METHODS AND APPARATUSES FOR PREDICTING THE EFFECTS OF ERYTHROPOIESIS STIMULATING AGENTS, AND FOR DETERMINING A DOSE TO BE ADMINISTERED

Abstract

The present invention relates to a method for predicting the concentration or the mass of hemoglobin or an approximation thereof, respectively, in a body fluid and/or an extracorporeal sample thereof of a patient at a later, second point of time, the patient having theoretically or in reality been administered a certain dose of an erythropoiesis stimulating agent at an earlier, first point of time. It relates further to a method for determining the dose of an erythropoiesis stimulating agent to be administered to a patient, to a method for determining whether a patient is affected by circumstances leading to the loss of hemoglobin, to corresponding devices and to an erythropoiesis stimulating medicament for use in the treatment of anemia.

Claims

1. (canceled)

2. A method for treating a patient with partial or zero renal function, the method comprising: administering a first dose of an erythropoiesis stimulating agent to the patient; measuring a hemoglobin concentration value of the patient via at least one sensor at a first time; measuring a bioimpedance of the patient at the first time; determining a hydration status of the patient based on the measured bioimpedance of the patient; correcting, by a programmable computer system and based on the hydration status, the hemoglobin concentration value measured via the at least one sensor for an overhydration of the patient to yield a corrected hemoglobin concentration value at the first time; detecting, by the programmable computer system and based on the corrected hemoglobin concentration value, a functional iron deficiency of the patient; calculating, by the programmable computer system and based on: (i) the corrected hemoglobin concentration value and (ii) a factor describing the detected functional iron deficiency of the patient, a second dose of the erythropoiesis stimulating agent that differs from the first dose and that will cause the patient to achieve a future hemoglobin concentration value at a target hemoglobin concentration value or in a target range of hemoglobin concentration values; and administering the second dose of the erythropoiesis stimulating agent to the patient.

3. The method of claim 2, wherein the measuring the hemoglobin concentration value comprises measuring the hemoglobin concentration value of an extracorporeal sample of a body fluid of the patient.

4. The method of claim 3, wherein the body fluid comprises blood.

5. The method of claim 3, wherein the body fluid comprises urine.

6. The method of claim 3, further comprising measuring a mass of hemoglobin in the body fluid of the patient at the first time.

7. The method of claim 2, wherein the erythropoiesis stimulating agent comprises erythropoietin.

8. The method of claim 2, wherein the erythropoiesis stimulating agent comprises iron.

9. The method of claim 2, wherein a target or target range for assessing a hemoglobin mass value or the hemoglobin concentration value is determined based on the hemoglobin mass value or the corrected hemoglobin concentration value.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0105] FIG. 1 shows a first apparatus comprising a controller for carrying out the method according to the present invention;

[0106] FIG. 2 shows a second apparatus comprising a controller for carrying out the method according to the present invention;

[0107] FIG. 3 shows the concept of two reference ranges;

[0108] FIG. 4 shows the concept of Hb regulation in health;

[0109] FIG. 5 shows the regulation of Hb with partial renal function (RF) or zero renal function;

[0110] FIG. 6 represents the fractional mass per gram of erythrocytes over the erythrocytes lifetime in days;

[0111] FIG. 7 shows the accumulation of Hb mass due to each new pulse of erythrocytes generated daily;

[0112] FIG. 8 represents the mass of Hb in circulation using an array of FIFO buffers;

[0113] FIG. 9 illustrating the erythroblast conversion over the transferrin saturation;

[0114] FIG. 10 reveals a pro-erythroblast transfer function;

[0115] FIG. 11 shows an expanded section of EPO to pro-erythroblast transfer function of FIG. 10;

[0116] FIG. 12 shows the gain of EPO as a function of the renal function;

[0117] FIG. 13 shows the derivation of the concentration of EPO appearing in the vascular space in a scheme;

[0118] FIG. 14 shows a scheme for intravenous infusion of EPO; and

[0119] FIG. 15 shows the renal function over the renal creatinin clearance.

DETAILED DESCRIPTION

[0120] FIG. 1 shows an apparatus 9 comprising a controller 11 for carrying out the method according to the present invention. The apparatus 9 is connected to an external database 13 comprising the results of measurements and the data needed for the method according to the present invention. The database 13 can also be an internal means. The apparatus 9 may optionally have means 14 for inputting data into the controller 11 or into the apparatus 9. Such data may be information about the mass, the volume, the concentration of Hb as is set forth above. Such data input into the apparatus 9 mayadditionally or insteadalso be information about the overhydration of the patient or an approximation thereof. The results of the prediction, evaluation, calculation, comparison, assessment etc. performed by the controller 11 and/or the apparatus 9 can be displayed on the monitor 15 or plotted by means of anot displayed but optionally also encompassedplotter or stored by means of the database 13 or any other storage means. The database 13 can also comprise a computer program initiating the method according to the present invention when executed.

[0121] In particular, the controller 11 can be configured for carrying out any method according to the present invention.

[0122] As can be seen from FIG. 2, for corresponding measurements, the apparatus 9 can be connected (by means of wires or wireless) with a bioimpedance measurement means 17 as one example of a means for measuring or calculating the hydration state or an overhydration state. Generally, the means for measuring or calculating the hydration state or an overhydration state can be provided in addition to the external database 13 comprising the results of measurements and the data needed for the method according to the present invention, or in place of the external database 13 (that is, as an substitute).

[0123] The bioimpedance measurement means 17 can be capable of automatically compensating for influences on the impedance data like contact resistances.

[0124] An example for such a bioimpedance measurement means 17 is a device from Xitron Technologies, distributed under the trademark Hydra that is further described in WO 92/19153, the disclosure of which is hereby explicitly incorporated in the present application by reference.

