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A Computational Model of Red Blood Cell Dynamics in Patients
with Chronic Kidney Disease
H. T. Banks1, Karen M. Bliss1, Peter Kotanko2, Hien Tran1
1Center for Research in Scientific Computation, North Carolina State University
2Renal Research Institute, New York
February 15, 2011
Abstract
Kidneys are the main site of production of the hormone erythropoietin (EPO) that is the
major regulator of erythropoiesis, or red blood cell production. EPO level is normally controlled
by a negative feedback mechanism in the kidneys, but patients with chronic kidney disease (CKD)
do not produce sufficient levels of EPO to maintain appropriate blood hemoglobin concentration.
A mathematical model, including interactions with iron and inflammation, is developed for ery-
thropoiesis in patients with CKD. Numerical solution methodologies and validation of numerical
results are discussed. Simulation results under varying conditions and treatment protocols are
presented.
Further author information: (Send correspondence to Karen M. Bliss.)
Karen M. Bliss: kmbliss@ncsu.edu
H. T. Banks: htbanks@ncsu.edu
Peter Kotanko: pkotanko@rriny.com
Hien Tran: tran@ncsu.edu
1
1 Introduction
It is estimated that 31 million Americans have chronic kidney disease (CKD). Among those, approx-
imately 330 thousand are classified as being in End-Stage Renal Disease (ESRD) and require dialysis
[17]. Dialysis is the bidirectional exchange of materials across a semipermeable membrane [2]. For
the purposes of this study, we consider only hemodialysis, where a patient’s blood is exposed to a
semipermeable membrane outside of the body.
In addition to regulating blood pressure and filtering waste products from blood, kidneys produce
a hormone called erythropoietin (EPO) that is the major regulator of erythropoiesis, or red blood cell
production. EPO level is normally controlled by a negative feedback mechanism in the kidneys, but
patients in ESRD do not produce sufficient levels of EPO to maintain blood hemoglobin concentra-
tion. Hemoglobin is the protein that gives red blood cells the ability to carry oxygen. Patients with
low hemoglobin concentration may present symptoms of anemia, such as decreased cardiac function,
fatigue, and decreased cognitive function.
In order to prevent anemia, patients typically receive recombinant human EPO (rHuEPO) in-
travenously to stimulate red blood cell production. However, treatment is far from perfect. In 2006,
only half of dialysis patients had a mean monthly hemoglobin greater than 11 grams per deciliter
[17], the desired minimum level set by the National Kidney Foundation [13].
Iron is required to produce hemoglobin, and iron deficiency can be an issue among patients
receiving rHuEPO therapy. Oral iron supplementation is often ineffective, so intravenous iron sup-
plementation has become a mainstay in many patients undergoing rHuEPO therapy [9].
Iron availability is negatively affected by inflammation level in the body. Most patients with
CKD have elevated levels of inflammation due to CKD and the presence of other medical issues (e.g.,
diabetes, hypertension, etc.) [10].
Our goals are (1) the development of a mathematical model for erythropoiesis of patients in ESRD
undergoing hemodialysis, taking into consideration the effects of EPO, iron level, and inflammation
level in the body, which has a reasonable degree of fidelity to the biological system, and (2) the
development of model-based control of the system. This note is a step toward the first of these two
goals.
2 Erythropoiesis
Erythropoiesis is the process by which erythrocytes, or red blood cells (RBCs), are formed. Erythro-
cytes transport oxygen and carbon dioxide between the lungs and all of the tissues of the body and
can be thought of as a container for hemoglobin [15], the protein that carries oxygen.
Erythrocytes are produced primarily from pluripotent stem cells in bone marrow. In the presence
of the cytokine named stem cell factor, hematopoietic stem cells divide asymetrically, producing a
committed colony-forming-unit (CFU) while maintaining the population of stem cells. The erythro-
cyte lineage shares the precursor CFU-GEMM (granulocyte, erythrocyte, macrophage, megakary-
ocyte) with other types of blood cells (white blood cells, platelets, etc.). The exact mechanisms
determining selection of lineage from this nodal point are not known [6].
Erythrocyte lineage continues as described in Figure 1: erythroid burst-forming unit (BFU-
E), erythroid colony-forming-unit (CFU-E), proerythrocyte, basophilic erythrocyte, polychromatic
erythroblast, orthochromatic erythroblast, reticulocyte, and erythrocyte. Cell division ceases with
the formation of the orthochromatic erythroblast. Division rate, death rate, and maturation rate are
influenced by the level of EPO [6]. This is described in more detail later.
Hemoglobin is synthesized beginning in the CFU-E stage, with the majority of synthesis occurring
2
CFU-GEMM
C
BFU-E
CFU-E
Proerythroblast
Basophilic erythroblast
Polychomatic erythroblast
Orthochromatic erythroblast
Reticulocyte
Erythrocyte
Figure 1: Erythropoiesis cell lineage.
in the polychromatic erythroblast stage. When the nucleus is extruded from the cell, the cell is
named a reticulocyte. Little hemoglobin synthesis happens at the reticulocyte stage, and synthesis
is completely absent in mature erythrocytes [6].
Reticulocytes begin to lose the adhesive proteins that hold them in the bone marrow. They de-
crease in size and begin to circulate in the blood. In healthy individuals, the time from proerythrob-
last to mature erythrocyte is approximately 7 days. Normal erythrocyte life span is approximately
120 days, at which time aging erythrocytes are enveloped by macrophages in the spleen.
3 Previous models
The process of erythropoiesis has been modeled in many physiological scenarios. In [14], rHuEPO
therapy is considered in healthy volunteers. This model incorporates the negative feedback to endoge-
nous EPO production. EPO is assumed to be cleared using Michaelis-Menten dynamics. A similar
model was used to fit data in rats [18]. Both of these models use delay instead of age-structured
modeling.
Both [3] and [4] use age-structured models, as does the model described in [11], which assumes
that the oldest mature erythrocytes will be destroyed, yielding a moving boundary condition. In [1],
EPO is assumed to accelerate maturation of cells undergoing erythropoiesis. Additionally, EPO is
assumed to be consumed during the process of erythropoiesis.
The model presented here is a significant departure from these models in that it incorporates the
effects of both iron plasma level and inflammation.
3
4 Model Overview
We use an age-structured model with three major classifications of erythroid cells in which the
structure variables µ, ν, and ψrepresent maturity levels, as shown in Figure 2.
EPO
Inflammation
Iron
RBC Progenitors,
P(t, µ)
Maturing RBCs,
M(t, ν)
Oxygen-carrying
capacity,
O(t, ψ)
hemoglobin in RBCs
recruitment rate
birth rate and death rate
maturation rate
death rate
EPO
clearance EPO produced
in the liver
and kidney
intravenous
EPO
treatment
iron
losses
intravenous
iron treatment
Figure 2: Model schematic.
P(t, µ) and M(t, ν) represent the number of progenitor cells and maturing hematopoietic cells,
respectively. O(t, ψ ) is a measure of the oxygen carrying capacity of circulating reticulocytes and
erythrocytes. For these cell classes, the second argument (e.g., µfor class P) is the structure
variable, maturity level in this case. We model EPO level, E, iron level, Fe, and a measure of overall
inflammation in the body, I. Time is measured in days.
Our state variables are
P=P(t, µ), M =M(t, ν ), O =O(t, ψ), E =E(t),and Fe =Fe(t).
Rate of exogenous EPO treatments, ˙
Eex,and rate of exogenous iron treatments, ˙
Feex,are input
functions, and hemoglobin concentration, Hb(t),is the output of the model.
