Two of our mice (1 and 7) were observed in aggressive interactions after day 42. Both mice subsequently showed typical physiological signs of aggression (reduced RBC concentration and elevated reticulocyte proportion [26]). We therefore did not include the data for these two mice after day 42.

The top panels of Figure 1A show the RBC concentration dynamics over the first 14 days (left panel: individual mice, right panel: mean±2sem) The bottom panels of Figure 1A show the reticulocyte proportion dynamics over the first 14 days (left panel: individual mice, right panel: mean±2sem). Figure 1B shows the same over the full 114 days.

The RBC concentration dynamics show several interesting and unexpected features. There is an initial loss of about 30% of RBCs to about 6.5×10⁹ cells/ml in the first day. RBC concentration then drops at a slower rate for another 2 to 5 days depending on the mouse. There is a lot of variation in RBC dynamics across the mice which causes the mean dynamics to appear flatter than in the individual mice data. From day 6 to day 12 average RBC concentration increases by about 0.37×10⁹ cells/ml/day, but then abruptly slows by an order of magnitude to about 0.03×10⁹ cells/ml/day. What appears to be unusual about these dynamics is that this abrupt change occurs when RBC concentration is still below equilibrium (the change occurs at an average RBC concentration of about 8×10⁹ cells/ml whereas average equilibrium concentration is about 8.5×10⁹ cells/ml). Average RBC concentration then rises through its equilibrium value to become polycythemic reaching a maximum of about 9.3×10⁹ cells/ml on day 48. There is a period of cell loss between days 48 and 60 which is probably the clearance of the large cohort of RBCs induced by anaemia reaching their maximum lifespan. Thereafter RBC concentration appears to be in equilibrium.

The reticulocyte proportion dynamics are similarly intriguing. Over the first 3 days there is a gradual rise in average reticulocyte proportion from 1.5% to 10%. The rate of increase rises over the next 2 days culminating in an average peak proportion of 30% although there is a lot of variation between mice. Over the next 15 days reticulocyte proportion declines back to normal values but with a series of unusual aperiodic spikes on days 5, 6, 9, 12 and potentially day 14. Average reticulocyte proportion shows an unexpected drop below normal by day 26—unexpected because RBC concentration appears to reach equilibrium values at this time (see inset Figure 1B). This is followed by a rapid return to normal by day 28. It then drops below normal again until day 54 (approximately correlating with the slowly rising polycythemic RBC concentration). It stays higher than normal until about day 83 after which it returns to normal. On day 113 there is again a rise.

Figure 2 shows the individual mice Epo dynamics in grey and their median in black. 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. The individual mice Epo data are difficult to interpret. However, several interesting patterns emerge from the aggregated data. Epo is generally undetectable until the third day, and then begins to rise over the next 3 days. From days 5 to 9, when we sampled twice per day, Epo concentration appears to exhibit diurnal oscillations with higher concentrations measured at 8am and lower at 3pm. Our data are consistent with mice being nocturnal, as humans have higher Epo concentrations in the afternoon compared to the morning [27]. After day 12, Epo concentration returns to undetectable levels. Oscillations may have occurred on days other than 5 to 9, but because we only sampled once per day diurnal oscillations would be undetectable.

To test if diurnal oscillations are indeed present, we performed a bootstrap analysis both within and across mice. For each mouse we counted the number of successive daily peaks and troughs in the Epo data (Table 1). We then bootstrapped without replacement each mouse's data 10⁵ times and counted the number of peaks and troughs in each simulated data set. This constructs a reference distribution of the number of peaks and troughs under the null hypothesis that Epo does not oscillate daily. Comparing the observed number of peaks and troughs to the reference distribution, we calculate a for the probability of observing at least as many peaks and troughs than was actually observed. These are given in Table 1 (). Except for mouse 5, the evidence for diurnal oscillations is weak. However, if we sum peaks and troughs across all mice there are 55 peaks and troughs in total. The 95% CI for the numbers of peaks and troughs across all mice under the null hypothesis is 24 to 45 (median 34), which gives a of 0.00051 (), thus strongly suggesting that Epo does oscillate diurnally. This difference in between tests on individual and aggregated data is due to the differences in statistical power of the tests which depend on sample size.

One of the five mice in the group-housed, PHZ-treated control experiment, and one of the five mice in the group-housed, non-PHZ-treated control experiment had anomalous RBC and reticulocyte dynamics and were removed from the analysis.

