MMath Project: Mathematical Analysis of the HCVModel

Mark RansleyUniversity of Surrey

Supervisors:

Philip J. Aston

Piet Van Der Graaf

September 1, 2011

Abstract

Mathematical models of the Hepatitis C Virus (HCV) enable quantitive and qualitative analysisof the viral, host and treatment dynamics. In Part I we examine the developments and adapta-tions of the HCV model along with the reasons for and consequences of doing so, in the processuncovering areas where the model could be further extended or improved. Using techniques fromdimensional analysis we investigate the over-parameterisation and parameter identifiability ofthe model along with ways this can be overcome.

In Part II we study the mechanisms of several medicines currently in development for whichlittle modelling and simulation work has been published. Based upon our simulatory resultswe suggest the most e↵ective ways to use and combine said treatments. Using of a populationmodel we conduct simulations with the aim of curing as many patients as possible.

In Part III we develop a novel method for creating patient-specific doses and courses oftreatment so as to minimise side e↵ects and improve the outcome of therapy.

Special thanks to Dr Piet Van Der Graaf1 for giving my work a context and purpose and formaking the professional environment a pleasure to work within, Dr Gianne Derks2 forintroducing me to the world of PKPD modelling, and Dr Philip Aston2 without whose

patience, care and dedication this project would not have been possible.

1Pfizer Research Laboratories, Sandwich2Department of Mathematics, University of Surrey

Introduction

Pfizer is the world’s largest research based pharmaceutical company and one of the most estab-lished entities in the healthcare industry. Founded in 1849 as a fine chemicals manufacturer thecompany has since expanded to develop, manufacture and distribute award winning medicinesfor a range of health conditions a↵ecting both humans and animals. Pfizer’s research park inSandwich, Kent is the company’s European research and development headquarters. Opened in1954, the research park is perhaps most notable as the birthplace of Viagra and Maraviroc andat its peak emploed some 3,000 sta↵ across the range of scientific disciplines.

On 2nd February 2011, it was announced that Pfizer would be withdrawing from the siteand would be closing down research with immediate e↵ect. A week later I began six monthsof industrial training there in the department of Pharmacometrics, where I was tasked withworking on the Hepatitis C Virus (HCV). The department’s workforce of around fifteen peopleare post-doctorate researchers mostly from pharmacological backgrounds. They use mathemat-ical and statistical techniques to construct models capable of describing and explaining certainillnesses and treatments, along with the underlying biological and chemical processes. Com-bining the models with observational data enables the department to conduct quantitative andqualitative analyses of the e↵ects of various pharmaceutical products, and to o↵er predictionsand suggestions regarding how these products could be used.

Despite withdrawing from the Sandwich site and closing most of the departments within it,Pfizer retained the entire Pharmacometrics team and so business in my department continuedas usual. Over the six months I was there I attended weekly meetings where members of theteam would take turns in presenting their work to their peers and would discuss the techniquesused. In the first few months of my time at Sandwich I also attended similar meetings withother departments and weekly statistical methods workshops, however as more departmentswere closed such activities ceased. Through attending the meetings I learned a great deal aboutdiseases, drug development and general industrial practice, and was surprised by the pleasant,relaxed environment Pfizer employees worked within.

With my supervisor Piet Van Der Graaf, I attended teleconferences with hepatologists, virol-ogists and PKPD3 modelling consultants regarding a new treatment for HCV. The confidentialityof the project along with the large amounts of money and research involved and the importanceof helping those a↵ected by the virus made this aspect of my industrial training particularlyexciting, as I felt my work was of a genuine value.

Throughout this project I present my work on the HCV models. My supervisor Piet waskeen on the idea of bringing a pure mathematician into the department, hoping that I would usetechniques not previously known to the team and provide a new perspective on viral dynamicsmodelling. To help direct my e↵orts, Piet posed the following three questions that I will attemptto answer in this report.

1. Can we learn anything about parameter identifiability in the HCV model? It was suggestedthat this question should be approached using techniques from dimensional analysis.

3Pharmacokinetics Pharmacodynamics (PKPD) is another term used to describe the work done by the Phar-macometrics team. PK is the analysis of what the body does to a drug, eg. clearance times, whilst PD denotesanalysis of the drug’s e↵ect on the body, eg. e�cacy.

1

2. Can we adapt the model to simulate and predict the e↵ectiveness of several therapiescurrently in development, used alone and in combination with other therapies, and proposethe best strategies for using them?

3. Would it be possible to devise a simple test to be carried out on patients in order todetermine how best to treat them on an individual basis?

2

Contents

I HCV Models and Dimensional Analysis 4

1 The Basic Model 51.1 Constructing the Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Non-Dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Stability at the Uninfected Steady State . . . . . . . . . . . . . . . . . . . . . . . 81.5 Stability at the Infected Steady State . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Other HCV Models 132.1 The Extended Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The First Aston Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 The Second Aston Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Quantifying the Mechanisms of Therapy . . . . . . . . . . . . . . . . . . . . . . . 232.5 The Snoeck Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Qualifying the Models 273.1 Fitting to Patient Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Discussion 30

5 Identifiability of parameters 33

II Prospective Therapies 35

6 Introduction - Population Modelling 36

7 SRB1 Entry Blocker 38

8 Direct Acting Antiviral Agents 40

9 Transfection Therapy (RNAi) 41

III Customised Therapeutics 47

10 4810.1 The need for individualised therapy . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.2 Determining the critical e�cacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.3 Determining the time required for cure . . . . . . . . . . . . . . . . . . . . . . . . 50

11 Conclusions and Further Work 52

3

Part I

HCV Models and DimensionalAnalysis

4

Chapter 1

The Basic Model

1.1 Constructing the Basic Model

Hepatitis C virus (HCV) is a blood-transmitted virus that a↵ects around 170 million peopleworldwide[1]. Once inside the liver, HCV infects liver cells (hepatocytes) and causes them toproduce more virus units (virions), forming a loop of infection that often results in cirrhosis,chronic liver disease and death. The liver is a highly active organ which continuously kills o↵its cells and produces new ones to stay as healthy as possible. Hence, in the absence of virus,we can model the time-evolution of the liver’s hepatocytes, T , as:

dT

dt= s� dT (1.1)

where s is the source, or rate of production of hepatocytes, and d is the death rate, proportionalto the number of hepatocytes present. Clearly when the hepatocyte level is constant we havethe steady state:

Tu

=s

d(1.2)

When the virus is present, there are now three dynamical quantities to be represented - theconcentration of healthy, or target, hepatocytes T , the concentration of infected hepatocytes Iand the concentration of virions V . They evolve together as follows:

dT

dt= s� dT � �V T

dI

dt= �V T � �I

dV

dt= pI � cV

where � is the rate at which virions infect hepatocytes upon coming into contact, � is the deathrate of infected cells, assumed to be di↵erent to the death rate of target cells due to the immunesystem, p is the production rate of virions within each infected cell, and c is the clearance rateat which the immune system gets rid of virions. During therapy, it is believed that the levels ofI and V can be brought down by decreasing the rates in the system at which they are produced.This leads us to the original model by Neumann et al.[2]:

dT

dt= s� dT � (1� ⌘)�V T (1.3)

dI

dt= (1� ⌘)�V T � �I (1.4)

dV

dt= (1� ✏)pI � cV (1.5)

5

where ✏ is the treatment’s e�cacy at blocking virion production, and ⌘ is its e�cacy at blockinghepatocyte infection. The model was constructed to ascertain which of the two antiviral mech-anisms was activated by Pegylated Interferon-↵-2a therapy, and what level of e�cacy resultedfrom various doses. In Chapter 3 we will show how the model can be fitted to patient’s data sothat the values of parameters such as ✏ may be deduced, and thus properties of the disease andtherapies inferred. The initial conditions are the infected steady state described later.

The system is nonlinear and as such has no closed form solution. However Neumann et al.[2]used the assumption that T remains roughly constant to eliminate (1.3) and solve the remainingequations to obtain an expression for V (t). They then used nonlinear least squares regressionto fit this expression to real patient viral load data, and thus ascertain the values of some ofthe parameters. However, this assumption is invalid as T does in fact vary, which we shall seeresults in a variety of di↵erent response profiles.

The units can be deduced from knowing (1.3) and (1.4) are in terms of cells/volume/dayand (1.5) is in terms of virions/volume/day, so all components in the equations must be inunits such that the overall system is in some concentration per day (Table 1).

Variable Name UnitT Target Cells IU/mlI Infected Cells IU/mlV Viral Load IU/mls Cell Production Rate IU/ml · day�1

d Cell Death day�1

� Infection Rate ml/IU · day�1

� Infected Cell Death day�1

p Virion Production day�1

c Clearance Rate day�1

⌘ Infection Blocking Rating N/A✏ Production Blocking Rating N/A

Table 1.1: Quantities and units of the HCV model

1.2 Non-Dimensionalization

Defining dimensionless variables

x =T

Tu

, y =I

Tu

, z =V

Tu

, ⌧ = t↵,

where ↵ can be �, d, p, or c, we can write

dx

d⌧=

dx

dT

dT

dt

dt

d⌧

for (1.3), and similarly for (1.4) and (1.5) in terms of y and z, thus obtaining

dx

d⌧=

1

Tu

(s� dxTu

� (1� ⌘)�xzT 2u

)1

↵dy

d⌧=

1

Tu

((1� ⌘)�xzT 2u

� �yTu

)1

↵dz

d⌧=

1

Tu

((1� ✏)pyTu

� czTu

)1

↵

6

Noticeably, every parameter from the original equations is still present. However, settingTu

= s/d (from (1.2)) and ↵ = d the equations simplify to

dx

d⌧= 1� x� (1� ⌘)

�s

d2xz

dy

d⌧= (1� ⌘)

�s

d2xz � �

dy

dz

d⌧= (1� ✏)

p

dy � c

dz

Defining dimensionless parameters

k1 =�T

u

d, k2 =

�

d, k3 =

p

d, k4 =

c

d, (1.6)

the original Neumann model (1.3) - (1.5) is written in dimensionless form as

dx

d⌧= 1� x� ⌘⇤k1xz (1.7)

dy

d⌧= ⌘⇤k1xz � k2y (1.8)

dz

d⌧= ✏⇤k3y � k4z (1.9)

where✏⇤ = 1� ✏, ⌘⇤ = 1� ⌘, (1.10)

with uninfected steady state

xu

=Tu

Tu

= 1, yu

=I0Tu

= 0, zu

=V0

Tu

= 0, (1.11)

and initial conditions given by the infected steady state

x(0) =Ti

Tu

= xi

, y(0) =Ii

Tu

= yi

, z(0) =Vi

Tu

= zi

,

which is derived in the following chapter. Note that whilst they could be absorbed into k1 andk3 we choose to keep the control parameters ⌘⇤ and ✏⇤ separate in (1.7), (1.8) and (1.9) so thatwe can vary them individually.

1.3 Steady States

Steady states are the equilibria of the system and as such have derivatives of zero. Clearly, thishappens in equations (1.7)-(1.9) when

x = 1, y = z = 0,

which is the uninfected steady state described in (1.11) and shown in Figure 1.2a. Followinginfection, the system will eventually reach a second equilibrium which is found by setting (1.9)equal to zero, and rearranging to obtain z in terms of other variables and parameters at thesteady state.

zi

= ✏⇤k3k4

yi

(1.12)

Note that prior to treatment, ✏ = ⌘ = 0 and hence ✏⇤ = ⌘⇤ = 1. Substituting (1.12) into (1.8)and setting the equation equal to zero we can solve for x

xi

=k2k4

k1k3✏⇤⌘⇤(1.13)

7

and then substituting both (1.13) and (1.12) into (1.7) we can solve for y

yi

=1

k2(1� x

i

) (1.14)

to obtain values for x, y and z that comprise the infected steady state (Figure 1.2b). Since(1.13), (1.14) and (1.12) contain control parameters ✏ and ⌘, this is not a single solution but anentire branch of infected steady states each with varying x

i

, yi

and zi

depending on ✏ and ⌘.We see in Figure 1.1 how the infected and uninfected branches of solutions react to changes ine�cacy. Setting ✏⇤ and ⌘⇤ to one, we assume that the resulting pre-treatment infected steadystate forms the set of initial conditions for modelling HCV infection:

x(0) =k2k4k1k3

y(0) =1

k2� k4

k1k3=

1

k2(1� x(0))

z(0) =k3k2k4

� 1

k1=

1

k1

✓1

x(0)� 1

◆

Clearly xi

is real and non-negative, since all of its components are, however yi

is non-negativeonly if:

1

k2� k4

✏⇤⌘⇤k1k3

i.e. as long as xi

1. Since xi

< xu

and xu

= 1, this is a reasonable assumption to make.Similarly, z

i

is non-negative provided:

✏⇤k3k2k4

� 1

⌘⇤k1

which, again, equates to xi

1.Note that in the original set all parameters and variables are non-negative, and in non-

dimensionalisation they are multiplied and divided by one another to obtain the new set, sothis must also be non-negative. Examining the infected steady state given by (1.13), (1.14) and(1.12) it is interesting to observe that:

lim✏,⌘!1

xi

= 1, lim✏,⌘!1

yi

= �1, lim✏,⌘!1

zi

= �1

Realistically this means that for perfect e�cacies, the infected steady state becomes an impos-sibility.

