ScattPort 17. Aug. 2009

**Erythrocyte shape**

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**Thomas Wriedt**

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To compute light scattering by the erythrocyte or red blood cell (RBC) various computer programs are available such as DDSCAT, ADDA, MMP3D and many FDTD programs [1]. To describe the shape of the erythrocyte the implicit equation given by Kuchel and Fackerell [2] is suitable:

(x^2+y^2+z^2)^2+P*(x^2+y^2)+Q*z^2+R=0

To generate a suitable triangulated shape model from this implicit equation different tools such as Maple, HyperFun and SurfX3D can be used. In the following we provide the input files for these tools such that you can generate you own RBC-shape using the tools. For the parameters of the equation we are using

P=-15.3; Q=42.0; R=-1.06;

given by Liu et al. [3] such that the dimensions of the shape files generated will be in micrometre. Maple and SurfX3D will generate AutoCAD .dxf files and HyperFun will generate Virtual Reality Modeling Language .wrl files. In the equation implemented in Maple the coordinates y and z have been interchanged such that the orientation of the erythrocyte is the same as with HyperFun and SurfX3D.

You may also use the online version of the Algebraic Surface Solver implemented in JavaView for visualisation of the erythrocyte shape and conversion into the wavefront .obj computer graphics format. You can also triangulated the shape generated. Enter this definition of the erythrocyte surface in the JavaView panel:

(x^2+y^2+z^2)^2+P*(x^2+y^2)+Q*z^2+R=0; P=-15.3; Q=42.0; R:=-1.06;

Once the surface has been created, drag the mouse in the display and hit the right mouse botton for the Control Panel. File save the shape to the .obj file format. You may use Mesh View to display the shape or for conversion into other computer graphics file formats.

The online Model Viewer will also give area and volume information on generated shape files. The extensions of the shape are D_{x}=D_{y}=7.77 µm, D_{z}=2.01. The surface area of the shape is 122.51 µm^{2} and the volume is 7.629 µm^{3.}

**References**

[1] T. Wriedt, J. Hellmers, E. Eremina, R. Schuh: Light scattering by single erythrocyte: Comparison of different methods. Journal of Quantitative Spectroscopy & Radiative Transfer, 100 (1-3) 444-456 (2006).

[2] Philip W. Kuchel, Edward D. Fackerell: Parametric-Equation Representation of Biconcave Erythrocytes. Bulletin of Mathematical Biology (1999) 61, 209–220.

[3] Changjun Liu, Dongwoo Sheen, and Kama Huang: A Hybrid Numerical Method to Compute Erythrocyte TMP in Low-Frequency Electric Fields. IEEE Transactions on Nanobioscience 2 (2003) 2, 104-109.