Introduction
The human adaptive immune response relies on a complex combination of cellular and
humoral immunity, mediated by T- and B-lymphocytes. Although vaccination aims to activate
both cellular and humoral immunity, vaccine induced immunity is typically evaluated
by means of the antibody titer, secreted by B-lymphocytes [1]. After encountering
antigens, B-cells are stimulated to proliferate and/or differentiate into memory B-cells
and plasma cells (PC). Memory B-cells permit a faster and more effective immune response
upon further exposures to the antigens, whereas PC are the main antibody-secreting
cells (ASC). Different antibody isotopes are present in human sera (IgM, IgA and IgG).
They each have relatively limited half-lives, with a maximum of 17.5–26.0 days for
Immunoglobulin G (IgG), which represent about 75% of the antibody isotopes in humans
[2], [3], [4]. Nonetheless, exposure to common viral and vaccine antigens has been
shown to induce a long-term humoral immune response, which illustrates that improving
our understanding of the mechanisms involved in the production and persistence of
antibodies remains a (relatively rarely explored) topic of fundamental scientific
interest [5].
Recently, Amanna and Slifka reviewed six plausible models describing the evolution
of the humoral immune response over time [2]. Four of these models were based on a
memory B-cell dependent process, assuming antibody production either due to chronic
or repeated infections, persisting antigen immune complexes on the surface of follicular
dendritic cells, or cross-reactive antigen stimulation [6], [7], [8], [9]. According
to the authors, none of these models is suitable to reproduce the evolution of antibody
levels with time after exposure to viral or vaccine antigens. In contrast with the
previous approaches, Amanna and Slifka [2] proposed two theoretical models considering
plasma cells as an independent B-cell subpopulation that is long-lived even in the
absence of replenishment by memory B-cells [5], [10]: the ‘plasma cell niche competition
model’ and the ‘plasma cell imprinted lifespan model’ [2]. There is strong evidence
that plasma cells can be long-lived when located in survival niches, especially in
bone marrow and to a lesser extent the spleen. These antibody-secreting cells could
be pivotal for the maintenance of humoral immunity [11], [12], [13], [14]. As suggested
by Radbruch et al.
[13], the first model was based on the assumption that there is competition between
resident and new migratory plasma cells for a finite number of survival niches. New
migratory plasma cells are unable to survive for long periods outside of these niches.
Since plasma cells accumulate in these niches due to new infections and reinfections
over time, the average age of plasma cells occupying the niches increases. Consequently,
the duration of the humoral response they induce should decay more rapidly with time.
The latter effect remains to be demonstrated [5]. The last model proposed by Amanna
and Slifka assumed an “imprinted” lifespan for antigen-specific plasma cells [2].
This model explicitly assumed no further division of plasma cells. In the absence
of replenishment of memory B-cells (due to reinfection or vaccine boosting), this
implies that serum antibody titers would be strongly related to the lifespan of antigen-specific
plasma cell populations. Hence, the antibody kinetics can be assumed to evolve over
three time-scales: the antibody lifespan, with an half-life ranging between 17.5 and
26 days, the short-lived plasma cell and long-lived plasma cell lifespans. However,
as noted by the authors, the imprinted lifespan model does not differentiate between
short lived-plasma cells and memory B-cell dependent mechanisms, such as the role
of persisting antigen stimulation in the early antibody kinetics, but provides insights
on the long-term persistence of antibodies after infection or vaccination and the
interplay between antibody titers and plasma cell kinetics. Although based on evidenced
immunological concepts, to our knowledge, Amanna and Slifka's models were not used
to analyze data and remained purely theoretical.