Group and individual housing had no significant effect on RBC concentration dynamics (compare green solid and blue dashed lines in Figure 3 top panel). However, individual housed mice had significantly lower reticulocyte proportions than group housed mice and did not exhibit reticulocyte spikes (bottom panel).

In the blood sampling control experiment mice were not given PHZ but were blood sampled as in the main experiment. Blood sampling had no effect on RBC concentration (black dot-dashed line in Figure 3, top panel). However, it produced a clear spike in reticulocyte proportion in three of the four mice on day 5, with potentially other spikes at other times in some mice (black dot-dashed line in Figure 3, bottom panel).

There is strong evidence () that polycythemia causes a reduction in reticulocyte production rate (hypothesis A1) rather than having no effect (hypothesis A2). This can be seen by comparing the dynamics of the two hypotheses. If polycythemia does not affect production rate, RBC concentration and reticulocyte proportion immediately return to equilibrium after the drop in RBC concentration after day 60 (see fit of mouse 6 in Figure 6A). However, if polycythemia does cause a reduction in production rate, the dynamics overshoot before returning to equilibrium (Figure 4B). The mean reticulocyte dynamics (Figure 1B) appear to support the latter rather then the former hypothesis, and this is consistent with experimental evidence [28],[29] as discussed in the section Production rate of bone marrow reticulocytes.

We cannot discriminate between a linear relationship between the difference in normal and anaemic RBC concentrations and reticulocyte production rate (A1), an exponential relationship () and a sigmoidal relationship (). This is demonstrated in Figure 6B: over the range of RBC concentrations () we observe for mouse 6, the fold-change in production rate () varies approximately linearly even for the model with an exponential relationship. Because we are still in the linear region of this response curve, the sigmoidal response does not improve the fit of the model.

We cannot discriminate between non-oscillatory (A1) and oscillatory production rate (). This does not mean that oscillatory production does not occur, just that any signal of oscillations is too weak to detect in our data.

There is no support for a time lag in the feedback between RBC concentration and reticulocyte production ().

Hypothesis B1 states that RBCs present in the circulation at the moment of PHZ treatment are immediately aged. This is consistent with our data because it accounts for the initial loss of about 30% of RBCs within the first day due to ageing of older RBCs past their maximum lifespan (see predicted dynamics of RBC age structure of mouse 6 in Figure 7). Hypothesis B1 also states that younger RBCs are aged more than older RBCs. This hypothesis is consistent with our data because it accounts for the reduction in RBC concentration from days 2 to 5 due to higher than normal clearance rate (see Figure 4A). (If no PHZ-induced lysis occurred during this time, RBC concentration would rise because production rate of reticulocytes would be greater than random clearance of RBCs.)

Hypothesis B2 states that RBCs present in the circulation at the moment of PHZ treatment lyse at a constant rate irrespective of their age. Thus this population decays exponentially and new RBCs entering the circulation are not aged by PHZ. This hypothesis is not supported ().

Hypothesis B3 states that PHZ lyse all circulating RBCs at a rate proportional to its concentration and that concentration decays exponentially over time. This hypothesis is not supported ().

A problem with hypotheses B2 and B3 is that they cannot account for the abrupt change in RBC loss between days 1 and 2. Hypothesis B4 overcomes this by adding an initial rapid loss to hypothesis B3. The addition of this mechanism, however, is also not supported as ).

We cannot discriminate between polycythemia causing a reduction in reticulocyte release rate from bone marrow (C1) or having no effect on (C2) release rate ().

An exponential (C1) rather than a linear (C3) relationship between the difference in normal and anaemic RBC concentrations and reticulocyte release rate is favoured (). This is demonstrated in Figure 6C which shows the prediction of this relationship for mouse 6. However, for a linear relationship, when RBC concentration rises above the threshold during polycythemia, release rate becomes 0 causing the reticulocyte proportion to be 0 until RBC concentration drops below this threshold; this is not observed in our data. Thus a linear relationship may be an adequate hypothesis, but its combination with polycythemia affecting release rate causes it not to be supported by the data. With no evidence for or against polycythemia causing a reduction in release rate, we must check the combined hypotheses of a linear relationship and polycythemia not affecting release rate. This hypothesis is not supported (), so we can conclude that an exponential relationship between the difference in normal and anaemic RBC concentrations and reticulocyte release rate is supported.

