# Modeling Anisotropic Electrical Conductivity of Blood: Translating Microscale Effects of Red Blood Cell Motion into a Macroscale Property of Blood

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- The RBCs are oriented such that the intermediate and major axes are in a plane of maximum shear stress, which we shall call a ‘shear plane’ in the following. There exist two perpendicular shear planes because the viscous stress tensor $\mathit{\tau}$ is symmetric. The intermediate and major axes of the RBCs are found to lie in the shear plane, which mostly contains the flow direction. The major axis of ellipsoidal tank-treading RBCs is found to be parallel to the flow direction;
- The intermediate axis is parallel to the vorticity vector of the flow. Also, in this case, the major axis of the RBCs is parallel to the flow direction. Figure 3 shows an idealized schematic of a tank-treading RBC in shear flows.

#### 2.1. Modeling RBCs Motion

**The eigenvector model**: The direction of the symmetry axis is determined by the normal vector of the shear plane, which contains or mostly contains the velocity vector. The normal vector can be computed using the eigenvectors of the viscous stress tensor;**The velocity–vorticity model**: the direction of the symmetry axis is determined by computing the cross product of vorticity and velocity, i.e., the so-called Lamb vector [28].

#### 2.1.1. Eigenvector Model

#### 2.1.2. Velocity–Vorticity Model

#### 2.2. Definition of the Conductivity Tensor

#### 2.3. Calculation of the Average Conductivities

Description | Symbol | Value | Units | References |
---|---|---|---|---|

Particle aspect ratio | $\lambda $ | $0.38$ | [-] | [7,10,20,30] |

Conductivity of the blood plasma | ${\sigma}_{\mathrm{pl}}$ | $1.3$ | $\mathrm{S}/\mathrm{m}$^{−1} | [31] |

Volume fraction of RBCs in the blood | H | 45 | % | [31] |

Short particle semiaxis | a | $1.52\times {10}^{-6}$ | $\mathrm{m}$ | [31] |

Long particle semiaxis | b | $4\times {10}^{-6}$ | $\mathrm{m}$ | [31] |

Membrane shear modulus | $\mu $ | ${10}^{-5}$ | $\mathrm{k}\mathrm{g}/{\mathrm{s}}^{2}$ | [7,32] |

Orientation/Disorientation constant | k | 1 | ${\mathrm{s}}^{-1/2}$ | [31,33] |

Dynamic viscosity of the blood plasma | ${\eta}_{\mathrm{pl}}$ | $4.8\times {10}^{-2}$ |
kg m^{−1}s^{−1} | [7,10,34] |

Blood density | ${\rho}_{\mathrm{bl}}$ | 1060 | $\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ | [35,36] |

^{−1}, as high beyond $16$ s

^{−1}, and as intermediate in-between.

#### 2.4. Computational Fluid Dynamics and Rheological Modeling

#### 2.5. Numerics

## 3. Results

^{−1}and the vessel diameter D is $0.04$ $\mathrm{m}$. With a hematocrit H of $45\%$, the kinematic viscosity of blood ${\nu}_{\mathrm{bl}}$, according to Equations (21) and (22), has a value of $4.59\times {10}^{-5}$ $\mathrm{m}$

^{2}/$\mathrm{s}$. The Reynolds number $Re$ is computed as follows:

#### Model Comparison

^{−1}, and the diameter D is $0.024$ $\mathrm{m}$. With a hematocrit H of $45\%$, the kinematic viscosity of blood ${\nu}_{\mathrm{bl}}$, according to Equations (21) and (22) is again $4.59\times {10}^{-5}$ $\mathrm{m}$

^{2}/$\mathrm{s}$. The Reynolds number $Re$ is computed as follows:

^{−1}and shear rate $\dot{\gamma}$ (b) in s

^{−1}. The velocity magnitude is zero at the wall due to the no-slip boundary condition at the wall, and it increases towards the center line of the aorta. The values of the shear rate $\dot{\gamma}$ close to the wall are higher than they are farther away from the wall. Also, in the vicinity of the inlet and the outlet of the aorta, the streamlines are analogous to a Poiseuille flow. However, in the arch itself, we observe that streamlines can move from the center towards the wall and vice versa due to the curvature of the geometry.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