Several mathematical models have been developed to study the long-term persistence
of vaccine-induced antibodies from serological follow-up surveys, using either the
general mean titer (GMT) or individual antibody titers as an outcome measure. Most
of these studies estimated the decay rate of antibodies assuming a simple exponential
decay or including rapid and slow components for decay depending on the time after
vaccination. Using these frameworks, long-term persistence (over 25 years) of hepatitis
A (HAV) vaccine-induced immunity was demonstrated [15], [16], [17], [18], [19]. Fraser
et al. [20] proposed a model accounting explicitly for B-cell population (antibody
secreting cells) kinetics and extended their model by differentiating an “activated”
and a memory B-cell subpopulation [20], [21]. In the present study, a mathematical
formulation of the “plasma-cell imprinted lifespan” model proposed by Amanna and Slifka
[2] was implemented and used to estimate long-term persistence of anti-HAV antibodies
from two 10-year follow-up studies in adults vaccinated with inactivated hepatitis
A vaccines.
Materials and Methods
Data
Two long-term follow-up datasets were used for parameter estimation. Healthy HAV-seronegative
adults aged between 18 and 40 years were enrolled after giving their written informed
consent [17]. The first dataset included 289 subjects vaccinated with 2 doses of Havrix™
1440 with 0-6 (109 individuals) or 0–12 months (180 individuals) vaccination schedules.
This inactivated hepatitis A vaccine, manufactured by SmithKline Beecham Biologicals
and introduced in 1994, was formulated to contain no less than 1440 ELISA units (El.U)
of hepatitis A antigen (strain HM175) per 1 ml dose, adsorbed onto 0.5 mg of aluminium
salts. Subjects received the vaccine in the right deltoid muscle. Various vaccination
schedules were shown to provide similar immune responses [22]. Blood samples were
taken in each participant before vaccination, to ensure seronegativity, as well as
between the primary and boosting doses, and after booster administration. In view
of our aim with the present study - the evaluation of long-term persistence of antibodies
after a full vaccination schedule, the dataset we use here is limited to time-points
after boosting, i.e. at 1, 12, 18, 24, 30, 36, 42, 48, 50, 66, 78, 90, 102, 114 and
126 months after boosting. The second dataset included 113 subjects vaccinated with
3 doses of Havrix™ 720 according to a 0-, 1-, 6-vaccination schedule [16], [23]. This
vaccine, which is the predecessor formulation of Havrix™ 1440, contained no less than
720 Elisa units per 1.0-ml dose. Blood samples were taken at 1, 6, 12, 18, 30, 42,
54, 66, 78, 90, 102 and 114 months after the booster dose (6 months). Antibody titration
was performed using an “in-house” ELISA inhibition assay [24]. Subjects with antibody
levels below 20 mIU/ml for the ELISA test were considered seronegative.
Mathematical models of antibody kinetics
The “plasma-cell imprinted lifespan” model accounting for the dynamics of plasma cell
(P) and antibody (A) populations was considered. The plasma cell population is divided
in two subpopulations according to their specific lifespan: short- and long-lived
plasma cells denoted by
and
, respectively. Assuming no renewal, plasma cell populations decline over time with
different decay rates according to their longevity. However, long-lived plasma cells
can survive for long periods of time residing in survival niches, mainly in the bone
marrow, and could consequently be considered as virtually steady [2], [13], [14].
Finally, assuming that the antibody lifespan is short relatively to plasma cell lifespan,
antibody kinetics can be considered to reflect the underlying kinetics of plasma cell
populations [2]. Owing to these different assumptions, three nested models were explored.
Complete model
The dynamics of plasma cell and antibody populations are described by the following
system of differential equations:
(1)
Where
and
represent the average decay rates of short-lived plasma cells, long-lived plasma cells
and antibodies, respectively;
and
are the production rates of antibodies by short- and long-lived plasma cells,
is the initial antibody level,
and
are the initial population sizes of short- and long-lived plasma cells.
This system has the following analytical solution:
(2)
where
and
Asymptotic model
Assuming that the lifespan of long-lived plasma cells is infinity, i.e.
, the asymptotic total antibody production rate is a constant different from zero
. Solution (2) then becomes
(3)
Plasma cell driven kinetic (PCDK) model
Assuming that the antibody lifespan is short relatively to plasma cell lifespan
, the antibody kinetics can be considered as being an immediate reflection of the
underlying kinetics of plasma cell populations [2]. Solution (2) amounts then to
(4)
where
and
.
