Iron metabolism can be described as a closed compartment system

The quasi-closed state of the iron system together with the ensuing internal steady-state makes into possible to simplify the non-linear structure to a system of ordinary linear differential equations. The dynamics of tracer motion depicts the statics of the underlying stationary flux-and-pool network. We could build on a number of previous attempts to model iron metabolism in this way [5–8, 38], reviews in [2, 38]. The novel aspect here is the detailed reversible balance in a network of peripheral tissues that were previously combined ad hoc to black boxes.

Iron metabolism is organized as temporal hierarchy on five time scales

The time hierarchy does not change appreciably between different iron statuses in normal mice (confirming the conclusion in [10]).

Iron turnover in the plasma compartment depends on the iron status

The concentration of transferrin-bound plasma iron in plasma is in the range of 100-200 μg/dL in the mouse (Additional file 1: Table S9). This is similar to other mammalian species (e.g. [12, 39–42]). The iron concentration tends to lower values in iron-deficient and to higher values in iron-loaded mice. The iron clearance from plasma defines a half-time of renewal in the range of 1-2 hours, again similar for species otherwise as different as Mus musculus and Homo sapiens. Rats [41] and dogs [5, 39] are also in the same range. In rats, however, iron deficiency does not lower the plasma concentration [41].

Iron distribution into body periphery is a three-level hierarchy of flux rates

The initial tracer concentration in plasma becomes rapidly cleared within a few hours after administration and stays at a low, but steady value afterwards. This coincides with the ascending tracer curves in the peripheral compartments (figures 2 and 3. The initial distribution is complete at the first time of measurement (12 h). The position of the maximum fixes the time point when plasma tracer is nearly washed out and the periphery begins to return some of the previously accumulated tracer iron into the plasma (figures 2 and 3. The continuous decrease of organ tracer content begins after 12-24 hours. It is an expression of the fact that "fresh" cellular iron is not only stored or channelled into biosynthesis, but also shows an appreciable back-flow into the plasma.

The descending branch of the peripheral tracer curves show that all tissues return the radioactivity into the plasma, unless they lose it by desquamation, which is the case for intestine and integument. This characteristic pattern proves that iron flux into the periphery and reflux into plasma take place simultaneously.

The quantitative level of all the superimposed fluxes can result only from a deeper analysis of the corresponding mathematical model. This analysis yields a set of fractional clearance parameters (table 1). From these values and by application of the steady-state assumption all iron fluxes can be estimated when the iron content of the central compartment is available. Data by Trinder et al. [10] contribute an estimate of the total plasma turnover clearance rate (Additional file 1: Table S9).

Three clusters of flux rates may be distinguished

Share of flux into tissues mirrors transferrin receptor expression

The clearance time of plasma iron is in the range of 1 h, largely independent of the plasma iron content and hence the state of the animal's iron supply. This linear kinetics suggests that the total population of TFR1 receptor molecules (responsible for most of the iron uptake) works far below its maximal capacity in all cells. The share of radioiron going into the body organs reflects this tissue-specific transferrin receptor expression. In contrast to the rather stable total clearance time the share of radioiron is dependent on the physiological state (Additional file 2: Figure S2). In the states of iron depletion and of normal iron supply more than two thirds of the plasma iron turnover is directed to the erythropoietic bone marrow and is rapidly incorporated into hemoglobin. This is, again, similar to other species [2, 13, 14, 31]. The corresponding fraction of tracer iron passes through the immature cells of the erythropoietic lineage until it reaches the erythrocyte compartment.

Tracer distribution iron-rich condition reflects the switch-over to the storage mode

The flux through the storage pathway into parenchymal organs increases from 25% to 49% of plasma turnover (table 1; Additional file 1: Table S8; visualized in Additional file 2: Figure S2 and S3). Stores are filled up in liver, kidney, spleen/RES (Additional file 1: Table S12), to a lesser extent also heart and skeletal muscle, but not integument and brain.