1.4 Stability at the Uninfected Steady State

The dimensionless equations (1.7) - (1.9) can be linearised about the uninfected steady state bydefining new variables

x0 =

2

4x� x

u

y � yu

z � zu

3

5 =

2

4x0

y0

z0

3

5

Clearly, the steady state in the new variables occurs at (0, 0, 0). Thus the system can be written,ignoring second and higher order terms, as

dx0

d⌧= J(x

u

, yu

, zu

)x0,

8

where J is the Jacobian Matrix:

J(xu

, yu

, zu

) =

2

4�1 0 �⌘⇤k10 �k2 ⌘⇤k10 ✏⇤k3 �k4

3

5 (1.15)

Computing eigenvalues for this Jacobian reveals them to be

�1 = �1

�2 =�(k2 + k4) +

p(k2 � k4)2 + 4✏⇤⌘⇤k1k3

2

�3 =�(k2 + k4)�

p(k2 � k4)2 + 4✏⇤⌘⇤k1k3

2

So it would appear, given all parameters must be non-negative, that �1 and �3 are alwaysnegative, and �2 is also negative forming a stable steady state (an attractor) when

k2 + k4 >p(k2 + k4)2 + 4✏⇤⌘⇤k1k3 � 4k2k4,

and �2 is positive forming an unstable steady state (a saddle point) when the inequality isreversed. Thus there exists a bifurcation point where �2 = 0, at which point ✏⇤⌘⇤ can beformulated as

✏⇤0⌘⇤0 =

k2k4k1k3

(1.16)

This is the critical e�cacy, where for e�cacies greater than it (✏⇤⌘⇤ < ✏⇤0⌘⇤0) we can expect

the system to stabilise around the uninfected steady state (a cure - Figure 1.2e). Note that from(1.10) it is important to distinguish between the e�cacies, ✏, ⌘ and the inhibition factors, ✏⇤,⌘⇤. In (1.13) and (1.14) we see ✏⇤ and ⌘⇤ occurring together as a pair, and so can write the thetotal inhibition as

⇠⇤ = ✏⇤⌘⇤

and similarly total e�cacy as⇠ = 1� ✏⇤⌘⇤

Also, it should be noted at this point that ✏, ⌘ 2 [0, 1] and as a result

✏⇤, ⌘⇤, ⇠, ⇠⇤ ⇢ [0, 1] (1.17)

9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3 x 106

Efficacy (ε)

Viru

s, H

epat

ocyt

e C

once

ntra

tion

(Iu/m

l) A

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100

102

104

106

108

Efficacy (ε)

Viru

s, H

epat

ocyt

e C

once

ntra

tion

(Iu/m

l) B

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3 x 106

Efficacy (η)

Viru

s, H

epat

ocyt

e C

once

ntra

tion

(Iu/m

l) C

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100

102

104

106

Efficacy (η)

Viru

s, H

epat

ocyt

e C

once

ntra

tion

(Iu/m

l) D

Figure 1.1: E�cacy dependent bifurcation between the steady states of the Neumann model.We vary ✏ between 0 and 1 in A (linear scale) and B (log scale) and we vary ⌘ similarly in C andD. T is represented in red, I in green and V in blue. The black dashed line is critical e�cacy(1.16). We examine both mechanisms separately to reveal the di↵erences in their e↵ects. Wesee that the infected steady state for V decreases linearly down to zero as we raise ✏, whichcorresponds with IFN treatment data, but we do not observe this with ⌘, leading us to believethat ✏ is the predominant mechanism in standard of care therapy. This e↵ect occurs becauseVi

(1.12) contains ✏ but not ⌘. Note that as critical e�cacy is approached from the left, theinfected steady state branch consisting of T

i

, Ii

and Vi

is stable, and beyond critical e�cacythe uninfected steady state is stable. This is shown by the stable branch being represented bysolid lines, and the unstable one by dashed. For this system we used s = 61000, d = 0.003,� = 1⇥ 10�7, � = 0.4, p = 15, c = 5.5, and found ✏0 = ⌘0 = 0.9278

1.5 Stability at the Infected Steady State

Using the infected steady state given in (1.13), (1.14), (1.12), the Jacobian is:

J(xi

, yi

, zi

) =

2

4�1� ⌘⇤k1zi 0 �⌘⇤k1xi

⌘⇤k1zi �k2 ⌘⇤k1xi0 ✏⇤k3 �k4

3

5

Computing the eigenvalues for this in Matlab produced three very long, complicated expres-sions that were not feasible to work with even after simplification and restoration to originalvariables. However, we have seen that there is a transcritical bifurcation between the two steadystates x

u

and xi

, (Figure 1.1) when ⇠⇤ = ✏⇤0⌘⇤0. This is because it is here that an eigenvalue

crosses the real axis. To determine whether this is the only local bifurcation in the system, weinvestigate the possibility of a complex eigenvalue crossing the imaginary axis (Hopf bifurcation)as the e�cacy varies along the infected steady state branch of solutions.

As � crosses the imaginary axis we have � = 0 + i⌦ and so we can evaluate these purely

10

0 50 100 150 200

100

102

104

106

108

Days

Vira

l Loa

d

a0 50 100 150 200

10−1

100

101

102

103

104

105

106

107

108

Days

Vira

l Loa

d

b

0 50 100 150 20010−1

100

101

102

103

104

105

106

107

108

Days

Vira

l Loa

d

c0 50 100 150 200

10−1

100

101

102

103

104

105

106

107

108

Days

Vira

l Loa

d

d

0 50 100 150 20010−1

100

101

102

103

104

105

106

107

108

Days

Vira

l Loa

d

e

Figure 1.2: Simulated concentrations|quantities of T |x (red), I|y (green) and V |z (blue) overa 200 day period in the case of (a) the uninfected steady state, (b) the pre-treatment infectedsteady state with ✏ = ⌘ = 0, (c) treatment where ✏ = 0.11, ⌘ = 0, (d) treatment where ✏ = 0,⌘ = 0.11, (e) treatment where ✏⇤ = 0.07 and ⌘⇤ = 0.07. Note that in the cases of unsuccessfultherapy (c) and (d), where total e�cacy ✏⇤⌘⇤ > ✏⇤0⌘

⇤0 the system settles in a di↵erent infected

steady state. Upon stopping treatment it will revert to the original one. In all graphs here,s = 8 ⇥ 105, d = 4.7 ⇥ 10�3, � = 0.3, c = 6, � = 6 ⇥ 10�7, p = 5.4 and hence k1 = 21482,k2 = 63.5, k3 = 1142.4, k4 = 1269.3 (5 s.f.)

11

imaginary eigenvalues as the solutions to

f(⌦) = det(J(xi

, yi

, zi

)� i⌦I) = 0 (1.18)

Since this complex-valued determinant is equal to zero, the individual real and imaginary partswill also be equal to zero:

=(f(⌦)) = ⌦

✓⌦2 �

✓✏⇤⌘⇤k1k3

k2+

✏⇤⌘⇤k1k3k4

◆◆(1.19)

= 0

which has the solutions

⌦ = 0 (1.20)

⌦2 = ✏⇤⌘⇤k1k3

✓1

k2+

1

k4

◆(1.21)

Similarly,

<(f(⌦)) =

✓k2 + k4 +

✏⇤⌘⇤k1k3k2k4

◆⌦2 + k2k4 � ✏⇤⌘⇤k1k3 (1.22)

= 0

Substituting (1.20) into (1.22) we see that a bifurcation occurs when ✏⇤⌘⇤ = ✏⇤0⌘⇤0 - i.e. the

original bifurcation point given in (1.16). However, substituting (1.21) into (1.22) we obtain aquadratic in ✏⇤⌘⇤. Hence

✏⇤⌘⇤ =�B ±

pB2 � 4AC

2A

where

A =k21k

23

k2k4

✓1

k2+

1

k4

◆, B = k1k3

✓(k2 + k4)

✓1

k2+

1

k4

◆� 1

◆, C = k2k4.

Clearly A,B (when expanded) and C are all positive, and so

B �pB2 � 4AC

and they are only equal in the case where either A or C are zero as a result of any of theparameters being zero, which does not make sense in an HCV infected system since they mustbe positive. Hence the e�cacies required to produce a Hopf bifurcation are negative, whichcontradicts our restrictions on e�cacy (1.17). In the case where B2 � 4AC > 0 we have realvalued but negative e�cacies which are not allowed, and conversely when B2 � 4AC < 0 wehave complex valued e�cacies, which also are not allowed. Hence a Hopf bifurcation does notoccur on the infected steady state branch, and so the transcritical bifurcation at ✏⇤0⌘

⇤0 is the only

one.

12

Chapter 2

Other HCV Models

2.1 The Extended Model

The Neumann model served its purpose in demonstrating that peg-IFN-↵-2a therapy worksthrough the ✏ mechanism rather than ⌘. The model also explained the biphasic decline observedin patients as being a result of virions being initially cleared at rate cV (when production isalmost entirely stopped by ✏) and then a shallower decline in viral load as the infected cells, I,which are still producing virions at a much reduced rate, are killed o↵ at rate �I. However insome cases a triphasic decline (see Chapter 3) was observed which the model could not describe,so clearly there were other processes at work and the model would need to be updated somehowto explain this.

Mechanistically, the liver is known to be a regenerative organ [6, 7], so that along with theturnover of hepatocytes described in (1.1) it will also actively restore itself to maximum capacityin the event of cell loss (eg. during transplantation) through proliferation. This mechanismapplies to both T and I and increases them proportionally to how far the liver is from itscapacity T

max

. Subsequently the Neumann model was extended to the following model byDahari et al. [4]:

dT

dt= s+ rT

✓1� T + I

Tmax

◆� dT � (1� ⌘)�V T (2.1)

dI

dt= (1� ⌘)�V T + rI

✓1� T + I

Tmax

◆� �I (2.2)

dV

dt= (1� ✏)pI � cV (2.3)

with uninfected steady state

Tu

=Tmax

2r

✓r � d±

r(r � d)2 +

4rs

Tmax

◆(2.4)

though we only take the additive solution, since Tu

must be positive. The infected steady state,due to the model’s quadratic form in T and I, is quite complicated and shall be saved until wehave reduced the number of parameters, since we now also have r and T

max

. As such we makethe following substitution:

x =T

Tu

, y =I

Tu

, z =V

Tu

, ⌧ = dt,

where it is important to note Tu

is (2.4) and not (1.2), to obtain the dimensionless system

dx

d⌧= k0 + k5x

✓1� x+ y

k6

◆� x� ⌘⇤k1xz (2.5)

13

dy

d⌧= ⌘⇤k1xz + k5y

✓1� x+ y

k6

◆� k2y (2.6)

dz

d⌧= ✏⇤k3y � k4z (2.7)

with new dimensionless parameters

k0 =s

dTu

, k5 =r

d, k6 =

Tmax

Tu

,

and k1 - k4 carried over from (1.6). The uninfected steady state for this model is

xu

=k62k5

k5 � 1±

r(k5 � 1)2 + 4

k0k5k6

!= 1, y

u

= zu

= 0,

and the infected steady state is

xi

=D ±

qD2 + 4k0k5

k6A2

2k5k6A2

, yi

= xi

(A� 1) + k6

✓1� k2

k5

◆, z

i

= ✏⇤k3k4

yi

,

where

A =✏⇤⌘⇤k1k3k6

k4k5, D = k2 +A(k2 � k5)� 1

Note that in solving the steady states for xu

and xi

we have two solutions as the systems arequadratic. However since x must be positive we need only consider the additive solution. Wederived x

u

from solving the system for x when y = z = 0, and we inferred from the factxu

= Tu

/Tu

that xu

= 1. Combining the latter property of xu

with the former, and substitutingx = 1, y = z = 0 into the system, we obtain the following relation

k0 �k5k6

+ k5 � 1 = 0 (2.8)

Again, evaluating the Jacobian at the uninfected steady state

J(xu

, yu

, zu

) = J(1, 0, 0) =

2

4k5 � 2k5

k6� 1 �k5

k6�⌘⇤k1

0 k5 � k5k6

� k2 ⌘⇤k10 ✏⇤k3 �k4

3

5 (2.9)

we can extract the eigenvalue �1 from the first column of J(xu

),

�1 = k5 � 2k5k6

� 1 (2.10)

which, substituting k5 from (2.8) can be written as

�1 = �k0 �k5k6

(2.11)

Clearly �1 is always negative. As �1 is the only non-zero entry in column one of J(xu

), we canfind �2 and �3 from the lower-right 2⇥2 sub-matrix.