We will make use of sigmoid functions throughout the model. An increasing sigmoid function
will be of the form
F(x) = Fmin −Fmax·ck
ck+xk+Fmax.
Note that when xis small, F(x) is close to Fmin,and when xis large, F(x) is close to Fmax .The
typical graph of such a function is depicted in Figure 3a. The values of cand kaffect the slope of
the curve and the location of the area of increase.
Similarly, a typical decreasing sigmoid function is depicted in Figure 3b and has the form
G(x) = Gmax −Gmin·ck
ck+xk+Gmin.
4
F(x)
Fmax
Fmin
x
(a) Generic increasing sigmoid function.
G(x)
Gmax
Gmin
x
(b) Generic decreasing sigmoid function.
Figure 3: Sigmoid function examples.
5 Iron
Iron is required to make hemoglobin, the protein that gives erythrocytes the ability to carry oxygen.
It is also the protein that gives erythrocytes their characteristic red color. If iron is not available
during erythropoiesis, the result is lighter-colored (hypochromic) erythrocytes with reduced capacity
to carry oxygen.
Control of iron in the body is a strictly regulated process, in part because there is no pathway
for the excretion of excess iron (Figure 4).
Blood Plasma
(iron is carried in transferrin)
Spleen
(removes old RBCs
and recycles iron
from the
hemoglobin)
Bone Marrow
(RBCs produced,
which contain iron in
their hemoglobin)
~20 mg iron/day
~20 mg iron/day
RBCs circulating in blood
Liver
(stores iron)
Iron losses
Iron from
diet
~2 mg iron/day
~2 mg iron/day
Figure 4: Iron cycle in healthy individuals.
5
When red blood cells age, they become enveloped by macrophages in the spleen. The iron from
their hemoglobin is then recaptured and sent to the bone marrow for use in making hemoglobin
for new erythrocytes. This recycling process is very efficient and is the main source of iron to
erythropoiesis [15]. In much smaller quantities, iron is absorbed from diet in the duodenum and can
be stored in the liver. The only losses to the system are from sweating, cells being shed, blood losses,
etc.
Iron is stored in the compound ferritin when it is within a cell, and in the compound transferrin
when it is in the blood plasma. The protein ferroportin is required to transport iron out of a cell and
into the plasma. The major regulator of this transport is the hormone hepcidin, which is produced
in the liver. Hepcidin binds to ferroportin and causes the complex to be absorbed into the cell,
effectively interrupting the transport of iron into the blood plasma, as depicted in Figure 5.
Hepcidin production is increased in the presence of certain cytokines which are released due to
inflammation in the body. It is thought that this might be a defense mechanism against foreign
organisms which may need iron to reproduce.
Since patients in ESRD commonly have other health problems (such as diabetes and hyperten-
sion), they often have higher than normal levels of inflammation. Thus, they may produce higher
than normal levels of hepcidin. As a result, even if there is enough iron in the body, it may not be
available for erythropoiesis because it cannot leave the cells and enter the plasma.
Current research suggests that EPO affects the interaction between cytokines and hepcidin. When
EPO level is sufficiently high, the effects of inflammation cannot be seen.
We model the amount, Fe, of iron in the blood plasma, in milligrams. We formulate a mass
balance involving the iron compartment (see Figure 6) as follows.
The main source of incoming iron to the compartment is recycled iron from the hemoglobin of
senescent erythrocytes that are enveloped by macrophages. We will develop class Oso that each
member in the class is assumed to contain exactly the same amount of iron. That is, the rate of iron
being recycled from class Ois kFe Rψf
0δO(ψ)O(t, ψ)∂ψ, where kFe is some proportionality constant
and δO(ψ) is the death rate of the cells in class O, explained in more detail later.
(iron stored
in ferritin)
(iron stored
in transferrin)
ferroportin
Fe
Fe
Fe
(a) Ferroportin is required for the transport of
iron out of cells.
ferroportin hepcidin
Fe
Fe
Fe
(b) Hepcidin is the major regulator of iron trans-
port out of cells.
Figure 5: Iron regulation at a cellular level.
6
EPO
Inflammation Iron
Oxygen-carrying
capacity,
O(t, ψ)
iron
losses
intravenous
iron treatment
Maturing RBCs,
M(t, ν)
Figure 6: Iron compartment.
The other main source of iron to the compartment is exogenous iron supplied as part of treatment.
We denote the rate of exogenous iron treatment by ˙
Feex(t)
A small amount of iron enters the system through absorption from diet and from storage in
the liver, and there are also iron losses (due to sweating, blood losses during blood draws and
hemodialysis, etc.). As described earlier, patients undergoing hemodialysis require iron supplements
intravenously. Therefore we assume that when we sum the iron losses and the iron entering the
system from diet and storage in the liver we obtain a net loss. Further, we will assume for an initial
model that the loss occurs at constant rate unless the current level of iron is small, in which case a
fraction of the iron is lost. That is, that rate of iron loss, ρFe,loss (Fe),is given by
ρFe,loss(Fe) = ρFe,const, Fe ≥Feth
ρFe,frac ·Fe, Fe < Feth.
This assumption will be revisited in future models, perhaps with greater losses when dialysis and
blood draws occur.
Next we need to account for iron leaving the compartment during erythropoiesis. For our first
model, we begin by making the assumption that iron enters red blood cells at the moment that a cell
matures from class Mto class O, which is the time that a cell leaves the bone marrow and begins
circulating. Red blood cells actually collect iron over the time period that they are in class M , but
the biochemistry of this process is not clearly understood. The assumption that all of the iron is
collected into a cell at one moment will certainly have to be revisited in future improvements of the
model.
In determining the amount of iron used during erythropoiesis, we first compute the amount of
iron that would be used if every cell leaving class Mwere to contain the appropriate amount of
hemoglobin so as to be at full oxygen-carrying capacity, i.e.,
Feneeded =kFeM(t, νf).(1)
In the presence of inflammation, even if there is enough iron in the plasma, it may not be available
to be used in erythropoiesis. For our initial model, we assume that there is an EPO threshold, E P Oth.
7
We assume that if EPO is above the threshold, the effects of inflammation can not be seen. That is,
we assume
Feavail =kFe,eff f(E, I)Fe, (2)
where
f(E, I ) =
(cFe,av)kFe,av
(cFe,av)kFe,av +IkFe,av , E < EP Oth
1, E ≥EP Oth .
Observe that in this model when EPO is greater than the threshold level, inflammation level
does not impact iron availability. However, when EPO level is lower than the threshold, the amount
of available iron depends on inflammation level–as f(E, I ) is close to one when inflammation is low
and close to zero when inflammation is high. The constant kFe,eff ,with 0 ≤kFe,eff ≤1,is an efficacy
constant that accounts for the fact that only a fraction of the iron in the plasma will actually be
available at the site of erythropoiesis at any given time.
The amount of iron actually used in erythropoiesis is therefore given by
Feused = min {Feneeded , Feavail},
= min {kFeM(t, νf), kFe,eff f(E, I)Fe}.(3)
We assume that the rate of iron leaving the iron compartment and entering class Ois proportional
to this quantity, Feused.That is,
ρFe→O=kρ,FeFeused.
