Models for the red blood cell lifespan

Abstract

The lifespan of red blood cells (RBCs) plays an important role in the study and interpretation of various clinical conditions. Yet, confusion about the meanings of fundamental terms related to cell survival and their quantification still exists in the literature. To address these issues, we started from a compartmental model of RBC populations based on an arbitrary full lifespan distribution, carefully defined the residual lifespan, current age, and excess lifespan of the RBC population, and then derived the distributions of these parameters. For a set of residual survival data from biotin-labeled RBCs, we fit models based on Weibull, gamma, and lognormal distributions, using nonlinear mixed effects modeling and parametric bootstrapping. From the estimated Weibull, gamma, and lognormal parameters we computed the respective population mean full lifespans (95 % confidence interval): 115.60 (109.17–121.66), 116.71 (110.81–122.51), and 116.79 (111.23–122.75) days together with the standard deviations of the full lifespans: 24.77 (20.82–28.81), 24.30 (20.53–28.33), and 24.19 (20.43–27.73). We then estimated the 95th percentiles of the lifespan distributions (a surrogate for the maximum lifespan): 153.95 (150.02–158.36), 159.51 (155.09–164.00), and 160.40 (156.00–165.58) days, the mean current ages (or the mean residual lifespans): 60.45 (58.18–62.85), 60.82 (58.77–63.33), and 57.26 (54.33–60.61) days, and the residual half-lives: 57.97 (54.96–60.90), 58.36 (55.45–61.26), and 58.40 (55.62–61.37) days, for the Weibull, gamma, and lognormal models respectively. Corresponding estimates were obtained for the individual subjects. The three models provide equally excellent goodness-of-fit, reliable estimation, and physiologically plausible values of the directly interpretable RBC survival parameters.

Appendices

Appendix 1: Weibull, gamma, and lognormal PDFs

The PDF of the Weibull distribution, denoted by $w(t;\alpha ,\beta )$, is given by

$$\begin{aligned} w(t;\alpha ,\beta )=\frac{\alpha }{\beta } \left( \frac{t}{\beta }\right) ^{\alpha -1}{\mathrm {e}}^{{-(t/\beta )}^\alpha }. \end{aligned}$$

(21)

The Weibull parameters $\alpha$ and $\beta$ are both $>$0 and are called the shape and scale parameters, respectively. The mean and variance for the Weibull are

$$\begin{aligned} \mu _w=\beta \Gamma \left( 1+\frac{1}{\alpha }\right) ,\ \sigma _w^2=\beta ^2\left[ \Gamma \left( 1+\frac{2}{\alpha }\right) -\left( \Gamma \left( 1+\frac{1}{\alpha } \right) \right) ^2\right] . \end{aligned}$$

(22)

The PDF of the gamma distribution with parameters $\alpha ,\beta > 0$ is

$$\begin{aligned} g(t;\alpha , \beta ) =\frac{1}{\Gamma (\alpha )\beta ^\alpha } t^{\alpha -1} {\mathrm {e}}^{-t/\beta } ,\ \ \ t > 0 \end{aligned}$$

(23)

($\Gamma$ denotes the Euler gamma function). Note that we write g instead of p for the PDF of the gamma distribution. The parameters $\alpha$ and $\beta$ are again called the shape and scale parameters, respectively. The mean and variance are

$$\begin{aligned} \mu _g=\alpha \beta ,\ \sigma _g^2 =\alpha \beta ^2. \end{aligned}$$

(24)

Let $X \sim {\mathrm {N}}(\alpha ,\beta ^2)$, i.e., normal with mean $\alpha$ and variance $\beta ^2$, and $Y = {\mathrm {e}}^X$. Then Y has the lognormal distribution with parameters $\alpha$ and $\beta$, and PDF

$$\begin{aligned} l(t;\alpha ,\beta )=\frac{1}{t\beta \sqrt{2\pi }}\exp \left( -\frac{(\ln t-\alpha )^2}{2\beta ^2}\right) ,\ t>0. \end{aligned}$$

(25)

The mean and variance for the lognormal are

$$\begin{aligned} \mu _l=\exp \left( \alpha +\frac{\beta ^2}{2}\right) , \sigma _l^2=\exp (2\alpha +\beta ^2)(\exp (\beta ^2)-1). \end{aligned}$$

(26)

The same letters $\alpha$ and $\beta$ are used for the Weibull, gamma, and lognormal parameters for brevity in presentation, but we emphasize that the parameters in the three distributions are entirely unrelated. The PDFs represent the full RBC lifespan distributions in the Weibull, gamma, and lognormal models respectively and the means correspond to the full RBC lifespan.

