Methods

Renal transplant recipients were included in the retrospective mathematical analysis if sufficient longitudinal data were available. All BKV plasma load measurements were performed in the Transplantation Virology laboratory of the University of Basel, Switzerland, as described elsewhere [3]. For the purpose of analysis, the limit of detection was set at 2.69 log₁₀ copies/mL. Error bars for individual data points were based on a coefficient of variation of 29.6, calculated from 168 polymerase chain reaction standard curves performed during this period

Viral growth rates, doubling times, clearance rates, and half-lives were calculated according to the following formulas: and where r denotes the exponential rate of viral increase (viral growth rate), c denotes the exponential rate of viral decrease (viral clearance rate), t ₂ denotes the viral doubling time, and t _1/2 denotes the viral half-life. V ₀ denotes the initial viral load, and V _t denotes the viral load after t time units. We used equations (1) and (3), solved for r and c respectively, when only 1 or 2 consecutive sampling intervals (<4 data points) were available. Otherwise, we fitted a straight line to log-transformed virus load data to obtain the slope and the 95% confidence interval (CI) using Mathematica (version 4.1; Wolfram Research). The generation time is an estimate of the time needed to complete a full replication cycle. For a virus, this can be calculated by where 1/δ is the average lifespan of an infected cell, and 1/c is the average lifespan of a virion. The basic reproductive ratio (R ₀) is a measurement of the efficacy of an intervention on the reduction in viral replication. R ₀ can be interpreted as the average number of secondary infected cells produced per primary infected cell [10]. For a lytically replicating virus such as BKV, which bursts from its host cell, we assumed a fixed delay of length d between the infection and viral burst. Thus, the formula is where s (the slope of the virus load curve plotted on a log_e scale) represents either the net growth rate, r (if s>0), or the net clearance rate, c (if s<0) [11]

Results

The BKV plasma load was determined in 3 renal transplant recipients undergoing allograft nephrectomy (figure 1A and table 1). Immunosuppressive regimens were discontinued in patient 3 but was continued in patient 1 for a simultaneous pancreas graft and in patient 2 to limit allosensitization before retransplantation. Many samples were obtained from patient 1 after surgical allograft removal, initially at 3-h intervals for the first 24 h and then daily for 4 days. After an initial 4-h increase in BKV plasma load, which was likely caused by surgical manipulations (the “washout phenomenon”), 2 consecutive intervals of viral decay could be observed. The first interval, which occurred 4–7 h after nephrectomy, yielded a clearance rate of 8.24 (±114%)/day, corresponding to an in vivo t _1/2 of 2 h, whereas the second interval, which occurred 7–10 h after nephrectomy, yielded a clearance rate of 15.4 (±61%)/day, corresponding to an in vivo t _1/2 of 1.1 h (range, 0.7–2.8 h). Averaging the data over the 6-h interval resulted in a clearance rate of 11.8 (±40%)/day (t _1/2, 1.4 h). In patient 2, the available BKV plasma loads were measured 1 week apart and yielded a clearance rate of 0.852 (±20%)/day (t _1/2, 20 h; range, 16–24 h). Fewer samples were obtained from patient 3; BKV plasma loads were determined 2 weeks apart. The clearance rate was 0.441 (±19%)/day, (t _1/2, 38 h; range, 32–47 h) In summary, fast (hours) or moderately fast (hours to 2 days) clearance rates were observed after allograft removal

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Figure 1

Longitudinal BK virus (BKV) load data and interventions. A After allograft nephrectomy. B After reduced immunosuppressive regimens. Vertical black arrows indicate the time of allograft removal. The horizontal arrow indicates the start of reduced immunosuppression. The dashed lines indicate the limits of detection. ↓, dose reduction; →, therapy change; ALG, antilymphocyte globulin; Aza, azathioprine; CsA, cyclosporine A; LeF, leflunomide; MMF, mycophenolate mofetil; MP, methylprednisolone pulse; Pred, prednisolone; Sir, sirolimus; Tac, tacrolimus

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Figure 2

Efficacy of reduced immunosuppressive regimens. The basic reproductive ratio (R ₀) is plotted during viral increase (white bars) steady state (light gray bars) and viral decrease (dark gray bars). For patient 4, the R ₀ during viral growth was 2.6 (“rooftop value”). The dashed horizontal line at R₀=1 indicates the critical threshold value separating viral growth from decrease