Thus, the mass balance in the iron compartment is given by
˙
Fe(t) = (rate in from class O) + (rate in intravenously)
−(rate out to class O)−(rate of iron losses)
=kFe Zψf
0
δO(ψ)O(t, ψ)dψ +˙
Feex(t)−ρFe→O−ρFe,loss(Fe).(4)
6 EPO
EPO is the primary regulator of erythropoiesis. It stimulates red blood cell production, differentiation
and maturation, and prevents apoptosis [6]. In healthy individuals, the majority of EPO production
occurs in the kidney. Sensors in the kidney monitor blood oxygen level. EPO production is increased
in response to low oxygen level and is decreased when oxygen level is high.
Patients in ESRD, whose kidneys have only minimal function, produce only a small basal level of
EPO in the kidney and liver [7]. Without intervention, patients can develop anemia; therefore, pa-
tients undergoing dialysis are commonly treated with intravenous rHuEPO. Two common rHuEPOs,
epoetin alfa and epoetin beta, share structural homology with endogenous EPO. Darbepoietin alfa,
the other major erythropoietic agent, is designed so that it has a longer half-life in-vivo. In this
model, we assume that darbepoietin is not the erythropoietic agent, and therefore we will not dis-
tinguish between rHuEPO and endogenous EPO with respect to their action. We assume that their
effects on erythropoiesis are identical.
8
EPO is measured in units of EPO. We assume the rate of endogenous EPO production in the
liver and kidney to be constant, and will denote it ρEP O,basal .
We will assume that EPO clearance is proportional to the amount present, although we could
consider Michaelis-Menten dynamics in future models. Finally, we also account for the rate of EPO
given via IV, denoted ˙
Eex(t).So we have
˙
E(t) = ρEP O,basal +˙
Eex(t)−1
t1/2
ln 2·E(t),
where t1/2is the half-life of EPO.
7 Inflammation
Inflammation affects two aspects of erythropoiesis, as depicted in Figure 2.
Even in patients without CKD, chronic inflammation can cause anemia, termed the anemia of
chronic disease. While the exact chemical pathways are not necessarily known, it is known that the
presence of inflammation can suppress erythropoiesis and may inhibit the action of EPO [16]. Since
EPO affects the birth and death rate of progenitors, we incorporate inflammation in the death rate
term associated with the progenitor cell class, P.
Inflammation level also impacts iron availability for erythropoiesis, as described previously. In-
flammation may cause an increase in ferritin production, which would cause iron to be retained within
cells, inhibiting the use of iron to make hemoglobin. Inflammation may also impair the ability of the
body to absorb dietary iron [16].
It is almost certain that inflammation affects these two aspects of erythropoiesis via completely
different chemical pathways. We assume that both aspects can be sufficiently described with some
overall measure of inflammation in the body. There are markers of inflammation, such as albumin
and C-reactive protein, which are often measured in patients undergoing dialysis. In future work, we
will investigate whether inflammation can be described as some combination of the levels of these
markers.
8 Class P(t, µ)
We group the progenitor cells (CFU-GEMM, BFU-E and CFU-E) in one class, P(t, µ). These cells
are affected by EPO level and inflammation level.
We make the following assumptions:
(i) There is a smallest maturity level, µ0= 0,and a largest maturity level, µf; i.e., 0 ≤µ≤µf.
(ii) The maturity rate depends on the EPO concentration and the maturity level [6]. For simplifi-
cation in our initial model, we assume that the maturity rate is constant:
dµ
dt =ρP.
(iii) The birth rate depends on EPO concentration [6] and the maturity level.
Regulation of erythropoiesis by EPO is focused on the progenitor class, and probably most
importantly the CFU-E. A rise in EPO level results in proliferation of CFU-E [6]. We will
9
EPO
Inflammation
RBC Progenitors,
P(t, µ) to class M
recruitment rate
birth rate and death rate
death rate
Figure 7: The progenitor cells, P(t, µ).
assume EPO affects all cells in class Pequally, independent of maturity level. We will model
the birth rate as an increasing sigmoid function,
βP(E) = βmin
P−βmax
P(cβ,P )kβ,P
(cβ,P )kβ,P +Ekβ,P +βmax
P.
(iv) The number of stem cells being recruited into the precursor cell population is directly propor-
tional to EPO level:
P(t, 0) = RPE(t).
It is reasonable to assume that recruitment is related to EPO level, as it is one of the hormones
that affects whether a stem cell will become an erythrocyte. Other hormones are certainly
involved as well, but the chemical pathway governing the differentiation of stem cells is still
largely unknown [6].
(v) The death rate depends on the concentration of EPO, the inflammation level, and the maturity
level, µ. We will simplify this for our first model to assume that death rate is not dependent
on maturity level.
EPO prevents apoptosis, or programmed cell death, of progenitor cells [6]. We use a decreasing
sigmoid function to describe this behavior.
Certain interferons, present under inflammatory conditions, can also cause death of progenitor
cells, specifically CFU-E [12]. We assume that the death rate of progenitor cells depends on
inflammation level, which is modeled by some increasing sigmoid function.
Finally, we assume that overall death rate is the sum of these two effects:
δP(E, I ) = δmax
P,E −δmin
P,E (cδ,P,E )kδ,P,E
(cδ,P,E )kδ,P,E +Ekδ,P,E +δmin
P,E
+δmin
P,I −δmax
P,I (cδ,P,I )kδ,P,I
(cδ,P,I )kδ,P,I +Ikδ,P,I +δmax
P,I .
10
Now we consider the rate of change in population from maturity level µto maturity level µ+ ∆µ.
rate of change in population on the interval (µ, µ + ∆µ) =
(rate of cells entering the interval) −(rate of cells leaving the interval)
+ (birth rate term) −(death rate term)
∂
∂t Zµ+∆µ
µ
P(t, ξ)dξ =ρPP(t, µ)−ρPP(t, µ + ∆µ)
+Zµ+∆µ
µ
βP(E)P(t, ξ)dξ −Zµ+∆µ
µ
δP(E, I )P(t, ξ)dξ
∂
∂t Zµ+∆µ
µ
P(t, ξ)dξ =−ρP[P(t, µ + ∆µ)−P(t, µ)]
+ [βP(E)−δP(E, I )] Zµ+∆µ
µ
P(t, ξ)dξ
Dividing by ∆µand then letting ∆µ→0,we obtain
∂
∂t P(t, µ) = −ρP
∂
∂µ P(t, µ)+[βP(E)−δP(E, I )] P(t, µ),
and we have the boundary condition
P(t, 0) = RPE(t).
9 Class M(t, ν)
Class M(t, ν) consists of immature hematopoietic cells: proerythroblasts, basophilic erythroblasts,
polychromatic erythroblasts, orthochromatic erythroblasts, and non-circulating reticulocytes (i.e.
those that still reside in the bone marrow). Cells are recruited from class Pand, upon maturation,
feed into class O. Their development is influenced by EPO concentration.
We make the following assumptions:
(i) There is a smallest maturity level, ν0= 0,and a largest maturity level, νf.That is, 0 ≤ν≤νf.
(ii) The maturation rate depends on the level of erythropoietin and the maturity level. However,
for our initial model, we assume that maturation rate does not depend on the maturity level.
EPO stimulates maturation [6], so we use an increasing sigmoid function for maturation rate,
ρM(E).
ρM(E) = ρmin
M−ρmax
M(cρ,M )kρ,M
(cρ,M )kρ,M +Ekρ,M +ρmax
M.
11
EPO
from class P Maturing RBCs,
M(t, ν) to class O
maturation rate
Figure 8: Maturing erythrocytes, M(t, ν ).
(iii) The birth rate depends on the maturity level, but for our first model, we assume birth rate is
a constant, ˜
βM.
