Hepatitis B is a DNA virus that infects liver cells and can cause both acute and chronic disease. It is believed that both viral and host factors are responsible for determining whether the infection is cleared or becomes chronic. Here we investigate the mechanism of protection by developing a mathematical model of the antibody response following hepatitis B virus (HBV) infection. We fitted the model to data from seven infected adults identified during acute infection and determined the ability of the virus to escape neutralization through overproduction of non-infectious subviral particles, which have HBs proteins on their surface, but do not contain nucleocapsid protein and viral nucleic acids. We showed that viral clearance can be achieved for high anti-HBV antibody levels, as in vaccinated individuals, when: (1) the rate of synthesis of hepatitis B subviral particles is slow; (2) the rate of synthesis of hepatitis B subviral particles is high but either anti-HBV antibody production is fast, the antibody affinity is high, or the levels of pre-existent HBV-specific antibody at the time of infection are high, as could be attained by vaccination. We further showed that viral clearance can be achieved for low equilibrium anti-HBV antibody levels, as in unvaccinated individuals, when a strong cellular immune response controls early infection.
Infection with hepatitis B virus (HBV) results in acute hepatitis followed by recovery in 85%–95% of human adults [1]. Recovery occurs when the organism mounts adequate immune responses against the virus. Such responses include production of protective, neutralizing antibodies against HBV surface antigen (HBsAg) [2], [3], activation of strong and diversified CD4 and CD8 T-cells [4], [2], expression of antiviral cytokines in the liver, such as gamma interferon and tumor necrosis factor alpha [5], [6], [7], [8], and generation of cells that are protected from reinfection [9], [10]. In contrast, progression to chronic HBV infection occurs predominantly in immuno-compromised adults and in unvaccinated infants [11]. Such individuals exhibit weak and inefficient humoral and cellular immune responses, resulting in continual virus replication and HBV surface antigenemia [12], [4]. Little is known about the relative contributions of different arms of the immune system, especially the roles of neutralizing antibodies in the onset and outcome of infection. The antibody response to HBV infection is difficult to study experimentally. Free antibody to surface antigen is not detected until after the resolution of HBV infection [13]. However, circulating immune complexes containing antibody and HBsAg are found in both acute and chronic HBV infections, suggesting that antibodies are produced much sooner than detected, and that they might play a role in the pathology of the disease [14], [15], [16], [17]. HBsAg-specific antibodies have neutralizing properties and mediate protective immunity [16]. Infection with hepatitis B virus results in the synthesis of a large number, probably of at least 1,000-fold, of “subviral particles” (SVPs) in relation to HBV particles [1], [18]. SVPs, which are produced by HBV infected cells, are particles that have HBV proteins on their surface, but do not contain nucleocapsid protein and viral nucleic acids and hence are non-infectious [19]. They exist in two main forms: spheres 25 nm in diameter and filaments 22 nm in diameter with variable lengths [20], [21], [22], [23]. The reasons for their overproduction and their contribution to HBV pathogenesis is still under investigation [24]. SVPs may influence the way the host reacts to HBV infection. They may induce tolerance during perinatal infection, thus delaying the rise of neutralizing antibodies. Additionally, the excess of subviral particles can serve as a decoy by adsorbing neutralizing antibodies and therefore delay the clearance of infection. In this paper, we aim to determine quantitative features of the antibody responses to virus and subviral particles following HBV infection. We build on basic chronic virus infection models [25], [26], [27], [28], [29], [30], [31] and determine the antibody characteristics that explain both the high peak and eventual viral clearance observed during acute hepatitis B infections [8]. We show that antibody responses can lead to viral clearance when the anti-HBV levels are high, as in vaccinated patients, and: (1) the rate of synthesis of hepatitis B subviral particles is slow; (2) the rate of synthesis of hepatitis B subviral particles is high but either anti-HBV antibody production is fast, the antibodies have high affinity, or the levels of pre-existent HBV-specific antibody at the time of infection are high. For lower anti-HBV antibody levels, as in unvaccinated patients, both cellular and humoral responses are needed in concert to clear acute HBV infection with the CD8 T cells controlling the initial burst of replication and the antibodies preventing virus rebound. The paper is structured as follows. In section Methods we develop the general model of antibody responses to viral and the subviral particles. In section Analytical results we analyze it analytically using asymptotic analysis techniques and in section Numerical results we present numerical results and compare these to data of primary HBV infection in humans, and present alternative models. We conclude with a discussion.
