## Serviços Personalizados

## Journal

## Artigo

## Indicadores

## Links relacionados

- Citado por Google
- Similares em SciELO
- Similares em Google

## Compartilhar

## Journal of the Brazilian Society of Mechanical Sciences

##
*versão impressa* ISSN 0100-7386

### J. Braz. Soc. Mech. Sci. v.22 n.4 Rio de Janeiro 2000

#### http://dx.doi.org/10.1590/S0100-73862000000400006

**Is there Correlation between the Turbulent Eddies Size and Mechanical Hemolysis ?**

**Marcos Pinotti **Departamento de Engenharia Mecânica. Universidade Federal de Minas Gerais. UFMG

Av. Antonio Carlos, 6627

31270-901 Belo Horizonte. MG. Brazil

Hemolytic profile of an artificial device chronically implanted in the cardiovascular system may represent the difference between the success and failure in its long-term performance. Last decades have witnessed efforts on the development of methods capable of predicting red blood cell damage in artificial organs. However, all of them have had limited success to predict hemolysis. The primary cause of this problem is that such models do not take into consideration structures of turbulent flow. The present paper demonstrates that microscopic measurable occurrences of the turbulent flow may be linked to red blood cell trauma.

This study suggests that if the smallest turbulent eddies dimension is under 10 m m hemolysis is not dependent on the exposure time and the red blood cells damage depends only on the dissipation of the turbulent energy in the erythrocyte membrane.

The analysis reported here opens the possibility of mapping the flow field in artificial assist devices based on the smallest eddy length scales. This is a promising new trend and should be considered in the designing requirements of the next generations of artificial organs.

Introduction

Last decades have witnessed efforts on the development of methods capable of predicting red blood cells (RBC) damage in artificial organs. Action of viscous shear on the cell membrane is claimed as the main cause of mechanical hemolysis (Rand, 1964). However, it was reported that models based on cellular damage using cone-plate or Couette viscometers have had limited success to predict hemolysis in complex flows (Schima *et al*., 1993). The primary cause of this problem is that such models do not take into consideration the microscopic structures of the flow. It is clear that viscous shear stress and exposure time play an important part on hemolysis issue but is there any other occurrence in the flow which influences red blood cell trauma?

The aim of the present paper is demonstrating that microscopic measurable occurrences of the turbulent flow may be also linked to red blood cell trauma.

The Energy Cascade and Blood Cell Trauma

Most studies of hemolysis continue to support a hypothesis first enunciated by Rand (1964) that red blood cell damage occurs when biaxial membrane strain exceeds a threshold value where pores open or the membrane tears.

Kramer (1970), Baldwin *et al*. (1994) and Jones (1995) have suggested that, based on the knowledge of the turbulent flow phenomenology, the characterization of the red cell trauma due to turbulence should be described by three essential parameters, namely the size of the turbulent eddies, the viscous shear stresses and the time in which RBC remain in regions of high turbulence. Recently, based on literature background, Pinotti (1996) has devised a methodology to determine zones with great potential of hemolysis in a centrifugal blood pump (Bio-pump), using a special circuit and a laser Doppler anemometer to allow noninvasive measurements of velocity and Reynolds stresses fields.

The main concept which supports the link between RBC damage in turbulent flow and the size of the smallest eddies is the Energy Cascade (Kolmogorov, 1941). The turbulent flow is characterized by three-dimensional flow structures known as turbulent eddies. Such eddies have different length scales, ranging from the order of magnitude of the flow path width to some hundredths of millimeter. The larger eddies are continuously transferring their energy to the smaller eddies through inertial interaction. At the same time viscosity effects together with dissipation becomes more and more important for the smaller eddies. The number of eddies per unit volume is assumed to grow to ensure that small eddies are as space filling as the large ones. Whenever the smallest eddies are the size of a red blood cell, they will be dissipated through interaction with the cell and transfer their energy to its membrane. Such dissipation causes the membrane to rupture with the consequent release of hemoglobin into the plasma. On the other hand, if the smallest eddies are greater than the erythrocyte, the cell will be convected away and its membrane will experience only the viscous shear stress due to its relative motion with respect to the turbulent fluid (Kramer, 1970; Baldwin *et al*., 1994; Jones, 1995).

The turbulent energy is introduced in the flow through the greatest eddies and it is dissipated in the microscopic level through the smallest eddies. Blood cell trauma, in this context, occurs due to interaction of the smallest eddies on the cell membrane and hemolysis will not depend on the exposure time. Conversely, if the smallest eddies are larger than RBC, then the effective turbulent viscosity should exceed that of the cells interior and the shear stress must be sufficiently high and imposed for a sufficient duration to cause cell lysis. This denotes that hemolysis, in this situation, is dependent on the exposure time.

Based on this idea, the evaluation of the smallest eddies dimension present in a blood flow would give information about the way the flow structures influence hemolysis potential. Such analysis could not be achieved by using shear stress-based models alone. The distinct red blood cell characteristic to exhibit tank-treading motion in shear flows makes it difficult to model RBC as bubbles (Fischer, 1980; Fung, 1981; Keller and Skalak, 1982). Hence, the analysis presented here points to a new trend on hemolysis evaluation.

