SIMULATION AND ANALYSIS OF FLOW IN DIFFERENTGEOMETRICAL ARRAYS OF A VASCULAR GRAFT

García J.*, González *, Briceño J.C. ** Group of Biomedical Engineering, Mechanical Engineering Department, University of Los Andes

Abstract

Blood flow dynamics has a fundamental role inthe success of vascular grafts. In particular, shearstresses and secondary flows are involved in plateletactivation and aggregation. In order to investigatethese three dimensional flow characteristics, a pro-gram was developed and applied to the results ofcomputational simulations. The program evaluatedshear stress and residence time values to get a level ofplatelet activation. The used geometries were a sim-plification of real geometries obtained by injection- corrosion method and angiographic images. Thepartial results show that shear stresses reached ab-normally high values in the anastomosis region, andpresence of secondary flows in the post - anastomosisregion.

Key words: Blood flow dynamics, wall shearstress, graft, platelet activation, computational fluiddynamics.

Introduction

Cardiovascular diseases are the principal cause ofdeath in the world. In 2005, 17.5 million people diedbecause of cardiovascular disease, being coronarycardiopaties and cerebrovascular events the princi-pal causes [6]. The group of biomedical engineering(GIB) has developed a regenerative vascular graftof porcine small intestine submucosa (SIS). The in-vivo experiments in animal veins showed a high rateof occlusion, therefore the study of flow is relevantto improve the design and implant protocol of thegraft. Several groups have proposed computationalevaluations of flow characteristics in cardiovasculardevices and pathologies. These studies showed thatshear stress along the particle trace was involved inblood damage, creating a blood damage index forhemolysis analysis. This index was based on shearstress and exposure time without checking plateletactivation [1]. Other studies analyzed the flow inaxisymmetric stenosis, and created a platelet activa-

tion index, based on shear stress and exposure time[2]. However, there is a shear stress threshold forplatelet activation independent of the exposure timeto each shear stress. Furthermore, most studies wereinterested in designing vascular grafts to improve theanastomosis in bypass surgery, not in a vessel re-placement (Termino - terminal anastomosis) [3, 4].As a result, in this work, flow through a simplifiedgeometry of real collagen graft in jugular vein of lep-orids was analyzed. In addition, animal experimentshave shown persistent occlusion of grafts. The oc-clusion taking into account flow variables only andthe possible relationship with physiological variableswill be explained. The real geometry is obtained inthe first minute after surgery and with help of an-giographic images, a group of simplified geometrieswas built with different rate between graft and ves-sel diameter. Computational simulation was basedon the finite volume method. Moreover a particledistribution at inlet of the model to be trajectorizedwas created. The results were exported to MATLABfor further analysis of platelet activation.

SIS Graft

GIB investigates invivo behavior of collagen regen-erative grafts tested in jugular veins of leporids. Theaverage diameter of the vein is 3mm and there werediscussions about the ideal graft diameter to improveflow conditions at the anastomosis. These groupcompleted works about the mechanical properties ofthe graft in different stages of regeneration [8, 16].Regarding the flow there are no precedent works.The moment to analyze the flow is the first post-implantation minute. It is in this moment whenphenomena like inflammation and vasoconstrictionoccur generating particular flow conditions. Theseconditions include high shear stress and secondaryflows, as potential causes of thrombogenicity andeventual graft occlusion.

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Objectives

General objective

• To simulate and analyze flow in different ge-ometries of the graft, varying the relation ofgraft-vessel diameter and the anastomosis di-ameter.

Specific objectives

• To acquire and create geometries.

• To simulate flow.

• To analyze flow variables: shear stress, resi-dence times and platelet activation.

Method

Acquisition and creation of geometries

A real geometry was acquired by using injection -corrosion technique. This technique consisted of in-jecting a polymer in the vessel, sealing it and waitingfor the hardening. Then the piece was immersed ina corrosive solution to acquire the internal geometry.Based on this geometry and angiographic images, agroup of geometries was created in Solid Edge (Syn-chronous Technology 2 - SIEMENS CAD software)with different rate of vessel - graft diameters anddifferent diameter of anastomosis, the vessel diam-eter was constant (3mm) and the rate varies from1 to 1.5, the last one was equivalent to the actualdiameter of graft in the animal model.