Standard mathematical models of viral infections consider dynamics and relations among uninfected target cells (), infected cells (), and virus () [25], [26], [27], [28], [29]. Briefly, these models assume that target cells become infected at a rate proportional to both the target cell concentration and the virus concentration, i. e. , at rate. Infected cells are thus produced at rate and die at rate (that includes immune system mediated killing). Virus is produced by infected cells at rate and is cleared at rate. These models have been adapted to describe hepatitis B virus infection where the target population is liver cells (hepatocytes) [30], [25], [31]. The model accounts for the liver' s ability to regenerate following injury [32], [33]. This regeneration, accomplished by several cycles of hepatocyte mitosis, is described by a logistic term with carrying capacity and maximal proliferation rate [9], [34]. Moreover, infected hepatocytes can be cured [10] and move back into the target population at rate [30], [31]. The dynamics of the system are governed by the following equations (1) where for acute infection, and. An analysis of this system predicts two outcomes [35], [36], [37]. The infection dies out whenand the infection takes off and leads to chronic hepatitis when This simple system, which lacks any explicit immune response, does not explain transient infections, where the liver gets infected, i. e. , , but the infection is eventually cleared, presumably by the immune system. Clearance of HBV infection occurs in 90% of adults infected with HBV [1]. While the role of the cellular immune responses has been studied both theoretically and experimentally [9], [34], [38], [39], less is known about the dynamics of the humoral immune response to HBV [40]. In the following section we investigate antibody responses by modifying system (1) to account for humoral immunity following HBV infection. To include the antibody response, we generalize the model given by Eq. (1) by considering seven populations, corresponding to target cells (), which are mostly or exclusively uninfected hepatocytes, productively infected cells (), free virus (), free subviral particles (), free antibody (), virus-antibody complexes (), and subviral particle-antibody complexes (). Since hepatocytes are in contact with the blood we assume, as above, that their infection can be described by a well-mixed system. Further investigation is needed to know whether spatial effects are important in HBV infection. For hepatitis C virus (HCV) infection, in which a much smaller fraction of cells become infected, spatial clustering of infected cells has recently been observed [41]. As in Eq. (1), we assumed that target cells are maintained through homeostasis described by the logistic term with carrying capacity and maximum proliferation rate, and become infected at a rate proportional to both the target cell concentration and the virus concentration, i. e. , at rate. Infected cells are thus produced at rate and die at rate (that includes immune system mediated killing). Upon infection, virus and subviral particles are produced at rates and, and cleared at rates and, respectively. We neglect the curing of infected cells by setting. With this simplification the basic reproductive number becomes. Previous papers [42], [43] have presented detailed models of B lymphocyte proliferation and differentiation into plasmablast, antibody producing plasma cells and memory cells after they encounter antigen. For simplicity, we ignore the details of B-lymphocyte dynamics and differentiation into antibody producing cells and assume that free antibody (), is produced at rate proportional to the antigen load, i. e, the viral and subviral concentrations, and is degraded at rate. Antibody is maintained after virus is cleared through antigen-independent homeostatic proliferation of memory B cells and long-lived plasma cells. In order to model this in a simple way, we add a logistic term to the antibody equation with maximum proliferation rate and carrying capacity. We show (in section 4. 3) that a model that explicitly includes B cell dynamics has behavior similar to this simpler model with antibody alone. Antigen elimination is facilitated by the formation of antigen-antibody complexes. We consider the reversible binding of free anti-HBsAg antibody (), to both free virus (), and subviral particles (), described by the reaction scheme (2) where and are the complexes formed between antibody and the viral and subviral particles, respectively. , are binding rate constants, and, are disassociation rate constants for antibody reacting to viral and subviral particles, respectively. We consider that complexes and are degraded at a constant rates and. Antibodies can also bind infected cells budding virus. In this model we consider this to occur