Appendix 2: Survival functions for Weibull, gamma, and lognormal lifespan distributions

For any lifespan distribution we have

$$\begin{aligned} \int _t^\infty \bar{P}(u)du=\int _t^\infty \int _u^\infty p(v)dv du. \end{aligned}$$

(27)

Interchanging the order of integration yields

$$\begin{aligned} \int _t^\infty vp(v)dv-t\bar{P}(t). \end{aligned}$$

(28)

The full survival functions (Eq. (1)), in the case of gamma, Weibull, and lognormal lifespan distributions are given by

$$\begin{aligned} \bar{G}(t;\alpha ,\beta )= & {} \int _t^\infty g(v;\alpha ,\beta )dv= \frac{1}{\Gamma (\alpha )\beta ^\alpha } \int _t^\infty v^{\alpha -1}{\mathrm {e}}^{-v/ \beta }dv,\end{aligned}$$

(29)

$$\begin{aligned} \bar{W} (t;\alpha ,\beta )= & {} {\mathrm {e}}^{-\left( \frac{t}{\beta }\right) ^\alpha }, \end{aligned}$$

(30)

and

$$\begin{aligned} \bar{L } (t;\alpha , \beta )= & {} 1-N\left( \frac{log(t)-\alpha }{\beta }\right) \end{aligned}$$

(31)

respectively, where N(x) is the CDF for the standard normal $Z\sim N(0,1)$. Thus using Eqs. (9) and (28), we obtain the residual survival function in the case of the gamma lifespan distribution

$$\begin{aligned} \bar{G}_{r}(t)=\bar{G}_{c}(t)=\bar{G}(t;\alpha +1,\beta )-\frac{t}{\alpha \beta }\bar{G}(t;\alpha , \beta ). \end{aligned}$$

(32)

Similarly, from Eq. (14) we obtain the excess lifespan survival function for the gamma lifespan distribution

$$\begin{aligned} \bar{G}_{e} (t)=\frac{\alpha \beta \bar{G}(t_e+t;\alpha +1,\beta )-(t_e+t)\bar{G}(t_e+t;\alpha ,\beta )}{\alpha \beta \bar{G}(t_e;\alpha +1, \beta )-t_e\bar{G}(t_e;\alpha ,\beta )} \end{aligned}$$

(33)

Given $\alpha$ and $\beta$, the survival functions in Eqs. (32) and (33) are readily calculated by numerical integration in most statistical packages, including MATLAB [59]; Eqs. (32) and (33) show that the survival functions of the residual lifetime and the current and excess lifespan distributions are no more difficult to calculate than the full survival function. It also follows from Eqs. (32) and (2) that the mean residual lifespan and mean current age are both equal to $(\alpha +1)\beta /2$: the integral from 0 to $\infty$ of $\bar{G} (t;\alpha +1,\beta )$ equals $(\alpha +1)\beta$ and the integral of $t\bar{G} (t;\alpha ,\beta )$ equals $\alpha \beta ^2(1+\alpha )/2$, as shown using integration by parts, $u = \bar{G}(t;\alpha ,\beta ),\ dv = tdt$.

The survival functions $\bar{G}_{r}$ and $\bar{G}_{c}$ of the residual lifetime and current age distributions, for the gamma lifespan distribution cannot be expressed in closed form, but there is a closed form when $\alpha$ is an integer $\ge$1. The gamma distribution with $\alpha = 2$ has been used [61] in an erythropoiesis model for chronic kidney disease patients.

