Chronic hepatitis B virus and liver fibrosis: A mathematical model

Hepatitis B virus (HBV) infection is a liver disorder that can result in cirrhosis, liver failure and hepatocellular carcinoma. HBV infection remains a major global health problem, as it affects more 350 million people chronically and kills roughly 600,000 people annually. Drugs currently used against HBV include IFN-α that decreases viremia, inflammation and the growth of liver fibrosis, and adefovir that decreases the viral load. Each of these drugs can have severe side-effects. In the present paper, we consider the treatment of chronic HBV by a combination of IFN-α and adefovir, and raise the following question: What should be the optimal ratio between IFN-α and adefovir in order to achieve the best ‘efficacy’ under constraints on the total amount of the drugs; here the efficacy is measured by the reduction of the levels of inflammation and of fibrosis? We develop a mathematical model of HBV pathogenesis by a system of partial differential equations (PDEs) and use the model to simulate a ‘synergy map’ which addresses the above question.


Introduction
Hepatitis B virus (HBV) infection is a liver disorder that can result in cirrhosis, liver failure, and hepatocellular carcinoma; it is one of the most prevalent infectious diseases associated with human liver diseases, including acute, fulminant and chronic hepatitis [1]. Despite the availability of HBV vaccine and the development of antiviral therapies, HBV infection remains a major global health problem. Chronic HBV affects more than 350 million people of whom roughly 600,000 die annually from HBV-related liver diseases [1][2][3].
To understand the virus biology and pathogenesis in HBV-infected patients, several animal models have been developed to mimic hepatic HBV infection and the immune response against HBV [1,4,5]. However, as observed in [1], the narrow host range of HBV infection and lack of a full immune response spectrum in animal models remain significant limitations. For this reason, it may be useful to develop a mathematical model of the progression of HBV, which includes the host response to treatment. In the present paper we introduce such a model and use it to study the effect of a combination therapy with two currently used drugs, interferon alpha (IFN-α) [5][6][7] and adefovir [8][9][10][11]. The model shows that it may be possible a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 to reduce HBV infection and viral load while, at the same time, decrease inflammation and control liver fibrosis.
Monotherapy of HBV infection with IFN-α decreases viremia, inflammation and the progression of liver fibrosis. However, only a small fraction of patients respond positively to treatment [5,6]. Monotherapy with adefovir decreases the viral load by blocking reverse transcriptase, an enzyme that is crucial for the reproduction of HBV in the body [8,12,13].
Both IFN-α and adefovir can have severe side-effects when administered alone, including brain disorder that may lead to a stroke, a heart attack or lung fibrosis for IFN-α [14], and kidney damage, a low amount of phosphate in the blood, a stroke or pancreatitis for adefovir [15]. A combination therapy with smaller dose may be as effective, or even more effective, than monotherapy with a larger dose if there is a 'positive synergy' between the two drugs. We shall use our mathematical model to determine the synergy between IFN-α and adefovir.
In their study of HBV infection in a humanized mouse model where the mice are inoculated with clinical isolates HBV in vivo, the authors in [16] showed that a persistence of liver disease due to HBV infection is associated with infiltration of M2 macrophages. They found that chronic HBV infection in the liver of humanized mice is associated with human hepatic stellate cells (HSCs) activation/infection and human liver fibrosis. It was also observed in [16] that infection of M1 and M2 macrophages with HBV results in increasing level of IL-10 and decreasing level of IL-12. Although the authors in [16] do not demonstrate that HBV infects macrophages, they show that inoculation of macrophage cultures with HBV results in macrophage activation, however the mechanism of activation is unknown. Since the possibility that HBV could infect macrophages remains to be conclusively proven, we do not assume that macrophage infection is a major source of virus as compared to HSCs.
A schematic diagram of our model is shown in Fig 1, where the nodes are defined in Table 1. We represent the cells by square nodes, but the fibrosis formation pathway by diamond nodes, and the virus and cytokines by circular nodes. The nodes with reddish colors (red and orange) indicate the variables that include and/or induce infection or fibrosis formation. the nodes with greenish colors (blue, green and silver) indicate the variables that counteract the infection. We shall use our model to compare the efficacy of treatment by a single agent (IFN-α or adefovir) with a combination, whereby the amount of each agent is less than in the monotherapy. We shall accordingly develop a 'synergy map' which addresses the question of achieving a desired target of efficacy in a combination therapy while using the smallest dose of these two drugs.