Tissue cells equilibrate influx and reflux of iron to maintain the iron pool

An adult mouse does not grow much during its life-time of ~2 years (if not killed before). Iron is taken up by cells with a time characteristic of a few days and must, therefore, be balanced by corresponding iron-release. Muscle, fat, heart, lungs, brain and testicles excrete iron into plasma or extravascular fluid. The influx of tracer is mediated by transferrin receptor. It is not clear from tracer data whether the export is mediated only by the ferroportin channel [11], or also via catabolism of heme-bound iron. Ferroportin is dominantly expressed in liver, duodenum, and macrophages, and to a lesser extent also in other tissues [43]. Ferroportin is not involved in the case of loss of whole cells (erythrocytes, intestine, and integument). The tracer data as used here cannot distinguish between export of iron and loss of whole cells. They yield only an estimate of the total flux out of the compartment.

Intracellular residence time of iron is longer than the life time of its protein "carriers"

The life-time of the iron-storage proteins (such as apoferritin/holoferritin) is in the range of one day in the liver [33]. Up to 4500 iron ions can be stored in one ferritin molecule [44], and become released on proteolytic ferritin degradation. The residence time of iron in the liver cell, however, is much longer-in the range of 1-2 weeks (Additional file 1: Table S11). This shows that iron released into the very small labile iron pool does not leave the cell, but is re-utilized. This slow export conforms to well-known data showing how slowly iron is mobilized from ferritin stores to replace iron losses, e.g. after phlebotomy (in men: [32, 45]). Intracellular iron stores are no inert long-term reserves, but are continuously turned over within the cell and may therefore be directed, in accordance with changing requirements, into the three competing pathways (biosynthesis, storage, export).

Readily accessible tissue iron pools are a fraction of the non-heme iron

These iron pools are stored in different subcompartments, mainly in non-heme form. The iron-loaded liver stores ferritin in the hepatocytes and a less mobilizable (hemosiderin?) form in the Kupffer cell [46–48]. The labelled and unlabelled iron data from whole organs do not permit to differentiate quantitatively between parenchymal and macrophage iron in such mixed cases. Tracer dynamics identifies iron pools that become quickly labelled. Their pool sizes have been estimated from the fractional plasma iron turnover and the tissue clearance rates (Additional file 1: Table S9 and table 1). Three groups may be distinguished. Red blood cells contain as hemoglobin the largest readily labelled iron pool (~ 300 μg Fe per mouse, about 50% of total haemoglobin-iron, see calculation in Additional file 1: Table S9). There is a second cluster of pools (integument, liver, bone marrow, skeletal muscles, skin), each containing about 20-40 μg Fe. In particular the hepatic iron pool is expandable in iron overload to reach a kinetic pool level of ~ 100 μg Fe, presumably in ferritin form. A still larger store can possibly accumulate on a longer time scale, which is not covered here. There are additional pools with an iron content (lungs, kidneys, intestine, heart, and spleen) of about 3 μg Fe each, which can moderately expand up to 4-14 μg (Additional file 2: Figure S4, lower left). Other organs, such as fat, testicles or brain, are not able to store more iron in overload. Additional file 1: Table S13 shows that in some tissues the readily accessible pools are only a fraction (6 to 40%) of cellular non-heme iron.

There are two kinetically distinct major iron pools in the mouse body

The total tracer-accessible iron amounts to ~400 μg (Additional file 1: Table S10). The residence time in the main compartments excluding intestine (Additional file 1: Table S11) is between 5 and 25 days. This comprises about 20% of the total iron (i.e. of ~2 mg per 25 g body [18]). The reminder is not readily accessible. The residence time of molecules here is ~200 days [18, 19].

Iron turnover occurs at similar rate in intestine and skin, but assignment to iron loss vs. iron reflux is only indirectly estimable

Physiologically iron enters the body via duodenal and (less) small-intestinal absorption in a tightly controlled way. It leaves the body by desquamation, exfoliation of epithelial cells, by blood losses, and to a lesser extent via bile and urine [49]. The net amount leaving the murine body is, according to literature references, 2 to 5 times larger than in other animals and man [18, 20, 40, 49–51]. A consequence of this higher excretion is that heavy iron-load is sometimes difficult to attain in mouse models.