Parameter estimation
A non-linear mixed effects model was used to estimate model parameters
as described by Snoeck et al.
[25]. Briefly, individual parameters are assumed to be log-normally distributed and
were used to predict the antibody titer in an individual
at a certain point in time
(
) [25]. The measured antibody-titers (
) were log10-transformed for the analysis with an additive residual-error:
The
values are assumed to be normally distributed with mean zero and variance
. Population parameters were estimated using MLE by the SAEM algorithm for the hierarchical
nonlinear mixed-effects model analysis using Monolix software (http://www.monolix.org)
[26].
A nonparametric bootstrap procedure was used to determine the 95% confidence intervals
of parameter estimates permitting the evaluation of the accuracy of parameter estimates.
One thousand bootstrap replicates were generated by resampling individual profiles
for each dataset. For each bootstrap replicate, each model was refitted to get an
estimate of the population parameters. The 95% confidence interval was constructed
from the 2.5th and 97.5th percentiles for each of the population parameters [27].
For each bootstrap replicate, long-term extrapolations of antibody decay were obtained,
resulting in predictions and 95% confidence intervals of the mean duration of vaccine-induced
immunity (antibody titers higher than 20mUI/ml), as well as the mean time for the
proportion of immune individuals to decrease down to 95% and 90%.
Alternative modeling assumptions: the power-law models
Fraser et al. [20] proposed an alternative to exponential distributions of decay rates,
assuming an heterogeneity in the decay rate of B-cells expressed by a gamma distribution.
This hypothesis led to the formulation of the so-called “conventional power-law” model
previously used to model antibody persistence [28], . In [20], this model was further
improved to account for two B-cell subpopulations leading to an “asymptotic model”,
assuming that a proportion of the B-cell population does not decrease, and a “full
model”, assuming a slower decay rate for a proportion of the B-cell population. Using
the notations in [20], these three models describing the antibody kinetics are given
by:
Conventional power-law model
Asymptotic power-law model
Full power-law model
where
is the
-transform of antibody titer at time
,
is the peak
-level,
and
represent the decay rates of short-lived and long-lived plasma cells, respectively,
and
is an arbitrary constant (often set to 0). Finally,
(
) is the relative level of antibodies produced in the long-term plateau. Using the
same methodology as previously described, parameters were estimated for each power-law
model.
Model diagnostic
AIC (Akaike Information Criterion) was used for model selection. As population based
diagnostics were not very informative, goodness of fit was assessed based on diagnostic
plots for the individual predictions (IPRED), and individual weighted residuals (IWRES)
by calculation of the ε-shrinkage [31].
Results
Parameter estimates are given in Table 1. For the complete model, the population average
antibody decay rates were close to 0.8 for both datasets (95% confidence intervals
[0.63, 1.34] and [0.65, 1.36] for the first and second datasets, respectively), corresponding
to a half-life of 26 days. Under the assumption of the asymptotic model, the average
decay rate obtained with the first dataset (0.75 [0.49, 1.10]) was slightly lower
than the one obtained with the second dataset (0.95 [0.68, 1.48]); these values remained
consistent with the literature (half-lives of 27.7 and 21.9 days, respectively) [2].
Using the individual estimates of the decay rate parameter provided by Monolix as
the mean of their posterior distribution [26], we performed Kruskal-Wallis tests to
investigate the difference between the kinetics at early time-points after the boosting
dose according to model assumptions (complete and asymptotic) and vaccine formulation
(Havrix™ 1440 and Havrix™ 720). Although no difference was found between the two models
(p = 0.84), a significant difference was shown between the two vaccines, with a higher
decay rate for the oldest vaccine (Havrix™ 720; p = 0.004). However, this difference
could also be due to the inclusion of the 6-month time-point in the second dataset
which allows for a decomposition of the kinetics according to the different population
time-scales. Moreover, under the asymptotic assumption (lowest AIC), the inter-individual
variability, estimated as the standard deviation of random-effects [26], was reduced
from 84% (Havrix™ 1440) with the first dataset to 61% for the second dataset (Havrix™
720). Note that when looking at the exclusion of random effects one by one, all were
significant (5% significance level) based on a 50∶50 mixture of a
and
distribution.
