- Email: [email protected]
Use of mathematical model for effectivity estimation of HIV infection therapies
il NOgrH- ~
U s e of a M a t h e m a t i c a l M o d e l for Effectivity E s t i m a t i o n of H I V Infection T h e r a p i e s Tom'£.~ H r a b a
Institute of Molecular Genetics Academy of Sciences of the Czech Republic 16637 Prague, Czech Republic and J a r o s l a v Dole~al
Institute of Information Theory and Automation Academy of Sciences of the Czech Republic 18208 Prague, Czech Republic Transmitted by F. E. Udwadia
ABSTRACT The previously developed mathematical model sinmlates the CD4 + lymphocyte dynamics in HIV infection considerably well. As the number of these cells is a good indicator of the infection progression, it was used to evaluate the effectivity of different therapeutic interventions. For chemotherapy simulation, both permanent and telnporary AZT administration was considered and the induced return of the CD4 + lymphocyte counts was analyzed. Similar analysis was performed for aetive and passive immunotherapy. The model offers also the possibility of simulating the CD4 + dynamics after depletion of CD8 + lymphocytes by antibodies. Even one simulated administration of anti-CD8 antibodies increases the CD4 + lymphocyte counts and prolongs the patient survival. HoWever, if cells involved in protective inmmnity are assumed to belong to the CD8 + category, anti-CD8 antibodies accelerate the decrease of CD4 + (:ells and thus shorten the patient survival.
Presented at tile Seventh Workshop oll Dynamics and Control, Ulm, Germany, July 17-20, 1994.
APPLIED MATHEMATICS AND COMPUTATION 78:153-161 (1996) © Elsevier Science Inc., 1996 655 Avenue of the America~, New York, NY 10010
0096-3003/96/$15.00 PII S0096-3003(96)00005-8
154
T. tIRABA AND J. DOLEZAL
INTRODUCTION The CD4 + lymphocyte depletion seems to be the major immune system lesion in HIV infection, and the number of these lymphocytes is a good prognostic indicator of the infection dynamics. As the developed mathematical model [1, 2] simulates the dynamics of CD4 + lymphocytes in HIVinfected individuals successfully [3, 4], it can also be used for simulation of different therapeutic intervention effects. In this paper we are presenting simulation results of zinovudin chemotherapy, active and passive anti-HIV immunotherapy, and anti-CD8 antibody administration. The latest treatment was suggested [5], because it seemed highly probable that a homeostatic mechanism regulating T-cell numbers did not discriminate between the loss of CD4 + or CD8 + lymphocytes. It was therefore assumed that activation of this mechanism by CD4 + lymphocyte depletion might be inhibited in HIV-infected individuals by increased numbers of CD8 + cells. In consequence, it was suggested to lower the CD8 + numbers by administering anti-CD8 antibodies and, in this way, to increase CD4 + lymphocyte level by a stronger influx of new T cells. The respective model could simulate the effect of anti-CD8 antibody administration, because it incorporated such a homeostatic mechanism regulating T-cell level [3]. MATHEMATICAL MODEL The model considers immature and mature CD4 + ( f i and P cells) and CD8 + lymphoeytes ( R and R cells). As normal values of R cells equal approximately two-thirds of those of P cells, it is assumed that also normal values correspond in a similar way to two-thirds of P cells. The sizes of these cell compartments at time t are described by Eqs. (1)-(4). The amount of HIV products at time t is given by Eq. (5). Finally, Eq. (6) gives the number of cytotoxic T cells specific for HIV ( C cells) at time t. In the model used in this article and in the preceding ones simulating therapeutic interventions in HIV infection, these cells both limit proliferation of HIV, as indicated in Eq. (5), and effect destruction of CD4 + cells presenting HIV products according to Eqs. (1)-(2): dP(t)
dt
Ip + f [ ( P o - P( t)) + ( R o - R ( t ) ) ] - Y p P ( t ) -
dP(t) dt
-Cpa( t ) C ( t) P( t),
P(O) = Po
Y p P ( t ) - "rpP(t) - c p a ( t ) C ( t ) P ( t ) ,
(1) P ( O ) = Po (2)
Modelling of HIV Infection Therapies
155
dR(t)
dt
~{Ip + f[( Po - P( t)) + ( R o - R ( t ) ) ] } -
dR(t) dt
¥~-R( t)
=
-
(r R + pR) R( t),
da( t) dt = a(t)[ KO - ~" - y C ( t ) ] ,
dO(t) dt
(3)
R(O)
= P0
(4)
(5)
a(O) = a o
I P(t) t ~
A(t) a ( t ) [ e I c + a C ( t ) ] ~ , ~ - - o ) - ( r c - p c )
C(t)'
c(o) = q,.
