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Some new model geometries for sickled erythrocytes. (English) Zbl 1177.53012

Six types of shapes of red blood cells observed in micrographs of blood smears of patients with sickle cell desease, are described in mathematical terms. Three of them, the deltoid, astroid, and hypocycloid are given in parametric form as \(x(t) = a(n\cos(t) +\cos(n\,t)),\, y(t) = a(n\sin(t) - \sin(n\,t))\) for \(n = 2, 3, 4\) respectively, where \(a\) corresponds to the actual size. Also, the witch of Agnesi \(y = a^3/(a^2 + x^2)\) can be given as a closed expression while two others, a five pointed star and a semi-circular are described in other terms. Expressions for the area of the regions surrounded by these curves are provided. This description can be used to classify and to identify types of cells, and to help in diagnosis and treatment regimes.

MSC:

53A04 Curves in Euclidean and related spaces
92C37 Cell biology
97G99 Geometry education
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