Quantitative analysis of mechanisms that govern red blood cell age structure and dynamics during anaemia

Abstract

Mathematical modelling has proven an important tool in elucidating and quantifying mechanisms that govern the age structure and population dynamics of red blood cells (RBCs). Here we synthesise ideas from previous experimental data and the mathematical modelling literature with new data in order to test hypotheses and generate new predictions about these mechanisms. The result is a set of competing hypotheses about three intrinsic mechanisms: the feedback from circulating RBC concentration to production rate of immature RBCs (reticulocytes) in bone marrow, the release of reticulocytes from bone marrow into the circulation, and their subsequent ageing and clearance. In addition we examine two mechanisms specific to our experimental system: the effect of phenylhydrazine (PHZ) and blood sampling on RBC dynamics. We performed a set of experiments to quantify the dynamics of reticulocyte proportion, RBC concentration, and erythropoietin concentration in PHZ-induced anaemic mice. By quantifying experimental error we are able to fit and assess each hypothesis against our data and recover parameter estimates using Markov chain Monte Carlo based Bayesian inference. We find that, under normal conditions, about 3% of reticulocytes are released early from bone marrow and upon maturation all cells are released immediately. In the circulation, RBCs undergo random clearance but have a maximum lifespan of about 50 days. Under anaemic conditions reticulocyte production rate is linearly correlated with the difference between normal and anaemic RBC concentrations, and their release rate is exponentially correlated with the same. PHZ appears to age rather than kill RBCs, and younger RBCs are affected more than older RBCs. Blood sampling caused short aperiodic spikes in the proportion of reticulocytes which appear to have a different developmental pathway than normal reticulocytes. We also provide evidence of large diurnal oscillations in serum erythropoietin levels during anaemia.

Figure 1. RBC and reticulocyte dynamics.

Individual mouse data (left panels) and mean±2sem (right panels) of RBC concentration (top panels) and reticulocyte proportion (bottom panels) of the 10 mice in the main experiment for A: first 14 days, B: full 114 days. Inset expands reticulocyte proportion from days 15 to 114.

Figure 2. Serum Epo concentrations.

Serum Epo concentrations for individual mice (grey lines) and median (black line) of main experiment. The two samples per day on days 5 to 9 suggest diurnal oscillations. Because we had to sample small volumes of blood (20 µl) our Elisa assay detection threshold is above normal Epo concentration, hence the many zeros in the data.

Figure 3. Control experiments.

Mean±1sem of RBC concentration (top panel) and reticulocyte proportion (bottom panel) of main experiment (red dotted lines, same as in Figure 1) sampling controls (black dot-dashed lines), individually-housed, PHZ-treated controls (blue dashed lines) and group-housed, PHZ-treated controls (green solid lines). Lines are slightly offset from each other to reveal extent of error bars.

Figure 4. Model fits.

Fits of best fitting model to RBC concentrations and reticulocyte proportions of individual mice of the main experiment. A: First 14 days, and B: all data. Grey regions are 95% posterior predictive intervals (95% of data should lie within this region), and solid lines are median solutions constructed from the 10⁴ samples of the posterior distribution.

Figure 5. Standardised residuals.

Assessment by standardised residuals of the model consisting of hypotheses A1, B1, C1, D1, E1 fitted to all data of the main experiment. Top panel: RBC concentration, bottom panel: reticulocyte proportion. Each cross represents the standardised residual of a time point for an individual mouse. These have an approximately normal distribution with mean 0 and variance 1. The solid line is the mean of the standardised residuals at each time point. The dashed lines define the approximate interval (Bonferroni corrected for multiple tests) within which we expect the mean to lie 95% of the time if the model were true. Note the nonlinear .

Figure 6. Hypothesis testing.

A: Hypothesis A2: polycythemia not reducing production rate does not capture RBC and reticulocyte dynamics after day 60 (shown for mouse 6). B: Hypothesis A1: Production rate varies almost linearly with anaemia even allowing for an exponential relationship (shown for mouse 6). C: Hypothesis C1: reticulocyte release rate is exponentially related to anaemia (shown for mouse 6). D: Hypothesis D2: fixed RBC lifespan does not capture RBC dynamics. E: Hypothesis D3: unlimited RBC lifespan does not capture RBC dynamics.

Figure 7. Predicted RBC age distributions.

Prediction of circulating RBC age distribution just before, just after, and at various days after, PHZ treatment for mouse 6. The black line shows the estimated median age distribution and the grey region the 95% posterior predictive interval calculated from the samples of the posterior distribution of the model consisting of hypotheses A1, B1, C1, D1, E1. On day 0 the age distribution is in equilibrium made up of about 1% reticulocytes. All RBCs are aged with PHZ treatment causing an approximately 30% drop in RBC concentration, and a higher than normal clearance rate for about 7 days. Anaemia induces early release of reticulocytes from bone marrow (day 5) and increased reticulocyte production. Random clearance of RBCs prevents a large drop in RBC concentration about 50 days after PHZ treatment (day 40). The system returns to equilibrium after about day 60.

Figure 8. Schematic of RBC age distributions.

Schematic of age distribution of reticulocytes in bone marrow and reticulocytes and normocytes in the circulation under normal equilibrium conditions. Reticulocytes are produced at a rate in bone marrow. These are released into the circulation at a rate . All reticulocytes are released by age , and have a maturation time of hours. RBCs are randomly cleared at a rate and have a maximum lifespan of days. Not to scale.

Figure 9. Parameter estimates.

Marginal distributions of parameters of model consisting of hypotheses A1, B1, C1, D1, and E1 for main, and housing control experiments.