Net iron losses cannot be measured by the tracer method as applied here. However, the fractional clearance rates (table 1) yield indirect information on iron fluxes through intestine and integument (Additional file 1: Table S8). Iron clearance of the epidermis integument is about 5% per day and that of the stomach-intestinal epithelium ca. 36% per day (calculated from table 1). From the fractional uptake from plasma one can calculate influx rates of ~1.7 μg per day into epidermis, and a sum of ~1.5 μg per day into intestine plus stomach (all for iron-adequate mouse, see Additional file 1: Table S8). These values are about 39% lower and higher in iron deficient and iron-load regimes, respectively. The data do not support a calculation of the rate of net iron loss through these compartments, because there may be a fraction that is recycled into plasma. The iron residence times for intestine are similar to the known exfoliation times of epithelium (3-5 days), which suggests that the main fraction goes into loss. For skin integument (iron residence about 40 days) such external information was not available.

Murine erythrocyte iron turnover has a random elimination component together with a lifespan-determined removal component

During one month after administration 60% of the tracer (40% in iron-loaded state) accumulates in the red blood cell compartment (figures 2, 3. The first quick uptake of ⁵⁹Fe reflects passage through bone marrow and incorporation into hemoglobin at a steady rate. The uptake reaches a saturation phase which is clearly visible in the RBC curve of figures 2 to 3. This behaviour proves the existence of a reflux caused by a random component of erythrocyte catabolism independent of the cell age. Without reflux iron would be further incorporated even at a very low plasma radioactivity. The erythron cycle transports (Additional file 1: Table S8) 15;19;14 μg Fe/d into bone marrow in iron-deficient, -adequate, and -loaded animals, respectively, of which 12;17;12 μg Fe/d pass through the RBC compartment back via into RES into plasma. This turnover rate is quantitatively analogous to ~ 25 mg Fe/d per 70 kg in iron-adequate humans.

The life span of mouse erythrocytes has been studied in mathematical detail by Horký et al. [52]. They also formulated an age-independent linear elimination component acting simultaneously with a lifespan-determined senescence process. Our elimination rate (between 0.03 and 0.06 d^-1) is somewhat higher than obtained in [52] (0.012d^-1). However, these estimates are not very reliable, as they stem from an indirect deduction. This applies also to the size of the "readily accessible" iron pool in red blood cells (300 μg instead of the 568 μg calculated from the hemoglobin pool of the mouse, see Additional file 1: Table S9).

The spleen is a mixed indicator of erythropoiesis and RES activity

The murine spleen is an erythropoietic organ [53]. Therefore, one subcompartment of iron in the spleen is expected to behave similar to iron in the bone marrow. Figures 2, 3, and Additional file 1: Table S5 to S7 show a similarly quick uptake phase in both bone marrow and spleen. The ratio of tracer iron content between both organs after 12 h is about 50 to 60 in adequate and iron-rich mice, and 20 in iron-deficient animals. Thus, the quantitative contribution of the spleen to total murine erythropoiesis is not high. Furthermore, the iron-deficient spleen loses iron as quickly as the bone marrow, reflecting the rapid flow into "iron-deficient" erythropoiesis. In contrast to the bone marrow, the iron-adequate, and even more so the iron-loaded spleen retains ⁵⁹Fe for long periods. This reflects a storage behaviour which is similar to that of RES cells in the liver and elsewhere. The spleen contains 5% and the liver 16% of the whole population of macrophages [54]. The RES system serves as scavenger to remove senescent erythrocytes together with their hemoglobin and colloidal iron from the circulation [8, 55]. Part of this RES iron is rapidly recirculated into plasma, thereby completing the iron-recirculation back to the erythrocyte pool. Except in iron deficiency, another part of the RES iron is stored as ferritin or hemosiderin [44].

The quantitative contribution of both spleen compartments to whole body iron turnover is low. The spleen is therefore an indicator, but not the main quantitative locus of the total erythropoietic and macrophage activity. In iron-deficiency splenic iron clearance is very rapid (15% d^-1, see table 1 and Additional file 2: Figure S1 and S2), while it is distinctly slower (down to 1.9% d^-1) in iron loaded mice. This may reflect distinct differences in the role of the spleen depending on the state of iron-repletion. A precise quantitative partition of splenic iron fluxes into a RES-and an erythron-fraction would require separation of the cells.

Experimental design for characterizing the iron status and the dynamic turnover of the C57BL6 mouse strain

The biochemical parameters yield a survey of the static of iron metabolism and its steady-state level. Ferrokinetics yields the fluxes. This full programme can be reduced, if in a particular situation preliminary analysis of data and their comparison with the mathematical model indicate that certain features of the iron status are not changed or are negligible.