(iv) The number of cells at maturity level ν= 0 is equal to the number of cells leaving the previous
stage:
M(t, 0) = P(t, µF).
(v) The death rate depends on the maturity level, νand on the iron level. To simplify, we assume
the death rate is a constant, δM.
As in the progenitor class, we can consider the rate of change in population from maturity level νto
maturity level ν+ ∆ν, then divide by ∆νand let ∆ν→0 to obtain
d
dtM(t, ν) = −ρM(E)∂
∂ν M(t, ν ) + h˜
βM−δMiM(t, ν).
Since we have made the assumption that the birth and death rates are both constant, it is clear
that they will not both be identifiable. We replace the difference ˜
βM−δMby the constant βM,which
then represents the net birth rate.
Hence, we have
∂
∂t M(t, ν) = −ρM(E)∂
∂ν M(t, ν ) + βMM(t, ν),
with the boundary condition
M(t, 0) = P(t, µF).
It is worth noting again that as this is our first model of the system, we have made the assumption
that iron level does not impact cell development until cells mature out of class Minto class O.
Specifically, we do not account for iron entering red blood cells throughout class Mand we ignore
any impact this would have on death rate in class M. Future versions of the model will need to
account for these interactions with the iron compartment.
12
10 Class O
Unlike the classes Pand M, class Odoes not represent the number of circulating reticulocytes
and mature erythrocytes, because knowledge of the number of cells alone does not give us enough
information to determine whether the cells contain the necessary amount of hemoglobin to carry
oxygen at full capacity.
Erythrocytes begin hemoglobinization at the polychromatic erythroblast stage (in class M).
They continue to acquire more hemoglobin throughout the orthochromatic erythroblast stage and
into the reticulocyte stage, until the reticulocyte leaves the bone marrow, at which time it ceases
hemoglobinization [15]. Hence, the oxygen carrying ability of a mature erythrocyte is determined by
how much hemoglobin is available during the time interval when that cell is in class M.
In order to initially simplify computations, we assume that a cell’s oxygen-carrying ability is
based solely on the availability of iron at the time that the cell matures out of class Mand begins
circulating in the blood. As previously noted, the biology does not support this formulation of the
problem, and this assumption will be reconsidered in future models.
Let us consider an example in order to elucidate this idea. Suppose we know that kFe = 0.2
mg/billion cells and that at some given time t, Feavail = 8 mg. Suppose also that at time tthere are
100 billion cells maturing out of class M; that is, M(t, νf) = 100.Then
Feavail < Feneeded =kFeM(t, νf) = 20 mg.
Then, per equation (3), Feused =Feavail = 8 mg, which is only 40% of the 20 mg that would be
needed for each cell maturing into class Oto have full oxygen-carrying capacity. Then the 100 billion
cells maturing into class Owould have, on average, only 40% oxygen-carrying ability. It would be
difficult to track both the number of circulating erythroid cells and the oxygen carrying capacity of
each. Instead, we think of the 100 billion cells with 40% oxygen-carrying ability as 40 billion cells
with 100% oxygen-carrying capacity. Hence, every “cell” in class Ois assumed to have full-oxygen
carrying capacity.
Now we present the assumptions we make about class O.
(i) We assume that there is a smallest maturity level, ψ0= 0,and a largest maturity level, ψf.
That is, 0 ≤ψ≤ψf.In the future, we may wish to allow ψfto vary.
(ii) The maturation rate of cells in this class is a function of the maturity level. We will further
assume, for simplification in this initial model, that the maturity rate is constant:
dψ
dt =ρO
(iii) The birth rate is zero. Cells at this stage mature but do not proliferate [6].
(iv) The number of members of class Oat maturity level ψ= 0 is equal to the number of cells
leaving the previous stage multiplied by the ratio of Feused and Feneeded:
O(t, 0) = Feused
Feneeded
·M(t, νf),
=Feused
kFeM(t, νf)·M(t, νf),
=1
kFe
Feused.(5)
As stated above, this assumption guarantees that each member of class Ohas full oxygen-
carrying ability.
13
(v) The death rate of cells in the class O(t, ψ),depends on the maturity level. We expect this to
be an increasing function, because macrophages envelop mainly aging adult erythrocytes [15].
Therefore, we will use the increasing sigmoid function
δO(ψ) = δmin
O−δmax
O(cδ,O)kδ,O
(cδ,O)kδ,O +ψkδ,O +δmax
O.
As in classes Pand M, we can generate the partial differential equation
∂
∂t O(t, ψ) = −ρO
∂
∂ψ O(t, ψ)−δO(ψ)O(t, ψ)
with boundary condition (5).
11 Hemoglobin Concentration
We have already assumed that hemoglobin exists only in erythrocytes in class O. We compute the
total number of members in class Oat a given time tby
Zψf
0
O(t, ψ)dψ. (6)
We previously made the assumption that each member of class Ohas exactly the same amount
of iron. Specifically, if we multiply the quantity (6) by kFe,we have the amount of iron (in mg)
circulating in erythrocytes at time t. We then multiply by a conversion factor to find the amount
of hemoglobin circulating. Then we need only divide by blood volume, BV (t),to determine the
hemoglobin concentration.
Blood volume is difficult to determine and varies greatly in patients undergoing dialysis. Patients
in ESRD are unable to clear fluids from their bodies. Fluids, for the most part, build up in the
patient’s body between dialysis treatments. Therefore, we assume that blood volume increases
linearly between dialysis treatments and decreases linearly during a dialysis treatment. Initially we
simulate patients undergoing dialysis (1) three times per week (i.e. Monday-Wednesday-Friday, or
MWF), or (2) every third day (ETD), as in Figure 9.
0 2 4 6 8 10
4.5
5
5.5
time, days
blood volume, liters
Blood Volume,
MWF dialysis treatment
0 2 4 6 8 10
4.5
5
5.5
time, days
blood volume, liters
Blood Volume,
every third day dialysis treatment
Figure 9: Blood volume over various treatment protocols.
14
Hence, hemoglobin concentration is a nonlinear function of the amount of iron circulating,
Hb(t) = kFe Rψf
0O(t, ψ)dψ
BV (t).
12 Modification to the Model
We now discuss how we produce a smooth approximation to the piecewise-defined function
f(E, I ) = fE<EP Oth , E < EP Oth
1, E ≥EP Oth
where
fE<E P Oth =(cFe,av)kFe,av
(cFe,av)kFe,av +IkFe,av
.
For our initial simulations, we assume that inflammation remains constant. Hence, for a given
inflammation level, fis a step function that oscillates between 1 and the constant 0 ≤fE<E P Oth ≤1.
Rather than choose the constants cFe,av and kFe,av,we choose two parameters 0 < f1, f0.5<1
such that when E < E P Oth,
f(E, 1) = f1and f(E , 0.5) = f0.5.
Thus,
(cFe,av)kFe,av
(cFe,av)kFe,av + 1kFe,av =f1(7)
and
(cFe,av)kFe,av
(cFe,av)kFe,av + (0.5)kFe,av =f0.5.(8)
Then we solve (7) and (8) for the constants cFe,av and kFe,av :
kFe,av =ln f0.5+ ln (1 −f1)−ln f1−ln (f0.5)
ln 2
and
cFe,av =1
1−f1ln 2
ln f0.5+ln (1−f1)−ln f1−ln (f0.5)
.
We solve the EPO differential equation for a given treatment protocol. Then we use the solution
to determine times ti=ti(E) where EPO moves from above E P Oth to below EP Oth and vice versa,
as in Figure 10.
