Cell Kinetic Model.

An ordinary differential equations model of bone marrow erythropoiesis was developed in the past (see [6]–[9], [21]). We briefly sketch this model for self-consistancy. The model consists of concatenated cell compartments describing the dynamics of the following bone marrow cell stages of erythropoiesis: haematopoietic stem cells (S), burst forming units - erythroid (BE), colony forming units - erythroid (CE), proliferating erythroblasts (PEB), maturing erythroblasts (MEB), reticulocytes (RET) and mature erythrocytes (ERY). Differentiation of cells is modelled by fluxes from one compartment to the next stage. Proliferation is modelled by amplification in the corresponding compartment (see [21]). Compartment sizes are modelled by balance equations of the formwhere is the amplification, the transit time, is the influx from the previous compartment, and the content of the compartment. The stem cell compartment has a different structure: Here, the percentage of self-renewing cell divisions and the percentage of proliferating cells is regulated by the bone marrow cell compartments (details see section A.2 in file S1). The stem cell compartment is identical to those used in other lineage models of our group [5], [11], [12].

Maturation is modelled by a set of concatenated first order transitions of the formwhere is the content of the -th subcompartment of , is the influx from the previous compartment and is the delay parameter with delay time . In [11] it has been shown that this results in Gamma-distributed delay times with expectation and variance . Later, this principle is not only applied to model maturation but also to delay chemotherapy action and (delayed) absorption of EPO injections.

The differential equations system is regulated by several feedback loops. Loop 1 and 2 are feedbacks regarding the stem cell self-renewal and proliferation [9], [22]. Loop 3 is mediated by the cytokine EPO which increases amplification () or shortens maturation time () in all bone marrow compartments except for the stem cells [7]. Endogenous EPO production is regulated by oxygen saturation. For a brief explanation of the model of endogenous EPO regulation see the file S1.

Quantities that depend on EPO concentration are typically regulated by so called Z-functions. The Z-function is a monotone function with minimum and maximum . It reads as follows(1)(2)where is the steady state value, is the sensitivity of under stimulation and is the EPO concentration relative to steady-state (See also [9], p. 69.)

A further model assumption is that in steady state most erythrocytes die dependent on age (i.e. similar to a first-in-first-out kinetic), but under stimulation, erythrocyte degradation occurs more randomly (i.e. exponential decay) [21]. Finally, iron metabolism is not explicitly modelled and it is assumed that iron supply is not a limiting factor. A complete set of model equations and parameters can be found in the file S1.

This model of bone marrow erythropoiesis was established on the basis of several experimental results for example on irradiation, bleeding, hypoxia and hypertransfusion in mice [9]. But so far, neither applications of EPO nor chemotherapy were considered. Due to its successes in explaining the above mentioned scenarios, we adopt the assumptions and equations of this cell kinetic model. They serve as a backbone of our comprehensive model in the sense that all other parts, i.e. pharmacokinetics and -dynamics of EPO and chemotherapy injections, are attached to this structure.

Pharmacokinetic Model of Erythropoietin.

A pharmacokinetic model of EPO was developed by Krzyzanski & Wyska [13] to explain dynamics of EPO serum concentrations in healthy volunteers after intravenous injections. Neither chemotherapy patients nor subcutaneous applications of different EPO derivatives were considered there. In brief, the model consists of three compartments: EPO in central serum (), EPO protein binding (peripheral compartment () and EPO bound to receptors (). EPO is eliminated at the rate from the central compartment. There is a first order exchange between the peripheral and the central compartment with the rates and respectively. EPO in central compartment can bind to free EPO receptors at the rate forming the drug-receptor complex . The complex dissociates with rate . Alternatively, EPO is internalised with rate . The parameter describes the receptor synthesis rate and is the degradation rate for EPO receptors [13].

Coupling this pharmacokinetic model with our cell kinetic model is achieved by three mechanisms:

Model assumption 1: There is an additional influx of endogenously produced EPO into the central compartment. Endogenous EPO is produced in accordance to the cell kinetic model.

Model assumption 2: The production of EPO receptors depends on the bone marrow cellularity, i.e. the compartments BE, CE, PEB, MEB, RET of the cell kinetic model.

Model assumption 3: Internalised EPO serves as the argument of all Z-functions of the cell kinetic model representing EPO regulations.