�2 = �1

2

k2 + k4 � k5 +

k5k6

+

r(k2 � k4 � k5 +

k5k6

)2 + 4✏⇤⌘⇤k1k3

!

�3 = �1

2

k2 + k4 � k5 +

k5k6

�r(k2 � k4 � k5 +

k5k6

)2 + 4✏⇤⌘⇤k1k3

!

14

For perfect e�cacies, ✏⇤ = ⌘⇤ = 0 and we find the Jacobian J(1, 0, 0) becomes upper triangular.Hence when this is the case the eigenvalues are the diagonal entries of J(1, 0, 0).

�2 = k5 �k5k6

� k2

�3 = �k4

Clearly �3 is also negative, but to show this for �2 we again substitute k5 from (2.8) to see that

�2 = 1� k0 � k2

Due to the enhanced immune response, we can assume that � > d, and since both parametershave been rescaled by d (� from (1.6) and d by implication from the dimensionless system) weknow that k2 > 1 and hence that �2 < 0. Hence for perfect e�cacies all eigenvalues are negativemaking the uninfected steady state an attractor. However, as in the Neumann model, �3 variesin sign depending on the value of ✏⇤⌘⇤. Hence at the point where �3 = 0 the stability of theuninfected steady state changes. We find that setting the determinant of J(x

u

) equal to zeroand solving for ✏⇤⌘⇤ gives the critical e�cacy at which point the bifurcation occurs.

✏⇤0⌘⇤0 =

k4(k2 +k5k6

� k5)

k1k3(2.12)

=k4(k2 � 1 + k0)

k1k3

This change in stability is most easily represented through the bifurcation diagrams (Figure2.1), which are structurally similar to those of the Neumann model (Figure 1.1). Observe thatagain, since k2 > 1 we can be certain that ✏⇤0⌘

⇤0 is positive, and therefore the bifurcation is

always possible for the defined range of e�cacies. When ✏⇤0⌘⇤0 � 1 the parameters are such that

the bifurcation occurs, and hence the virus clears, without the need for therapy. This has beenobserved in patients and is termed spontaneous clearance, though in many such cases HCVinfection goes unnoticed and hence unreported.

The Extended model can fit triphasic decline profiles, and it was observed by Dahari [4] thatthis profile emerges in the model (and thus in patients) when at the start of therapy, I > T ,otherwise biphasic declines occur. It should be noted that biphasic and triphasic declines occurwhen the system is moving to the uninfected steady state, ie. for greater than critical e�cacies.For other e�cacies, the eigenvalues become complex causing the system to oscillate about theinfected steady state resulting in a variety of di↵erent behaviours (see Chapter 3).

2.2 The First Aston Model

The details of liver regeneration are still somewhat ambiguous, since there are multiple mech-anisms involved, namely hepatocyte mitosis and stem cell repair. There are two mechanismsrepresenting regeneration in the extended HCV model, s and the pair of logistic growth termspreceded by r, though clearly the e↵ect of each one di↵ers since s is a linear term and the rterm is quadratic in T and I. To further simplify the model and possibly learn more about theunderlying behaviour we consider the evolution of the total number of hepatocytes, H, in theDahari model:

H = T + IdH

dt=

dT

dt+

dI

dt

= s+ rH

✓1� H

Tmax

◆� dT � �I (2.13)

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

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2

2.5

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106

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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1.5

2

2.5

3 x 106

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104

106

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Figure 2.1: E�cacy dependent bifurcation between the steady states of the Dahari model. Wevary ✏ between 0 and 1 in A (linear scale) and B (log scale) and we vary ⌘ similarly in C andD. T is represented in red, I in green and V in blue. The black dashed line is critical e�cacy.As with the Neumann model (Figure 1.1) we see a bifurcation at critical e�cacy, where theinfected branch becomes unstable (shown here as the point where the infected branch intersectsthe uninfected branch) and the system stabilises at the uninfected state for greater than criticale�cacies. Strikingly, due to the proliferation of I, we find that the viral load actually increasesslightly along the infected branch for ⌘, though again as we approach and exceed critical e�cacythe viral load tends to zero. Note that e�cacies lower than ✏ ⇡ 0.7 cause such a small decrease inviral load on the log scale that the viral profile could be classed as a null response (see Chapter3). Parameter values used were s = 61000, d = 0.003, � = 4⇥ 10�8, � = 0.139, p = 20, c = 10,r = 0.65, T

max

= 2⇥ 107 and found ✏0 = ⌘0 = 0.9131

It has been suggested by Philip Aston (unpublished material) that since r is a regenera-tive mechanism, s may be unnecessary, and that the proliferation term need not be quadraticprovided it slows as the liver approaches T

max

. Hence (2.13) becomes

dH

dt= r

✓1� H

Tmax

◆� dT � �I

Under this model all hepatocyte production is bounded by Tmax

. Yet when we consider the twotypes of cell individually, the proliferation of T is proportional to the amount of T present andthe proliferation of I is proportional to the amount of I. Hence in order to extract equationsfor T and I from the H equation, the proliferation term must be weighted, giving the system

dT

dt= r

T

H

✓1� H

Tmax

◆� dT � ⌘⇤�V T (2.14)

dI

dt= ⌘⇤�V T + r

I

H

✓1� H

Tmax

◆� �I (2.15)

dV

dt= (1� ✏)pI � cV (2.16)

16

Note that since we have divided by the fundamental quantity concentration in the regenerationterms, the units of r must be changed to concentration/day to accommodate this. The steadystates of this system are

Tu

=rT

max

r + dTmax

, Iu

= Vu

= 0,

and

Ti

=r

d� �+

c�

�✏⇤⌘⇤p+

cr

Tmax

�✏⇤⌘⇤p(2.17)

Ii

=Tmax

cr

cr + c�Tmax

� Ti

�✏⇤⌘⇤pTmax

� Ti

(2.18)

Vi

= ✏⇤p

cIi

(2.19)

Since the term s is absent, there is no constant influx of healthy hepatocytes, and so thesystem also admits a third steady state of pure infection:

Tp

= 0

Ip

=rT

max

r + �Tmax

Vp

= ✏⇤prT

max

c(r + �Tmax

)

It will be shown in Chapter 3 that the model can describe real patient data, though theparameter values required for it to do this are somewhat di↵erent to those in previous models.The dimensionless form of the new system is

dx

d⌧= x

k̂5

x+ y� k̂5

k6� 1� ⌘⇤k1z

!(2.20)

dy

d⌧= ⌘⇤k1xz + y

k̂5

x+ y� k̂5

k6� k2

!(2.21)

dz

d⌧= ✏⇤k3y � k4z (2.22)

where

x =T

Tu

, y =I

Tu

, z =V

Tu

, ⌧ = dt

k1 =�T

u

d, k2 =

�

d, k3 =

p

d, k4 =

c

d, k̂5 =

r

dTu

, k6 = Tu

Tmax

.

The uninfected steady state for this system is

xu

=k̂5k6

k̂5 + k6= 1, y

u

= zu

= 0.

We find that through the equation for xu

we are able to write k6 in terms of 1 and k5 and thuseliminate it from the system giving

dx

d⌧= x

k̂5

x+ y� k̂5 � ⌘⇤k1z

!

dy

d⌧= ⌘⇤k1xz + y

k̂5

x+ y� k̂5 � k2 + 1

!

dz

d⌧= ✏⇤k3y � k4z

17

with infected steady state

xi

=k4(k2 + k̂5 � 1)

✏⇤⌘⇤k1k3+

k̂51� k2

yi

=k4k̂5

k4(k2 + k̂5 � 1)� ✏⇤⌘⇤k1k3xi� x

i

zi

= ✏⇤k3k4

yi

= ✏⇤

k̂5

k3(k4(k2 + k̂5 � 1)� ✏⇤⌘⇤k1k3xi)� x

i

!

Evaluating the Jacobian we find that

J(xu

, yu

, zu

) = J(1, 0, 0) =

2

4�k̂5 �k̂5 �⌘⇤k10 1� k2 ⌘⇤k10 ✏⇤k3 �k4

3

5

From here we can calculate the eigenvalues as

�1 = �k̂5

�2 = �1

2

⇣k2 + k4 � 1 +

p(k2 � k4 � 1)2 + 4✏⇤⌘⇤k1k3

⌘

�3 = �1

2

⇣k2 + k4 � 1�

p(k2 � k4 � 1)2 + 4✏⇤⌘⇤k1k3

⌘

Clearly �1 is always negative, as is �2 provided k2 + k4 � 1 > 0, which is satisfied as in Chapter2.1 under the reasonable assumption that k2 > 1. As in the Neumann and Snoeck models, thesign of the third eigenvalue depends upon the value of ✏⇤⌘⇤. Since the determinant is the productof the eigenvalues, we find that solving det J(x

u

, yu

, zu

) = 0 for ✏⇤⌘⇤ yields the critical e�cacyfor which �3 = 0 to be

✏⇤0⌘⇤0 =

k4(k2 � 1)

k1k3.

This expression is always positive (and hence allowable) assuming k2 > 1, and as in the previouschapter, describes spontaneous clearance when ✏⇤0⌘

⇤0 > 1.

The dimensionless form of the pure-infected steady state branch is

xp

= 0

yp

=k̂5

k2 + k̂5 � 1

zp

= ✏⇤k3k4

yp

=✏⇤k3k̂5

k4(k2 + k̂5 � 1)

where yp

and zp

are positive since we assume k2 > 1. Evaluating the Jacobian at this steadystate,

J(xp

, yp

, zp

) =

2

4k2 � ✏⇤⌘⇤S � 1 0 0

✏⇤⌘⇤S � k̂5 � k2 + 1 1� k̂5 � k2 00 ✏⇤k3 �k4

3

5

where

S =k1k3k̂5

k4(k2 + k̂5 � 1)

Since J(xp

, yp

, zp

) is lower triangular, the eigenvalues are the diagonal entries. Evidently theeigenvalues from J2,2 and J3,3 are negative (�2 being inferred from the positivity of y

p

), and wesee that J1,1 is negative, making the pure-infected steady state an attractor, when

✏⇤⌘⇤ <k2 � 1

S(2.23)

18

and there exists a bifurcation between the infected and pure-infected steady states when bothsides of the above expression are equal. Note that this bifurcation point is always positive(and hence allowable) using the k2 > 1 assumption, and that if its value is negative then thepure-infected steady state is impossible.

