Gene therapy of T helper cells in HIV infection: Mathematical model of the criteria for clinical effect

Bulletin of Mathematial

Biology, Vol. 59, No. 4, pp. 725-745, Elsevier 0 1997 Society

for Mathematical 0092-8240/97

1997

Science

Inc.

Biology

$17.00 + 0.00

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GENE THERAPY OF T HELPER CELLS IN HIV INFECTION: MATHEMATICAL MODEL OF THE CRITERIA FOR CLINICAL EFFECT OLE LUND, OLE S0GAARD LUND, GREGERS GRAM, SUSANNE DAM NIELSEN, KRISTIAN SCHP)NNING, JENS OLE NIELSEN and JOHN-ERIK STIG HANSEN* Laboratory for Infectious Diseases, Hvidovre Hospital, University of Copenhagen, Hvidovre, DK-2650 Denmark (Email:

[email protected])

ERIK MOSEKILDE Department of Physics, Technical University of Denmark, Lyngby, DK-2800 Denmark (Email:

[email protected])

The paper presents a mathematical analysis of the criteria for gene therapy of T helper cells to have a clinical effect on HIV infection. The analysis indicates that for such a therapy to be successful, it must protect the transduced cells against HIV-induced death. The transduced cells will not survive as a population if the gene therapy only blocks the spread of virus from transduced cells that become infected. The analysis also suggests that the degree of protection against disease-related cell death provided by the gene therapy is more important than the fraction of cells that is initially transduced. If only a small fraction of the cells can be transduced, transduction of T helper cells and transduction of haematopoietic progenitor cells will result in the same steady-state level of transduced T helper cells. For gene therapy to be efficient against HIV infection, our analysis suggests that a 100% protection against viral escape must be obtained. The study also suggests that a gene therapy against HIV infection should be designed to give the transduced cells a partial but not necessarily total protection against HIV-induced cell death, and to avoid the production of viral mutants insensitive to the gene therapy. 0 1997 Society for Mathematical Biology

1. Introduction. At present, no curative treatment of HIV-infected persons exists. Various drugs can reduce the viral load for a period. Two weeks to one year after initiation of treatment, however, resistant HIV mutants can be detected (Larder et al., 1989; Pauza and Streblow, 1995). Gene therapy is a promising alternative approach to the treatment of HIV infection (Yu et al., 1994). This technique is based on inserting an anti-viral gene in some of the patients

cells. The first trials with gene therapy

against

*Author to whom correspondence should be addressed. 725

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HIV have been evaluated recently (Woffendin et al., 1996), and several trials are underway (Crystal, 1995). HIV mainly infects T helper cells, and the depletion of the population of these cells is a hallmark of the HIV infection. As reviewed earlier (Yu et al., 19941, gene therapy approaches against HIV infection can be divided into two types: 1) intracellular immunization, and 2) immunotherapy. The first category, intracellular immunization (a term proposed by Baltimore, 19881, aims at genetically manipulating viral target cells in such a way that they become protected against de novo infection or are rendered unable to produce new virions following infection. The principle of intracellular immunization against HIV-l has been demonstrated in cell lines by ribozyme- or anti-sense constructs targeted against the genome of HIV-l (Sarver et al., 1990; Yamada et al., 1994; Sczakiel and Pawlita, 1991; Chatterjee et al., 1992; Cohli et al., 1994), and by the expression of mutated HIV-l proteins capable of interfering with the production of new viruses in a negative transdominant way (Malim et al., 1989; Hope et al., 1992; Trono et al., 1989). The other main group of gene therapy approaches, immunotherapy, aims at potentiating the immune system in order to repress viral replication. This group includes such diverse strategies as DNA vaccination (Wang et al., 1993) and infusion of CD8 cells genetically manipulated to target gp120 expressing cells by a chimeric T-cell receptor-associated molecule containing the extracellular domain of CD4 (Roberts et al., 1994). In the present study, we have confined the analysis to approaches of intracellular immunization. In the clinical trials in HIV-infected persons, the objectives have been to induce an anti-viral, immune-stimulating or cell-killing mechanism (Yu et al., 1994; Crystal, 1995; Walker et al., 1996; Nabel et al., 1994). Other trials have investigated the survival of gene-marked T helper cells in HIV-infected persons (Walker et al., 1993). Mathematical models have been used to investigate the dynamics of the T helper cell population (Sidorov and Romanyukha, 1993; De Boer and Perelson, 1994; De Boer and Perelson, 1995) and its interaction with HIV in vivo and in vitro (Layne et al., 1989; Dimitrov et al., 1993; Lund et al., 1995). A variety of studies have illustrated how HIV can influence the immune response against other pathogens (Cooper, 1986; Reibnegger et al., 1987; Reibnegger et al., 1989; Anderson and May, 1989; McLean and Kirkwood, 1990; McLean and Nowak, 1992), and how the dynamics of an immune response against multiple epitopes might lead to complex dynamical patterns in the concentrations of cytotoxic T lymphocytes and HIV (Nowak et al., 1995b; Nowak and McMichael, 1995). Mathematical models have also been proposed to explain the progression of the disease (Nowak et al., 1990; Hraba et al., 1990; Nowak et al., 1991; Nowak and May, 1991; Nowak, 1992; Nowak and May, 1993; De Boer and Boerlijst, 1994; Schen-