(6)
Here Ie is the influx of P cells, i.e., the rate (all rates are in days -1) of differentiation of P cells from stem cells, Yr is the rate of maturation of .P cells into P cells, and rp is the rate of natural death of P cells; the quantities YR and r n are defined in a fully analogical way. Further, f is the amplifying coefficient of the linear feedback effect of P a n d / o r R cell decrease on the influx of _P and R cells at time t. The quantity Ep a(t) C(t) is the rate of elimination of P cells due to the amount of HIV products a(t) and the number of cytotoxic T (:ells C-~t) at time t. Analogously, c~,a(t)C(t) is the rate of elimination of P cells. The value % is the function of the infectious dose of HIV, 0 characterizes the growth rate of HIV, and y is the rate of inactivation of HIV products mediated by cytotoxic C cells. The maturation of these cells from their precursors is assumed to be dependent on the encounter with HIV products and the effect of HIV-specific helper T cells. I c is the influx of C cell precursors, ~ their maturation rate, a the proliferation rate of C cells under the antigenic stinmlation by HIV products and helper T-cell influence, and r c their natural death rate. Helper T-cell effect on maturation and proliferation of C cells is expressed by the ratio P(t)/Po; the coefficient u is introduced to characterize the intensity of this helper effect. Effects of theiapeutic interventions are described by the following parameters: K, chemotherapeutic inhibition factor of HIV proliferation; A, immune response enhancing factor increasing active anti-HIV immunity; ~', HIV elimination rate by passive immunization effected by administration by anti-HIV antibodies; Pt¢ and Pc elimination rates of CD8 + ( R and C cells, respectively) by anti-CD8 antibodies.
156
T. HRABA AND J. DOLEZAL
If not otherwise s t a t e d the model p a r a m e t e r s in s i m u l a t i o n runs are selected as follows: ~p = 0.2, Tp = 0.01, YR = 0.2, TR = 0.01, ~'c = 0.01, Ip = 1.0, I c = 0.2, P0 = 5.0, P,, = 100.0, R 0 = 3.33, R 0 = 66.7, C O = 0.0, % = 0.0005, f = 0.01, c~ = 0.1, e = 0.144, ~ / = 0.7, 0 = 0.02, v = 2.0. O n l y the m a t u r e CD4 + l y m p h o c y t e s ( P cells) are assulned to be susceptible to HIV products, i.e., ?e = 0.0, c r = 20.0. As a rule, p a r a m e t e r e is used for final a d j u s t m e n t of the respective s i n m l a t i o n run. If no t h e r a p e u t i c i n t e r v e n t i o n s are a s s u m e d (K = 1.0, X = 1.0, ~" = 0.0, p~ = 0.0, P c = 0.0), the resulting CD4 + curve characterizes best fit of the observed clinical d a t a [ 6 ] - - s e e the c o m p a r i s o n [4]. F o r the sake of comparison this curve is included in all further p r e s e n t e d figures; before t h e t r e a t m e n t starts, it is identical to the p r e s e n t e d t h e r a p y s i m u l a t i n g curve, while after it this curve continues as a broken line. M a t u r e CD4 + l y m p h o cytes ( P cells) counts are given in t h e subsequent figures as percentage of P0, i.e., n o r m a l CD4 + cell counts before t h e HIV infection.