[0125] The bioimpedance measurement means 17 may comprise various electrodes. In FIG. 2, only two electrodes 17a and 17b are shown which are attached to the bioimpedance measurement means 17. Additional electrodes are, of course, also contemplated.

[0126] Each electrode implied can comprise two or more (sub-)electrodes in turn. Electrodes can comprise a current injection (sub-)electrode and a voltage measurement (sub-) electrode. That is, the electrodes 17a and 17b shown in FIG. 2 can comprise two injection electrodes and two voltage measurement electrodes (i.e., four electrodes in total).

[0127] Generally spoken, the means for measuring or calculating the hydration state or an overhydration state can be provided by means of weighing means, a keyboard, a touch screen etc. for inputting the required data, sensors, interconnections or communication links with a lab, any other input means, etc.

[0128] Similarly, the apparatus 9 may have means 19 for measuring or calculating means for obtaining a value reflecting the mass, the volume or the concentration of the substance that can again be provided in addition to the external database 13 already comprising the results of measurements and the data needed for the method according to the present invention, or in place of the external database 13 (that is, as an substitute).

[0129] The means 19 can be provided as a weighing means, a keyboard, touch screen etc. for inputting the required data, sensors, interconnections or communication links with a lab, a Hb concentration probe, any other input means, etc.

[0130] Again, it is noted that the figures relate examples showing how one embodiment according to the present invention may be carried out. They are not to be understood as limiting.

[0131] Also, the embodiments according to the present invention may comprise one or more features as set forth below which may be combined with any feature disclosed somewhere else in the present specification wherever such combination is technically possible.

[0132] In the following, three different ways to achieve the target of the present invention are described in detail. They should, however, not be understood as intended to limit the present invention in any way.

[0133] A first embodiment of the present invention is called the straight forward approach and will be explained in the following without making reference to any figure. The straight forward approach comprises or consists of the following steps, findings or considerations:

[0134] In the straight forward approach, if one or more pre-measured (i. e., measured before carrying out the method according to the present invention) concentration values or mass values of hemoglobin (short also: Hb) are found to be below the target range suggested by guidelines or requested by the physician in charge for a particular patient based on prior art knowledge, the prescription of the patient's dose of ESA (or EPO, in particular) is stepwise increased, particularly in form of a mental or academic act, e. g., in form of hints regarding to the prescription of EPO.

[0135] Also, Hb concentration or mass values obtained by frequent measurements (in certain embodiments these are values that had been obtained before and/or after every dialysis treatment, that is, not as part of the present method) are considered in the present straight forward approach.

[0136] Further, values reflecting the patient's fluid overloadwhich in particular have been obtained from measuring on a irregular or regular basis prior to carrying out the present methodare considered. The values reflecting the patient's fluid overload may have been obtained at the moment in which also the Hb concentration or mass values have been obtained.

[0137] In another step, some or all of the measured Hb values are computationally or mathematically corrected for the fluid overload to obtain the normohydrated Hb levels.

[0138] Also, in yet another step, measures are requested or contemplated at least for correcting the fluid status as another mental or academic stepit may at least be intended to correct the fluid status to levels of, e. g., TAFO (time averaged fluid overload, see also below)=0.3-0.5 litre (L). It is noted that a step of correcting the fluid status by therapy is in certain embodiments of the present invention not part of the method according to the present invention, whereas in others it may be. A correction of the fluid status may, of course, be contemplated or take place independently from the method described in here. TAFO, the time averaged fluid overload, means the time averaged fluid overload indicating the average fluid status over one weekthus compensating for the peak overhydration before the haemodialysis session and the possible dehydration after the haemodialysis session. The TAFO can also be used when haemodialysis and peritoneal dialysis patients are comparedthe fluid status measured in peritoneal dialysis patients is very close to the TAFO. Studies have shown that a TAFO of 0.3-0.5 L could be achieved.

[0139] A highly suitable time for carrying out this correction may in certain embodiments be in the order of the time lag between EPO dose change and expected start of change in the Hb mass. Usually, this time lag is about 20 days. Hence, in some embodiments, the correction is carried out about 20 days after the last administration of exogenous EPO. EPO has to be understood as one example of a erythropoiesis stimulating agent only.

[0140] A curve (a line or a 1st order function) is in some embodiments fitted in a suitable diagram (e.g., normohydrated Hb over time) through the normohydrated Hb values and extrapolated.

[0141] In any way, the straight forward approach may finally comprise a check if and when this extrapolation will meet the prior art Hb target or target range. The slope of the Hb rise will depend on the EPO concentration. The life time of the erythrocytes will determine which steady state Hb level will be reached. In this respect, it is noted that certain diseases or physical states can influence the red blood cell (RBC) lifetime in a negative waye.g., acute inflammatory situations can decrease the RBC lifetime significantly, as is stated in literature. In generally healthy subjects, the average lifetime of RBC can be assumed to be constant around 120 days. Chronic kidney disease (CKD) patients might however have shorter RBC lifetimes.

[0142] One of the advantages provided by the straight forward approach according to the present invention is that is does not necessarily need a sophisticated model accounting for the EPO or Hb kinetics. Hence, this approach is easy to handle and does neither need particular effort nor computing resources.

[0143] A device intended for carrying out the method described above with regard to the straight forward approach will comprise means for at least each step to be carried out.

[0144] With respect to FIG. 3, a second embodiment of the present invention, called a simplified Hb control based on normohydrated Hb, will be explained in the following.

[0145] A major feature of this second embodiment is defining a target range based on the normohydrated Hb. In the following, this target range is called a second target range irrespective of whether there is also a first target range or not so as to emphasize that this second target range may be different from a commonly observedand therefore referred to as firstHb target range proposed by, e.g., guidelines.

