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Modeling Red Blood Cell and Iron Dynamics in Patients
with Chronic Kidney Disease
H. T. Banks1, Karen M. Bliss2and Hien Tran1
1Center for Research in Scientific Computation
North Carolina State University
Raleigh, NC 27695
2United States Military Academy
West Point, NY 10996
February 10, 2012
Abstract
Chronic kidney disease causes a slow loss of kidney function over time and can even-
tually lead to End Stage Renal Disease, where a patient must undergo dialysis to remove
fluids and wastes from the body. These patients also suffer from a lack of the hormone
erythropoietin (EPO), produced in the kidneys, that stimulates red blood cell (RBC)
production. Without intervention, patients suffer from anemia. Patients are treated
with both EPO and iron in order to stimulate RBC production.
We develop a partial differential equation model for RBC dynamics using two struc-
ture variables, one for age and one for cellular iron endowment. We couple this with
a set of ordinary differential equations modeling iron dynamics. We take into account
the effects of both inflammation and neocytolysis, which are known to affect patients
undergoing treatment.
Keywords: mathematical model, mathematical biology, erythropoiesis, erythrocyte, red
blood cell, chronic kidney disease, dialysis, iron, neocytolysis, hepcidin, EPO, hemoglobin
e-mails: htbanks@ncsu.edu, kmbliss@ncsu.edu, tran@ncsu.edu
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Modeling Red Blood Cell and Iron Dynamics in Patients with Chronic
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14. ABSTRACT
Chronic kidney disease causes a slow loss of kidney function over time and can even- tually lead to End
Stage Renal Disease, where a patient must undergo dialysis to remove uids and wastes from the body.
These patients also su er from a lack of the hormone erythropoietin (EPO), produced in the kidneys, that
stimulates red blood cell (RBC) production. Without intervention, patients su er from anemia. Patients are
treated with both EPO and iron in order to stimulate RBC production. We develop a partial di erential
equation model for RBC dynamics using two struc- ture variables, one for age and one for cellular iron
endowment. We couple this with a set of ordinary di erential equations modeling iron dynamics. We take
into account the e ects of both in ammation and neocytolysis, which are known to a ect patients undergoing
treatment.
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ABSTRACT
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Standard Form 298 (Rev. 8-98)
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1 Introduction
It is estimated that 31 million Americans have chronic kidney disease (CKD). Among those,
approximately 330 thousand were classified as being in End-Stage Renal Disease (ESRD) and
required dialysis [42] to remove wastes and fluids from the blood. 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 mecha-
nism in the kidneys, but patients in ESRD do not produce sufficient levels of EPO to maintain
blood hemoglobin concentration. Hemoglobin is the protein that gives red blood cells the abil-
ity 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) intravenously 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 [42], the desired minimum level set by the National Kidney Foun-
dation [29].
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
supplementation has become a mainstay in many patients undergoing rHuEPO therapy [25].
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.) [26].
We develop a model for red blood cell populations using two structure variables, one for
age-structuring and one to account for a cell’s iron endowment. We couple these partial
differential equations with a system of ordinary differential equations that models the iron
cycle.
2 Previous models
The process of erythropoiesis has been modeled in several physiological scenarios. In [31],
rHuEPO therapy is considered in healthy individuals. This model incorporates the negative
feedback to endogenous EPO production. EPO is assumed to be cleared using Michaelis-
Menten dynamics. A similar model was used to fit data in rats [46]. Both of these models use
delay instead of age-structured modeling.
Both [6] and [7] use age-structured models, as does the model described in [27], which
assumes that the oldest mature erythrocytes will be destroyed, yielding a moving boundary
condition. In [2], 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 incorpo-
rates iron. Additionally, we have accounted for the effects of inflammation and neocytolysis,
a down-regulating mechanism by which red blood cells are selectively destroyed when blood
2
oxygen level rises.
The authors have previously presented a more simplified version of this model in [5, 8],
including simulations. The simplified model is significantly less sophisticated; assumptions
were made there so that simulations would be more tractable.
3 Overview of Red Blood Cell Production
Erythrocytes are produced primarily from stem cells in bone marrow. In the presence of
certain hormones, stem cells divide asymmetrically, producing a committed colony-forming-
unit (CFU) while maintaining the population of stem cells. Erythrocyte lineage continues
as depicted in Figure 1: erythroid burst-forming unit (BFU-E), erythroid colony-forming-
unit (CFU-E), proerythrocyte, basophilic erythrocyte, polychromatic erythroblast, orthochro-
matic erythroblast, reticulocyte, and erythrocyte.
CFU-GEMM
C
BFU-E
CFU-E
Proerythroblast
Basophilic erythroblast
Polychomatic erythroblast
Orthochromatic erythroblast
Reticulocyte
Erythrocyte
Figure 1: Erythropoiesis cell lineage.
Hemoglobin is synthesized beginning in the basophilic erythrocyte stage, with the majority
of synthesis occurring 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 [20]. Reticulo-
cytes begin to lose the adhesive proteins that hold them in the bone marrow. They decrease
in size and begin to circulate in the blood. In healthy individuals, erythrocyte life span is
approximately 120 days, at which time aging erythrocytes are enveloped by macrophages in
the spleen.
3
In a healthy individual, an increase in blood oxygen level is detected by the kidneys, which
in turn decrease production of EPO. Down-regulation of the RBC production is achieved by
the increased death rate of RBC progenitors due to this decrease in EPO. However, it takes
18 to 21 days for cells to mature and be released into circulation, so this down-regulation has
a long delay before it actually affects blood oxygen level. Additionally, down-regulation would
only cause the circulating red cell mass to drop by about 1% per day [32].