Although the Weibull survival function (30) has a simple form, the corresponding residual survival function does not. The integral in Eq. (9) can be expressed in terms of the gamma survival function:

$$\begin{aligned} \int _t^\infty {\mathrm {e}}^{-\left( \frac{u}{\beta }\right) ^\alpha }du=\frac{\beta }{\alpha } \int _{\left( \frac{t}{\beta }\right) ^\alpha }^\infty v^{\left( \frac{1}{\alpha }-1\right) } {\mathrm {e}}^{-v} dv = \beta \Gamma \left( 1+\frac{1}{\alpha }\right) \bar{G}\left( t^\alpha ;\frac{1}{\alpha },\beta ^\alpha \right) . \end{aligned}$$

(34)

The residual, current age, and excess lifespan survival functions, $\bar{W}_{r}$, $\bar{W}_{c}$, and $\bar{W}_{e}$, can now be written for the Weibull model as follows:

$$\begin{aligned} \bar{W}_{r}(t;\alpha ,\beta )\,=\,& {} \bar{W}_{c}(t;\alpha ,\beta )\,=\,\bar{G}(t^\alpha ;1/\alpha ,\beta ^\alpha ),\ {\mathrm {and}}\end{aligned}$$

(35)

$$\begin{aligned} \bar{W}_{e}(t;\alpha ,\beta )= & {} \frac{\bar{G}((t_e+t)^\alpha ;1/\alpha ,\beta ^\alpha )}{\bar{G}(t_e^\alpha ;1/\alpha ,\beta ^\alpha )}. \end{aligned}$$

(36)

Using Eqs. (9), (14), (28) and (31), the residual, current age, and excess lifespan survival functions, $\bar{L}_{r}$, $\bar{L}_{c}$, and $\bar{L}_{e}$ for the lognormal model can be written as follows:

$$\begin{aligned} \bar{L}_{r}(t;\alpha ,\beta )= \bar{L}_{c}(t;\alpha ,\beta )= 1-N\left( \frac{log(t)-\alpha }{\beta }-\beta \right) -\frac{t}{\mu _l} \bar{L}(t;\alpha ,\beta ),\ {\mathrm {and}} \end{aligned}$$

(37)

$$\begin{aligned} \bar{L}_{e}(t;\alpha ,\beta )=\frac{ 1-N \left( \frac{log(t_e+t)-\alpha }{\beta }- \beta \right) -\frac{(t_e+t)}{\mu _l}\bar{L}(t_e+t;\alpha ,\beta )}{ 1-N\left( \frac{log(t_e)-\alpha }{\beta } -\beta \right) -\frac{t_e }{\mu _l}\bar{L} (t_e;\alpha ,\beta )}. \end{aligned}$$

(38)

From Eqs. (2) and (9) the mean residual lifespans for the gamma, Weibull, and lognormal distributions are given by $\mu _{g,r}=\frac{(\alpha +1)\beta }{2}$, $\mu _{w,r}=\frac{\beta \Gamma (2/ \alpha )}{\Gamma (1/ \alpha )}$, and $\mu _{l,r} =\frac{\exp (\alpha +3\beta ^2/2)}{2}$, respectively (using Eq. (11)); these are obviously different from the corresponding mean full lifespans (Appendix 1). We also note that, in equilibrium, the steady state residual lifespan distribution has PDF $p_{r,ss}(t) =\bar{P}(t)/\mu$, which is the same as the transient residual lifespan distribution. The steady state full lifespan has PDF $p_{ss}(t) = tp(t)/\mu$ [60]. Thus the mean residual and mean full lifespans in steady state are respectively $(\sigma ^2+\mu ^2)/2\mu$ (Eq. (11)) and $(\sigma ^2+\mu ^2)/ \mu$. In steady state, therefore, the mean residual lifespan is one half the mean full lifespan. For the gamma, Weibull, and lognormal the steady state mean full lifespans equal $\mu _{g,ss}=(\alpha + 1) \beta$, $\mu _{w,ss}=2\beta \Gamma (2/ \alpha )/ \Gamma (1/ \alpha )$, and $\mu _{w,ss}=\exp (\alpha + 3\beta ^2/2)$, respectively. For an individual in stable condition, these distributions are what would be seen for RBCs in the circulation.