[0146] The second target range of the normohydrated Hb will in most cases be higher than (first) guideline values for Hb measuredthe guidelines refer to a typical predialytic situation to set up the target, a situation in which the patient is ca. 2 L fluid overloaded.

[0147] In the second embodiment or model, the Hb concentration is measured on a frequent basis. The fluid status is also measured on a frequent basis. In certain embodiments, frequent refers to, e. g., once weekly or even at every treatmentin acute patients the measurement frequency could be several times per day.

[0148] In accordance with the present invention, the control of Hb is based on the second reference range (Hb normohydrated).

[0149] The second reference range could also allow comparing different dialysis centres with different fluid stati for reasons of quality controls. Currently, it is difficult to compare the Hb concentrations between treatment centersmost centers have currently different policies towards the fluid management, and the measured Hb concentrations (found before or after dialysis) will be influenced by this effect. If, however, normohydrated Hb values are compared between the centers, the fluid effect is already compensated; therefore, the comparison is much easier.

[0150] FIG. 3 reveals the concept of having two reference rangesone for the normohydrated Hb, one for the Hb concentration range as suggested by presently observed guidelines, or to have even only one reference range, the one for the normohydrated Hb.

[0151] As can be seen from FIG. 3, which shows the development of the concentration of Hb [g/dl] of a particular dialysis patient over time t [in days (d)]. A measured Hb curve 31 (solid curve) depicts the measured Hb concentration over time. A normohydrated Hb curve 33 (bold curve) depicts the measured Hb concentration after having the Hb concentration values corrected for fluid overload (therefore, curve 33 is called normohydrated).

[0152] FIG. 3 further reveals a first target range 35, also called the target range for Hb as measured. It may be set according to guidelines presently considered. In FIG. 3, the first target range 35 is only shown for illustrating the differences and advantages of the model according to the present embodiment of the present invention when compared with the prior art methods. FIG. 3 reveals also a second target range 37 as proposed by means of the present invention for the corrected Hb concentration or the normohydrated Hb, respectively, see curve 33.

[0153] As can be seen from FIG. 3, a simplified model prediction, expressed by a dotted model prediction curve 39 is assessed by means of second target range 37 provided for the predicted Hb concentration values.

[0154] In FIG. 3, the increased dose 40 of ESA, applied on day 0 has lead to an increase both of measured Hb curve 31 and normohydrated Hb curve 33.

[0155] As can further be seen from FIG. 3, the corrected or normohydrated Hb curve 33 is much smoother when compared to measured Hb curve 31. Hence, the temptation for the physician to amend the ESA dose without need due to given fluctuations of the Hb concentration such as to be seen at reference numeral 41 depicting a maximum Hb value measured on day 240 is much lower when the normohydrated Hb curve 33 and its position within the second target range is considered instead of measured Hb curve 31 and its corresponding first target range 35.

[0156] As can further be seen from FIG. 3, also due to the smoother characteristic of normohydrated Hb curve 33, when compared to measured Hb curve 31, events such as the occurrence of blood loss due to, e. g., bleeding, the onset of which is depicted at reference numeral 43 on day 320, are earlier discovered as is readily understood by means of FIG. 3.

[0157] In FIG. 3, arrow 45 shows a difference between the model prediction curve 39 and the measured normohydrated Hb curve 33. As was explained above, in the particular example of FIG. 3, this difference or deviation finds its reason in an internal bleeding.

[0158] A third embodiment of the present invention is called by the inventors a Hb kinetic model and will be explained in the following with respect to the FIGS. 4 to 14.

[0159] The general idea of the model may be described as follows:

[0160] In health, receptors that monitor oxygen delivery control the secretion of erythropoietin (also referred to as EPO being an erythrocyte stimulating agent, ESA) that regulates the production of pro-erythroblasts from colony forming units. Depending on the level of available iron (e. g., measured with the TSAT level) the content of heme or hemoglobin in the red blood cells (RBC) is influenced. TSAT has no impact on the number of RBC; rather, it is needed to fully equip the RBC with hemoglobin. Low TSAT and high ferritin levels are, e. g., a marker of possible iron deficiency. The supply and destruction of erythrocytes ultimately determines the Hb mass that is maintained in a subject. Normally, when Hb levels decrease this causes an increase in EPO concentration to stimulate more production of erythrocytes.

[0161] FIG. 4 shows the concept of Hb regulation in health. The underlying model may be explained as follows:

[0162] As can be seen from the operator depicted at reference numeral 51, the Hb regulation in health is controlled also by means of a feedback loop 53.

[0163] The production of erythrocytes is stimulated from a baseline endogenous EPO production 55 only. The total intravasal or intravascular, respectively, EPO concentration (or the part hereof that can be measured from, e. g., venous blood samples) is indicated by reference numeral 57. The EPO concentration is influenced by EPO kinetics, indicated by reference numeral 58.

[0164] Depending on a function f1 describing the production of pro-erythroblasts from a given EPO concentration, pro-erythroblasts 59 are produced/generated in the model in question.

[0165] A given saturation 61 of transferrin (TSAT) in the body has a certain influence via another function f2 on the generation rate G*.sub.Hb(t) of hemoglobin 64 produced. TSAT will not influence the number of Hb containing cells (HCT) but the heme (hemoglobin) content of these cells. This effect is modeled by a quality indicator Q, which is dependent on TSAT.

[0166] A finally resulting Hb mass, indicated as M.sub.Hb(t) is also influenced by erythrocyte kinetics 65. The ratio u/v allows calculating the corresponding Hb concentration Hb(t) and vice versa. Vp is calculated because the EPO is distributed in the plasma volume. Vp is influenced by the fluid status (hydration)the rate of change of Vp is thus relevant for the EPO dosinge. g., if Vp goes down, the concentration of EPO will go upsee also equation 34 and FIG. 5.