RESULTS AND DISCUSSION F i r s t t h e effect of c h e m o t h e r a p y t r e a t m e n t is s i m u l a t e d , i.e., the administ r a t i o n of zinovudin (AZT), the drug t h a t limits HIV proliferation. This is reflected in the above model b y i n t r o d u c t i o n of t h e proliferation rate inhibition factor K ill Eq. (5), which is used to decrease HIV proliferation rate 0 (K = 1.0, u n i n h i b i t e d case; K = 0.0, t o t a l inhibition). F i g u r e 1 d e p i c t s t h e effect of p e r m a n e n t a d m i n i s t r a t i o n of different doses of A Z T . T h e t r e a t m e n t s t a r t s 2 years post-infection. A high dose of A Z T
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FIG. 1. T i l e e f f e c t o f p e r m a n e n t A Z T t r e a t m e n t K = 0.25; c u r v e 2, K = 0.85; c u r v e 3, K = 0.97.
t [year ]
started
iO
2 y e a r s p o s t - i n f e c t i o n : c u r v e 1,
Modelling of HIV Infection Therapies
157
(K = 0.25, curve 1) returns CD4 + lymphocyte value to almost the normal one and this new steady-state level is maintained. A smaller dose (K = 0.85, curve 2) increases treatment start-up values by nearly 20% and reaches also a steady-state level. A very low dose of AZT (K = 0.97, curve 3) is not sufficient to increase the current CD4 + lymphocyte level or to maintain the treatment start-up values and only slows down the decline of CD4 + lymphocytes. However, only a slightly different value of K = 0.95 (not depicted) keeps CD4 + lymphocyte count practically on the treatment start-up value for all remaining simulation period. If the AZT administration is started 5 years post-infection (Fig. 2), thc two higher AZT doses (K = 0.25 and K = 0.85, curves 1 and 2, respectively), increase CD4 + lymphocyte values to the same steady-state levels as in the previous case, when the treatment is started 2 years post-infection (Fig. 1). Tile low AZT dose of Fig. 1 (K = 0.97, not depicted in Fig. 2) has practically no effect. Even a somewhat higher AZT dose (K = 0.9, Fig. 2, curve 3) cannot restore any steady-state level and CD4 + lymphocyte decline is slowed down only. In general, the recovered CD4 + lymphocyte counts started to decline again, when AZT treatment was discontinued (temporary AZT therapy). Fairly similar results are obtained, when active or passive anti-HIV immunotherapy is simulated. Active immunotherapy is reflected in the model by stimulation factor X in Eq. (6), which is used to increase generation of C cells. For passive immunotherapy factor ~" introduced in Eq. (5) represents HIV elimination by administered anti-HIV antibodies. Figure 3 depicts active immunotherapy (~ = 2.0) started 3 years postinfection. Permanent application (curve 1) leads to a substantial increase in
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T h e effect o f p e r m a n e n t
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K = 0.25: c u r v e 2, K = 0.85; c u r v e 3, K = 0.9.
-
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158
T. HRABA AND J. DOLEZAL tO0
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50
0 0
5
t
Cyear:]
[0
FIc, 3. The effect of d u r a t i o n of active a n t i - H I V i m m u n o t h e r a p y s t a r t e d 3 years postinfection (A = 2.0): curve 1, p e r m a n e n t ; curve 2, 1 year; curve 3, 30 days.
CD4 + cell values and to establishment of a steady state. When the treatment is terminated after 1 year (curve 2), CD4 + cell count reaches almost the value of this steady state and then starts to decline again with considerable prolongation of survival. Treatment lasting 30 days (curve 3) effects only a small increase in CD4 + lymphocytes, and their decline starts again immediately after termination of the therapy with a much smaller prolongation of survival. A very similar picture was obtained for passive immunotherapy (~ = 0.01) started also 3 years post-infection (Fig. 4). Generally speaking, it can be concluded that such treatments, if intensive enough to eradicate the HIV, the CD4 + lymphocyte counts return to
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Fie. 4. T h e effect of d u r a t i o n of passive a n t i - H I V i m m u n o t h e r a p y s t a r t e d 3 years post-infection (~" = 0.01): curve 1, p e r m a n e n t ; curve 2, 1 year; curve 3, 30 days.