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Table 1

Calculated BK virus (BKV) clearance rates and half-lives in vivo

In 12 patients treated with a reduction in immunosuppressive regimens, a detailed analysis was performed of 16 intervals of viral decay (figure 1B and table 1). In patient 4, a clearance rate of 1.32 (±89%)/day was calculated between days 66 and 67 (line not shown; t _1/2, 13 h; range, 7–119 h). When we fitted a straight line to the data for days 84–151, an average clearance rate of 0.076 (95% CI, 0.093–0.06) was calculated, corresponding to a t _1/2 of 9 (95% CI, 7–12) days. In patient 5, 2 nonconsecutive intervals of viral decay were observed after dosages of cyclosporine A (CsA) and mycophenolate mofetil (MMF) were reduced: the first, between days 83 and 84, yielded a clearance rate of 3.01 (±39%)/day (line not shown; t _1/2, 5.5 h; range, 4–9 h); the second, between days 112 and 126, yielded a clearance rate of 0.515 (±16%)/day (t _1/2, 33 h; range, 28–39 h) with biweekly sampling. In patient 6, dosages of CsA, MMF, and prednisone (Pred) were reduced, and MMF was later replaced by azathioprine (Aza). BKV plasma loads were measured weekly and yielded a clearance rate of 0.750 (±22%)/day (t _1/2, 22 h; range, 18–29 h). In patient 7, 2 nonconsecutive intervals of viral decay were observed after the reduction of dosages of tacrolimus (Tac), MMF, and Pred. The first, between days 115 and 125, yielded a clearance rate of 0.592 (±20%)/day (t _1/2, 28 h; range, 23–35 h), and the second, between days 363 and 370, yielded a clearance rate of 0.546 (±31%)/day (t _1/2, 31 h; range, 23–44 h). When we fitted a straight line to the data for days 125–300, the average clearance rate was 0.024/day (95% CI, 0.032–0.017/day), with a t _1/2 of 28 days (95% CI, 22–42 days). In patient 8, Aza was continued and Tac was replaced by CsA, which yielded a clearance rate of 0.06/day (95% CI, 0.035–0.088/day), with a t _1/2 of 12 days (range, 8–20 days). In patient 9, switching from a low-dose regimen of Tac, sirolimus (Sir), and Aza to one of Sir and Aza only resulted in a clearance rate of 0.21/day (95% CI, 0.18–0.23/day), with a t _1/2 of 3.3 days (range, 3–4 days). In patient 10, switching from a regimen of CsA and MMF to CsA and Aza, combined with a methylprednisolone pulse around day 220, yielded a clearance rate of 0.094/day (95% CI, 0.035–0.16/day), with a t _1/2 of 7 days (range, 4–20 days). In patient 11, a low-dose Tac and Sir regimen was changed to one of Aza and Sir, which yielded a clearance rate of 0.083/day (95% CI, 0.025–0.14/day), with a t _1/2 of 8 days (range, 5–28 days). In patient 12, a triple regimen of CsA, Pred, and Sir was changed to a low-dose CsA and Pred regimen, which yielded an average clearance rate of 0.093/day (95% CI, 0.12–0.066/day), with a t _1/2 of 7 day (range, 6–11 days). In patients 13, 14, and 15, a regimen of Tac, MMF, and Pred was switched to a dual regimen of Sir and Pred. The average clearance rate in these patients was 0.046/day (95% CI, 0.1–0.014/day), with a t _1/2 of 15 days (95% CI, 7–50 days). Although there was no statistically significant difference between the clearance rates of patients 1–3 (allograft removed) and those of patients 4–7 (immunosuppressive regimens modified), the rates were significantly longer in patients 8–15 (2-sided P<.01, Mann-Whitney U test). This difference was partially due to methodological differences, because linear regressions provide average clearance rates. In summary, BKV clearance rates were moderately fast (hours to days) or slow (several days) in renal transplant recipients after immunosuppressive regimens were reduced (table 1)