Neocytolysis has been observed in astronauts entering space [4, 43] and in high altitude
dwellers who descend to sea level [28, 34]; all have higher aggregate RBC mass than needed.
Their RBC mass was decreased by 10-15% in a few days, which could not be explained by
simple down-regulation of RBC production.
Neocytolysis, believed to be caused by a drop in EPO level, is a physiological process
that aids in this control of the red cell mass by causing the selective destruction of young
circulating RBCs [11, 32]. Since these cells were circulating and contributing to blood oxygen
level, their death has a much faster effect on blood oxygen level than the down-regulation of
the production cycle. Neocytolysis has been determined to contribute to the anemia of renal
disease [3, 33], as patients undergoing therapy have constantly fluctuating EPO levels.
4 RBC Model
Cells in the RBC lineage divide, differentiate and die under the influence of several hormones.
Figure 2, a modification of a figure in [22], depicts different aspects of the system we will model.
This figure contains possible variables that one might use in formulating mathematical models
for red blood cell production. In particular, this figure suggests five cell classes:
•P1: early BFU-E
•P2: late BFU-E, CFU-E, proerythroblasts, and early basophilic erythroblasts
•P3: late basophilic erythroblasts and early polychromatic erythroblasts
•P4: late polychromatic erythroblasts, orthochromatic erythroblasts, and non-circulating
reticulocytes
•P5: mature erythrocytes and circulating reticulocytes.
We discuss each compartment individually in Sections 4.2 through 4.6, but it should be
noted that the original system chosen to name the erythroid cell progression was based on
distinguishing characteristics when samples were taken, stained, and viewed through a mi-
croscope. It is not surprising that we choose our five classes of cells differently as we focus
on the interactions with EPO and iron and the location of the cells (in the bone marrow or
circulating).
The current RBC lineage model is represented in Figure 3. We introduce the model for
the iron compartments in Section 5.
4
Figure 2: Stages of red blood cell production. We use the following short-hand notations:
BFU-E = burst-forming unit–erythroid, CFU-E = colony-forming unit–erythroid, ProEB =
proerythroblast, BasoEB = basophilic erythroblast, PolyEB = polychromatic erythroblast,
OrthoEB = orthochromatic erythroblast, Retic = reticulocyte, RBC = red blood cell.
Figure 3: Model schematic. Note that the states variables are P1(t, µ1), P2(t, µ2), P3(t, µ3, γ),
P4(t, µ4, γ), P5(t, µ5, γ ) and EP O(t).The iron model is described in Section 5.
5
4.1 Derivation of Model Equations
We use a model with two structure variables, one for maturity level and one for cellular iron
level. The structure variables µiand γare unitless and represent the maturity level and iron
state, respectively, of cells. Thus, P3(t, µ3, γ) is the number of cells (in billions) in population
P3at time tthat have maturity level µ3and iron state γ. Each class is assumed to have a
maximum maturity level, (µi)f.
Throughout, we use βto represent a birth/proliferation rate, δto represent a death rate,
and ρto represent other rates (maturation rates, rates of treatment, etc.).
We will assume that the iron state structure variable γvaries from 0 to γf= 2, where
γfull =γf
2= 1 represents the amount of iron the “typical” RBC contains in a healthy individual
(this quantity is to be estimated later). We will assume that when a cell divides, each daughter
cell inherits half of the parent cell’s iron. Therefore, when a cell with iron state γdivides, it
leaves iron state γand two cells enter iron state γ
2.Similarly, for each cell in iron class 2γthat
divides, two cells enter iron class γ.
We derive equations for those classes that incorporate iron by considering, at some time
t, an incremental “area” of size ∆µby ∆γ, with 0 ≤γ≤γf
2= 1,as in Figure 4.
Figure 4: Maturation and hemoglobinization processes.
We treat this region as a arbitrary elemental compartment and consider the time rate of
change of the cell population in the compartment.
6
∂
∂t Zγ+∆γ
γZµ+∆µ
µ
P(t, ξ, ζ )dξdζ = 2(rate entering iron class γfrom class 2γ)
−(rate of cells leaving iron class γ)
−(death rate) + (rate of maturation in)
−(rate of maturation out)
+ (rate of hemoglobinization in)
−(rate of hemoglobinization out)
= 2 Zγ+∆γ
γZµ+∆µ
µ
βP (t, ξ, 2ζ)dξdζ
−Zγ+∆γ
γZµ+∆µ
µ
βP (t, ξ, ζ)dξdζ
−Zγ+∆γ
γZµ+∆µ
µ
δP (t, ξ, ζ)dξdζ
+Zγ+∆γ
γ
ρP (t, µ, ζ)dζ −Zγ+∆γ
γ
ρP (t, µ + ∆µ, ζ )dζ
+Zµ+∆µ
µ
hP (t, ξ, γ)dξ −Zµ+∆µ
µ
hP (t, ξ, γ + ∆γ)dξ,
where 0 ≤µ≤µf,βis the proliferation rate, δis the death rate, ρis the maturation rate,
and his the hemoglobinization rate (equivalent to the rate of iron uptake), which generally
depends on the plasma iron level Fepl.
