Numerical simulation of hemolysis: a comparison of Lagrangian and Eulerian modelling

Transcription

1 Modelling in Mediine and Biology VI 361 Numerial simulation of hemolysis: a omparison of Lagrangian and Eulerian modelling S. Pirker 1, H. Shima 2 & M. Stoiber 2 1 Johannes Kepler University, 4040 Linz, Austria 2 Medial University Vienna, 1090 Vienna, Austria Abstrat In rotary blood pumps the degree of hemolysis is of ruial importane. Therefore, it would be desirable to aount for flow indued hemolysis by means of numerial simulations in an early design stage. In this paper two numerial simulation methods for the simulation of hemolysis are proposed. Hereby, a Lagrangian approah based on traking distint erythroytes is ompared to an Eulerian approah in whih the erythroytes are treated as a ontinuous speies. Both models aount for the loal instantaneous stress as well as the stress exposure time. The basi funtionality and onsisteny of the above models is shown by simple test geometry. In a seond step the Eulerian method is applied for the simulation of unsteady hemolysis in a rotary blood pump. Keywords: CFD-simulation, rotary blood pump, hemolysis, Lagrangian modelling, Eulerian modelling, unsteady hemolysis. 1 Introdution In reent years numerial flow simulations (CFD) have proven to be a valuable tool in designing artifiial organs, i.e. rotary blood pumps (e.g. [1-3]. On one hand lassial design riteria like pumping effiieny, pressure head and hydrauli losses an be addressed. One the other side flow indued hemolysis an be investigated. In this paper two mathematial methods, namely the Lagrangian and the Eulerian approah, for the simulation of hemolysis are introdued and ompared to eah other. The prinipal funtionality of the models is shown by a simple WIT Transations on Biomediine and Health, Vol 8, 2005 WIT Press ISSN (on-line)

2 362 Modelling in Mediine and Biology VI ase study. In a seond step the simulation of hemolysis during unsteady operation modes is addressed. All modelling is realised within the framework of the ommerial CFD pakage Fluent [4]. The simulations are performed on Dell workstations and typial alulation times range from some minutes for the first ase study up to several days in the ase of the semi-axial pump. 2 Hemolysis Lagrangian modelling Reent researh work [5-13] agrees that hemolysis is at least influened by both instantaneous shearing stresses as well as exposure time. Thus, the history of stress loading of an erythroyte is said to influene the probability of ell damage. A ommon approah for the simulation of hemolysis is traing the paths of distint erythroytes in a Lagrangian frame of referene. In this Lagrangian model the path of an individual erythroyte is determined by the loal flow onditions. In this modelling onept any bakward influene is negleted as well as interations between erythroytes. In the numerial model every trajetory represents a whole thread of distint ells and is haraterised by a partile mass flow rate, m p, as skethed in Figure 1. pd p m + p Dp m γ t a) b) b ery m D(x).. b ery m. D( x) γ t + Figure 1: Integration of the blood damage potential in (a) Lagrangian and (b) Eulerian modelling. During traing the path of a partiular erythroyte the time-dependent stress loading an be monitored. In this model blood is assumed as a Newtonian fluid so that the instantaneous loal stress σ an by represented by the shearing rate γ and a onstant visosity µ, σ = µ γ. WIT Transations on Biomediine and Health, Vol 8, 2005 WIT Press ISSN (on-line)