Because high-level BKV replication accompanies host cell lysis in the release of infectious progeny, we estimated the loss of BKV-infected renal cells in vivo. Based on the data in table 1 and those of previous clinical studies [3, 8], we assumed a BKV plasma load of 10⁴–10⁶ copies/mL. This amounted to a total of 2.5×10⁷–2.5×10⁹ virions in a plasma compartment of 2.5 L. When we applied the fast clearance rate (t _1/2, 1 h; table 1), only 1–150 virions/day remained uncleared. Thus, in a steady-state situation, almost the entire BKV plasma load was generated every day. When we applied the moderately fast clearance rate (t _1/2, 20 h), 1.09×10⁷–1.09×10⁹ virions were produced and cleared every day. To estimate the impact of this high in vivo turnover, we assumed a BKV burst size of 1 × 10³–1 × 10⁴ virions/lytically replicating host cell [12], which translates to 1×10³–1×10⁶ cells/day lysed through BKV replication alone. If this estimate of the number of infected cells is reasonably accurate, 4×10⁴–4×10⁸ BKV viruses are released from lysed tubular epithelial cells every hour

Increases in BKV plasma load are of particular interest, because this may reflect the spread of BKV infection and the extent of tissue at risk. We found a median growth rate of 0.18/day (t ₂, ∼4 days), with the fastest being 0.90/day (t ₂, 18 h) in patient 4 and the slowest being 0.019/day (t ₂, 37 days) in patient 3 (table 2). In the densely sampled patient 1, a transient increase in BKV plasma load was noted in the first 4 h after nephrectomy, which was unlikely to have resulted from increased replication but may have been caused by surgical manipulations, as was described for Epstein-Barr virus load after resection of nasopharyngeal carcinoma [13]

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Table 2

Calculated BK virus (BKV) growth rates and doubling times in vivo

On the basis of the rapid BKV clearance rate in vivo, corresponding to a t _1/2 of ∼2 h, and the intracellular delay d (the viral eclipse phase between infection and the bursting of progeny virions) in vitro of 48 h [14], we calculated a BKV generation time of ∼50 h. For a lytically replicating virus such as BKV, if one assumes a fixed value of d between infection and the bursting of progeny virions, the R ₀ is calculated by R₀=exp(sd) [11]. Applying a d of ∼48 h, we calculated R ₀ during viral growth (R₀>1) and decrease (R₀<1) after the respective interventions (figure 2). Three examples are discussed in more detail. In patient 2, the R ₀ during the viral growth and decrease stages was 1.05 and 0.97, respectively. Therefore, switching from Tac and AZA to CsA and Aza around week 45 after transplantation had a 7% efficacy (R₀=1.04–0.97). In patient 4, we observed a sharp increase in BKV plasma load after methylprednisolone around day 80 (R₀=2.6) that was followed by a decrease in BKV plasma load over the course of 60 days (R₀=0.86), which indicated a 66% efficacy of switching from Tac, Aza, and Pred plus methylprednisolone to low-dose Tac and leflunomide (LeF) but 14% efficacy relative to that observed for the Tac, Aza, and Pred regimen before antirejection treatment with methylprednisolone. In patient 5, we observed a reduction in the R ₀ of 26% (from 1 to 0.74) after the change in regimen around day 105 that did not persist; the R ₀ returned quickly to ∼1, although a lower level of BKV replication was maintained. In patient 6, we observed a reduction in the R ₀ of 78% (from ∼1 to 0.22) around day 95. In patient 11, viral growth (R₀=1.34) was observed after a steady state (R₀=1) and treatment of a steroid-refractory interstitial rejection with coexisting PVAN with anti-lymphocyte globulin and a switch from a Tac and Aza regimen to low-dose Tac and Sir. Viral growth decreased (R₀=0.85) after the switch to low-dose Sir and Aza. For patients 7–15, the respective efficacies were as follows: patient 7, 13%; patient 8, 11%; patient 9, 34%; patient 10, 37%; patient 11, 30%; patient 12, 17%; patient 13, 9%; patient 14, 10%; and patient 15, 8%. The mean (median) efficacy for all patients and regimens was 27% (17%). Under the assumption of an efficacy of 20% for reduced immunosuppressive regimens in a model patient with a quasi steady-state (R₀=1) BKV plasma load of 1×10⁵ copies/mL, a viral generation time of 2 days, and a detection limit of 500 copies/mL, it would take ∼7 weeks for the BKV plasma load to decrease to below the limit of detection. For a BKV plasma load of 1×10⁷ copies/mL, it would take 13 weeks for the virus load to decrease to below the limit of detection

