a structured population model for erythropoiesis

A Structured Population Model for

ErythropoiesisDoris H. Fuertinger 1, Franz Kappel 1, Peter Kotanko 2, Nathan W. Levine 2,

Stephan Thijssen 2

1 Institute for Mathematics and Scientific Computation,University of Graz (Austria)

2 Renal Research Institute New York, New York

A Model for Erythropoiesis SiMCRT-2011, Kobe, November 1 – 3, 2011 – p. 1/25

Problem statement

In 2006 ∼ 1.4 million patients received dialysis treatment(≈21% in the US, ≈17% in the EU).

Almost all patients with chronic kidney disease are affectedby chronic anemia, i.e., the patients do not have enough redblood cells (RBC) in order to maintain sufficient oxygensupply for tissues and organs.

The main reason for this is the failure of the kidneys toproduce the hormone erythropoietin (EPO), which controlsthe production of new RBCs

Anemia not only reduces physical and neurocognitivecapacity, but also leads to left ventricular dilation andmyocardial ischemia. Cardiovascular complications are themost common causes of death among dialysis patients.


Problem statement

In order to reduce the mortality rate among dialysis patientsit is necessary to compensate the loss of EPO production indialysis patients by administration of erythropoiesisstimulating agents (ESA).

ESA treatment regimens are usually determined based onthe prior experience of the physician and on establishedguidelines.

In order to meet strict requirements concerning the numberof RBCs in the patients blood it is necessary to developpredictive models of erythropoiesis under an ESA treatmentregimen.


ReferencesBelair, J., Mackey, M. C., Mahaffy, J. M., Age-structured and two-delay models forerythropoiesis, Mathematical Biosciences 128 (1995), 317 – 346.

Mahaffy, J. M., Belair, J., Mackey, M. C., Hematopoietic model with moving boundarycondition and state dependent delay: Applications in erythropoiesis, Journal of TheoreticalBiology 190 (1998), 135 – 146.

Mahaffy, J. M., Polk, S. W., Roeder, R. K., An age-structured model for erythropoiesisfollowing a phlebotomy. Tech. Rep. CRM-2598, Department of Mathematical Sciences, SanDiego State University, San Diego, CA 92182-0314, (1999).

Adimy, M., Crauste, F., Ruan, S., Modelling hematopoiesis mediated by growth factors withapplications to periodic hematological diseases, Bulletin of Mathematical Biology 68 (2006),2321 – 2351.

Crauste, F., Pujo-Menjouet, L., Genieys, S-, Molina, C., Gandrillon, O., Adding self-renewalin committed erythroid progenitors improves the biological relevance of a mathematicalmodel of erythropoiesis, Journal of Theoretical Biology 250 (2008), 322 – 338.

Ackleh, A. S., Deng, K., Ito, K., Thibodeaux, J., A structured erythropoiesis model with

nonlinear cell maturation velocity and hormone decay rate, Mathematical Biosciences 204

(2006), 21 – 48.


References

Banks, H. T., Cole, C. E., Schlosser, P. M., Tran, H. T., Modelling and optimal regulation oferythropoiesis subject to benzene intoxication, Mathematical Biosciences and Engineering 1(2004), 15 – 48.

Loeffler, M., Pantel, K., Wulff, H., Wichmann, H. E., A mathematical model of erythropoiesisin mice and rats. Part 1: Structure of the model, Cell and Tissue Kinetics 22 (1989), 13 – 30.

Wichmann, H. E., Loeffler, M., Pantel, K., Wulff, H. H., A mathematical model oferythropoiesis in mice and rats. Part 2: Stimulated erythropoiesis, Cell and Tissue Kinetics22 (1989), 31 – 49.

Wulff, H., Wichmann, H. E., Pantel, K., Loeffler, M., A mathematical model of erythropoiesisin mice and rats. Part 3: Suppressed erythropoiesis, Cell and Tissue Kinetics 22 (1989), 51– 61.

Roeder, I., Quantitative stem cell biology: Computational studies in the hematopoieticsystem, Current Opinion in Hematology 13 (2006), 222 – 228.

Roeder, I., Loeffler, M., A novel dynamic model of hematopoietic stem cell organization

based on the concept of within-tissue plasticity, Experimental Hematology 30 (2002), 853 –

861.A Model for Erythropoiesis SiMCRT-2011, Kobe, November 1 – 3, 2011 – p. 5/25

Erythropoiesis

Diagram of the different cell stages during erythropoiesis. The changing width of the triangles

labeled "EPO" and "IRON" represents the receptor densities on the cells.


Cell classes

Basic modeling assumption: There is always sufficient ironavailable to satisfy the demands.