We rearrange these to obtain
∂
∂t Zγ+∆γ
γZµ+∆µ
µ
P(t, ξ, ζ )dξdζ = 2 Zγ+∆γ
γZµ+∆µ
µ
βP (t, ξ, 2ζ)dξdζ
−Zγ+∆γ
γZµ+∆µ
µ
[β+δ]P(t, ξ, ζ )dξdζ
−Zγ+∆γ
γ
ρ[P(t, µ + ∆µ, ζ )−P(t, µ, ζ)] dζ
−Zµ+∆µ
µ
h[P(t, ξ, γ + ∆γ)−P(t, ξ, γ)] dξ,
7
then divide by ∆µ∆γ:
∂
∂t 1
∆γZγ+∆γ
γ
1
∆µZµ+∆µ
µ
P(t, ξ, ζ )dξdζ=2
∆γZγ+∆γ
γ
1
∆µZµ+∆µ
µ
βP (t, ξ, 2ζ)dξdζ
−1
∆γZγ+∆γ
γ
1
∆µZµ+∆µ
µ
[β+δ]P(t, ξ, ζ )dξdζ
−1
∆γZγ+∆γ
γ
ρP(t, µ + ∆µ, ζ )−P(t, µ, ζ)
∆µdζ
−1
∆µZµ+∆µ
µ
hP(t, ξ, γ + ∆γ)−P(t, ξ, γ)
∆γdξ.
When we allow ∆µ→0 and ∆γ→0,we obtain
∂
∂t P(t, µ, γ)=2βP (t, µ, 2γ)−[β+δ]P(t, µ, γ )−∂
∂µ ρP (t, µ, γ)
−∂
∂γ hP (t, µ, γ ),0≤γ≤γf
2= 1,(1)
the general form of the equations governing cells in classes P3through P5.Note that the rates
β, δ, ρ and hcould be functions that depend on µ, γ or other states in the system.
By our assumptions, the maximum iron state is γf= 2,so cell division cannot yield a cell
with iron state γ > γf
2= 1.Thus, for cells with iron states greater than γf
2= 1,we omit the
birth rate term associated with iron state 2γ, as below:
∂
∂t P(t, µ, γ) = −[β+δ]P(t, µ, γ )−∂
∂µ ρP (t, µ, γ)−∂
∂γ hP (t, µ, γ ),γf
2= 1 ≤γ≤γf= 2.
(2)
Thus our generic balance laws for the P3through P5compartments can be written (using
the characteristic function χfor the interval [0,1], i.e., χ[0,1](γ) = 1 when γ∈[0,1] and
χ(γ) = 0 otherwise) as
∂
∂t P(t, µ, γ) = χ[0,1] (γ)2βP (t, µ, 2γ)−[β+δ]P(t, µ, γ)−∂
∂µ ρP (t, µ, γ)
−∂
∂γ hP (t, µ, γ ),0≤γ≤2.(3)
Cells in classes P1(t, µ1) and P2(t, µ2) do not incorporate iron, so they are functions of
only time and maturity level. The general form of the partial differential equations governing
these classes, derived in [5], is
∂
∂t P(t, µ) = [β−δ]P(t, µ)−∂
∂µ ρP (t, µ),(4)
where βis the proliferation rate, δis the death rate, and ρis the maturation rate.
8
4.2 Class P1(t, µ1)
Class P1consists of early BFU-E. These are the most immature cells that are committed to
the erythroid lineage. We make the following assumptions about cells in this class.
1. The maturation rate is constant, ρ1.
2. BFU-E differentiate into CFU-E in approximately seven days [35, 38]. BFU-E begin
expressing EPO receptors (EPORs), which means that, in time, they do become influ-
enced by EPO. Hence, we allow for cells to reside in class P1for three days (during
which they are not influenced by EPO), while late BFU-E are in class P2(where they
are under the influence of EPO). Therefore we choose the maximum maturity level to
be (µ1)f= 3.
3. The majority of cell proliferation happens in later classes [22, 23], so we set birth rate
equal to death rate for cells in this class.
Hence, for class P1,equation (4) becomes
∂
∂t P1(t, µ1) = −ρ1
∂
∂µ1
P1(t, µ1) (5)
with initial condition
P1(0, µ1) = Pinit
1(µ1)
and boundary condition at µ1= 0
P1(t, 0) = Pbdy
1(t).
4.3 Class P2(t, µ2)
Class P2consists of late BFU-E, CFU-E, proerythroblasts, and early basophilic erythroblasts.
We make the following assumptions about cells in class P2.
1. The maturation rate is constant, ρ2.
2. Cells reside in this class for twelve days [17, 35], and therefore (µ2)f= 12.
3. During the time cells reside in class P2,they undergo approximately 6 cell divisions
[17, 18, 19, 22]. We assume that the birth rate is a constant, β2.
4. Cells in this class express EPORs [23, 37], and this interaction of EPO with EPOR is the
most important control point for erythropoiesis [17, 36, 47]. Cells at this stage depend
absolutely on EPO for their survival and will undergo apoptosis in its absence [23, 38].
Even in healthy individuals, complete survival of progenitors would require an EPO level
much higher than normal level; the normal production rate of RBCs represents survival
of only a minority of progenitor cells [22]. Therefore we choose the death rate to be a
decreasing function of the EPO level:
δ2(EP O ) = δmax
2−δmin
2·cδ2kδ2
cδ2kδ2+EP O kδ2
+δmin
2.
9
Thus, using equation (4), the state equation for class P2is given by
∂
∂t P2(t, µ2)=[β2−δ2(EP O)]P2(t, µ2)−ρ2
∂
∂µ2
P2(t, µ2),(6)
with initial condition
P2(0, µ2) = Pinit
2(µ2)
and boundary condition at (µ1)f= 3, µ2= 0
P2(t, 0) = P1(t, 3).
