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Current Drug Targets – Infectious Disorders 2003, 3, 329-344 329 Mathematical Approaches in the Study of Viral Kinetics and Drug Resistance in HIV-1 Infection V. Müller* and S. Bonhoeffer Ecology & Evolution, ETH Zürich, ETH Zentrum NW, 8092 Zürich, Switzerland Abstract: We review some crucial aspects of drug therapy and viral resistance that have been investigated within the framework developed for the modelling of virus kinetics. First, we give a general overview on the use of mathematical models in the field of HIV research. We seek to identify the factors that determine the steady state virus load and show that stable reductions during antiviral therapy are difficult to explain within the standard model of virus dynamics. We discuss possible extensions that enable the models to account for the moderately reduced virus loads during non-suppressive treatment and argue that the residual viremia under suppressive treatment can probably be attributed to the survival of long-lived infected cells, rather than to new rounds of replication. Next, we address the emergence of resistance during suppressive therapy and demonstrate that the resistant virus is more likely to be present already at the start of treatment than to be generated during therapy. The appearance of resistance after a prolonged period of initial suppression indicates that drug efficacy is not continuously maintained over time. We investigate the potential risks and benefits of therapy interruptions. Considering the effect of recombination, we argue that it probably decelerates, rather than accelerates the evolution of multidrug-resistant virus. We also review state-of-the-art methods for the estimation of fitness, which is crucial to the understanding of the emergence of resistance during therapy or the re-emergence of wild type upon the cessation of therapy. Key Words: HIV-1, drug resistance, mathematical modelling, viral fitness. 1. INTRODUCTION Drug resistance poses a major challenge to the treatment of HIV-1 infection. The response to the challenge is twofold: while new drugs are needed to combat viruses that are resistant to existing drugs, it is also important to devise optimal strategies for the application of the available tools, to minimise the risk of the emergence of resistance in the first place. Understanding the biological processes that underlie drug resistance is clearly crucial for the latter aspect, and this is where mathematical modelling can help. Mathematical models have been instrumental in the elucidation of the dynamical nature of HIV-1 infection [1-6]. The fitting of simple models to experimental data has revealed that relatively stable virus loads reflect a dynamic equilibrium between extremely fast production and destruction of cells and virus particles, rather than true latency. In this review, we first give an overview of the general modelling framework used to describe the withinhost dynamics of the disease. We investigate what factors might determine the steady-state virus load, which varies several orders of magnitude between patients and has a strong effect on disease outcome [7]. In particular, we address how antiviral therapy and drug resistance influence the virus load and demonstrate that the effect of new drug *Address correspondence to this author at ELTE Department of Plant Taxonomy and Ecology, Pázmány Péter Sétány 1/C, 1117 Budapest, Hungary; Tel: (+41) 1 6327105; Fax: (+41) 16321271E-mail: [email protected] 1568-0053/03 $41.00+.00 classes under development can be implemented in the standard modelling framework without major modifications. Next, we consider whether resistance is more likely to emerge due to mutations occurring during drug treatment or due to the low-level presence of mutants prior to therapy. Recombination is also considered to be relevant to the appearance of multidrug resistance—we present recent results showing that the effect of recombination is strictly dependent on the fitness relations of single and multidrugresistant mutants. In the next section, we review modelling results on the risk of the emergence of drug resistance during structured therapy interruptions (STIs). Under what conditions do the benefits outweigh the risks? The replicative capacity of resistant mutants is also highly relevant for the emergence and maintenance of drug resistance. In the final section we discuss recent methods for the estimation of viral fitness. 2. THE BASIC MODEL In the basic model of viral dynamics we consider uninfected target cells, T, productively infected cells, I, and virus particles, V. Uninfected cells are produced at a constant rate, σ, from a pool of precursor cells and die at rate δTT. Virus reacts with uninfected cells to produce infected cells at rate βTV, and infected cells die at rate δI. Virus is produced from infected cells at rate pI and is cleared at rate cV. Finally, we prefer a slight modification to the “basic scheme” and introduce a factor f to describe the probability that a newly infected cell progresses to virus production [8] (and Bonhoeffer et al. in press). This gives rise to the following system of ordinary differential equations: © 2003 Bentham Science Publishers Ltd. 330 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4 dT = σ δTT βTV, dt dI = f βTV δI, dt dV = pI cV dt (1-3) The scheme of the model is shown in Fig. (1). Clearly, such a simple model cannot cover the true complexity of HIV-1 infection, but it can capture some important aspects of the disease and has led to fundamental insights into HIV replication kinetics in vivo [1, 2]. It can also serve as a starting point for the inclusion of further biological details. As a general rule, the complexity of a model should be kept at the minimum that is necessary to answer the questions that inspired the creation of the model. The equations are built upon biological hypotheses of the infection, and these diverge increasingly as more of the true biological complexity is considered. For each detail, a choice has to be made, and this will constrain the applicability of the results obtained with the model. The basic model of HIV-1 infection reflects a consensus of the field, and further details are subject to controversies about alternative hypotheses. The properties of the model have been discussed extensively elsewhere [4, 5, 9, 10]. Here we provide an overview with an emphasis on discussing possible extensions to the model. A major advantage of mathematical modelling is that the initial assumptions that limit the validity of the results are clearly defined by the unambiguous mathematical formulation. The basic model has several important simplifying assumptions. It assumes that all processes occur in a wellmixed, spatially homogeneous system, and the rate of new infections is proportional to the abundance of virus particles and susceptible cells. This limitation can be partially overcome by considering several spatial compartments [1113], but ordinary differential equations cannot describe spatial heterogeneity within the compartments. Another Müller and Bonhoeffer assumption is the homogeneity of the cellular and viral pools. The basic model considers a single virus-producing cell population, productively infected CD4+ T lymphocytes, which arise by the infection of a single target cell type, activated CD4+ T cells. This simplification is justified in untreated individuals, in whom this cell type is responsible for the majority of virus production. Under therapy, the population of productively infected cells dwindles fast, but the decline of the virus level decelerates after the first few weeks, indicating that other cell types are responsible for the residual virus production. Models of long-term treatment are therefore extended by the description of long-lived infected cells and latently infected cells [14, 15]. Apart from considering several cell types, some models have also implemented heterogeneity within a single cell type by employing a continuously distributed parameter (death rate) instead of a fixed value [15, 16]. The virus is also described as a single, homogeneous population in the basic model. This approach is justified as long as only the total abundance of virus (virus load) is considered and the processes can be characterized by the average parameters of the swarm (quasispecies) of diverse virus variants within the patient. For instance, not all virus particles might be infectious, but the infectivity parameter, β, in the model provides an infection rate averaged over the total population of infectious and non-infectious viruses. If there is need to keep track of diversity or specific individual variants (e.g. wild type and drug-resistant forms, see below), the virus load equation can be duplicated to describe each variant separately. In practice, the whole diversity of the viral quasispecies cannot be followed with this method, but this is not needed for the modelling of general virus dynamics. In some models of disease progression, virus variants are distinguished on the basis of their antigenic properties: each virus variable describes the variants that can be targeted by a particular CTL clone [17, 18]. A further general assumption of the basic model is that infection occurs instantaneously, newly infected cells start to produce virus immediately after infection. This simplification does not affect the behaviour of the model when the system is at steady state, but it might Fig. (1). The scheme of the basic model of HIV-1 infection. Susceptible target cells arise from a pool of precursor cells at rate, σ, and die at rate δTT. Virus reacts with uninfected cells to produce infected cells at rate βTV, but only a fraction f of newly infected cells progress to virus production. Infected cells die at