Modeling of non-stationary RBC survival: Effect of erythropoietin on RBC mortality vs. production in rats.

An age-structured population model was applied to describe red blood cell (RBC) survival data in normal rats treated with recombinant human erythropoietin (rHuEPO). The primary purpose was to estimate the mean lifespan and age of RBCs from a study employing the random labeling technique. The second objective was to assess an impact on the RBC survival a hypothetical rHuEPO inhibitory effect on the RBC elimination.

RBC survival is an important determinant of many physiological processes and clinical diagnostic markers. Experimental techniques to measure survival of erythrocytes have been employed for over a half century. They are based on random or cohort labeling techniques which allow one to calculate the mean lifespan and age of RBCs from a series of blood samples collected over time [1, 2]. A variety of statistical models have been developed to describe the survival curves [3, 4, 5]. Recently, a pharmacodynamic model for RBC survival has been proposed [6]. An aging process has been a subject of mathematical models of age-structured cell populations [7]. We have adopted the latter to account for a time evolution of the erythrocyte age density n(t,a) as a turnover process controlled by the production rate of cells of age a kin(t)d(a), and the elimination rate of cells of age a m(t,a)n. The conditional mortality rate m(t,a) was determined by from the RBC lifespan distribution. Three types of probability density functions for RBC lifespan distribution were tested: generalized Gamma, Weibull [8], and Dornhorst [9]. The model was used to derive an equation for the survival curve that was fitted to placebo treated RBC stationary survival data for normal rats. The parameter estimates were further to simulate RBC survival in rats treated with rHEPO (non-stationary data).  The Dornhorst distribution permitted explicit solutions for n(t,a) that have been used to simulate the RBC survival curves when m(t,a) was represented as a product of the steady-state conditional mortality rate and a time dependent function representing a drug effect. Similarly, kin(t)  was represented as a product of the RBC production rate constant and a time dependent function representing a drug effect.

Three groups of normal male Wistar rats (n=3 per group, 350 g) received placebo (Group A) and treatment with rHuEPO (Epogen, Amgen Inc.) consisted of IV injections of 450 IU/kg t.i.w. for two weeks (Groups B and C). Survival studies were performed using water-soluble biotin as a tag for RBC. Biotinylated cells were detected by streptavidin conjugated to R-phycoerythrin analyzed by flow cytometry. Blood samples of 2.5 ml were drawn from rats, biotinylated and injected back into the same animals within 1 h. This resulted in labeling about 1-3 % of circulating RBC. Blood samples (~100 ml) were drawn after 24 h daily for the first week, then three times a week until the signal reached a limit of quantification. The RBC labeling with biotin was done at the time of the first dose for Groups A and B, and after 14 days for Group C. The survival data were represented as the fraction of surviving cells. The fraction of survived cells predicted by the model was fitted to the data using NONMEM 6 (Icon Inc.) first order conditional estimation with the proportional residual error. The parameters of the lifespan distribution were used to simulate the RBC lifespan and age distributions, as well as the non-stationary survival curves by MATLAB 7.7 (MathWorks Inc.).

All three studied lifespan distributions resulted in similar fittings of the survival data for Group A. The standard errors of the parameter estimates were less than 0.07% of their values. The mean lifespan estimates were 43.2±6.3, 42.2 ±6.7, and 34.6±8.4 days for Gamma, Weibull, and Dornhorst lifespan distributions, respectively.  Analogous values for the mean age were 23.3±2.7, 23.1 ±2.7, and 22.3±2.8 days. One way ANOVA showed no difference between the means of lifespan and age estimated by these three methods. Simulations of the RBC survival assuming only a stimulatory drug effect on kin(t) showed a noticeable difference between survival curves for Groups A and C and no difference for Groups A and B. Simulations of the inhibitory drug effect on the RBC mortality rate indicated a difference between survival curves for Groups A and B, but no difference for Groups A and C.

In conclusion, our study showed that the age-structured population model is particularly useful in extracting information about the means of lifespan and age distributions from the random labeling RBC survival study. A typical probability function for RBC lifespan distribution characterized by two parameters can be inferred from the fitting of the survival data.  An arbitrary shape of this distribution has a little impact on the estimates of the mean values. Simulations showed that a survival study performed simultaneously with rHuEPO administration is most informative about the rHuEPO effect on the mortality rate, whereas a survival study that follows rHuEPO administration is dominated by the rHuEPO effect on the RBC production.

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Wojciech Krzyzanski

  • University at Buffalo