# Mathematical Modeling of Cytotoxic Lymphocyte-Mediated Immune Response to Hepatitis B Virus Infection

Authors: Changjiang Long; Huan Qi; Sheng-Hu Huang
Journal: Journal of Biomedicine and Biotechnology
 (2008)

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## Abstract

Nowak's model of the human immunodeficiency virus (HIV) infection has been extensively and successfully used to simulate the interaction between HIV and cytotoxic lymphocyte- (CTL-) mediated immune response. However, this model is not available for hepatitis B virus (HBV) infection. As the enhanced recruitment of virus-specific CTLs into the liver has been an important novel concept in the pathogenesis of hepatitis B, we develop a specific mathematical model analyzing the relationship between HBV and the CTL-mediated immune response, and the indicator of the liver cell damage, alanine aminotransferase (ALT). The stability condition of the complete recovery equilibrium point at which HBV will be eliminated entirely from the body is discussed. A different set of parameters is used in the simulation, and the results show that the model can interpret the wide variety of clinical manifestations of HBV infection. The model suggests that a rapid and vigorous CTL response is required for resolution of HBV infection.


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## Body

## 1. Introduction

Infection with the hepatitis B virus (HBV) is a major
health problem, which can lead to cirrhosis and primary hepatocellular
carcinoma (HCC). More than 2 billion people alive today have been infected by HBV. The population of
HBV carrier is about 400 million, of whom 75% are located in Asia. Accordingly, HBV causes approximately 1 million deaths each year worldwide. In China
alone, nearly 15 million new
infections occur annually, more than 30
million people are chronically infected, and more than 350 thousand of them die each year from cirrhosis and
HCC.In order to find an efficient way to prevent and treat the
infection, it is of great importance to understand the immunopathogenesis of
HBV. Although molecular techniques have provided fundamental insight into the
fine detail of the interaction between HBV and immune system, many biologically
important questions are not primarily concerned with the molecular mechanisms
of immune recognition but with the population dynamics of the immune response. Mathematical
models are always needed to answer these questions. Studies on humans infected with HIV-1 and macaques infected
with simian immunodeficiency virus (SIV) show that cytotoxic lymphocytes (CTLs) are critical in controlling virus
replication [1]. Virus mutants of human immunodeficiency
virus (HIV) and SIV are able to escape the dominant CTL response and become the
major replicating strains in vivo [1]. In contrast to HIV and many other
viruses, cell culture systems that allow efficient in vitro infection and passaging
of virus are not available for HBV, which is hepatotropic and noncytopathic. Recent
studies on HBV pathogenesis in animal models demonstrated that the enhanced
recruitment of virus-specific CTLs into the liver cells is critical for the
pathogenesis of both HBV infection and hepatocellular carcinoma [2, 3]. The most common mathematical model in HIV infection is
presented by Nowak to explain the dynamics of CTL-mediated host immune response
to HIV and the pathogenesis of AIDS [4].
Mathematical models, which are not based on CTL-mediated host response, have
been proposed for modeling HBV or HCV
infection and evaluating the effectiveness
of antiviral therapy [5–8]. Even the models describing the
interactions between host
immune response and virus were built
to explain the mechanism of acute hepatitis [9–11]. However,
these models fail to explain the various outcomes of HBV infection [12]. A new mathematical model for
dissecting the role of CTL in HBV diseases is needed for the following reasons.(1)
Being different kinds of viruses, HIV is cytopathic virus and HBV is a noncytopathic virus [12], that is, cells infected by HBV will not be killed by virus
directly, cellular function and lifespan of HBV-infected hepatocytes are
almost the same as that of the uninfected cells in vitro [13]. The death rate of noncytopathic
virus-infected cells in the absence of immunity equals that of uninfected
target cells [14]. The lifespan
of HBV-infected cells varies greatly in vivo which is mainly due to the strength of the anti-HBV
CTL response [15].  CTL will not only kill but also cure the infected hepatocytes by a nonlytic effector mechanism [16, 17]. The
effect of CTL response should be considered in the model.(2)
The non-CTL models
ignore the kinetics of hepatocyte replication. Actually, both uninfected and
infected hepatocytes can replicate at the same rate [13]. Infected cells are generated not only from normal
cells infected by HBV, but also from replication of its own [18, 19].(3)
In the non-CTL models,
the equilibrium abundance of infected cells depends only on the immunological
parameters [6]; but in fact the characteristic of HBV also has
great influence on the result [12].(4)
The value of alanine
aminotransferase (ALT) in the blood stream is generally taken to be an
indicator of the liver cell damage [9]. ALT was
not included in the non-CTL models.


