Last modified: 2014-03-28

#### Abstract

The oxygen flow in humans and other higher animals depends on the erythrocyte-to-blood volume ratio, the hematocrit. It is physiologically favourable when oxygen transport rate is maximum. If the hematocrit were too low, too few erythrocytes could transport oxygen. If it were too high, the blood would be very viscous, so that oxygen flow would again be reduced. We here calculate that hematocrit value that is optimal for transporting a maximum amount of oxygen per time. This can be done by using the Hagen-Poiseuille law and considering the dependence of blood viscosity on the hematocrit. Different empirical or theoretically derived formulas for the dependence of viscosity on the concentration in a suspension have been proposed in the literature. We check which formulas lead to the best agreement between the theoretical and observed values. We show that especially a formula proposed by Svante Arrhenius (1917) is very appropriate for this purpose, leading to an optimal value of 40 %. This conforms very well to the observed values in humans and many other species.

The results are valid and useful in spite of considerable simplifications such as considering blood as a Newtonian fluid and neglecting the deformation, orientation and aggregation of erythrocytes. Also the prediction that the ratio between the viscosities of the blood and blood plasma at high shear rates nearly equals Euler's constant (2.718) is in good agreement with observed values.

Finally, we discuss possible extensions of the theory. We give an explanation for the difference in hematocrit between genders. Moreover, we derive the theoretical optimal hematocrit for persevering divers among marine mammals such as seals to be around 65 %, in excellent agreement with the values observed in several species.

This theoretical analysis has important implications for understanding human and animal physiology since oxygen transport is a crucial factor for medicine and physical performance. In particular, our results imply that blood doping in sports is physiologically questionable.

**References**

S. Arrhenius: The viscosity of solutions. *Biochem. J*. 11 (1917) 112–133.

H. Stark, S. Schuster: Comparison of various approaches to calculating the optimal hematocrit in vertebrates. *J. Appl. Physiol.* 113 (2012) 355-367.

S. Schuster, H. Stark: What can we learn from Einstein and Arrhenius about the optimal flow of our blood? Biochim. *Biophys. Acta – Gen. Subj. *1840 (2014) 271–276.