Mathematical models for population dynamics of HBV transmission have been developed [56][57][58][59][60][61][62][63]. In [62], the authors used a SEIR-type mathematical model by a system of ordinary differential equations (ODEs) to study the dynamics of HBV infection under administration of vaccination and treatment. A compartmental mathematical model by a system of partial differential equations (PDEs) to predict the dynamics of HBV transmission and evaluate the long-term effectiveness of the vaccination program was developed in [63]. Another class of models that consider uninfected hepatocytes, infected hepatocytes and free virus have been developed to analyze changes in HBV levels during drug therapy [56][57][58][59][60][61]. Techniques in [58,59] where used in [56,60,61] to develop mathematical models by a system of ordinary differential equations to understand the factors that govern HBV infection dynamics, and to analyze the immune mechanisms responsible for viral clearance in the case of acute HBV infection. They found that the viral load initially increases exponentially, then decreases and seems to approach a plateau after some time. Interestingly, they obtained that the number of infected cells, the number of uninfected cells and the number of CD8 + T cells increase in acute HBV infection, leading to clearance of the viral load. A more recent model by Friedman and Hao [64] uses reaction-diffusion equations to study the evolution of fibrosis in the liver, without specifically targeting the cause of the disease.

Mathematical model
In this section, we develop a mathematical model of liver fibrosis due to infection by HBV. In our model that builds on the model in [64], infection takes place in some region O of the liver. The variables used in the model are given in Table 1. These variables satisfy a system of PDEs in O.
Equation for macrophage density. The equation for the density of healthy M1 macrophages, not yet including a source, is given by where 10 ; The term r Á (M 1 χ C 3 rC 3 ) is the chemotactic effect of C 3 on M1 macrophages; χ C 3 is the chemotactic coefficient. M2 macrophages can become M1 macrophages under the influence of IFN-γ and TNF-α, a process resisted by IL-10 [65][66][67], and M1 macrophages can become M 2 macrophages under the influence of IL-4 and IL-13 [29,68]. The fourth term on the righthand side is the infection of M1 macrophages by external virus, V e . Infected cells produced chemokine CCL3 [45,52,53] in order to attract immune response. Monocytes circulate in the blood. They are attracted to the liver tissue by CCL3 [31,54], and then they differentiate into either M1 or M2 macrophages [69][70][71]. We denote M 10 and M 20 the densities of the monocytes from the vascular system which differentiate into M1 and M2 macrophages, respectively. Accordingly, on the boundary of each blood capillary, there is an influx of M1 macrophages into the tissue, wherebðC 3 Þ is an increasing function of C 3 . Using a homogenization method as in [52], we can replace the boundary conditions on the blood capillaries by a source term in the tissue, which we take to be bðC Hence the final equation for the density of M1 macrophages is the following: The density M 1i of infected M1 macrophage satisfies the following equation: where we assume that the death rate of M 1i is increased due to the viral load represented by The density of healthy M2 macrophages satisfies an equation similar to Eq (1): The third, fourth and fifth terms in the right-hand side of Eq (3) are complementary to the corresponding terms in Eq (1). The density M 2i of infected M2 macrophages satisfies an equation similar to Eq (2): where we assume that the death rate of M 2i is increased due to the viral load μ V i2 V i2 . Equations for hepatic stellate cells (HSCs) density. In homeostasis, HSCs have a source A H and death rate μ H . In HBV, the proliferation of the healthy population of HSCs is enhanced by PDGF and TGF-β [45,53,72] and is reduced when they become infected by the extracellular virus. Hence the equation for H is the following: The infected HSCs satisfy the equation where the death rate of infected HSCs is increased due to the viral load μ V iH V iH . Equations for T cells density. Th1 cells are activated from naive T cells (T 0 ) by contact with M1 macrophages, under IL-12 environment [33]; this process is resisted by IL-10 and IL-13 [17,73]. IL-2 enhances the proliferation of Th1 cells [17,65]. Hence the equations for the densities of Th1 and Th2 cells are given as follows: T 0 cells in the liver may also be activated into Th2 cells (T 2 ) by contact with M 2 in an I 4 environment [36]. The activation of T 2 is antagonized by T 1 [36,40,41].
