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Physiologically Structured Population Model of Intracellular Hepatitis C Virus Dynamics
Wojciech Krzyzanski Xavier Woot de Trixhe
Filip De Ridder An Vermeulen
PAGE 2013 Glasgow, UK
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• Overview of current models of HCV infection aiming at describing drug effects on intracellular viral replication.
• Basics of physiologically structured population models.
• Introduction of PSP model of HCV dynamics.
• Comparison between PSP model and current models.
Outline
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Overview
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Life Cycle of Hepatitis C Virus
HCV replication requires the formation of protein complexes: Replication Units
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Standard Viral Kinetics Model:
Guedj et al. Proc Natl Acad Sci U S A. 110:3991-6 (2013).
I- infected cells, T- target cells, V – circulating HCV.
The virion production rate:
Drug inhibits the production of virions
ssT)(T =0 ssI)(I =0 ssV)(V =0
VTdTsdtdT
β−−=
IVTdtdI
δ−β=
cVpI)(dtdV
−ε−= 1
pI
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• Discrete transition from uninfected to infected states.
• Total vRNA production is proportional to I.
• Drugs do NOT affect intracellular viral replication.
Standard Model Assumptions:
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Intracellular Cell Infection (ICCI) Model:
Guedj and Neumann, J. Theor. Biol. 267:330-340 (2010).
dTbVTTT
ITrdtdT
max
−−
+−= 1
IbVTdtdI
δ−=
cVRIdtdV
−ρ=
UU
URdt
dU
max
γ−
−β= 1
R)(U)(dtdR
µ+ρ−ε−α= 1
ssT)(T =0 ssI)(I =0 ssV)(V =0
ssU)(U =0 ssR)(R =0
U - Replication unit in a single infected cell synthesizing viral RNA R. The virion production rate: Drug blocks RNA production.
RIρ
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• Mean-field assumption: R(t) is same for all cells.
• No distinction between new and ‘mature’ infected cell
ICCI Model Assumptions
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Multiscale Model:
a – is age of infected hepatocytes. R(a) is the vRNA in a hepatocyte of age a. The virion production rate: Drug affects synthesis of R and the assembly and secretion of virons.
Guedj et al. Proc Natl Acad Sci U S A. 110:3991-6 (2013).
∫∞
0
),(),( dataItaRρ
R)1(R)1(aR
tR
s ρε−−κµ−αε−=∂∂
+∂∂
α
IaI
tI
δ−=∂∂
+∂∂
VT)t,(I β=0 )a(I),a(I ss=0
VTdTsdtdT
β−−=
)a(R),a(R ss=0 10 =)t,(R
cVda)t,a(I)t,a(R)(dtdV
s −ρε−= ∫∞
0
1
ssT)(T =0 ssV)(V =0
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• Short term kinetics: – Target cells are constant throughout the study
• Effective treatment: – Production of new infected cells is negligible
• Solution:
Simplifying Assumptions
ssT)t(T ≡
IVTss δ<<β
( ) ( ) ( )( )( ) ( )( )))()1((exp)(exp
)(exp)(exp )(exp)(
00
0000
ttttcCttttcBttcVtV
s −+−+−−−−+−−−−−+−−=
µκρεδδ
( ) ( )( )µκ+ρε−+−δµκ+ρε−δ+αδ+ρ+µε−ε−α
−µκ+ρε−+−δ
ε−=
µκ+ρε−−δδ+αδ+ρ+µε−ε−α
= αα
)(c)()())()((c
)(c)(cVC
)()c)(())()((cVB
ss
s
s
s
s
s
1111
11
111
00
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Multiscale Model: Patterns of Viral Response.
Viral load decline from baseline in a patient treated with daclatasvir (black ) compared with the decline seen in a patient treated with 10 MIU IFN (blue ) , and the corresponding best-fit model prediction (solid lines) using the multiscale model.
Guedj et al. Proc Natl Acad Sci U S A. 110:3991-6 (2013).
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Physiologically structured population (PSP) models
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• Individual characteristic that can vary between the subjects.
• Distinguishes individuals according to certain physiological traits.
• Examples of physiological structures: - Age - Size (length, body weight)
Structure
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• Let s be a structure.
The density of s in the population at time t is define as:
Number of subjects in population at time t:
If s is amount of substance in a subject (e.g. mass of a marker in a subject) then the total amount of s in the population is
Density
∫∞
=0
),()( dsstntN
∫∞
⋅=0
),()( dsstnststot
tat time Δs)(s and sbetween structure of subjects#)s,( +=∆stn
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• i-state is a vector of structures x = (x1,…, xm) such that: – It fully determines the population dynamical properties of a subject. – The future of the subject is fully determined by its i-state and the
environmental history.
