Static model of stent based on the model of nonlinear hyperelastic rod

Abstract

Stent je mrežica koja zatvara oblik cjevčice, sa primjenom u medicini. Klinički ishodi procedure dobrim dijelom ovise o mehaničkim svojstvima stenta koji se implantira. Mehanička svojstva stenta ovise o obliku, načinu proizvodnje i materijalima od koji je napravljen. U ovom radu razvijen je matematički model koji se može primijeniti kod onih stentova koji ima oblik mrežice sastavljene od bridova (štapića), a gdje su krajevi bridova međusobno fiksno spojeni. Spojevi krajeva raznih bridova nazivamo vrhovi. Kako bi se smanjili troškovi i ubrzao razvoj potrebno je procijeniti mehanička svojstva stenta još u fazi dizajna stenta. U tu svrhu provode se računalne simulacije koje su do sada počivale na 3D modelima elastičnosti ili elastoplastičnosti. Takve su simulacije izrazito zahtjevne za suvremena računala. Nedavno je razvijen linearan 1D model elastičnog stenta, tako da računalne simulacije koje se na njemu temelje zahtjeva ju znatno manje računalnih resursa. Međutim, rezultati linearnog modela opravdani su samo kod malenih deformacija. Novi nelinearan 1D model elastičnog stenta, koji je u ovom radu razvijen i opravdan, može biti temelj za razvoj efikasnih numeričkih algoritama koji će biti točni i za veće deformacije. Novi model stenta temelji se na poznatom 1D nelinearnom hiperelastičnom modelu štapa Scardia (2006), koji je formuliran kao problem minimizacije funkcionala tzv. unutarnje energije. U modelu stenta se prirodno koristi model štapova koji predstavljaju bridove mreže. Model dopušta savijanje i torziju bridova. Postojeći model štapa ovdje je poopćen na manje regularne geometrije, a zatim uparen s rubnim uvjetima. Postojanje rješenja rubnog problema pokazano je uz pomoć standardnih matematičkih tehnika varijacijskog računa i nekih novih geometrijskih rezultata o krivuljama u prostoru. Dalje je korištenjem teorije $\Gamma$ -konvergencije pokazana jedna vrsta neprekidne ovisnosti rubnog problema štapa o njegovim parametrima. Ovaj rezultat stabilnosti može biti važan u praktičnim računalnim simulacijama jer otklanja mogućnost da malene promjene u geometriji stenta uzrokuju drastično različita mehanička svojstva. Nadalje, taj rezultat omogućuje aproksimaciju krivulja u računalnim simulacijama, npr. sa p o dijelovima ravnim elementima. Model stenta iskazan je u minimizacijskoj formulaciji, koja prirodno proizlazi iz formulacije modela štapa. Pokazano je kako se model može poopćiti i na druge strukture sastavljene o d štapova (npr. građevine sastavljene o d tankih štap ova). Matematički rezultati sastoje se od dokaza postojanja rješenja i stabilnosti minimizacijske zadaće u odnosu na geometriju koju stent zauzima. Rezultati za stent prirodno se nadovezuju na pokazane rezultate za štap. Osim spomenutih matematičkih metoda korišteni su i drugi elementi: iz realne analize, topologije, geometrije i teorije grafova. U izdvojenom odjeljku diskutirana je sila i predloženo kako provesti mehaničko testiranje stenta i eksperimentalnu validaciju prikazanog matematičkog modela.Classical elasticity: Even though the current bio engineering literature models stent exclusively as a single 3D elastic body, we approach stent modeling by simulating slender stent struts as 1D nonlinear rods. Simulating slender stent struts using 3D approaches (3D Finite Elements) is computationally very expensive and time consuming. This makes testing of a large number of stents for optimal stent design computationally prohibitive, and often times produces simulation results with poor accuracy. This is where our approach could prove advantageous: it could speed up the computation by the order of magnitude while keeping the accuracy of results, even when deformations are large. The approach has been applied in Tambača et al. (2010) to model equilibrium deformation within the scope of the linearized elasticity. We are not aware of any research related to the 1D stent modelling using 1D nonlinear rod models. Thus this work is original indeed. Main results: Nonlinear 1D stent models are based on the use of 1D model for struts plus junction conditions on vertices. We take the 1D nonlinear bending-torsion model for curved rods that was rigorously derived from nonlinear three-dimensional elasticity in Scardia (2006). This nonlinear curved rod model is derived by $\Gamma$–convergence techniques applied to the elastic energy and for the middle curve of the undeformed curved rod parameterized by a $C^2$ function. This is what we do: 1. generalize formulation of the 1D model of elastic curved rods to include Lipschitz middle curves, 2. prove existence of a solution of the boundary value problem for the generalized formulation of the nonlinear curved rod model, 3. prove continuity (stability) of the model with respect to the geometry of the undeformed rod, 4. formulate the 1D model for general structure that consists of rods and prove the existence of the solution, 5. formulate the 1D model for stent and prove the existence of the solution, 6. prove continuity (stability) of the stent model with respect to the geometry of the stent, 7. investigate the loads that can be replicated in an experiment and used in this stent model. Junction of two curves is not necessarily smooth. Complex junctions app ear naturally in stents: for example where three or more rods join together. Thus, in order to build a stent model, one first needs to reformulate the 1D model to be well defined for the Lipschitz curved rods. Analysis of the properties of the rod model serves as the good introduction in analysis of the more complex stent models. The results of this analysis also have their own merit. Continuity with respect to undeformed geometry is important feature of the model. It