## 2. Materials and Methods

### 2.1. Mathematical Models

The model contains six variables, that is, uninfected
hepatocytes (X), infected hepatocytes (Y), total host hepatocytes (N=X+Y), free
virus (V), a CTL response (Z), and ALT. The changes of population over time can be described by a system of
differential equations.The corresponding mathematics
equations are(1)dXdt=F(N)X−d1X−b1XV+k1YZ,dYdt=F(N)Y+b1XV−d1Y−(k1+k2)YZ,dVdt=k3Y−d3V,dZdt=(g4+k4YZ)(1−ZZmax⁡)−d4Z,dALTdt=k5(d1X+d1Y+k2YZ)−d5ALT,F(N)=d1N3,N=X+Y,X0=1,Y0=0,V0=0,Z0=g4d4,ALT0=k5d1d5d1,N0=1, where F(N) is the natural growth rate of hepatocytes. It
is a monotonically decreasing function [20]. We took F(N)=d1/N3 [13]; F(N)=d1 when N=1 (without loss of generality, we take the cell
and virus concentrations to be scaled such that in the uninfected system the
total cell concentration is N=1 [18, 19]).Both uninfected and infected hepatocytes replicate at a
rateF(N) and die at a rate d1,
while uninfected ones are infected by virus at a rate b1XV. The CTL response can activate two different pathways to eliminate a virus, either by killing the infected cells or by eliminating the
virus from within the cell without killing it. Infected cells are
assumed to be killed by the CTL
response at a rate k2YZ and be cured by the CTL response at a rate k1YZ.
Infected cells produce free virus at a rate k3Y and free virus particles are removed at a rate d3V. CTLs proliferation can be described by two terms g4 and k4YZ, where g4 represents antigen-independent proliferation
and k4YZ represents antigen-dependent
proliferation. CTLs decay at a rate d4Z. ALT is generated by the dead hepatocytes at a
rate k5 and decay at a
rate d5. All the variables
and parameters of the above are
nonnegative.


### 2.2. Equilibrium States Analysis

There are three possible steady
states: Hepatocytes are not infected—the uninfected state,
all the hepatocytes are infected—wholly infected
state, and the coexisting state—the uninfected and the infected
hepatocytes coexist. As these states are
too complex to
analyze, we only discuss the linear ability of the most concerned
state—uninfected state.As the equation ofZ is a logistic model, it
will make the expression of the analysis result very complex. To get a meaning
result easy for comparing with the clinical conclusion, we ignore the
limitation of Zmax⁡ as we only discuss small disturbance to the initial
state (the maximum number of specific T-cells can be 106 times of its initial number [20]).As ALT is just an
indicator of hepatic injury and do not influence other variables, it is not
included in the analysis progress for simplicity.Then the equations of the system can be
rewritten as(2)dXdt=F(N)X−d1X−b1XV+k1YZ,dYdt=F(N)Y+b1XV−d1Y−(k1+k2)YZ,dVdt=k3Y−d3V,dZdt=g4+k4YZ−d4Z,F(N)=d1N3,N=X+Y.For the uninfected state(X=1,Y=0,V=0,Z=g4/d4), as it is an equilibrium state, it should satisfy dX/dt=dY/dt=dV/dt=dZ/dt=0, so get the coordinates (1,0,0,g4/d4). The correspondence
Jacobian matrix is
(3)[F′(N)F′(N)+k1g4d4−b100−(k1+k2)g4d4b100k3−d300k4g4d40−d4].The characteristic equation is(4)(λ−F′(N))(λ+d4)(λ2+(d3+Δ)λ+d3Δ−b1k3)=0, where
(5)Δ=(k1+k2)g4d4,∂N∂X=∂(X+Y)∂X=1;∂N∂Y=∂(X+Y)∂Y=1,∂F(N)∂X=∂F(N)∂N∂N∂X=F′(N),∂F(N)∂Y=∂F(N)∂N∂N∂Y=F′(N),F′(N)=∂F(N)∂N=∂(d1/N3)∂N=−3d1N4=−3d1.Since−F′(N)=3d1>0,d4>0,d3>0,Δ>0,d3+Δ>0, according to the Routh-Hurwitz
criterion [21], if d3Δ−b1k3>0,
that is,
(6)(k1+k2)g4d4>b1k3d3,   there
will be linear stability with
respect to perturbations in (1,0,0,g4/d4). The left-hand side of the equation represents the ability of the
immune system, k1 represents the nonlytic ability of CTLs and k2 represents the lytic ability of CTLs. Also, g4/d4 is the initial value of CTLs. The right-hand side
of the equation represents the ability of HBV, b1 represents the infective ability of HBV, k3 represents the multiplication
ability of HBV, and d3 represents the death rate of HBV. If the left part is bigger than the right
part, it means that the immune system is strong enough to eliminate the
infection otherwise HBV can invade the body and exist for a long time.