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zle, 1994). Recently, mathematical models have successfully described the dynamics of T helper cells and HIV after treatment with anti-HIV drugs (McLean et al., 1991; McLean and Nowak, 1992; Frost and McLean, 1994; De Boer and Boucher, 1996). In this study, we shall focus on the dynamics of gene-transduced cells. We analyze the effects of transduction of T helper cells or haematopoietic progenitor cells with vectors aimed at increasing the lifespan of the T helper cells or ensuring that, if infected, the transduced cells do not produce any infectious progeny virus. We determine the parameters which should be measured in gene-marking trials, and establish the criteria for a positive outcome of therapeutic trials.

2. T Helper Cell Dynamics. T helper cells are produced by a combination of production via proliferation of haematopoietic progenitor cells and peripheral proliferation of mature T helper cells (Sprent and Tough, 1994; Tough and Sprent, 1994; Mackall et al., 1993; Mackall et al., 1995). Approximately 2% of the lymphocytes in humans are in the blood, the remainder being in other compartments such as the lymph nodes (Trepel, 1974). In humans, lymphocytes stay on the average approximately 30 min in the blood (Schick et al., 1975a; Schick et al., 1975b). For rats, it has been found that they circulate between the blood and other compartments one-two times per day (Smith and Ford, 1983). After reinfusion of transduced cells, 98% of the cells are therefore expected to disappear rather quickly from the bloodstream. If the drop in the concentration of transduced cells is significantly larger than this, it might indicate that the cells were not in a viable condition at the time of reinfusion, or that they were removed by an unspecific immune response. We consider a population of T helper cells represented by the normalized concentration Tt. The normalization expresses the cell concentration relative to that of a normal healthy person with a T helper cell count of lOOO/~l blood. The T helper cells are produced by a combination of proliferation of haematopoietic progenitor cells at a rate A, and by proliferation of peripheral T helper cells at a rate TtA,(T,). The dynamics of the normalized concentration of T helper cells Tt can be described by

dT, -g- = A, + T,A,(T,)- s,T,

(1)

where S, is the per cell death rate for T helper cells. Estimates for the half-life ln2/6, for T helper cells in non-HIV-infected persons vary markedly. From studies of the decay rate of T cells with chromosome damage in patients having received radiotherapy, the lifespan of T helper