159
Modelling o f H I V Infection Therapies
normal values. When the treatment does not stop HIV proliferation completely, then in most cases CD4 + lymphocyte counts stabilize at steady-state levels, lower than normal values, but often higher than the treatment start-up values. Such values will be the closer to the normal ones, the more intensively treatment is applied. Only when the treatment intensity is low a n d / o r the treatment is applied rather late post-infection, no new intermediate CD4 + lymphocyte steady state is established and their decline is only slowed down. This picture is observed when the treatment is assumed to be permanent. However, AZT treatment loses its effectivity after some time and neither immunotherapy can be expected to be effective permanently. Therefore the cases with temporary therapy applications are more appropriate from a clinical point of view on such simulation results. It is of interest that in most cases of temporary treatment simulation the survival prolongation is longer than the duration of treatment. An analogical simulation picture to chemotherapy and anti-HIV immunotherapy is also observed when small doses of anti-CD8 antibodies are injected to lower moderately the increased CD8 + lymphocyte counts in HIV-infected persons. This is effected by factor PR in Eq. (4). The case of anti-CD8 treatment started 3 years post-infection ( Pn = 0.01) is in Fig. 5. A substantial effect requires prolonged application of the treatment (permanent, curve 1; 1 year, curve 2; 30 days, curve 3). These results are in agreement with the expectations of [6]. This happens only when C cells are not CD8 + cells ( P c = 0.0). However it is highly probable that C cells belong to this T-cell subpopulation and are destroyed by anti-CD8 antibodies. This being the case, the decreased protective immunity effects the accelerated drop of CD4 + cell numbers, i.e.,
too
CO4 +
2
50
0 0
5
t [year J
tO
FIG. 5. The effect of duration of anti-CD8 administration started 3 years post-infection ( Pc = 0.0, PR = 0.01): curve 1, permanent; curve 2, 1 year; curve 3, 30 days.
160
T. H R A B A AND J. D O L E Z A L £oo
.
.
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0
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shortening of survival. As can be seen in Fig. 6 ( P c = 0.01, PR = 0.01) this shortening is very small when the treatment lasts only 30 days (curve 2), but it is substantial when the treatment is permanent (curve 1). Effect of the treatment lasting 1 year is practically identical to that of the permanent one (not depicted).
CONCLUSIONS The described mathematical model belongs among those attempting to simulate substantial processes in HIV infection, especially the virus and CD4 + lymphocyte dynamics [7, 8]. The results presented above show applicability of the model to simulation of broad spectrum of HIV infection treatments. Actually, as far as chemotherapy simulation is concerned, our results are analogous to those of other authors [7]. The major difference between their and our approach lies in the primary aims to be achieved. In our case it is the maximal agreement between observed clinical and simulated values. Other authors concentrate more on mathematical characterization of processes involved in infection. REFERENCES 1 2
T. H r a h a a n d J. Dole~al, M a t h e m a t i c a l model of CD4 + lymphocyte depletion in HIV infection, Folia Biol. (Praha) 35:159-163 (1989). T. H r a h a a n d J. Dole~al, C o n t r i b u t i o n of m a t h e m a t i c a l modelling to elucidation of AIDS pathogenesis, J. Biol. Systems 4:349-362 (1993).
Modelling of HIV Infection Therapies 3 4
5 6
7 8
161
T. Hraba, J. Dole~,al, and S. Celikovsk~, Model-based analysis of CD4 + lymphocyte dynamics in HIV infected individuals, Immunobiology 181:108-118 (1990). J. Dole~al and T. Hraba, Model-based analysis of CD4 + lymphocyte dynamics in HIV infected individuals. II. Evaluation of the model based on clinical observations, Immunobiology 182:178-187 (1991). L.M. Adleman and D. Wofsy, T-cell homeostasis: Implications in HIV infection, J. AIDS 6:144-152 (1993). W. Lang, H. Perkins, R. E. Anderson, R. Royce, N. Jewell, and W. Winkelstein, Jr., Patterns of T lymphocyte changes with human immunodeficiency virus infection: From seroconversion to the development of AIDS, J. AIDS 2:63-69 (1989). A. Perelson, D. E. Kirschner, and R. de Boer, Dynamics of HIV infection of CD4 + T cells, Math. Biosci. 114:81-125 (1993). M.A. Nowak, R. M. Anderson, A. R. McLean, T. F. W. Wolfs, J. Goudsmit, and R. M. May, Antigenic diversity thresholds and the development of AIDS, Science 254:963-969 (1991).