### 2.3. Simulation

By combining the various results
derived in the previous section we can deduce an appropriate parameter set for the
simulation of the model.The
parameters (1 time unit = 1 day) are set up as follows:d1=0.002 (average life of hepatocyte is about 500 d [13]); d3=0.58 (estimated average half-life of free virions
is about 1.2 d [22]); d4=0.2 (the mean life of CTL is 4–6 d [20]); d5=0.25 (average half-life of ALT is between 0.5–5 [13]); Zmax⁡=0.001 there
should be no more than 108 HBV-specific CTL in the entire body and there are approximately 1011 infected hepatocytes in the human liver [23], as we took N=1, so Zmax⁡=108/1011);
ALT0=16 (The normal
range for ALT is between 0–40).


## 2.1. Mathematical Models

The model contains six variables, that is, uninfected
hepatocytes (X), infected hepatocytes (Y), total host hepatocytes (N=X+Y), free
virus (V), a CTL response (Z), and ALT. The changes of population over time can be described by a system of
differential equations.The corresponding mathematics
equations are(1)dXdt=F(N)X−d1X−b1XV+k1YZ,dYdt=F(N)Y+b1XV−d1Y−(k1+k2)YZ,dVdt=k3Y−d3V,dZdt=(g4+k4YZ)(1−ZZmax⁡)−d4Z,dALTdt=k5(d1X+d1Y+k2YZ)−d5ALT,F(N)=d1N3,N=X+Y,X0=1,Y0=0,V0=0,Z0=g4d4,ALT0=k5d1d5d1,N0=1, where F(N) is the natural growth rate of hepatocytes. It
is a monotonically decreasing function [20]. We took F(N)=d1/N3 [13]; F(N)=d1 when N=1 (without loss of generality, we take the cell
and virus concentrations to be scaled such that in the uninfected system the
total cell concentration is N=1 [18, 19]).Both uninfected and infected hepatocytes replicate at a
rateF(N) and die at a rate d1,
while uninfected ones are infected by virus at a rate b1XV. The CTL response can activate two different pathways to eliminate a virus, either by killing the infected cells or by eliminating the
virus from within the cell without killing it. Infected cells are
assumed to be killed by the CTL
response at a rate k2YZ and be cured by the CTL response at a rate k1YZ.
Infected cells produce free virus at a rate k3Y and free virus particles are removed at a rate d3V. CTLs proliferation can be described by two terms g4 and k4YZ, where g4 represents antigen-independent proliferation
and k4YZ represents antigen-dependent
proliferation. CTLs decay at a rate d4Z. ALT is generated by the dead hepatocytes at a
rate k5 and decay at a
rate d5. All the variables
and parameters of the above are
nonnegative.