1D | One-dimensional |

2D | Two-dimensional |

3D | Three-dimensional |

CFD | Computational fluid dynamics |

CVD | Cardiovascular disease |

CT | Computed tomography |

ICG | Impedance cardiography |

IPG | Impedance plethysmography |

MRI | Magnetic resonance imaging |

RBC | Red blood cell |

## References

- Semelka, R.C.; Armao, D.M.; Elias Junior, J.; Huda, W. Imaging strategies to reduce the risk of radiation in CT studies, including selective substitution with MRI. J. Magn. Reson. Imaging
**2007**, 25, 900–909. [Google Scholar] [CrossRef] [PubMed] - Kanal, E.; Barkovich, A.J.; Bell, C.; Borgstede, J.P.; Bradley, W.G.; Froelich, J.W.; Gilk, T.; Gimbel, J.R.; Gosbee, J.; Kuhni-Kaminski, E.; et al. ACR Guidance Document for Safe MR Practices: 2007. Am. J. Roentgenol.
**2007**, 188, 1447–1474. [Google Scholar] [CrossRef] - Badeli, V.; Ranftl, S.; Melito, G.M.; Reinbacher-Köstinger, A.; von der Linden, W.; Ellermann, K.; Biro, O. Bayesian inference of multi-sensors impedance cardiography for detection of aortic dissection. COMPEL—Int. J. Comput. Math. Electr. Electron. Eng.
**2021**, 41, 824–839. [Google Scholar] [CrossRef] - Badeli, V.; Melito, G.M.; Reinbacher-Köstinger, A.; Bíró, O.; Ellermann, K. Electrode positioning to investigate the changes of the thoracic bioimpedance caused by aortic dissection—A simulation study. J. Electr. Bioimpedance
**2020**, 11, 38–48. [Google Scholar] [CrossRef] [PubMed] - Wiegerinck, A.I.P.; Thomsen, A.; Hisdal, J.; Kalvøy, H.; Tronstad, C. Electrical impedance plethysmography versus tonometry to measure the pulse wave velocity in peripheral arteries in young healthy volunteers: A pilot study. J. Electr. Bioimpedance
**2021**, 12, 169–177. [Google Scholar] [CrossRef] - Reinbacher-Köstinger, A.; Badeli, V.; Bíró, O.; Magele, C. Numerical Simulation of Conductivity Changes in the Human Thorax Caused by Aortic Dissection. IEEE Trans. Magn.
**2019**, 55, 1–4. [Google Scholar] [CrossRef] - Hoetink, A.; Faes, T.; Visser, K.; Heethaar, R. On the Flow Dependency of the Electrical Conductivity of Blood. IEEE Trans. Biomed. Eng.
**2004**, 51, 1251–1261. [Google Scholar] [CrossRef] - Jaspard, F.; Nadi, M.; Rouane, A. Dielectric properties of blood: An investigation of haematocrit dependence. Physiol. Meas.
**2003**, 24, 137. [Google Scholar] [CrossRef] - Edgerton, R.H. Conductivity of Sheared Suspensions of Ellipsoidal Particles with Application to Blood Flow. IEEE Trans. Biomed. Eng.
**1974**, BME-21, 33–43. [Google Scholar] [CrossRef] - Gaw, R.L.; Cornish, B.H.; Thomas, B.J. The Electrical Impedance of Pulsatile Blood Flowing Through Rigid Tubes: A Theoretical Investigation. IEEE Trans. Biomed. Eng.
**2008**, 55, 721–727. [Google Scholar] [CrossRef] - Maxwell, J.C. A Treatise on Electricity and Magnetism; Clarendon Press: Oxford, UK, 1873. [Google Scholar]
- Rayleigh, L. LVI. On the influence of obstacles arranged in rectangular order upon the properties of a medium. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1892**, 34, 481–502. [Google Scholar] [CrossRef] - Fricke, H. A Mathematical Treatment of the Electric Conductivity and Capacity of Disperse Systems I. The Electric Conductivity of a Suspension of Homogeneous Spheroids. Phys. Rev.