U s e of a M a t h e m a t i c a l M o d e l for Effectivity E s t i m a t i o n of H I V Infection T h e r a p i e s Tom'£.~ H r a b a
Institute of Molecular Genetics Academy of Sciences of the Czech Republic 16637 Prague, Czech Republic and J a r o s l a v Dole~al
Institute of Information Theory and Automation Academy of Sciences of the Czech Republic 18208 Prague, Czech Republic Transmitted by F. E. Udwadia
ABSTRACT The previously developed mathematical model sinmlates the CD4 + lymphocyte dynamics in HIV infection considerably well. As the number of these cells is a good indicator of the infection progression, it was used to evaluate the effectivity of different therapeutic interventions. For chemotherapy simulation, both permanent and telnporary AZT administration was considered and the induced return of the CD4 + lymphocyte counts was analyzed. Similar analysis was performed for aetive and passive immunotherapy. The model offers also the possibility of simulating the CD4 + dynamics after depletion of CD8 + lymphocytes by antibodies. Even one simulated administration of anti-CD8 antibodies increases the CD4 + lymphocyte counts and prolongs the patient survival. HoWever, if cells involved in protective inmmnity are assumed to belong to the CD8 + category, anti-CD8 antibodies accelerate the decrease of CD4 + (:ells and thus shorten the patient survival.
Presented at tile Seventh Workshop oll Dynamics and Control, Ulm, Germany, July 17-20, 1994.
APPLIED MATHEMATICS AND COMPUTATION 78:153-161 (1996) © Elsevier Science Inc., 1996 655 Avenue of the America~, New York, NY 10010
0096-3003/96/$15.00 PII S0096-3003(96)00005-8
154
T. tIRABA AND J. DOLEZAL
INTRODUCTION The CD4 + lymphocyte depletion seems to be the major immune system lesion in HIV infection, and the number of these lymphocytes is a good prognostic indicator of the infection dynamics. As the developed mathematical model [1, 2] simulates the dynamics of CD4 + lymphocytes in HIVinfected individuals successfully [3, 4], it can also be used for simulation of different therapeutic intervention effects. In this paper we are presenting simulation results of zinovudin chemotherapy, active and passive anti-HIV immunotherapy, and anti-CD8 antibody administration. The latest treatment was suggested [5], because it seemed highly probable that a homeostatic mechanism regulating T-cell numbers did not discriminate between the loss of CD4 + or CD8 + lymphocytes. It was therefore assumed that activation of this mechanism by CD4 + lymphocyte depletion might be inhibited in HIV-infected individuals by increased numbers of CD8 + cells. In consequence, it was suggested to lower the CD8 + numbers by administering anti-CD8 antibodies and, in this way, to increase CD4 + lymphocyte level by a stronger influx of new T cells. The respective model could simulate the effect of anti-CD8 antibody administration, because it incorporated such a homeostatic mechanism regulating T-cell level [3]. MATHEMATICAL MODEL The model considers immature and mature CD4 + ( f i and P cells) and CD8 + lymphoeytes ( R and R cells). As normal values of R cells equal approximately two-thirds of those of P cells, it is assumed that also normal values correspond in a similar way to two-thirds of P cells. The sizes of these cell compartments at time t are described by Eqs. (1)-(4). The amount of HIV products at time t is given by Eq. (5). Finally, Eq. (6) gives the number of cytotoxic T cells specific for HIV ( C cells) at time t. In the model used in this article and in the preceding ones simulating therapeutic interventions in HIV infection, these cells both limit proliferation of HIV, as indicated in Eq. (5), and effect destruction of CD4 + cells presenting HIV products according to Eqs. (1)-(2): dP(t)
dt
Ip + f [ ( P o - P( t)) + ( R o - R ( t ) ) ] - Y p P ( t ) -
dP(t) dt
-Cpa( t ) C ( t) P( t),
P(O) = Po
Y p P ( t ) - "rpP(t) - c p a ( t ) C ( t ) P ( t ) ,
(1) P ( O ) = Po (2)
Modelling of HIV Infection Therapies
155
dR(t)
dt
~{Ip + f[( Po - P( t)) + ( R o - R ( t ) ) ] } -
dR(t) dt
¥~-R( t)
=
-
(r R + pR) R( t),
da( t) dt = a(t)[ KO - ~" - y C ( t ) ] ,
dO(t) dt
(3)
R(O)
= P0
(4)
(5)
a(O) = a o
I P(t) t ~
A(t) a ( t ) [ e I c + a C ( t ) ] ~ , ~ - - o ) - ( r c - p c )
C(t)'
c(o) = q,.