## 2.2. Equilibrium States Analysis

There are three possible steady
states: Hepatocytes are not infected—the uninfected state,
all the hepatocytes are infected—wholly infected
state, and the coexisting state—the uninfected and the infected
hepatocytes coexist. As these states are
too complex to
analyze, we only discuss the linear ability of the most concerned
state—uninfected state.As the equation ofZ is a logistic model, it
will make the expression of the analysis result very complex. To get a meaning
result easy for comparing with the clinical conclusion, we ignore the
limitation of Zmax⁡ as we only discuss small disturbance to the initial
state (the maximum number of specific T-cells can be 106 times of its initial number [20]).As ALT is just an
indicator of hepatic injury and do not influence other variables, it is not
included in the analysis progress for simplicity.Then the equations of the system can be
rewritten as(2)dXdt=F(N)X−d1X−b1XV+k1YZ,dYdt=F(N)Y+b1XV−d1Y−(k1+k2)YZ,dVdt=k3Y−d3V,dZdt=g4+k4YZ−d4Z,F(N)=d1N3,N=X+Y.For the uninfected state(X=1,Y=0,V=0,Z=g4/d4), as it is an equilibrium state, it should satisfy dX/dt=dY/dt=dV/dt=dZ/dt=0, so get the coordinates (1,0,0,g4/d4). The correspondence
Jacobian matrix is
(3)[F′(N)F′(N)+k1g4d4−b100−(k1+k2)g4d4b100k3−d300k4g4d40−d4].The characteristic equation is(4)(λ−F′(N))(λ+d4)(λ2+(d3+Δ)λ+d3Δ−b1k3)=0, where
(5)Δ=(k1+k2)g4d4,∂N∂X=∂(X+Y)∂X=1;∂N∂Y=∂(X+Y)∂Y=1,∂F(N)∂X=∂F(N)∂N∂N∂X=F′(N),∂F(N)∂Y=∂F(N)∂N∂N∂Y=F′(N),F′(N)=∂F(N)∂N=∂(d1/N3)∂N=−3d1N4=−3d1.Since−F′(N)=3d1>0,d4>0,d3>0,Δ>0,d3+Δ>0, according to the Routh-Hurwitz
criterion [21], if d3Δ−b1k3>0,
that is,
(6)(k1+k2)g4d4>b1k3d3,   there
will be linear stability with
respect to perturbations in (1,0,0,g4/d4). The left-hand side of the equation represents the ability of the
immune system, k1 represents the nonlytic ability of CTLs and k2 represents the lytic ability of CTLs. Also, g4/d4 is the initial value of CTLs. The right-hand side
of the equation represents the ability of HBV, b1 represents the infective ability of HBV, k3 represents the multiplication
ability of HBV, and d3 represents the death rate of HBV. If the left part is bigger than the right
part, it means that the immune system is strong enough to eliminate the
infection otherwise HBV can invade the body and exist for a long time.


## 2.3. Simulation

By combining the various results
derived in the previous section we can deduce an appropriate parameter set for the
simulation of the model.The
parameters (1 time unit = 1 day) are set up as follows:d1=0.002 (average life of hepatocyte is about 500 d [13]); d3=0.58 (estimated average half-life of free virions
is about 1.2 d [22]); d4=0.2 (the mean life of CTL is 4–6 d [20]); d5=0.25 (average half-life of ALT is between 0.5–5 [13]); Zmax⁡=0.001 there
should be no more than 108 HBV-specific CTL in the entire body and there are approximately 1011 infected hepatocytes in the human liver [23], as we took N=1, so Zmax⁡=108/1011);
ALT0=16 (The normal
range for ALT is between 0–40).


## 3. Results

### 3.1. Acute Hepatitis

HBV can cause acute hepatitis, resulting in short-term inflammation of
the liver before the immune system is able to remove the virus from the body. In
acutely infected patients who successfully control the virus, the immune response that the patients produce against the viral proteins is
polyclonal, multispecific; and the
virus is eliminated from the blood and liver entirely. If the maximum damage and the maximum concentration of free virus are low,
the disease may come and go without any symptoms, otherwise severe clinical
symptoms will be observed.The parameters during the simulation of acute hepatitis are set up as
follows:d1=0.002;b1=3E−5;k1=1E6;k2=1E3;b3=0.1;k3=800;d3=0.58;g4=1E−8;k4=16;d4=0.2;k5=2000;d5=0.25;X0=1;Y0=0;V0=10;Z0=5E−8;ALT0=16. As the cellular
immune response is vigorous
and (k1+k2)(g4/d4)=0.0500>b1(k3/d3)=0.0414, so the immune system is strong enough to eliminate all the infected cells and
virus. The number of virus is at its peak at the beginning and then decrease.
HBV can only infect a small
number of cells. The number of
infected cells (4.53×10−4) peaked at 7 days after infection. As the
number is so small that the level of serum ALT and the number of total hepatocytes almost stay steady when the infected cells are eliminated  by the CTLs, viral
clearance occur rapidly and efficiently
with little evidence of liver disease. Simulation results
are shown in Figure 1.Acute hepatitis.
(a)(b)(c)(d)(e)(f)When the(k1+k2)(g4/d4)−b1⁢(k3/d3)<0, the cellular immune response is not able to eliminate all the infected hepatocytes, so the virus will be persistent. In this
case, if the immune response is strong and many infected cells are killed, it
will be chronic hepatitis B. If the immune response is very weak, there will be
no symptom. If the virus has a strong
infectious capability and the immune response is vigorous but not enough to resolve the
infection, it will be fulminant hepatitis.