We approximate fwith
fs(E, I , t) = hshift +X
i
h(i, I)Hs
ti(t),
15
EPO level
time (days)
EPOth
t1t2t3t4t5t6
units of EPO
Figure 10: Determining the times tiwhere E(t) = E P Oth.
a linear combination of smoothed “heaviside” functions of the form
Hs
ti(t) = 1
2+1
2tanh (kheavy (t−ti))
=1
1 + e−2kheavy (t−ti).
Choice of the parameter kheavy determines the steepness of the approximation to each jump dis-
continuity. The coefficients h(i, E ) depend on (i) whether EPO level is passing from above E P Oth
to below or vice versa, and (ii) the value of the quantity fE<EP Oth ,which depends on the level of
inflammation.
Figure 11 shows an example of a function f(EPO three times per week, inflammation =0.5) with
two smooth approximations, kheavy = 15 and kheavy = 5.
This formulation yields a function fsthat is smooth, approximates f, and has a smooth derivative.
We replace fwith fsthroughout the model and therefore we use the parameters f1, f0.5and kheavy
in place of cFe,av and kFe,av.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.6
0.8
1
time (days)
f(E,I) (unitless)
f(E,I) and Smooth Approximations of f(E,I)
kheavy = 15
kheavy = 5
f(E,I)
Figure 11: A smooth approximation of the function f(E, I ).
16
13 Model Summary
In summary, we have the system
∂
∂t P(t, µ) = −ρP
∂
∂µ P(t, µ)+[βP(E)−δP(E, I )] P(t, µ),(9)
∂
∂t M(t, ν) = −ρM(E)∂
∂ν M(t, ν ) + βMM(t, ν),(10)
∂
∂t O(t, ψ) = −ρO
∂
∂ψ O(t, ψ)−δO(ψ)O(t, ψ),(11)
˙
Fe(t) = kFe Zψf
0
δO(ψ)O(t, ψ)dψ +˙
Feex(t)−ρFe→O−ρFe,loss (12)
˙
E(t) = ρEP O,basal +˙
Eex(t)−1
t1/2
ln 2·E(t),(13)
with boundary conditions
P(t, 0) = RPE(t),(14)
M(t, 0) = P(t, µF),(15)
O(t, 0) = 1
kFe
Feused,(16)
and initial conditions
P(0, µ) = P0(µ),(17)
M(0, ν) = M0(ν),(18)
O(0, ψ) = O0(ψ),(19)
Fe(0) = Fe0,(20)
E(0) = E0.(21)
Hemoglobin concentration is a nonlinear function of the amount of iron circulating,
Hb(t) = kFe Rψf
0O(t, ψ)dψ
BV (t).(22)
Hence, we have a nonlinear coupled system of ordinary and partial differential equations with
nontrivial boundary coupling with the following auxiliary equations.
17
βP(E) = βmin
P−βmax
P(cβ,P )kβ,P
(cβ,P )kβ,P +Ekβ,P +βmax
P(23)
δP(E, I ) = δmax
P,E −δmin
P,E (cδ,P,E )kδ,P,E
(cδ,P,E )kδ,P,E +Ekδ,P,E +δmin
P,E
+δmin
P,I −δmax
P,I (cδ,P,I )kδ,P,I
(cδ,P,I )kδ,P,I +Ikδ,P,I +δmax
P,I (24)
ρM(E) = ρmin
M−ρmax
M(cρ,M )kρ,M
(cρ,M )kρ,M +Ekρ,M +ρmax
M(25)
δO(ψ) = δmin
O−δmax
O(cδ,O)kδ,O
(cδ,O)kδ,O +ψkδ,O +δmax
O(26)
fs(E, I , t) = hshift +X
i
h(i, I)Hs
ti(t) (27)
Hs
ti(t) = 1
1 + e−2kheavy (t−ti)(28)
ρFe,loss(Fe) = ρFe,const, Fe ≥Feth
ρFe,frac ·Fe, Fe < Feth (29)
ρFe→O=kρ,FeFeused (30)
Feneeded =kFeM(t, νf).(31)
Feavail =kFe,eff f(E, I)Fe, (32)
Feused = min {Feneeded, Feavail }(33)
14 Parameter Value Considerations
•Treatment Protocol:
We perform simulations for two different “typical” treatment protocols: (1) a patient who goes
in for dialysis every third day (ETD) and (2) a patient on a Monday, Wednesday, and Friday
(MWF) treatment schedule. In both cases, dialysis is assumed to occur over a four-hour period
during which time 5000 units of EPO are assumed to be administered at a constant rate. For
those on the ETD schedule, iron is administered every ninth day; those on the the MWF
schedule receive iron every Monday. We assume a standard preparation of 62.5 mg iron per
administration.
•EPO:
The half-life t1/2of EPO is estimated to be 25 hours [15]. We assume the rate of EPO produced
by the body, ρEP O,basal ,is 100 units of EPO per day, chosen to be small relative to the amount
provided intravenously.
•Iron: For this set of simulations, we assumed that the net amount of exogenous iron entering
the system is equal to the net amount of iron losses in the system. For example, for a patient
on MWF treatment schedule, exogenous iron treatment is 62.5 mg of iron every seventh day;
therefore we assume that the rate of iron losses to be 62.5/7 mg iron per day.
18
•Blood volume:
Typical adult blood volume is between 4.5 and 5 liters. We assume that blood volume reaches
its minimum, 4.5 liters, at the end of the four hours of dialysis. For a patient undergoing ETD
treatment, we assume blood volume increases linearly to its maximum, 5 liters, just before they
start a dialysis treatment. This is also true for a patient on MWF treatment, except that we
assume the blood volume increases further, to 5.3 liters, over the weekend.
•Maturity Levels: Based on the literature [6], we assume µf= 3 and νf= 2.In healthy
individuals, red blood cells have an average life span of approximately 120 days. In patients in
ESRD, the life span of red blood cells is significantly shorter, so we assume that the maximum
maturity level in class Ois ψf= 120.
•kFe : In a healthy individual, each red blood cell (RBC) contains approximately 270 million
hemoglobin molecules (CITE). We use basic stoichiometry to determine kFe as follows:
kFe =270 ×106Hg molecules
1 RBC ·109RBCs
1 billion RBCs ·4 iron atoms
1 Hg molecule
·1 mol iron
6.022 ×1023 iron atoms ·55.845 grams iron
1 mol iron ·103mg iron
1 gram iron
= 0.10015 mg iron / billion RBCs
•Other parameters: The remaining parameters were given nominal values that produced ex-
pected numbers of cells in classes Pand M, and appropriate Hb concentrations. These param-
eters could be expected to vary among individuals. The remaining nominal parameter values
we use appear in Table 2 in Appendix A.
15 Numerical Solution Methodologies
We solve our system in a sequential manner, beginning with (13). We then solve (9) numerically, us-
ing the solution of (13) in the boundary condition, (14). Similarly, use this solution in the boundary
condition (15) to solve (10), and we use the solution of (10) when we solve (11) and (12) simultane-
ously. We solve using Matlab’s ode23t solver.
For equations (9), (10) and (11), we use a Galerkin finite element method. We outline this
procedure here for class Ponly, but the same procedure is used for classes Mand O.