4.4 Class P3(t, µ3, γ)
Class P3consists of late basophilic erythroblasts and early polychromatic erythroblasts. We
assume the following.
1. The maturation rate is constant, ρ3.
2. Cells reside in this class for 1 day [17, 35], and therefore (µ3)f= 1.
3. Cells in this stage continue to proliferate [17], with a constant rate of β3= ln 2 days−1,
which corresponds to one cell division per day.
4. Cells in class P3continue to express EPORs, but the level of expression declines sig-
nificantly as cells mature through this class [12]. Hence, cells in this class become less
dependent on EPO for survival. We model this as
δ3(µ3, E P O) = δmax
3(µ3)·cδ3kδ3
cδ3kδ3+EP O kδ3
where δmax
3(µ3) is a decreasing function, which we assume to be affine for this model
given by
δmax
3(µ3) = −m3µ3+b3,where m3>0, b3−m3·(µ3)f= 0.
The function δ3(µ3, E P O),for a fixed maturity level µ3,is a decreasing sigmoid function
of EPO. That is, at a fixed maturity level, as EPO increases, the death rate decreases. If
we consider instead a fixed EPO level, then increasing maturation level causes a decrease
in the maximum death rate. That is, low EPO affects younger cells in this class more
than it affects more mature cells. Both the fixed maturity level and fixed EPO level
phenomena are depicted in Figure 5.
5. Cells in class P3express transferrin receptors (Tfr) and begin the process of taking in iron
and synthesizing hemoglobin [17, 22]. The rate of hemoglobinization (or equivalently
iron uptake) is a function of iron level Fepl (in the blood plasma) and iron state γ(of a
cell). A full description of the hemoglobinization rate, h3(Fepl, γ),along with definition
of the parameter Fepl appears in Section 4.7.
10
Figure 5: Death rate in class P3is a function of both maturity level, µ3,and EPO level. For
a fixed EPO level, death rate decreases as maturity level increases. For a fixed maturity level,
death rate is larger when EPO is small. These plots were generated with parameter values
m3= 0.2, b3= 0.2, cδ3= 3000 and kδ3= 20.
Hence, for class P3,equation (3) becomes
∂
∂t P3(t, µ3, γ) = χ[0,1] (γ)2β3P3(t, µ3,2γ)−[β3+δ3(µ3, E P O)]P3(t, µ3, γ)−ρ3
∂
∂µ3
P3(t, µ3, γ)
−∂
∂γ h3(Fepl, γ)P3(t, µ3, γ),0≤γ≤2,(7)
with initial condition
P3(0, µ3, γ) = Pinit
3(µ3, γ)
and boundary conditions at (µ2)f= 12, µ3= 0
P3(t, 0, γ) = P2(t, 12), γ = 0
0, γ 6= 0
and
P3(t, µ3,0) = P2(t, 12), µ3= 0
Pbdy,γ
3(t, µ3), µ36= 0.
4.5 Class P4(t, µ4, γ)
Class P4contains late polychromatic erythroblasts, orthochromatic erythroblasts, and non-
circulating reticulocytes, the last stages of erythroid cells residing in the bone marrow. We
make the following assumptions about cells in class P4.
11
1. The maturation rate is constant, ρ4.
2. Cells reside in this class for 2 days [17, 35], and therefore (µ4)f= 2.
3. Cells in this class have stopped proliferating [17, 22]. Thus, β4= 0.
4. It is well-documented that red blood cells have a shorter life span in individuals with
iron deficiency [10, 13, 14, 44]. Hence, we assume that any RBCs in class P4that are
severely iron deficient have increased mortality.
We begin by defining cδ4(µ4) as an increasing function of µ4.For simplicity, we choose
an affine function,
cδ4(µ4) = m4µ4+b4, m4>0, b4≥0,
with the added restriction that 0 < cδ4(µ4)<(µ4)ffor all µ4.
We assume now that death rate depends on maturity level and cellular iron state, as
below :
δ4(µ4, γ) = δmax
4−δmin
4·cδ4(µ4)kδ4
cδ4(µ4)kδ4+γkδ4
+δmin
4.
Notice that when γis relatively small, δ4(µ4, γ) is close to δmax
4,and when γis relatively
large, δ4(µ4, γ) is close to δmin
4.However, the “relativity” is affected by cδ4(µ4) when
γ < γcrit,4.As cells mature through class P4(i.e., as µ4increases), then cδ4(µ4) increases,
which means that the death rate for a given iron level γalso increases (we choose
kδ4>1). Hence, a cell having only a little iron when it arrives in class P4is less likely
to die than a cell with that same level of iron that is about to mature out of class P4.
Figure 6 depicts δ4(µ4, γ) over varying µ4and γ.
5. Cells in class P4continue to collect iron and synthesize hemoglobin [17, 22] at a rate
h4(Fepl, γ),explained further in Section 4.7.
Thus, using equation (3), the state equation is given by
∂
∂t P4(t, µ4, γ) = −δ4(µ4, γ)P4(t, µ4, γ)−ρ4
∂
∂µ4
P4(t, µ4, γ)
−∂
∂γ h4(Fepl, γ)P4(t, µ4, γ),(8)
with initial condition
P4(0, µ4, γ) = Pinit
4(µ4, γ)
and boundary conditions at (µ3)f= 1, µ4= 0
P4(t, 0, γ) = P3(t, 1, γ )
and
P4(t, µ4,0) = P3(t, 1,0), µ4= 0
Pbdy,γ
4(t, µ4), µ46= 0.