### 3.2. Fulminant Hepatitis

The parameters during the simulation of
fulminant hepatitis are set up as follows:d1=0.002;b1=0.01;k1=100;k2=900;b3=0.1;k3=1000;d3=0.58;g4=1E−10;k4=20;d4=0.2;k5=2000;d5=0.25;X0=1;Y0=0;V0=1E−5;Z0=5E−8;ALT0=16.The mortality of fulminant hepatitis is up to60%∼90%. In this case, the virus rapidly replicates and infects every
hepatocyte in the liver. Most infected hepatocytes are destructed by the CTLs,
resulting in severe liver dysfunction. Simulation results of fulminant
hepatitis are shown in Figure 2.Fulminant hepatitis.
(a)(b)(c)(d)(e)(f)The
initiation value ofV0 is set as 10−5(106/1011), as we assume transfusing 100 mL of blood from an inactive
carrier whose serum HBV DNA
level was about 104 copies/mL would deliver  1×106(100×104) particles of HBV into the body.As shown in Figure2, hepatitis would outbreak
sharply 7 days after the virus entered the body of a host. As the virus has a
strong infectious capability (b1=0.01) and replicates
rapidly (k3=1000),  90% of the hepatocytes in the liver will get infected within two days. The total number of HBV particles peaked at 794×1011 (equals serum HBV DNA level =2.6×1010 copies/mL). As the cellular immune response is rapidly elicited (k4=20), CTLs soon
arrive the maximum
value. The cytopathic effect of CTLs is more powerful than its noncytopathic
effect [(k2=900)>(k1=100)]. Most
infected cells are killed by CTL directly and this would lead to serious liver
necrosis, and the level of ALT starts to rise sharply, it peaks at 1223 at 7 day.


### 3.3. Acute–Turn-Chronic
Hepatitis

The parameters during the simulation of acute–turn-chronic
hepatitis are set up as follows:d1=0.002;b1=0.005;k1=6000;k2=1000;b3=0.1;k3=600;d3=0.58;g4=1E−10;k4=16;d4=0.2;k5=2000;d5=0.25;X0=1;Y0=0;V0=1E−7;Z0=5E−8;ALT0=16.HBV infection can become a chronic infection when the immune system cannot fight off the
virus within six months after infection. It will establish a chronic, lifelong
infection in the liver, and will have an enormously increased risk of
developing liver cancer. It is well known that the T cell response is much less
vigorous in chronically infected patients than it is during acute infection.
Simulation results of acute–turn-chronic
hepatitis are shown in Figure3.Acute–turn-chronic
hepatitis.
(a)(b)(c)(d)(e)(f)The initiation value ofV0 is set as 10−7(104/1011),
as we assume transfusing 1 mL of blood from an inactive carrier whose serum HBV DNA level was about 104 copies/mL would deliver
1×104=104 particles
of HBV into the body.The incubation period of hepatitis B caused by the virus or blood transfusions is about  14 to 180
days. As shown in Figure3, hepatitis would outbreak 16 days after the virus
entered the body of a host. Eighty percent of the hepatocytes in the liver would
get infected within 16 days, total
number of HBV peaked at 371 (equals serum HBV DNA level = 1.2×1010 copies/mL). As the cytopathic effect of CTLs is
so powerful (k2=1000), many infected cells were killed by CTLs, and the level of ALT peaked to 569 at 19 day.


### 3.4. Chronic Hepatitis without Acute Phase

The
parameters during the simulation of chronic hepatitis without acute phase are
set up as follows:d1=0.002;b1=0.2;k1=100;k2=40;b3=0.1;k3=0.8;d3=0.58;g4=1E−10;k4=30;d4=0.2;k5=2000;d5=0.25;g6=1E−10;X0=1;Y0=0;V0=1E−8;Z0=5E−8;ALT0=16.Many chronically infected people show little
or no clinical signs. The HBV-specific immune response is too weak to eliminate
HBV from all infected hepatocytes, but it is strong enough to continuously destroy
HBV-infected hepatocytes, maybe resulting in progressive tissue damage and even
cancer. Simulation results of chronic hepatitis without acute phase are shown
in Figure4.Chronic hepatitis.
(a)(b)(c)(d)(e)(f)The incubation of the Hepatitis
B is about 42 to180 days,
average 180 days. The initiation value ofV0 is set as 10−8(103/1011),
as we assume that a hypodermic needle carrying HBV-contaminated blood which circulates through the body will spread  103 virions. As shown in Figure 4, hepatitis would outbreak 51 days after the virus
entered the body. Maximally 35% of the hepatocytes in the liver would get
infected, total number of HBV peaked
at 0.56 (equals serum HBV DNA
level = 1.8×107 copies/mL). The level of ALT reaches its peak value (104) at 117 day. The system will arrive its steady state at 175 day, 15% of
the hepatocytes in the liver are infected cells, and total number of HBV is 0.24 (equals serum HBV DNA
level = 8×106 copies/mL) and the ALT
level is 48.