Let
0 = µ1< µ2<· · · < µNP=µf
be a uniform partition of NP−1 subintervals, each of length hP=µf
NP−1.We define NPpiecewise
linear continuous functions
φj, j = 1,2, . . . , NP,
which we will call trial solution functions, by
φj(µ) =
µ−µj−1
hP
, µj−1≤µ≤µj,
µj+1 −µ
hP
, µj≤µ≤µj+1,
0, µ < µj−1or µ>µj+1.
We will also use a set of test functions ˜
φj.Typically, the test functions are identically the same
as the trial solution functions, but choice of these test functions is discussed later.
19
We make a weak formulation of (9) by multiplying by the jth test function and integrating over
all maturity levels:
Zµf
0
∂
∂t P(t, µ)˜
φj(µ)dµ =−ρPZµf
0
∂
∂µ P(t, µ)˜
φj(µ)dµ +Zµf
0
[βP(E)−δP(E, I )] P(t, µ)˜
φj(µ)dµ.
We assume that the solution P(t, µ) has the form
P(t, µ) =
NP
X
i=1
ai(t)φi(µ),(34)
and then manipulate the resulting equation to obtain
NP
X
i=1
a0
i(t)Zµf
0
φi(µ)˜
φj(µ)dµ =−ρPaNP(t)φj(µf) + ρPO(t, 0)φj(0)
+
NP
X
i=1
ai(t)"ρPZµf
0
φi(µ)˜
φj
0(µ)dµ
+ [βP(E)−δP(E, I )] Zµf
0
φA
i(µ)˜
φj(µ)dµ#.(35)
We can let jrange from 1 to NPto yield a system of NPordinary differential equations for the
coefficients ai(t).We solve this system, and reconstitute our solution using (34).
In order to validate our code, we implement a forcing function strategy. For class P, for example,
we solve a modified version of (9):
∂
∂t ˜
P(t, µ) = −ρP
∂
∂µ ˜
P(t, µ)+[βP(E)−δP(E , I)] ˜
P(t, µ) + F(t, µ).(36)
We choose a function such as ˜
P∗(t, µ) = 10e−t/2+ 15e−µ/3,which is smooth and decreases to
zero with increasing time and maturity level, then determine the forcing function Fthat guarantees
that ˜
P∗is the exact solution of (36). We solve (36) numerically and compare our solution with the
known exact solution.
It is well known that the solution to (9) will propagate along its characteristic curves, which we
can think of as a “wave front.” When we use standard linear splines φjas both the trial solution
functions and the test functions, we introduce error at this wave front, which is propagated in time.
In Figure 12a, we see that the error can become large and that standard linear splines are insufficient
to resolve the solution, as described in [8]. (We will discuss the error that is similar in both Figures
12a and 12b later.)
In order to alleviate this problem, we use a Petrov-Galerkin finite element method, also known
as upwinding. We continue using linear spline elements φjfor the trial solution functions, but for
the test functions we use second-order functions of the form φj+ωχj,where
χj(µ) =
(µ−µj−1)(µj−µ)
h2, µj−1≤µ≤µj,
−(µ−µj)(µj+1 −µ)
h2, µj≤µ≤µj+1,
0, µ < µj−1or µ>µj+1.
Figure 13 provides an example of standard test elements with varying levels of the upwinding pa-
rameter ω. Note that ω= 0 corresponds to no upwinding, or standard linear spline elements.
20
(a)
(b)
Figure 12: Error with and without upwinding. Exact solution is of the order 102.
When we solve (36) numerically using nonzero values of ω, and compare our solution with the
exact solution, we see significant improvement, as in Figure 12b.
We note that the small error seen in both Figure 12a and Figure 12b (that propagates along a
linear characteristic from t= 0 to approximately t= 3) is due to a high order discontinuity between
the boundary condition and initial condition at (t, µ) = (0,0).This error diminishes with use of a
finer mesh on the structure variable.
In order to determine an appropriate value for the parameter ω, we fix the number of elements
and solve (36). Figure 14 shows the error using several values for the upwinding parameter.
0
1
Test Elements φj + ωχj
ω = 0
ω = 0.5
ω = 1
μjμj+1
μj-1 h
Figure 13: Test basis elements, with varying values of the upwinding parameter ω.
21
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2x 10−5
maturity level, µ
error
Error at t = 15 days for various levels of upwinding
ωP = 1
ωP = 1.5
ωP = 2
ωP = 2.5
ωP = 3
ωP = 3.5
Figure 14: Effect of varying ωPon error between numerical and exact solution at t= 15 for N= 256
spatial elements. Exact solution is of the order 102.
We note that the error is of the same order for several values of the parameter and we choose to
continue our simulations with ωP= 2.5 as the upwinding parameter for class P.
As one final validation of our code, we sequentially increase the number of splines elements by a
factor of two to confirm that the numerical solution converges to the exact solution and to observe
the rate of convergence, which is essentially quadratic. The results appear in Table 1.
We repeat this process of validating the code for classes Mand O. The results appear in Appen-
dices B and C.
Table 1: Convergence of Solution–Maximum Error at t= 15 with ωP= 2.5 for an increasing number
of splines. Exact solution is of the order 102.
NPMaximum error (Max Error for NP)/(Max Error for 2NP)
4 0.1298 5.2451
8 0.0247 4.5190
16 0.0055 4.2395
32 0.0013 4.1153
64 3.1384e-04 4.0566
128 7.7365e-05 4.0280
256 1.9207e-05 4.0140
512 4.7850e-06 1.7820
1024 2.6851e-06
22
16 Numerical Results and Discussion
An example of our numerical results is shown next for a patient undergoing treatment on the ETD
schedule with inflammation level 0.5. As expected, in Figure 15 we see the patient’s EPO level
0 1 2 3 4 5 6 7 8 9
0
1000
2000
3000
4000
5000
6000
time, days
units of EPO
EPO, ETD treatment
Figure 15: EPO level over time for a patient with inflammation = 0.5 undergoing ETD treatment.
increases every third day when exogenous EPO is provided, and decays between treatments.
In Figure 16, we observe that the boundary condition at µ= 0 for class Pmimics the shape of
the EPO plot because the recruitment rate is directly proportional to the EPO level.
Figure 16: Number of cells in class Pfor a patient with inflammation = 0.5 undergoing ETD
treatment.
Cells in class Pmature and divide at a rate of approximately one division per day. Cells in class
Pmature into class M, which is pictured in Figure 17.
Comparing Figure 16 and 17, we see that, as expected, P(t, µf) = M(t, 0) for all t. Cells in class
Mdivide at a rate of approximately one division per day before they mature into class O.
23
Figure 17: Number of cells in class Mfor a patient with inflammation = 0.5 undergoing ETD
treatment.
As we consider Figure 18, it is worth noting that there has been a change in the time scale. This
plot actually closely resembles the plots for classes Pand M, but appears much different because
results are shown over a time period of 120 days as opposed to just two or three days; this is not an
example of the “noise” we discussed previously. The boundary condition, O(t, 0),is not identical to
M(t, νf) in Figure 17 because the number of cells maturing in to class Ois also dependent on how
much iron is available. Note also that cells in class Ono longer divide; they simply mature and die.
Figure 18: Number of cells in class Ofor a patient with inflammation = 0.5 undergoing ETD
treatment.
24
0 5 10 15 20 25
50
60
70
80
90
100
110
time, days
iron, mg
Iron Level (other than in RBCs)
Figure 19: Iron level for a patient with inflammation = 0.5 undergoing ETD treatment.
The amount of iron (other than that being carried in RBCs), in Figure 19, is seen to increase
greatly when exogenous iron is introduced (on days 0 and 9). Small increases are due to iron being
recycled from RBCs that have died.