Equations for fibroblast (f) and myofibroblast (m) densities. Hyaluronic acid (H A ) enhances the proliferation of fibroblasts [44]. Fibroblasts are transformed into myofibroblasts by G and T β [30,46,47,74,75]. Hence equations of f and m are: Equations for ECM (ρ) and scar (S) densities. The ECM consists primarily of fibrillar collagens and elastins, but it includes also fibronectins, lamina and nitrogen that support the matrix network by connecting or linking collagens [49]. For simplicity we represent the ECM by the density of collagens. ECM is produced by f, m and by HSCs [47,48]. The production of ECM by m is enhanced by T β [47,50]. We assume that MMP degrades the ECM at a rate proportional to M P ρ. Hence the equation of ρ is as follows: where 1 À r Fibrotic diseases are characterized by excessive scarring due to excessive production and deposition of ECM and disruption of normal healthy protein cross-linking. MMP disrupt collagen cross-linking and increases scarring in cases of excessive collagen concentrations [51]. Accordingly we model the growth of a scar as, in [51], by the formula where ρ Ã is the concentration of collagen in normal healthy tissue. Equations for cytokines and other proteins. PDGF (G) is produced by M 2 [31,32]. Hence, Hyaluronic acid (H A ) is produced by HSCs and is degraded by sinusoidal epithelial cells [42,43]. Hence H A satisfies the following equation where, for simplicity, the degradation of H A is taken at a constant rate.
IL-1β is secreted by healthy and infected M 2 , and this process is inhibited by I α and I 10 [17,18,20,24]. Hence, IL-2 is secreted by T 1 [17]; hence, IL-4 is secreted by T 1 , M 2 and M 2i [25,26], so that the equation of I 4 is given by IL-6 is produced by healthy and M 1 macrophages, a process enhanced by TNF-α [17][18][19]. Hence, IL-10 is produced by M 2 and M 2i [17][18][19][20], and by T 2 [27,28]; so that @I 10 @t À D I 10 DI 10 ¼ λ I 10 M 2 M 2 þ λ I 10 M 2i M 2i þ λ I 10 T 2 T 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} production À m I 10 I 10 |ffl ffl{zffl ffl} degradation ð19Þ IL-12 is produced by M 1 and M 1i , a process inhibited by IL-10 and IL-13 [17,20,21]. Hence IL-13 is produced by M 2 and M 2i [17][18][19][20], and by T 2 [27][28][29]. Hence the equation of I 13 is given as follows: IFN-α is produced by T 1 and T 2 , a process resisted by IL-1β [34], so that IFN-γ is secreted by T 1 [17,20,35], a process resisted by IFN-α [23]. Hence IFN-γ satisfies the equation TNF-α is secreted by M 1 , M 1i and M 2i , a process resisted by IL-10 [22] and IL-13 [23]. Hence the equation of T α is given by TGF-β is produced by M 2 and M 2i , a process enhanced by IL-13 [30,75]. Hence, MMP and TIMP are produced by M 2 and M 2i macrophages [31,32], and depleted by binding to each other [73]. Hence the equations for M P and T P are given, respectively, as follows: @T P @t CCL3 is produced by infected M 1 and M 2 macrophages, as well as infected HSCs [45,52,53]. CCL3 is degraded naturally, but is also lost when internalized by M 1 [64]. Hence C 3 satisfies the equation: Equations for intracellular (V i1 , V i2 , V iH ) and extracellular (V e ) viruses. In order to explain the dynamics of the viral loads, we introduce the following notations: C will denote any of the cells M 1 , M 2 or H, and C i will denote their infected states. V i will denote any of the intracellular viral loads V i1 , V i2 or V iH . We assume that the growth rate of V i is proportional to C i , and is resisted by I α , T α , T β and I γ , with the effect of I γ being modulated by I 6 [20,35,55], thus reducing the growth rate coefficient λ by a factor ð1 þ I a =K a Þð1 þ T a =K T a Þð1 þ T b =K T b Þð1 þ T g =ðK g þ I 6 ÞÞ. There is also an increase of V i at a rate proportional to CV e when external viruses are internalized by C cells [20]. We assume that the rate of this increase is λ CC i N V CV e , where N V is a dimensionality coefficient which is of order of magnitude of (mass of 1 virus)/(mass of 1 cell). We assume that when one infected cell, C i , dies at rate μ C i = μ C (1 + μ V i V i ), N viruses are released. We conclude that the density of intracellular viruses in C i satisfies the equation Hence, we have the following equations for V i1 , V i2 , V iH and V e : Note that the diffusion coefficients of intracellular viruses were taken the same as the diffusion coefficients of their hosts. Note also that we have included the factor 1 1þA=K A in the first terms on the right-hand sides of Eqs (29)- (31). This factor accounts for the control of the viral load within infected cells by a drug, such as adefovir, that inhibits the growth of the virus. If the drug is not given, then A = 0 and the factor 1 1þA=K A may be dropped. V e is increased when infected cells die and release the virus that they contained. There is a reduction in V e when cells internalize extracellular viruses and become infected. Hence, However, we note that the simulation results with different initial conditions do not appreciably change after a few days.