### 3.5. Recurring Hepatitis

The parameters during the simulation of recurring hepatitis are set up as follows:d1=0.002;b1=0.0002;k1=1000;k2=2500;b3=0.1;k3=400;d3=0.58;g4=1E−10;k4=2.1;d4=0.2;k5=2000;d5=0.25;X0=1;Y0=0;V0=1E−6;Z0=5E−8;ALT0=16.When the viral concentration is at its lowest,
the patient may be diagnosed as complete recovery; but the virus never completely disappears and an apparent reinfection will soon appear. This
recurrence will last for years. The simulation results of recurring hepatitis
are shown in Figure5.Recurring
hepatitis.
(a)(b)(c)(d)(e)(f)


### 3.6. Asymptomatic Chronic Hepatitis

The parameters during the simulation of asymptomatic
chronic hepatitis are set up as follows:d1=0.002;b1=0.0001;k1=1;k2=4;b3=0.1;k3=100;d3=0.58;g4=1E−10;k4=10;d4=0.2;k5=2000;d5=0.25;X0=1;Y0=0;V0=1E−7;Z0=5E−8;ALT0=16.Vertical
transmission of HBV results in milder hepatitis in patients with no symptoms.
The virus establishes itself in this immunologically immature population and is
tolerated so that there will be no adequate immune response. Neonatal tolerance
is probably responsible for both the lack of an antiviral immune response and the viral persistence
after mother–infant
transmission. This is the most common antecedent of persistent HBV infection
worldwide. The simulation results of asymptomatic chronic hepatitis are shown
in Figure6.Asymptomatic
hepatitis.
(a)(b)(c)(d)(e)(f)


## 3.1. Acute Hepatitis

HBV can cause acute hepatitis, resulting in short-term inflammation of
the liver before the immune system is able to remove the virus from the body. In
acutely infected patients who successfully control the virus, the immune response that the patients produce against the viral proteins is
polyclonal, multispecific; and the
virus is eliminated from the blood and liver entirely. If the maximum damage and the maximum concentration of free virus are low,
the disease may come and go without any symptoms, otherwise severe clinical
symptoms will be observed.The parameters during the simulation of acute hepatitis are set up as
follows:d1=0.002;b1=3E−5;k1=1E6;k2=1E3;b3=0.1;k3=800;d3=0.58;g4=1E−8;k4=16;d4=0.2;k5=2000;d5=0.25;X0=1;Y0=0;V0=10;Z0=5E−8;ALT0=16. As the cellular
immune response is vigorous
and (k1+k2)(g4/d4)=0.0500>b1(k3/d3)=0.0414, so the immune system is strong enough to eliminate all the infected cells and
virus. The number of virus is at its peak at the beginning and then decrease.
HBV can only infect a small
number of cells. The number of
infected cells (4.53×10−4) peaked at 7 days after infection. As the
number is so small that the level of serum ALT and the number of total hepatocytes almost stay steady when the infected cells are eliminated  by the CTLs, viral
clearance occur rapidly and efficiently
with little evidence of liver disease. Simulation results
are shown in Figure 1.Acute hepatitis.
(a)(b)(c)(d)(e)(f)When the(k1+k2)(g4/d4)−b1⁢(k3/d3)<0, the cellular immune response is not able to eliminate all the infected hepatocytes, so the virus will be persistent. In this
case, if the immune response is strong and many infected cells are killed, it
will be chronic hepatitis B. If the immune response is very weak, there will be
no symptom. If the virus has a strong
infectious capability and the immune response is vigorous but not enough to resolve the
infection, it will be fulminant hepatitis.