rate δI; virus is produced from infected cells at rate pI and is cleared at rate cV. Mathematical Approaches in the Study of Viral Kinetics Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4 become important when perturbations (e.g. drug therapy) are considered. In the latter case, the assumption can be relaxed by the use of delay differential equations [19, 20]. Whereas the assumptions listed above concern general structural properties of the basic model and their possible modifications, further biological details can also be introduced into any specific process described by the model. Each parameter can in fact reflect a complex biological process, which would make it a function of other parameters and variables. This approach has been explored in detail elsewhere [8], where we have suggested the term “process function” to capture this potential for hidden complexity. Indeed, each parameter can be replaced by a function to accommodate further biological details in the model. For example, the process function σ by which target cells are produced could be written as σ =aQ to represent activation of quiescent cells, Q, with an activation rate a [21]. The loss function of target cells, δT, could then be written explicitly as a sum of death and reversion to resting state. Note that in such a model increasing activation of CD4+ target cells can alone result in their depletion, which is now a favoured hypothesis of HIV pathogenesis [22-25]. An important extension is the implementation of an HIVspecific immune response that can react to varying levels of infection. In the basic model the effect of the immune response is hidden in the parameters, its strength is fixed. To allow for an adaptable immune response, a new variable has to be introduced to describe the level of the HIV-specific effector cells [6, 26], e.g.: dE = αEI dt δEE (4) where α is the immune responsiveness, indicating the ability of effector cells to respond by expansion to the presence of infected cells, and δE is the death rate of these cells. The process function approach can then help to implement the action mechanism of the effector cells within the frame of the basic model. If cytotoxic cells kill virus producing cells, the death rate of infected cells can be expanded by the killing term: δ = δ1 + kE, where δI is the intrinsic death rate and k is the efficiency parameter of effector cells. However, if newly infected cells can be killed before the onset of virus production, then effector cells reduce the probability of progressing to virus-producing state: f = 1 +1kE . Further options might involve non-cytotoxic mechanisms that influence the infection rate, β, or the rate of virus production, p. The immune responsiveness, α, and the effector efficiency, k, can also be functions of the target cell or infected cell levels to indicate CD4-help [27, 28] or saturated stimulation [6, 8, 26], respectively. In all, the basic model has immense potential to accommodate further biological details. Note that the expansion of parameters into process functions does not affect the validity of the steady state results obtained by the basic model. However, the new results obtained by the expanded forms depend on the precise form of the expansion. The basic model assumes that parameters are constant in time. This assumption is justified if the period of observation is sufficiently short, such that changes in the parameters can be neglected, as is the case in the first few 331 days to weeks after initiation of therapy. To accommodate the description of long-term dynamics in the context of the basic model, the parameters need to be interpreted as process functions that may change their values over time. Finally, even the basic model can be simplified further if this fits the purpose of the study. For instance, if there is no need to keep track of the virus level explicitly, the equation of the virus load (Eq 3) can be dropped and the rate of new infections can be made a function of the infected cell load directly, i.e. β*TI instead of βTV. This quasi-steady state assumption is justified because the dynamics of the free virus population are considerably faster than those of the infected cells, and consequently the free virus can be assumed to be always proportional to the infected cell population (i.e. β*I = βV). We will see examples for both the expansion and the simplification of the basic model at the discussion of particular applications below. 3. STEADY STATE Without treatment, virus load and the number of infected and uninfected cells settle to a relatively stable level. This situation can be well approximated by the steady state of the mathematical models. Because differential equations describe the rate of change of the variables, a steady state corresponds to setting all equations to zero, i.e. to no change in any of the variables. In the basic model, this procedure yields two solutions. One equilibrium corresponds to the uninfected state with no virus and T = σ / δT uninfected target cells. The other reflects the infected steady state with levels: ^ I = σf δ p ^ ^ δT c , ^ δc V = c I, T = βp βfp (5) If virus is introduced into an uninfected population, the system will approach one of the two steady states depending on the parameters. The criterion can be summarised in the notion of the basic reproductive number, R0 [29, 30], which is defined as the average number of infected cells that a single infected cell produces when placed into an uninfected cell population. In the basic model, it is obtained as R0 = βσfp / δδ T c . If R0 < 1, the system approaches the uninfected equilibrium and the infection dies out. If R0 > 1, the system approaches the infected steady state, and infection is established in the host. There are no unambiguously documented cases in which the virus would have been cleared completely, which indicates that R0 is almost always above one. It is an open question whether highly exposed uninfected individuals have an R0 below one, which would indicate systemic resistance to the virus, or can prevent the virus from accessing the main target cell population by a form of local defence at the site of viral entry. An interesting problem arises if one introduces an explicit description of the HIV-specific immune response, as given in Eq. (4) [6, 26]. The steady state level of infected and uninfected cells and virus load assumes a new form: ^ δ p ασc I = αE , V^ = c I,^ T^ = αδT c + βδE p (6) 332 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4 The steady state level of HIV-specific effectors will depend on the precise form of the effector function. The structurally different steady states in Eqs. (5-6) are difficult to reconcile with each other [6, 8]. However, the transition between the two forms can be established by allowing for more biological details in the immune responsiveness, following the method of process functions. We have shown that in untreated patients the HIV-specific immune response is likely to be largely insensitive to changes in the infected cell level due to saturation in the presence of excess antigen and competition between effector cells [8]. This implies that the level of the HIV-specific immune response can be regarded as constant, and needs not to be described by a variable. The basic model thus provides a sufficient description of an untreated, stable infection. Note, however, that by influencing the process functions of the basic model, the parameters of the immune response still have a role in the setting of the steady state levels. The debate about the factors that determine the steady state is far from academic. The viral “set point”, which is equivalent to the steady state virus load in the model, is a strong prognostic marker of disease progression [7]. The set point virus load can vary more than a thousand fold between patients [31] and a higher virus load implies faster disease progression in untreated infection. Remarkably, despite the immense differences between individuals, the set point is very stable within each patient and the virus load tends to return to baseline after perturbations, e.g. when therapy is discontinued [32-35]. This confirms that the set point is indeed strictly determined by the biological parameters of the system. It is of paramount importance to determine which host and viral factors contribute to the setting of the steady state and hence also the rate of disease progression. Briefly, we have found that variation in the virus load correlates strongly with variation in the level of infected cells (Bonhoeffer et al., in press), as has been hypothesized before [6, 26]. Variation in the level of infected cells, in turn, probably reflects variation in the net production of susceptible target cells (Bonhoeffer et al., in press). At closer inspection, this variation can be dissected into variation in multiple host and virus parameters [8]. We have used the process function approach to expand those “parameters” that had the largest variation as estimated in a well-studied patient set. Those parameters or variables that show little variation can be treated as constants. By this method we have found that most of the variation in the virus load can be broken down into variation in the total production of target cells, the infection rate and the parameters that determine the level of the HIV-specific immune response. Note that the immune response is, as mentioned before, typically saturated at a constant level, but this level varies between patients and can thus affect the steady state virus load. Finally, disease progression in this modelling framework manifests itself as a slow shifting of the steady state towards higher virus load and lower CD4 count, which implies a change in the parameters that set the steady state. 