Results
All the computations are done using Python 2.7.6. The parameter values of the model equations are estimated in Section A in S1 File and are listed in Tables C-F in S1 File. The techniques used for the simulations are also described in Section B in S1 File. The increase in IL-10 and decrease in IL-12 suggest that the immune system is not strong enough to confront the infection. Cytokines produced by Th1 cells tend to reduce the load of intracellular viruses, while cytokines produced by Th2 cells work in the opposite direction. It is therefore interesting to note that, since T 2 dominates T 1 , the viral loads (mostly represented by the intracellular HBV) are continuously increasing. Our model also shows that the density of fibroblasts decreases monotonically while the density of myofibroblasts increases, which enhances the production of ECM; the transition from fibroblast to myofibroblast is induced by PDGF and TGF-β, and it increases in time. The concentrations of MMP and TIMP also increase in time. The scar continues to grow, as a result of the increase in MMP and ECM.
We can also simulate the spatial variations of the variables, however we are interested, in this paper, only in the average profiles of the variables. One is tempted to use the simpler ODE model to compute averages. We note however that the diffusion coefficients of cytokines and extracellular viruses are several orders of magnitude larger than the diffusion coefficients of cells. For this reason the ODE system cannot adequately represent the spread of the infection, and hence, the averages of the variables. Indeed, in the simulation of the ODE system (not shown here) the infected macrophages decrease, rather than increase; healthy HSCs are increasing, rather than decreasing, and the intracellular and extracellular viruses decrease and stabilize, rather than monotonically increase, as in Fig 2.

Treatment
In this section, we consider two drugs that are commonly used for the treatment of HBV, namely IFN-α and adefovir [5,6,12,13]. Treatment with IFN-α in chronic HBV adult patient ranges between 16-52 weeks, while treatment with adefovir is continuous for 2-3 years [39,76]. In oncology, it is found that in case of intermittent therapies in which the drug amount administered per unit time is constant, the response is significantly influenced by the amount of drug given and the period of time within which the drug is given [77][78][79]. Moreover, if negative side effects are ignored, then by doubling the amount of drug and correspondingly shortening the period of treatment (keeping the total amount of drug fixed fixed) we may get better results in reducing the infection [65]. Assuming that the total amount of each drug is fixed, we use the mathematical model to evaluate the efficacy of combination therapy with IFN-α and adefovir. We start the treatment after 90 days of control and administer the drugs for 90 days.
Treatment with IFN-α. In our model, injection of IFN-α is represented as a source term, c α (t), in Eq (22), which then takes the form Treatment with adefovir (A). We denote the source of the drug by c A (t). The drug is diminished by absorption by infected cells, and by natural degradation. Hence, the equation of adefovir, A, is given by Combination therapy with IFN-α and adefovir. Combination therapy with IFN-α and adefovir is modeled by using Eqs (33) and (34) together with System (1)- (32). Adefovir is given daily in one or two tablets [80], and its effective time is about 10 hours [81]. For simplicity, we assume a daily average constant level c A taking IFN-α is given in shots three times a week, or in slow injection, as pegylated IFN-α, once a week [82]. We assume that it is given once a week at level c α , so that after t days its level reduces to c a e À st ; for some s > 0; and at the end of the week (t = 7 days) its level is negligible, taking so that σ = 1. The average level during the week is We wish to simplify the simulations by replacing the intermittent treatment of injections at level 7c α by a constant level of the drug at the level c α , that is, by In order to justify this simplification, we simulated in Figs 3 and 4 the average densities of all the variables with treatment (IFN-α-orange, adefovir-green and combination-red) and without treatment when the drugs are given intermittently (Fig 3) and continuously (Fig 4). We see that the profiles of the virus and scar are almost the same under Figs 3 and 4. Hence, for simplicity, we shall use the scheme for Fig 4, that is, Eqs (35) and (36), to compute the efficacy and synergy maps.