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cells has been estimated to be months to years (Buckton et al., 1967; Bogen, 1993; McLean and Nowak, 1992; McLean and Michie, 1995). Somewhat lower estimates are obtained from studies with labelled T cells in mice (Sprent and Basten, 1973; Tough and Sprent, 1994). In studies of the decay rate of T cells after the administration of hydroxyurea which kills cycling cells, the half-life of lymphocytes in mice has been estimated to be as short as two-three days (Freitas et al., 1986). These variations in the estimated lifespans may arise because the actual lifespans of T lymphocytes differ from man to mouse. It is also possible that variations in the experimental conditions may influence the measured proliferation and death rates. In this study, we have chosen to use an intermediate value for the natural death rate a,, for T helper cells in normal non-HIV-infected persons S, = S,, = O.Ol/day, which at steady state (7” = 1) corresponds to a total supply of A = A, + h,(l) = O.Ol/day. The fraction of T helper cells that is produced by the proliferation of haematopoietic progenitor cells is not known. It has been reported that the output from thymus is low in adult mice compared to young mice. Moreover, labelling of lymphocytes in adult mice newly formed with BrdU is only slightly slower in adult thymectomized mice (mice where the thymus has been removed) than in control mice (Tough and Sprent, 1994). After thymectomy, the concentration of T lymphocytes drops to 60% of the concentration in sham thymectomized mice. On the basis of estimates of the turnover rate of T cells and the output from thymus, it has been estimated that the thymic output can account for only 17% of the renewal of T cells in adult mice (Stutman, 1986). Here, we will assume that the fraction c of T helper cells produced from the bone marrow at steady state (Tt = 1) constitutes one fifth (c = 0.2) of the total production of T helper = O.O02/day and h,(l) = (lcells corresponding to A, = CA = 0.2 x 0.01 c)A = 0.8 X 0.01 = O.O08/day. A Michaelis-Menten type function is used to describe the proliferation rate of the T helper cells: h,(T,) = ((1 - c)a)/(b + Tt>. This functional form has been derived by De Boer and Perelson (De Boer and Perelson, 1995) based on the hypothesis that the different clones compete for antigen. Assuming that A,(T’) is n times higher at T’= 0, we have a = n&/(n - 1) and b = l/(n - 1). With IZ= 100 and S, = O.Ol/day, this gives a = 0.01 and b = 0.01. This choice of parameters corresponds to a lymphocyte proliferation rate of up to O.S/day when the number of T helper cells is very low. This proliferation rate is similar to that observed for haematopoietic progenitor cells (Tough and Sprent, 1994) and activated lymphocytes, and might be close to the maximal biologically possible proliferation rate for these cells. In the following, we first discuss which parameters one can obtain from gene-marking trials, and subsequently predict the effects of gene therapy.

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3. Gene Marking. Insertion of genes can be performed by applying the genes naked or encapsulated in liposomes (transfection), which can give rise to a transient expression of the gene. Alternatively, replication-deficient viruses (vectors) can be used to insert the gene (transduction). Transduction by retroviral vectors can give rise to a permanent expression as these vectors integrate into the host cell genome (Crystal, 1995). The treatment can be performed by harvesting the relevant cells from the patient and transducing them EXuiuo before reinfusing the cells. Altematively, the gene preparation can be injected into the patient (in uivo). After en: uivo gene-marked cells have been reinfused into the patient, or they have been gene-marked in viuo, the cells are expected to quickly redistribute between the blood and other compartments of the body. To describe the temporal development of the concentration of the transduced T helper cells T’ and the non-transduced T helper cells T hereafter, we write dT z = (1 -+A,

+ TA,(T,) - 6,T

(2)

dT’ dt =fAc + T’XJT,) - S;T’. Here, f is the fraction of the haematopoietic progenitor cells that have been transduced. f = 0 thus corresponds to the marking of T helper cells. Tt = T + T’ is the total normalized concentration of cells. This expression for Tt ensures that the steady-state T helper cell concentration before and after gene marking is equal if the gene marking influences neither the proliferation nor the death rate of the cells. Marked haematopoietic progenitor cells will give rise to new T helper cells at a rate fA,, assuming that the gene-marking protocol does not alter the number of haematopoietic progenitor cells in the patient significantly. A population of marked T helper cells will differ from the non-marked cells in that they are not produced by the haematopoietic progenitor cells (f = 0), and that they might have altered proliferation and death rates. The observed decay rate of transduced T helper cells in gene-marking trials 8, will equal &CT,) - 8;. Thus, the decay rate is a function of the effect of the inserted gene on their proliferation and death rates, as well as the fraction of T helper cells produced by peripheral proliferation. In T helper cell gene-marking trials, it will not be possible to distinguish between these factors. Fortunately, it might not be important to do so. If the death rate of the T helper cells in a gene therapy trial can be reduced by S,, the transduced cells will survive as a population, irrespectively of the mechanisms that determine 8,. In HIV-infected persons, the per-cell death rate is increased due to the killing of T helper cells by HIV. Based on the observed increase in T helper