4. DRUG THERAPY Drug treatment perturbs the steady state. The effect of drugs can be modelled by adjusting the parameters of the Müller and Bonhoeffer models. Reverse transcriptase inhibitors (RTIs) block new infections, which can be modelled as a reduction in the infection rate, such that the infection rate becomes β’ = (1ε)β, where ε is the efficacy of the treatment. Inhibitors of the viral protease (PIs) render newly produced virus noninfectious, which can be modelled by introducing a new equation for non-infectious virus. Note that the effect of the two drug classes is very similar due to the short life span of virus particles: infectious particles produced before the start of protease treatment disappear quickly, which stops new infections with only a negligible delay compared to the direct inhibition by RTIs. To simplify the analysis, it is often assumed that drugs act with perfect efficacy, i.e. the infection rate is set to zero or all newly produced virions are non-infectious. In this simplified case, virus load decays exponentially, at a rate determined largely by the death rate of virus producing cells, δ [1-3]. This has allowed the estimation of the death rate and thus the average life span of infected cells. However, intensified therapy regimens result in faster initial decay, which implies that the efficacy of common drug combinations cannot be perfect [36, 37] and previous studies have overestimated the life span of virus producing cells, which is now estimated to be one day at the beginning of treatment. If imperfect efficacy is considered in the models, it can be indeed shown that the initial rate of virus decay becomes approximately the product of the true death rate and the efficacy, and thus decreases with decreasing efficacy [5]. After the first few weeks of therapy the decline of plasma viremia begins to decelerate, indicating a change in the cell population that is responsible for the bulk of virus production [14]. Productively infected CD4+ T cells dwindle quickly, and give way to other cell types with smaller initial numbers but a longer life span, such as persistently and latently infected cells. As mentioned before, the basic model can be expanded to describe also these populations. Remarkably, the deceleration of virus decline seems to be gradual, indicating a range of life spans within the virus producing cell population. We and others have proposed that this phenomenon might reflect the heterogeneity of latently infected cells with respect to the activation and death rates [15, 38]. Latently infected cells are memory CD4+ T lymphocytes that have a long life span, unless they are reactivated by antigenic stimulation. Some latently infected cells might be specific for common antigens and might thus have a high probability to receive stimulation and then die quickly; other cells might be specific for rare antigens and thus have a lower rate of re-activation and death. This hypothesis has also received some experimental support [38]. A consequence of decelerating virus decline is that, after prolonged therapy, the average life span of virus producing cells cannot be clearly defined, and hence predictions for the time required for total eradication of the virus have a high degree of uncertainty. The estimation of the rate of virus clearance has also proven to be notoriously difficult. The steep decline of viremia after the initiation of antiviral therapy reflects mostly the death rate of virus producing cells, and the estimation of virus clearance is unreliable from these data [20]. Other experimental settings based on plasma apheresis or infusion Mathematical Approaches in the Study of Viral Kinetics Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4 333 have yielded conflicting results [39-41]. Part of the discrepancies can be explained by models that distinguish between decay in the lymphoid tissue and in the blood plasma compartment [13]. The decline of infectivity in the blood plasma probably reflects the decay rate of virus in the lymphoid tissues, whereas the estimate obtained by plasma apheresis reflects the faster decay rate in the blood. However, the even faster rate estimated in the infusion experiments still remains unexplained (De Boer et al., in preparation). Beyond estimating the turnover parameters of the infection from the decline of viremia, mathematical modelling has also provided interesting results concerning the long-term outcome of therapy. In terms of the basic model, there are two possible outcomes. The changes in viral parameters due to therapy can either lower R0 below one, and thus drive the virus to extinction, or R0 remains above one despite the reduction and a new infected steady state is attained. Remarkably, it is not always straightforward to distinguish between the two scenarios. Accumulating evidence indicates that HIV is likely to persist for life even in the face of effective suppressive treatment [38, 42-46]. However, it is not clear whether this low-level persistence results from ongoing low-level replication (corresponding to a new infectious steady state) or from the survival of latently or persistently infected cells. In the second case, the virus can persist even though the new steady state would be free of virus, because the uninfected steady state is not attained in the lifetime of the patient. In the age of HAART, the virus load in most patients who do not harbour drug-resistant virus variants is suppressed below the limit of detection of standard assays, which makes the study of virus dynamics more difficult. However, studies in earlier days have shown that suboptimal therapy, e.g. a combination of lamivudine and zidovudine can result in stable reductions in the virus load 10 to 100fold below the pre-treatment level [47]. Suppression is achieved through a reduction in the infection rate of the virus: the direct inhibition of the reverse transcriptase by the drugs and the reduced replicative capacity of the emerging lamivudine-resistant mutant are both likely to act on the infection rate. Partial suppression can sometimes be maintained also in patients who develop resistance mutations under current combination therapy [48-50]. Moreover, patients who acquire multidrug-resistant virus in the primary infection have a lower untreated viral set point [51], which also confirms that resistance has a price for the virus in terms of lower replicative capacity. These observations suggest that reductions in the infection rate can indeed result in a stable infected steady state with a lower virus load. Remarkably, however, the standard models of HIV-infection cannot account for this plausible scenario. It has been shown that reductions in the infection rate (or, more generally, in the basic reproductive number) either result in the extinction of the virus or they hardly affect the virus load at all [52, 53]. Fig. ( 2) depicts the dependence of the steady state virus load on the efficacy of treatment, as obtained from the basic model. The virus load is stable until the critical efficacy (required for the eradication of the virus) is approached, and the transition from the infected steady state to no infection occurs in a very narrow parameter range. The intuitive Fig. (2). Steady state virus load as a function of treatment efficacy in the basic model. reason for this phenomenon is that in response to the reduced infectivity of the virus the level of target cells increases, which compensates for the decreased infection rate. Thus, treatment affects the target cell level, but not the virus load, unless the virus gets close to extinction. It has been shown that the difference between the critical efficacy and the realized efficacy of a certain treatment cannot be greater than the factor of reduction in the steady state virus load [53]. For instance, a 100-fold reduction (factor of reduction: 0.01) implies that the efficacy can be at most 0.01 less than the critical efficacy. Such a close match is very unlikely given that the efficacy can vary between zero and one. As discussed above, the suppression achieved by current combination therapy might be consistent with the case when the virus would be driven to extinction after a time exceeding the life span of an individual. However, the low steady state observed with suboptimal or failing therapy, or transmitted mutant virus, cannot be accounted for in the frame of the basic model. The implementation of the HIVspecific immune response as in Eq. (4) cannot solve the paradox, either. In this case, drug treatment reduces the level of the anti-HIV immune response and does not affect the virus load at all until the response is lost completely. As this occurs very close to the critical efficacy that drives also the virus to extinction, the virus load converges very steeply to zero beyond this point, and the overall behaviour is very similar to that observed in the basic model. Several solutions have been proposed to overcome the problem. Low but detectable steady state virus load can be explained if virus induces the death of uninfected cells, or if the proliferation of HIV-specific effector cells depends on the level of infected cells, but not on the current effector population size [52]. Both cases predict a linear dependence of the virus load on treatment efficacy. This can explain the low virus load observed under suboptimal therapy and with multidrug-resistant viruses, but the extremely low virus load under suppressive therapy would still require an efficacy very close to the critical value. In the models, a steady state virus load around the limit of detection (50 copies ml-1) can only be attained if either the death rate of infected cells is assumed to depend on the number of these cells, or there is a compartment where the drugs cannot penetrate [53]. In the former case, the death rate is