3 Modelling in Mediine and Biology VI 363 In three-dimensional flows the shearing rate γ is the square root of the seond invariant of the rate of deformation tensor. If the stress exeeds a ertain minimum ativation level the model assumes that the ell will turn into an ativated state desribed by a damage potential D p. If the stress loading ontinues the damage potential is inreased by integrating the urrent strain rate over time. The simplest possibility to realise this integration on a omputational grid is to sum up the produt of the finite volume s strain rate, γ, with the partile s residene time in that very omputational ell, t. In Figure 1 the inoming partile of a distint partile thread is desribed by an initial damage Potential D - p. During integration the damage potential rises to D - p + σ t until the thread s partiles leave the very finite volume. If this damage potential D p reahes a ritial value the ell s membrane ruptures and the very erythroyte will be irreversibly damaged. In this proposed model the evaluation of global hemolysis is not arhived by ounting the damaged erythroytes in a huge set of introdued ones. Instead, the damaged ells are assumed to emit their hemoglobin into the blood s serum. In the mathematial model this is done by defining m p as an adequate soure term for the liquid phase s hemoglobin. Thereafter, the emitted hemoglobin is traed by an additional transport equation in an Eulerian frame of referene. As a onsequene the hemolysis in the omputational domain an be evaluated by simply looking at the alulated serum s free hemoglobin onentration at the domain's outlet. 3 Hemolysis Eulerian modelling If the distributed erythroytes neither interfere with eah others nor influene the main blood flow they an be treated as a passive ontinuous speies. In this ase an Eulerian equivalent of the above Lagrangian hemolysis model is possible. In this ontinuous model the flow rate of the erythroytes is given by the blood flow rate, m b, times the mean erythroyte onentration, ery. To aount for stress loading one again a damage potential is defined. In ase of the Eulerian model this damage potential is given by a salar field D(x). This damage potential field an alternatively be interpreted as a onentration field of ativated erythroytes. If in some finite volume the flow indued stress exeeds the ativation level the damage potential D(x) will be inreased. This transformation is ahieved by volumetri soure terms. If several stress related finite volumes are passed suessively the onentration of the ativated erythroytes will inrease aordingly. In a similar way to above Lagrangian integration the ativation state will be summed up until a rupture limit is reahed. WIT Transations on Biomediine and Health, Vol 8, 2005 WIT Press ISSN (on-line)

4 364 Modelling in Mediine and Biology VI p p γ t b In the above equation the integration step of the Lagrangian model is ompared to the Eulerian model. The left hand side desribes the total hange in the damage potential of all Lagrangian threads of erythroytes passing a omputational ell with a loal strain rate γ. In the right hand side the flux of the Eulerian damage potential field is inreased aordingly. If the ell membrane ruptures the ativated erythroyte speies emits hemoglobin into the blood s serum like in the Lagrangian modelling. So an estimation of the degree of hemolysis is given one again by the onentration of the serum s free hemoglobin at the outlet of the omputational domain. 4 Case study 1 simple stenosis onfiguration m = m In the simple stenosis situation of Figure 2 blood passes two onstritions. In Figure 2a the orresponding veloity field is given. Based on the flow field the strain rate and therewith the stress distribution in the liquid is obtained as given in Figure 2b. A distint erythroyte passing this geometry will experiene several distint stress maxima orresponding to the shearing rate aused by the two onstritions. A typial stress loading history is given in Figure 2. Based on above modelling onept most of the blood ells will be ativated in the wall near region shortly before the first onstrition. Some survive the first maximum stress but rupture shortly behind the seond one. So applying this model the event of irreversible damage, say ell membrane s rupture, do not oinide with the moment of maximum stress loading. An extreme single stress event an lead to blood damage as well as a series of medium stresses experiened during a long exposure time. In both modelling onepts, the estimation of hemolysis is based on an integration of stress and exposure time. In the Lagrangian approah this summation is done along erythroytes trajetories while in the Eulerian approah a ontinuous salar field is integrated. In Figure 3 the results of both hemolysis modelling approahes are given. Keeping in mind the unertainties of the physiologial mehanism of hemolysis both models deliver reasonable and onsistent results. Obviously the basi modelling onept of integrating instantaneous stress times exposure time an be realised in a Lagrangian frame of referene as well as in an Eulerian one. From a omputational point of view the Eulerian model has some obvious advantages over the Lagrangian. The Lagrangian model requires several thousands of erythroytes to be traed in order to reeiving reliable and repeatable results. Therefore, this approah turns out to be very CPU intensive in tehnial appliations. Another drawbak for the Lagrangian model are numerial unertainties in the integration of the distint trajetories. During alulation erythroytes might be trapped in boundary layers espeially in narrow gap flows and adulterate the results. ery γ t WIT Transations on Biomediine and Health, Vol 8, 2005 WIT Press ISSN (on-line)