Interplay between the three steady states is seen in the bifurcation diagram Figure 2.2. Thepure-infected steady state is disquieting, for clearly once reached there is no chance of regaininghealthy hepatocytes and purging the virus. Increasing ✏ will reduce the viral load all the timetreatment is present, but will never reduce the number of infected cells as they now proliferateindependently, and the ⌘ mechanism has no e↵ect at all.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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Efficacy (ε)Log 10

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DC

A B

Figure 2.2: Bifurcation diagrams for the first Aston model. T is represented in red, I in greenand V in blue. Diagrams A and C vary ✏ and ⌘ respectively and plot the steady states (thevalues they will eventually settle at, given a particular e�cacy) of T , I, and V . Diagrams Band D plot the same relationships on the log10 scale. For each e�cacy the stable states areshown in solid lines and the unstable ones in dashed. With this particular set of parameters(d = 0.01, � = 5.68⇥ 10�1, � = 0.2, p = 10, c = 0.73, r = 71200 and T

max

= 5.12⇥ 109) we findthat all three steady states are stable on some intervals in the range of permissible e�cacies.Thus there are two bifurcation points in total, ✏

c

= ⌘c

(the dashed vertical line) and ✏p

= ⌘p

(the dotted vertical line). Behaviour at ✏c

and ⌘c

is similar to that already discussed under theNeumann and Dahari models. However, we see that with the pure-infected steady state, highvalues of ✏ when maintained will reduce the steady state value of V , but not of I since at thisstate I sustains itself through proliferation and no de-novo infection is required. It should alsobe observed that ⌘ has no e↵ect on the pure-infected steady state for this reason. We see thatthe bifurcations occur at the intersection of steady states, notably at ✏

c

and ⌘p

where Ti

= Tu

,Ii

= Iu

= 0 and Vi

= Vu

= 0. However, moving from the pure-infected steady state to theinfected steady state at ✏

p

and ⌘p

requires T to move from zero to positive (most apparent indiagrams B and C) which is not possible under this model, without some external interventionsuch as transplant.

2.3 The Second Aston Model

Observing regeneration data in non-HCV livers [12] it is apparent that the model must allow forthe regeneration to slow o↵ as T

max

is approached. However it may not be necessary to bound

19

the proliferation by Tmax

, instead the e↵ect could be expressed through having the proliferationterm inversely proportional to the number of hepatocytes

dH

dt=

r̂

H� dH

whereby the properties of r and Tmax

have e↵ectively been combined to form r̂ which is inconcentration2/day. This leads to the second Aston Model

dT

dt=

r̂T

(T + I)2� dT � ⌘⇤�V T (2.24)

dI

dt= ⌘⇤�V T +

r̂I

(T + I)2� �I (2.25)

dV

dt= ✏⇤pI � cV (2.26)

with steady states

Tu

= ±r

r̂

d, I

u

= Vu

= 0

Ti

=c�

�✏⇤⌘⇤p� �✏⇤⌘⇤pr̂

c(d� �)2

Ii

= ±pc2�r̂ � T

i

�c✏⇤⌘⇤pr̂

c� � Ti

�✏⇤⌘⇤p� T

i

Vi

= ✏⇤p

cIi

Note that for Tu

and Ii

it only makes sense to take the positive solution. The dimensionlessform of this model is

dx

d⌧= x

k̂5

(x+ y)2� 1� ⌘⇤k1z

!

dy

d⌧= ⌘⇤k1xz + y

k̂5

(x+ y)2� k2

!

dz

d⌧= ✏⇤k3y � k4z

where

k1 =�T

u

d, k2 =

�

d, k3 =

p

d, k4 =

c

d, k̂5 =

r̂

T 2u

d.

Note that if we substitute Tu

into the above expression for k̂5 we find that k̂5 = 1, and will usethis substitution throughout the following analysis. The steady states for this system are

xu

= 1 yu

= zu

= 0

xi

=k2k4

✏⇤⌘⇤k1k3� ✏⇤⌘⇤k1k3

k4(k2 � k1)2

yi

= ±pk2k24 � ✏⇤⌘⇤k1k3k4xik2k4 � ✏⇤⌘⇤k1k3xi

� xi

zi

= ✏⇤k3k4

yi

20

with the pure-infected steady state

xp

= 0

yp

=

r1

k2

zp

= ✏⇤k3k4

yp

The Jacobian, evaluated at the uninfected steady state, is

J(xi

, yi

, zi

) =

2

4�2 �2 �⌘⇤k10 1� k2 ⌘⇤k10 ✏⇤k3 �k4

3

5

from which we may calculate the eigenvalues as

�1 = �2

�2 = �1

2(k4 + k2 + 1 +

p(k4 � k2 + 1)2 + 4✏⇤⌘⇤k1k3)

�3 = �1

2(k4 + k2 + 1�

p(k4 � k2 + 1)2 + 4✏⇤⌘⇤k1k3)

Again, �1 and �2 are always negative, and �3 becomes negative too for e�cacies greater thanthe critical e�cacy

✏⇤0⌘⇤0 =

k4(k2 � 1)

k1k3

which is the same as that for the first Aston Model (2.23). Evaluating the determinant of theJacobian at the pure-infected steady state

J(xp

, yp

, zp

) =

2

4k2 � ✏ ⇤ ⌘⇤S � 1 0 0✏⇤⌘⇤S � 2k2 �2k2 0

0 ✏⇤k3 �k4

3

5

where

S =k1k3k4pk2

and solving for ✏⇤⌘⇤ reveals the pure-infected steady state to be an attractor for e�cacies lessthan the pure-critical e�cacy

✏⇤p

⌘⇤p

=k4(k2 � 1)

pk2

k1k3

We will see in Chapter 3 that both Aston models can fit the various patient response profiles,and so it is possible that the liver, HCV infection and treatment can be described without s.However, without this component we found that the models can permit the pure infected steadystate, from which a patient will never recover from HCV infection. We see in Figures 2.2 and2.3 that even perfect e�cacies would curb virion production all the time therapy is present,but that the infected cells, which swell up and are the cause of the cirrhosis and hepatocellularcarcinoma, remain since they proliferate of their own accord and healthy cells cannot be formed.

Whilst the complete inability to recover from pure-infection could explain the null responseprofile of Chapter 3, cases of the pure infected steady state have never been explicitly observed, sowe can conclude from studying the Aston models that either healthy cells can be formed duringthe mitosis of infected cells, which seems unlikely given the cell’s DNA has been modified, orthat there is indeed a second source of hepatocyte production such that the number of healthycells cannot decay to zero. This extrahepatic progenitor has been speculated to reside in the

21

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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4

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1102

103

104

105

106

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(Iu/m

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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4

6

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103

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A B

C D

Figure 2.3: Bifurcation diagrams for the second Aston model. T is represented in red, I ingreen and V in blue. Diagrams A and B display the steady states on the linear and log scalesrespectively as ✏ varies, and similarly diagrams C and D display the steady states as ⌘ varies.Parameter values used were d = 0.0008, � = 1e � 08, � = 0.04, p = 39, c = 1.2 and r =1350000000

22

bone marrow or the Canals of Hering, and to come into play when usual hepatic regenerationis insu�cient in maintaining healthy liver function [7]. Hence, though s may not play as largea part in hepatocyte turnover as the mitotic processes represented by r, this parameter is incertain scenarios crucial and HCV should not be modelled without it.

2.4 Quantifying the Mechanisms of Therapy

The most longstanding component in the Standard of Care (SOC) for HCV is currently aweekly injection of PEG-Interferon alpha-2a/b (peg-IFN); an enhanced variant of the interferonsproduced by the immune system. Using regression analysis (Fig. 3.1A and Appendix A) wesee that in HCV therapy with peg-IFN, the biphasic decline typical of successful treatmentis due to ✏ rather than ⌘. This corroborates the hypothesis that peg-IFN suppresses virionproduction/release [9].

Since the e�cacy of peg-IFN can be improved through increasing dose, we can formulate ✏in terms of dose and individual susceptibility, through the E

max

model

✏ = ✏max

Dosen

Dosen + EDn50

(2.27)

where ✏max

is the maximum e�cacy, in this case 1, ED50 is the dose required to attain 50% e�-cacy in an individual, and n is the Hill Coe�cient, a constant that determines the intensity of therelationship between dose and e�cacy and varies from 1-4 [10]. Thus for more thorough phar-macokinetic understandings and analyses of peg-IFN treatment, the control parameter shouldbe ED50 rather than simply ✏.

Another antiviral used in HCV treatment is Ribavirin (RBV), which is believed to causegenetic mutations during the viral replication stage. With enough RBV, mutations occur thatrender virions noninfectious at rate ⇢, so whilst not actively participating in the HCV cycle,they still contribute to the viral load when samples are taken. Hence a new compartment wasintroduced by Dixit et al. [11]:

dT

dt= s� dT � �V

I

T (2.28)

dI

dt= �V

I

T � �I (2.29)

dVI

dt= (1� ⇢)(1� ✏)pI � cV

I

(2.30)

dVNI

dt= ⇢(1� ✏)pI � cV

NI

(2.31)

V = VI

+ VNI

(2.32)

Observe that ⌘ has been done away with since it appears, from a range of fits that will beexamined in Chapter 3 that the antiviral mechanism of IFN can be described entirely by ✏.

It is interesting to note that the antiviral e↵ect of RBV is negligible when used as monother-apy, yet over a 48 week treatment course it enhances the rate of SVRs1 when used in combina-tion with moderate doses of peg-IFN from 45% to 77% [11]. This is due to the fact that onlywhen peg-IFN has reduced the viral load can RBV cause enough mutations per virion to elicitVI

) VNI

. The current model does not represent this condition, so we propose the followingmodel for Ribavirin e�cacy

⇢ =RBV

↵V +RBV(2.33)

1Sustained virological response (SVR) is defined as undetectable viral load 24 weeks after therapy cessation.Once SVR has been attained a patient is said to be cured of HCV

23

where RBV is the dose of Ribavirin, and ↵ is a scaling constant that has yet to be deter-mined. The Ribavirin e�cacy is now inversely proportional to the viral load, which would proveimportant were we to simulate scenarios where the IFN dose is reduced.

2.5 The Snoeck Model

The Snoeck Model [5] uses equations (2.28) - (2.32) combined with Dahari’s regeneration mech-anism to obtain the following system.

dT

dt= s+ rT

✓1� T + I

Tmax

◆� dT � �V

I

T

dI

dt= �V

I

T + rI

✓1� T + I

Tmax

◆� �I

dVI

dt= (1� ⇢)(1� ✏)pI � cV

I

dVNI

dt= ⇢(1� ✏)pI � cV

NI

V = VI

+ VNI

which has steady states

Tu

=Tmax

2r

✓r � d+

r(r � d)2 +

4rs

Tmax

◆Iu

= VIu

= VNIu

= 0

Ti

=1

2

�r2D

A2+

r(r2D

A2)2 +

4rsTmax

A2

!

Ii

= Ti

✓A

r� 1

◆+ T

max

� �Tmax

r

VIi

= ✏⇤⇢⇤p

cIi

VNIi

= ✏⇤⇢p

cIi

where

A =✏⇤⇢⇤p�T

max

c, D =

Tmax

r2[A(r � �) + r(d� �)], ⇢⇤ = (1� ⇢)

Non-dimensionalising the model, we obtain

dx

d⌧= k0 + k5x

✓1� x+ y

k6

◆� x� ⌘⇤k1xzI

dy

d⌧= ⌘⇤k1xzI + k5y

✓1� x+ y

k6

◆� k2y

dzI

d⌧= ⇢⇤✏⇤k3y � k4zI

dzNI

d⌧= ⇢✏⇤k3y � k4zNI

where

k0 =s

dTu

, k1 =�T

u

d, k2 =

�

d, k3 =

p

d, k4 =

c

d, k5 =

r

d, k6 =

Tmax

Tu

24

The dimensionless model has steady states

xu

=k62k5

k5 � 1±

r(k5 � 1)2 + 4

k0k5k6

!= 1

yu

= zu

= 0

xi

=D ±

qD2 + 4k0k5

k6A2

2k5k6A2

yi

= xi

(A� 1) + k6

✓1� k2

k5

◆

zIi

= ⇢⇤✏⇤k3k4

yi

zNIi

= ⇢✏⇤k3k4

yi

where

A =✏⇤⇢⇤k1k3k6

k4k5D = k2 +A(k2 � k5)� 1

Equations (2.34), (2.34) and (2.34) and their dimensionless counterparts are the same asequations (2.1), (2.2) and (2.3) from the Dahari model, except (2.34) has the additional controlparameter ⇢ which decouples a fraction of V to form (2.34). As such the relation (2.8) still holds,and the critical e�cacy ✏⇤0⇢

⇤0 is the same as ✏⇤0⌘

⇤0 given in (2.12). The ⇢ mechanism e↵ectively just

increases the ✏ inhibition multiplicatively, however the additional equation for VNI

is requiredto describe the measured viral load.