## 3.2. Fulminant Hepatitis

The parameters during the simulation of
fulminant hepatitis are set up as follows:d1=0.002;b1=0.01;k1=100;k2=900;b3=0.1;k3=1000;d3=0.58;g4=1E−10;k4=20;d4=0.2;k5=2000;d5=0.25;X0=1;Y0=0;V0=1E−5;Z0=5E−8;ALT0=16.The mortality of fulminant hepatitis is up to60%∼90%. In this case, the virus rapidly replicates and infects every
hepatocyte in the liver. Most infected hepatocytes are destructed by the CTLs,
resulting in severe liver dysfunction. Simulation results of fulminant
hepatitis are shown in Figure 2.Fulminant hepatitis.
(a)(b)(c)(d)(e)(f)The
initiation value ofV0 is set as 10−5(106/1011), as we assume transfusing 100 mL of blood from an inactive
carrier whose serum HBV DNA
level was about 104 copies/mL would deliver  1×106(100×104) particles of HBV into the body.As shown in Figure2, hepatitis would outbreak
sharply 7 days after the virus entered the body of a host. As the virus has a
strong infectious capability (b1=0.01) and replicates
rapidly (k3=1000),  90% of the hepatocytes in the liver will get infected within two days. The total number of HBV particles peaked at 794×1011 (equals serum HBV DNA level =2.6×1010 copies/mL). As the cellular immune response is rapidly elicited (k4=20), CTLs soon
arrive the maximum
value. The cytopathic effect of CTLs is more powerful than its noncytopathic
effect [(k2=900)>(k1=100)]. Most
infected cells are killed by CTL directly and this would lead to serious liver
necrosis, and the level of ALT starts to rise sharply, it peaks at 1223 at 7 day.


## 3.3. Acute–Turn-Chronic
Hepatitis

The parameters during the simulation of acute–turn-chronic
hepatitis are set up as follows:d1=0.002;b1=0.005;k1=6000;k2=1000;b3=0.1;k3=600;d3=0.58;g4=1E−10;k4=16;d4=0.2;k5=2000;d5=0.25;X0=1;Y0=0;V0=1E−7;Z0=5E−8;ALT0=16.HBV infection can become a chronic infection when the immune system cannot fight off the
virus within six months after infection. It will establish a chronic, lifelong
infection in the liver, and will have an enormously increased risk of
developing liver cancer. It is well known that the T cell response is much less
vigorous in chronically infected patients than it is during acute infection.
Simulation results of acute–turn-chronic
hepatitis are shown in Figure3.Acute–turn-chronic
hepatitis.
(a)(b)(c)(d)(e)(f)The initiation value ofV0 is set as 10−7(104/1011),
as we assume transfusing 1 mL of blood from an inactive carrier whose serum HBV DNA level was about 104 copies/mL would deliver
1×104=104 particles
of HBV into the body.The incubation period of hepatitis B caused by the virus or blood transfusions is about  14 to 180
days. As shown in Figure3, hepatitis would outbreak 16 days after the virus
entered the body of a host. Eighty percent of the hepatocytes in the liver would
get infected within 16 days, total
number of HBV peaked at 371 (equals serum HBV DNA level = 1.2×1010 copies/mL). As the cytopathic effect of CTLs is
so powerful (k2=1000), many infected cells were killed by CTLs, and the level of ALT peaked to 569 at 19 day.


## 3.4. Chronic Hepatitis without Acute Phase

The
parameters during the simulation of chronic hepatitis without acute phase are
set up as follows:d1=0.002;b1=0.2;k1=100;k2=40;b3=0.1;k3=0.8;d3=0.58;g4=1E−10;k4=30;d4=0.2;k5=2000;d5=0.25;g6=1E−10;X0=1;Y0=0;V0=1E−8;Z0=5E−8;ALT0=16.Many chronically infected people show little
or no clinical signs. The HBV-specific immune response is too weak to eliminate
HBV from all infected hepatocytes, but it is strong enough to continuously destroy
HBV-infected hepatocytes, maybe resulting in progressive tissue damage and even
cancer. Simulation results of chronic hepatitis without acute phase are shown
in Figure4.Chronic hepatitis.
(a)(b)(c)(d)(e)(f)The incubation of the Hepatitis
B is about 42 to180 days,
average 180 days. The initiation value ofV0 is set as 10−8(103/1011),
as we assume that a hypodermic needle carrying HBV-contaminated blood which circulates through the body will spread  103 virions. As shown in Figure 4, hepatitis would outbreak 51 days after the virus
entered the body. Maximally 35% of the hepatocytes in the liver would get
infected, total number of HBV peaked
at 0.56 (equals serum HBV DNA
level = 1.8×107 copies/mL). The level of ALT reaches its peak value (104) at 117 day. The system will arrive its steady state at 175 day, 15% of
the hepatocytes in the liver are infected cells, and total number of HBV is 0.24 (equals serum HBV DNA
level = 8×106 copies/mL) and the ALT
level is 48.


