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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