Rather than find the bifurcation point ✏0, Snoeck et al. [5] define the basic reproductionnumber R0 as the number of newly infected hepatocytes that will arise from one infected cell inan otherwise healthy liver. They state that this is

R0 ⇡sp�

dc�(2.34)

and that its magnitude will determine whether the disease will increase (R0 > 1), clear withouttherapy (R0 < 1) or is in steady state (R0 = 1). They then introduce

R1 = (1� ✏t

)R0

as the reproductive ratio in presence of therapy with total e�cacy ✏t

, and infer from above thattherapy will induce cure if

R1 = (1� ✏t

)sp�

dc�< 1

Hence there is a transcritical bifurcation point when they are equal, in which case

✏⇤c

=dc�

sp�=

k2k4k0k1k3

which is the same as the dimensional critical e�cacy given by the Neumann model (1.16). Thisis clearly a simplification, since the Snoeck model is very similar to the Dahari model, and aspointed out above has the same critical e�cacy. However, taking ✏0 from Neumann requires lessparameter estimation, and the fact it approximates the critical e�cacy under more advancedmodels will prove highly useful in Part III.

Snoeck et al. [5] add a number of extra conditions to the model so as to make the simulationsbased upon it match their study data. They account for the fact that tests cannot detect thepresence of virions for concentrations below the lower limit of quantification (LLOQ), which iscurrently 50IU/ml, and so a detected absence of virions crucially does not necessarily correspond

25

to an actual clearance of infection. Since the only variable that can be frequently measuredwithout destroying part of the liver is V = V

I

+VNI

, this explains why the virus may return evenafter a patient appears cured. To simulate these relapses the model displays any V < LLOQ asLLOQ yet continues to simulate what is going on below this limit. Thus the model can emulatethe detected viral loads of a patient experiencing relapse.

The model also uses the discrete nature of cells to define the cure bound at one infectedhepatocyte. When the simulated value of I drops below this level, virion production, or p, is setto zero. We argue that this definition of cure should be amended to include the extra conditionthat when V < 1, there can be no de novo infection of T , and only when both of these conditionsare met is cure inevitable.

The pharmacokinetics of IFN are accounted for by, rather than just setting ✏ = 0 at the endof therapy, defining

✏⇤end

= (1� e�(t�t

end

)✏) (2.35)

so that when therapy is stopped the e�cacy decays exponentially with decay constant as theIFN is cleared from the body. Since the inclusion of PK is quite a departure from the previousmodels, we will not include this mechanism in the forthcoming data assimilation. Likewise, wewill also keep the e�cacy of Ribavirin constant, setting ⇢ = 0.5 to avoid unreasonable estimatesof ✏.

26

Chapter 3

Qualifying the Models

3.1 Fitting to Patient Data

In order to determine whether each of the proposed models are good or not, it seems reasonableto say that a model qualifies only if it can fit all data collected from real HCV patients.

Using data from [5] we see that patients can be categorised into the following viral loadprofiles defined by Snoeck et al. [5].

1. Sustained Virological Response (SVR)

2. Partial Virological Response (PVR)

3. Relapse

4. Breakthrough

5. Null Response

Here the data is sampled often only once every four weeks but over a period of 72 weeks, makingthis long term data. However looking at short term data from [14], [4] and [2] where the viralload has been measured several times a day usually for 2-8 weeks, the literature acknowledgesthree profiles.

1. Biphasic Decline

2. Triphasic Decline

3. Null Response

A model that adequately describes HCV therapy should be able to fit data from every profilelisted. In Figure 3.1 we attempt to fit each model to each profile using MATLAB. During thesimulations we include the cure bound and LLOQ conditions from Snoeck et al. [5], with ✏ and⇢ set to zero before and after treatment, except in the Snoeck model where (2.35) is used aftertreatment.

The MATLAB programme we made is built upon MATLAB’s unconstrained nonlinear op-timisation function fminsearch. It varies the set of parameters to try and minimise the sum ofsquares of errors (SSE)

nX

i=1

(log10Mi

� log10 V (ti

))2

where Mi

is the ith measured viral load from a given profile, and V (ti

) is the correspondingsimulated viral load at the time the measurement was taken. When the minimum of thisfunction has been located, the model is assumed to fit the data as well as possible, and the SSEalong with the parameters required to achieve this value are recorded.

27

3.2 Assumptions

It is assumed, as in all the literature, that patients have reached the infected steady state priorto the start of treatment, so this is calculated from (and thus varies with) the parameters usingthe expressions given in previous chapters.

Since the lifespan of a healthy hepatocyte is known to be around 300 days [15], we could fixd = 1/300 day�1 across all the models. When this assumption was not made the optimisationoften returned unreasonable estimates for d. We can also make the assumption, for the appro-priate models, that T

max

is roughly the same across all patients, since we assume they are allmature adults, and so could fix the value T

max

= 18.5 ⇥ 106 cells/ml from [19]. From here wecan use the uninfected steady state to calculate s = 61.7⇥ 103 cells/ml/day, and fix this valuewhere appropriate. As stated in Chapter 2.5 we set ⇢ = 0.5 and = 0 in the Snoeck model.Noting that in Snoeck et al. [5] the value of p is fixed to 25.1 without giving any justificationwhereas in fact the value of p is not currently known.

In a recent letter to Nature [16], Harel Dahari challenged the value of r in the Snoeck paper[5], saying that it was too small. Snoeck et al. [5] used the values of s, d and T

max

listed aboveto fit the following equation for a healthy liver to data from a liver donor taken from Pomfretet al. [12].

dT

dt= s+ rT

✓1� T

Tmax

◆� dT

Tu

=Tmax

2r

✓r � d+

r(r � d)2 +

4rs

Tmax

◆

T (0) =Tu

2

where T (0) represents half the liver having been donated.Subsequently they fixed r to 0.00562, however upon fitting the equation to regeneration

data from a separate study [17], Dahari estimated the proliferation rate r to be considerablyhigher (r = 0.24) in agreement with his previous papers [4] [6]. Hence when fitting the Daharimodel, we applied the constraint that r � 0.2 whilst fixing it to 0.00562 in the other models,the implications of which will be investigated in the discussion. By comparison, it should benoted that in a separate study by Philip Aston (unpublished material) the first Aston modelwas fitted to the same data from Pomfret [12] and gave a value of r = 5.4⇥ 10�8.

One important property of the dimensionless models is that the variables have been rescaled,in this case so that T

u

! 1 across the models. As a consequence the parameters are rescaledas well, which we have seen simplifies the equations by e↵ectively eliminating some parameters(reducing them to 1). With everything operating around variables ⇡ O(1) it becomes evidentwhether a particular term is significant or not. Snoeck’s value of r appears to be very small,almost enough to make the regeneration term insignificant, however when the correspondingdimensionless parameter is evaluated

k5 =r

d

=0.00562

0.003= 1.873 (3.1)

its e↵ect is revealed to be similar in magnitude to several of the other parameters (k0 = 1.067,k6 = 0.9657) and not as negligible as one might first presume.

28

0 50 100 150 200 250 300 350 400 450 500100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

A

0 50 100 150 200 250 300 350 400 450 500100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

B

0 50 100 150 200 250 300 350 400 450 500100

101

102

103

104

105

106

107

108

109

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

C

0 50 100 150 200 250 300 350 400 450 500100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

D

0 50 100 150 200 250 300 350 400 450 500100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

E

−10 0 10 20 30 40 50 60100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

F

Figure 3.1: The set of viral load profiles being fitted by each model. (A) SVR, (B) PVR,(C) Relapse, (D) Breakthrough, (E) Null Responder, (F) Triphasic Decline. The models arerepresented by (Magenta) Neumann, (Cyan) Dahari, (Green) Aston 1, (Red) Aston 2, (Blue)Snoeck. Measured viral load data is represented by black dots and the LLOQ (50 IU/ml) by thehorizontal black dashed line. Treatment begins at t = 0 and ends with the vertical black line.Viral load data simulated below the LLOQ is represented with dotted lines for each model. Thefull list of parameters and SSEs is contained in Appendix A

29

Chapter 4

Discussion

We see from Figure 3.1 that every model appears to be able to at least approximate the responseprofiles, sometimes fitting very accurately. The average error for a model is computed bysumming its errors across all response profiles and dividing by the total number of data points.

Model Neumann Dahari Aston 1 Aston 2 SnoeckAverage Error (Steady State) 0.2934 0.8055 0.3151 3.3660 1.1605

Average Error (Variable T (0), I(0)) 0.0972 0.8652 0.0961 0.1720 0.0923

Table 4.1: Viral Load / Model Errors

Profile SVR PVR Relapse Breakthrough Null TriphasicAverage Error (Steady State) 7.9176e-25 0.7653 2.7498 1.1442 0.9297 2.3045

Average Error (Variable T (0), I(0)) 8.4875e-25 0.1845 0.8172 0.5289 0.3134 1.1452

Table 4.2: Viral Load / Model Errors

It is interesting to note that in profile C the viral load decreases without the steep first phasetypical of IFN therapy, and none of the models can represent this. Similarly in profile E theviral load actually increases after the onset of treatment, which the models can only approximatewith a straight line. The latter observation implies either that the e�cacy is negative, whichis certainly not allowed, or that the parameters are actually variables, which is not very usefuleither, or that the system was still ascending to the infected steady state at the start of therapy.The third scenario has led us to question the assumption that patients always begin therapy withtheir infection having stabilised. Subsequently we carried out a second data assimilation, fixingV (0) to the initial measurement, and allowing the optimiser to vary T (0) and I(0). The resultsare shown in Figure 4.2 and the average errors are shown in Row 3 of Tables 2 and 3. Note thatdue to time constraints, no parameters were fixed during this second fitting. However, in Chapter5 it will be shown that equivalent results can be gained through fixing certain parameters.

To understand the behaviour of the entire system under both assumptions about the startingconditions, we see from Figure 4.1 that at the initiation of therapy, t = 0, I is still increasingand T is decreasing. Note that the e�cacy for this profile is very small, but importantly it isstill non-negative.

Clearly, with the exception of SVR which already has an extremely low average error, the datais represented significantly better when the system does not begin in equilibrium. Interestingly,the Dahari model can not explain PVR, relapse, breakthrough and null responses. Hence, inTable 3 where we have computed the average error for each profile we have omitted the Daharimodel.

30

The reason for this is that, as mentioned in Chapter 3.2, the Dahari model has a relativelyhigh value of r compared to the structurally similar Snoeck model (where r is several ordersof magnitude smaller but as discussed with (3.1) by no means insignificant). This high valueallows for T and I to reach steady state more rapidly, and allows for the quasi-steady stateresponsible for triphasic decline, where I and V remain constant until T > I (see Dahari etal. [4]). However, this rapid stabilisation prevents the predator-prey type oscillations (Figure1.2c) responsible for PVR, relapse and breakthrough. This is apparent in Figures 3.1 and 4.2.This can also be explained through stability analysis since the oscillations we observe are aboutthe infected steady state (where they will eventually settle) and are a consequence of complexeigenvalues about this equilibrium. The critical value of r (or any parameter) can be formulatedsuch that two of the three eigenvalues change between being real and complex conjugate pairs,though since formulating this involves calculating the discriminant of the cubic determinant, itis too complicated to include here.

0 100 200 300 400 500 600100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

Figure 4.1: The Snoeck model fit-ting the Null response. Solu-tions with variable starting val-ues are shown in solid lines andthose starting with the infectedsteady state in dashed. The in-dividual variables are representedby (Blue) T , (Green) I and (Red)T .

31

0 50 100 150 200 250 300 350 400 450 500100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

A

0 50 100 150 200 250 300 350 400 450 500100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

B

0 50 100 150 200 250 300 350 400 450 500100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

C

0 50 100 150 200 250 300 350 400 450 500100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

D

0 50 100 150 200 250 300 350 400 450 500100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

E

0 10 20 30 40 50 60100

102

104

106

108

1010

time (days)

log 10

HC

V lo

ad (I

u/m

l)

F

Figure 4.2: The set of viral load profiles fitted by each model, this time allowing T (0) and I(0)to be optimised, and fixing V (0) to the first measured viral load. The profiles and colour codingare the same as those in Figure 3.1.