Therapy with continuous drug delivery. From Fig 4, we see that IFN-α reduces the viral load by 30.36%, which agrees with reported clinical results by Paul and Han [83]; adefovir reduces by 42.28% and the combination reduces by 57.66%. The reduction in the scar is 20.41% by IFN-α, 28.24% by adefovir, and 37.8% by the combination, although the scar still keeps increasing. We also see that the drugs (in monotherapy and in combination therapy) effectively reduce the inflammation by considerably reducing the densities of infected phagocytic cells (M 1i , M 2i and H i ), and the pro-inflammatory cytokine TNF-α. The pro-inflammatory cytokines IL-6 and IL-12 already have decreasing profiles in the control case, making the effect of drugs on them less significant. We observe in Fig 4 that the drugs have little (or no) effect on the profiles of Th1 and Th2 cells, and hence also on IFN-γ and IL-2 which are secreted by T cells.
We proceed to consider the effect of therapy for a range of doses of adefovir and IFN-α. We denote by S X (180) and V i,X (180) the densities of S and V i , respectively, at day 180 under administration of drug X, where X is either adefovir alone, or IFN-α alone, or their combination. We denote by S(180) and V i (180) the densities of S and V i , respectively, at day 180 in the control case (no drugs), and define two efficacy functions: Both IFN-α and adefovir can have severe side-effects when administered alone. These sideeffects could be aggravated if both drugs are given simultaneously with the same doses as in monotherapy. Thus, there is the need to find a strategy for combination therapy that maximizes the efficacy while reducing the sides effects. The dose-response relationship of the amount of drugs to the severity of negative side-effects has been extensively studied [84][85][86]: at higher doses, undesired side-effects appear and grow stronger with increases in the dose.
We compare treatment of combination therapy (c α , c A ) with monotherapy I α and monotherapy A. For monotherapy I α , we take (1 + θ I α )c α , and for monotherapy A we take (1 + θ A )c A , with θ I α > 0, θ A > 0, to reflect the highest toxicity expected when combining the two drugs. If E S (c α , c A ) is larger than both E S ((1 + θ I α )c α , 0) and E S (0, (1 + θ A )c A ), then we say that the 'synergy' in scar density reduction for the combination (c α , c A ) is positive, and otherwise, we say that the synergy is negative [87]. Similarly, if E V i (c α , c A ) is larger than both E V i ((1 + θ I α )c α , 0) and E V i (0, (1 + θ A )c A ), then we say that the synergy in viral load reduction for the combination (c α , c A ) is positive, and otherwise, we say that the synergy is negative [87]. More generally, we define the synergy in scar density reduction σ S = σ S (c α , c A ) by the formula (as in [87]): Then σ S (c α , c A )>0 (positive synergy) if the combination (c α , c A ) reduces the level of scar density more than either one of the single agents (1 + θ I α )c α or (1 + θ A )c A . Negative synergy occurs in the reverse case where instead of a combination therapy with (c α , c A ) we achieve better reduction of the level of scar density if we apply only one drug, (1 + θ I α )c α or (1 + θ A )c A . Similarly we define the synergy in reducing the viral load by the formula: There are negative side-effects such as lactic acidosis in the case of treatment with adefovir, and serious depression in the case of IFN-α [88]. Since the mechanisms of these side-effects are not well understood, and since it is difficult to evaluate and compare the severity of these side-effects, we take for definiteness θ I α = θ A = 2; this choice is arbitrary and could be made more precise as more clinical data become available that will inform on the side-effects associated with IFN-α and adefovir, separately or in combination. Fig 6 shows the 'synergy map' associated with Model (1)-(32) where we compute the values for the synergy as in Eq (38) related with the drugs IFN-α (I α ) and adefovir (A). The color columns assign a "synergy number" (SN) to any level of adefovir and IFN-α; SN varies from −0.16 to 0.64 for the level of scar density, and from −0.15 to 0.69 for the viral load. Given any level of c α , as c A initially increases so does the SN, until c A reaches a level c Ã A which depends on c α ; i.e., c Ã A ¼ c A ðc a Þ. Thereafter, the SN begins to decrease as c A increases. The biological explanation of the color patterns in the synergy maps can be traced to the fact that the virus proliferation is inhibited by the product ð1 þ I a =K a Þð1 þ A=K A Þ, as seen in Eqs (29)- (31). This product is highly correlated, after scaling, to the product (1 + c α )(1 + c A ). Viewing this product as representing the efficacy of the combined therapy, the resulting expression for the synergy is then a function of the following type: We now observe that f(c α , c A ) is an increasing function in c A if c A < c α and a decreasing function in c A if c A > c α , and similarly, f(c α , c A ) is increasing in c α if c α < c A and is decreasing in c α if c α > c A ; and this behavior is reflected in the synergy maps. In both synergy maps Fig 6A and 6B, SN is negative when c A < 1.25 × 10 −5 g/cm 3 per day, or c α < 0.15 × 10 −8 g/cm 3 per day. Negative synergy indicates that it is more beneficial to use monotherapy with either 3c α or 3c A instead of combination therapy with (c α , c A ). However this conclusion does not take into account negative side-effects.

