An evolutionary approach to Wall Shear Stress prediction in a grafted artery

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Abstract

Restoring the blood supply to a diseased artery is achieved by using a vascular bypass graft. The surgical procedure is a well documented and successful technique. The most commonly cited hemodynamic factor implicated in the disease initiation and proliferation processes at graft/artery junctions is Wall Shear Stress (WSS). WSS distributions are predicted using numerical simulations as they can provide quick and precise results to assess the effects that alternative graft/artery junction geometries have on the WSS distributions in bypass grafts. Validation of the numerical model is required and in vitro studies, using laser Doppler anemometry (LDA), have been employed to achieve this. Numerically, the Wall Shear Stress is predicted using velocity values stored in the computational cell near the wall and assuming zero velocity at the wall. Experimentally obtained velocities require a mathematical model to describe their behavior. This study employs a grammar based evolutionary algorithm termed Chorus for this purpose and demonstrates that Chorus successfully attains this objective. It is shown that even with the lack of domain knowledge, the results produced by this automated system are comparable to the results in the literature.

Introduction

Since their inception, Evolutionary Algorithms (EAs) have been seen as a step towards automatic problem solving. They have already been shown to be effective in Preprint submitted to Elsevier Science 23 August 2003 domains as diverse as mathematical modeling [19], data mining [6], automatic generation of computer programs [4], [8], timetable scheduling [7] and electronic circuit design [5]. As Evolutionary Computation methods are bottom up methods, they do not, as such, require problem specific (be it theoretic or practical) information. This distinguishes them from domain specific problem solving techniques and provides us with a different approach to attack the problem.

The problem investigated in this paper relates to perhaps the most serious of issues, life itself. Most of us are familiar with the idea of occluded, stenosed or blocked arteries. Any such blockage, in any part of the human body, can result in disease in the body part with possible amputation resulting. Fortunately, most cases of this nature are treatable by a surgical procedure where a bypass graft (a man made artery) is used to allow the blood to flow past the blockage. Various procedures are used for different occluded arteries in different parts of the body. We will concentrate on limb extremities where it is well known that such grafting has been successful in the restoration of blood supply (see [10], [12], [15]).

Bypass grafts have moderate long-term patency rates, which indicate how long the graft can remain disease free. These patency rates vary with surgical technique and also from one surgical group to the next. It is widely agreed that the endto-side graft/artery junctions created during bypass surgery result in flow patterns that place abnormal Wall Shear Stress (WSS) distributions in the endothelial cells which occur on the bed of the junction and normally cover the inner surface of blood vessels [16]. Therefore, the most commonly cited hemodynamic factor responsible for the initiation of disease and proliferation processes at graft/artery junction beds is Wall Shear Stress. Hence, the computation of the Wall Shear Stress distribution on a graft/artery beds is important in determining the role of hemodynamics in the disease formation process after the installation of a bypass graft. The bypass graft is connected to the artery at a certain angle, to investigate the optimal angle which would result in flow patterns that are least damaging to the endothelial cells, is the main focus of the research being undertaken in the development of bypass grafts. The Wall Shear Stress prediction requires a velocity profile for the values measured near the wall and assuming zero velocity at the wall. This requires a mathematical model which fits the experimentally observed values for the velocities.

This paper demonstrates the use of a position independent, grammar based evolutionary algorithm, Chorus [17] for the problem. As is the case with Genetic Programming (GP) [4] and its variants such as Paterson’s approach [20] and Grammatically-based Genetic Programming by Whigham [21], Chorus belongs to the same class of EAs which evolve computer programs to solve the given problem. Following a biological metaphor, where the genetic print out is responsible for the observable traits of an organism, Chorus recognizes a distinction between genotype and phenotype. A genotype–phenotype mapping process dictates the translation from the genetic material to the observable behavior, which in our case represents a candidate solution to the problem. As with [20], [21] and Grammatical Evolution (GE) [13], [18], Chorus makes use of the context free grammars represented in Backus Naur Form (BNF) to describe the language in which the computer programs are produced. The use of grammars gives the user the flexibility of selecting the language of his choice. It should be noted that, usually, only a subset of a high level language is used. A complete high level programming language specification is not needed and may lead to unnecessarily large search space for most of the problems.

The paper first gives a brief introduction to the concept of artificial evolution, as all EAs can be seen as simplified simulations of very complex process of evolution going in nature. It is followed by a description of a context free grammar in Backus Naur Form, which is used to specify the language we want our solutions to be in. We then describe the Chorus system and the process involving the mapping from a genotype to phenotype is discussed, with an example. Section 6 discusses the problem domain and the experimental results, to be followed by conclusions. The problem domain selected for this paper focuses on the research work being undertaken by scientists engaged in the development and analysis of the grafting process in arteries (see [9], [15], [23]).

Section snippets

Artificial evolution

The process of evolution by natural selection is seen as a procedure of finding individuals that fit their environment. A simplified simulation extrapolates this phenomenon to automated problem solving. A pool of candidate solutions is created for a particular problem. It is subjected to a process of refinement derived from a subset of the principles of natural evolution. It is analogous to a population of individuals undergoing adaptation to match the requirements of an environment. Every

Backus Naur Form

Backus Naur Form is a notation for describing grammars. A grammar is represented by a tuple {N,T,P,S,}, where T is a set of terminals, i.e. items that can appear in legal sentences of the grammar, and N is a set of non-terminals, which are interim items used in the generation of terminals. P is the set of production rules that map the non-terminals to the terminals, and S is a start symbol, from which all legal sentences may be generated. All legal sentences are produced by modifying the start

The Chorus System

Chorus [17] is an automatic programming system based coarsely on the manner in which enzymes regulate the metabolism of a cell. Chorus belongs to the same family of algorithms as Grammatical Evolution [13], [18], and shares several characteristics with it. In particular, the output of both systems is governed by a BNF grammar as above, and the genomes, variable length binary strings, interpreted as 8 bit integers, are used to produce legal sentences from the grammar. The integers are read to

Genetic operators

The binary string representation of the genotypes of the individuals effectively provides a separation of search and solution spaces [22]. The genotypes that are directly subjected to the genetic operators come from the search space where the evolutionary search process looks for the right set of codons which can be mapped to high quality solutions from the solution space. This is different from tree based GP [4] where no distinction is made between genotype and phenotype. This separation

Blood flow problems in grafted arteries

In this section we demonstrate one of the multi faceted applications of the Chorus system. As mentioned at the onset, our aim is to address the problem of computing the Wall Shear Stress. This study is concerned with lower limb arteries and the stenoses (reduction in internal cross-sectional area) and occlusions (blockages) that occur there. As mentioned earlier, a graft is used to bypass the occluded or stenosed region. A typical graft is illustrated in Fig. 1 where we see the proximal

Conclusions

In this paper we have shown the application of a grammar based evolutionary algorithm, Chorus, as an aid in solving real life problems. The results obtained by Chorus are not only slightly better than the standard methods employed by researchers engaged in this work, they are innovative as well. This can be attributed to the bottom up approach where the system assumes little familiarity with the problem domain and user experience. Such an approach can be extremely useful in giving an unbiased

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