provides also a justification for our generalized model in case of less regular middle curve geometries mentioned above. Similar analysis for the linear Naghdi shell model for shells with little regularity is performed in Blouza, A. Le Dret (2001). This continuity is done using curves of the same and of different lengths. Obtained continuity property is important in order to simplify numerical approximation of the model. Formulation of the stent model is the starting point for the stent modeling. From the mathematical point of view the existence of the solution is important in order to have well posed problem. Continuity of the stent model with respect to the geometry is important from several points of view. It can be viewed as a stability result which is important in any kind of modeling. It also can be used to approximate the rod geometry by piecewise linear geometry which is easier to discretize. Methodology: The formulation of this model is by minimization of the total energy functional on a suitably chosen set of admissible deformations. For this part of the project we will apply direct methods of the calculus of variations. The formulation of the boundary value problem of the nonlinear bending-torsion rod model for specific loads can be described as a minimization problem for $R \in W^{1,2}(0, l; SO(3))$ on a suitably chosen set of functions including boundary conditions. The columns of $R$ are the tangent, normal and binormal, i.e., the Frenet frame, to the rod’s deformed middle curve. As $R(s) \in SO(3)$ the rod is inextensible and unshearable. The strain in the model from Scardia (2006) is given as the difference $R^T.\dot{R}-Q^T.\dot{Q}$ , where the columns of $Q$ form the Frenet frame of the undeformed geometry. This formulation requires at least $Q \in W^{1,2}(0, l; SO(3))$. However, the rotation $R$ can be viewed as a rotation $P$ applied at the undeformed geometry, i.e., $R(s) = P(s).Q(s), s \in [0, l]$. A simple calculation shows that the boundary value problems can be easily reformulated in terms of the ’transformation rotation’ $P$. Such formulations have no derivatives on $Q$ and the models are now well formulated for $P \in W^{1,2}(0, l; SO(3))$ and any measurable $Q$ with values in $SO(3)$ almost everywhere, i.e., $Q \in L^1(0, l; SO(3))$. This implies that the new formulation includes Lipschitz middle curves. For example, the new formulation is well defined for undeformed geometries with corners. This is in agreement with the one-dimensional model of junction of rods Tambača & Velčić (2012). As a consequence of general theory, $\Gamma$-limit functional is lower semicontinuous and if it is bounded from below on a compact set it attains minima on the set. However, in Scardia (2006) no loads and boundary conditions are prescribed and additionally, as mentioned above, we have reformulated the model. We will prove the existence of the boundary value problem of the new formulation of the model by classical direct methods of calculus of variations. In the case of the boundary value problem for rods clamped at both ends the most difficult part, due to the inextensibility of the rod, is to show that the set of admissible functions is nonempty. Let us consider a sequence of geometries described by $Q_n \in L^2$ that converge to $Q$ in $L^2$. For the model with both ends clamped we will show that the sequence of total energy functionals associated with $Q_n$, in the appropriate topology (in which minimizers converge), $\Gamma$–converges to the total energy functional associated with $Q$, in case $Q$ is somewhat special. As a consequence, limit points of a sequence of any minimizer of energy for the geometry $Q_n$ are minimizers of the energy for the limit geometry $Q$. To prove this we build a complex result about approximation of the deformed geometry with precise estimates. In the case of rod clamped at only one end the situation is more simple as no special geometry result is necessary. The key step in construction of strongly convergent approximation sequence for use in the lim sup inequality of the $\Gamma$-convergence is based on the following result: for undeformed geometry $Q$, deformation rotation $P$, two endpoints $v_0, v_1$ of the curve given by $P.Q$ we get that for all $\tilde{Q}, \tilde{v}_0, \tilde{v}_1$ such that $\| Q–\tilde{Q} \|, \| \tilde{v}_0–v_0 \| , \| \tilde{v}_1–v_1 \|$ are small enough there is $\tilde{P}$ with the same values at ends as $P$, such that $\tilde{P}.\tilde{Q}$ connects $\tilde{v}_0$ and $\tilde{v}_1$ and furthermore which is close enough to $P$ in $W^{1,2}$. We prove it using the inverse function theorem with precise estimates, see Xinghua (1999). Finally, we use nonlinear bending-torsion curved rod model to model stent struts and more genereal structures. Junction conditions at vertices are given similarly as in the linear case by: continuity of the displacement of the middle curve of the struts joining in the vertex and continuity of the rotation of the cross-section of the struts joining in the vertex (this means that deformation rotation is well defined for vertices). Nonemptyness of the admissible function set is trivial as we suppose that the stent is already built and the reference position satisfies the junction conditions. Then the existence result follows using classical methods of calculus of variations. We obtain the continuity of the stent model with respect to geometry by using $\Gamma$–convergence of the total energy functional, see e.g. Braides (2002) for details on $\Gamma$–convergence. The most delicate limsup result is obtained based on the geometrical approximation lemma stated earlier for one rod, and some delicate analysis. One hard case of continuity result is solved by introducing the notion of equivalence between stents