Biologic variability of CD4 cell counts between visits was explicitly captured by allowing for all possible transitions among CD4 count strata. Measurement error (laboratory variation) of CD4 counts was assumed to be negligible in our model. First, only a low variability for small CD4 counts exists (10). Second, the CD4 ranges created reduce much of this uncertainty. Third, the SHCS population is large so that possible minor discrepancies were therefore assumed to cancel each other out.
To compare the two collection periods, Markov chain summaries were computed over a 3-year period (six cycles). This implies two cycles of extrapolation for data gathered from 1996 to 1997, which was necessary to achieve two comparable follow-up periods. We multiplied the transition matrix by itself once per cycle to derive the estimates of interest. These summaries included the probability of developing symptomatic AIDS prior to the end of each cycle and the expected number of AIDS-free months using half-cycle correction (7).
We additionally used the 1996 to 1997 data set for a Markov chain probability analysis over a 5-year period to display possible future situations. The Markov cycle tree was developed and analyzed by cohort simulation (matrix multiplication) using the computer programs DATA version 3.0.16 (TreeAge software, Williamstown, MA, U.S.A.) and S-Plus version 4.0 (MathSoft Inc., Seattle, WA, U.S.A.).
We extracted data from the SHCS database to construct 5 × 5 transition matrices containing the probabilities of switching from one CD4 stratum to another or developing AIDS, given a specified CD4 stratum. Individuals were only considered for this analysis when they were without symptomatic AIDS, had at least one recorded CD4 count <100 cells/mm3 after January 1, 1993, and had at least one follow-up. In the SHCS, at an approximate 6-month interval, a new assessment of the CD4 count and clinical assessment for the eventual presence of symptomatic AIDS disease is made. We excluded follow-up intervals that were either longer than 9 months apart or shorter than 3 months apart. Individuals contributed to the model from the first date characterized by a CD4 count <100 cells/mm3. Ascending to a CD4 count >100 cells/mm3 after that date was permissible. Depending on the number of follow-ups, each patient could contribute repeated 6-month observations.
Maximum likelihood estimates were generated by pooling the recorded 6-month transitions in the respective cells of the transition matrix. We pooled follow-up data recorded before and after December 31, 1995 separately to estimate the probabilities during the two periods. We generated the transition matrices using SAS software version 6.11 (Cary, NC, U.S.A.).
Efron's bootstrap was used to compute confidence intervals (CIs) for each summary (11). Each bootstrap replication involved resampling from each row of the observed matrix. The same row total of transitions were sampled from a multinomial distribution in which the probabilities are the observed row percentages. The 95% CIs were generated based on 5000 samples using S-Plus version 4.0 (MathSoft Inc.).
We supplemented the 95% confidence limits with a sensitivity analysis of worst-case and best-case scenarios expected to provide extremely conservative ranges. For each estimated probability in the base-case transition matrix, we selected the 97.5 and 2.5 percentiles to construct worst-case and best-case scenarios. The 2.5 percentiles for probabilities describing a transition into a more advanced immunosuppressed state or the symptomatic AIDS state, and the 97.5 percentiles for probabilities describing a transition into the prior or a higher CD4 stratum, were used to define a best-case scenario. These probabilities were then normalized to assure a transition matrix row probability of one. The 97.5 percentiles for probabilities describing a transition into a more advanced immunosuppressed state or the symptomatic AIDS state, and the 2.5 percentiles for probabilities describing a transition into the prior or a higher CD4 stratum, were used to define a worst-case scenario. These probabilities were then normalized to ensure a transition matrix row probability of one.
A total of 1027 individuals (mean age, 35.2 years; range, 17.6-75.6 years) contributed to 2634 pairs of 6-month observations from 1993 to 1995, and 681 individuals (mean age, 34.4 years; range, 17.7-74.5 years) contributed to 2077 pairs of 6-month observations from 1996 to 1997. The latest follow-up date was February 2, 1998. These 6-month observations were pooled to construct the two five-state transition matrices shown in Table 1.