Experimental design

The mathematical model presented here has been derived from our own experimental data [9]. In brief, male young adult mice (C57BL6 strain; 18-20 g) underwent a 5-week period (growth to 25 g) with a diet controlled for iron content (iron content of diet induction of deficiency: 6 mg/kg; for adequate supply: 180 mg/kg; for iron overload 25000 mg/kg). The experiment was started by intravenous administration of ionic radioactive tracer (Fe59 nitrate in complex with nitrolotriacetic acid) The single tracer dose contained about 0.285 μg per mouse, which is less than 2% of plasma iron and in the range of 0.01% of body iron. At certain intervals between 12 hours and 28 days animals (n = between 3 and 7) were sacrificed, blood was collected and their main organs were dissected, weighed, and their iron status (non-heme iron) was measured. Hematocrit and haemoglobin content of blood was measured, and aliquots were separated into plasma and red blood cell compartment. The weight of organs and tissue samples was measured and normalized to the whole body (25 g). The ambient Fe59 content of organs and blood compartments was measured by scintillation counting and also converted to a whole body value. The tissue contents of tracer were corrected by subtraction of the tracer in the residual blood, by a calculation scheme that was derived from parallel model experiments with Fe59-labeled erythrocytes as indicator [9].

All tracer data were corrected for the decay of radioactivity during the experiment by normalizing them with the help of the radioactivity of the injection solution measured at the same time.

Data concerning the total iron clearance rate could be taken from the literature [10], as they were obtained on control mice of the same strain under very similar dietary regimes.

General structure of iron metabolism in the mouse body

Mammalian organisms absorb iron from the intestinal tract, mainly via the duodenal epithelium, and loose it predominantly by exfoliation of intestinal epithelium, by desquamation of the skin, by occasional or repeated loss of blood, and to a lesser extent via excretion of a non-reabsorbed fraction of bile and urine. Duodenal absorption transfers iron into plasma where it is bound to transferrin. Transferrin-bound iron is distributed to peripheral tissues in accordance with their expression level of the transferrin receptor (TFR1). This stream into the periphery can be measured after injection of radioactive ⁵⁹Fe into plasma as rate of appearance of the tracer in the periphery. This intake of iron into cells is balanced by an out stream back into plasma of similar strength, mediated by iron export protein ferroportin [11].

This general scheme of iron distribution into the periphery and its reflux into plasma is adequately represented by a "mammillary" compartment system with reversible flux (figure 1). Transferrin-bound iron equilibrates quickly (< 1 day) between plasma and extravascular fluid [2, 12]. Together they are the central compartment. Other tissues form peripheral compartments. The erythropoetic compartment of bone marrow has a high expression level of the transporter TFR1 and rapidly integrates iron into hemoglobin [13, 14]. Both fluxes as well as the filtering-out of senescent blood cells are irreversible. In contrast, the expression level of ferroportin in bone marrow is low. This allows us to model the iron pathway from plasma over bone marrow, erythrocyte pool, and RES (emphasized by thick arrows in figure 1) as a circular irreversible flux without reflux. A smaller flux from bone marrow into the Reticulo Endothelial System (RES) has to be included. It represents partly spleen erythropoiesis in mice and partly "ineffective erythropoiesis" [15].

Some of the peripheral compartments have two outlets, one leading back into the central compartment and the other one leaving the system by loss of cells or excretion. The clearance of radioactive tracer can be estimated in these cases, but not the partition between these two outlets. Loss of tracer from the body is difficult to measure, and the plasma curve is not sufficiently sensitive to small variants of back-flux from those tissues. For parameter estimation an a priori decision had to be made about which of the double outflows is quantitatively the more important. In the intestine and integument, with the exception of the duodenum, iron losses from the body are assumed to be the main route. This reflects the rapid exfoliation of intestinal epithelium (around 4 days), as well as the slower, but of larger volume, desquamation of skin and integumental adnexes. In the duodenum the ⁵⁹Fe influx from plasma counters the physiological uptake of unlabelled iron from the lumen. In erythrocytes, liver, and kidney it was decided that the main pathway balancing iron uptake is reflux into plasma rather than loss out of the body.