modified as δ’=δIω. However, the effect is very sensitive to the parameter ω, which expresses the strength of the density dependence. The presence of a drug sanctuary is therefore the most plausible hypothesis at 334 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4 present that accounts for a low steady state virus load comparable to the virus load in well-suppressed patients. In the models, this can be described by duplicating the equations of the basic model (Eqs. (1-3.)) to describe virus load, uninfected and infected target cells in the two compartments separately. The connection between the two compartments is established through an exchange of virus particles. Note that to maintain the extremely low viremia observed in the main compartment, the drugs have to act very efficiently. Otherwise, the virus spilling out of the protected compartment could initiate new rounds of infection in the main compartment and thus result in detectable viremia. We are thus left with two possibilities. If the low virus load after prolonged suppressive therapy reflects a true steady state, there must be a sanctuary site where the drugs cannot take effect. Alternatively, residual virus loads below the detection limit are maintained by long-lived cells, rather than ongoing replication and thus do not reflect a true steady state. In both cases, the moderately reduced virus load during suboptimal or failing therapy, or after primary transmission of multidrug-resistant virus, can be accounted for by virusinduced killing of uninfected cells or an HIV-specific cytotoxic response. What is the take-home message? In either case, the basic model requires extensions to explain the changes in the virus load during long-term antiviral therapy. Each extension reflects a possible biological process, and our present knowledge does not allow us to decide which scenario is correct. However, the predictions of the models can be used to test the underlying biological hypothesis. For example, if unambiguous evidence for ongoing virus replication were found in well-suppressed patients, the model would predict the existence of drug sanctuaries, which should eventually be found. Conversely, a prolonged failure to find drug sanctuaries would strongly argue against the role of ongoing virus replication in the maintenance of viremia. New rounds of infection might still occur, but they cannot maintain a stable level of viremia according to the model results. It has indeed been observed that occasional viral ‘blips’ can slow down the long-term virus decay in some patients [46, 54], but are not needed for the persistence of residual viremia. Since no drug sanctuaries have been identified as yet, we would risk the prediction that viral persistence after longterm suppressive treatment is a consequence of the long-term survival of infected cells, rather than a result of recurring rounds of new infection. The background for the reduced steady state virus load attained during suboptimal therapy is more ambiguous. There is evidence for the virus-induced killing of uninfected cells, for example by chronic hyperactivation [55, 56]. However, there is also strong evidence for the role of the virus-specific cytotoxic immune responses in the control of viremia [57-60]. Moreover, in previous work we have shown that the activation of the immune response is likely to be saturated in untreated chronic infection, which accounts for a similar behaviour as in the implementation that has been shown to explain low virus steady states [8]. Saturation due to competition between effector cells corresponds to the independence of immune activation of the current level of the effector population as in [52]. Indeed, the model developed in [8] can Müller and Bonhoeffer also account for moderately reduced steady state virus loads. The low steady state observed under suboptimal therapy and with multidrug-resistant virus might thus reflect a combination of the two possible factors. Finally, we consider the effect of the new classes of inhibitors that might become available for therapy in the foreseeable future. Fusion inhibitors, chemokine analogues and the inhibitors of the viral integrase all block new infections, and thus act on the infection rate, β, in the models. The implementation and the predicted effect of the new treatments are thus identical to that of RTIs if only one drug class is considered. However, a combination of drugs from different classes can have tremendous impact, as the effect of drugs acting on subsequent steps of the viral life cycle is probably multiplicative. In this case, the infection rate becomes β’ = (1-ε1)(1-ε2)(1-ε3)(1-ε4)β, where the parameters ε1-4 denote the efficacy of the RTI, fusion inhibitor, integrase inhibitor and coreceptor antagonist arm of a drug combination. Drug classes not included in the combination can simply be regarded as having zero efficacy. The first clinical trials have indeed demonstrated that adding a fusion inhibitor to standard combination therapy can considerably improve the degree of virus control [61-63]. 5. DRUG RESISTANCE Antiviral drugs can suppress the replication of wild type HIV, but mutations can render the virus resistant to the drugs. Monotherapy in the early days of HIV treatment invariably led to the fast emergence of resistant virus and the rebound of viremia. Even current combination therapy can fail due to the evolution of multidrug-resistant virus variants, and a large fraction of patients have been “cycled” through many drug regimens. It is of paramount importance to identify the sources of resistance and devise optimal strategies to prevent its development. Mathematical modelling can be a useful tool also in this context. The simplest implementation of drug resistance is to duplicate the equation for infected cells to describe wild type and resistant virus separately. As the emphasis is on the spread of resistance and the explicit implementation of virus levels is not relevant in this context, we can simplify the model by making the rate of new infections a function of the infected cell levels (see Basic Model). Thus we write: dT = σ δT T dt β1 TI1 β2 TI2 , dI1 = (1 µ )β1 TI1 + µβ2 TI2 δI1, dt dI2 = µβ 1TI1 + (1 µ)β2 TI2 δI2. dt (7-9) I1 and I2 denote the populations of cells infected with wild type and resistant virus, respectively. To be consistent with the detailed studies in the field [64-69], we have abandoned the factor of infected cells that progress to virus production (in effect, we fix f = 1). Mutations occur at rate µ. We assume that the two variants differ only in their infection Mathematical Approaches in the Study of Viral Kinetics rates, which are denoted by β1 and β2, respectively. The basic reproductive number of the two variants can then be written as R1 = β1σ / δδ T and R2 = β2σ / δδT . Under therapy, ‘resistance’ and ‘sensitivity’ can be defined on the basis of the reproductive number. A virus variant is resistant if it can maintain a steady state virus load in spite of therapy, which corresponds to the criterion R2 > 1. Sensitive virus would eventually be pushed to extinction, i.e. R1 < 1 during therapy. The ‘wild type’, on the other hand, is defined as the dominant variant in drug naïve untreated patients, which implies that in the absence of therapy the sensitive wild type virus has the highest basic reproductive number, i.e. R1 > R 2 > 1. (Note, that here we make also the assumption that the basic reproductive number of the resistant virus is not lower in the absence then in the presence of therapy). Importantly, the basic reproductive number depends on both host and viral factors; ‘resistance’ is therefore a property of the hostvirus system, not just that of the virus. The outcome of treatment depends on its strength. Weak drugs might be unable to push the basic reproductive ratio of the wild type below one. In the narrow sense, even the wild type is resistant in this case. Zidovudine monotherapy reflects such a situation: it has been shown that the first rebound of viremia after the start of therapy is actually composed of wild type virus [70, 71]. As mentioned before, the resurgence is fuelled by the increasing abundance of target cells. Even in this case, however, mutant viruses can have higher infection rates in the presence of the drug, i.e. R2 > R1 > 1. In this case the initial resurgence is followed by the replacement of wild type virus with mutants of increasing drug resistance, as has indeed been observed during longterm zidovudine therapy [72, 73]. The sequential emergence of resistant variants has been described by mathematical models [71, 74]. More efficient drugs can lower the basic reproductive number of the wild type below one, while still allowing the replication of the resistant mutants such that R2 > 1 > R1. In this case the wild type decays exponentially and the resistant mutant is selected for. Stronger drugs result in faster replacement of the wild type [64]. Interestingly, the basic model predicts that the benefit of a larger initial reduction in the virus load is cancelled out by the faster resurgence. That is, the long-term benefits of therapy in terms of increased CD4 count or decreased virus load are independent of the efficacy of the drugs on the wild type virus, as long as virus is not completely suppressed. Finally, in the optimal scenario the drugs can suppress all virus variants such that 1 > R2 > R1. In this case the virus is suppressed and eradication is prevented only by replication in drug sanctuaries (where R1 or R2 > 1) or the survival of cells infected before the start of therapy. This might be the situation with current combination therapy, contingent upon perfect adherence. An important question in the dynamics of drug resistance is whether the resistant variants responsible for drug failure are