Hypothesis D1 states that there is random clearance of RBCs until they reach their maximum lifespan (about 50 days) at which they are immediately cleared. This hypothesis gives age distributions as shown in Figures 7 and 8. This hypothesis is supported for two reasons. i) Under equilibrium conditions, input rate of RBCs from bone marrow is equally matched with clearance rate. The two mechanisms that clear RBCs are random clearance and reaching the maximum lifespan. Now, treatment with PHZ lyse all RBCs present at the time of administration within about a week (see Figure 7). The post-treatment cohort of RBCs only experience random clearance until they reach their maximum lifespan. Therefore, until this time, RBC input rate into the circulation is slightly higher than their clearance rate causing a slowly rising polycythemia which we observe in our data. If there were no random clearance of RBCs (hypothesis D2) their concentration would grow by about 0.5×10⁹ cells/ml each day (equal to ) once RBC concentration had reached equilibrium at about day 12. We observe growth an order of magnitude lower than this at 0.03×10⁹ cells/ml/day. One might argue that the resulting polycythemia could cause a reduction in reticulocyte production rate to this level. But, to achieve a growth rate of 0.03×10⁹ cells/ml/day would require an order of magnitude reduction in reticulocyte production rate causing reticulocyte proportion to fall to about 0.15%. This though, is not consistent with our data which falls to just 1% during polycythemia. The fit of hypothesis D2 (shown in Figure 6D) confirms this, and so this hypothesis is not supported by the data (). ii) If there is no immediate clearance of RBCs of around 50 days old (hypothesis D3) we do not observe a drop in RBC concentration at this time (Figure 6E). This hypothesis is not supported ().

Hypothesis E3 tests if reticulocyte spikes can be produced intrinsically in our base model. This hypothesis is not supported (). Sampling-induced reticulocytes must therefore be produced by some unknown mechanism not explicitly included in our model. We can test how these reticulocytes enter the erythropoietic system. Hypothesis E2 tests if spikes are produced by large transient increases in the production rate of reticulocytes in the bone marrow. This hypothesis is not supported () because the maturation age of reticulocytes is about 2–4 days, so spikes would be observed for that length of time, instead we observe spikes lasting for less than a day. This observation led us to suggest hypothesis E1 in which sampling-induced reticulocytes enter directly into the circulation.

The marginal distribution of the standard deviation of RBC measurement error , is 0.53×10⁹ cells/ml with a 95% CI of 0.49 to 0.58×10⁹ cells/ml.

Figure 9 shows the marginal distributions of the parameters of the model consisting of hypotheses A1, B1, C1, D1 and E1. The base model (Equations 1–6) has six parameters, the reticulocyte maturation time , the maximum reticulocyte residence time in bone marrow (equal to ), the normal reticulocyte production rate , the normal reticulocyte release rate , the circulating RBC clearance rate , and their maximum lifespan .

The 95% CIs of the marginal distributions of contain 0 and do not go above about 10 hours (for the main experiment mice at least, which have more data and therefore give more accurate parameter estimates). This suggests i) that normocytes do not remain in bone marrow (which would occur if which we allowed it to do), and ii) that, in normal conditions, most reticulocytes are released as, or at most about half a day before, they mature. This is similar to sheep [30], but not humans in which reticulocytes appear to be released about a day before they mature [31].

For most mice the distribution of is not well defined, taking values between 2 and 5 days. It also exhibits strong negative correlation with (not shown) because both parameters determine the concentration of circulating reticulocytes under normal conditions (Equation 23). So a change in can be compensated for with a change in .

The marginal distributions of exhibit a large variation across mice from 7.9×10⁻⁴ (95% CI: 7.1×10⁻⁴, 8.5×10⁻⁴) hr⁻¹ for mouse 9 to 0.021 (0.015, 0.026) hr⁻¹ for mouse 4. Taken together, the broad estimates in and suggest that our data do not allow us to precisely determine the age distribution of reticulocytes in bone marrow and the circulation. More precise priors on one of these parameters would improve the precision of the other. We have not been able to find estimates for either or in mice in the literature. In humans is thought to be about 4 days [32], and in rats about 3 days [18].

We took a narrow prior for maximum RBC lifespan () to improve identifiability of and . The mean of 50 days corresponds roughly to the drop in RBC concentration and previous estimates [33]. There is very little difference between the marginal distributions of and their prior. Therefore our data poorly identify .

The marginal distributions for normal RBC concentration , although variable between mice, are precise. It is interesting that the mice observed in aggressive interactions (1 and 7) have low estimated .