12
Figure 6: Death rate in class P4is a function of both maturity level, µ4,and cellular iron
state, γ. For a fixed maturity level µ4,death rate decreases as iron state γincreases. That
is, for a given maturity level, cells with little iron are more likely to die. For a iron state
γ < γcrit,4,death rate is increases as maturity level increases. These plots were generated with
parameter values δmin
4= 0.1, δmax
4= 0.5, m4= 0.38, b4= 0,and kδ4= 12.
4.6 Class P5(t, µ5, γ)
Cells in class P5are mature erythrocytes and circulating reticulocytes. For cells in this class,
we make the following assumptions.
1. The maturation rate is constant, ρ5.
2. The average life span of a red blood cell in a patient with chronic kidney disease is about
70 days, significantly shorter than for healthy persons [24, 39, 41]. We set the maximum
maturity level to be (µ5)f= 100 days−1,but the death rate is set such that virtually no
cells reach the maximum maturity level.
3. As they have no nuclei, cells in this stage do not proliferate [17], and therefore β5= 0.
4. Death rate in this class is assumed to have four components: death due to low cellular
iron, death due to aging, death due to neocytolysis, and death due to blood loss.
(a) Death due to low cellular iron. As in the previous class, cells that lack a full
complement of iron have increased mortality. As in class P4,we have
cδ5,γ (µ5) = m5µ5+b5, m5>0,
13
and
δ5,γ(µ5, γ) = δmax
5,γ −δmin
5,γ ·cδ5,γ (µ5)kδ5,γ
cδ5,γ (µ5)kδ5,γ +γkδ5,γ
+δmin
5,γ .
(b) Death due to aging. Senescent (aged) erythrocytes are enveloped and destroyed in
the spleen [17], so death rate in this class is also a function of maturity level. This
process occurs independent of cellular iron state:
δ5,µ(µ5) = δmax
µ5−δmin
µ5·cδµ5kδµ5
cδµ5kδµ5+µ5kδµ5
+δmin
µ5
(c) Death due to neocytolysis. Neocytolysis targets RBCs between 14-21 days old (i.e.,
for maturity level 14 ≤µ5≤21) [4, 32]. These cells have increased death rate when
EPO level drops, but it is not yet known whether the mechanisms that cause this
increase in death rate are related to a large drop in EPO level (i.e., a large negative
rate of change of EPO level) and/or simply a low EPO level. We account for both
in this model, with tuning parameters α1and α2to account for the relative effects.
We begin with the assumption that neocytolysis is a response to low EPO level.
We choose a decreasing sigmoid function for the maximum death rate:
δmax
5,neo,1(EP O ) = δmax,1
5,neo ·cneo
δ5kneo
δ5
cneo
δ5kneo
δ5+EP O kneo
δ5
.
Then the death rate is given by
δ5,neo,1(µ5, E P O) = δmax
5,neo,1(EP O )1
1 + e−2kh,1(µ5−15) −1
1 + e−2kh,1(µ5−20) ,
the sum of smoothed heaviside functions. Note that when kh,1is chosen large
enough, the death rate is zero when µ5<14 and µ5>21.Also, for maturity levels
14 ≤µ5≤21,when EPO level is small, the maximum death rate is large. Figure
7 depicts the output of the death rate function δ5,neo,1(µ5, E P O),for both the case
of a fixed maturity level with varying EPO and the case of fixed EPO with varying
maturity level.
For the assumption that neocytolysis is a response to large negative rate of change
of EPO, we choose a decreasing function for the maximum death rate:
δmax
5,neo,2d
dtEP O=δmax,2
5,neo 1−1
1 + e−2kh,3(d
dt EP O−ρE P O,crit ).
(Note here that we do not use a sigmoid function (as we did for δmax
5,neo,1(EP O ))
because the area of interest includes negative values of d
dt EP O .) The death rate,
14
Figure 7: Low-EPO neocytolysis death rate in class P5is a function of maturity level µ5
and EPO level. In the first plot, we fix EPO and show the death rate over varying maturity
level. Notice that neocytolysis affects cells such that 14 ≤µ5≤21.Also, as EPO increases,
the effects of neocytolysis diminish. This fact is demonstrated again in the second plot. To
produce these plots, we chose parameter values δmax,1
5,neo = 0.2, cneo
δ5= 2000, kneo
δ5= 20 and
kh,1= 10.
given by
δ5,neo,2µ5,d
dtEP O
=δmax
5,neo,2d
dtEP O"1
1 + e−2kh,2(µ5−15) −1
1 + e−2kh,2(µ5−20) #,
is plotted in Figure 8 for both the constant EPO rate with varying maturity case
and the constant maturity level with varying EPO rate case.
Then the death rate due to neocytolysis is given by
δ5,neo µ5, E P O, d
dtEP O=α1δ5,neo,1(µ5, E P O) + α2δ5,neo,2µ5,d
dtEP O.
(d) Death due to blood loss. We make a separate death rate to account for losses such
as blood lost during blood draws, denoted δ5,loss(t, µ5). The iron from these losses
is not recycled.
15
Figure 8: Large negative EPO rate neocytolysis death rate in class P5is a function of maturity
level µ5and EPO rate. Note that neocytolysis targets cells with maturity level 14 ≤µ5≤21.
As EPO rate attains large negative values, the death rate due to neocytolysis increases. To
produce these plots, we chose parameter values δmax,2
5,neo = 0.2, kh,3= 0.0075, ρEP O,cr it = 750
and kh,2= 5.