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cell counts after anti-viral chemotherapy, the death rate for T helper cells in HIV-infected persons has been estimated to be O.OS/day (Ho et al., 1995a; Wei et al., 1995). Assuming that the death rate for T helper cells 6,, in the absence of HIV equals O.Ol/day, a gene therapy might thus at most reduce the death rate of the transduced cells by O.O4/day. It must be noted, however, that the death rate of T helper cells in HIV-infected persons might be lower than the above-mentioned O.O5/day since the observed increase in peripheral blood T helper cell counts in HIV patients after anti-HIV treatment could be due to redistribution of these cells (Moiser, 1995; Sprent and Tough, 1995; Dimitrov and Martin, 1995). Many gene therapies aimed at decreasing HIV-associated cell death are designed to block HIV infection of T helper cells. It is important to notice that if the T helper cells in HIV-infected persons are killed mainly by mechanisms other than direct HIV infection (e.g. autoimmunity), a gene therapy of the T helper cells will, of course, have to be designed to block these mechanisms.

4. Gene Therapy. The model described by equations (2) and (3) was used to simulate the steady-state concentrations of T helper cells after transduction of haematopoietic progenitor cells (Fig. 1). In these calculations, the death rate of the non-transduced T cells was set to O.OS/day to simulate an increased mortality of these cells due to, for example, an adenosine deaminase (ADA) deficiency or HIV infection. In our model, this death rate corresponds to a steady-state T helper cell count of approximately 200/~1 blood. We have simulated the effect of gene therapy by setting the death rate of the transduced cells 6; to a lower value than the death rate of the non-transduced cells, and have assumed that the transduced and the non-transduced cells have the same proliferation rate (A, = A’,). If a small fraction of the haematopoietic progenitor cells is transduced with a gene that reduces the per-cell death rate of their T helper cell progeny to O.Ol/day, our model predicts that the derived T helper cells will proliferate and cause the T helper cell count to be almost normalized at a level of ((1 - c>a - 6#/S;. = 1 - c = 0.8. According to our model, this level will depend mainly on the proportion c of the T cells which in normal healthy persons are produced by haematopoietic progenitor cells. If the transduced cells have a death rate of O.O2/day, i.e. twice that of T helper cells in normal healthy individuals, the model predicts that the T helper cell level will be stabilized at a level approximately equal to 40% of that found in normal healthy individuals. Much attention has been given to increasing the fraction of cells that become transduced. Our calculations show that the survival advantage of the T helper cells, that the transduced haematopoietic progenitor cells give rise to, might be more important than the fraction of the haematopoietic progenitor cells that become transduced. However, the fraction of haema-

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topoietic progenitor cells transduced is important for the time it takes before the T cell count reaches the steady-state level (data not shown). Figure 2a shows simulations of gene therapy of T helper cells. The normalized steady-state T helper cell concentration after gene therapy is plotted as a function of the concentration of these cells before gene therapy. Low T helper cell concentrations associated with disease are simulated by increasing the death rate 6, of the non-transduced cells. In these simulations, we have assumed that the transduced cells can have a lower death rate than the non-transduced cells, but have a proliferation rate equal to that of the non-transduced cells. The pre/post gene therapy T helper cell levels are shown for different values of the percentage of protection against disease-associated cell death. The percentage protection defined as ((8, - 6&)/C&- - a,,)) X 100% might be estimated by in vitro studies. It equals 0% if the transduced cells have the same death rate as the non-transduced cells (8, = Sk>, and 100% if the transduced cells have the same death rate as the T helper cells in normal healthy persons (6, = S,,). The results shown in Fig. 2a are also representative for a gene therapy where a small fraction (f c 1) of the haematopoietic progenitor cells have been transduced as the production of gene-modified T helper cells from the

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MATHEMATICAL MODEL OF GENE THERAPY FOR HIV INFECTION

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haematopoietic progenitor cells f., will then be negligible. Figure 2b shows a similar simulation, but with 100% of the haematopoietic progenitor cells being transduced. This results in somewhat (approximately l/(1 - cl = 1.25 times) higher T helper cell concentrations than a gene therapy directed against T helper cells. 5. Gene Therapy of HIV Infection.