The marginal distributions of reticulocyte production rate and RBC clearance rate are correlated because reticulocyte production rate must match RBC clearance rate in equilibrium. The marginal distributions of vary between 1.31×10⁷ (1.25×10⁷, 1.44×10⁷) cells/ml/hr for mouse 10 and 2.94×10⁷ (2.77×10⁷, 3.13×10⁷) cells/ml/hr for mouse 3. The marginal distributions of are, in general, quite broad, varying between 0.00122 (0.00112, 0.00132) hr⁻¹ for mouse 8 and 0.00347 (0.00295, 0.00395) hr⁻¹ for mouse 1. Mouse 1 has a high modal clearance rate, and this is reflected in its flat RBC concentration after day 12 (Figure 4B) rather than the increasing concentration seen in the other mice. This is probably partly an artifact of the shorter time series for this mouse. Mouse 3 has a similarly high clearance rate. Again this is reflected in its relatively flat RBC concentration after day 12. In other modelling studies RBC clearance rate is assumed to be around 0.02 to 0.025 day⁻¹ (0.001 hr⁻¹) because RBC lifespan in mice is about 40 to 50 days.

The modal value of the time to delay before increased production rate , is about 2.5 days. This corresponds to just before the rapid increase in reticulocyte proportions are observed (Figure 1) and when Epo concentrations rise above detection (Figure 2). However, it appears that a delay is only required to adequately explain the data of some of the mice; the 95% CI of in the majority of mice contains 0. This is probably because only 2 or 3 data points provide estimates for .

The marginal distributions for the scaling factor between anaemia and fold-increase in RBC production rate vary between 0.28×10⁻¹⁰ (0.21×10⁻¹⁰, 0.37×10⁻¹⁰) (cells/ml)⁻¹ for mouse 3, and 3.3×10⁻¹⁰ (3.0×10⁻¹⁰, 3.5×10⁻¹⁰) (cells/ml)⁻¹ for mouse 4.

The PHZ-induced change in age of RBCs of age 0, , are well defined within mice and vary between 41 and 45 days across mice. The PHZ-induced change in age of RBCs of age , , is less well defined; for most mice it is between 2 and 9 days with a mean of roughly 3 days.

The marginal distributions for the scaling factor between anaemia and fold-increase in reticulocyte release rate vary between 0.70×10⁻⁹ (0.65×10⁻⁹, 0.74×10⁻⁹) (cells/ml)⁻¹ for mouse 10 and 2.24×10⁻⁹ (1.74×10⁻⁹, 2.93×10⁻⁹) for mouse 4.

The maturation time of sampling-induced reticulocytes in the circulation is quite variable, from as little as 4 hours in mice 9 and 10, up to about 2 days in other mice. The marginal distribution for mouse 4 is quite broad, most likely reflecting the lack of reticulocyte spikes for this mouse (Figure 4B). The fold-increase in sampling-induced reticulocyte production rate is quite broad varying between 2.5 (1.9, 3.2) for mouse 1 and 12 (11, 13) for mouse 10.

The marginal distributions of the PHZ-treated control mice are shown in Figure 9. Most of the parameters have similar distributions as the main experiments, although some are slightly wider due to fewer data for these mice. The most striking difference in parameter estimates is the PHZ-induced ageing parameter .

Initially we used the same priors for the control experiments as we did for the main experiment. We found that RBC clearance rate , was not well defined in control mice (data not shown) because the RBC dynamics had not returned to equilibrium by the end of these experiments. This affected the estimation of the PHZ-induced ageing parameters and because these depend on the equilibrium RBC age distribution which depends on . A potential difference between the main and the control experiments could have been the efficacy of the PHZ administered because these experiments were done with different batches of PHZ. Such a difference could affect the PHZ-induced ageing of RBCs. Therefore, to be able to compare PHZ-induced ageing between the main and control experiments we based the prior of for the control experiments on its estimated marginal distributions from the main experiment. The prior we chose was N(0.0015, 0.0005). The posterior estimates shown in Figure 9 clearly demonstrate that is smaller for most mice in the control experiments. We therefore suggest that the efficacy of the PHZ administered in these experiments was weaker than for the main experiment. This could explain why mice in the main experiment were more anaemic and had higher reticulocyte proportions than control mice (Figure 3).