## 3.5. Recurring Hepatitis

The parameters during the simulation of recurring hepatitis are set up as follows:d1=0.002;b1=0.0002;k1=1000;k2=2500;b3=0.1;k3=400;d3=0.58;g4=1E−10;k4=2.1;d4=0.2;k5=2000;d5=0.25;X0=1;Y0=0;V0=1E−6;Z0=5E−8;ALT0=16.When the viral concentration is at its lowest,
the patient may be diagnosed as complete recovery; but the virus never completely disappears and an apparent reinfection will soon appear. This
recurrence will last for years. The simulation results of recurring hepatitis
are shown in Figure5.Recurring
hepatitis.
(a)(b)(c)(d)(e)(f)


## 3.6. Asymptomatic Chronic Hepatitis

The parameters during the simulation of asymptomatic
chronic hepatitis are set up as follows:d1=0.002;b1=0.0001;k1=1;k2=4;b3=0.1;k3=100;d3=0.58;g4=1E−10;k4=10;d4=0.2;k5=2000;d5=0.25;X0=1;Y0=0;V0=1E−7;Z0=5E−8;ALT0=16.Vertical
transmission of HBV results in milder hepatitis in patients with no symptoms.
The virus establishes itself in this immunologically immature population and is
tolerated so that there will be no adequate immune response. Neonatal tolerance
is probably responsible for both the lack of an antiviral immune response and the viral persistence
after mother–infant
transmission. This is the most common antecedent of persistent HBV infection
worldwide. The simulation results of asymptomatic chronic hepatitis are shown
in Figure6.Asymptomatic
hepatitis.
(a)(b)(c)(d)(e)(f)


## 4. Discussion

The
diversity of clinical syndromes and disease manifestations associatedwith
HBV infection strongly suggests that the clinical outcome of this
infection is determined by host-virus interactions, especially the quality and vigor of the antiviral
immune response produced by the infected host. Most perinatal HBV infections become persistent, presumably due to a suboptimalcellular immune response that
destroys some of the infected hepatocytes and does not purge the
virus from the remaining infected hepatocytes. It thereby permits the
persisting virus to trigger a chronic indolent necroinflammatory liver
disease that sets the stage for development of HCC. In contrast, most of the adult
onset HBV infections resolve, presumably due to the polyclonal,
multispecific cellular immune response that the patients produce
against the viral proteins [24].The qualitative analysis and
simulation results suggest the
following pattern.If the cellular immune response is vigorous
and satisfied(k1+k2)g4/d4>b1k3/d3,
the immune system is strong enough to eliminate the infection. Otherwise, chronic
hepatitis appears.If the virus with strong infectious capability
(b1 is large) replicates rapidly
(k3 is large), most hepatocytes in the liver get infected, resulting in massive
liver necrosis due to the strong CTL response. The outcome will be fulminant
hepatitis.If the virus with weak infectious capability replicates slowly, the CTL response to HBV is rapid (k4 is large) and vigorous (k1+k2 is large) enough to eliminate the virus from the blood and liver entirely. The
outcome will be acute hepatitis. If the maximum damage and the maximum
concentration of free virus are low, the disease may come and go unnoticed, otherwise severe clinical symptoms will be observed.If the immune system defends against HBV with a
weak killing ability (k1+k2 is small) and weak CTL level (k4 is small), the infected cells cannot be cleared out entirely. The outcome will
be chronic hepatitis with little or no clinical signs.This model
is able to account for the different outcomes of HBV infection. However, this
model can be further improved to explain why many patients with acute hepatitis,
whose hepatocytes are almost all infected, can recover, and why many patients
with chronic hepatitis B can get rid of HBV when getting older. For the future
studies, the model will be applied to fit clinical data for the evaluation of
immune states and virus characteristic,
thus providing information about the potency of antiviral therapies and
guiding the development of optimal drug dosages and regimens.



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*Source: 743690-2008-02-13.xml*