To simulate the effect of gene therapy of HIV infection, we use a simple Lotka-Volterra like model. Similar models have previously been proposed for describing HIV infection (McLean and Kirkwood, 19901, and have been capable of explaining the dynamics of the virus and T helper cell populations after anti-HIV drug therapy (McLean et al., 1991; McLean and Nowak, 1992; Frost and McLean, 1994; De Boer and Boucher, 1996). dT z = (1 -f)h,

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T and I are the uninfected and infected non-transduced T cells, respectively. The primes signify transduced populations, and Tt = T + T’ is the total concentration of uninfected cells. Since HIV infection during the silent period after the initial peak in HIV and before the onset of AIDS is most active in the lymph nodes (Pantaleo et al., 19931, we interpret the equations as describing the infection process in this compartment. To compare the results of the model with data obtained from blood samples, we need to assume that the distribution between these two compartments is constant. Since the lifespan of free virus (6 hr) is much shorter than the lifespan of infected cells (2 days), the virus population will reach a quasi-steady state approximately 48/6 = 8 times faster than the population of infected cells (Perelson et al., 1996). It is therefore a good approximation to assume that the number of free virions is proportional to the number of infected cells (De Boer and Boucher, 19%). In a model based on these lifespans, it is not necessary to model the virus explicitly, even if the number of new infections are thought to be proportional to the number of virions since it will be a good approximation to assume that the infection rate is proportional to the number of infected cells.

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The parameter s is the susceptibility to infection of the transduced cells relative to the non-transduced cells. It has been assumed that the transduced cells have a gene inserted to ensure that, upon infection, they do not produce any infectious progeny virus. If this is not the case, terms of the form - /3T”, p TI’, -s/I T’I’ and sp T’I’ should be added to the right-hand side of equations (4), (5), (6), and (7), respectively. In the absence of transduced cells, the concentrations of non-infected and infected cells at steady state will attain the values T = a,,//3 and I = (A, + T&(T) - G,T)/PT, respectively. In the above model, there is no progression of the HIV disease. During the approximately ten years of HIV infection, before AIDS develops, the concentration of T helper cells in the blood decreases slowly. As a first approximation, we will assume that the T helper cell count in the absence of gene therapy can be assumed to be constant on a short time scale (= 1 year). In our model, a small decrease of the infectiousness of the virus p will cause a decrease in the number of infected cells, but the number of infected cells will be stabilized at a lower level as the number of target cells T increases (McLean and Nowak, 1992; De Boer and Boucher, 1996). For the virus to be eradicated, dI/dt must be negative even when the number of uninfected T cells is increased to the level found in normal healthy persons, i.e., Tt = 1. In our model, this corresponds to p < 8,. This criterion can also be formulated as: the number of times N, that the infectivity must be decreased in order to eradicate the infection in a given patient equals the T helper cell count before HIV infection divided by the present T helper cell count. If the concentration of non-transduced T cells is decreased from its steady-state value, dI/dt will be negative (De Boer and Boucher, 1996). This can happen if the transduced cells compete strongly for the “space” in the immune system. If the transduced T cells have a gene that provides them with some protection against HIV infection, they will have a survival advantage as long as HIV is present. In the absence of HIV, the transduced cells do not have a survival advantage, and hence decay until they constitute a fraction of the T cells corresponding to the fraction of the haematopoietic progenitor cells f that has been transduced. The model described above was used to simulate the temporal development in the normalized concentrations of T helper cells after gene therapy (Fig. 3). In the se calculations, 8, = O.S/day, corresponding to the infected cells having an average lifespan of approximately 2 days (Perelson et al., 1996), p has been set to 2.5, which results in a steady-state normalized concentration of uninfected T helper cells equal to 0.2. In a simulation where the transduced cells are infected as easily as the non-transduced cells (S = l), the former will not survive as a population (Fig. 3a). After gene therapy, a transient increase in the T helper cell count and a drop in viral load are seen.