We assume that these phenomena occur independently and that the resultant death rate
for class P5is the sum of these four components,
δ5µ5, γ, E PO, d
dtEP O=δ5,γ (µ5, γ) + δ5,µ (µ5)
+δ5,neo µ5, E P O, d
dtEP O+δ5,loss(t, µ5).
5. Only reticulocytes at this stage are still able to collect iron, and at a smaller rate than
previous cell classes [9, 15, 17]. As before, the hemoglobinization rate, h5(Fepl, γ),for
cells in class P5is discussed in Section 4.7.
Using equation (3), cells in class P5are governed by the equation
∂
∂t P5(t, µ5, γ) = −δ5µ5, γ, EP O, d
dtEP OP5(t, µ5, γ)−ρ5
∂
∂µ5
P5(t, µ5, γ)
−∂
∂γ h5(Fepl, γ)P5(t, µ5, γ),(9)
with initial condition
P5(0, µ5, γ) = Pinit
5(µ5, γ)
and boundary conditions at (µ4)f= 2, µ5= 0
P5(t, 0, γ) = P4(t, 2, γ )
16
and
P5(t, µ5,0) = P4(t, 2,0), µ5= 0
Pbdy,γ
5(t, µ5), µ56= 0.
4.7 Hemoglobinization
Cells in classes P3, P4and P5participate in iron uptake for the purposes of synthesizing
hemoglobin. As noted earlier, we use γas a unitless structure variable that indicates the level
of iron in a given cell. Moreover, recall that 0 ≤γ≤2,where γfull = 1 represents the amount
of iron the “typical” RBC contains in a healthy individual. Finally recall when a cell divides,
we assume that each of the daughter cells receives half of the parent cell’s iron endowment.
In order to determine the rate of iron moving from the iron compartment and entering
RBCs, we compare the rate of iron required for cells to be hemoglobinized at the maximum rate
with the rate of iron available in the blood plasma (i.e., iron in the blood plasma compartment).
Then we can determine if there is sufficient iron available to hemoglobinize at the maximum
rate or if hemoglobinization is occurring in an iron-restricted fashion.
We assume that when there is an abundance of iron available, cells can obtain a full com-
plement of iron in three days [17]. Cells begin the hemoglobinization process in class P3.Cells
reside in this class for one day and they undergo cell division at a rate of 1 division per day.
After cells transition out of class P3,they no longer undergo cell division (i.e., cells in classes
P4and P5do not divide). While we model hemoglobinization and proliferation as continuous
processes, the following considerations in discrete time nodes indicate that the maximum rate
of iron uptake should be set to 2
5γfull days−1so that a cell could attain a full iron complement
in three days’ time.
day 0: γ= 0
day 1 before cell division: γ=2
5γfull =2
5
day 1 after cell division: γ=1
5γfull =1
5
day 2: γ=1
5γfull +2
5γfull =3
5
day 3: γ=3
5γfull +2
5γfull = 1
Cells in class P4can uptake iron at this rate until their iron state reaches γf ull,at which
point they stop taking in iron. Reticulocytes in class P5(i.e., cells in class P5with maturity
level 0 ≤µ5≤2) are circulating in the blood. While they are still capable of iron uptake
until they reach iron state γfull = 1,the rate of uptake is smaller [17], so we multiply the rate
for class P4by the constant 0 < k5<1.Therefore we define the maximum iron uptake rate
17
functions hmax
3(γ), hmax
4(γ),and hmax
5(µ5, γ),depicted in Figure 9, by
hmax
3(γ) = 2
5
hmax
4(γ) = 2
5·γfullkh,4,max
γfullkh,4,max +γkh,4,max =2
5·1
1 + γkh,4,max
hmax
5(µ5, γ) = k5hmax
4(γ),0≤µ5≤2
0, µ5>2.
Then the rate of iron needed if every cell becomes hemoglobinized at its maximum rate is
ρFe,needed = (max rate iron needed in class P3) + (max rate iron needed in class P4)
+ (max rate iron needed in class P5)
=kFe"Zγf
0Z1
0
2
5P3(t, µ3, γ)dµ3dγ
+Zγf
0Z2
0
hmax
4(γ)P4(t, µ4, γ)dµ4dγ
+Zγf
0Z2
0
hmax
5(µ5, γ)P5(t, µ5, γ )dµ5dγ#,
where kFe is the amount of iron in one billion red blood cells with a full iron endowment.
We compare this with the rate of iron available for erythropoiesis at time t. We assume
that the iron available for erythropoiesis is proportional to amount of iron in the blood plasma
compartment. Hence, the rate of iron availability is given by
ρFe,avail =kFe,eff Fepl(t),
Figure 9: Maximum iron uptake rates where 0 ≤µ5≤2.
18
where 0 < kFe,ef f ≤1 days−1.
We define the iron availability rate fraction, 0 < kFe,avail ≤1,to be
kFe,avail =(1, ρFe,needed ≤ρFe,avail,
ρFe,avail
ρFe,needed
, ρFe,needed > ρFe,avail.
Then the actual rates of iron uptake are given by
hi(Fepl, γ) = kFe,availhmax
i(γ), i = 3,4,5.
4.8 EPO compartment
While healthy individuals produce EPO in response to a decrease in blood oxygen, patients
whose kidneys have only minimal function produce only a small basal level of EPO in the
kidney and liver [21]. They receive rHuEPO intravenously during their dialysis treatments.