Parameters and determine the duration and magnitude of the spikes in reticulocyte proportion. For mouse 4 in the main experiment and all the individually housed mice, the 95% CIs of the product of these two parameters contains 0. This correlates with the observation that these mice showed weak or no spikes.

Epo production is known to increase exponentially with falling RBC concentration [41],[42], but the functional relationship between serum Epo concentration and production rate is not known. Sigmoidal [1],[9] and linear [11] responses have been used in previous models. We considered three hypotheses which relate RBC concentration to reticulocyte production rate: exponential (hypothesis A1), linear (hypothesis A3) and sigmoidal (hypothesis A6). See below for their formulation.

When oxygen supply exceeds demand, as for instance during repeated transfusions of RBCs causing polycythemia (higher than normal RBC concentration), circulating reticulocyte concentration decreases due to production of fewer reticulocytes in bone marrow [28],[29]. We consider two hypotheses: polycythemia reduces (hypothesis A1), or has no effect on (hypothesis A2), production rate.

There is evidence of diurnal oscillations in circulating reticulocyte concentration and mitotic activity in bone marrow in rats and mice [43]. Our data and other work [44] suggest diurnal oscillations in Epo concentration in mice. It is therefore suggestive that diurnal oscillations in Epo could cause oscillations in production rate. We therefore tested two hypotheses: with (hypothesis A4), and without (hypothesis A1), diurnal oscillations in production rate.

In our data, two pieces of evidence suggest there may be a delay of several days from administration of PHZ to increased production rate: the rapid rise in reticulocyte proportion around day 3 (Figure 1A and 1B), and Epo concentration generally below detectability until about day 3 then rapidly rising (Figure 2). Hypothesis A1 tests this idea by only allowing increased production after some time . Hypothesis A5, however, tests another idea that may explain the data: production rate at time depends on RBC concentration at some earlier time . This idea has been used in other modelling studies (e.g., [20]). The mathematical formulations of all these hypotheses are detailed below.

Hypothesis A1 Increased reticulocyte production is delayed after PHZ treatment. Production rate depends instantaneously and exponentially on the difference between normal and anaemic RBC concentrations.

This means (the anaemia-induced fold-change in reticulocyte production rate over normal) in Equation 1 has the form(7)where is the delay between PHZ administration and increase in production rate, is a scaling factor and , the normal RBC concentration.

Hypothesis A2 As hypothesis A1, but production rate remains normal during polycythemia ().

This means has the form(8)

Hypothesis A3 As hypothesis A1, but production rate depends linearly on the difference between normal and anaemic RBC concentrations.

This means has the form(9)The maximum value is taken to prevent production rate becoming negative during polycythemia.

Hypothesis A4 As hypothesis A1, but reticulocyte production rate exhibits diurnal oscillations

This means has the form(10)where is the amplitude and the initial phase of the oscillations.

Hypothesis A5 As hypothesis A1, but with a lag in the feedback between reticulocyte production rate and RBC concentration instead of a delay from PHZ administration to increased production rate.

This means has the form(11)where is lag time.

Hypothesis A6 As hypothesis A1, but production rate depends sigmoidally on the difference between normal and anaemic RBC concentrations.

This means has the form(12)where is the maximum fold-change in reticulocyte production rate over normal.

Phenylhydrazine (PHZ) has been used for over 100 years to induce haemolytic anaemias in experimental animals [25] through RBC lysis. Remarkably, studies on whether PHZ alters the age distribution of RBCs have not been previously undertaken. We test four hypotheses for the action of PHZ on circulating RBCs; we assume it has no effect on bone marrow reticulocytes.

It has been shown in rabbits that PHZ is eliminated slowly from the body; 60% of a 50 mg/kg dose is removed in 10 days [45]. However, it is unknown if PHZ lyse only those RBCs present in the circulation when it is administered (hypothesis B2) or if in addition it lyses RBCs that enter the circulation after it is administered (hypothesis B3). For hypothesis B2 we assume that RBCs are lysed at a constant rate, and for hypothesis B3 we assume that lysis rate is proportional to PHZ concentration which decays exponentially over time.

These two hypotheses cause a smooth, exponential decay in RBC concentration. Our data suggest, however, an abrupt change from a fast to a slow lysis rate sometime before 24 hrs post PHZ administration (Figure 1A). We test this hypothesis (hypothesis B4) by removing a proportion of RBCs from the circulation just after PHZ administration.