32

Chapter 5

Identifiability of parameters

We saw through dimensional analysis that the models are all over parameterised, and we alsosee this in the optimisation. Clearly a specific set of parameters will cause a model to match adata set, but we find that a di↵erent set of parameters will cause an equally good fit.

s d � � p c r Tmax

✏ SSE61700 0.003 1.108⇥10�8 0.0906 22.6 3.6108 0.00562 18.5⇥106 0.6953 1.079861700 0.003 6.5906⇥10�9 0.1152 10 1.3912 0.00562 18.5⇥106 0.4442 1.0798

Table 5.1: Parameters used in the Snoeck model

The two parameters p and ✏⇤ appearing uniquely as a product across all models providesinfinite solutions to the equations, since for any solution, rescaling p by a factor ↵ and ✏⇤ by afactor ↵�1 causes the scalings to cancel, in other words changes in p can be compensated for bychanges in ✏ to provide an equally good fit. More precisely, we find that

(1� ⌫✏)↵p = (1� ✏)p

where

⌫ =✏� 1 + ↵

↵✏

Clearly if either p or ✏ is fixed then such a scaling cannot happen, and since ✏ is a controlparameter it would be wise to fix p.

We took equations (1.3), (1.4) and (1.5) with ⌘ = 0 and considered the e↵ect of scaling timeby a factor ↵�1

1 . If every parameter (except the control parameter ✏) is scaled by ↵1 then thescaling factor cancels throughout, leaving the original 3 equations.

dT

dt/↵1= ↵1

dT

dt= ↵1s� ↵1dT � ↵1�V T

= (1.3)dI

dt/↵1= ↵1

dI

dt= ↵1�V T � ↵1�I

= (1.4)dV

dt/↵1= ↵1

dV

dt= ↵1✏

⇤pI � ↵1cV

= (1.5)

Hence there exists a family of solutions to the three equations. This is fairly trivial, e↵ectivelyshowing that if the profiles of T , I and V are scaled along the time axis, we can expect the

33

parameter values to all be scaled inversely. We considered next scaling the three dependentvariables T , I, and V by a factor ↵2.

d↵2T

dt= ↵2

dT

dt= s� d↵2T � �↵2

2V T

d↵2I

dt= ↵2

dI

dt= �↵2

2V T � �↵2I

d↵2V

dt= ✏⇤p↵2I � c↵2V

Rescaling s by ↵2 and � by ↵�12 the equations again reduce to the original form, revealing a

second family of solutions. This family illustrates that rescaling the constant (s) and squaredterm coe�cient (�) results in the dependent variables being scaled inversely.

Applying both ↵1 and ↵2 scalings simultaneously and defining ↵1 = d�1 and ↵2 = T�1u

results in the dimensionless system given by (1.7), (1.8) and (1.9). We observed since values fors and d are known and have little interpatient variation, fixing d prevents the ↵1 and fixing sprevents the ↵2 scalings from being allowed. Note that also through observing V (t) the ↵1 and↵2 scalings are not permitted. Eliminating the ↵ family by fixing p, all parameter families arerestricted and the remaining parameters (�, �, c and ✏) become identifiable.

The fact only one of the three variables (V) is measurable allows for a fourth family ofsolutions, which is evident if we allow our unknowns T and I, and the parameters s and p to bescaled by ↵3.

d↵3T

dt= ↵3

dT

dt= s� d↵3T � �V ↵3T

d↵3I

dt= ↵3

dI

dt= �V ↵3T � �↵3I

dV

dt= ✏⇤p↵3I � cV

Hence without fixing s or p, multiple fits to a specific viral profile would be possible since T andI could be taking any values whilst remaining unobserved.

When fitting the model to viral load data, V and t are e↵ectively known and cannot be scaled,so the ↵1 and ↵2 families of solutions cannot occur, and thus fixing s and d is not necessary,although s should still be fixed to prevent ↵3 scaling. However fixing p is still necessary toidentify ✏. Conversely when simulating patient data, fixing s and d to known values will preventscaling of t and V and thus make the simulation realistic.

To conclude this section on parameter identifiability, we see that through fixing s, d andp and through measuring V all possible types of scalings, which result in families of solutions,are prevented, leaving the remaining parameters identifiable. Throughout all of Part 1 we haveseen that when these parameters are not fixed, scaling by certain dimensional quantities allowsfor the model to be reduced to dimensionless form where there are fewer parameters, whichconsequently simplifies some qualitative analysis. However in scaling the variables we see thatthey become incompatible with real data.

34

Part II

Prospective Therapies

35

Chapter 6

Introduction - Population Modelling

Currently, the most tested treatment available for HCV is a combination of peg-IFN and RBV.This standard of care therapy is known to illicit SVR in ⇡ 50% of patients with HCV Genotype1 (G1), which has ⇡ 70% prevalence, and ⇡ 80% for the remaining genotypes (Gn1) [5]. Tocompound this dissatisfying cure rate, the side e↵ects of peg-IFN, which can include severedepression, hair loss, fever and anorexia, are often unbearable and patients have been known todiscontinue treatment for this reason. Those who do not fall into the cure demographic abovehave had no alternative means of eradicating the virus until recently, with a variety of new typesof drug being currently in development.

In this section we will investigate the e↵ectiveness of certain developmental therapies throughtheir perceived mechanisms of action and using population modelling. In Snoeck et al. [5] theSnoeck model was fitted using the Monolix software to viral load profiles from 2, 100 HCVpatients. With such a large number of profiles, the median value and variance of each parametercould be calculated for the study population as a whole, and also for sub-populations such aspatients with HCV Gn1 or patients that experienced a viral breakthrough, to try and understandwhy certain responses occurred. By using the population parameters, we were able to runn simulations and see how a typical population of n HCV carriers would respond to noveltreatments.

To avoid the over-parameterisation described in Chapter 5, Snoeck et al. only computed thevariance of �, c, EC50 (by implication ✏) and R0 (by implication �) whilst holding all the othervalues fixed. The variance values given in Snoeck et al., termed inter-individual variability, werelisted as percentages of the coe�cient of variation.

Snoeck et al. stated that the variable parameters were log-normally distributed, so weconstructed a distribution of the population parameters as follows

�i

= ✓�

e⌘�i

�i

= ✓�

e⌘�i

ci

= ✓c

e⌘ci

Emax

i

= ✓E

max

e⌘Emax

i

where ✓x

is the median value for parameter x, and ⌘x

is a value drawn from the normal dis-tribution with µ = 0 and �2 = CV

x

(the coe�cient of variance percentage in decimal form).To obtain the distribution for � we used the distribution for R0 and then transformed it using(2.34).

We simulated therapy for 100 patients with HCV G1 (a higher value for ✓delta

distinguishesGn1) and the results are shown in Figure 6.1. In this simulation 55% of patients achieved SVR,which is close to the established value of 50%, and is most likely higher due to the model havingcomplete adherence (none of our patients drop out of therapy). We will now add additionalfactors to the model to incorporate developmental therapies.

36

0 100 200 300 400 500 600100

102

104

106

108

1010

Days

Vira

l Loa

d IU

/ml

55% SVR

Figure 6.1: Population simulation for 100 patients undergoing standard of care therapy for 340days. Cure rates agree with real world observations. The dashed line indicates the limit ofdetection. We observe that every profile in Chapter 3 is present in this population.

37

Chapter 7

SRB1 Entry Blocker

We have seen that IFN operates through the ✏ mechanism, and ⌘ has been disregarded through-out the model qualification of previous chapters. It is likely that ⌘ was considered by Neumannet al. [2] as their model was based on earlier HIV models by Perelson [18], for which entryinhibiting therapies were in early development (Maraviroc is an entry inhibitor for HIV, FDAapproved in 2007). Molecules that exhibit this e↵ect do so through binding to protein receptorsof host cell that would otherwise bind to the virus resulting in infection.

Currently compounds in development such as ITX-5061 from iTherX are able to inhibitHCV binding, provoking a renewed interest in the e↵ectivity of ⌘. In Part I we conducted muchof the model analysis with ⌘ present and observed that the two antiviral mechanisms could becombined multiplicatively to cause a system bifurcation (1.16).

Hence entry inhibitor therapies could be used in combination with SOC to increase the totale�cacy to greater than critical values, or could allow for a reduced (or even unnecessary) dose ofIFN. Phase II trials in ITX-5061 are under way but the ED50 and population parameters of thisnew class of drug have not been released. However, using our population model we simulatedentry-inhibitor e↵ectiveness across three scenarios for various ⌘ values (held constant throughoutthe population) and plotted the predicted SVR rates. The three scenarios considered were

1. Entry-inhibitor monotherapy

2. Entry-inhibitor with SOC

3. Entry-inhibitor with RBV alone

In Figure 7.1 (black) it is apparent that even for perfect e�cacies (⌘ = 1) the SVR rate for entryinhibitor monotherapy is only 86%, implying that 48 weeks is not long enough for the infectedcells to be eradicated in all patients. However, depending on the safety profile of entry inhibitortherapy, patients may be able to endure the course of treatment for longer than with SOC. Thelonger cure time is a result of virion production still occurring at rate pI during ⌘ therapy, soprofiles do not exhibit the steep first phase of decline seen with ✏ when virions are cleared atrate ⇡ c whilst production is reduced.

In Figure 7.1 (green) we ran a population simulation for each ⌘ value, this time combined withstandard of care IFN/RBV treatment. Clearly, even for low ⌘ there is a noticeable improvementon the cure rate of SOC alone. However the cure rate is still not 100%, indicating that thetreatment period needs to be extended. On the same diagram (blue) we present the resultsof simulations where the duration of IFN/RBV was reduced to 36 weeks whilst that of entry-inhibitor was extended to 85, noting that in this case 100% of subjects attain SVR for ⌘ = 1.When the length of IFN/RBV therapy was reduced to 16 weeks (red) and entry-inhibitor wasused with RBV alone (gold), the SVR rate lowered considerably for all but the highest (� 0.9)⌘ e�cacies.

38

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Efficacy (η)

% o

f Pop

ulat

ion

acie

ving

SVR

Figure 7.1: Percentage of patients achieving SVR with (black) 48 weeks of entry-inhibitingmonotherapy (⌘), (green) 48 weeks of SOC (IFN and RBV) with entry inhibitor (⌘), (gold) 48weeks of RBV with entry inhibitor, (blue) 36 weeks of SOC with 85 weeks of entry inhibitor(⌘), (red) 16 weeks of IFN/RBV with 85 weeks of entry inhibitor (⌘). For each e�cacy 100patients were simulated, and we see from the SVR rates that there is large interpatient variationin critical e�cacy.

39

Chapter 8

Direct Acting Antiviral Agents

Protease, helicase nucleoside and polymerase inhibitors, taken orally, target certain structuresused by HCV for intracellular replication, and subsequently can be modelled using ✏. The newclass of drugs o↵ers higher values of ✏ than IFN, however due to the heterogeneity of the virus,even within a specific genotype, there are treatment-resistant strains of the virus created througherrors in the replication cycle that are formed before and during therapy. Despite the low errorrate in HCV RNA copying (10�5 per virion), the sheer number of virions produced (1012 perday) makes the emergence of resistant strains possible. This new type of viral rebound has beenobserved in trials of Telaprevir, a protease inhibitor that was FDA approved in April 2011.

There are two sources of resistant virus - production from Ires

, that is hepatocytes infectedwith resistant virus V

res

and also mutations that occur within hepatocytes infected by theordinary wild-type virus. Hence the model can be extended as follows

dT

dt= s+ rT

✓1� T + I

wt

+ Ires

Tmax

◆� �(V

wt

+ Vres

)T

dIwt

dt= �V

wt

T + rIwt

✓1� T + I

wt

+ Ires

Tmax

◆� �I

wt

dIres

dt= �V

res

T + rIres

✓1� T + I

wt

+ Ires

Tmax

◆� �I

res

dVwt

dt= (1� µ)(1� ✏)pI

wt

� cVwt

dVres

dt= pI

res

+ µ(1� ✏)pIwt

� cVres

Where µ is the mutation rate from wild-type virus to any strain resistant to that particulartherapy. There have been many variants on this model, some simpler, such as the MerckVaniprevir study by Poland et al [20] where the proliferation terms are omitted, and some morecomplicated such as the extended model by Guedj and Neumann [21] that includes intracellularreplication dynamics to explain the entire range of new viral load profiles that can occur withDAAs.

There are currently scores of DAAs in development, each a↵ecting di↵erent enzymes usedfor viral production. We propose that modelling the use of ”drug cocktails” consisting of acombination of DAAs and entry inhibitors would reveal ways to overcome the resistance problem,though modelling the interplay of resistant and target strains of virus would become complicatedas the number of DAAs increased.