5 Modelling in Mediine and Biology VI 365 a) b) 1,5 Strain Rate (1/s) 1,2 0,9 0,6 γ at 0,3 ) 0,0 5,00 0,00 0,03 0,06 0,09 0,12 Damage potential (.) 4,00 3,00 2,00 1,00 d) 0,00 0,00 0,03 0,06 0,09 0,12 Length (m) Figure 2: First test ase with (a) veloity magnitude field (v max = 0.3 m/s) and (b) strain rate distribution ( γ max = /s) as well as () strain rate history and (d) damage potential integration along a Lagrangian erythroyte path. WIT Transations on Biomediine and Health, Vol 8, 2005 WIT Press ISSN (on-line)

6 366 Modelling in Mediine and Biology VI a) b) Free hemoglobin onentration (.) a) b) Radius (m) Figure 3: Conentration of the serum s free hemoglobin at the domain s outlet as result of (a) Lagrangian and (b) Eulerian modelling. Another advantage of the Eulerian model is that unsteady simulations of hemolysis are possible without any modifiation of the mathematial model. An unsteady evaluation of hemolysis an easily be done by monitoring the timedependent onentration of the serum s free hemoglobin at the domain s outlet. Therefore, with help of this model the blood damage of a pulsing flow situation whih e.g. ours in implanted rotary blood pumps an be studied in detail. 5 Case study 2 rotary blood pump While the first ase study onentrates on a omparison between the Lagrangian and the Eulerian hemolysis model the seond ase study emphasises on the possibility of unsteady hemolysis simulation by means of the Eulerian model. The omputational domain of the semi-axial pump onsidered in this ase study starts at the input flange and ends at the pressure side ring volume of a semi-axial rotary blood pump. For sake of simpliity only one blade setion of the impeller is onsidered. Blood is treated as a Newtonian fluid of onstant properties (ρ= 1013 kg/m 3, µ = 3.4 kg/m.s). At the pump inlet the flow veloity is fixed while on the ring outlet a onstant stati pressure is assumed. WIT Transations on Biomediine and Health, Vol 8, 2005 WIT Press ISSN (on-line)

7 Modelling in Mediine and Biology VI 367 C Figure 4: Wall shear stress plotted on the rotating impeller showing ritial regions (C) at the blade s leading edges. Figure 5: Colour abrasion experiments and simulated oil traes on the impeller. The orientation of the abrasion and the oil traes at the blade s trailing edge are given by the bold lines. As a result of these simulations the time-dependent fully three-dimensional veloity and pressure distribution are obtained. Based on the flow situation the distribution of shearing stresses an be evaluated. In Figure 4 the wall shear stress is given on the impeller. The highest stresses an be observed at the leading edge of the impeller blades. Beause of the simplified omputational domain global simulation results like pressure head and volumetri flow rate annot diretly ompared to measurements. Therefore, a olour abrasion experiment was performed in order WIT Transations on Biomediine and Health, Vol 8, 2005 WIT Press ISSN (on-line)

8 368 Modelling in Mediine and Biology VI to depit loal flow phenomena. In Figure 5 the orientation of the olour abrasion pattern agree well with the simulation of a orresponding wall oil flow. The basi funtionality of the Eulerian hemolysis model is shown during two unsteady pump operation modes. First, a stepwise inrease in rotation speed is onsidered while seondly, the pump experienes a pulsing inflow. Revolutions per min. (1/min) e Hemoglobin Soure (kg/s) 2.5e-8 2.3e-8 2.1e-8 1.9e Time(s) Figure 6: Simulated hemoglobin prodution in semi-axial LVAD pump during stepwise inrease in rotational speed. In Figure 6 the serum s hemoglobin prodution, thus the amount of hemolysis is plotted versus time. During the stepwise inrease in the pump s rotation speed, the shear stress loading of the passing blood ells rises. As a onsequene the global hemolysis of the pump is inreased. An important onsideration is the behaviour of an implanted left ventrile assistant devie (LVAD) in pulsing flow onditions. If a positive pressure pulse is defined at the sution inlet of the pump, the overall pump s pressure head dereases. At onstant rotation speed the primary flow rate through the pump inreases while any seondary leakage bak flows alm down. From the viewpoint of hemolysis this operation mode is advantageous beause both stress loading and exposure time are redued. In Figure 7 the serum s hemoglobin prodution is plotted during an inoming positive pressure pulse. It learly depits that hemolysis is signifiantly redued in that operation mode. WIT Transations on Biomediine and Health, Vol 8, 2005 WIT Press ISSN (on-line)