Flow simulation

The simulation was performed in the commercialsoftware Fluent (Versión 12.0, ANSYS). The geome-tries were meshed with tetrahedral elements chang-ing the growth rate. Pressure convergence waschecked increasing the number of calculated ele-ments and calculating the head loss pressure in eachsimulation. Boundary conditions were pressure out-let at the end of the geometry (P = 666Pa) anda paraboloid velocity profile at inlet with averagevelocity (V = 0.1m

s ) for the worst high velocitycase. The convergence criterion for simulation was10−6 for residuals values (Continuity and velocities).

Newtonian fluid with viscosity µ = 3.5cP was con-sidered. These conditions put the problem in lami-nar regimen with Re = 90. Streak lines were usedto determinate shear stress and exposure time, a La-grangian tracking approach was employed with dis-placement of each streak line being computed usingforward Euler integration of the velocity over a timeinterval. The streak lines began at the geometryinlet in user defined position (Figure 1) and not allended at the outlet, some streak lines entered in vor-texes. Hydrodynamic conditions kept the plateletsaway from the vessel wall. The starting point forplatelets at the inlet had a minimum distance fromthe vessel wall of 0.1mm taking into account hydro-dynamic behavior.

Figure 1: Starting points for platelets at inlet

Flow analysis

The results of streak lines were exported to MAT-LAB (R), and the stress tensor (ST ) at a given pointin space was constructed as follows:

ST =

σxx τxy τxzτyx σyy τyzτzx τzy σzz

(1)

τxy = τyx = µ

�∂v2∂x1

+∂v1∂x2

�(2)

τxz = τzx = µ

�∂v3∂x1

+∂v1∂x3

�(3)

τyz = τzy = µ

�∂v3∂x2

+∂v2∂x3

�(4)

2

σzz = −p2

3µ∇.V + 2µ

∂v3∂x3

(5)

σyy = −p2

3µ∇.V + 2µ

∂v2∂x2

(6)

σxx = −p2

3µ∇.V + 2µ

∂v1∂x1

(7)

where σ is the normal stress and τ is the shearstress.

Once the shear stresses were determined, theEigen values were found for every point of all tra-jectories, providing the 3D principal stresses, andthe maximum shear stress:

τmax = σ3D =(σmax − σmin)

2(8)

The obtained data was processed to calculate alevel of platelet activation (LPA) [2]:

LPA =�

τ ti (9)

where τ is the maximum shear stress calculatedby equation 8, and ti is the exposure time exportedfrom the simulation.

The program creates a group of symbolicplatelets simulating the whole blood volume, a num-ber corresponding to trajectories is randomly se-lected, and randomly placed in each trajectory, forsymbolic platelets, the program calculates the LPAand finds the maximum shear stress. A plateletwas activated if the LPA is higher than 1.7Pa ∗ sor if τmax is higher than 10.5Pa [2]. These valueswere considered for young platelets, the capabilityto become activated decreases with the age of theplatelet [5]. An age factor is randomly given to eachplatelet, the older platelet is activated under LPA =10, 5Pa ∗ s and τmax = 21Pa or higher values. Theinformation was saved and the platelets returned tothe blood volume, which was randomly mixed. Adegeneration factor was included for platelets, ran-domly replacing some of them in each cycle for newplatelets and replacing the older ones. This cyclewas repeated ten times to calculate the percentageof activated platelets. The complete process was re-peated ten times to get an average to eliminate therandom effect. Results were analyzed and related topossible physiological factors that could acceleratethe activation and occlusion process.

Results and analysis

Acquisition and creation of geometries

The principal characteristic of the geometries is theinfluence of the sutures that creates a parachuteshape in the anastomosis region (Figure 2). Thegeometries created have six differents relation of di-ameters graft - vessel DG

DVfor 1.0 to 1.5, and three

different relation of diameters anastomosis - vesselDADV

(Figures 3 and 4).