40

Chapter 9

Transfection Therapy (RNAi)

Several biotech companies have developed a potential HCV treatment based upon RNA-interferencemechanisms, which allow for particular genes to be ”silenced”. It is speculated that the viralvector, administered intravenously, penetrates both healthy and infected hepatocytes and si-lences certain genes involved in HCV infection, thus rendering these cells immune to the virus.It is likely that the viral vector may only be administered once due to immune response, butdevelopers have promised very high (up to 100%) transfection rates. Hence we adapted themodel to include transfected cells

dT

dt= s+ rT

✓1� T + I + T

trans

+ Itrans

Tmax

◆� dT � ⌘⇤�V T

dI

dt= ⌘⇤�V T + rI

✓1� T + I + T

trans

+ Itrans

Tmax

◆� �I

dTtrans

dt= �dT

trans

dItrans

dt= ��I

trans

dV

dt= ✏⇤pI � cV

withT (0) = T

i

, I(0) = Ii

, V (0) = Vi

, Ttrans

(0) = 0, Itrans

(0) = 0

to describe transfection by RNAi. When the viral vector is administered, the simulations are”paused” and restarted with the following initial conditions

T (RNAi) = (1� �)T (pre

RNAi)

I(RNAi) = (1� �)I(pre

RNAi)

Ttrans

(RNAi) = �T (pre

RNAi)

Itrans

(RNAi) = �I(pre

RNAi)

V (RNAi) = V (pre

RNAi)

wherepre

RNAi represents the moment before the e↵ect of RNAi (assumed to be instantaneous)and � is the fraction of cells that are transfected. Note that V remains una↵ected by this therapy.We conducted a population simulation for the most optimistic scenario, 100% transfection, ofRNAi as monotherapy. The results are shown in Figure 9.1.

It has been found through simulation that RNAi when combined with SOC achieved the bestoutcome when RNAi was administered at week 16 of SOC, with assumed 100% transfection.However, the outcome was an increase in SVR rate from 53% (SOC alone) to 75% (SOC +

41

0 100 200 300 400 500 600

100

104

108

10−4

Time (days)

Vira

l loa

d (IU

/ml)

A

0 100 200 300 400 500 600100

104

108108

Time (days)

Hep

atoc

yte

(T)

Con

cent

ratio

n (IU

/ml)

B

0 100 200 300 400 500 600

100

104

108

10−4

Time (days)

Infe

cted

Cel

l (I)

Con

cent

ratio

n (IU

/ml)

C

0 100 200 300 400 500 600100

104

108

Time (days)

Tran

sfec

ted

Cel

l (T

trans

+Itra

ns)

Con

cent

ratio

n (IU

/ml)

D

20% SVR

Figure 9.1: RNAi monotherapy with 100% transfection in 100 patients. Both healthy andinfected hepatocytes (B and C) are removed completely at t = 10 and a universal drop isobserved in the viral load (A). In the transfected cell population (D) we see a biphasic declineas I

trans

and then Ttrans

undergo exponential decay. A small decrease in transfected cells onthe log scale allows for a rapid resurgence in T , so that the remaining virions may restart theinfection process. It should be noted that in a population simulation of patients with no therapywhatsoever, around 8% attain SVR. This is termed spontaneous clearance and is a result ofR0 1 (see (2.34)).

RNAi). Since SOC combined with DAAs has been shown to elevate SVR rates to around90% [20], these initial simulations were disappointing, though perhaps RNAi could enhance theSOC+DAA cure rate to near 100%.

However, these simulations employed a rigid dosing strategy, where all patients were givenRNAi at the same time. We simulated an individualised dosing strategy where each patient’sviral load was frequently monitored, and the RNAi was administered either the moment theirviral load dropped below the limit of quantification, or appeared to have zero derivative (i.e. wasabout to undergo a viral breakthrough). Again, 100% transfection was assumed. The resultsare shown in Figure 9.2.

42

0 100 200 300 400 500 600100

101

102

103

104

105

106

107

108

109

1010

Days

Vira

l Loa

d IU

/ml

76% SVR

Figure 9.2: Viral load profiles for 100 patients undergoing individualised SOC+RNAi therapywith 100% transfection. RNAi is administered either when the viral load reaches 50 IU/ml, or thedecline appears to have stopped, indicating a breakthrough. The cure rate is an improvement onthat of the optimum fixed-administration strategy described previously. Simulating this conceptfor 1000 patients gives a cure rate of 81.3%

There are two factors that combine to result in the predicted increases in SVR rate beingrelatively small.

1. There are still virions in the system

2. Due to the model parameter s, there is always a source of new hepatocytes, which we canpresume are not transfected and thus are prone to infection.

Hence despite the fact that at the moment of transfection there are no target or infected cells,new target cells will be formed and if there are still virions present infection may resume. Thesimulations show that it is rare for the entire viral load to be cleared before de novo infectionoccurs.

At the moment of transfection the will be up to a 100% reduction in virion production, soalthough SOC improves cure rates by reducing the viral load pre-transfection, it would seemunnecessary immediately after administration of RNAi as it is the viral load remaining frombefore transfection that resets the HCV cycle. Hence, we considered using the SRB1 entryblocker with RNAi to restrict the degree to which left over virions may infect remaining ornewly formed target cells. Results are shown in Figure 9.3.

One flaw in the RNAi model is that it neglects to include proliferation of the transfectedcells. Though they are included in the proliferation terms of the T and I equations, and thusslow the regeneration of HCV susceptible cells, once transfected the cells simply die o↵ at theirrespective rates. Since we do not currently know how transfected cells proliferate, we speculatedthe following possibilities

43

0 100 200 300 400 500 600100

102

104

106

108

1010

Days

Vira

l Loa

d IU

/ml

99% SVR

Figure 9.3: RNAi with 100% transfection, administered at the optimum time on an individualbasis (see Fig. 9.2), in combination with entry-inhibitor ⌘ = 0.95 until day 336, at which pointa single relapse is observed. Administering RNAi at 16 weeks as in Exprimo simulations gave acure rate of 95%.

1. RNAi can only transfect the cells present at its administration. Following the initialconversion, T

trans

and Itrans

will divide to form T and I respectively.

2. RNAi permanently alters the cell’s genes - transfection is passed on through mitosis.

Clearly the first scenario does not engender a higher SVR rate, since it allows for T , I and as aresult V to reappear even more rapidly. The second scenario seems more optimistic - indeed wesee in Figure 9.4 that the e↵ects of transfection can last for months, indicating that they couldbe passed from cell to cell.

The lifespan of an individual hepatocyte is around 5 months, so the Figure 9.4 does notnecessarily prove transference of RNAi - the observations may have been taken specifically fromhepatocytes known to be transfected. However, the fact that the data was collected in vivoimplies that the e↵ect is observed across the entire hepatocyte population, and contradicts theratio of transfected to vulnerable cells predicted by the original RNAi model, shown in Figure9.5. Subsequently we adapted the model so that the equations for T

trans

and Itrans

containproliferation terms, and conducted a population simulation of RNAi as monotherapy, shown inFigure 9.6.

We found that including the e↵ects of proliferative mitosis on transfected cells did not im-prove the cure rate significantly from the previous models. This is because the proliferationterms are inversely proportional to the total number of cells in the liver, causing rapid regrowthin the advent of hepatocyte loss (eg. liver donation) but remaining small when the liver isnear equilibrium. As no cells have been lost through transfection, proliferation remains small,yet there is still a continuous supply of healthy hepatocytes (s) enabling T to be restored, andinfection to resume, despite transfected cells initially dominating the total liver volume. It ispossible from Figure 9.4 that the full e↵ects of RNAi are not currently reflected in the model.

44

Figure 9.4: From an RNAi biotech (compound names changed to RNAi): The graph abovedemonstrates the ability of RNAi to target and inhibit three separate regions of the HCV genomesimultaneously in mice for more than two months after a single administration. The first threebars in each group represent inhibition of three di↵erent regions of the HCV genome by RNAi;percent inhibition is calculated relative to a negative control (the final bar in each group) andrepresents total reduction in a readily detectable reporter fused to HCV genetic sequences.

Through the set of RNAi simulations presented in this section, we observed that SVR ratescould be improved by administering RNAi at particular times dependent upon the individualresponses to SOC, and that the RNAi-SRB1 combination therapy presented in Figure 9.3 en-genders the highest success rate of all strategies condsiered. However since the variance onED50

SRB1 is not known, a value of ⌘ = 0.95 was maintained throughout the population. Weran a similar simulation fixing ✏ = 0.95 and found the SVR rate to be 97%, indicating that ⌘ isnot intrinsically a better mechanism for combining with RNAi.

What is apparent from Figure 9.3 when contrasted with the high ⌘ values in Figure 7.1 isthat when e�cacies are greater than critical, the use of RNAi will reduce the time required forI ! 0. Hence RNAi appears to be beneficial in cases where patients would achieve SVR wereconventional therapy maintained for long enough, but due to therapy being stopped undergo arelapse. In Part III we devise a method for obtaining the critical e�cacy in an individual andthe length such an e�cacy must be maintained to achieve SVR, and we posit that RNAi couldbe used to reduce this length.

45

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

Time after transfection (days)

Frac

tion

of to

tal h

epat

ocyt

es

that

are

tran

sfec

ted

Figure 9.5: Following simulated transfection of 100% of vulnerable hepatocytes at t = 0, thefraction of transfected to total hepatocytes was plotted over 600 days. Parameter values usedwere the population medians found in [5]. The dashed line at day 63 demonstrates that thepercentage of total hepatocytes exhibiting transfection di↵ers greatly to the value reported inFigure 9.4. This implies that the model for transfected cells is incorrect.

0 100 200 300 400 500 600

100

105

Days

Vira

l Loa

d (V

i + V

ni) IU

/ml

0 100 200 300 400 500 600100

102

104

106

108

Days

Targ

et H

epat

ocyt

eC

once

ntra

tion

(T) I

U/m

l

0 100 200 300 400 500 600

100

105

Days

Infe

cted

Hep

atoc

yte

Con

cent

ratio

n IU

/ml

0 100 200 300 400 500 600100

102

104

106

108

Days

Tran

sfec

ted

Cel

lC

once

ntra

tion

IU/m

l

23% SVR

A B

C D

Figure 9.6: Population simulation of RNAi as monotherapy with 100 subjects, 100% transfectionand proliferation of transfection. The results di↵er very little from Figure 9.1 and it is likely theslight increase in SVR rate is merely chance.

46

Part III

Customised Therapeutics

47

Chapter 10

10.1 The need for individualised therapy

It is clear from the 50% SVR rate in HCV G1 patients that the Standard of Care is failing manypeople. Though the designated therapy has undergone some changes since the discovery ofHCV in 1989 (e.g. the inclusion of Ribavirin) the treatment o↵ered is only adapted to maximisesuccess across the population, and not on an individual basis.

With many new therapies set to become available in the near future, it is clear the currentSOC will eventually become obsolete, and that a means of deciding which treatments shouldbe given to which patients will be required. The EuResist Network has already developed acomputerised system for determining the most e↵ective drug combinations for patients carryingHIV, and there are projects such as the ”Dynamically Individualised Treatment for HCV”,chaired by A. Neumann, that aim to create such a system for HCV.

We defined two conditions for therapeutically driving an individual’s HCV infection to SVR

1. The patient must be given a su�ciently strong dose

2. The dose must be maintained for long enough to facilitate eradication of V and I

Due to safety concerns, it is not practical to simply give every patient the strongest dose possiblefor a prolonged period, so we will now investigate whether it is possible to find the minimumdose and time required to cure an individual of HCV.

10.2 Determining the critical e�cacy

In Part I it was shown that for every set of patient parameters there is a corresponding equilib-rium of T , I and V about which the infection will eventually stabilise, termed the infected steadystate. It was also shown that for every set of parameters there is a corresponding critical e�cacyfor which e�cacies greater than this value will facilitate cure if maintained for long enough.In Figure 1.1B it was observed that for each e�cacy less than the critical value given by theindividual’s parameters, there is a corresponding equilibrium on the branch of infected steadystates. It is also apparent from Figure 1.1A that V

i

is a linear function of ✏ so the relationshipbetween them can be determined from knowing two values.

It is relatively simple to measure the value of V in a patient at any time, however we haveseen that lim

t!+1 V (t) = Vi

so it would be better to measure Vi

as a function of e�cacy,lim

t!+1 V (✏) = Vi

(✏) in order to identify what value of ✏ will give Vi

(✏) 0.Through measuring a patient’s viral load on two separate occasions with no therapy present,

we can determine whether they are at the infected steady state:

If V (t1) = V (t2)

Then V (✏ = 0) = Vi

(0)

=sp� � dc�

�c�

48

=sp

�c� d

�= A+B (10.1)

where the infected steady state is the dimensional form of (1.12). Note we used the state fromthe Neumann model for simplicity.