[0167] A number of the items referred to with regard to FIG. 4 is explained below in more detail.

[0168] In contrast, in case of disease there may be only partial renal function and some of the ability to regulate Hb physiologically is lost (K.sub.EPO is reduced), see FIG. 5 showing the regulation of Hb with partial renal function (RF) or zero renal function, requiring administration of exogenous EPO. With zero renal function there is no physiological regulation of Hb because the feedback loop is hampered. In either case, EPO levels usually need to be supplemented with exogenous EPO 71 and/or with iron 72, which is, e.g., IV administered. The feedback loop 53 then has to be re-established or substituted, e.g., by measuring Hb on a monthly basis in the clinic, e. g., by means of assessing a blood sample, and by adjusting the exogenous or administered EPO dose accordingly. Plasma kinetics are depicted at 74, BV(t) indicates the blood source.

[0169] Now with the advent of technology such as the means 73 for monitoring the body volume of the patient and means 75 for measuring the Hb concentration or mass comprised by the patient's body, the possibility exists for more frequent monitoring. The means 73 for monitoring the body volume could be used fortnightly, for example, to obtain volume information. The means 75 for measuring the Hb concentration can easily monitor Hb at every treatment. Such means are well known to the skilled person. Examples of means suitable for the purposes discussed in this paragraph are disclosed, for example, in FIG. 1 and FIG. 2.

[0170] Using the means 73 and 75 as feedback sensors contributes to determining the exogenous EPO and/or iron level that both allows the target Hb to be achieved whilst maintaining reasonable Hb stability by means of ESA control 76 (with ESA referring to EPO and iron in the example of FIG. 5).

[0171] In this model according to the third embodiment according to the present invention, the erythrocyte lifetime and Hb mass may be determined as is explained below:

[0172] Erythrocytes have a lifetime of typically 120 days before being destroyed and much of the constituent components being recycled. The lifetime of erythrocytes is normally distributed as shown in FIG. 6 representing the fractional mass per gram of erythrocytes being eliminated from the patient's circuit or body. As can be seen from FIG. 6, the first RBC produced on day 0 are eliminated on day 67, the last ones on day 174; the mean lifetime is 120 days. Knowing the typical lifetime span may allow in certain embodiments to calculate the production rate (if Hb is known). This applies at least for the steady state.

[0173] In order to grasp how the variable lifetime affects the accumulation of hemoglobin mass, it is convenient to consider a discrete production process, even though the production of erythrocytes is continuous.

[0174] Each day, a pulse of erythrocytes are generated, represented by the daily generation rate G.sub.Hb(t). Within this pulse, sub units of erythrocytes are generated with specific lifetimes such that

[00001]$\begin{array}{cc}{G}_{\mathrm{Hb}}\left(t\right)=\underset{i=1}{\overset{N}{.Math.}}.Math..Math.g_i\left(t\right),& \mathrm{Eq}.Math..Math.1\end{array}$

where N is the number of lifetime elements considered in the distribution and i is the lifetime index. i is a unit of timeit may refer to hours, days, week, and the like. In the case of days, i may run from, e. g., 1 to 180.

[0175] FIG. 7 shows the accumulation of Hb mass due to each new pulse of erythrocytes generated daily. The sum of the sub units with specific lifetimes, gi, under the pulse distribution is referred to as the daily generation rate.

[0176] The value of the sub units gi may be calculated from a Gaussian function:

[00002]$\begin{array}{cc}{g}_i\left(t\right)=\frac{G_{\mathrm{Hb}}\left(t\right)}{\sqrt{2.Math.{}^2}}.Math.e^{\frac{-{\left({\mathrm{eLT}}_i-\right)}^2}{2.Math.{}^2}}& \mathrm{Eq}.Math..Math.2\end{array}$

where is the mean erythrocyte lifetime and is the standard deviation of the distribution and therefore the erythrocyte lifetime (eLT) has the range

3eLT+3Eq 3

[0177] Taking three standard deviations of the mean lifetime allows 99.7% of the distribution to be considered whilst allowing reducing computational overhead. Setting =120 days and =15 days gives a typical distribution.

[0178] The mass of Hb, M.sub.Hb(t), can be calculated by considering an array of FIFO (First-In-First-Out) buffers, the length of each buffer representing a specific erythrocyte lifetime, see FIG. 8 representing the mass of Hb in circulation using an array of FIFO buffers with buffers FIFO.sub.1 to FIFO.sub.n forming a FIFO memory 81. In FIG. 8, different erythrocyte lifetimes are depicted as eLT.sub.1 to eLT.sub.n. A pulse distributer is represented by gi and the Hb mass sum 83 is computed on basis of one sum 85 related to each FIFO buffer weighted by .

[0179] As each pulse of erythrocytes is generated as shown in FIG. 7, the pulse passes through the FIFO buffers. The total mass 83 of Hb in circulation is the sum of all storage elements FIFO.sub.1 to FIFO.sub.n in the FIFO buffer.