at a small rate and we neglect it. Based on the scheme (2) and the assumptions above we construct the following equations of virus-host interaction (3) where, , , , , and. The total concentration of viral DNA is described by (4) and the total concentration of anti-HBsAg antibody is given by (5) We developed a set of mathematical models of the antibody response to hepatitis B viral infection and tested whether viral clearance is possible in the presence of an excess number of subviral particles. Subviral particles are non-infectious but have HBsAg on their surface and thus bind anti-HBV antibody. If they bind enough anti-HBV antibody they have the potential to counter the antibody response. We used the models and data from seven acutely infected patients to determine important parameters that describe virus and antibody dynamics. Models of HIV have considered the antibody' s effect on virus indirectly by modeling opsonization through enhanced viral clearance and neutralization through decreased virus infectivity [64], [65]. Others considered in detail the interaction between virus and antibody that leads to complex formation [64]. Here, for the case of HBV, we modeled complex formation between antibody and both viral and subviral particles. This antibody model suggests that viral clearance is highly dependent on the characteristics of the antibody response: equilibrium antibody level, affinity, antigen-dependent and -independent antibody growth rates. The antibody model (7) predicts that, for the same antibody dynamics, virus clearance can occur for low SVP∶virus ratios but not for high SVP∶virus ratios. However, viral clearance can be achieved regardless of the SVP∶virus ratio if enough HBsAg-specific antibody is present at the time of infection or the antigen-dependent/-independent antibody growth rate is high enough. If we consider, as in clinical observations, that a healthy individual who produces an excess of -particles very likely clears the infection, than virus clearance occurs in the absence of pre-existent antibody when the antibody population' s doubling time is faster than days (Fig. 9 left panel). Moreover, the virus can still be cleared for slow antibody expansion if vaccine induced antibodies are present at the time of infection [66]. It is thought that following hepatitis B vaccination and boosting, patients with anti-HBsAg levels of mIU/ml (mg/ml) or higher are protected from infection [67], [68], [69]. In our study, if we assume an antibody population' s doubling time of 16. 6 days, the presence of HBsAg-specific antibody levels higher than mg/ml leads to virus clearance (Fig. 9 right panel). We assumed that the antibody' s carrying capacity is fixed at the maximum antibody levels observed after vaccination and boosting. Moreover, we assumed that the initial inoculum is low and fixed among all patients. Under these conditions we obtained virus clearance in the absence of pre-existent antibody in six out of seven patients. The predicted antibody levels needed for clearance, however, are higher than observed clinically in unvaccinated individuals, where free antibodies are usually only detected after HBV is nearly cleared. If the predicted free antibody levels are decreased substantially (through decrease of activation parameters and or through decrease of the carrying capacity) then pre-existent antibodies are required for protection (Figs. S1 and S2 in Text S1). The antibody model (7) predicts that more than of hepatocytes are infected at the peak of acute infection, lower than previously estimated [9]. These cells are replaced through homeostasis and the maximum liver loss at any one time ranges between. The length of time the continuous liver death and replacement occurs is dependent on the antibody dynamics, such as the antigen-dependent and -independent growth rates and, and the initial values, . Rapid liver cell turnover can lead to accumulation of mutations in the host genome that could result in genetic alterations, chromosomal rearrangements, activation of oncogenes, inactivation of tumor suppressor genes, and ultimately to hepatocellular carcinoma as seen in many patients with chronic hepatitis [70]. Model (7) assumed that one antibody is sufficient to neutralize a virion. We relaxed this assumption by developing a multivalent model (27) that accounted for multiple binding events. We tested whether the observed dynamics change when multiple antibodies bind a virion and consequently lower virus infectivity and/or enhance virus clearance. We found that such antiviral activity has an effect on the size (but not the shape) of free and bound virus during the second phase decay but not on viral peak (Fig. 6). Another antiviral response that we tested was the effect of increasing the ratio between the clearance of immune complexes and that of free