Administering a small dose of ✏ based therapy (IFN), here termed ✏1, will then cause theviral load to stabilise at a new value on the branch of infected steady states, as shown in Figure10.1. The value of V is then

Vi

(✏1) =(1� ✏1)sp� � dc�

�c�

= ✏⇤1sp

�c� d

�= ✏⇤1A+B (10.2)

From Figure 1.1 we note that ✏ = ✏c

causes the system to stabilise about the uninfectedsteady state, ie. V

u

= Iu

= 0, and that at critical e�cacy the two steady states intersect. Hence

Vi

(✏c

) =(1� ✏

c

)sp� � dc�

�c�= ✏⇤

c

A+B

= Vu

= 0 (10.3)

Therefore

✏⇤c

= �B

A(10.4)

From (10.1) and (10.2) we can formulate

✓Vi

(0)Vi

(✏1)

◆=

1 1

(1� ✏1) 1

�✓AB

◆

which can be solve for A and B,

✓AB

◆=

"� 1

✏1

1✏1

(1�✏1)✏1

� 1✏1

#✓Vi

(0)Vi

(✏1)

◆

in order to formulate (10.4) in terms of known quantities.

✏c

= 1� ✏⇤1Vi

(0)� Vi

(✏1)

Vi

(0)� Vi

(✏1)(10.5)

In real patients, we cannot control ✏ and instead vary the dose, which is modelled as inputto the e�cacy function (2.27). However this brings a new control parameter D into the problemand a new unknown ED50, which varies from patient to patient. This presents us with threeequations in four unknowns, and so in order to calculate the unknown values, a second and thirddose (resulting in a fourth and fifth equation but no additional unknowns) must be applied. Thisleads us to the following problem

A+B � Vi

(0) = 0 (10.6)✓1� D

c

Dc

+ ED50

◆A+B = 0 (10.7)

✓1� D

j

Dj

+ ED50

◆A+B � V

i

(Dj

) = 0 (10.8)

49

0 50 100 150 200 250 300105

106

107

Time (days)

Vira

l Loa

dε = 0

ε = 0.5

Figure 10.1: Administering ✏ = 0.5 at day 0 results in V (and also T and I though they cannotbe observed) stabilising at a new infected steady state that is closer to the uninfected one (SeeFigure 1.1). The critical e�cacy for this particular set of parameter values is ✏

c

= 0.9976.E�cacies greater than ✏

c

will cause V an I to stabilise at negative values - interpreted as thecure of HCV.

where j = 1, 2, 3It is then possible to calculate ED50 as

ED50 =D1D2(Vi

(D1)� Vi

(D2))�D1D3(Vi

(D1)� Vi

(D3)) +D2D3(Vi

(D2)� Vi

(D3))

D1(Vi

(D2)� Vi

(D3)) +D2(Vi

(D3)� Vi

(D1)) +D3(Vi

(D1)� Vi

(D2))(10.9)

which may be substituted into (10.7) and (10.8) to obtain (10.5) from which the criticaldose may be calculated. This method allows calculation of the dose that will induce cure beforefull therapy commences, however it depends upon the infection stabilising under a constant dosetwice. This requires frequent monitoring since the time required for stabilisation varies, and maytake longer than the 48 weeks of SOC, as observed in breakthrough profiles. Hence we suggestbeginning with a small dose such as 0.1µg, so as to minimise the disturbances from V

i

(0).

10.3 Determining the time required for cure

There are two conditions required for a patient to be cured of HCV.

1. The viral load V must be zero

2. The infected cell count I must be zero

With either of the above quantities present infection may resume, so a greater than criticale�cacy must be maintained long enough for both to decay to zero. Since the decay is exponential,they will asymptote to zero in the model but never reach it. Hence we instead define clearanceof I at I < 1/13.5 ⇥ 103 as there are 13.5l of hepatocyte containing plasma in the body.We assume that virions occupy the same volume, and thus virion clearance is achieved whenV < 1/13.5⇥103. For greater than critical e�cacies, it has been observed that a biphasic declinein the viral load ensues all the time that e�cacy is held constant. A triphasic decline may alsooccur, in which case the patient should be monitored until the flat ’shoulder’ phase ends, for itis the third phase of decline that is of most interest.

50

On the log scale, the decline of V is bilinear, with the first phase being roughly proportionalto c and the second to � [2]. Since the second phase is linear on the log scale, we can take twoseparate viral load measurements when this phase is in process and extrapolate to find the timeat which V = 0, as shown in Figure 10.2.

0 50 100 150 200 250 30010−4

10−2

100

102

104

106

Time (days)

Vira

l Loa

d C

once

ntra

tion

IU/m

l

Figure 10.2: Decline profiles for V (blue) and I (green) for greater than critical e�cacy. Thesolid blue line represents the variable being frequently measured. The dotted blue line illustrateshow the second phase can be extrapolated to determine the time when there are no virions in thesystem, indicated by the black solid line at 13.5⇥ 10�3 IU/ml. The entirety of I is representedwith a dashed line since it cannot be measured. The limit of quantification is represented by theblack dotted line. Measurements are taken at 10 and 30 days, indicated by white circles, afterwhich the viral load can be estimated by the dashed blue line. Note how easily the viral loadcan be estimated below the LOQ during bilinear decline.

We still require that I reaches < 13.5 ⇥ 10�3 in order to declare cure, however measuringthe concentration of I requires removing the liver, so is not practical. However it was shown inChapter 9 that there are treatments in development that momentarily eliminate most or possiblyall of I through transfection. Such therapies were shown to be ine↵ective when used alone sinceany remaining virus can reinitiate the HCV cycle, and even administering RNAi the momentthe viral load falls below the Limit of Quantification provides a less than satisfying cure rate asthere is still enough virus to facilitate relapse.

Using the techniques presented in this chapter it is possible to determine the time at whichV has been eliminated. Administering RNAi at this time could allow for I to also be eliminated,resulting in cure from HCV.

51

Chapter 11

Conclusions and Further Work

In this report all three questions posed by Piet van der Graaf in the Introduction have beenanswered. The HCV model and its key modifications have been reviewed, and we have suggestedways in which it could be further adapted such as the advanced Ribavirin model (2.33). Wehave shown which parameters should be fixed to known values according to the way in which themodel is being used and shown through non-dimensionalisation the relative significance of theparameter r, adding a strong argument to a current debate in one of the leading science journals[16]. We have also used non-dimensionalisation to simplify the model for stability analysis andmake inferences about constraints such as non-negativity of parameters and variables.

We have constructed a population model to predict the outcome and o↵er suggestions onuse of the SRB1 entry inhibitor and RNAi therapies based on the limited information thatis currently known about them, and we have shown how to ascertain the dose required tocure an individual using any of the virion inhibitor therapies (IFN and DAAs). We have alsochallenged the assumption that patients begin therapy with their infection having stabilised,and through the Aston models we have shown that there must be multiple mechanisms forhepatocyte formation both in the models and in reality, for without the parameter s we see acontradiction in the permission of the pure-infected steady state.

As should be evident through reading this project report, I have learned a tremendousamount during the course of my industrial training. I have learned new mathematical techniquessuch as non-dimensionalisation and have built upon my previous knowledge of topics such asODEs and stability analysis. I have also learned a great deal about numerical methods and haveadvanced my ability to tackle problems in Matlab to the point where I now feel comfortableusing the software to attempt any problem that requires it. I have also been introduced tonew programmes such as Monolix for population analysis and Berkeley Madonna for visualisingnumerical solutions with ease and e�ciency.

Applying the techniques learned at university to real problems has been very rewardingindeed and has strongly enhanced my mathematical confidence and intuition. This was evidencedtowards the end of my placement when I independently constructed a model of a separate processto help a biology intern collecting stomach/duodenum data.

Pharmaceutical mathematics is a highly dynamic field, where new information from observa-tional scientists enables more useful theoretical analysis from teams such as Pharmacometrics.When the ED50 and safety guidelines of the SRB1 compound becomes known, the model ofChapter 7 can be improved to give a realistic prediction of how patients will respond. As dis-cussed in Chapter 9, the mechanisms of RNAi are still closely guarded and it is likely that thetransfection model will need to be completely rebuilt to reflect the findings of Figure 9.4.

It would be interesting to study the degree to which the techniques developed in Part IIIwork in vivo. Another aspect we have neglected to include throughout our work is the pharma-cokinetics - we have assumed that e�cacy remains constant during treatment whereas a fullycomprehensive model would take into account ligand clearance and dosing frequency. We suggest

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adapting the model and the work conducted in Part III to include such e↵ects.It has been claimed by Philip Aston that the time taken for I to be eliminated under

su�ciently strong therapy can also be inferred through measuring V , so someone wishing tofurther develop the methods of Part III may wish to investigate this. It is clear that as we headtowards individualised HCV therapy with a plethora of drugs to chose from, building a completemodel that encompasses IFN, RBV, SRB1 DAAs and RNAi will be necessary. Strategies devisedfrom such a model could ultimately lead to highly e↵ective personalised therapies with the aimof eliminating the virus completely.

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Bibliography

[1] World Health Organisation HCV fact-sheet, http://www.who.int/mediacentre/factsheets/fs164/en

[2] Neumann, A.U, et al. 1998. Hepatitis C Viral Dynamics in Vivo and the Antiviral E�cacyof Interferon-alpha Therapy. Science 282 (5386), 103-107.

[3] Aston, P.J, et al. 2011. Mathematical Analysis of the PKPD Behaviour of MonoclonalAntibodies: Predicting in vivo Potency.

[4] Dahari, H et al. 2007. Modeling HCV dynamics: Liver regeneration and critical drug e�-cacy.

[5] Snoeck E. et al. 2010. A Comprehensive HCV Kinetic Model Explaining Cure

[6] Dahari, H et al. 2005a. Second hepatitis C replication compartment indicated by viraldynamics during liver transplantation. J. Hepatol. 42 (4), 491-498

[7] Fausto, N., 2004. Liver regeneration and repair: hepatocytes, progenitor cells and stemcells. Hepatology 39 (6), 1477-1487

[8] www.hepatologytextbook.com/hep chapt13.pdf

[9] Guo et al. 2004. Mechanism of the Interferon alpha response against HCV replicons. Virol-ogy

[10] Talal A. et al. 2006. Pharmacodynamics of PEG-IFN-alpha di↵erentiate HIV/HCV Coin-fected SVRs from non-SVRs. Hepatology

[11] Dixit N. et al. 2004. Modelling how Ribavirin improves response rates in HCV infection.Nature

[12] Pomfret E.A. et al. 2003. Liver regeneration and surgical outcome in donors of right-lobeliver grafts. Transplantation

[13] Dahari, H. et al. 2007. Triphasic decline of HCV RNA during antiviral therapy. Hepatology

[14] Herrmann, E. et al. 2003. E↵ect of Ribavirin on HCV kinetics in patients treated withPEG-IFN. Hepatology

[15] MacSween’s Pathology of the Liver, 5th Edition. 2006

[16] Dahari H. et al. 2011. Hepatocyte proliferation and Hepatitis C Virus kinetics during treat-ment. Clinical Pharmacology and Therapeutics 89 3, 353-354

[17] Naladin S. et al. 2004. Volumetric and functional recovery of the liver after right hepatec-tomy for living donation. Liver Transplantation

[18] Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M. & Ho, D.D. HIV-1 dynamicsin vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271,15821586 (1996).

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[19] Colombatto,P.et al.Sustainedresponsetointerferon-ribavirin combination therapy predictedby a model of hepatitis C virus dynamics using both HCV RNA and alanine aminotrans-ferase. Antivir. Ther. (Lond.) 8, 519530 (2003).

[20] Poland, B. Viral Dynamics Modeling and Simulation of the HCV Protease In-hibitor MK-7009 (Vaniprevir) with Peg-Interferon and Ribavirin http://www.go-acop.org/sites/default/files/webform/posters/PolandAcoP2011PosterHCV 1.pdf

[21] Guedj, J., & Neumann, A. U. (2010). Understanding hepatitis C viral dynamics with direct-acting antiviral agents due to the interplay between intracellular replication and cellularinfection dynamics. Journal of Theoretical Biology, 267(3), 330-340. Elsevier.

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