This assumption is satisfied for healthy subjects, but not formost dialysis patients.

Five different population classes of cells are considered:

BFU-E (Burst-forming Unit Erythroids)

CFU-E (Colony-forming Unit Erythroids)

erythroblasts

marrow reticulocytes

erythrocytes (including blood reticulocytes)


BFU-E cellsp(t, µp) . . . population density for the BFU-E cells, where

µp is the maturity of a BFU-E cell (= age),∫ µp

2

µp1

p(t, µp) dµp =number of cells with

maturity in [µp1, µ

p2]

∂

∂tp(t, µp) +

∂

∂µpp(t, µp) = βpp(t, µp),

p(t, 0) = S0,

p(0, µp) = p0(µp),

βp . . . constant proliferation rateS0 . . . constant rate at which stem cells commit to the

erythroid lineagep0(µ

p) . . . population density at t = 0, 0 ≤ µp ≤ µpmax = 7 d


CFU-E cells

q(t, µq) . . . population density of the CFU-E cells, whereµq is the maturity of a CFU-E cells (= age)

∂

∂tq(t, µq) +

∂

∂µqq(t, µq) = (βq − αq(E(t))) q(t, µq),

q(t, µqmin) = p(t, µp

max),

q(0, µq) = q0(µq),

βq . . . constant proliferation rate

αq(E(t)) . . . rate of apoptosis depending on the EPO

concentration in plasma

q(t, µqmin

) = p(t, µpmax

) . . . rate at which cells leave the BFU-E cell

stage and enter the CFU-E cell stage

q0(µq) . . . population density at t = 0, 7 = µq

min≤ µq ≤ µq

max= 13


Precursor cells: erythroblasts

r(t, µr) . . . population density for the erythroblastpopulation at time t whereµr is the maturity of an erythroblast

∂

∂tr(t, µr) +

∂

∂µrr(t, µr) = βrr(t, µr),

r(t, µrmin) = q(t, µq

max),

r(0, µr) = r0(µr),

βr . . . constant proliferation rater(t, µr

min) = q(t, µqmax) . . . rate at which cells leave the

CFU-E stage and enter the erythroblast cell stager0(µ

r) . . . population density at t = 0,13 = µr

min ≤ µr ≤ µrmax = 18


Precursorcells: marrowreticulocytes

s(t, µs) . . . population density for the erythroblastpopulation at time t, whereµs is the maturity of an erythroblast.

∂

∂ts(t, µs) + vs(E(t))

∂

∂µss(t, µs) = −αs

0s(t, µs),

vs(E(t))s(t, µsmin) = r(t, µr

max),

s(0, µs) = s0(µs),

vs(E(t)) . . . maturation velocity depending on EPO, i.e.,(d/dt)µs = vs(E(t))

αs0 . . . rate of ineffective erythropoiesis


Precursorcells: marrowreticulocytes

vs(E(t))s(t, µsmin) = r(t, µr

max) . . . rate at which cellsleave the erythroblast cell stage and enter the reticulocytecell stage,

r0(µs) . . . population density at t = 0,

18 = µsmin ≤ µs ≤ µs

max = 21


Erythrocytes

m(t, µm) . . . population density of erythrocytes at time t,where µm is the population density oferythrocytes

∂

∂tm(t, µm) +

∂

∂µmm(t, µm) = −αm(E(t), µm)m(t, µm),

m(t, 0) = vs(E(t))s(t, µsmax),

m(0, µm) = m0(µm),

m(t, 0) = vs(E(t))s(t, µsmax) . . . rate at which

reticulocytes enter the blood stream

m0(µm) . . . population density at t = 0,

0 ≤ µm ≤ µmmax = 120


Erythrocytes

αm(E(t), µm) . . . mortality rate incorporating the effectof neocytolysis and the random daily breakdown oferythrocytes:

αm(E(t), µm) =

αmr + min

( cE

E(t)kE, bE

)

for E(t) < τE,

µmn,min

≤ µm ≤ µmn,max

αmr otherwise,

αmr . . . random daily breakdown

bE , cE , kE are constants

τE . . . threshold for E(t) beneath which neocytolysis istriggered


Erythrocytes

[µmn,min, µm

n,max] . . . age-interval where neocytolysis ispossible µm

n,min = 14 days and µmn,max = 21

Apoptosis rate αq(E(t)) for CFU-E cells:

αq(E(t)) =a1 − b1

1 + ek1E(t)−c1

+ b1, (sigmoid function)