present at the start of therapy or arise during treatment [64]. Resistant variants can appear by random mutations even in the absence of therapy [75-77]. If their replicative capacity is much lower than that of the wild type, they cannot persist and the mutants present at any time point are the result of recent mutations. The frequency of such mutants Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4 335 is thus equivalent to the rate of the mutation that creates them from the wild type. If the selective disadvantage of the mutant is not profound, the mutant can persist at a higher equilibrium frequency. If the infection rate of the mutant in the absence of therapy is written as β 2 = (1 − s)β1 , where s is the selective disadvantage of the mutant, then the pretreatment frequency can be approximated as µ/s [67]. This gives the frequency of one-point mutations. However, to escape current combination therapy the virus has to evolve several mutations. If all mutation rates are equal and all intermediate mutants have the same selective disadvantage, the frequency of two-point mutants is 2(µ/s)2, while that of three-point mutations is around 6(µ/s)3. With an estimated mutation rate of µ = 3×10-5 [78] a mutant that has a selective disadvantage of 0.01 can have a frequency around 3×10-3 prior to the start of therapy. Considering an estimated 107108 infected cells in a chronically infected patient [79], oneand two-point mutations might be relatively abundant and even three-point mutations are likely to be present prior to therapy. Such frequencies are not easy to detect in vivo, but there is some experimental support for the pre-existence of one-point mutants [76, 80] Higher selective disadvantage will, of course, decrease the probability of pre-existence. Importantly, the selective disadvantage of the intermediate one- and two-point mutations also has a large impact on the pre-existence of fully resistant mutants. A defective intermediate creates a barrier to the generation of resistant virus. Note that in differential equation models the mutants are always present at some low concentration. As an approximation, the mutant can be considered not present when its population size is below one. Alternatively, stochastic models can directly describe the probabilities of transitions, such as the appearance of mutations. This approach yields results that are compatible with the approximation by deterministic differential equation systems [68]. The probability of de novo emergence of resistance depends on the number of new cell infections during therapy. If the efficacy of treatment is 100%, no cells are infected and the probability of emergence is zero. The number of newly infected cells in the case of imperfect efficacy cannot be calculated precisely, but an analytical approximation based on the modelling framework presented here has been performed [69]. The results indicate that in the case of suppressive therapy the number of cells infected during treatment is smaller than the number present at the start of treatment. This also holds for the number of mutants produced. Interestingly, the relation between the probability of pre-existence and emergence is independent of the mutation rate, because the two probabilities increase comparably with the mutation rate. This conclusion is also insensitive to changes in other parameters. A further important result is that drugs specifically targeted at the wild type but less effective against intermediate sensitive strains can actually increase the risk of the emergence of resistance. The reason for this is that the suppression of the wild type leads to an increase in the target cell level which benefits the intermediate virus forms [66, 69]. It is thus not necessarily the best strategy to treat with a regimen specifically optimised against the wild type virus. 336 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4 Because under suppressive therapy pre-existence is more likely than emergence, and the former depends on the number of infected cells, the risk of drug resistance is the lowest when the infected cell population is the smallest. This supports the ‘hit early and hard’ strategy of treatment, to attack the virus when the infected cell load is still low. However, primary infection comprises a transient peak of the virus load and the infected cell population, therefore also the presence of resistance mutations is more probable in this period [68]. Considering resistance, optimally timed therapy should thus start after the decline of primary viremia. We have concluded that the emergence of resistance under suppressive therapy is less likely than its preexistence. Considering that combination therapy in drugnaïve patients is almost invariably successful for a considerable time, one might ask the question: how can multidrug resistance evolve at all? It is not present at the start of therapy and it should be even less likely to appear later. The key is that this result assumes continuous efficient suppression of virus replication. If new rounds of replication occur, even transiently in ‘blips’, the probability of emergence increases. This problem will be discussed in the next section. Stochastic simulations indicate that multidrug resistance evolves in a stepwise manner through intermediate forms [68]. This brings up the possibility that recombination between one-point mutants could accelerate the development of fully resistant virus. Recombination is prevalent in retroviruses [81, 82]. Virions produced by cells harbouring multiple proviruses can contain RNA copies from two different infecting strains. When such a “heterozygous” virion infects the next cell, recombination during reverse transcription can give rise to recombinant provirus. The high rate of recombination per replication cycle (the reverse transcriptase alternates on average about 3 times between the two genomic RNA strands [81, 82]), combined with the high prevalence of multiply infected cells [83] allows for a very high rate of recombination at the cell population level. The high prevalence of recombination in lentiviruses has also gained support by the analysis of sequence sets from SIVinfected macaques [84]. However, we have shown that the effect of recombination on the pre-existence and emergence of resistance depends on the exact relations between the infection rates of the various mutants (Bretscher et al., in press). If the replication capacity of a one-point mutant is closer to that of a fully resistant double mutant than to that of the fully sensitive wild type, then recombination will indeed accelerate the emergence of resistance during therapy. However, if the viruses carrying one resistance mutation are still strongly suppressed by the remaining drugs of the combination therapy, then recombination will actually slow down the emergence of resistance. Recombination can not only create but also break up favourable gene combinations, depending on the exact circumstances. As the latter scenario is more likely, we arrive at the counterintuitive conclusion that recombination might actually slow down the evolution of drug resistance in this system. The pre-existence of double mutant also depends on the relative replication capacity of the intermediate mutant. However, in the absence of drugs it is not clear whether the intermediate mutant is closer to the Müller and Bonhoeffer double mutant or to the wild type in replication capacity. The direction of the effect on pre-existing frequencies can therefore not be predicted yet. A further interesting question is the disappearance or maintenance of resistance after drug treatment is stopped. As we have discussed before, in the absence of therapy the original wild type virus is likely to have a higher replicative capacity than the resistant mutant that emerges during therapy. Accordingly, the resurgence of wild-type is often observed when therapy is stopped [49, 50, 85-87]. Generally, it is reasonable to expect that the disadvantage of drugresistant virus in the absence of therapy is smaller than its advantage during treatment. Antiviral drugs suppress the replication of wild type virus drastically, which resistant viruses evade. The mutations responsible for resistance might reduce the general replicative capacity of the mutant virus, but this effect is probably much weaker than the effect of drugs on wild type virus. The selection pressure driving the outgrowth of wild type after the cessation of therapy is thus probably smaller than that exerted by the drugs during the primary emergence of resistance after the start of treatment. Yet the number of resistance mutations often begins to decrease as soon as six weeks after the interruption of treatment [49, 50]. This suggests that wild type virus is not reconstructed by backmutations, but is persistently present also during therapy, probably in the “archive” of latently infected cells [88]. Indeed, in patients who have been infected by multidrug-resistant virus in the first place and have never had wild type virus, the resistant virus can persist for as long as five years in the absence of treatment [51]. In this case, the genetic background of the primary resistance mutations might also inhibit the appearance of wild type virus (Fig. (3)). Multidrug-resistant virus is often the result of long evolution under therapy, and the replicative capacity impaired by primary resistance mutations can be partially restored or sometimes even increased above the initial level by secondary mutations [48, 89, 90]. However, the secondary mutations that are optimised for the mutant reverse transcriptase or protease might function less efficiently with the wild type of these enzymes. In such a situation, the wild type is not selected for even in the absence of therapy. Reversion to wild