Finally, Figure 20 shows the resulting hemoglobin concentration. The desired range, 11 to 13
g/dL, is also shown.
0 5 10 15 20 25
10.5
11
11.5
12
12.5
13
time, days
Hg, g/dL
Hemoglobin Concentration in g/dL
Figure 20: Hemoglobin concentration for a patient with inflammation = 0.5 undergoing ETD treat-
ment.
25
Figure 21 shows similar results for a patient at inflammation level 0.5 undergoing MWF treatment.
0 2 4 6 8 10 12 14
1000
2000
3000
4000
5000
6000
7000
time, days
units of EPO
EPO, MWF treatment
0 2 4 6 8 10 12 14 16 18 20
40
50
60
70
80
90
100
110
time, days
iron, mg
Iron Level (other than in RBCs)
0 2 4 6 8 10 12 14 16 18 20
10
10.5
11
11.5
12
12.5
13
time, days
Hg, g/dL
Hemoglobin Concentration in g/dL
Figure 21
26
0 2 4 6 8 10 12 14 16 18
9
9.5
10
10.5
11
11.5
12
12.5
13
time, days
hemoglobin conc, g/dL
Hb concentration with varying inflammation
infl = 0
infl = 0.25
infl = 0.5
infl = 0.75
infl = 1
0 2 4 6 8 10 12 14 16 18
20
40
60
80
100
120
140
160
180
time, days
mg of iron
Iron (other than in RBCs) with varying inflammation
infl = 0
infl = 0.25
infl = 0.5
infl = 0.75
infl = 1
Figure 22: Hemoglobin concentration and iron with varying inflammation, ETD treatment.
0 2 4 6 8 10 12 14
8.5
9
9.5
10
10.5
11
11.5
12
12.5
13
time, days
hemoglobin conc, g/dL
Hb concentration with varying inflammation
infl = 0
infl = 0.25
infl = 0.5
infl = 0.75
infl = 1
0 2 4 6 8 10 12 14
20
40
60
80
100
120
140
160
time, days
mg of iron
Iron (other than in RBCs) with varying inflammation
infl = 0
infl = 0.25
infl = 0.5
infl = 0.75
infl = 1
Figure 23: Hemoglobin concentration and iron with varying inflammation, MWF treatment.
We also present some results of varying the inflammation level for both ETD (Figure 22) and
MWF treatment (Figure 23).
It should be noted here that the solutions are dependent on the initial conditions, and therefore
careful choice of initial conditions must be made in order to produce results that biologically rea-
sonable. For example, if one starts with an an initial condition O(t, 0) that is large, then the iron
being carried in those cells eventually ends up in the iron compartment, which in turn affects the
recruitment rate into class O. As a result, it is possible, for example, to produce a set of simulations
such that the hemoglobin concentration for a patient with inflammation level 0.5 is actually higher
than for a patient with a lower inflammation level.
For each treatment protocol, we also compare iron needed, Feneeded and iron available, Feavail,
over time at various levels of inflammation so that we might understand the dynamics of iron in
determining the number of cells maturing in to class O(see Figures 24 and 25). It is worth noting
again that the solutions (in particular iron available) are dependent on the initial conditions.
27
0 2 4 6 8 10 12 14 16 18
0
20
40
60
80
100
120
time, days
mg of iron
Iron available Compared with Iron Needed, inflammation = 0
Available
Needed
0 2 4 6 8 10 12 14 16 18
0
20
40
60
80
100
120
time, days
mg of iron
Iron available Compared with Iron Needed, inflammation = 0.25
Available
Needed
0 2 4 6 8 10 12 14 16 18
0
20
40
60
80
100
120
time, days
mg of iron
Iron available Compared with Iron Needed, inflammation = 0.5
Available
Needed
0 2 4 6 8 10 12 14 16 18
0
20
40
60
80
100
120
time, days
mg of iron
Iron available Compared with Iron Needed, inflammation = 0.75
Available
Needed
0 2 4 6 8 10 12 14 16 18
0
20
40
60
80
100
120
time, days
mg of iron
Iron available Compared with Iron Needed, inflammation = 1
Available
Needed
Figure 24: Feneeded and Feavail with varying inflammation, ETD treatment.
17 Summary
The model presented is capable of describing patients over a broad range of conditions, including
various inflammation levels and treatment protocols. Parameter values were prescribed per the
literature, when available, and numerical testing validates the use of our code in solving the model.
Some numerical results are presented, making note that the results are highly dependent on the
initial conditions, which are estimated.
This model makes significant simplifications with regard to how iron is assimilated into red blood
cells. As such, the next step will be to revisit the model to determine a more biologically reasonable
way to incorporate iron, perhaps through the use of another structure variable which will account
for the amount of iron in a cell.
This paper did not attempt to address the concept of iron homeostasis, but it is also becoming
more and more evident that understanding iron homeostasis will play a role in this model’s ability
to predict the outcomes of treatment protocols. It appears that hepcidin production is regulated,
28
0 2 4 6 8 10 12 14
0
10
20
30
40
50
60
70
80
90
100
time, days
mg of iron
Iron available Compared with Iron Needed, inflammation = 0
Available
Needed
0 2 4 6 8 10 12 14
0
10
20
30
40
50
60
70
80
90
100
time, days
mg of iron
Iron available Compared with Iron Needed, inflammation = 0.25
Available
Needed
0 2 4 6 8 10 12 14
0
10
20
30
40
50
60
70
80
90
100
time, days
mg of iron
Iron available Compared with Iron Needed, inflammation = 0.5
Available
Needed
0 2 4 6 8 10 12 14
0
10
20
30
40
50
60
70
80
90
100
time, days
mg of iron
Iron available Compared with Iron Needed, inflammation = 0.75
Available
Needed
0 2 4 6 8 10 12 14
0
10
20
30
40
50
60
70
80
90
100
time, days
mg of iron
Iron available Compared with Iron Needed, inflammation = 1
Available
Needed
Figure 25: Feneeded and Feavail with varying inflammation, MWF treatment.
at least in part, by the rate of erythropoiesis [5], but hepcidin has only recently been discovered as
the major regulator of iron homeostasis, and therefore its action is only partially understood as the
body of literature on hepcidin is relatively small.
This model does address, for the first time, the roles inflammation and iron play in red blood cell
dynamics, which are known to have a significant impact on red blood cell dynamics in patients with
CKD.
Acknowledgements
This research was supported in part by Grant Number R01AI071915-07 from the National Insti-
tute of Allergy and Infectious Diseases.
29
References
[1] A. Ackleh, K. Deng, K. Ito, and J. Thibodeaux. A structured erythropoiesis model with non-
linear cell maturation velocity and hormone decay rate. Mathematical Biosciences, 204(1):21 –
48, 2006.
[2] Suhail Ahmad. Manual of Clinical Dialysis, Second Edition. Springer Science + Business Media,
LLC, New York, 2009.
[3] H. T. Banks, C. Cole, P. Schlosser, and H. T. Tran. Modeling and optimal regulation of erythro-
poiesis subject to benzene intoxication. Mathematical Biosciences and Engineering, 1(1):15–48,
2004.
[4] J. Belair, M. Mackey, and J. Mahaffy. Age-structured and two-delay models for erythropoiesis.
Mathematical Biosciences, 128(1-2):317 – 346, 1995.
[5] Tomas Ganz. Iron homeostasis: Fitting the puzzle pieces together. Cell Metabolism, 7:288–290,
2008.