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Figure 3. (Continued). (c) A large fraction (80%, corresponding to f- 0.8) of the haematopoietic progenitor cells, but no T helper cells were transduced with a gene giving the T helper cells a survival advantage (s = 0.5). (d) The renewal of the T helper cells is dependent on peripheral proliferation only (modelled by setting c to zero). The T helper cells have been transduced with a gene that gives them a survival advantage (S = 0.5). At time t = 0, 10% of the T helper cells, but no haematopoietic progenitor cells were transduced. In these simulations, p = 2.5, S, = 0.01, 6, = 0.5, c = 0.2 and a = b = 0.01.

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If the transduced cells have a survival advantage, they will cause a stable increase in the number of T helper cells (Fig. 3b). However, in our model, the virus will be eradicated only if either a large fraction (80%) of the haematopoietic progenitor cells are transduced (Fig. 3c), or the renewal of the T helper cells is dependent on peripheral proliferation only (Fig. 3d, where c is set to zero>. These simulations indicate that a transduced population of T helper cells will survive as a population, and will give rise to an increased T helper cell count only if the inserted gene provides protection against infection (Fig. 3a vs Fig. 3b). However, the virus might still persist due to the production of non-transduced T helper cells from the haematopoietic progenitor cells (Fig. 3b vs Fig. 3c and 3d). Gene therapies aiming at reducing the production of virus from infected cells by inserting a gene that codes for a transdominant protein might decrease the death rate of the infected cells 8,. In a recent study, it has been shown that expression of an inhibitory rev protein prolongs the survival of T helper cells in HIV-infected persons (Woffendin et al., 1996). The effect of such a strategy on the T helper cell count will depend on how much the lifespans of the uninfected and infected transduced cells are affected by an immune response against the transdominant protein. In the calculations above, we have not included the effect of HIV mutants which are or become resistant to the effects of the inserted gene. To simulate this, an equation representing a population of cells producing virus that is insensitive to the transduced genes dl,/dt = P(T + T’)I, - &I, was added to the set of equations (4)-(71, and terms of the form - PTI, and - PT’I, were added to the right-hand side of equations (4) and (61, respectively. The simulation (Fig. 4) shows that, at first, the transduced cells increase in numbers. This creates a selective pressure on HIV, causing a shift from the gene-therapy-sensitive strain to the gene-therapy-resistant strain after approximately 100 days. The steady-state level of infected cells after gene therapy equals the level before, the only difference being that the genetherapy-sensitive virus has been replaced by a gene-therapy non-sensitive one. This, in turn, means that the transduced cells no longer have a selective advantage, and their numbers will therefore start to decrease. The predicted changes in viral load and T helper cell count are smaller than what is observed using chemotherapy against HIV (Ho et al., 1995a; Wei et al., 1995). This is due to the small fraction of cells that is transduced initially. The simulation underscores that great effort must be made to avoid viral escape for changes in T helper cell counts and viral load to be observed. It also shows that the development in the percentage of resistant virus might be the most sensitive measure of anti-viral effect. Progression can be included in models of HIV infection in a number of ways (De Boer and Perelson, 1996). One can assume, for example, that the

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Figure 4. Temporal development of the normalized concentrations of transduced and non-transduced, uninfected and infected T helper cells. In this simulation, the transduced T helper cells cannot be infected by the gene-therapy-sensitive strain (s = 0.0); no haematopoietic progenitor cells have been transduced (f = 0). Initially, 1% of the infected cells produce virus which is non-sensitive to the inserted gene. Other parameters are as described in Fig. 3.

infectivity of the virus increases (increasing p), or that the immune response against HIV decreases (increasing p or decreasing 8,). It is also possible that the cells begin to proliferate slower because they are exhausted, due to the gradual destruction of the lymph nodes (decreasing A,), or because of a lower production of T cells associated with the gradual destruction of the thymus (decreasing A,). To study the effect of including progression of the disease in the model, we added an equation dp/dt =pI, describing the development of the infectiousness of the virus to the set of equations (4)-(7). It is here assumed that the infectiousness p increases at a rate proportional to the number of infected cells. To simulate the decline in T helper cells seen in HIVinfected persons, we chose k(O) = 0.7 and p = 0.04” since this leads to a 40% drop in the number of T helper cells during the initial infection, and a further drop to 20% of the initial level during the first 10 years of infection as seen in HIV-infected persons. Simulations with this set of equations show that a gene therapy similar to that of Fig. 3b will have a relatively long lasting, but not permanent, effect if progression is included in the model (Fig. 5). It has been found that the clearance rate of HIV and HIV-infected cells after chemotherapy was not significantly correlated with the T helper cell