virus. Originally we assumed that the immune complexes are cleared four times faster than the free virus, as in HIV [55]. In the virus clearance region, an increase in this ratio led to the decrease of free virus during the second phase decay (Fig. S4 in Text S1). An unresolved issue is whether the assay used to measure patients HBV levels (the Amplicor HBV Monitor Test [57]) measures only free virus, as we assumed, or both HBV DNA in free virus and immune complexes. In this regard, the patient HBV DNA was recovered from serum samples by a chemical denaturant method [56], which has lower yield than techniques that use both chemical methods and proteolytic enzymes [71]. The latter should be more efficient at breaking apart virus-antibody complexes. We compared the estimates for the other parameters with those from our previous hepatitis B study that used the same patient data [9]. We found that the median infectivity rate is higher, and the viral production rate is smaller in the antibody model (7). Our best estimates predict viral clearance in the first six patients and viral persistence in patient 7. This is consistent with clinical results which report that patient 7 was immunosuppressed and developed chronic disease. Data fitting shows that he has the lowest infected cell clearance rates. The antibody model (7) shows that in order for the virus to be cleared this patient would need to have one or a combination of the following: (1) low subviral particles production together with a high pre-existing antibody level; (2) antibody of high affinity; (3) high antigen-dependent and/or -independent antibody growth; (4) increased clearance of virus-antibody complexes; (5) increased loss of infected cells. One important finding was that to obtain clearance of HBV, as observed in patients 1 to 6 and a majority of acutely infected adults, one needs high levels of antibody, higher than usually found in clinical observation. An alternative hypothesis for the effect of antibody is that they help clear the infection, once cellular immunity controls the initial burst of replication. The CD8-antibody model shows that cytotoxic effects lead to the initial viral control, and antibodies prevent re-infection. Previously, we had shown that a refractory/immune state of target cells could preserve the integrity of the liver while preventing re-infection [9]. Here we argue that antibodies could have a similar effect, with dynamics compatible with the observed levels and timing of antibodies in patients. Which of these effects is dominant, or if potentially both contribute to control of infection, needs further experimental studies. One limitation of model (30) is that, for patients 2 to 7, killing of infected hepatocytes by CD8 T cells leads to more than liver loss. Therefore non-cytotoxic CD8 T cells effects leading to infected cells being cured and refractory to reinfection are needed to preserve liver integrity. In summary, we have developed a set of models of hepatitis B infection that give insight into the opposing roles of antibody and subviral particles in the resolution of acute HBV infection. In particular, we showed that even when the virus produces a large number of subviral particles, as a decoy against antibody protection, viral clearance can still be achieved when pre-existing immunity induced through vaccination or cross immunity leads to the presence of high antibody levels early in infection. However, in individuals with low initial antibody levels, as in most unvaccinated individuals and in individuals without prior exposure to HBV, antibodies could have more of a mop up function, clearing the infection and preventing viral resurgence after a cellular immune response controls the initial infection.
Hepatitis B vaccine induces life-long protection in vaccinated individuals. In the absence of vaccination, however, hepatitis B virus can cause both self-limiting and chronic disease. We investigate whether antibodies against hepatitis B play a role in virus clearance. We developed a mathematical model that describes the production of antibodies to both infectious virus and non-infectious subviral particles (with hepatitis B surface proteins, but no nucleic acids) and compared the model to patient data. We predict that high levels of antibodies, either pre-existing, as in vaccinated individuals, or through fast expansion, can control the infection and lead to viral clearance. However, when the antibody levels are more similar to those observed in a clinical context, cellular immune responses are needed to control the virus and antibodies act only in late stages to aid in viral clearance.
infectious diseases
medicine and health sciences
mathematics
applied mathematics
biology and life sciences
immunology
physical sciences
immune response
Antibody Responses during Hepatitis B Viral Infection