The proportion of disease-free HIV-infected patients with a CD4 count <100 cells/mm3 for whom antiretroviral triple combination therapy was prescribed increased from 5% in 1995, to 26% in 1996, and to 70% in 1997.
The probability of being in each state was calculated over six cycles (36 months) using transition matrices A and B (Table 1), and over 10 cycles using transition matrix B (Fig. 2). The transition matrices A and B refer to transition probabilities derived from the observation periods 1993 to 1995 and 1996 to 1997, respectively. When we applied transition matrix A, 804 of 1000 patients (95% CI, 761-842) starting in CD4 stratum 0 to 49 cells/mm3 would be expected to develop symptomatic AIDS by 3 years, so the probability of AIDS by 3 years is 0.804 (Fig. 2A1). When we used transition matrix B, 398 of 1000 (95% CI, 329-466) would be expected to develop symptomatic AIDS by 3 years. When we applied transition matrix B over a 5-year period (3 years beyond the range of the directly observed data) the expected number with AIDS was 481 (95% CI, 402-557; Fig. 2B1). The corresponding estimates for a 1000-patient cohort starting in CD4 strata 50 to 74 cells/mm3 and 75 to 99 cells/mm3 are also shown in Figure 2.
Using the transition matrices (Table 1), expected AIDS-free survival over a 3-year period was calculated for a cohort starting in each of the four CD4 strata (Table 2). Regardless of the initial CD4 stratum, over a 3-year period, an expected gain of ∼10 AIDS-free months is anticipated since the introduction of antiretroviral triple combination therapy. For each computed summary with transition matrices A and B, the 95% CIs do not overlap, suggesting a statistically significant difference (p < .05). As shown in Table 2, the expected AIDS-free survival in the worst-case scenario on the basis of transition matrix B is always higher than the best-case estimate derived from transition matrix A, for any stratum. Transition matrix B was also used to develop AIDS-free survival estimates over a 5-year period as shown in Table 2. We perceived this was the maximum realistic time horizon for extrapolation in this highly dynamic and changing field.
Since the introduction of antiretroviral combination therapies, several cohort studies of HIV-infected individuals have reported a considerable decline in AIDS-related mortality and incidence of AIDS (3-5). Analyses from cohort studies are usually based on survival analysis and hazard models with adjustment for relevant prognostic cofactors. In the present study, we have chosen an alternative approach and have analyzed the SHCS database by Markov population modeling (7). Such an approach may present several advantages. This technique allows the researcher to model prognosis over time in a more dynamic way and is particularly useful for simulating chronic disease conditions (12,13). The history of a disease can be accurately represented with transitions in and out of various health states occurring over a sequence of fixed time intervals. The absence of memory in a Markov model is known as the Markovian assumption (6). Albeit few biologic systems strictly adhere to this assumption, its approximate correctness and feasibility make Markov models a recommended method for prediction in chronic diseases (6,13).
Our analysis shows an average 10-month gain in AIDS-free survival over 36 months of simulation when we use data collected from 1996 to 1997 instead of data collected from 1993 to 1995 (Table 2). This holds true for any of the three initial CD4 strata. These results imply one year of extrapolation for data collected from 1996 to 1997. We regard this as a rather conservative estimate. First, matrix B includes contributions from individuals without antiretroviral triple therapy during that period. Furthermore, the percentage of individuals taking the three-drug regimen will likely increase further in later years. Finally, the increasing knowledge about drug resistance patterns will lead to a more efficient use of drug combinations.