(6)
Here Ie is the influx of P cells, i.e., the rate (all rates are in days -1) of differentiation of P cells from stem cells, Yr is the rate of maturation of .P cells into P cells, and rp is the rate of natural death of P cells; the quantities YR and r n are defined in a fully analogical way. Further, f is the amplifying coefficient of the linear feedback effect of P a n d / o r R cell decrease on the influx of _P and R cells at time t. The quantity Ep a(t) C(t) is the rate of elimination of P cells due to the amount of HIV products a(t) and the number of cytotoxic T (:ells C-~t) at time t. Analogously, c~,a(t)C(t) is the rate of elimination of P cells. The value % is the function of the infectious dose of HIV, 0 characterizes the growth rate of HIV, and y is the rate of inactivation of HIV products mediated by cytotoxic C cells. The maturation of these cells from their precursors is assumed to be dependent on the encounter with HIV products and the effect of HIV-specific helper T cells. I c is the influx of C cell precursors, ~ their maturation rate, a the proliferation rate of C cells under the antigenic stinmlation by HIV products and helper T-cell influence, and r c their natural death rate. Helper T-cell effect on maturation and proliferation of C cells is expressed by the ratio P(t)/Po; the coefficient u is introduced to characterize the intensity of this helper effect. Effects of theiapeutic interventions are described by the following parameters: K, chemotherapeutic inhibition factor of HIV proliferation; A, immune response enhancing factor increasing active anti-HIV immunity; ~', HIV elimination rate by passive immunization effected by administration by anti-HIV antibodies; Pt¢ and Pc elimination rates of CD8 + ( R and C cells, respectively) by anti-CD8 antibodies.
156
T. HRABA AND J. DOLEZAL
If not otherwise s t a t e d the model p a r a m e t e r s in s i m u l a t i o n runs are selected as follows: ~p = 0.2, Tp = 0.01, YR = 0.2, TR = 0.01, ~'c = 0.01, Ip = 1.0, I c = 0.2, P0 = 5.0, P,, = 100.0, R 0 = 3.33, R 0 = 66.7, C O = 0.0, % = 0.0005, f = 0.01, c~ = 0.1, e = 0.144, ~ / = 0.7, 0 = 0.02, v = 2.0. O n l y the m a t u r e CD4 + l y m p h o c y t e s ( P cells) are assulned to be susceptible to HIV products, i.e., ?e = 0.0, c r = 20.0. As a rule, p a r a m e t e r e is used for final a d j u s t m e n t of the respective s i n m l a t i o n run. If no t h e r a p e u t i c i n t e r v e n t i o n s are a s s u m e d (K = 1.0, X = 1.0, ~" = 0.0, p~ = 0.0, P c = 0.0), the resulting CD4 + curve characterizes best fit of the observed clinical d a t a [ 6 ] - - s e e the c o m p a r i s o n [4]. F o r the sake of comparison this curve is included in all further p r e s e n t e d figures; before t h e t r e a t m e n t starts, it is identical to the p r e s e n t e d t h e r a p y s i m u l a t i n g curve, while after it this curve continues as a broken line. M a t u r e CD4 + l y m p h o cytes ( P cells) counts are given in t h e subsequent figures as percentage of P0, i.e., n o r m a l CD4 + cell counts before t h e HIV infection.