In this model, we assume the use of a bioidentical rHuEPO; we will not distinguish between
rHuEPO and endogenous EPO with respect to their action.
We assume the rate of endogenous EPO production in the liver and kidney to be constant.
We will assume that EPO clearance is proportional to the amount present. Thus we have
d
dtEP O(t) = ρEP O ,endog +ρE P O,exog(t)−ln 2
t1/2
EP O (t),(10)
with initial condition
EP O (0) = E0,
where ρEP O,endog is the (assumed constant) rate of endogenous EPO production, ρEP O,exog (t)
is the rate of exogenous EPO provided during treatment, and t1/2is the half-life of EPO.
5 Iron Model
We now introduce the second portion of our model, the iron compartments. 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 [16, 45]. 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 [40]. In
much smaller quantities, iron is absorbed from diet and is transferred in and out of storage in
the liver. The only losses to the system are from sweating, cells being shed, blood losses, etc.
The major regulator of iron homeostasis is the hormone hepcidin, which is produced in
the liver [16, 30, 45]. Hepcidin’s role in iron homeostasis has only been recently studied; the
19
(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
transport out of cells.
Figure 10: Iron regulation at a cellular level.
mechanisms and pathways leading to the production and action of hepcidin are only beginning
to be understood.
The protein ferroportin is required to transport iron out of a cell and into the plasma.
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 10. Hence, an
increase in hepcidin level causes iron to remain in cells and not be released into the blood
plasma (where it could be used in RBC production).
Hepcidin production is increased in response to high iron level in the blood plasma and/or
liver and is decreased in response to erythropoietic activity [16, 30]. Production is also in-
creased in the presence of certain cytokines which are released due to inflammation in the body
[16, 45]. 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,
they often have higher than normal levels of inflammation. Thus, they may produce higher
than normal levels of hepcidin and 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 [1].
We propose a model of ordinary differential equations for the iron model using the con-
ceptual model given in Figure 11.
5.1 Hepcidin
We choose H(t) to represent plasma hepcidin level at time t, with
0≤H(t)≤1.
A hepcidin level of 0 indicates little or no hepcidin present. In this situation, iron can pass
unencumbered across a cell membrane and enter the iron in the blood plasma compartment.
20
Figure 11: Iron Model. Note that the states variables are H, Feli, Fepl and Fere.
A hepcidin level of 1 indicates a maximum hepcidin level; when hepcidin is at its maximum
level, iron cannot cross a cell membrane to enter the blood plasma. In order to ensure that
0≤H≤1,we use characteristic functions that are “turned on” when a condition is true and
“off” when it is false. For example,
χ{H<1}(t) = χ{τ|H(τ)<1}(t) = 1, H (t)<1,
0, H(t)≥1.
We assume the following:
1. Hepcidin is produced in response to high levels of iron; how this is measured by the body
is unclear. We assume that production is increased in response to high levels in both
the liver and in the plasma. The relative size of these two responses can be adjusted by
way of the coefficients kFe,pl and kFe,li :
rate of increase in hepcidin due to high level of iron in plasma = kFe,plFepl(t)χ{H <1},
rate of increase in hepcidin due to high level of iron in liver = kFe,liFeli(t)χ{H <1},
with 0 ≤kFe,pl, kFe,li ≤1.The characteristic functions in these terms allow hepcidin level
to increase to H= 1 in response to high level of iron in the plasma or liver, but do not
allow hepcidin level to increase above H= 1.
2. Hepcidin production is increased in the presence of inflammation, but the specific path-
ways that cause this response are still under investigation. We assume that the input
I(t) is a measure of the overall inflammation state of the body (a broad simplification
that should be revisited in future model iterations) such that 0 ≤I≤1.Therefore,
rate of increase in hepcidin due to inflammation = kinf lI(t)χ{H<1},
21
where 0 ≤kinf l ≤1.
3. While hepcidin is actually “consumed” as it is used to prevent the action of ferroportin,
we initially assume a clearance proportional the amount present.
Hence, we model hepcidin level by
dH
dt (t) = kinflI(t)χ{H <1}(t) + kFe,plFepl(t)χ{H <1}(t) + kFe,liFeli(t)χ{H <1}(t)−kHH(t)χ{H>0}(t),
(11)
where 0 ≤kH≤1.
5.2 Iron in the reticuloendothelial system
The reticuloendothelial system consists of macrophages in the spleen and other cells that
collect/envelop old and damaged cells and reclaim their iron. This includes any cell in the red
blood cell lineage that dies, as well as liver cells that become senescent and/or damaged.
Based on our model for RBC dynamics, the rate of iron in to the reticuloendothelial system
from RBC lineage is
kFe"Zγf
0Z1
0
δ3(µ3, E P O)γP3(t, µ3, γ)dµ3dγ
+Zγf
0Z2
0
δ4(µ4, E P O)γP4(t, µ4, γ)dµ4dγ
+Zγf
0Z100
0δ5,γ(µ5, γ) + δ5,µ (µ5)
+δ5,neo µ5, E P O, d
dtEP OγP5(t, µ5, γ)dµ5dγ#,
where kFe is the amount of iron in one billion RBCs with a “normal” iron endowment. Notice
that for class P5we do not recycle iron from deaths due to blood losses.
The liver serves as a storage area for iron. Iron residing in liver cells that die is recaptured
and ends up in the reticuloendothelial system. Hence, we assume that
rate of iron in to the reticuloendothelial system from the liver = kli→reFeli(t),
where 0 ≤kli→re ≤1.