PHZ is known to react with haemoglobin to produce free radicals that cause peroxidation of RBC lipid membranes leading to lysis [46]. This led us to consider the hypothesis that PHZ does not kill RBCs outright, but prematurely ages them (hypothesis B1). In addition, we consider the hypothesis that PHZ might age younger RBCs more than older RBCs, perhaps because younger RBCs are more metabolically active and thus more vulnerable to oxidative damage. The mathematical formulations of these hypotheses are detailed below.

Hypothesis B1 RBCs present in the circulation when PHZ is administered are prematurely aged. RBCs entering the circulation after PHZ treatment are not aged by PHZ. The change in a RBC's age is linearly correlated with its age at time of treatment.

Just after , RBC ages are transformed to(13)where is the change in age of RBCs of age 0, is the change in age of RBCs of age , and is the maximum RBC lifespan. Any RBCs aged past the maximum lifespan are immediately cleared.

Hypothesis B2 RBCs present in the circulation when PHZ is administered lyse at a constant rate. RBCs entering the circulation after PHZ treatment are not aged.

Equation 2 is modified by addition of the term(14)where is the PHZ-induced lysis rate.

Hypothesis B3 All RBCs in the circulation (pre and post PHZ treatment) lyse at a rate proportional to PHZ concentration which itself decays exponentially.

Equation 2 is modified by addition of the term(15)where is the initial PHZ-induced lysis rate, and the decay rate of PHZ.

Hypothesis B4 As hypothesis B3, but with immediate lysis of a proportion of RBCs present when PHZ is administered.

Equation 2 is modified as for hypothesis B3. In addition, just after , RBC age distribution is transformed to(16)where is the proportion of RBCs lysed just after treatment.

We test if release rate is linearly (hypothesis C3) or exponentially (hypothesis C1) related to the difference in normal and anaemic RBC concentrations. We test if polycythemia reduces (hypothesis C1) or has no effect on (hypothesis C2) release rate. Finally we test the combined hypotheses of a linear relationship and polycythemia having no effect on release rate (hypothesis C4).

Hypothesis C1 Reticulocyte release rate from bone marrow depends instantaneously and exponentially on the difference between normal and anaemic RBC concentrations.

This means (the anaemia-induced fold-change in reticulocyte release rate over normal) in Equations 1 and 2 has the form(17)where is a scaling factor.

Hypothesis C2 As hypothesis C1, but polycythemia does not reduce release rate.

This means has the form(18)

Hypothesis C3 As hypothesis C1, but release rate depends linearly on the difference between normal and anaemic RBC concentrations.

This means has the form(19)The maximum is taken to prevent the release rate becoming negative during polycythemia.

Hypothesis C4 As hypothesis C3, but polycythemia does not reduce release rate.

This means has the form(20)

We wish to test if RBCs are cleared randomly and independently of age (hypothesis D3), or if they are all cleared at a fixed age (hypothesis D2), or if they are cleared randomly up to a fixed age and then cleared immediately (hypothesis D1).

Hypothesis D1 RBCs are cleared at rate until age , any remaining are then immediately cleared.

No modification to Equation 2 is required.

Hypothesis D2 No clearance of RBCs until they reach age , they are then rapidly cleared.

This means in Equation 2.

Hypothesis D3 RBCs are cleared at rate .

This means in Equation 2.

Spikes in reticulocyte proportion have not previously been documented in the literature. This could be because sampling is rarely undertaken more frequently than once per day, so a spike lasting for less than 24 hours would not be observed. Or it could be because spikes have previously been interpreted as experimental noise.

Our control experiments strongly suggest that reticulocyte spikes are induced by blood sampling and not by any intrinsic response to anaemia. Moreover, we have found no biologically plausible mechanism that can produce reticulocyte spikes at non-periodic intervals. Therefore, we model them phenomenologically (that is, without any mechanistic explanation) as a series of pulsed inputs of reticulocytes either into bone marrow (hypothesis E2) or directly into the circulation (hypothesis E1). We also test if our base model can provide an adequate fit to our data without including reticulocyte spikes (hypothesis E3). This phenomenological model can be thought of as a filter, filtering out nuisance data in order to extract information from the underlying data. Nevertheless, these spikes do change RBC and reticulocyte dynamics, so we may be able to make inferences about their mechanistic cause.

Hypothesis E1 Sampling-induced reticulocytes enter directly into the circulation. Their production rate and maturation time may be different to normal reticulocytes.