RESULTS AND DISCUSSION F i r s t t h e effect of c h e m o t h e r a p y t r e a t m e n t is s i m u l a t e d , i.e., the administ r a t i o n of zinovudin (AZT), the drug t h a t limits HIV proliferation. This is reflected in the above model b y i n t r o d u c t i o n of t h e proliferation rate inhibition factor K ill Eq. (5), which is used to decrease HIV proliferation rate 0 (K = 1.0, u n i n h i b i t e d case; K = 0.0, t o t a l inhibition). F i g u r e 1 d e p i c t s t h e effect of p e r m a n e n t a d m i n i s t r a t i o n of different doses of A Z T . T h e t r e a t m e n t s t a r t s 2 years post-infection. A high dose of A Z T
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t [year ]
started
iO
2 y e a r s p o s t - i n f e c t i o n : c u r v e 1,
Modelling of HIV Infection Therapies
157
(K = 0.25, curve 1) returns CD4 + lymphocyte value to almost the normal one and this new steady-state level is maintained. A smaller dose (K = 0.85, curve 2) increases treatment start-up values by nearly 20% and reaches also a steady-state level. A very low dose of AZT (K = 0.97, curve 3) is not sufficient to increase the current CD4 + lymphocyte level or to maintain the treatment start-up values and only slows down the decline of CD4 + lymphocytes. However, only a slightly different value of K = 0.95 (not depicted) keeps CD4 + lymphocyte count practically on the treatment start-up value for all remaining simulation period. If the AZT administration is started 5 years post-infection (Fig. 2), thc two higher AZT doses (K = 0.25 and K = 0.85, curves 1 and 2, respectively), increase CD4 + lymphocyte values to the same steady-state levels as in the previous case, when the treatment is started 2 years post-infection (Fig. 1). Tile low AZT dose of Fig. 1 (K = 0.97, not depicted in Fig. 2) has practically no effect. Even a somewhat higher AZT dose (K = 0.9, Fig. 2, curve 3) cannot restore any steady-state level and CD4 + lymphocyte decline is slowed down only. In general, the recovered CD4 + lymphocyte counts started to decline again, when AZT treatment was discontinued (temporary AZT therapy). Fairly similar results are obtained, when active or passive anti-HIV immunotherapy is simulated. Active immunotherapy is reflected in the model by stimulation factor X in Eq. (6), which is used to increase generation of C cells. For passive immunotherapy factor ~" introduced in Eq. (5) represents HIV elimination by administered anti-HIV antibodies. Figure 3 depicts active immunotherapy (~ = 2.0) started 3 years postinfection. Permanent application (curve 1) leads to a substantial increase in
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T h e effect o f p e r m a n e n t
i
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K = 0.25: c u r v e 2, K = 0.85; c u r v e 3, K = 0.9.
-
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s t a r t e d 5 y e a r s p o s t - i n f e c t i o n : c u r v e 1,
158
T. HRABA AND J. DOLEZAL tO0
CO4+ [Z] 2
50
0 0
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t
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[0
FIc, 3. The effect of d u r a t i o n of active a n t i - H I V i m m u n o t h e r a p y s t a r t e d 3 years postinfection (A = 2.0): curve 1, p e r m a n e n t ; curve 2, 1 year; curve 3, 30 days.
CD4 + cell values and to establishment of a steady state. When the treatment is terminated after 1 year (curve 2), CD4 + cell count reaches almost the value of this steady state and then starts to decline again with considerable prolongation of survival. Treatment lasting 30 days (curve 3) effects only a small increase in CD4 + lymphocytes, and their decline starts again immediately after termination of the therapy with a much smaller prolongation of survival. A very similar picture was obtained for passive immunotherapy (~ = 0.01) started also 3 years post-infection (Fig. 4). Generally speaking, it can be concluded that such treatments, if intensive enough to eradicate the HIV, the CD4 + lymphocyte counts return to
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'
'
1
I
.
.
.
.
CO4+ 2
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.
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.
.
.
, 5
, \ N ~ -
,
t I:~lear :]
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Fie. 4. T h e effect of d u r a t i o n of passive a n t i - H I V i m m u n o t h e r a p y s t a r t e d 3 years post-infection (~" = 0.01): curve 1, p e r m a n e n t ; curve 2, 1 year; curve 3, 30 days.
