Genetic Algorithm Search for Stent Design Improvements

Transcription

1 Genetic Algorith Search for Stent Design Iproveents K. Tesch, M.A. Atherton & M.W. Collins, South Bank University, London, UK Abstract This paper presents an optiisation process for finding iproved stent design using Genetic Algoriths. An optiisation criterion based on dissipated power is used which fits with the accepted principle that arterial flows follow a iniu energy loss. The GA shows good convergence and the solution found exhibits iproved perforance over proprietary designs used for coparison purposes. Introduction Stents are etallic cage like structures (Figure 1) that are inserted into an artery blocked by calcified plaque (stenosis). Stents differ significantly in shape, cross-sections, and other details, which affect the haeodynaics of the blood flow through the treated region. Fig. 1. Palaz-Schatz stent Early designs were woven fro round-section wire but current designs are laser-cut cylinders enabling a wider range of designs. In its unexpanded for a stent is delivered on an angioplasty balloon and receives its final shape after expansion beyond its elastic liit. Stent Pattern Description Stent design is deceptively siple but the degrees of freedo of the patterns ean there are thousands of potential solutions. Also these shapes can be rather coplicated and therefore it is difficult to describe all the shapes using siple descriptions. In this study the stent patterns were represented by five ain features as described below. Fro Figure 1 the stent is seen to coprised of strut arranged to ake the patterns. Three values of strut thickness were used ranging for 0.08 to 0.10 (Figure ) Fig.. Strut Thickness, d 1

2 The shape of the strut cross-section was represented as a ratio of thickness to wih (Figure 3) based upon the actual thickness value used fro Figure. d x d d x 1.5 d d x 1.5 d Fig. 3. Strut Thickness: Wih ratio Generally the stent patterns are syetrical therefore a skew paraeter was introduced in order to explore asyetry. A skew value of 0.5 represents syetrical pattern and value of 0.9 produces asyetry through bringing the two peaks of the pattern closer together (Figure 4) Fig. 4. Pattern Skew Repeating Pattern deterines whether the next stents segent is erely a copy or it is a irror iage of the existing segent. If a segent is copied then an artificial link ust be added to join the two segents together. A irror operation does not need any linking eleents because segents are joined naturally (Figure 5). copy irror Fig. 5. Repeating Pattern Shape Order (Figure 6) defines the degree of curvature of the segents. In this particular case 1 st order produces a sharp definition and nd order produces a soother definition. second first Fig. 6. Shape Order

3 The five variables described above allow to coverage of a relatively large design space. However it is only a siplification and one could iagine a ore coplex description. Table 1. Stent pattern description Nae Range Strut Thickness d () Strut Thickness:Wih ratio 1:1-1:1.5 Pattern Skew Repeating Pattern Copy/Mirror Shape Order 1 st and nd As a first approxiation, the design was paraeterised according to Table 1 in order to siplify the search. For coputational purposes, it is convenient to work with a partial odel of a stent rather than a full 3D odel Physics Blood flow behaviour ay be described by Navier-Stokes and continuity equations (Equations 1). d U 0 (1) du g σ The general for has too any unknown variables. We need another equation known as Newtonian hypothesis. It describes a linear relation between stress tensor σ and strain rate tensor D σ p Uδ D 3 where strain rate tensor is given by 1 : U U T D (3) Having 10 scalar forulas (1 scalar 1 vector 1 syetrical tensor) we still have 1 unknown variables: U,U,U,,,,,,,p,. We ay treat blood as an, x y z xx yy zz xy xz yz incopressible ediu then const, = var (4) The 1 scalar forulas can be rewritten now as follows U 0 du p U (5) D There are 4 equations and 5 unknown variables: U,U, U,p,. It is necessary to decide whether to treat blood as Newtonian or non-newtonian. A non-newtonian behaviour allows a relationship between viscosity and velocity gradients according to the following power law n1 k (6) where k and n are constants. e.g. blood n 0. 61, k kg s and shear strain rate for a general 3D case D: D (7) Finally, the fifth non-linear equation is established, which describes the non-newtonian behaviour of blood x y z () 3