type would require simultaneous backmutation in the primary and secondary resistance genes, which has a very low likelihood. The evolution of wild type virus is thus inhibited by the low replicative capacity of the intermediate virus variants that form the link between the fully resistant virus and the pure wild type, both of which contain well co-adapted genes. Recombination in this case also acts to slow down the spread of wild type virus. To understand the dynamics of replacement between different virus types, one thus needs estimates on the relative replicative capacity of the two virus types and also of the intermediate forms between them. Replicative capacity is described by the concept of fitness, which we review in Section 7. Besides fitness, the speed of evolution also depends on the mutation rate. Different types of point mutations are generated at very different rates [78, 91] which affects both the pre-existence and the emergence of resistant variants. For instance, the M184I mutant is generated by a high probability guanine→adenine mutation and is therefore Mathematical Approaches in the Study of Viral Kinetics Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4 337 Fig. (3). Fitness relations and the direction of evolution in the presence (A) and absence (B) of drug treatment. Rows contain wild type (WT) or mutant (MUT) alleles of the primary resistance loci, and columns contain the alleles of compensatory loci that can restore replication capacity in the presence of primary resistance mutations. The direction of evolution is indicated by arrows. During therapy (A), primary mutations confer a selective advantage over the wild type, and compensatory mutations provide a further benefit in viruses with primary mutations. Evolution in this case is therefore unidirectional from wild type to primary resistance and then to the double mutant genotype. In the absence of therapy (B) we show fitness relations assuming that compensatory mutations co-adapt to primary resistance mutations and thus perform better with the mutant than with the wild type allele of the primary resistance loci even in the absence of therapy. This creates a barrier to the reversion of the MUT/MUT genotype to wild type, and thus the cessation of therapy cannot reverse the direction of evolution. more likely to pre-exist than most other resistance mutations. Moreover, the mutation rate can also be influenced by interactions between the virus and the infected cell type [9295], by the direct effects of drug treatment [96-98] and by resistance mutations in the reverse transcriptase gene [96]. Finally, note that the framework developed in this section is independent of the action mechanism of the drugs. The conclusions are thus also valid for the new classes of drugs that are currently under development. Indeed, the emergence of resistance has already been documented during monotherapy (in phase one clinical trial) with the fusion inhibitor enfuvirtide (T-20) [99]. Moreover, primary viruses exhibit considerable variability in their susceptibility to T-20 before the start of treatment [100, 101]. The presence of less susceptible variants can be regarded as an example for the pre-existence of resistance. An interesting property of entry inhibitors is that their effect depends on the target cell tropism of the virus. X4 and R5 viruses use the CXCR4 and the CCR5 coreceptor of target cells, respectively. Each chemokine analogue can only block one of the two coreceptors and will thus affect only one virus type. Such drugs might therefore be able to influence the evolution of the target cell tropism of the virus population. This possibility has been explored in detail with the help of mathematical modelling [102]. Compared to the basic model, this study has duplicated the target cell equation to distinguish between naïve and memory CD4+ T cells. The former were assumed to carry CXCR4 only, while memory cells carry both CXCR4 and CCR5. The study explored the outcome of competition between virus variants with different affinity towards the two coreceptors, and then investigated the effect of coreceptor antagonists in this context. In the model CXCR4 inhibitors select for the coexistence of R5 and X4 viruses, and cannot drive X4 virus to extinction, because R5 viruses cannot utilise the resource of naïve target cells. The effect of CCR5 inhibitors depends on whether X4 variants are present at the start of therapy. If they are not, CC5 inhibitors do not facilitate the initial appearance of X4 variants. However, if X4 variants pre-exist, then treatment with CCR5 inhibitors accelerates the switch, because X4 viruses can infect both memory and naïve cells. This advises caution in the use of CCR5 inhibitors because X4 viruses have been associated with accelerated disease progression [103]. Similarly to drug-resistant and drug-sensitive virus variants, X4 and R5 viruses can also persist in the latent reservoir, and may thus never be truly lost [104]. This implies that evolution driven by coreceptor antagonist treatment can immediately reverse its direction when treatment is stopped. 6. STRUCTURED THERAPY INTERRUPTIONS The outgrowth of wild type virus in the absence of drugs, as has been described in the previous section, is one of the potential benefits that inspired the idea of controlled therapy interruptions. Reversion to wild type restores sensitivity to the previously employed drugs, which is especially important in patients who have accumulated resistance against most available drugs. Another hope was that intermittent peaks of viremia would boost the immune responses directed against HIV and facilitate the immune control of the infection even in the absence of further therapy. The method of structured therapy interruptions (STIs), which involves a protocol of alternating periods on and off therapy, was devised to avoid the inhibition of the immune responses by the fast growing virus, and allow for a gradual build-up of HIV-specific immunity. Besides the potential benefits, however, this approach carries the danger that renewed virus replication during treatment interruptions might generate drug-resistant mutants. The risks and benefits must therefore be assessed by careful experimental and theoretical studies. The modelling framework described in the preceding sections can be used to investigate both sides of the coin. Previously, we have argued that the probability of the appearance of resistance mutations depends on the number of cells that become newly infected. The relative risk of the emergence of resistance thus depends on the extent of virus replication occurring during treatment interruptions. Unless 338 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4 virus replicates to levels approaching the set point virus load maintained before the start of antiviral therapy, the probability of emergence is lower than that of pre-existence before therapy [105]. However, the probability increases with the height of the peaks and with the cumulative time off therapy [105, 106]. Furthermore, the comparison of relative risks used to predict the source of resistance during therapy cannot be used as a guideline in this case. Even if mutations are generated with a lower probability during STIs than before the initial start of therapy, this gives a new chance for the virus. Patients who do not harbour pre-existing mutants, run a new risk during STIs, which should be evaluated on absolute, rather than relative criteria. Clearly, phenotypic resistance needs the fixation of the resistance mutations after their initial appearance. However, resistant mutants can only increase in frequency when the levels of the drugs are low enough to allow their replication but high enough to give them a selective advantage over the wild type (reviewed in [107]). Since multidrug-resistance develops most probably by the stepwise accumulation of multiple mutations, the initial mutants are resistant to one drug only, and are suppressed under combination therapy. The time window for the selection of resistant variants is thus probably short during each treatment cycle. However, a modelling study has shown that partially resistant variants can increase in frequency unless they have a large selective disadvantage in the absence of therapy [106]. The model used in the study was an expanded version of Eqs. (7-9.), involving also latently infected cells and virus particles. The growth of mutant virus is facilitated also by the abundance of target cells at the beginning of each interruption period. In contrast to chronic infection, there is no competition between mutant and wild type viruses at this stage. This free unlimited growth also implies that appearing resistance mutations can be preserved even if they do not have a selective advantage during STI. In such a case, the first appearance of the mutant will establish its “pre-existence”, which enables the mutant to grow immediately once the conditions become favourable for it (cf. Drug Resistance). In summary, the appearance of multidrug resistance is improbable during STIs, but partial resistance might develop, especially if the virus load is allowed to reach higher levels and/or the protocol is continued for a long time. Resistance mutations have indeed been detected in some patients under STI [108] (Metzner et al., in press), which might reflect the growth of pre-existing mutants in some cases [109]. However, Metzner et al. have detected minor populations of the M184V and L90M mutations in 14 and 3 out of 25 subjects who have been on their first combination therapy before the STI and had no prior history of suboptimal treatment. In this case, the mutations probably arose de novo during the interruptions. Furthermore, even if phenotypic resistance is not detected, hidden pre-existence