Equation 2 is modified by the additional term(21)where is the production rate of sampling-induced reticulocytes, is their maturation time in the the circulation, is the Dirac delta function, is the time after PHZ administration that spike appears, is the duration of spike and the number of spikes.

Hypothesis E2 Sampling-induced reticulocytes are produced in bone marrow. Their maturation time is the same as normal reticulocytes but their production rate may be different.

Equation 3 is modified by the additional term(22)

Hypothesis E3 No sampling-induced reticulocytes.

No modifications to Equations 1 and 3 are required.

Close to the end of the main experiment we used seven of the mice to determine the error structure and variance of the RBC concentration measurements. For each mouse, four 2 µl blood samples were taken and coulter counted. If the measurement error variance is , then the difference between any pair of samples from a mouse has expectation 0 and variance . For each mouse there are two independent pairs, and therefore, with 7 mice, 14 independent pairs in total. The errors were found to be normally distributed (Anderson-Darling test [47]) and the estimate of was 0.61×10⁹ (95% CI: 0.44×10⁹, 0.84×10⁹) cells/ml.

We further refined our estimate of using the main experiment data during the model fitting. Our prior on was taken as normally distributed with mean 0.61×10⁹ and standard deviation 0.1×10⁹ (based on the estimated 95% CI of ).

For each smear, the number of reticulocytes and normocytes were counted from a random sample of 500 RBCs (except at the first time point for mice 1 and 2 when 2000 RBCs were counted, and at the first time point for the other mice when 1000 RBCs were counted). If the true proportion of reticulocytes in mouse at time is then the number of counted reticulocytes , is Binomially distributed with parameters and , 1000 or 2000.

The models were numerically implemented in discrete time with a timestep of 1 hr. We used an Evolutionary Monte Carlo (EMC) method [48] to sample the posterior density. Parameters were estimated separately for each mouse, except the RBC concentration measurement error which was simultaneously estimated across all mice. We did not estimate any hyper parameters.

The parameter was not directly estimated, but was derived by solving the following equation for using the bisection method with tolerance 10⁻⁵(23)where is the initial circulating reticulocyte concentration which was also estimated. This equation was derived by considering the total concentration of reticulocytes in bone marrow and circulation () and subtracting from that the concentration of reticulocytes in bone marrow. The equation is approximate because we have not included random clearance of reticulocytes, but this effect will be small.

The likelihood of a model with parameter vector given the data for mouse is, up to a constant of proportionality,(24)where is the number of data points for mouse , is the coulter counter measured RBC concentration at time point , is the number of RBCs counted on a smear (500, 1000 or 2000), is the number of reticulocytes counted, is the RBC measurement error, is the model solution of RBC concentration at time , is the model solution of reticulocyte concentration at time , and is the model solution of reticulocyte proportion at time . Most of the priors on the parameters were uniformly distributed, their ranges roughly based on pre-model fitting estimates. Based on previous estimates, the prior on the maximum RBC lifespan was [33] which helps identifiability of reticulocyte production rate and RBC clearance rate .

The Markov chain Monte Carlo jumping distributions were multivariate normal with covariance matrices estimated from a empirical covariance matrices calculated during an adaptive phase [49]. The scale of the covariance matrices were tuned to give an acceptance rate between 20 and 40% [50]. Our inferences are based on 10⁴ samples, thinned from 2×10⁶ iterations of a non-adaptive Markov chain with the first half of the chain discarded as burn-in.

We assess model adequacy with standardised residual plots. By overlaying plots for each mouse on one graph, the power of the plots to reveal poor fit is improved. We also plot the fits to the data with 95% posterior predictive intervals. These intervals predict where 95% of data would lie given the model being true and the posterior distribution. A poor fitting model would have significantly less than 95% of the data lying within this interval. These intervals are constructed by simulating the model using the parameter vectors from the Markov chain and generating replicate data sets with the known error structures and variances. At a given time point the 95% interval is taken to lie between the 2.5 and 97.5 percentiles of the distribution of replicated data.

To compare models we use Bayes factors [51]. We compare all models to the model consisting of hypotheses A1, B1, C1, D1 and E1. To calculate Bayes factors we need to calculate the marginal likelihood of a model, [52]. The Bayes factor of model 1 and model 2 is(25)It is more convenient to work in decibans (tenths of a power of 10)(26)A dB<5 suggests almost no difference between the models, a dB between 5 and 10 suggest substantial evidence in favour of model 1 and a dB>10 is strong evidence in favour of model 1 [51].