[0180] The total mass 83 is thus:

[00003]$\begin{array}{cc}{M}_{\mathrm{Hb}}\left(t\right)=\underset{i=1}{\overset{N}{.Math.}}.Math..Math.\underset{=0}{\overset{{\mathrm{eLT}}_i}{.Math.}}.Math..Math.g_i\left(t-\right).Math..Math..Math.T.Math..Math.t>& \mathrm{Eq}.Math..Math.4\end{array}$

[0181] The mass however changes depending only on the mass entering and leaving the FIFO buffer at a given time instant. Hence the mass is preferably computed even more efficiently as

[00004]$\begin{array}{cc}{M}_{\mathrm{Hb}}\left(t\right)=\underset{i=1}{\overset{N}{.Math.}}.Math..Math.g_i\left(t\right).Math..Math..Math.T.Math..Math.t<={\mathrm{eLT}}_i.Math..Math.\mathrm{and}& \mathrm{Eq}.Math..Math.5\\ M_{\mathrm{Hb}}\left(t\right)=\underset{i=1}{\overset{N}{.Math.}}.Math..Math.g_i\left(t\right).Math..Math..Math.T-g_i\left(t-{\mathrm{eLT}}_i\right).Math..Math..Math.T.Math..Math.t>{\mathrm{eLT}}_i& \mathrm{Eq}.Math..Math.6\end{array}$

Erythrocyte 64 Production

[0182] The processes involved in erythrocyte production, which is referred to as being part of the erythrocyte kinetics 65 of FIG. 4 and FIG. 5, encompass three main elements, namely the production of pro-erythroblasts from a given EPO concentration based on function f1 of FIGS. 4 and 5, the loading of the pro-erythroblasts with hemeinfluenced by TSAT (f2)and a production transport delay 63.

[0183] In most practical cases, the production rate of erythrocytes 64 can be determined easily from measurements. Also, the production rate may be calculated from the Hb concentration (known), the Gauss distributed life time (also known, e. g., from literature); based on this, the production rate may be calculated in a reverse manner. Once calculated, the production rate can be compared to data known from literature. With regard to the production rate, the model can be adapted to the particular patient.

[0184] The present invention contributes to calculating the EPO concentration that is necessary to support the given production rate. Following three processes, shown in FIG. 4 and FIG. 5 as f1, f2 and the production time constant 63, are of relevance to do so for certain embodiments. Therefore, they are explained in the following in the reverse order, or, with reference to FIG. 4 or 5, from right to left:

Production Time Constant 63

[0185] The production of erythrocytes involves several processes from the time when EPO doses changes until a change in the generation rate may be observed (or takes place). According to Guyton, after a change in EPO doses, new erythrocytes do not appear in the circulation for 2 to 4 days and the maximum rate of new production is observed after 5 days or more. This effect may be characterized by a first order lag with a time constant, of the order of 2.5 days. This could be a transport delay, but we are simplifying it with a first order lag.

[00005]$\begin{array}{cc}{G}_{\mathrm{Hb}}\left(t\right)=\frac{1}{}.Math.{}_0^t.Math.G_{\mathrm{Hb}}^{*}\left(t\right)-G_{\mathrm{Hb}}\left(t\right).Math.\mathrm{dt}& \mathrm{Eq}.Math..Math.7\end{array}$

[0186] G.sub.Hb(t) represents the generation rate of erythrocytes. G*.sub.Hb(t) is used to denote the production rate before the first order lag and G.sub.Hb(t) after the first order lag. In other words, G.sub.Hb(t) lags behind G*.sub.Hb(t) and in steady state G.sub.Hb(t)=G*.sub.Hb(t).

Transferrin Saturation 61 (TSAT)

[0187] The loading of the pro-erythroblasts 59 with heme depends on the availability of ironwhich can be simplified by using TSAT. Thus, TSAT defines the quality of the erythrocytes; it states or accounts for how much hemoglobin is in the RBC. It is assumed that the relation between TSAT and k.sub.Q (being the quality indicator of the RBC) is non-linear requiring some saturation effect. This is modeled with an exponential function as a first proposal. Thus, the generation of Hb, G.sub.Hb(t), is related to the generation rate of pro-erythroblasts G.sub.HCT(t) via f1 and the loading of heme into the RBC (f2).

f.sub.2(t)=G*.sub.Hb(t)=G.sub.HCT(t)(1e.sup.kQ)Eq 8

[0188] In the inverse form the generation of pro-erythroblasts is:

[00006]$\begin{array}{cc}{G}_{\mathrm{HCT}}\left(t\right)=\frac{G_{\mathrm{Hb}}^{*}\left(t\right)}{\left(1-e^{-k_Q}\right)}& \mathrm{Eq}.Math..Math.9\end{array}$

where k.sub.Q is the constant describing the filling of the erythrocytes with hemedepending on the transferrin saturation rateguidelines (e. g., European best practise guidelines) propose a TSAT>20% threshold for dialysis patients (see arrow in FIG. 9).

Pro-Erythroblast 59 Generation

[0189] The rate of pro-erythroblast production depends on the concentration of EPO in the vascular space. For the purposes of derivations that follow, the concentration of vascular EPO will be denoted by [EPOagg]v. The subscript v distinguishes vascular (or plasma) concentrations of EPO from subcutaneous concentrations as discussed later regarding the topic of exogenous EPO administration. The subscript agg indicates the aggregate concentration of EPO formed from both endogenous and exogenous sources. Based on its origin, below, EPO is denoted respectively as [EPOe]v and [EPOx]v originating from different sources. This differentiation appears helpful because endogenous EPO can have a different half life than exogenous EPO.

[0190] To avoid confusion, the following argument is developed with the variable [EPOagg]v, regardless of whether it is solely endogenous EPO or a combination of both endogenous and exogenous EPO. The physiological range of [EPOagg]v within the plasma volume is 10 to 30 [u/L]. In extreme cases, the concentration can increase 100 to 1000 fold in a healthy subject. Whether this translates to a proportional increase in pro-erythroblast production rate seems unlikely as can be seen from the following argument:

[0191] Firstly the mass of Hb in circulation in steady state is the product of mean erythrocyte lifetime and production rate and this is also equal to the product of the current Hb and blood volume, i.e.,

M.sub.Hb(ss)=BV.Math.Hb=G.sub.Hb.Math.Eq 10

with SS indicating the steady state.