of mutations might be established due to the unlimited growth of virus during short interruptions. Partially resistant viruses might also enter the latent reservoir [106]. Importantly, during repeated cycles of interruption, recurrent growth of partially resistant viruses gives an opportunity for the accumulation of further mutations and the development of multidrug-resistance. This process is likely to occur in Müller and Bonhoeffer uncontrolled “interruptions” due to non-adherence. Irregular pill intake can also lead to out of phase fluctuations in the concentration of different drugs, which also facilitates the consecutive acquisition of resistance mutations. As we have concluded in the previous section, this is probably the most important path to multidrug resistance. The importance of pre-existence is also underlined by the results of STI studies. Not only does the wild type outgrow resistant mutants during interruptions, but also the mutants return when continuous therapy is re-initiated [110, 111]. The growth of partially resistant variants during STIs might also be attributed to pre-existence in some cases [109]. It seems that once a variant had been present in the virus pool, it will remain archived in the pool of latently infected cells, and can re-grow under favourable selecting conditions. This property abolishes the benefit of STIs in re-establishing drug sensitivity. As soon as therapy is restarted, resistant mutants that existed before the STIs can resurge—the virus “remembers” all of its previous states. The other objective of STI studies has been to enhance the immune control of the infection, ideally to a point where drug treatment can be stopped completely. Indeed, treatment interruptions can induce an increase in the size and breadth of the HIV-specific immune responses [34, 112-116]. Furthermore, the growth rate of the virus during the interruptions decreases over the consecutive STI cycles [35, 113, 117]. Given that the target cells are abundant at the beginning of the interruptions, it has been hypothesised that the reduced growth rate might reflect the effect of a boosted immune response. Unfortunately, in chronically infected patients the increased immune responses and slower viral growth rate are not associated with enhanced control of the virus [35, 117, 118]. Virus levels increase in most patients during the interruption periods [34, 112, 113]. After the end of STIs some patients maintain reduced virus load for as long as 12 months without therapy [112], others, however, attain levels comparable to the pre-treatment baseline [114, 118]. A better immune control of the virus is achieved only if treatment is initiated during primary infection and then stopped [116, 119-121]. These phenomena can be understood in terms of the steady states of the models. In the basic model (Eqs. (1-3)), multiple parameters determine the steady state virus load in the absence of treatment. As we have shown, therapy disturbs this steady state and either establishes a new equilibrium, or puts the virus onto an asymptotic course towards extinction. However, when therapy is stopped, all levels return to their pre-treatment value, and the basic model returns to the baseline virus load preceding treatment. Going through cycles of interruptions cannot change this behaviour, unless the parameters (process functions) are altered and this change persists after the end of STIs. The basic model can thus explain the effects of STIs observed in chronically infected patients, but not in acutely infected patients. To account for sustained effects one needs an extended version of the model that can have two alternative infected steady states with a possibility for early treatment to push the system from one steady state to the other. We can devise such a model extension by analysing the mathematical possibilities to achieve such behaviour and/or by considering Mathematical Approaches in the Study of Viral Kinetics biologically plausible scenarios. In the former approach we describe the rate of change of the infected cell population with a simplified version of Eq. (2), again assuming that the level of free virus follows that of infected cells at a quasi steady state: dI / dt = f βTI − δI . In the absence of therapy the system settles to a steady state, which implies that the rate of change will be zero, the positive production term and the negative death term are equal and balance each other perfectly. During therapy, the level of susceptible target cells, T, increases. If all other parameters remain unchanged, this will increase the input term of infected cells and thus result in a positive growth rate and an increased virus load when the treatment blocking new infections is stopped [105]. To maintain the suppression of the infection after the cessation of therapy, either the infection rate, β, or the fraction of infected cells progressing to virus-production, f, has to decrease, or the death rate of infected cells, δ, has to increase during therapy. Having identified the mathematical possibilities, we seek a biological explanation that could be responsible for them. Potentially, all three effects can be achieved if the level of HIV-specific effector cells increases. By killing infected cells these cells can reduce the fraction progressing to virus-production or increase the death rate of virus-producing cells. Alternatively, non-cytotoxic effector cells might reduce the infection rate. However, in the standard models of HIV-specific effector cells, e.g. in Eq. (4), the proliferation of the cells depends on antigenic stimulation by HIV. Suppression of the viremia results in reduced effector levels, which is consistent with observations on chronically infected patients entering therapy [122]. To account for the sustained control of viremia in patients who were treated during acute infection, one requires a mechanism either to increase the activation rate of effector cells (immune responsiveness) or to decrease their death rate [105]. The dependence of the effector response on CD4-help has been proposed to account for the former possibility [28, 123-125]. The basic model (Eqs. (14)) has been expanded by an equation for the precursors of the effector cells, and the proliferation of precursors was assumed to depend on both the infected cell load and the level of CD4 target cells. In this model, an increase in overall CD4 counts due to virus suppression also increases the level of HIV-specific CD4 help, which might improve either the proliferation or the effector function of HIVspecific CD8 T cells. There are two stable steady states in the absence of therapy: one with a preserved HIV-specific helper response, a strong CD8 effector response and low, controlled virus load; the other with a severely reduced helper population and high virus load in spite of the induced CD8 response. Early antiviral therapy during acute infection can preserve the CD4 population until the CD8 effector cells proliferate to a level sufficient to control the virus [28, 123]. Interestingly, the analysis of a generalised model has demonstrated that in such a system a single phase of welltimed drug therapy should be able to push the system from the uncontrolled to the controlled steady state and thus establish sustained immunity [126]. The failure of therapy to do so in chronic HIV infection indicates that the HIV specific immune response might be lost irrevocably at this stage. We note, however, that too long therapy can restore the system to the initial susceptible state with too few Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4 339 effector cells to prevent the growth of the virus into the uncontrolled steady state once therapy is stopped [105]. Similar results are obtained if we assume that the death rate of effector cells increases with the virus load due to some inhibitory effect exerted on the cells by HIV [105]. Writing the death rate as a process function, e.g. as δE’ = δE + I / ( K + I ) to account for a saturating impairment at high infected cell load, also allows for the existence of two stable steady states. The behaviour of this model is similar to that of the CD4 help version. Finally, a further concern about treatment interruptions is that they might refill the latent reservoir. It has been shown that this process is not likely to be relevant if the kinetics of latently infected cells is similar to that observed during chronic infection [105]. Virus replication during interruptions could refill the latent pool only if it reached a level comparable to the pre-treatment baseline. However, the turnover of latently infected cells might be much faster during chronic infection, which is accompanied by immune hyperactivation, than during therapy and even the STI cycles [38]. In this case, low levels of virus replication during STIs could indeed rapidly refill the reservoir. In all, treatment interruptions cannot restore drug sensitivity and they can only contribute to immune control if the virus load has a strong and reversible effect on the HIVspecific effector cells. However, interruptions impose a small but with time increasing risk for de novo emergence of drug resistance, and might contribute to the re-seeding of the latent reservoir. 