The compartments in figure 1 represent organs and tissues. They are not kinetically homogenous, as every organ consists of cell types that may differ in their iron metabolism. Tracer content in such compartments and intercompartment fluxes represent therefore a weighted mean over different cell types. Mixture compartments of this type are liver, spleen and muscle.

Numerical Scales of Pools and Turnover Rates

The generic model of figure 1 has to be quantitatively specified. The assumed reference organism of all data in this paper is an adult mouse of 25 g body mass; all other references are transformed to this scale. Comparing relevant data from other species involves a scale factor of approximately 10 from mouse to rat and 2500 to 3000 from mouse to humans. Compartments are envisaged as iron aggregates ("pools") the iron content of which is a systemic variable, expressed in units of μg iron per animal. Fluxes into and out of a compartment are expressed in units of μg iron per animal per day.

The systemic structure of this model is specified by "content" data ("concentration" of iron in its various biochemical forms in the different compartments), and by "turnover" data, usually obtained from tracer studies. Both data sets must be scaled up to the whole organism. The model must have a mathematical structure that reflects statics and dynamics. Its parameters are to be estimated from the empirical data.

Mathematical structure of the model

where

C_i ( ≥ 0) - "pool size": iron content of the i-th compartment, i = 1...n (number of pools) - chosen scale is μg per body

v_ij ( ≥ 0) - rate of iron (in-) flux from compartment j to i (j = 1...n; j ≠ i) (μg per body per day)

v_ji ( ≥ 0) - rate of iron (out-)flux from compartment i to j (μg per body per day)

v_io ( ≥ 0) - rate of iron flux from outside the system into compartment i (μg per body per day); only influx into duodenum of "cold" iron has a value > 0

v_oi ( ≥ 0) - rate of iron flux from compartment i out of the system (μg per body per day; values > 0 only for intestine, stomach, integument).

Clearance mode of model description and derivation of motion equations

where a term k_io * x_o has been dropped because re-absorption of excreted radioactivity can be neglected, if the tracer has been applied to the plasma compartment. The k-values for non-existing fluxes (out of the system or non-reversible) are set to zero. The system of differential equations describing the tracer motion is written down in Additional file 3.

Residence time

Ferrokinetic experiments on mice

The phenomenology of iron metabolism is measured by injection of radioactive iron into the central compartment. In mathematical terms this introduces the boundary conditions for the system (equation 3). Since the uptake of iron via the natural way (guts) is slow compared to the inner exchange rates, the radioactive tracer (by mass being less than 2% of total plasma iron) has to be introduced at time t = 0 intravenously in the form of transferrin-bound iron or of ionic salt that equilibrates quickly with transferrin-bound iron. Mixing in the central (= intravascular) compartment can be assumed to be quick, and is described as a "boundary condition" by x_plasma (t = 0) = 100 (%). All other compartments have initial condition x(t = 0) = 0.

In iron-deficient mice the tracer content of the spleen dropped to zero after a short time and even reached apparent negative values. This is clearly an overcorrection for blood-related ⁵⁹Fe, as the spleen contains a large extravascular blood pool that cannot be removed completely by perfusion. The true tracer content of the splenic tissue proper was obviously close to zero at those time points (see Additional file 1: Table S2). Therefore, we set these small negative values equal to zero. This manipulation did not appreciably change the fractional clearance parameter of the iron-deficient spleen.

Contribution of organs and tissues to whole body mass

An important scale factor is the fractional contribution of each organ or tissue to the mass of the whole body. Additional file 1: Table S4 provides data for organ weights of C57BLC mice litter mates from our previous publication [9]. We include the estimates of plasma and blood volume of the whole body corrected for the bias of peripheral venous measurement [16].

Iron content in tissues and organs and its interval of fluctuation

which assumes that the measurements of organ mass and iron content are independent random variables.

Averaged tracer content in the intestine

As the precise organ weights of intestinal subsections, such as ileum/coecum/colon were not established, we replaced the tracer concentrations in Additional file 1: Table S1 to S3 by a mean ascribed to the intestine as a whole. The standard deviation of this derived quantity was calculated according to formula 5.

Normalization of tracer content of compartments to a common scale

where t is measured in days.