[0192] Taking a typical blood volume of 50 dl, an Hb of 14 g/dl leads to

[00007]$\begin{array}{cc}{G}_{\mathrm{Hb}}=\frac{\mathrm{BV}.Math.\mathrm{Hb}}{}=5.833.Math..Math.g.Math.\text{/}.Math.\mathrm{day}& \mathrm{Eq}.Math..Math.11\end{array}$

[0193] In steady state G.sub.Hb equals G.sub.ery. Therefore, if the transferrin saturation TSAT is 20%, then the corresponding generation of pro-erythrocytes from Eq 9 is 11.68 ml/day

[0194] A 100 fold increase in G.sub.pro to 1.168 kg/day is highly unlikely and a 1000 fold increase in G.sub.pro to 11.68 kg/day is completely implausible. This would be the basis therefore for a saturation function, modeled in the simplest form as an exponential of the form:

[00008]$\begin{array}{cc}{G}_{\mathrm{HCT}}=G_{\mathrm{HCT\_max}}.Math.\left(1-e^{-k.Math.{\left[{\mathrm{EPO}}_{\mathrm{agg}}\right]}_v}\right)& \mathrm{Eq}.Math..Math.12\end{array}$

[0195] Taking the middle of the physiological range is an EPO concentration in the vascular space [EPOagg]v of 20 u/L and assume this can increase 100 fold. Assuming therefore that 2000 u/L leads to 99.99% of the maximum production rate of pro-erythrocytes then

10.9999=e.sup.2000kEq 13

From which k=0.002

[00009]$\begin{array}{cc}{G}_{\mathrm{HCT}}.Math.\max =\frac{11.68}{1-e^{-k20}}298.Math..Math.\mathrm{ml}.Math.\text{/}.Math.\mathrm{day}& \mathrm{Eq}.Math..Math.14\end{array}$

[0196] FIG. 10 reveals a pro-erythroblast transfer function f1, illustrating Pro_ery [g/d, i.e., gram/day] over [EPOagg]v in unit per millilitre [u/ml] (ml relates to the blood volume). In FIG. 10, the normal physiological range is referred to by reference numeral 101.

[0197] 298 ml/day peak production rate might be just plausible, but it should be recalled that this represent an extreme condition. For normal physiological ranges even with significant deviations of 10 fold, the function is largely linear.

[0198] FIG. 11 shows an expanded section of EPO to pro-erythroblast transfer function f1. The linear function is a good approximation even for a 10 fold increase in EPO.

Baseline Endogenous EPO Production 55

[0199] Using the same principles as above, it is possible to estimate the baseline endogenous EPO production rate. In this case, the source of EPO is clear and therefore use of the variable [EPOe]v is appropriate. The easiest way to achieve this is to consider a healthy subject that then develops anemia due to fluid overload. The blood volume is increased by an extra litre from 5 L to 6 L, and the patient's steady state Hb falls to 8 g/dl.

[0200] In some embodiments, it is assumed that mean erythrocyte lifetime remains normal at 120 day. In certain embodiments, it is assumed that TSAT is maintained in the healthy range of 40% by suitable iron therapy.

[0201] By combining Eq 9 and Eq 11, the production rate of pro-erythroblasts is given by:

[00010]$\begin{array}{cc}{G}_{\mathrm{HCT}}\left(t\right)=\frac{\mathrm{Hb}.Math.\mathrm{BV}}{.Math.\left(1-e^{-K_Q}\right)}& \mathrm{Eq}.Math..Math.15\end{array}$

[0202] Substituting above values, the production rate of pro-erythroblasts is 6.66 g/day. Rearranging Eq 12 yields:

[00011]$\begin{array}{cc}{\left[{\mathrm{EPO}}_e\right]}_v=\frac{-1}{k}.Math.\ln \left(1-\frac{G_{\mathrm{HCT}}}{G_{\mathrm{HCT\_Max}}}\right)& \mathrm{Eq}.Math..Math.16\end{array}$

[0203] Thus, the EPO concentration corresponding to 8 g/day pro-erythroblast generation rate is 13.61 u/L. L relates to the plasma volume.

[0204] The baseline flux of EPO (generation rate) into the vascular system F.sub.EPO.sub._.sub.baseline must balance the flux out of the vascular system caused by liver clearance. The flux out is the product of EPO concentration and liver clearance:

F.sub.EPO.sub._.sub.baseline=[EPO.sub.e].sub.v.Math.K.sub.LiverEq 17

[0205] From which F.sub.EPO.sub._.sub.baseline=0.01361.75=0.0238 U/min (see later sections for determination of liver clearance). In other words in the absence of physiological regulation, baseline endogenous EPO flux supports a steady state Hb of 8 g/dl. This is valid for a subject with a normal blood volume of 50 dL. Subjects with higher and lower BVs will need to be scaled accordingly. Therefore,

[00012]$\begin{array}{cc}{F}_{\mathrm{EPO\_baseline}}=\frac{\mathrm{BV}}{50}.Math.0.0238.Math..Math.U.Math.\text{/}.Math.\min & \mathrm{Eq}.Math..Math.18\end{array}$

[0206] Note that F.sub.EPO.sub._.sub.baseline has been shortened to F1 in FIGS. 13 and 14.

Physiologically Controlled EPO Production

[0207] Wherever it is referred to physiologically controlled EPO production, the overall productionincluding the kidneyis contemplated. Baseline endogenous EPO production relates, however, to any endogenously produced EPO that stems from any organ (including, for example, the liver) but the kidneys.