MATHEMATICAL

MODEL OF GENE! THERAPY FOR HIV INFECTION

739

0.8 E 0.7 0.8 I 0.5 0.4 / ,’

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Figure 5. Temporal development of the normalized concentrations of T helper cells with and without gene therapy. In the simulation with gene therapy, the transduced T helper cells have a survival advantage (S = 0.5). The progression rate constant p = 0.04. Initially, at time t = 0, the infectiousness variable B = 0.7, the normalized concentration of uninfected cells T(O) = 1.0 and the normalized concentration of infected cells I(O) = 4.0 x lo-“. 1% of the T helper cells, but no haematopoietic progenitor cells were transduced at time t = 3000 days. Other parameters are as described in Fig. 3.

count (Ho et al., 1995a). In the simulations above, we have therefore only modelled the negative effects of an increasing T helper cell count, namely that the number of target cells which HIV can infect increases. However, a persistent increase in the T helper cell count might also cause an increase in the ability of the immune system to clear HIV. The model thus provides a minimum estimate of the effect of gene therapy. 6. Discussion. Preliminary results indicate that when transduced T cells are infused into HIV-positive persons, their number remain fairly constant for the initial 12 weeks in most persons. Hereafter, different patterns of persistence and decline are seen (R. Walker, personal communication). In our terminology, this corresponds to the observed decay rate of transduced T helper cells S, being approximately zero initially. This indicates that the renewal of T helper cells depends mainly on peripheral proliferation, and that the inserted gene does not alter the proliferation or death rate of the transduced cells. In patients treated with the new and efficient anti-HIV drugs, the half-life of the peripheral blood mononuclear cells containing a nonresis-

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tant provirus has been reported to be as long as SO-100 days (Wei et al., 1995). Since the turnover rate of T helper cells in the same studies was reported to be as high as S%/day, this indicates that these cells proliferated at a rate of approximately 4%/day. This indication, that the T cells in HIV-infected persons are mainly renewed by peripheral proliferation, is also supported by the preliminary results that the cells after anti-HIV drug therapy display a number of activation markers (Ho et aE., 1995b). Recently, the results of the first human trials with gene therapy against the fatal genetic disorder adenosine deaminase (ADA) deficiency, which results in severe combined immunodeficiency @CID), were published (Bordignon et al., 1995; Blaese et al., 1995). It will be useful to compare the results of these studies with studies of gene marking of cells in HIV-infected persons. Previous studies (Bordignon et al., 1995) studied the effect of transducing an ADA gene with two different vectors, one into haematopoietic progenitor cells and another into lymphocytes. Initially, most of the lymphocytes with the ADA gene were derived from the transduced lymphocytes. One year after discontinuation of treatment, however, an increasing fraction of the transduced cells was derived from haematopoietic progenitor cells. As the regeneration of lymphocytes in children is known to be much more dependent on thymus than in adults (Mackall et al., 1995), and as the ADA deficiency might affect the lifespan of the haematopoietic progenitor cells as well as that of the T cells (Bordignon et al., 1995), this development might not occur in HIV-infected persons. The hypothesis that the transduced haematopoietic progenitor cells in the ADA-deficient patients have a survival advantage is supported by the fact that, 16 months after discontinuation of the treatment, as many as 17-25% of the haematopoietic progenitor cells were transduced. Kohn et al. (1995), who studied gene therapy of CD34 + cells in neonates with an ADA deficiency, also found a higher frequency of transduced cells among the haematopoietic progenitor cells than among the peripheral blood T cells. Another study (Blaese et al., 1995) showed the transduction of the ADA gene into lymphocytes only. Their data showed that the cells in one of their two patients continued to express the inserted gene with a frequency of almost one copy per cell, even 545 days after the last treatment. This indicates that the transduced cells have had a survival advantage, and that, in the absence of competition from T cells derived from transduced haematopoietic progenitor cells, they could repopulate the immune system. In the present study, we have identified a number of parameters that can be measured in gene-marking trials, and have discussed their significance for the outcome of therapeutic trials. The most significant parameters are: 1) the fraction of T helper cells that are produced by the haematopoietic progenitor cells, and 2) the death rate of the transduced cells. These parameters can be estimated by fitting the data from gene-marking studies