Materials and Methods

In order to demonstrate that it is possible to correlate the size of the turbulent eddies to the potential trauma to red blood cells it was employed data obtained by the author (Pinotti, 1996; Pinotti and Paone, 1996) and data extracted from the literature. Most of the data reported in the references was not organized in terms of turbulent eddy size, however such information was obtained by employing equations derived from turbulence flow governing equations.

The dimension of the smallest eddies (Ls) in a turbulent flow (also called Kolmogorov length scale) may be related to the local large-scale quantities by the following relation (Panton, 1984).

Where Le is the characteristic length of the flow (for example, the tube diameter in tube flow) [m], n is the fluid kinetic viscosity [m^{2} s^{-1}] and u_{0} is the local RMS velocity [m s^{-1}].

Alternatively, it may be more convenient calculate Ls as a function of the Normal Reynolds stress (s_{n}):

where

Once u_{0} may be related to the Reynolds normal stress, Eq.(1) and Eq.(2) are equivalent. Equation (1) is useful in evaluating the eddies length scales obtained by laser Doppler anemometer (LDA) or Pulsed Doppler Ultrasound (PDU), since the RMS velocity values may be taken directly from the instrument processors (Pinotti, 1996; Hasenkam *et al*., 1988). Equation (2) were employed to calculate Ls in situations which RMS velocity values were not reported.

Results and Discussion

The correlation between the length scale of the smallest eddies and hemolytic potential may be checked by plotting the large-scale dimension in terms of the Normal Reynolds stress for two distinct situations described in the literature: the intentionally induced RBC damage to study the erythrocyte membrane strength (Forstom, 1969; Blackshear *et al*., 1966; Sallam and Hwang, 1984; Sutera and Mehrjardi, 1975) and the hopefully avoided hemolysis in investigations about the flow field inside artificial heart valves and ventricles (Baldwin *et al*., 1994; Pinotti, 1996; Jin and Clark, 1993; Pinotti and Paone, 1996). The coordinating points displayed in the Fig.1 were obtained from the literature. Two distinct regions were possible to be identified by employing the smallest eddies length scale criterion. The curve which separates the two regions was obtained by substituting Ls = 10 mm, r = 1000 kg m^{-3} and n = 4 x 10^{-6} m^{2} s^{-1} in Eq.(2) and then by performing a parametric plot by varying Le and s_{n}. The value of 10 mm was used because it represents the approximate dimension of a stretched red blood cell.

Table 1 completes the information provided in Fig.1 by listing the calculated smallest eddies length scales (Ls) for each point displayed in the figure. It is worth pointing out that, although each condition displayed in Fig.1 and Table 1 carries its own hemolytic potential, the situations in which the calculated Ls is less than 10 mm do appear in the exposure time-independent hemolytic region of Fig.1. On the other hand, points with Ls greater than 10 mm appear in the exposure time-dependent hemolytic region.

Figure 1 makes it possible to identify the two hemolytic pathways in function of the blood passage size (large-scale dimension, Le) and Normal Reynolds Stress. Such information is useful, in the practical point of view, because hemolytic profile of an artificial organ or implantable device may be assessed by the means available to the device designer.

The model, as proposed here, has two major limitations. It does not take into account the intermittence phenomena of turbulent flow neither consider the fact of the measurement axes of Normal Stresses to be coincident or not to the principal axes.

The intermittence phenomena of the turbulent flow will affect the characteristic of the turbulent eddies to be space filling (Frisch, 1995). Considering that RBC trauma will occur in the outer area of the turbulent eddies and according to the Prandtl's turbulence hypothesis, the number of affected cells is proportional to 1/Le. Hence, the effect of the smallest eddies (smaller than 10 mm) in the flow will not be 100% lethal, but proportional to the ratio between eddies volume and the volume of the region where the dissipation is occurring.

The use of two dimensional principal stress analysis with two-component velocity data obtained from measurements misaligned with the plane of maximum mean flow shear (Pinotti and Rosa, 1995) can underpredict maximum normal stress by as 15% (Fontaine et al., 1996). Therefore, the dimension of the dissipation eddies may be, eventually, even smaller than those reported in Table 1.

Conclusions

Designing procedures of artificial organs, such artificial ventricle or prosthetic heart valves should consider the approach based on turbulent scale analysis in order to check potential regions of hemolysis. The value of such analysis is to provide the link between easily obtained data of the flow structure and the way blood cells are damaged due to turbulence.

The methodology described in the present paper suggests that, depending on the level of the blood flow turbulence, hemolysis is not a function of exposure time anymore and eddies dissipation in the cells membrane is the dominant phenomenon of red blood cells trauma. Therefore, as it may be concluded from this study, there is correlation between the turbulent eddy size and mechanical hemolysis.

Additional work is required to understand the exact interaction between the smallest scale eddies and the RBC damage, as well as the relationship between the measurable flow occurrences (velocity RMS fluctuations) and the percentage of hemoglobin release.