All three assumptions are natural extensions of the model. A schematic illustration is shown in figure 2. The first assumption translates in the following (adapted) equation of the central compartment:(3)where is a function of external EPO applications specified later. The function of endogenous EPO production is defined as follows:(4)where equals one in steady state, so that equation 3 equals zero. depends on the oxygen partial pressure and the number of circulating red blood cells (details see section A.2 in the file S1 or [7]–[9]).

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Figure 2. Pharmacokinetic model of EPO.

Boxes represent compartments. Arrows represent actions or flows. The pharmacokinetic model of erythropoietin was adapted from [13].

https://doi.org/10.1371/journal.pone.0065630.g002

Dynamics of the peripheral compartment and the EPO-receptor complex are described by the following equations [13]:

The specific elimination of EPO depends on the number of EPO receptors. According to assumption 2, we assume that the dynamics of EPO-receptors can be calculated on the basis of the bone marrow content of the cell kinetic model, namely the compartments , , , , and . The different receptor densities of these cell states are modelled by weighting factors , , , , and . The highest weighting factor is applied for CFU-E due to the highest number of EPO receptors [23]. We further assume that the density of EPO receptors declines with further maturation. Hence and . Mature erythrocytes are assumed to show no specific binding of erythropoietin [23], [24]. Hence(5)where is the number of EPO receptors relative to steady-state.

We now derive the initial values of the differential equations. At first, we set.where IU/l is the basic level of endogenous EPO, and denotes the distribution volume of EPO [13]. In consequence, steady-state concentration of EPO in central compartment is equal to the observed (mean) serum concentration.

Initial values of the other equations are also derived from steady-state conditions:

According to [13], we set which implies that the parameter is not free because from equation 5 it follows that .

The coupling of the pharmacokinetic model and the cell kinetic model is completed by using the relative internalised EPO () as argument of the Z-functions (equation 2) of all parameters regulated by EPO, namely amplifications () and transition times ()

Model of EPO Injections.

External EPO applications are modelled by an additional influx into the central compartment of the EPO pharmacokinetic model (equation 3). Different modes of EPO application were tested in clinical practice resulting in different absorption kinetics. While intravenous injections can be simply modelled by pulse functions (see equation 6 below), subcutaneous injections are characterised by delayed and incomplete absorption.

Kota et al. [14] proposed a model of Darbepoetin absorption in sheeps explaining absorption kinetics of subcutaneous EPO applications at different sites. Different models were considered for different sites of application. We adapted model proposal B of Kota et al. for our purposes. EPO absorption is determined by two processes, directly to the bloodstream or indirectly via the lymphatic system. Both processes are delayed. There is also a loss of EPO at the injection site and in the lymphatic system. In comparison to model proposal B of Kota et al. we made the following simplifications (see also figure 3):

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Figure 3. Structure of the EPO injection model with two ways of EPO absorption.

The model is in analogy to [14]. Boxes denote the compartments, arrows represent EPO fluxes between the compartments.

https://doi.org/10.1371/journal.pone.0065630.g003

Model assumption 4: First order absorptions of EPO into central compartment or lymphatic system are neglected, i.e. we only assume delayed absorptions.

Model assumption 5: Different EPO derivatives and injection sites are modelled by the same equations but different parameters.

This results in the following equations. The general EPO injection function can be written as a sum of pulse functions(6)is the Heaviside-function, are time points at which EPO at dose is administered. The duration of injection () is set to five minutes. The applied dose is normalised to this injection time by the factor . is given by the administered dose in IU/kg. EPO dynamics at the injection site (subcutaneous tissue) are then described by:(7)where is the efflux constant towards the central compartment, is the efflux constant towards the lymphatic system and represents a first order loss term of EPO resulting in a reduced bioavailability. The direct efflux from subcutaneous compartment into peripheral blood is delayed by four delay compartments, i.e.(8)with the settings

and

finally enters the central EPO compartment of the PK model (equation 3, see also equation 12). Analogously, the lymphatic absorption is modelled by a delay function with four compartments:(9)with the settings

and

where enters the central lymph compartment. Thus, EPO dynamics in the lymphatic compartment are given by

(10)(11)where is the absorption constant of the transition between the lymphatic and the central compartment. The constant represents another loss term resulting in further reduction of the bioavailability via this route of absorption. Summarising the effluxes of the direct and the lymphatic way of absorption yields(12)for subcutaneously administered EPO and(13)for intravenous injections. Due to differences in the lymphatic flow at different anatomical regions [14], [25], [26], one cannot assume that the same parameters of the injection model are valid for all injection sites considered. In contrast to [14] we do not assume differences in the structure of the model in dependence on the injection site. Rather than this, EPO derivatives (Alfa, Beta, Delta, Darbepoetin Alfa) and injection sites used in our clinical data sets were modelled by different parameter sets of the same injection model (assumption 5). For this purpose, we split available injection modi into 9 groups for which different parameter sets are assumed (see later).