(a)

(b)

Figure 2: Suture influence in injection - corrosionmodel (a), global shape in angiography image (b)

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Figure 3: Simplified geometries, three different anas-tomosis diameters a) DA

DV= 0.5, b) DA

DV= 0.66, c)

DADV

= 0.86

Figure 4: Simplified geometries, group with DADV

=

0.66 and different relation of diameter DGDV

Flow simulations

The convergence analysis starts with a calculationof 50.000 tetrahedral elements and the number ofelements was duplicated until 1.6 ∗ 106. The simu-lation results showed that the adopted model with0.8 ∗ 106 elements and a refined model with 1.6 ∗ 106element has less than 1% difference between headloss pressure (Figure 5). The mesh was controlledby varying the growth rate. The paraboloid bound-

ary condition of velocity at inlet is shown in figure6.

Figure 5: Problem convergence graphic

Figure 6: Paraboloid velocity inlet contour

The flow phenomenon observed in each geometryhas particular elements to evaluate. Therefore, it isimportant to reduce the shear stresses and to avoidthe secondary flows. The general results present anacceleration at the anastomosis and secondary flowsafter the anastomosis, consequences of the reductionand expansion of diameter. The exception is thesmall relation of graft - vessel diameter in the biggeranastomosis, D = 2.5mm. In these cases, the geom-etry has low perturbation of flow, but in the others

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geometries the phenomenon change completely. Thesize of the secondary flows increases with the rela-tion of diameters and the inverse of the anastomosisdiameter. These secondary flows have a low rota-tional speed when the relation of diameters is large.Consequently, low speeds could promote aggregationof platelets when groups of activated platelets fall inthese zones. Figures 7 - 12 show the pathlines for thesmaller and the bigger relation of each anastomosisdiameter.

Figure 7: Pathlines Dgraft

Dvessel= 1 and D = 2,5mm

Figure 8: Pathlines Dgraft

Dvessel= 1.5 and D = 2,5mm

Figure 9: Pathlines Dgraft

Dvessel= 1 and D = 2mm

Figure 10: Pathlines Dgraft

Dvessel= 1.5 and D = 2mm

Figure 11: Pathlines Dgraft

Dvessel= 1 and D = 1,5mm

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Figure 12: Pathlines Dgraft

Dvessel= 1.5 and D = 1,5mm

Flow analysis

Particle trajectories were exported from the compu-tational calculation with values of velocity and resi-dence time. This information was organized to workparticle by particle and the shear stress is calculatedusing the equation 8. An example of trajectories isshown in figure 13 where all trajectories start at inletand almost everyone escape for outlet, some of themfall in secondary flow zones, and recirculate until athreshold of 10000 steps of trajectory calculation.

Figure 13: Example of streak lines

For each geometry LPA and maximum shearstress were calculated and the percentage of acti-vated platelets was found for each cycle. The resultsof activated platelets do not show a general depen-dency with relation of diameters. However, evaluat-ing each anastomosis diameter, a weak relation be-tween activated platelets and relation of diameters

was found. In anastomosis with a rate smaller thanDADV

= 0.66, the number of activated platelets by cy-cle increase with the relation of diameters, and whenthe anastomosis diameter decreases, this behavior ismore evident (Figure 14).

(a)

(b)

(c)

Figure 14: Graphic of activated platelets vs num-ber of cycles, anastomosis diameter a) DA

DV= 0.5,

b) DADV

= 0.66, c) DADV

= 0.86, legend at right sideindicates DG

DV

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Histograms were constructed for LPA and τmax

in each geometry and cycle (Figures 15 and 16). InDADV

= 0.86, there is no activation by τmax, just byLPA. In the other two anastomosis diameters theprincipal activator is shear stress.