Acknowledgements

CNPq (Proc. n. 300556/97-7) supported the present work.

References

Baldwin, JT, Deutsch, S, Geselowitz, DB, Tarbell, JM., 1994, "LDA Measurements of Mean Velocity and Reynolds Stress Fields within an Artificial Heart Ventricle". TRANS ASME - J. Biomechanical Engineering, Vol.116, pp.190-200. [ Links ]

Blackshear, PL, Dorman, FD, Steinbach, JH, Maybach, EJ, Singh, A, Collingham, RE., 1966, "Shear Wall Interaction and Hemolysis", TRANS. ASAIO,Vol.l12, pp.113-120. [ Links ]

Fischer, T.M., 1980, "On the energy dissipation in a tank-treading human red blood cell". Biophysics Journal, Vol.32, pp. 863-868. [ Links ]

Fontaine, A A, Ellis, J T, Healy, T M, Hopmeyer, J, Yoganathan, A P., 1996, "Identification of peak stresses in cardiac prostheses. A comparison of two-dimensional versus three-dimensional principal stress analysis". ASAIO J., Vol.42, n.3, pp.154-163. [ Links ]

Forstom, RJ., 1969, "A New Measure of Erythrocyte Membrane Strength: The Jet Fragility Test", Ph.D. Thesis, University of Minnesota, Minneapolis. [ Links ]

Frisch, U., 1995, "Turbulence". Cambridge University Press, Cambridge, UK. [ Links ]

Fung, Y.C., 1981, "Biomechanics - Mechanical Properties of Living Tissues". Springer-Verlag, New York. [ Links ]

Hasenkam, JM, Nygaard, H, Giersiepen, M, Reul, H, Stödkilde-Jorgensen, H, 1988, "Turbulent Stress Measurements Downstream of Six Mechanical Aortic Valves in a Pulsatile Flow Model", Journal of Biomechanics, Vol.21, pp.631-645. [ Links ]

Jin, W, Clark, C., 1993, "Experimental Investigation of Unsteady Flow Behaviour within a Sac-Type Ventricular Assist Device (vad)", Journal of Biomechanics, Vol.26, pp.697-707. [ Links ]

Jones, AS., 1995, "A Relationship Between Reynolds Stresses and Viscous Dissipation: Implications to Red Cell Damage", Annals of Biomedical Engineering, Vol.23, pp.21-28. [ Links ]

Keller, S.R., Skalak, R., 1982, "Motion of a tank-treading ellipsoidal particle in a shear flow", Journal of Fluid Mechanics, Vol.120, pp. 27-47. [ Links ]

Kolmogorov, AN., 1941, "The local structure of turbulence in incompressible viscous fluid for very large Reynolds number", Dokl. Akad. Nauk. SSSR, Vol.30, pp.301-305. [ Links ]

Kramer, C., 1970, "Studies on Flow Induced Mechanical Haemolysis". Advisory Group for Aerospace Research and Development Conference Proceedings on Fluid Dynamics of Blood Circulation and Respiratory Flow - North Atlantic Treaty Organisation, Vol.65, Naples. [ Links ]

Panton, RL., 1984, "Incompressible Flow", John Wiley and Sons, New York. [ Links ]

Pinotti, M., 1996, "Escoamento no Interior de um Dispositivo Centrífugo Utilizado em Circulação Extracorpórea", Ph.D. Thesis, Universidade Estadual de Campinas, Campinas-SP. [ Links ]

Pinotti, M, Paone, N., 1996, "Estimating Mechanical Blood Trauma in Centrifugal Blood Pump: LDA Measurements of the Mean Velocity Field", Artificial Organs, Vol.20,pp.546-552. [ Links ]

Pinotti, M., Rosa, E.S., 1995, "Computational prediction of hemolysis in a centrifugal ventricular assist device". Artificial Organs, v.19, n.3, p.267-273. [ Links ]

Rand, RP., 1964, "Mechanical Properties of the Red Cell Membrane. II. Viscoelastic Breakdown of the Membrane", Biophysical Journal, Vol.4, pp.303-316. [ Links ]

Sallam, AM, Hwang, NHC., 1984, "Human Red Blood Cell Hemolysis in a Turbulent Shear Flow: Contribution of Reynolds Shear Stresses", Biorheology, Vol.21, pp.783-797. [ Links ]

Schima, H., Muller, MR, Tsangaris S, Gheiseder, G, Schlusche C, Losert, U, Thoma, H, Wolner, E., 1993, "Mechanical blood traumatization by tubing and throttles in In Vitro pump tests: Experimental results and implications for hemolysis theory", Artificial Organs, Vol.17, n.3, pp.164-170. [ Links ]

Sutera, SP, Mehrjardi, MH., 1975, "Deformation and Fragmentation of Human Red Blood cells in turbulent shear flow", Biophysical Journal, Vol.15, pp.1-10. [ Links ]

Manuscript received: April 1998. Technical Editor: Angela Ouívio Nieckele.