**1924**, 24, 575–587. [Google Scholar] [CrossRef] - Visser, K.R. Electric properties of flowing blood and impedance cardiography. Ann. Biomed. Eng.
**1989**, 17, 463–473. [Google Scholar] [CrossRef] - Ulbrich, M.; Mühlsteff, J.; Leonhardt, S.; Walter, M. Influence of physiological sources on the impedance cardiogram analyzed using 4D FEM simulations. Physiol. Meas.
**2014**, 35, 1451. [Google Scholar] [CrossRef] - Voss, F.; Korna, L.; Leonhardtb, S.; Walterb, M. Modeling of flow-dependent blood conductivity for cardiac bioimpedance. Int. J. Bioelectromagn.
**2021**, 23, 21. [Google Scholar] - Jafarinia, A.; Badeli, V.; Melito, G.M.; Müller, T.S.; Reinbacher-Köstinger, A.; Hochrainer, T.; Biro, O.; Ellermann, K.; Brenn, G. False lumen thrombosis in aortic dissection and its impact on blood conductivity variations—An application for impedance cardiography. In Proceedings of the Book of abstract, Young Investigators Conference, Valencia, Spain, 7–9 July 2021; p. 297. [Google Scholar]
- Badeli, V. Modelling and Simulation of Aortic Dissection by Impedance Cardiography. Ph.D. Thesis, Graz University of Technology, Graz, Austria, 2021. [Google Scholar]
- Badeli, V.; Jafarinia, A.; Melito, G.M.; Müller, T.S.; Reinbacher-Köstinger, A.; Hochrainer, T.; Brenn, G.; Ellermann, K.; Biro, O.; Kaltenbacher, M. Monitoring of false lumen thrombosis in type B aortic dissection by impedance cardiography – A multiphysics simulation study. Int. J. Numer. Methods Biomed. Eng.
**2023**, 39, e3669. [Google Scholar] [CrossRef] - Melito, G.M.; Müller, T.S.; Badeli, V.; Ellermann, K.; Brenn, G.; Reinbacher-Köstinger, A. Sensitivity analysis study on the effect of the fluid mechanics assumptions for the computation of electrical conductivity of flowing human blood. Reliab. Eng. Syst. Saf.
**2021**, 213, 107663. [Google Scholar] [CrossRef] - Fischer, T.M.; Stöhr-Liesen, M.; Schmid-Schönbein, H. The Red Cell as a Fluid Droplet: Tank Tread-Like Motion of the Human Erythrocyte Membrane in Shear Flow. Science
**1978**, 202, 894–896. [Google Scholar] [CrossRef] [PubMed] - Goldsmith, H.L.; Marlow, J.; MacIntosh, F.C. Flow behaviour of erythrocytes—I. Rotation and deformation in dilute suspensions. Proc. R. Soc. Lond. Ser. B. Biol. Sci.
**1972**, 182, 351–384. [Google Scholar] [CrossRef] - Keller, S.R.; Skalak, R. Motion of a tank-treading ellipsoidal particle in a shear flow. J. Fluid Mech.
**1982**, 120, 27–47. [Google Scholar] [CrossRef] - Bitbol, M. Red blood cell orientation in orbit C = 0. Biophys. J.
**1986**, 49, 1055–1068. [Google Scholar] [CrossRef] - Schmid-Schönbein, H.; Wells, R. Fluid Drop-Like Transition of Erythrocytes under Shear. Science
**1969**, 165, 288–291. [Google Scholar] [CrossRef] - Abkarian, M.; Viallat, A. Vesicles and red blood cells in shear flow. Soft Matter
**2008**, 4, 653–657. [Google Scholar] [CrossRef] - Minetti, C.; Audemar, V.; Podgorski, T.; Coupier, G. Dynamics of a large population of red blood cells under shear flow. J. Fluid Mech.
**2019**, 864, 408–448. [Google Scholar] [CrossRef] - Lamb, H. Hydrodynamics; Cambridge University Press: Cambridge, UK, 1932. [Google Scholar]
- Millikan, R.A.; Bishop, E.S. Elements of Electricity: A Practical Discussion of the Fundamental Laws and Phenomena of Electricity and Their Practical Applications in the Business and Industrial World; American Technical Society: Sacramento, CA, USA, 1917. [Google Scholar]
- Goldsmith, H.L. Flow-induced interactions in the circulation. In Advances in the Flow and Rheology of Non-Newtonian Fluids; Rheology Series; Elsevier: Amsterdam, The Netherlands, 1999; Volume 8, pp. 1–62. [Google Scholar] [CrossRef]