7. ESTIMATION OF VIRAL FITNESS Fitness in a broad sense is used as a synonym for replicative capacity or growth rate in virology, especially in the context of measuring and comparing the growth rate of different virus variants. The stricter definition of the term depends on the particular methods used for its estimation. “Fitness” values from different studies are therefore only comparable if they were obtained with the same method. In the preceding sections we have shown that the direction and speed of viral evolution during and after therapy, and also the composition of the virus population before therapy depends on the relative replicative capacity, or fitness, of the competing virus variants under the given circumstances. The probability that a resistant mutant is present at the start of therapy depends on its fitness relative to that of the drug sensitive wild type virus in the absence of the drugs. The short-term outcome of treatment depends on the fitness relations in the presence of the drugs. The evolution of multidrug-resistant virus is influenced by the fitness of the intermediate variants and a similar effect shapes the reversion of complex mutants to wild type in the absence of therapy. To understand and possibly predict viral evolution, one thus needs to know the fitness relations of the relevant virus variants. For detailed reviews on viral fitness see [127]. Fitness is sometimes quantified on the basis of direct biochemical measurements of enzyme activity [89, 128] or virus growth kinetics in pure cultures that contain only one variant [50, 89, 129]. Experimental protocols have been reviewed recently in [130]. However, the assays that operate 340 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4 on single virus variants cannot take the interactions between different virus strains into account. Many studies have therefore investigated the relative growth kinetics of two virus variants growing in competition either in vivo [131134] or in vitro [128, 135-139]. The interpretation of such experiments requires careful mathematical considerations [140, 141]. A measure of fitness is typically derived by plotting the ratio of the two competing variants on a logarithmic scale against time and estimating the linear slope of this graph (Fig. (4a)). The underlying mathematical model is written as follows: dW = [r(t) δ]W(t), dt dM = [(1 + s)r(t) δ]M(t). dt (10-11) We use the notation employed in fitness estimations, but note that these equations can be derived from the standard modelling framework. W and M denote the cells infected with wild type or mutant virus, respectively, and thus correspond to I1 and I2 in Eqs. ( 8-9). Since we are concerned with the relative abundance of the two variants, the level of virus particles needs not be modelled explicitly. A further simplification from the basic model is that the growth rate of the infected cells is described by a single parameter, r, which might be a function of time. Assuming a quasi steady state for the virus, this generalised growth rate corresponds to r(t) = βfpT(t)/c in the basic model and is thus a complex function of viral parameters and target cell availability. It is assumed that the different virus variants have the same death rate, δ, and differ in the growth rate by a factor, s. This factor, the so-called selection coefficient, denotes the relative fitness difference. As a slightly confusing practice, in the modelling of drug treatment s is typically used to describe the selective Müller and Bonhoeffer disadvantage of one variant compared to the other and is thus interpreted as a reduction in the growth rate (or in the infection rate, which is a linear component of the growth rate). Note that the only difference is the sign of the factor: a negative selection coefficient corresponds to a positive selective disadvantage. From Eqs. (10,11) the slope of the logarithmic plot of the mutant to wild type ratio at time t can be derived as r(t)s. Importantly, this measure reflects the absolute, rather than the relative fitness difference between the two variants. Under better conditions for growth, both variants can grow faster (higher r) and the absolute fitness difference estimated by the slope of the logarithmic plot will also be greater. This implies that estimates obtained under different growth conditions cannot be compared with each other. The selection coefficient, s, which gives the relative fitness difference, eliminates all factors that affect the replication of both variants and thus enables a better comparison between data. To calculate the selection coefficient, s, from the absolute fitness difference, one needs an estimate for the replication rate, r, under the given experimental conditions. Since this is usually not available, many researchers have simply divided the absolute fitness difference by the generation time of the virus to obtain an estimate for the selection coefficient [131, 132, 135, 136, 138]. However, the reciprocal of the generation time equals the replication rate only if the populations are at steady state, which certainly does not hold for the standard growth competition assays in vitro. Moreover, the growth rate might change over the duration of a single experiment, e.g. due to the progressing depletion of susceptible target cells in the culture. Fluctuations in the growth rate can also occur in vivo. In such a case, the logarithmic plot is no longer linear, and a simple division by the growth rate cannot solve the problem (Fig. (4b)). Denoting the log mutant to wild type ratio by h = ln(M/W) and the log wild type virus load by w = ln(W), one Fig. (4). The estimation of fitness from competition experiments. The logarithm of the ratio of mutant to wild type virus is monitored over time. In the conventional method (A), the fitness difference between the competing variants is estimated as the slope of the line fitted to the data points by linear regression. However, the expected change in the log ratio of mutant to wild type is linear only if the basic growth rate of the two variants is constant over the duration of the experiment (B). If the growth conditions that affect both variants change during the experiment, simple linear regression cannot provide an appropriate fit to the data. Mathematical Approaches in the Study of Viral Kinetics can derive the following relation between the selection coefficient and the values of h and w at the time-points t = 0 and t = T [141]: h(T) - h(0) = s (w(T) - w(0) + δ T). The advantage of this equation is that it allows one to estimate the selection coefficient without explicit knowledge of the replication rate, and thus circumvents the problem of a timeand patient-dependent replication rate. A shortcoming of the method is that it takes only two data points into account. However, this problem can be circumvented by a non-linear estimation method, which also provides confidence intervals for the estimated selection coefficient [140]. The data needed are the total virus load and the fraction of mutant virus at each time point, and the death rate of virus-producing cells. An implementation of the method is freely accessible at http://www.eco.ethz.ch/fitness.html. 8. CONCLUDING REMARKS We have attempted to give an overview of the most important applications of mathematical modelling in drug therapy and resistance, within the field of HIV-infection. Beyond discussing particular problems our intention was also to give an introduction to the methodology of modelling. We have shown how the complexity of the biological system must first be reduced to the bare bones, which can then be fleshed out according to the varying demands of particular problems. We have demonstrated the power and the limitations of the basic, or “standard” model of viral dynamics. The first extension was required to account for stable low viral steady states. We have concluded that the moderately reduced virus loads during non-suppressive therapy and in patients infected with multidrug-resistant virus can only be explained in the models if we implement the virus induced killing of uninfected cells or a cytotoxic response that is induced proportional to the level of infected cells. This “requirement” of the model implies that at least one of these two processes are relevant in the setting of the steady state, i.e. of the clinical status of patients. Similarly, the stable maintenance of virus loads below detection level is only possible in the models if we extend the basic scheme with a drug sanctuary compartment. Since such a compartment has not been identified yet, we are inclined to use the conclusion obtained by modelling to argue against the existence, or at least the relevance, of drug sanctuaries. The second extension of the model was required to keep track of drug resistant and sensitive virus variants separately for the study of drug resistance. Quantitative modelling has demonstrated that the resistant mutants that emerge during suppressive therapy are more likely to arise by the selection of pre-existing mutants than by de novo generation during therapy. However, the reliable suppression of viremia in drug naïve patients initiating therapy, which is typically maintained for months if not years, indicates that there are typically no pre-existing mutants that could grow in the face of current combination treatments. The modelling results then suggest that resistant mutants should be even less likely to appear during treatment. In the light of these predictions, the prevalence of eventual treatment failure due to emerging multidrug resistance is puzzling. Combining the modelling results with common sense we conclude that the reason for Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4 341 emerging resistance probably lies in temporal (and perhaps spatial) troughs in drug concentrations, which relax suppression and allow for new rounds of replication. It is of vital importance to elucidate the causes of such drops in the availability of drugs. A probable candidate in many cases is non-adherence, which therefore comes at a very high price: non-adherent patients can accumulate resistance to consecutive drug regimens, eventually developing superresistant viruses. Non-adherence to the first regimen is probably a strong predictor of non-adherence to later therapy options. 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