Maturation velocity vs(E(t)) of marrow reticulocytes:

vs(E(t)) =a2 − b2

1 + e−k2E(t)+c2

+ b2


Feedback control loop via EPO

Eendin (t) . . . rate at which endogenous erythropoietin is

released from the kidneys

Eendin (t) =

a3 − b3

1 + ek3M (t)−c3

+ b3,

M(t) = 10−8M(t)/TBV . . . scaled erythrocyte“concentration”

M(t) =∫ µm

max

0 m(t, µm) dµm . . . total number oferythrocytes

TBV . . . total blood volume


Feedback control loop via EPO

Dynamics of endogeneous EPO in plasma:

Eend =1

TBVEend

in (t) − cenddegE

end(t),

Eend(t) . . . endogenous EPO concentration in plasma

Eendin . . . rate at which EPO is released by the kidneys

cenddeg . . . degradation rate of endogenous EPO


Exogeneous EPOEex(t) concentration of the exogenous hormone, i.e., of anESA

Eex(t) =1

TBVEex

in (t) − cexdegE

ex(t),

Eexin (t) . . . rate at which the exogenous hormone is

administered

cexdeg . . . rate at which the exogenous hormone is

degraded

Intravenous administration:Eex

in (t) = Eex0 (t)δt0(t)

Overall concentration E(t) of EPO in plasma:

E(t) = Eex(t) + Eend(t)


Structure of the model

Diagram of the model


High altitude dwellers

0 5 10 15 20 25 30 3520

22

24

26

28

30

32

34

36circulating RBC (# × 1012)

time [days]

tota

l pop

ulat

ion

−11.3 %

−14.5 %

Changes in the number of circulating RBCs in a high altitude dweller descending to alower altitude: model output with (solid line) and without (dashed line) neocytolysis duringdays 0 – 35.

Rice, L., Ruiz, W., Driscoll, T., Whitley, C. E., Tapia, R., Hachey, D. L., Conzales, G. F.,Alfrey, C. P., Neocytolysis on descent from altitude: A newly recognized mechanism forthe control of red cell mass, Annals of Internal Medicine 134 (2001), 652 – 656


Blood donation

0 20 40 60 80 100 120 140 160 1805

10

15erythropoietin concentration

time [days]

EP

O [m

U/m

l]

0 20 40 60 80 100 120 140 160 18040

60

80reticulocytes in the marrow (# × 1010)

time [days]tota

l pop

ulat

ion

0 20 40 60 80 100 120 140 160 180

23

24

25

circulating RBC (# × 1012)

time [days]

tota

l pop

ulat

ion 36 days

Simulation of donation of 550 ml blood. The mean recovery period according toPottgiesser et al. is 36 days (indicated in the bottom graph by a vertical line).

Pottgiesser, T., Specker, W., Umhau, M., Dickhuth, H. H., Roecker, K., Schumacher, Y.O., Recovery of hemoglobin mass after blood donation, Transfusion 48 (2008), 1390 –1397


Blood donation

0 10 20 30 40 50 6022

22.5

23

23.5

24

24.5

25

25.5


time [days]

tota

l pop

ulat

ion

Simulation of donation of 550 ml blood. Data points taken from Pottgiesser et al.


Blood donation

0 50 100 150 200 250 300 350 40022.5

23

23.5

24

24.5

25

25.5


time [days]

tota

l pop

ulat

ion

Simulation over a long time interval after blood donation, model output without neocytol-ysis (dashed line) and with neocytolysis (solid line).


Preoperative administration

0 5 10 15 20 25 3025

25.5

26

26.5

27

27.5

28

28.5

29

29.5circulating RBC (# × 1012)

time [days]

tota

l pop

ulat

ion

Preoperative administration of Epoetin-α. Data points are taken from Feagan et al.(marked by stars). Model simulations are presented for the low-dose group (4×20000U once a week) (dashed line) and the high-dose group (4×40000 U once a week) (solidline).

Feagan, B. G., Wong, C. J., Kirkley, A., Johnston, D. W., Smith, F. C., Whitsitt, P., Wheeler,S., Lau, C. Y., Erythropoietin with iron supplementation to prevent allogeneic blood trans-fusion in total hip joint arthroplasty, Annals of Internal Medicine 133 (2000), 845 – 854


Preoperative administration

0 5 10 15 20 25 300

500

1000erythropoietin concentration

time [days]

EP

O [m

U/m

l]

0 5 10 15 20 25 300

0.02

0.04neocytolysis

time [days]

deat

h ra

te

0 5 10 15 20 25 3050

100

150reticulocytes in the marrow (# × 1010)

time [days]

tota

l pop

ulat

ion

Preoperative administration of Epoetin-α, low-dose group (4×20000 U once a week)(dashed line), high-dose group (4×40000 U once a week) (solid line).


a structured population model for erythropoiesis

Documents