- Gaw, R.L. The Effect of Red Blood Cell Orientation on the Electrical Impedance of Pulsatile Blood with Implications for Impedance Cardiography. Ph.D. Thesis, Queensland University of Technology, Brisbane City, Australia, 2010. [Google Scholar]
- Evans, E. New Membrane Concept Applied to the Analysis of Fluid Shear- and Micropipette-Deformed Red Blood Cells. Biophys. J.
**1973**, 13, 941–954. [Google Scholar] [CrossRef] - Bitbol, M.; Quemada, D. Measurement of erythrocyte orientation in flow by spin labeling. Biorheology
**1985**, 22, 31–42. [Google Scholar] [CrossRef] [PubMed] - Merrill, E.W. Rheology of blood. Physiol. Rev.
**1969**, 49, 863–888. [Google Scholar] [CrossRef] [PubMed] - Hinghofer-Szalkay, H. Method of high-precision microsample blood and plasma mass densitometry. J. Appl. Physiol.
**1986**, 60, 1082–1088. [Google Scholar] [CrossRef] [PubMed] - Jafarinia, A.; Müller, T.S.; Windberger, U.; Brenn, G.; Hochrainer, T. A Study on Thrombus Formation in Case of Type B Aortic Dissection and Its Hematocrit Dependence. In Proceedings of the 6th World Congress on Electrical Engineering and Computer Systems and Sciences (EECSS’20), Prague, Czech Republic, 13–15 August 2020. [Google Scholar] [CrossRef]
- Carreau, P.J. Rheological Equations from Molecular Network Theories. Trans. Soc. Rheol.
**1972**, 16, 99–127. [Google Scholar] [CrossRef] - Beris, A.N.; Horner, J.S.; Jariwala, S.; Armstrong, M.J.; Wagner, N.J. Recent advances in blood rheology: A review. Soft Matter
**2021**, 17, 10591–10613. [Google Scholar] [CrossRef] - Weller, H.G.; Tabor, G.; Jasak, H.; Fureby, C. A Tensorial Approach to Computational Continuum Mechanics Using Object-Oriented Techniques. Comput. Phys.
**1998**, 12, 620–631. [Google Scholar] [CrossRef] - Brown, B. Electrical impedance tomography (EIT): A review. J. Med. Eng. Tech.
**2003**, 27, 97–108. [Google Scholar] [CrossRef] - Lee, C.A.; Paeng, D.G. Effect of particle collisions and aggregation on red blood cell passage through a bifurcation. Microvasc. Res.
**2009**, 78, 301–313. [Google Scholar] [CrossRef] - Fontes, A.; Fernandes, H.P.; Barjas-Castro, M.L.; Thomaz, A.A.d.; Pozzo, L.d.Y.; Barbosa, L.C.; Cesar, C.L. Red blood cell membrane viscoelasticity, agglutination, and zeta potential measurements with double optical tweezers. In Proceedings of the Imaging, Manipulation, and Analysis of Biomolecules, Cells, and Tissues IV, San Jose, CA, USA, 21–26 January 2006; Volume 6088, pp. 296–305. [Google Scholar] [CrossRef]
- Lee, C.A.; Paeng, D.G. Numerical simulation of spatiotemporal red blood cell aggregation under sinusoidal pulsatile flow. Sci. Rep.
**2021**, 11, 9977. [Google Scholar] [CrossRef] - Giannokostas, K.; Dimakopoulos, Y. TEVP model predictions of the pulsatile blood flow in 3D aneurysmal geometries. J. Non-Newton. Fluid Mech.
**2023**, 311, 104969. [Google Scholar] [CrossRef] - Giannokostas, K.; Moschopoulos, P.; Varchanis, S.; Dimakopoulos, Y.; Tsamopoulos, J. Advanced constitutive modeling of the thixotropic elasto-visco-plastic behavior of blood: Description of the model and rheological predictions. Materials
**2020**, 13, 4184. [Google Scholar] [CrossRef] [PubMed] - Wtorek, J.; Polinski, A. The contribution of blood-flow-induced conductivity changes to measured impedance. IEEE Trans. Biomed. Eng.
**2005**, 52, 41–49. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**An idealized schematic of how RBCs align in the flow direction, creating channel-like paths near the vessel wall. The green arrows indicate the passage of the electrical current.