9 Modelling in Mediine and Biology VI 369 Total Pressure at Inlet (Pa) e Hemoglobin Soure (kg/s) 2.0e-8 1.5e-8 1.0e-8 0.5e Time (s) Figure 7: Simulated hemolysis in semi-axial LVAD pump during an inflow pressure pulse. 6 Conlusion In this study two mathematial methods for the simulation of hemolysis are introdued. While in the Lagrangian model distint erythroytes are traed the Eulerian model treats the erythroytes as a ontinuous speies. Both models take into aount the instantaneous stress as well as the exposure time. In a first ase study the two models are ompared in a simple stenosis onfiguration. On prinipal both the Lagrangian and the Eulerian approah deliver omparable results. Nevertheless, the Eulerian model is by far more effetive by means of omputational osts. In a seond ase study unsteady hemolysis phenomena in a semi-axial rotary blood pump are studied with the help of the Eulerian model. Pulsatile inflow onditions are addressed as well as hanges in the pump s operation point. Up to this point two mathematial approahes have been ompared to eah other and their prinipal funtionality has been shown. They are by no means validated by real measurements what should be heked in near future. WIT Transations on Biomediine and Health, Vol 8, 2005 WIT Press ISSN (on-line)

10 370 Modelling in Mediine and Biology VI Referenes [1] Burgreen G.W. et al., Computational Fluid Dynamis as a Development Tool for Rotary Blood Pumps. Artifiial Organs, 25(5), pp , [2] Bertram C.D., Qian Y. and Reizes J.A., Computational Fluid Dynamis Performane Predition for the Hydrodynami Bearings of a VentrAssist Rotary Blood Pump. Artifiial Organs, 25(5), pp , [3] Chung M.K.H. et al., Impeller Behavior and Displaement of the VentrAssist Implantable Rotary Blood Pump. Artifiial Organs, 28(3), pp , [4] Fluent, Fluent 6.1 User Guide. Fluent In., Lebanon, USA, [5] Paul R. et al., Shear Stress Related Blood Damage in Laminar Couette Flow. Artifiial Organs, 27(6), pp , [6] Yeleswarapu K.K., A Mathematial Model for Shear-Indued Hemolysis. Artifiial Organs, 19(7), pp , [7] Bludszuweit C., Model for a General Mehanial Blood Damage Predition. Artifiial Organs, 19(7), pp , [8] Bludszuweit C., Three-Dimensional Numerial Predition of Stress Loading of Blood Partiles in a Centrifugal Pump. Artifiial Organs, 19(7), pp , [9] Arora D. et al., A Tensor Based Measure for Estimating Blood Damage. Artifiial Organs, 28(11), pp , [10] Apel J. et al., Assessment of Hemolysis Related Quantities in Miroaxial Blood Pumps by Computational Fluid Dynamis. Artifiial Organs, 25(5), pp , [11] Garon A. and Farina M-I., Fast Three-dimensional Numerial Hemolysis Approximation. Artifiial Organs, 28(11), pp , [12] Goubergrits L. and Affelt K., Numerial Estimation of Blood Damage in Artifiial Organs. Artifiial Organs, 28(11), pp , [13] Grigioni M. et al., The Power-law Mathematial Model for Blood Damage Predition. Artifiial Organs, 28(5), pp , WIT Transations on Biomediine and Health, Vol 8, 2005 WIT Press ISSN (on-line)