This describes the long-term rate of iron loss from the murine body [18, 19]. Bonnet et al. [20] give a somewhat lower rate of ~ 0.004 per day. The process of normalization smoothed the ups and downs of determined tracer contents. It indicates a loss of about 13% of the injected tracer over an experimental period of one month, which is in keeping with previous estimates (10-15%) [21].

Parameter estimation strategy

Model calculation and parameter estimation have recently undergone standardization and adaption to modern computing capacity [22–24]. We have applied a simplified simulation strategy, which was tailored to our needs. Incorporating actual data, the parameters of a model were point-estimated by optimization of fit-criteria. Such criteria measure the mathematical "distance", i.e. the squared difference between data (weighted according to precision) and a prediction obtained by simulation under a hypothesized parameter vector. The best-fitting parameter vector minimizes this distance. The "weighted non-linear least-squares"-method is an example for this strategy.

Fit Criteria

Data of fluctuating precision may be ranked by weight factors. We decided to use the inverse of the standard deviation of the measurements at a given time in the respective organs (called fval_chi_sqr in table 1). This led to better fits (estimated by inspection) than uniform weighting which overemphasizes high concentrations, or the use of the inverse of the variance which overemphasizes very low concentrations.

Quality of final fit

As an intuitive measure of the quality of the final fit we chose the root of mean of squared weighted deviation between prediction and the mean of measurements (table 1; "sq root of mean weighted squared dev").

Scatter interval of parameter estimates

Fluctuant data make the point estimates of parameters uncertain. We initially tried to estimate the related uncertainty by expanding a quadratic surface around the "optimal fit" and projecting its domain on the parameter scales. This is the usual maximum-likelihood method of estimation. However, since the real surface of the fit landscape was not approximately quadratic, and since the distribution of the data scatter did not harmonize with the assumptions of the theory, we transformed the scatter interval of measurements into a corresponding interval of realistic parameter estimates. This is a less assumption-laden strategy.

For this purpose we generated sets of new data by use of the Monte-Carlo resampling method. By randomization we obtained data that had the same mean and standard deviations as the actually measured data. The simulated data were subjected to the same parameter estimation and this yielded statistical information of the parameter uncertainty. We document the parameter set of the best fit together with an upper and lower bound obtained from a sextile-truncated sample of the empirically found parameter variation. In the case of a Gaussian distribution the interval thus defined would be twice the standard deviation.

Parameter identifiability

Some parameters or sets of parameters are not identifiable by measurement in a model of given design. For instance, if ferrokinetic measurements are available only for the time course of changes in plasma concentration of tracer, the total clearance rate can be adequately assessed, but not the distribution into the network of body compartments. In contrast, if the first measurement was obtained after most of the tracer has left the plasma compartment, the relative distribution to the various organs can be assessed, but the flux dynamics of this distribution cannot. Even if parameters seem to be identifiable in the Laplace domain [25], the error structure of the data will make the ensuing algebraic treatment ill-conditioned and will expand the range of parameter estimates. They become meaningless. We tackled this problem by studying the parameter space using the alternating conditional expectation algorithm (ACE) [26, 27].

The essence of this strategy is to explore the total domain with all parameter values that support an acceptable fit, i.e. the criterion value of this fit is sufficiently close to the optimum. This is done by running a large number of parameter estimations from a large range of starting points. The result is a set of sometimes widely differing parameter vectors that yield very similar optimal values of the fit criterion. Interdependent and hence non-identifiable parameter combinations can be addressed by the information furnished by the ACE method. On the basis of this analysis one or more suitable parameters are chosen to be fixed or held in definite proportion to each other so that the other parameters became identifiable. The choice of suitable parameters for this reduction of parameter redundancy and of their fixed values is not always easy and requires a theoretical understanding of the dynamic structure of the model. Maiwald et al. [28] have developed software ("Potter's wheel") the tools of which support this computer-time-intensive study very effectively. Specific assumptions made to remove strong parameter intercorrelation will be mentioned explicitly below.

Calculation of absolute flux rates

The iron pool of the plasma plus ECF is calculable from the iron concentration and the plasma volume plus the accessible volume of ECF

Estimation of peripheral pool size

If the data support estimates of the two rate constants (k_{i_plasma} defining the early phase and k_{plasma_i} characterizing the late phase of tracer distribution) and of the plasma/ECF iron content, then the size of the "accessible" pool C_i may be inferred.