159
Modelling o f H I V Infection Therapies
normal values. When the treatment does not stop HIV proliferation completely, then in most cases CD4 + lymphocyte counts stabilize at steady-state levels, lower than normal values, but often higher than the treatment start-up values. Such values will be the closer to the normal ones, the more intensively treatment is applied. Only when the treatment intensity is low a n d / o r the treatment is applied rather late post-infection, no new intermediate CD4 + lymphocyte steady state is established and their decline is only slowed down. This picture is observed when the treatment is assumed to be permanent. However, AZT treatment loses its effectivity after some time and neither immunotherapy can be expected to be effective permanently. Therefore the cases with temporary therapy applications are more appropriate from a clinical point of view on such simulation results. It is of interest that in most cases of temporary treatment simulation the survival prolongation is longer than the duration of treatment. An analogical simulation picture to chemotherapy and anti-HIV immunotherapy is also observed when small doses of anti-CD8 antibodies are injected to lower moderately the increased CD8 + lymphocyte counts in HIV-infected persons. This is effected by factor PR in Eq. (4). The case of anti-CD8 treatment started 3 years post-infection ( Pn = 0.01) is in Fig. 5. A substantial effect requires prolonged application of the treatment (permanent, curve 1; 1 year, curve 2; 30 days, curve 3). These results are in agreement with the expectations of [6]. This happens only when C cells are not CD8 + cells ( P c = 0.0). However it is highly probable that C cells belong to this T-cell subpopulation and are destroyed by anti-CD8 antibodies. This being the case, the decreased protective immunity effects the accelerated drop of CD4 + cell numbers, i.e.,
too
CO4 +
2
50
0 0
5
t [year J
tO
FIG. 5. The effect of duration of anti-CD8 administration started 3 years post-infection ( Pc = 0.0, PR = 0.01): curve 1, permanent; curve 2, 1 year; curve 3, 30 days.
160
T. H R A B A AND J. D O L E Z A L £oo
.
.
.
.
i
'
'
,
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50
. . . .
o
0
x~ 5
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,
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FIG. 6. The effect of duration of anti-CD8 administration started 3 years post-infection ( Pc = 0.01, PR 0.01): curve 1, permanent; curve 2, 30 days. =
shortening of survival. As can be seen in Fig. 6 ( P c = 0.01, PR = 0.01) this shortening is very small when the treatment lasts only 30 days (curve 2), but it is substantial when the treatment is permanent (curve 1). Effect of the treatment lasting 1 year is practically identical to that of the permanent one (not depicted).
CONCLUSIONS The described mathematical model belongs among those attempting to simulate substantial processes in HIV infection, especially the virus and CD4 + lymphocyte dynamics [7, 8]. The results presented above show applicability of the model to simulation of broad spectrum of HIV infection treatments. Actually, as far as chemotherapy simulation is concerned, our results are analogous to those of other authors [7]. The major difference between their and our approach lies in the primary aims to be achieved. In our case it is the maximal agreement between observed clinical and simulated values. Other authors concentrate more on mathematical characterization of processes involved in infection. REFERENCES 1 2
T. H r a h a a n d J. Dole~al, M a t h e m a t i c a l model of CD4 + lymphocyte depletion in HIV infection, Folia Biol. (Praha) 35:159-163 (1989). T. H r a h a a n d J. Dole~al, C o n t r i b u t i o n of m a t h e m a t i c a l modelling to elucidation of AIDS pathogenesis, J. Biol. Systems 4:349-362 (1993).
Modelling of HIV Infection Therapies 3 4
5 6
7 8
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T. Hraba, J. Dole~,al, and S. Celikovsk~, Model-based analysis of CD4 + lymphocyte dynamics in HIV infected individuals, Immunobiology 181:108-118 (1990). J. Dole~al and T. Hraba, Model-based analysis of CD4 + lymphocyte dynamics in HIV infected individuals. II. Evaluation of the model based on clinical observations, Immunobiology 182:178-187 (1991). L.M. Adleman and D. Wofsy, T-cell homeostasis: Implications in HIV infection, J. AIDS 6:144-152 (1993). W. Lang, H. Perkins, R. E. Anderson, R. Royce, N. Jewell, and W. Winkelstein, Jr., Patterns of T lymphocyte changes with human immunodeficiency virus infection: From seroconversion to the development of AIDS, J. AIDS 2:63-69 (1989). A. Perelson, D. E. Kirschner, and R. de Boer, Dynamics of HIV infection of CD4 + T cells, Math. Biosci. 114:81-125 (1993). M.A. Nowak, R. M. Anderson, A. R. McLean, T. F. W. Wolfs, J. Goudsmit, and R. M. May, Antigenic diversity thresholds and the development of AIDS, Science 254:963-969 (1991).