Mathematical analysis of multistage population
balances for cell growth and death
Margaritis Kostogloua, María Fuentes-Garíb, David García-Münzerb, Michael C. Georgiadisc,
Nicki Panoskaltsisd, Efstratios N. Pistikopoulosb, Athanasios Mantalarisb
aDepartment
of Chemistry, Aristotle University of Thessaloniki, Thessaloniki, Greece
bDepartment of Chemical Engineering Imperial College London, London, UK
cDepartment of Chemical Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece
dDepartment of Haematology, Imperial College London, Harrow, Middlesex, UK
Presenter: [email protected] , tel: +30 2310994184
INTRODUCTION
Normal cell proliferation occurs following a four-step cycle, known as the cell cycle: G1 phase, when
INDICATIVE RESULTS
cells grow and stock up on nutrients; S phase, when their DNA is duplicated; G2 phase, when DNA
is checked for duplication errors and M phase, when the cell divides giving birth to two new cells.
9
2
These can in turn enter the cell cycle or stay in a quiescent state (G0) until needed. The cell cycle is
t=0.2
8
inherently regulated by the scheduled expression of cyclins intracellularly, which activate their
t=0.4
B
7
partner cyclin-dependent kinases (cdks) in a phase-specific manner leading to progression-related
1.5
events. More specifically, the concentration of cyclins E and B typically fluctuates and reaches a
F(x,t)
D
6
threshold in G1 and G2 phases respectively as cells come closer to transitioning to the next phase.
t=0.6
t=0.8
E
N
5
Cyclin concentration and DNA content are thus excellent candidates for use as progress variables
1
C
4
within G1 (cyclin E), S (DNA) and G2 (cyclin B). Cell cycle modelling is of relevance for two specific
0.5
3
applications: leukaemia proliferation according to chemotherapy response and phase-specific
A
2
production of antibodies in industrial cell lines. Population balance models (PBMs) are particularly
suitable as they account both for inter- and intra- phase growth, allowing for distributed phases
0
1
0
0.5
1
1.5
detail. The three governing equations of this model are composed by growth and transition terms. A
2.5
0
3
0.2
0.4
Figure 1. Evolution of the total number of cells N for several
shapes of the initial cell distribution function (conditions:
one equation analog, zero width transition zone).
0.6
x
kt
which give greater accuracy. A specific three stage biologically supported population balance model
which has been recently proposed to simulate evolution of several cell cultures is studied here in
2
0.8
1
Figure 2. Snapshots of the cell distribution function F(x,t)
during a doubling period. The behavior of F is periodic.
The initial distribution is this of case C (conditions: one
equation analog, zero width transition zone)..
one equation analogue of the multistage model is formulated and it is solved analytically in the selfsimilarity domain. The effect of initial conditions at the system evolution is studied numerically. The
three model equations are then considered by using asymptotic and numerical techniques.
1.4
0.5
t=0.5
t=1.5
t=3.5
t=9.5
1.2
A=0
0.4
THE CELL CYCLE:
λ/R
F(x,t)
1
1
0.8
A=0.5
0.3
A=1
0.6
0.2
A=1.5
0.4
0.1
0.2
0
0
MAIN PART
0.5
1
1.5
0
2
0
x
0.2
0.4
0.6
Denoting as G(x,t), S(y,t), M(z,t) the cell number distributions at the stages G1, S, G2
1
R /R
.
The mathematical model
0.8
2
Figure 3. Snapshots of the cell distribution function F(x,t) at
the middle of several doubling periods. The behavior of F is
non-periodic. The initial distribution is this of case E
(conditions: one equation analog, finite width transition zone).
1
Figure 4. Exponential growth factor λ dependence on
transition and growth parameters (included in normalized
parameters R1, R2, A) for the three stage model with finite
width transition zone.
respectively where x, y, z are the corresponding cell content variables, the cell evolution
problem can be described by the following set of equations and boundary conditions:
1.2
40
Boundary conditions
Equations
∂G(x, t)
∂G(x, t)
+ kg
=
−rg (x)G(x, t)
∂t
∂x
∂S(y, t)
∂S(y, t)
+ ks
=
0
∂t
∂y
35
1
∞
B
k g G(0) = 2∫ rm (z)M(z, t)dz
30
C
0.8
0
∞
k sS(0) = ∫ rg (x)G(x, t)dx
N
α
25
D
B
0
C
15
∂M(z, t)
∂M(z, t)
+ km
=
− rm (z)G(z, t)
∂t
∂z
k m M(0) = k SS(2)
D
0.6
20
0.4
10
A
0.2
A
5
where kg, km, ks are the content growth rate at the three stages and rg, rm the transition
functions for the two stages. In case of the transition functions is of Dirac delta form the
problem is characterized as having a zero width transition zone.
Description of the performed analysis
•A one equation analog of the model is developed and examined first as a prorotype of the
0
0
0
1
2
3
4
kt
5
0
0.5
1
1.5
2
kt
Figure 6. Evolution of the first stage partition factor α for
Figure 5. Evolution of the total number of cells N for several
several initial cell distributions. The system behavior is
shapes of the initial cell distribution function (conditions: three periodic (conditions: three stages model, zero width transition
stages model, zero width transition zone).
zone).
model allowing analytical solutions.
•Closed form asymptotic solutions for both models (one and three equations) are derived.
CONCLUSIONS
•A finite difference technique is developed to allow transient solutions under general form of
the transitions functions.
•In the particular case of zero width transition functions, an analytical treatment based on the
principle of superposition of fundamental solutions is possible.
•A lumped model of the cell evolution process is analyzed and its results is compared to
those of the full model.
Tested initial conditions
-One equation analog: The case A corresponds to F(x,0)=δ(x) (Dirac delta), the case B to F(x,0)=2x,
the case C to F(x,0)=2-2x the case D to F(x,0)=2/3(1-x)+2/3 and the case E to F(x,0)=1.
- Three stages model: The case A corresponds to uniform distribution in first stage, the case B to
uniform distribution in third stage, the case C to uniform distribution in all the three stages and the
case D to uniform distribution in first and third stage.
• The one equation analog is capable to reveal several features of the three stages model offering
increased analytical insight
• The system ends to a large time asymptotic solution which can be steady or oscillating.
• The type of the asymptotic solution (steady or oscillating) and the time needed for its approach are
depended on the shape of the initial cell distribution and on the width of the interstage transition zone.
• Oscillating asymptotic behaviour is observed only in case of zero width transition zones and
discontinous initial cell distribution.
• Lumped models are capable to approximate the asymptotic behaviour only in the abscence of
oscillations. In any case the approach to the asymptotic behaviour is poorly approximated by lumped
models.
ACKNOWLEDGMENT: This work is supported by ERC-BioBlood (no. 340719), ERC-Mobile Project (no. 226462), by the EU 7th Framework Programme
[MULTIMOD Project FP7/2007-2013, no 238013] and by the Richard Thomas Leukaemia Research Fund.