Discussion

Mathematical models have contributed considerably to the understanding of viral infections in vivo, including those with hepatitis viruses, cytomegalovirus, and HIV-1 [15]. The common hallmark of these entities is progressive organ compromise through persistent viral replication in the setting of immune dysfunction. Although polyomavirus infections have not been studied so far, it is clear that the newly recognized PVAN shares these characteristics: it is a chronic progressive disorder accompanied by high BKV plasma loads in intensely immunosuppressed renal allograft recipients. In clinical practice, PVAN is still viewed as a slowly progressing disease, although the mean time from diagnosis to allograft failure is only 11 months. Our analysis indicated that rather rapid dynamics of BKV replication underlie the course of PVAN. Although effective BKV-specific antivirals are still lacking, the decrease in BKV plasma load after allograft nephrectomy provided a unique opportunity to obtain the first minimal estimates of BKV clearance as being fast (t _1/2, 1–2 h) or moderately fast (t _1/2, 20–38 h). In accordance with the data, we assumed that the replication base for BKV had been removed from the patient’s body with the allograft nephrectomy. Of note, any residual replication would render these rates even faster. In patients with reduced immunosuppressive regimens, BKV clearance was moderately fast or slow, with t _1/2 ranging from 6 h to 17 days. This variability is not unexpected, given that different interventions were used and a number of complex factors may underlie this net decrease, including the individual net state of immunosuppression and the quality and quantity of BKV-specific immune effectors [16]

On the basis of the study of renal biopsies by Randhawa et al. [12], who reported that BKV-infected renal cells release, on average, 6000 virions, and on the viral clearance rates that we calculated, we estimated the daily tubular epithelial-cell loss resulting directly from BKV replication to be 1×10³–1×10⁶ cells. However, the overall impact on allograft function is likely to be underestimated, because this describes the average cytopathic aspect of PVAN without regarding the impact of focal tubular damage per nephron over the total of 1×10⁶ renal nephrons and ignores other pathologic aspects of PVAN, such as immune-mediated damage

We propose the basic R ₀ as a measure to estimate the efficacy of complex interventions. The R ₀ is intimately related to viral fitness in an individual transplant recipient [10, 15, 17]. The efficacies of reduced immunosuppression varied from 7% to 78% (mean, 28%; median, 22%), which are comparable to those of cidofovir and LeF in vitro [18]. Because it is independent of the plasma viral load, R ₀ may prove to be an important and much-needed in vivo measurement for the evaluation of complex combined antiviral interventions in future clinical studies. If one assumes an average efficacy of 20% for an immunosuppressive regimen in a model patient with 1×10⁵ or 1 × 10⁷ BKV copies/mL at steady state, it would take ∼7 or 13 weeks, respectively, for the BKV plasma load to decrease to below the limit of detection. These kinetics suggest that, for clinical management, biweekly monitoring in patients with reduced immunosuppression may be sufficient. However, we note that, after reducing immunosuppressive regimens, there may be a delay of 4–10 weeks in some patients before the BKV plasma load starts to decrease (e.g., patients 13–15 in figure 1B )

The limitations of our study include the varying sampling density, the relatively small sample size, and its retrospective nature. However, the results were derived from careful analysis of multiple intervals of viral decay. The estimated kinetics of BKV in PVAN lie in between the extremes of simian immunodeficiency virus [19] and HIV [17, 20], on the one side, and hepatitis B virus [21, 22], on the other (table 3), and are comparable to those of hepatitis C virus in chronic infection [20, 23, 24]. However, the smaller number of susceptible tubular epithelial host cells, in addition to the complex situation present with allograft, may explain the faster progression to end-stage kidney failure in PVAN, compared with that in chronic hepatitis C infection

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Table 3

Contrasting BK virus (BKV) kinetics with other known viral kinetics

In conclusion, we report the first evidence (to our knowledge) of fast and moderately fast replication kinetics of BKV in renal transplant recipients. Our results emphasize organ cell damage as a major pathogenetic factor in PVAN. Finally, the fast BKV dynamics may explain the high frequency of BKV genome rearrangements, which are unusual for DNA viruses and may be important for immune evasion, antiviral resistance, and development of cancer