Reticuloendothelial cells require ferroportin to transfer iron from within the cell into the
blood plasma. Hepcidin negatively regulates this transport; when hepcidin level is high, the
amount of iron that can leave liver cells and enter the blood plasma is low. We assume that
rate of iron out of the reticuloendothelial system in to the blood plasma = kre→pl(1−H(t))Fere(t),
where 0 ≤kre→pl ≤1.
22
Thus, we have
dFere
dt =kFe"Zγf
0Z1
0
δ3(µ3, E P O)γP3(t, µ3, γ)dµ3dγ
+Zγf
0Z2
0
δ4(µ4, E P O)γP4(t, µ4, γ)dµ4dγ
+Zγf
0Z100
0δ5,γ(µ5, γ) + δ5,µ (µ5) + δ5,neo µ5, E P O, d
dtEP OγP5(t, µ5, γ)dµ5dγ#
+kli→reFeli −kre→pl(1 −H)Fere.(12)
5.3 Iron in the Liver
Iron enters the liver (for storage) from the blood plasma when the level of iron in the blood
plasma is high and leaves storage when the level of iron in the blood plasma is low. We model
these rates with sigmoid functions as follows:
rate of iron into liver from the blood plasma compartment = Kpl→li (Fepl(t)) Fepl(t),
where
Kpl→li (Fepl) = kmax 1−(cpl→li )kpl→li
(cpl→li)kpl→li + (Fepl)kpl→li !
and
rate of iron out of the liver into the blood plasma
=Kli→pl (Fepl(t)) (1 −H(t))Feli(t)
where
Kli→pl (Fepl) = kmax (cli→pl )kli→pl
(cli→pl)kli→pl + (Fepl)kli→pl !.
Notice that 1−Happears in the expression for rate of iron into the blood plasma compartment
from the liver compartment because hepcidin inhibits the action of ferroportin in allowing the
transport of iron out of the liver. The functions kpl→li (Fepl) and kli→pl (Fepl) are depicted in
Figure 12.
When cells in the liver die, the iron in those cells is collected by cells in the reticuloen-
dothelial system. We model this as
rate of iron out of the liver in to the reticuloendothelial system = kli→reFeli(t).
Hence, the time rate of change of iron in the liver is given by
dFeli
dt =Kpl→li (Fepl(t)) Fepl(t)−Kli→pl (Fepl(t)) (1 −H(t))Feli(t)−kli→reFeli(t).(13)
23
Figure 12: Rate coefficients for iron in and out of storage (i.e., iron into the liver from the
blood plasma and iron into the blood plasma from the liver).
5.4 Iron in the Blood Plasma
Iron in the blood plasma is the main source of iron for erythropoiesis. Iron added to blood
plasma during intravenous treatment is denoted Feex(t).Iron also enters the blood plasma
from the reticuloendothelial system, the liver, and the intestines from diet. Since each of these
requires ferroportin to transfer the iron from within the cell to the plasma, the associated rates
of transfer are all subject to the negative regulatory effect of hepcidin.
Iron enters the intestines from a patient’s dietary intake. We define
ρiron,diet = maximum rate of iron coming from diet. mg/day
Iron in intestinal cells must be transported into the blood plasma via ferroportin. Hence, the
rate of iron into the blood plasma is limited by the hepcidin level. Thus, we assume
rate of iron into blood plasma from diet = kdiet(1 −H)ρiron,diet,
where 0 ≤kdiet ≤1.
We define
F eex(t) = rate of iron into plasma from intravenous treatment
and
ρFe,pl,loss(t) = rate of iron lost via blood plasma losses.
The rates of iron (i) into blood plasma from the reticuloendothelial system, (ii) into the
blood plasma from the liver, (iii) into the liver from the blood plasma, and (iv) into the red
blood cell lineage were all described previously. Thus, we have the following equation for iron
24
in the blood plasma:
dFepl
dt = rate of iron into plasma from diet
+ rate of iron into plasma from reticuloendothelial system
+ rate of iron into plasma from liver
+ rate of iron into plasma from intravenous treatment
−rate of iron into liver from plasma
−rate of iron into RBC lineage from plasma
−rate of iron lost via blood plasma losses
=kdiet(1 −H)ρiron,diet
+kre→pl(1 −H)Fere(t)
+Kli→pl(1 −H)Feli(t)
+F eex(t)
−Kpl→li(Fepl)Fepl(t)
−kFe [h3(Fepl, γ) + h4(Fepl, γ) + h5(Fepl, γ )]
−ρFe,pl,loss(t).(14)
6 Summary
We have developed a model for both red blood cell and iron dynamics in patients in ESRD
undergoing hemodialysis. In particular, our model consists of the RBC precursor and cir-
culating fragmentation equations (5)-(9) with appropriate initial and boundary conditions,
coupled with (10)-(14). We use structure variables to account for the age and iron endowment
of cells in the red blood cell lineage in the fragmentation equations for P1through P5. The
model also permits investigation of the effects of neocytolysis and inflammation on red blood
cell dynamics. The coupled fragmentation and EPO, iron, hepcidin with inflammation in-
put are nontrivial to solve. We are currently carrying out simulations on both the continuous
model presented here, and a corresponding discrete time version, in verification and validation
studies.
Acknowledgments:
This research was supported in part by the National Institute of Allergy and Infectious Disease
under grant NIAID 9R01AI071915. The authors would like to thank Dustin Kapraun, Peter
Kotanko, Franz Kappel and Doris Furtinger for stimulating and fruitful conversations during
the course of their efforts.
25
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