[0208] Simple proportional control is assumed such that a flux denoted by proportional control factor F3 is generated that is proportional to the difference between the Hb set point and the measured Hb. This may be represented as:

F.sub.3=(Hb.sub.setHb).Math.K.sub.EPO.sub._.sub.gainEq 19

where K.sub.EPO.sub._.sub.gain represents the proportional gain. K.sub.EPO.sub._.sub.gain can be easily determined in health as it is the gain that is required to restore Hb back to normal levels within a time frame of x days, following blood loss. Where a subject has partial renal function a reduced value of K.sub.EPO.sub._.sub.gain can be expected. This can be easily modeled with the expression:

K.sub.EPO.sub._.sub.gain=K.sub.EPO.sub._.sub.Max.Math.(1.Math.e.sup.k.sup.rf.sup..Math.R)Eq 20

where R (an enhancement, without dimension, see also FIG. 12) is the renal function and k.sub.rf is the decay constant.

[0209] FIG. 12 shows the gain of K.sub.EPO as a function of the renal function in [%].

EPO Kinetics 65

Subcutaneous Administration of EPO

[0210] The derivation of the concentration of EPO appearing in the vascular space applies the scheme shown in FIG. 13. The dynamics of the EPO are determined according to two pools namely the subcutaneous compartment 131 and the vascular compartment 133, the mass transfer coefficient K.sub.SCV of EPO (from subcutaneous to the vascular space) and the clearance K.sub.liver of EPO by the liver. The baseline endogenous EPO concentration [EPO.sub.e].sub.v, see reference numeral 135, is factored in the present model by function F1, its physiological regulation, see reference numeral 137, is factored in the present model by function F2.

[0211] For example, the mass transfer coefficient K.sub.SCV may be 0.005 ml/min. Although 0.005 ml/min is preferred by the inventors, K.sub.SCV is, however, not limited to this value. Rather, any suitable value for K.sub.SCV may also be used. Suitable values have been found by the inventors to be in the range between 0.0001 ml/min to 0.1 ml/min, further in the range between 0.001 ml/min and 0.01 ml/min and in the range between 0.0025 ml/min and 0.0075 ml/min.

[0212] A steady baseline generation of endogenous EPO is assumed as explained earlier. Note the concentration of EPO is denoted by the use of square brackets to differentiate concentration from mass.

[0213] FIG. 13 shows the parameter that influence the subcutaneous administration 139 of EPO according to the 2-pool-model.

[0214] The mass of EPO in the vascular compartment 133 is the sum of the endogenous and exogenous components of EPO:

M_EPO.sub.agg.sub._.sub.v=[EPO.sub.e].sub.v.Math.V.sub.p+[EPO.sub.x].sub.v.Math.V.sub.pEq 21

[0215] To avoid unwieldy expressions in the following derivations, let M=M_EPOagg_v, Ce=[EPOe]v and Cx=[EPOx]v. However, it will be necessary to switch the use of the original variable name and the substitutions for clarity. Rewriting Eq 21 with substitutions made:

M=C.sub.e.Math.V.sub.p+C.sub.x.Math.V.sub.pEq 22

[0216] As Ce and Cx and Vp are all variables then a change in the mass of EPO is

[00013]$\begin{array}{cc}.Math..Math.M=\frac{M}{C_e}.Math..Math..Math.C_e+\frac{M}{C_x}.Math..Math..Math.C_x+\frac{M}{V_p}.Math..Math..Math.V_p& \mathrm{Eq}.Math..Math.23\end{array}$

[0217] From which the rate of change of mass with respect to time is

[00014]$\begin{array}{cc}\frac{.Math..Math.M}{.Math..Math.t}=\frac{M}{C_e}.Math.\frac{.Math..Math.C_e}{.Math..Math.t}+\frac{M}{C_x}.Math.\frac{.Math..Math.C_x}{.Math..Math.t}+\frac{M}{V_p}.Math.\frac{.Math..Math.V_p}{.Math..Math.t}& \mathrm{Eq}.Math..Math.24\end{array}$

[0218] Therefore

[00015]$\begin{array}{cc}\frac{.Math..Math.M}{.Math..Math.t}=V_p.Math.\frac{.Math..Math.C_e}{.Math..Math.t}+V_p.Math.\frac{.Math..Math.C_x}{.Math..Math.t}+\left(C_e+C_x\right).Math.\frac{.Math..Math.V_p}{.Math..Math.t}& \mathrm{Eq}.Math..Math.25\end{array}$

[0219] In the limit as t.fwdarw.0 then

[00016]$\begin{array}{cc}\frac{\mathrm{dM}}{\mathrm{dt}}=V_p.Math.\frac{{\mathrm{dC}}_e}{\mathrm{dt}}+V_p.Math.\frac{{\mathrm{dC}}_x}{\mathrm{dt}}+\left(C_e+C_x\right).Math.\frac{{\mathrm{dV}}_p}{\mathrm{dt}}& \mathrm{Eq}.Math..Math.26\end{array}$

[0220] Assuming instantaneous mixing of EPO within the vascular compartment, then applying simple mass balance principles, the fluxes of EPO (mass transferred per unit time) is

[00017]$\begin{array}{cc}\frac{\mathrm{dM}}{\mathrm{dt}}=F_1+F_2+F_3-F_4& \mathrm{Eq}.Math..Math.27\end{array}$

[0221] The flux F1 represents the baseline endogenous EPO generation rate. Note that F3 is the case for physiologically regulated EPO flux where some degree of renal function (may be assessed by means of the renal creatinin clearance function, depicted as C in FIG. 15) is assumed. In chronic renal failure this may be set to zero. The flux F2 depends on the concentration of EPO within the subcutaneous and vascular pools. However, more specifically, it is assumed that the concentration gradient between the two pools is not affected by the EPO composition in the vascular space, but merely the aggregate concentration, i.e.:

[00018]$\begin{array}{cc}{F}_2=-V_{\mathrm{SC}}.Math.\frac{d\left({\left[{\mathrm{EPO}}_x\right]}_{\mathrm{SC}}\right)}{\mathrm{dt}}=\left({\left[{\mathrm{EPO}}_x\right]}_{\mathrm{SC}}-{\left[{\mathrm{EPO}}_{\mathrm{agg}}\right]}_V\right).Math.K_{\mathrm{SCV}}& \mathrm{Eq}.Math..Math.28\end{array}$

[0222] Note the minus signs indicating the direction of flux out of the subcutaneous pool.