Chemotherapy Model.

We recently proposed a model of cytotoxic chemotherapy action on the granulopoietic system [5], [11], [12]. This concept is translated to our erythropoiesis model. It is based on the following assumptions:

• Chemotherapy results in a reversible and transient depletion of the compartments S, BE, CE, PEB, MEB, and RET. None of the other compartments or model parameters are affected by chemotherapy applications [11].
• The damaging effect is delayed, modelled by four subcompartments in analogy to equations 8, 9 [51]. This implies that the damaging effect is zero at time of injection and maximal at a later time point which can be specified by the delay parameter.
• The toxic effect is specific for the cell stages and for different drugs or drug combinations. It is quantified by corresponding sets of toxicity parameters [2].
• The toxicity is higher for the first administration of a chemotherapeutic drug (first cycle effect) which is modelled by a factor multiplied to all toxicity parameters for the first therapy cycle [11].
• Different cytotoxic drugs damage independently of each other [11]. This allows adding toxicity functions of chemotherapeutic drugs applied in combinations.

These assumptions are extensively discussed in [5], [11], [12]. We now present the corresponding equations. The infusion of chemotherapeutic drugs is again modelled by Heaviside functions(14)where is the number of chemotherapy cycles and is the duration of chemotherapy application. The delayed effect of the drugs is again modelled by four compartments(15)with

The output-function is multiplied by the toxicity parameters of the single compartments , , , , , and respectively. Since both, our former model of granulopoiesis and the present model of erythropoiesis use the same stem cell compartment, the parameters , , and were taken from the granulopoiesis model described in [5], for consistency. All other toxicity parameters are unknown and were estimated from clinical data (see below). If multiple cytotoxic drugs are applied, corresponding toxicity functions are added resulting in an overall toxic effect which is cell stage and chemotherapy specific. In complete analogy to our granulopoiesis model, the effect of chemotherapy is included into the balance equations of the bone marrow cell compartments by a first-order loss term. Hence, the schematic compartment equation 1 has the form

Numerical methods for simulation.

Implementation and simulation of the model were performed using MATLAB 7.5.0.342 (R2007b) with SIMULINK toolbox (The MathWorks Inc., Natick, MA, USA). Numerical integrations of the equation system were performed using the variable step solver from Adams and Bashford included in the SIMULINK toolbox (ode113).

Estimation of parameters.

Data on bone marrow erythropoiesis are hardly available for humans. Consequently, one can find only a few hints regarding values of certain model parameters in the literature. The majority of model parameters must be estimated indirectly via fitting the predictions of the model to clinical data after various therapeutic interventions. This applies for parameters of all submodels: the cell kinetic model, the PK model, the injection model and the chemotherapy model.

Even if parameter estimates of the submodels were provided in general, it was necessary to adapt these parameters in view of the greater data base to be explained with our model. For example, the injection model was developed for sheeps and must be re-parametrised for humans. The original PK model [13] was developed for intravenous EPO-Alfa injections but not for (subcutaneous) injections of other EPO derivatives. The cell kinetic model never considered external EPO applications so far. This implies that new pharmacodynamic parameters must be estimated.

The effects of chemotherapy on this cell kinetic model were also not studied in the former publications. Modelling different chemotherapy regimens is of particular concern since it requires estimation of corresponding toxicity parameters. In general, these parameters describe the toxicity of a drug combination, since cytotoxic drugs are usually applied as polychemotherapy including several drugs at the same time. However, by comparing different chemotherapies, it is sometimes possible to estimate the toxicity of single drugs.