Discussion
Hepatitis B virus (HBV) infection is one of the most prevalent infectious diseases associated with human liver. Despite the availability of a vaccine, HBV remains a global health problem, which affects more than 350 million people annually, with 600, 000 deaths resulting from HBV-related liver diseases. One of the hallmarks of the disease is the development of liver fibrosis. Anti-HBV drugs currently in use include adefovir, which is anti-viral therapy, and IFN-α, which is decreases viremia, inflammation and liver fibrosis. The treatment with each of these drugs typically extends over a period of many months, and incurs severe negative sideeffects; kidney damage, low amount of phosphate in the blood, stroke and pancreatitis for adefovir, brain disorder, heart attack and lung fibrosis for IFN-α.
In the present paper, we considered the effect of these two drugs on slowing the growth of liver fibrosis associated with HBV. With the aim of using the least amount of drugs, we asked the following question: Is a combination of adefovir and IFN-α at a certain level more effective in reducing the fibrotic scar than a monotherapy with either adefovir or IFN-α at an 'appropriate' increased level? If so, we say that the combination therapy is synergetic. We developed a mathematical model by a system of PDEs and used the model to address this question. The model includes cells and cytokines that significantly affect the fibrosis of the liver in HBV. Some of the parameters were taken from the literature, while others were estimated under some assumptions; simulations of the model validated these assumptions.
The main result of the paper is the "synergy maps" in Fig 6. The color columns assign a "synergy numbers" (SN) to any level of adefovir (c A ) and IFN-α (c α ). Given any level of c α , as c A initially increases so does each SN, until c A reaches a level c Ã A which depends on c α ; i.e., c Ã a ¼ c a ðc A Þ. Thereafter, the SN begins to decrease as A increases. We suggest that the dosage of IFN-α should be related to adefovir so that the proportion should lie as closed as possible to both solid curves in Fig 6. In Fig 6, negative synergy means that it is more beneficial to treat the patient with either IFN-α or adefovir as a single agent, rather than with combination of the two drugs. In the definition of efficacy we compared treatment of combination therapy (c α , c A ) to monotherapy with either (1 + θ I α )c α or (1 + θ A )c A , where we took θ I α = θ A = 2. This choice was quite arbitrary. If it turns out that the negative side-effects in the combined therapy are much more severe than the negative side-effects of a monotherapy with either IFN-α and adefovir, then it would be more appropriate to increase the values of θ A and θ I α in the synergy map. On the other hand, if the negative side-effects from the combination are not significantly more severe than the negative side-effects from either one of the two drugs, then it would be appropriate to decrease the chosen values of θ A and θ I α in the synergy map. A meta-analysis for the comparison of efficacy in the combination IFN-α-adefovir versus IFN-α in monotherapy is done in [89]. In this work that does not include the effect of drug toxicity, the authors show that the efficacy of IFN-α plus adefovir combination therapy is superior to IFN-α monotherapy. However, combined clinical data on efficacy and toxicity are quite limited at this time. Hence, our model should be viewed as setting up a computational framework, to address the question of optimal efficacy in combination therapy with IFN-α and adefovir.
Supporting information S1 File. Supporting information-Chronic hepatitis B virus and liver fibrosis: A mathematical model. Parameter estimates (Section A in S1 File), parameter values (Tables C-F in S1 File) and numerical methods used (Section B in S1 File). (PDF)