[0223] The flux through the liver depends on concentration and the clearance rate, but differs depending on the half life of the EPO component. Consequently two liver time constants are defined for endogenous and exogenous components. This may be represented as

F.sub.4=[EPO.sub.e].sub.v.Math.K.sub.liver.sub._.sub.e+[EPO.sub.x].sub.v.Math.K.sub.liver.sub._.sub.xEq 29

[0224] Combining Eq 26 and 27 and rearranging for the endogenous rate of concentration change leads to

[00019]$\begin{array}{cc}\frac{{\mathrm{dC}}_e}{\mathrm{dt}}=\frac{F_1+F_2+F_3-F_4-V_p.Math.\frac{{\mathrm{dC}}_x}{\mathrm{dt}}-\left(C_e+C_x\right).Math.\frac{{\mathrm{dV}}_p}{\mathrm{dt}}}{V_p}& \mathrm{Eq}.Math..Math.30\end{array}$

and for the exogenous EPO concentration:

[00020]$\begin{array}{cc}\frac{{\mathrm{dC}}_x}{\mathrm{dt}}=\frac{F_1+F_2+F_3-F_4-V_p.Math.\frac{{\mathrm{dC}}_e}{\mathrm{dt}}-\left(C_e+C_x\right).Math.\frac{{\mathrm{dV}}_p}{\mathrm{dt}}}{V_p}& \mathrm{Eq}.Math..Math.31\end{array}$

and from Eq 28

[00021]$\begin{array}{cc}\frac{{d\left[\mathrm{EPO}\right]}_{\mathrm{SC}}}{\mathrm{dt}}=\frac{\left({\left[\mathrm{EPO}\right]}_V-{\left[\mathrm{EPO}\right]}_{\mathrm{SC}}\right).Math.K_{\mathrm{SCV}}}{V_{\mathrm{SC}}}& \mathrm{Eq}.Math..Math.32\end{array}$

[0225] The half life of endogenous EPO when broken down by the liver is 8 hrs. This may be related to the liver clearance, K.sub.liver.sub._.sub.e as follows

[00022]$\begin{array}{cc}\frac{K_{\mathrm{liver\_e}}}{V_p}=\frac{\ln \left(2\right)}{t_{\mathrm{h\_e}}}& \mathrm{Eq}.Math..Math.33\end{array}$

Where the plasma volume Vp is

V.sub.p=(1Hct).Math.BVEq 34

[0226] Thus converting to minutes and assuming a blood volume of 5000 ml and Hct of 0.36 then

[00023]$\begin{array}{cc}{K}_{\mathrm{liver\_e}}=\frac{\ln \left(2\right)}{860}.Math.5000.Math.\left(1-\mathrm{Hct}\right)=4.62.Math..Math.\mathrm{ml}.Math.\text{/}.Math.\min & \mathrm{Eq}.Math..Math.35\end{array}$

[0227] Exogenous liver clearance K.sub.liver.sub._.sub.x can be determined in the same manner. With some ESA agents the half life is 21 hrs leading to a K.sub.liver.sub._.sub.x value of 1.76 ml/min. The time of the peak concentration of [EPOagg]v in the blood after EPO administration is governed by the mass transfer coefficient K.sub.SCV and the liver clearance. This peak occurs typically 5-24 hours after injection. Taking a value of 12 hours, K.sub.SCV can be determined easily from simulation.

Intravenous Administration of EPO

[0228] The scheme for intravenous infusion (iv) is shown in FIG. 14. The derivation of the kinetic is similar to that of subcutaneous, but simplified without the presence of a flux F2.

[0229] FIG. 14 shows the EPO kinetics of intravenous application (IV). The IV kinetics are identical to subcutaneous kinetics of FIG. 13 with F2 set to zero,

[00024]$\begin{array}{cc}\frac{{\mathrm{dC}}_e}{\mathrm{dt}}=\frac{F_1+F_3-F_4-V_p.Math.\frac{{\mathrm{dC}}_x}{\mathrm{dt}}-\left(C_e+C_x\right).Math.\frac{{\mathrm{dV}}_p}{\mathrm{dt}}}{V_p}& \mathrm{Eq}.Math..Math.36\\ \frac{{\mathrm{dC}}_x}{\mathrm{dt}}=\frac{F_1+F_3-F_4-V_p.Math.\frac{{\mathrm{dC}}_e}{\mathrm{dt}}-\left(C_e+C_x\right).Math.\frac{{\mathrm{dV}}_p}{\mathrm{dt}}}{V_p}& \mathrm{Eq}.Math..Math.37\end{array}$

[0230] F2 is set to zero as it makes no sense to consider a flux of EPO as such, which is administered as an impulse. Instead the effect of IV injection is to reset the [EPOagg]v to a new value caused by the mass of exogenous EPO injected. Thus at the time instant of EPO injection

[00025]$\begin{array}{cc}{\left[{\mathrm{EPO}}_{\mathrm{agg}}\right]}_v.Math.\left(t+1\right)=\frac{{\left[{\mathrm{EPO}}_{\mathrm{agg}}\right]}_v.Math.\left(t\right).Math.V_p+{\left[{\mathrm{EPO}}_x\right]}_{\mathrm{ivi}}.Math.V_{\mathrm{inj}}}{V_p+V_{\mathrm{inj}}}& \mathrm{Eq}.Math..Math.38\end{array}$

[0231] V.sub.inj is typically 1 ml for injection. While this crucially important in setting the dose of EPO administered, it clearly has negligible effect on the plasma volume.