4 n1 : D k D (8) Forulas (5) and (8) ay be solved nuerically using Finite olue Method or Finite Eleents Method. However for either approach the stent geoetry first ust be created and then discretised (Figure 7). Fig. 7. CFD Mesh exaple The descretisation process and esh quality are crucial to the accuracy of results. Therefore special care ust be taken near the stent surfaces where velocity gradients are relatively high. Different types of eleent were ipleented in order to iprove esh quality and the convergence. Objective Fitness Function One could iagine a lot of different fitness functions. It is believed that Wall Shear Stress plays the ost iportant role in bio-edical flows []. However as WSS is distributed along the surface it cannot be directly used as a perforance easure in driving a search algorith that needs a nuber(s). Indeed two different WSS distributions cannot be copared directly, yet it is possible to define such a perforance based on that distribution. It should be borne in ind that if a nuber is generated fro the 3D distribution then inforation is always lost. Dissipated power is introduced here as an alternative perforance easure. Let us consider a for of the Gibbs equation that has the shape of an energy equation: de p Ts d T (9) where T is teperature and and we neglect the heat conductivity we have Ts is the intensity of entropy production. As blood is incopressible de (10) In other words the dissipated energy causes an increental change of internal energy. The effect of this energy dissipation is an increase of local teperature. Intensity of entropy production ay be calculated fro the velocity field (for an incopressible ediu). Ts = σ : U (11) It can be proved that not all of the work in Equation 1 is converted into kinetic energy dl dl = de Ts dxdydz (1) S kin The energy that is not converted, is dissipated. Therefore dissipated power is defined as follows N := Ts dxdydz (13) or dissipated energy diss diss t Ts E := Ts dxdydz (14) Miniising such an objective fitness function (Equation 13) helps us to search for a stent shape with the sallest possible energy losses. 4

5 Dissipated power [W thickness copy shape Paraeter Encoding Two alleles were used for both ratio and thickness in encoding the GA chroosoe. As these variables have only three values and the alleles can represents four values ( ) then a duy level was necessary. Therefore the fourth value in the allele cobination was ade equal to the third value. Table. Nae Range Step Type Bits Strut Thickness d () Floating Strut Thickness:Wih ratio Floating Pattern Skew Floating 3 Repeating Pattern Boolean 1 Shape Order 1-1 Integer 1 The full paraeter encoding for the stent chroosoe is shown below. The total length of chroosoe is 9, which represents 5 variables. ratio skew Fig. 8. Paraeter encoding The above encoding presents us with 9 51 possibilities. It is necessary to ention that not all of the are unique because of the duy level, which eans there are 88 unique cases. Optiisation alues and Results Due to the relatively short chroosoe length and the high cost of calculation, the population size was chosen to be 10. Crossover probability was set at 0.75 and utation probability at 0.0 to avoid local extree convergence. The total nuber of generations passed to convergence was 9. During the calculation process 15 utations and 38 crossovers were produced. The total nuber of unique stent shapes tested was 8, which is about 10% of the whole design space. The tournaent ethod of selecting individuals for crossover was used for two reasons. Firstly the roulette ethod was unsuitable for dealing with a iniised objective function and also it needed function scaling to avoid rando wandering. Secondly our experience suggested it as the superior approach. A tournaent size of was used Average Miniu Population Fig. 9. Convergence 5

6 The optial solution obtained fro GA optiisation process is shown below. Fig. 10. The GA optial solution The set of design variables defining the optial solution is shown in Table 3. Discussion & Conclusions Table 3. Nae alue Strut Thickness d () 0.08 Strut Thickness:Wih ratio 1:1.5 Pattern Skew 0.5 Repeating Pattern Mirror Shape Order 1 st The GA is liited in ters of its inability to interpolate the discrete values of design variables explored. However, the big advantage of the GA ethod was seen to be in its quick convergence. One could ask if the GA found the true optiu, which of course cannot be answered without knowing all the possible solutions. Indeed the GA solution copared favourably with that fro another solution. It is clean that optial shape depends on the objective fitness function. Dissipated power prefers shape that looks siilar to a Palaz-Schatz design. Yet other perforance easure would presuably give different results. To answer that question it is necessary to introduce a way of classifying shape perforance based on WSS. This will be investigated in our future work. References [1] Goldberg, D.E., Genetic Algoriths in Search Optiization and Machine Learning, Addison Wesley, (1989). [] Sigwart, U., Coronary stents: will they survive?, pp in 'Coronary Stents' (Eds. U. Sigwart and G. I. Frank). Springer-erlag (199). [3] Haupt R.L., Haupt S.E. Practical Genetic Algoriths John Wiley & Sons, Inc. (1998) 6