Our models offer some additional interesting insights that warrant further comment. For example, when an asymptomatic 1000-patient cohort starts in one of the three CD4 strata <100 cells/mm3 using transition matrix A (1993-1995), after two cycles (12 months), between 26% (initial CD4 stratum 75-99 cells/mm3) and 46% of patients (initial CD4 stratum 0-49 cells/mm3) will already have developed an AIDS-defining disease. Moreover, the proportion of AIDS-free patients in the lowest CD4 stratum with 0 to 49 cells/mm3 will be expected to be highest after two cycles when starting in CD4 stratum represented by 75 to 99 cells/mm3, and after one cycle when starting in CD4 stratum 50 to 74 cells/mm3. Quite to the contrary, however, when matrix B (1996-1997) is used, in a cohort that starts in the most severely immunosuppressed state with CD4 counts 0-49 cells/mm3, the proportion of patients with CD4 counts >100 cells/mm3 is dominating after two cycles (12 months). After the first two cycles (12 months), the number of per-cycle subjects who are expected to develop an AIDS-defining disease has decreased dramatically. At 42 months, the number of patients that are expected to have developed AIDS will exceed the expected number of patients with a CD4 count of >100 cells/mm3.
In our analysis, we focused on those at greatest risk for AIDS and have exclusively analyzed patients with a CD4 count <100 cells/mm3 and without symptomatic AIDS. The inclusion of fewer immunosuppressed subjects in our model would have resulted in an unfocused view not adequately representing the benefit of new therapy regimens to those who are at greatest risk for AIDS. In our model, the risk of AIDS was conditioned on the patients' CD4 cell counts. We did not consider blood HIV RNA levels in our model, although our methods could be applied. In our cohort of severely immunosuppressed study subjects, the CD4 cell count has a higher predictive value than blood HIV RNA levels (14).
Several arguments indicate that our model may quite accurately reflect the present and near-future development of AIDS in the HIV-infected population with advanced immunosuppression in Switzerland. We did not exclude those patients who did not receive antiretroviral triple combination therapy, and therefore, our model shows how the general HIV-infected disease-free population behaves. Because patients enrolled in the SHCS represent ∼70% of all cases reported to the Swiss Federal Office of Health, the model allows us to estimate the real-time population dynamics in general. The remainder of HIV-infected patients in Switzerland not enrolled in the SHCS are mostly treated by physicians in private practice. Detailed study data are thus not available for these patients. Nonetheless, we believe that the presented estimates on the future development of AIDS may be of interest to countries with a health care system similar to that of Switzerland. In Switzerland, access to antiretroviral triple combination therapy is offered to all HIV-infected individuals with Swiss health insurance coverage.
In conclusion, the comparison of disease progression among individuals at highest risk for AIDS before and after the introduction of antiretroviral triple combination therapy in Switzerland documents a significant improvement in AIDS-free survival. The projected estimates of disease dynamics indicate prolonged AIDS-free survival and therefore a higher quality of life in individuals in far advanced disease stages.
The members of the Swiss HIV Cohort Study are M. Battegay (Co-Chair of the Scientific Board), E. Bernasconi, Ph. Bürgisser, M. Egger, P. Erb (Chairman of the "Laboratories" group), W. Fierz, M. Flepp (Chairman of the "Clinics" group), P. Francioli (President of the SHCS, Centre Hospitalier Universitaire Vaudois, CH-1011-Lausanne), H. J. Furrer, P. Grob, B. Hirschel (Chairman of the Scientific Board), L. Kaiser, B. Ledergerber, R. Malinverni, L. Matter, M. Opravil, F. Paccaud, G. Pantaleo, L. Perrin, W. Pichler, J-C. Piffaretti, M. Rickenbach (Head of Data Center), P. Sudre, J. Schupbach, A. Telenti, and P. Vernazza.
Acknowledgments: This study was supported in part by Swiss HIV Cohort Study grant #235. P. Sendi was supported by an unlimited research fellow grant from Merck Sharp & Dohme-Chibret, Switzerland. The authors thank two anonymous reviewers for their insightful comments.
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Keywords:© 1999 Lippincott Williams & Wilkins, Inc.
AIDS epidemiology; Markov chains; Anti-HIV agents; Disease progression; Switzerland