MATHEMATICAL

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to the solutions of equations (2) and (3). The inactivation rate for the inserted gene might also be an important parameter. For the transduced cells to survive as a population in gene therapy trials, their death rate has to be decreased corresponding to the observed decay rate of these cells in a gene-marking trial. We have used our model to predict the T helper cell count at steady state after gene therapy as a function of the T helper cell count before gene therapy, and the percentage protection against disease-related cell death that the gene therapy provides. We found that the percentage protection was more important for the resulting T helper cell concentration than the initial number of cells transduced. According to our model, transducing all haematopoietic progenitor cells in a patient will only give rise to an approximately 1.25 times higher steady-state T helper cell count than transduction of a low fraction of the haematopoietic progenitor cells or of the T helper cells. The analysis of our model indicates that an afferent effector (inhibiting infection of transduced cells) does not have to protect the cells 100% in order to give the transduced cells a survival advantage. This is important since it might not be possible to develop a non-immunogenic effector that has such a high efficiency. Fragments of. endogenously produced proteins can be presented to the immune system by the MHC I molecule. Using genes that code for a protein to protect the cells against infection may therefore not be feasible because the transduced cells might be considered foreign by the immune system. A survival advantage for the transduced cells might be obtained by inserting an effector gene that codes for an RNA molecule such as an anti-HIV ribozyme (Yamada et al., 1994) or an antisense molecule directed against HIV genes. RNA effecters might not be as efficient as protein effecters, and protein effecters might therefore be used to inhibit the spread of HIV from cells already infected with HIV. An afferent RNA effector could be used in combination with an efferent effector (inhibiting the spread of virus from infected, transduced cells) designed to avoid the selection of gene therapy resistant virus. The fast mutation rate of HIV is a common obstacle to all attempts to treat HIV infection. Since the HIV reverse transcriptase has an error rate of approximately lop4 errors/nucleotide, and the size of the HIV genome is approximately 10 kb, HIV makes approximately one error per replication cycle (Roberts et al., 1988; Preston et al., 1988). Recent estimates suggest that approximately 1 - 10 x lo9 HIV particles are produced per day (Ho et al., 1995a; Wei et al., 1995; Nowak et al., 1995a; Perelson et al., 1996) so any single mutant is probably produced many times every day (Coffin, 1995). For this reason, a combination of several drugs is seen as one of the promising approaches to inhibit HIV infection (Glanz, 1996). Preliminary results from ongoing trials show that the viral load in some cases can be

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reduced to undetectable levels for as long as 24 weeks (Glanz, 1996). Likewise, it is thought that the problems concerning mutants that become resistant to the gene therapy might be minimized by directing the therapy against more than one target, or against targets where HIV cannot mutate without losing infectivity (Yu et al., 1994). If resistant mutants arise, gene therapy will have a lower effect on the T helper cell levels and on the viral load than a chemotherapy with a comparable protection against viral escape. This is because only a small fraction of the T helper cells can be transduced initially. Viral clearance might require that a major fraction of the haematopoietic progenitor cells is transduced. It might therefore not be possible, via gene therapy or by other means, to clear an HIV infection entirely. However, even a transient decrease in viral load might have a beneficial effect, and if the viral load is kept low for an extended period, it might give the immune system time to regenerate. The authors wish to thank Rob J. de Boer and Alan S. Perelson for sharing their unpublished manuscripts. This work was supported by the Danish 1991 Pharmacy Foundation, the Danish National Research Foundation, the Danish Science Research Council, the Danish Medical Research Council, the John and Birte Meyer Foundation and the Foundation of 17.12.81.

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Received 10 September 1996 Revised version accepted 6 February 1997