again to kinetic energy as te blood reverses direction, forming a new fluid ammer and restarting a cyclic reverberation. Tis oscillation is gradually damped by te resistance outflow condition and viscous forces in te fluid being directly modeled. Te reverberation of te fluid ammer impact on te closing valve is te source of te S2 eart sound, marking te beginning of diastole [09, 0]. However, te flow rate oscillation tat follows from te rigid artery assumption is observed to be muc smaller or completely absent in uman aortas [, 2]. Tis is consistent wit our elastic-wall computations, wic sow tat an elastic artery as a compliance effect and can distend to absorb a fluid ammer impact and dissipate te initial kinetic energy to surrounding tissues and interstitial fluids (modeled ere troug damping. Te artery s absorption of fluid ammer impacts on te valve greatly reduces te maximum strains (and tus stresses observed in te leaflets, as sown in Figure 9. Remark 4 Te strains sown in Figure 9 are te maximum in-plane principal Green Lagrange strain (MIPE, te largest eigenvalue of E. We coose to plot te strains on te aortic side of te leaflets to include contributions from bot stretcing and bending. Evaluation of strain at te sell midsurface, ξ3 = 0, would only display te membrane contribution. Fig. 9: Leaflet oscillation and te igest MIPE during te cardiac cycle for te rigid and elastic arterial wall cases. Te strains are evaluated on te aortic side of te leaflets. Te maximum MIPE on te plots are for te rigid-wall case and for te elastic-wall case. Remark 5 Note tat te effect we demonstrate ere is not te full Windkessel effect. Direct simulation of te Windkessel effect would require a muc larger network of arteries downstream of te valve, and te final outflow from tese arteries sould be relatively constant [8]. We instead demonstrate tat te elasticity of te arteries directly adjacent to a eart valve can significantly impact its dynamics, especially at te point of valve closure, were maximum strains occur, and sould terefore not be neglected in simulations. We recommend combining tis tecnology wit a lumped-parameter Windkessel model of arteries furter downstream, but we ave applied a simple resistance boundary condition in tis present work to more clearly igligt te effect of arterial FSI witin te directly-simulated domain. We now examine te details of te fluid and structure solutions obtained from te elastic-artery computation. Figure 0 sows several snapsots of te details of te fluid solution fields and Figure sows te deformations and strain fields of te leaflets at several points during te cardiac cycle. As te valve opens during systole, we see transition to turbulent flow. We also see tat te leaflets remain partially in contact wile opening. Te snapsot at t = 0.35 s illustrates te fluid ammer effect tat is evident in te flow rate. After 0.62 s, te solution becomes effectively ydrostatic. Te strain near te commissure points at t = 0.35 s is sligtly iger tan at t = 0.7 s. Tis is due to te effect of

13 Fluid structure interaction analysis of bioprostetic eart valves: Significance of arterial wall deformation 3 t = 0.03 s t = 0.05 s t = 0. s t = 0.26 s t = 0.33 s t = 0.34 s t = 0.35 s t = 0.62 s Fig. 0: Volume rendering of te velocity field at several points during a cardiac cycle. Te time t is syncronized wit Figure 7 for te current cycle. t = 0.03 s t = 0.26 s t = 0.35 s t = 0.05 s t = 0.33 s t = 0.62 s t = 0. s t = 0.34 s t = 0.86 s Fig. : Deformations of te valve from te FSI computation, colored by te MIPE evaluated on te aortic side of te leaflet. Note te different scale for eac time.

14 4 Ming-Cen Hsu et al. Fig. 2: Relative wall displacement between opening (t = 0.24 s and closing (t = s pases. te fluid ammer striking te valve as it initially closes. Tis penomenon is usually neglected by bot quasi-static and pressure-driven dynamic computations, as neiter accounts for te inertia of te fluid [07, 3]. Te FSI solution also sows tat te geometrical symmetry of te initial data is not preserved, wic is typical for turbulent flow. Tis result underscores te importance of computing FSI for te entire valve, witout symmetry assumptions. In Figure 2, te models are superposed in te configurations corresponding to te opening (t = 0.24 s and closing pases (t = s for better visualization of te relative arterial wall displacement results. 4 Conclusions We presented a computational framework for FSI wic combines a recently proposed variational immersedboundary metod [2] and te traditional ALE tecnique. We applied tis ybrid ALE/immersed-boundary framework to simulate a bioprostetic eart valve implanted in an artery tat is allowed to deform in te model. Our computations demonstrate tat te variational immersed-boundary metod for FSI remains effective for eart valve analysis wen te background fluid mes undergoes relatively mild deformations, corresponding to te expansion and contraction of an elastic artery. Furter, we find tat arterial wall deformation contributes significantly to te realism of BHV simulation results. It damps out oscillations in te flow rate and valve deformation during te closing pase, leading to flow profiles tat more closely resemble tose observed in practice [, 2]. Te igest strain on te valve, occurring at te point of valve closure, is muc lower wen wall elasticity is considered. Tis difference in peak strain between te rigid-artery and elastic-artery computations suggests a potential future researc direction: it indicates tat arterial stiffness could be an important variable to consider in computational studies of structural fatigue in BHVs. Aterosclerosis and BHV leaflet deterioration are known to be correlated [4], altoug te prevailing ypotesis, wic we do not purport to refute in tis work, is tat tese penomena ave a sared etiology rater tan a cause-and-effect relationsip. One conspicuous sortcoming of our simulations is te relatively simple material model of te valve leaflets. Te St. Venant Kircoff material used in tis work does not accurately reflect some of te properties of biological materials [94,5]. In te present application, te largest strains are primarily tensile, avoiding te St. Venant Kircoff material s most significant patology: instability under compression. However, its tensile beavior does not exibit te exponential stiffening caracteristic of soft tissues [6, 7]. Te introduction of a more realistic soft tissue material model will allow for meaningful comparison of te valve s deformations wit detailed geometrical data collected in te flow loop experiments of Iyengar et al. [8] and Sugimoto et al. [9]. Acknowledgements Y. Bazilevs was supported by te NSF CAREER Award No T. J. R. Huges was supported by grants from te Office of Naval Researc (N , te National Science Foundation (CMMI-00007, and SINTEF (UTA wit te University of Texas at Austin. M. S. Sacks was supported by NIH/NHLBI grants R0 HL08330 and HL9297, and FDA contract HHSF P. D. Kamensky was partially supported by te CSEM Graduate Fellowsip. We tank te Texas Advanced Computing Center (TACC at te University of Texas at Austin for providing HPC resources tat ave contributed to te researc results reported in tis paper. References. F. J. Scoen and R. J. Levy. Calcification of tissue eart valve substitutes: progress toward understanding and prevention. Ann. Torac. Surg., 79(3: , P. Pibarot and J. G. Dumesnil. Prostetic eart valves: selection of te optimal prostesis and long-term management. Circulation, 9(7: , R. F. Siddiqui, J. R. Abraam, and J. Butany. Bioprostetic eart valves: modes of failure. Histopatology, 55:35 44, S. Hales. Statical Essays: Containing Haemastaticks: Or, an Account of Some Hydraulick and Hydrostatical Experiments Made on te Blood and Blood-Vessels of Animals. W. Innys and R. Manby; T. Woodward, London, O. Frank. Die Grundform des arteriellen Pulses. Erste Abandlung. Matematisce Analyse. Zeitscrift für Biologie, 37: , 899.