To reduce the number of parameters to be fitted, we used the following approach:

• We made simple model assumptions to reduce the complexity of equations, and with it, the number of unknown parameters.
• Parameters for which experimental data are available were taken from the literature, either directly or by defining plausible ranges.
• Parameter values of the cell kinetic model and the chemotherapy model which were determined by previous modelling works were kept constant. This applies for parameters of the stem cell regulation and chemotherapy toxicity.
• Different EPO derivatives and injection sites were grouped according to expected similarities regarding absorption kinetics. Parameters of the injection model were assumed to be constant within these groups but may differ between groups. Nine such groups were identified (see below). PK parameters were assumed to be constant for endogenous EPO and the derivatives Alfa, Beta, Delta but are assumed to be different for Darbepoetin.

Parameters of our model were determined by a stepwise fitting procedure. First, pharmacokinetic and -dynamic parameters (PD) of EPO and parameters of the cell kinetic model were estimated by comparing the predictions of the model with available time series data after application of EPO into healthy volunteers. Then, data of different chemotherapy regimens are used to identify corresponding toxicity parameters. Parameters are kept constant in the following fitting process. Fitting is achieved by optimising the agreement of model and data via the condition.(16)where is the solution of the differential equation system at time for the parameter set . Here, describe the range of time points for which data are available. The function represents the interpolated curve of patients medians for a given data set. We used the logarithm of data due to log-normally distributed cell counts and cytokine concentrations. The left hand side of equation (16) is referred as the fitness function in the following. The simultaneous fit of more than one data set is realised by adding corresponding fitness functions.

We used (1+3)-evolutionary-strategies with self-adapting mutation step size to find optimal parameter sets as good as possible [27], [28]. Evolutionary strategies are non-deterministic optimisation algorithms adopting principles of evolution, namely mutation, realisation and survival of the fittest. (1+3) is an evolution strategy with one possibly immortal parent having three children in each generation (see [27], [28] for further details). The weighting factors of equation 5 were fitted with the restriction , (see [23]). For the toxicity parameters we claim that higher doses correspond to the same or larger toxicity parameters. These restrictions are fulfilled by rejecting corresponding violations during the fitting process.

A complete parameter list for our model is provided in the file S1. We also performed an extensive sensitivity analysis in order to determine the identifiability of all model parameters. This is achieved by estimating the deterioration in fitness after changing the parameters. Results can also be found in the file S1.

Data Sets.

Available data sets comprised time course data of haemoglobin, haematocrit, reticulocyte counts, percentage of reticulocytes, red blood cell counts or serum concentration of EPO after single or multiple applications of four different EPO derivatives (Alfa, Beta, Delta, Darbepoetin Alfa) to healthy volunteers at a large variety of injection sites. These data were taken from the literature. Automated tools were used to extract the data from the figures as precisely as possible (software “ycasd”, developed by A. Gross, publication in progress [29]). Additionally, time series data of haemoglobin in patients treated with chemotherapy and with or without EPO support were available. A total of 12 chemotherapies were considered. For most of them we have access to raw patient data due to collaborations with the German High Grade Non-Hodgkin's-Lymphoma Study Group and the German Hodgkin's Lymphoma Study Group. For our purposes, we censored patient data at the time of first erythrocyte transfusion. Patient data of the same chemotherapy arm, cycle number and cycle day were pooled in order to obtain median time courses under the different therapies. Data sets of patients with anaemia due to kidney failure were intentionally not considered so far since this clinical situation requires other model assumptions. An overview of all available data sets is presented in table 1.

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Table 1. Available data sets.

https://doi.org/10.1371/journal.pone.0065630.t001

Not all of these scenarios were utilised for parameter fitting. A few scenarios were kept in reserve for model validations (see results). Since our model describes the dynamics of reticulocytes and erythrocytes, a set of transformation formulas are required to compare model results with available dynamics of haemoglobin (HB) or haematocrit (HCT). Baseline HB values showed a large variance between the heterogeneous populations in the data sets. Since relative compartment sizes were modelled, comparison with data can be achieved by normalising values to baseline. We assume proportionality of HB and the numbers of reticulocytes and erythrocytes. Thus,(17)

Analogously, we use the baseline RBC as normalisation factor and calculate the changes in this cell count by(18)

Percentages of reticulocytes and the haematocrit were determined byHere, is the amount of plasma which is assumed to be constant for all scenarios. Most of our data sets comprise time series of either percentage of reticulocytes (14 scenarios) or HB (22 scenarios).