[6] L. Israels and E. Israels. Erythropoiesis: an overview. In Erythropoietins and Erythropoiesis:
Molecular, Cellular, Preclinical and Clinical Biology, pages 3–14. Birkh¨auser Verlag, 2003.
[7] W. Jelkmann. Erythropoietin: structure, control of production, and function. Physiological
reviews, 72(2):449–489, 1992.
[8] Shengtai Li and Linda Petzold. Moving mesh methods with upwinding schemes for time-
dependent pdes. Journal of Computational Physics, 131:368–377, 1997.
[9] I. Macdougall. Use of recombinant erythropoietins in the setting of renal disease. In S.G. Elliot
G. Molineaux, M.A. Foote, editor, Erythropoietins and Erythropoiesis: Molecular, Cellular,
Preclinical and Clinical Biology, pages 153–159. Birkh¨auser Verlag, 2003.
[10] I. Macdougall and A. Cooper. Erythropoietin resistance: the role of inflammation and pro-
inflammatory cytokines. Nephrol. Dial. Transplant., 17:39–43, 2002.
[11] J. Mahaffy, J. Belair, and M. Mackey. Hematopoietic model with moving boundary condi-
tion and state dependent delay: applications in erythropoiesis. Journal of Theoretical Biology,
190(2):135–146, 1998.
[12] R. Means Jr. and S. Krantz. Inhibition of human erythroid colony-forming units by interfer-
ons alpha and beta: differing mechanisms despite shared receptor. Experimental Hematology,
24(2):204–208, 1996.
[13] National Kidney Foundation Kidney Disease Outcomes Quality Initiative. KDOQI Clinical
Practice Guidline and Clinical Practive Recommendations for Anemia in Chronic Kidney Dis-
ease: 2007 Update of Hemoglobin Target. American Journal of Kidney Diseases, 50(3):471–530,
2007.
[14] R. Ramakrishnan, W. Cheung, M. Wacholtz, N. Minton, and W. Jusko. Pharmacokinetic and
pharmacodynamic modeling of human erythropoietin after single and multiple doses in healthy
volunteers. The Journal of Clinical Pharmacology, 44:991–1002, 2004.
[15] A. Simmons. Basic Hematology. Charles C Thomas, Springfield, Illinois, 1973.
30
[16] P. Stenvinkel. The role of inflammation in the anaemia of end-stage renal disease. Nephrology,
Dialysis, Transplantation, 16(Suppl 7):36–40, 2001.
[17] U.S. Renal Data System. USRDS 2008 Annual Data Report: Atlas of Chronic Kidney Disease
and End-Stage Renal Disease in the United States. National Institutes of Health, National
Institute of Diabetes and Digestive and Kidney Diseases, Bethesda, MD, 2008.
[18] S. Woo, W. Krzyzanski, and W. Jusko. Pharmacokinetic and pharmacodynamic modeling of
recombinant human erythropoietin after intravenous and subcutaneous administration in rats.
Journal of Pharmacology and Experimental Therapeutics, 319(3):1297–1306, 2006.
31
Appendices
A Model Parameter Values
Table 2: Model Parameters and Units
Parameter Units Parameter value
ρPday−11
βmax
Pday−10.2
βmin
Pday−10.1
cβ,P unitless 5250
kβ,P unitless 5
δmax
P,E day−10.03
δmin
P,E day−10
cδ,P,E unitless 1800
kδ,P,E unitless 6
δmax
P,I day−10.05
δmin
P,I day−10
cδ,P,I unitless 0.75
kδ,P,I unitless 7
RPbillions of cells/unit EPO 0.018
ρmax
Mday−11.2
ρmin
Mday−11
cρ,M unitless 5500
kρ,M unitless 10
βMday−10.25
ρOday−11
δmax
Oday−10.13
δmin
Oday−10
cδ,O unitless 80
kδ,O unitless 7
kFe,eff unitless 0.65
f1unitless 0.3
f0.5unitless 0.6
kheavy unitless 5
EP Oth units of EPO 3000
kρ,Fe day−10.993
ρFe,frac day−10.3
Feth mg 15
32
B Code validation for class M
As in class P, we note that if we don’t use upwinding, the error can propagate in time and overwhelm
the solution, as demonstrated by comparing the numerical solutions for class M(using the forcing
procedure described perviously) with and without upwinding, as in Figures 26a and 26b.
(a) (b)
Figure 26: Error with and without upwinding. Exact solution is of the order 102.
In order to determine an appropriate value for the parameter ωM,we fix the number of elements
and compare the error using several values for the upwinding parameter.
0 0.5 1 1.5 2 2.5 3
0
1
2
3
4
5
6
7
8
9x 10−6
maturity level, μ
error
Error at t = 15 days for various levels of upwinding
ωM = 0.5
ωM = 1
ωM = 1.5
ωM = 2
ωM = 2.5
ωM = 3
Figure 27: Effect of varying ωMon error between numerical and exact solution at t= 15 for N= 256
spatial elements. Exact solution is of the order 102.
We note that the error is of the same order for several values of the parameter and we choose to
continue our simulations with ωM= 2 as the upwinding parameter for class M.
As before, we sequentially increase the number of splines elements by a factor of two to confirm
that the numerical solution converges to the exact solution. The results appear in Table 3.
33
Table 3: Convergence of Solution–Maximum Error at t= 15 with ωM= 2 for an increasing number
of splines. Exact solution is of the order 102.
NMMaximum error (Max Error for NM)/(Max Error for 2NM)
4 0.0614 5.4641
8 0.0112 4.5726
16 0.0025 4.2623
32 5.7693e-04 4.1259
64 1.3983e-04 4.0617
128 3.4427e-05 1.0000
256 3.4427e-05 16.1836
512 2.1273e-06 2.4047
1024 8.8464e-07
C Code validation for class O
Unlike classes Pand M, at time t= 15 days, in Figure 28 we are not able to see the error over-
whelming the solution without the use of upwinding.
(a) (b)
Figure 28: Error with and without upwinding. Exact solution is of the order 102.
It appears there may not be an advantage to using upwinding in this case. We continue in-
vestigating by fixing the number of elements and comparing the error using several values of the
upwinding parameter ωO.
We note that the error is of the same order for several values of the parameter, including for no
upwinding. We choose to continue our simulations with upwinding (for consistency with the other
classes), using ωO= 3.5 as the upwinding parameter for class O.
As before, we sequentially increase the number of splines elements by a factor of two to confirm
that the numerical solution converges to the exact solution. The results appear in Table 4.
34
0 20 40 60 80 100 120
−6
−4
−2
0
2
4
6
x 10−4
maturity level, ψ
error
Error at t = 15 days for various levels of upwinding
ωO = 0
ωO = 1
ωO = 2
ωO = 3
ωO = 3.5
ωO = 4
Figure 29: Effect of varying ωOon error between numerical and exact solution at t= 15 for N= 1201
spatial elements. Exact solution is of the order 102.
Table 4: Convergence of Solution–Maximum Error at t= 15 with ωO= 3.5 for an increasing number
of splines. Exact solution is of the order 102.
NMMaximum error (Max Error for NO)/(Max Error for 2NO)
8 4.6086 2.6825
16 1.7180 3.1906
32 0.5385 3.5164
64 0.1531 3.7385
128 0.0410 3.8632
256 0.0106 3.9299
512 0.0027 3.9645
1024 6.8052e-04 3.9821
2048 1.7089e-04 3.9911
4096 4.2819e-05
35