**Figure 2.**A triaxial ellipsoidal particle (

**a**) characterized by a short ($2a$), an intermediate ($2c$), and a long ($2b$) axis and an oblate spheroid (

**b**) with a short ($2a$) and two equal long axes ($2b$). See Table 1 for the values of a and b.

**Figure 3.**An idealized schematic of a tank-treading ellipsoidal RBC near a vessel wall with a high shear rate. The RBC is assumed to be an ellipsoidal particle with two equal long axes with the length $2b$ and one short axis of length $2a$. The shear plane, shown in gray, is the plane of maximum shear stress containing the velocity vector $\mathit{u}$. The vorticity vector $\mathbf{\omega}$, is shown in a blue vector. The curved blue arrow indicates the cavity flow of the cytoplasm. The green arrows on the RBCs membrane indicate the local membrane speed due to the tank-treading motion.

**Figure 4.**3D representation of the unit basis vectors. The unit basis vectors ${\mathit{e}}_{1}$, ${\mathit{e}}_{2}$, ${\mathit{e}}_{3}$ are the eigenvectors of the viscous stress tensor $\mathit{\tau}$ and correspond to maximum, intermediate, and minimum eigenvalues, respectively. The plane in gray is the shear plane. The unit basis vector ${\mathit{e}}_{\alpha}$ is normal to the shear plane. The angle $\theta $ is the angle between velocity vector $\mathit{u}$ and $\frac{1}{\sqrt{2}}({\mathit{e}}_{1}+{\mathit{e}}_{3})$.

**Figure 5.**The fraction of aligned RBCs, computed by $f\left({\dot{\gamma}}_{\mathrm{max}}\right)$, versus the maximum shear rate ${\dot{\gamma}}_{\mathrm{max}}$.

**Figure 6.**Color contour of (

**a**) velocity magnitude $\left|\mathit{u}\right|\times {10}^{2}$ in $\mathrm{m}$ s

^{−1}, and (

**b**) shear rate $\dot{\gamma}$ in s

^{−1}.

**Figure 7.**Color contours of (

**a**) ${\sigma}_{\alpha}$ in $\mathrm{S}$ m

^{−1}, (

**b**) ${\sigma}_{\beta}$ in $\mathrm{S}$ m

^{−1}, and (

**c**) anisotropic indicator $\eta $.

**Figure 8.**Color contours of (

**a**) velocity magnitude $\left|\mathit{u}\right|\times {10}^{3}$ in $\mathrm{m}$ s

^{−1}and (

**b**) shear rate $\dot{\gamma}$ in s

^{−1}. The color contours are illustrated in the cross-sections (A–D) and on the streamlines.

**Figure 9.**Color contours of (

**a**) ${\psi}_{\alpha}$ in degrees and (

**b**) ${\psi}_{\sigma}$ in degrees. The color contours are illustrated in the cross-sections (A–D) and in the volume of the aorta. ${\psi}_{\alpha}$ is the angle between ${\mathit{e}}_{\alpha}^{\mathrm{EV}}$ and ${\mathit{e}}_{\alpha}^{\mathrm{VV}}$ vectors resulting from the two models. ${\psi}_{\sigma}$ is the angle between the conductivity tensors ${\mathit{\sigma}}^{\mathrm{EV}}$ and ${\mathit{\sigma}}^{\mathrm{VV}}$ resulting from the two models.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jafarinia, A.; Badeli, V.; Krispel, T.; Melito, G.M.; Brenn, G.; Reinbacher-Köstinger, A.; Kaltenbacher, M.; Hochrainer, T.
Modeling Anisotropic Electrical Conductivity of Blood: Translating Microscale Effects of Red Blood Cell Motion into a Macroscale Property of Blood. *Bioengineering* **2024**, *11*, 147.
https://doi.org/10.3390/bioengineering11020147

**AMA Style**

Jafarinia A, Badeli V, Krispel T, Melito GM, Brenn G, Reinbacher-Köstinger A, Kaltenbacher M, Hochrainer T.
Modeling Anisotropic Electrical Conductivity of Blood: Translating Microscale Effects of Red Blood Cell Motion into a Macroscale Property of Blood. *Bioengineering*. 2024; 11(2):147.
https://doi.org/10.3390/bioengineering11020147

**Chicago/Turabian Style**

Jafarinia, Alireza, Vahid Badeli, Thomas Krispel, Gian Marco Melito, Günter Brenn, Alice Reinbacher-Köstinger, Manfred Kaltenbacher, and Thomas Hochrainer.
2024. "Modeling Anisotropic Electrical Conductivity of Blood: Translating Microscale Effects of Red Blood Cell Motion into a Macroscale Property of Blood" *Bioengineering* 11, no. 2: 147.
https://doi.org/10.3390/bioengineering11020147