The evolution in time of LPA in platelets circu-lating through the graft (Figure 15) has a particularbehavior. In general, the LPA values are higher athigh DG

DV, but the relation of the evolution and val-

ues with the anastomosis diameter does not have a

increasing or decreasing tendence. The LPA is theproduct of two variables, shear stress and residencetime, which are dependent on each other, becauseboth depend on velocity. If shear stress is high, thevelocity also has to be high, thus, the residence timeis going to be shorter. This perspective explains theevolution of LPA. In DA

DV= 0.86 the velocity does

not have an important increase at the anastomosisregion, thus, the residence time is longer and theshear stress values are higher than normal by geo-metric conditions.

(a) DADV

= 0.86, DGDV

= 1 (b) DADV

= 0.66, DGDV

= 1 (c) DADV

= 0.5, DGDV

= 1

(d) DADV

= 0.86, DGDV

= 1.5 (e) DADV

= 0.66, DGDV

= 1.5 (f) DADV

= 0.5, DGDV

= 1.5

Figure 15: Histograms LAP, legend at right side indicates number of cycles

In DADV

= 0.66, velocity at anastomosis is higherthan velocity in last geometry and the residence timeis shorter. On the other hand, the secondary flowzones are larger, so there are larger regions of highshear stress between main accelerated flow and sec-ondary flows.

In DADV

= 0.5, velocity at anastomosis are higherthan the other geometry velocities, and the residencetime shorter. In addition, the secondary flow zonesincrease their size again.

In general there are three important variablesasociated with LPA, two of which are directly asoci-ated: residence time and shear stress. The otheroneis from the observation of flow: secondary flow zonessize. The results suggest that the relationship of theLPA with the anastomosis diameter has a maximumof between DA

DV= 0.5 and DA

DV= 0.86, because in the

histograms the rate DADV

= 0.66 always has highervalues, where the three variables generate a highertotal.

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(a) DADV

= 0.86, DGDV

= 1 (b) DADV

= 0.66, DGDV

= 1 (c) DADV

= 0.5, DGDV

= 1

(d) DADV

= 0.86, DGDV

= 1.5 (e) DADV

= 0.66, DGDV

= 1.5 (f) DADV

= 0.5, DGDV

= 1.5

Figure 16: Histograms maximum shear stress, legend at right side indicates number of cycles

Histograms of shear stress have a relationshipwith the rate of diameters and the anastomosis di-ameter. If the reate of the diameters increases, thehistograms of shear stress register higher values andmore population in high values. This happens be-cause the size of secondary flow zones increase andthe acceleration of flow at graft outlet is higher (Fig-ure 16). The shear stress increases with the inverseof the anastomosis diameter because the velocity atanastomosis increases and these high velocity valuesare closer to the wall.

Discussion

LPA and shear stress were the two activating factorstaken into account, LPA has low values at relationsDADV

= 0.86 DADV

= 0.5, and shear stress has low val-ues at DA

DV= 0.86. In general, these two factors have

minimum values at small relation of diameters. Theanalysis suggests a relation of diameter DG

DV= 1 for

the size of the secondary flow zones and an anasto-mosis diameter closer to vessel diameter.

When a platelet is activated its morphologychanges to get more chances of aggregation withother platelets and adhesion to the vascular wall.

Moreover, collagen, the main graft material, is rec-ognized by platelets, for example in the presence ofinjury, and this gets them activated starting a co-agulation process. If enough platelets become ag-gregated, the coating starts and the possibility ofadhesion to the graft wall increases. Physiologicalfactors help flow variables to occlude the graft, butthe largest number of platelets and coagulation fac-tors are present 7 to 10 days after injury. This periodof time is important because the occlusion could oc-cur during those conditions.

Conclusion

In this simplified model, the analysis of flow vari-ables and the activation of platelets suggest an im-portant role of geometry, especially the anastomosisdiameter, in the phenomenon of occlusion in colla-gen vascular grafts. This analysis could be relatedto physiological coagulation factors and increase theprobability of occlusion. An improvement in implan-tation method is recommended, trying to reduce theinfluence of sutures, avoiding the reduction of diam-eter at the anastomosis. Another important momentof flow evaluation is the period between 7 to 10 days

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after surgery, when an inflammatory response hap-pens and the number of platelets increase to a max-imum. Possible future work can include a plateletaggregation model to make a more accurate predic-tion of graft occlusion.

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