• The time evolution of i-state is described by a system of ODEs
‘Individual’ i-state
)x,t(gdtdx
=
Metz and Diekmann, The Dynamics of Physiologically Structured Populations, 1986.
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• The p-state is a distribution or density function n(t,x) such that: – The size and composition of the structured population is uniquely
determined.
• Evolution of p-state:
‘Population’ p-state
),(),(),(),)((),( xtnxtxtbxtgndivt
xtn λ−=+∂
∂
de Roos , A gentle Introduction to Models of Physiologically Structured Populations, 1987. 1987
∑= ∂∂
=m
i i
i
x)ng()gn(div
1
Influx rate Per capita mortality rate (hazard function)
Divergence of i-state
Impact of i-state progression on the p-state distribution
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• A PSP model describes dynamics of a population in terms of the behavior of its constituent individuals. It consists of: – i-state equations
– p-state equations
– Boundary conditions
– Initial conditions
Physiologically Structured Population Model
)x,t(n)x,t()x,t(b)x,t)(gn(divt
)x,t(nλ−=+
∂∂ Ω∈> x ,t 0
)x,t(gdtdx
= Ω∈> x ,t 0
)x,t()x,t(n)x,t(g)x( α=⋅ν Ω∂∈> x ,t 0
)x(n)x,(n 00 = Ω∈ x
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• Each infected hepatocyte contains a certain amount of vRNA which makes the R(t) a physiological structure. – i-state:
– p-state: density of vRNA in infected hepatocytes i(t,R)
– Conditions:
R-Structured HCV Model
)R(gR)(R)(dtdR
s =ρε−−κµ−αε−= α 11
( ) iR
i)R(gti
δ−=∂
∂+
∂∂
∫∞
=0
),()( dRRtRitRtot
cVR)(dtdV
tots −ρε−= 1
VTdTsdtdT
β−−=
)t(T)t(V)t,(i β=0 )R(i),R(i ss=0 ssV)(V =0 ssT)(T =0
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R - intra-cellular vRNA. i(R,t) - distribution of vRNA over infected hepatocytes (time=t) The virion production rate: Drug affects synthesis of R and the production of virons.
R-Structured HCV Model
totRρ
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ODE Model of HCV Dynamics
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VTdTsdtdT
β−−=
IVTdtdI
δ−β=
cVR)(dtdV
tots −ρε−= 1
totstot R))((I)(
dtdR
δ+ρε−+κµ−αε−= α 11
ssT)(T =0 ssI)(I =0 ssV)(V =0 totsstot R)(R =0
The new model differs from the standard model by presence of Rtot(t) that determines the viron production rate:
pI)( ε−1 tots R)( ρε−1vs.
standard new
The p-state can be integrated over R resulting in a simplified ODE model:
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• Triple exponential: exp(-ct); exp(-δt); exp(-...t)
Solutions under Simplifying Conditions
( ) ( ) ( )( )( ) ( )( )tctC
tctBctVtV
s ))1((expexp expexp exp)( 0
µκρεδδ+−+−−−+
−−−+−=
( ) )()c(
))()((cVBs
s
µκ+ρε−−δδ+ρ+µε−ε−
= α
111
0
( )( )µκ+ρε−+−δµκ+ρε−δ+ρ+µε−ε−
−µκ+ρε−+−δ
ε−= α
)(c)())()((c
)(c)(cVC
ss
s
s
s
1111
11
0
δ+αα
missing compared to the multiscale model solution
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Comparison with current models
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Viral Load Time Course: Multiscale vs. PSP Model
α = 40, t0 = 0 α = 1, t0 = 0
Simulated time courses of viral loads in two patients treated with daclatasvir and INF using multiscale and PSP models. For the original parameters the time courses overlap. They slightly differ if α =1.0.
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Time Courses of V, I, and Rtot
Simulated time courses of V, I, and Rtot in two patients receiving treatment with daclatasvir (left) and INF (right).
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Conclusions • Inclusion intracellular vRNA dynamics in models of HCV infection permits more
adequate quantification of drug effects for direct-acting antiviral agents.
• Structure population models integrate in a natural way a single cell model with the total amount of vRNA in infected cells.
• R-structured population model provides almost identical description of the viral load dynamics as the multiscale model.
• PSP model is simpler and more mechanistic than the multiscale model.
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This works was initiated and supported by a grant from Janssen Research & Development, a division of Janssen Pharmaceutica NV, Beerse, Belgium.
Acknowledgments