10.1371/journal.pcbi.1002418.t001
Table 1
Parameter estimates according to the modeling assumptions: complete, asymptotic or
plasma-cell driven kinetics (PCDK) model (95% confidence intervals determined using
bootstrap percentile intervals).
Population parameter estimates (CI)
Havrix™ 1440 dataset
Havrix™ 720 dataset
Parameters
Complete Model
Asymptotic Model
PCDK Model
Complete Model
Asymptotic Model
PCDK Model
Φ
s
(1e3 mIU/ml* Month−1)
1.12 (0.81, 2.20)
1.04 (0.55, 1.71)
-
1.00 (0.65, 1.37)
0.97 (0.68, 1.72)
-
Φ
l
(1e3 mIU/ml* Month−1)
0.54 (0.43, 0.92)
0.51 (0.33, 0.75)
-
0.26 (0.20, 0.59)
0.40 (0.20, 0.65)
-
βs
(1e3 mIU/ml)
-
-
3.38 (2.95, 3.96)
-
-
5.56 (3.89, 8.01)
βl
(1e3 mIU/ml)
-
-
0.84 (0.70, 0.97)
-
-
1.43 (1.15, 1.71)
μs
(Month−1)
0.069 (0.062, 0.080)
0.07 (0.058, 0.074)
0.14 (0.12, 0.16)
0.014 (0.011, 0.026)
0.02 (0.013, 0.028)
0.76 (0.51, 1.04)
μl
(Month−1)
1.8e−6 (5.2e-7, 7.8e-6)
-
1.5e−3 (3.03e-5, 2.3e−3)
9.8e−4 (1.4e−4, 1.3e−3)
-
8.1e−3 (6.1e−3, 9.8e−3)
μA
(Month−1)
0.79 (0.63, 1.34)
0.75 (0.49, 1.10)
-
0.82 (0.65, 1.36)
0.95(0.68, 1.48)
-
A0
(1e3 mIU/ml)
7.79 (6.38, 12.21)
7.60 (5.90, 10.66)
-
8.62 (6.32, 14.6)
9.26 (6.27, 15.41)
-
AIC
−1626.63
−1630.63
−1354.10
−346.2
−346.35
−308.16
ε-shrinkage (%)
16
16
13
18
17
13
Testing whether
is significantly different from 0 was done using a likelihood ratio test for which
the asymptotic null distribution is a 50∶50 mixture of a
and
distribution [32], [33]. The estimate of
was not found to be significantly different from 0 with the complete model, meaning
that the lifespan of long-lived plasma cells cannot be estimated and this subpopulation
could be considered constant. Nevertheless, the inclusion of a supplementary data-point
in the early stage of the kinetics (6 month post-boosting; Havrix™ 720 dataset) permitted
to improve the estimation accuracy for the long-lived plasma-cells decay rate, decreasing
substantially the relative standard error (RSE) of the estimate from 2e4% for the
first dataset to 231% for the second (data not shown). Discarding the additional 6-month
data point from the second dataset, the asymptotic model resulted in estimates of
the antibody decay rate close to the one obtained with the first dataset (data not
shown). This result suggests that more time points during the first year would allow
estimating the three time scales using the complete model. The third model (PCDK)
assumed that the antibody decay rate can be ignored relative to the plasma-cell kinetics,
leading to an “adiabatic” formulation. For both datasets, the time scales obtained
for the short- and long-lived plasma cell lifespan differ by two orders of magnitude.
For the first dataset (Havrix™ 1440), the estimated lifespan of short-lived plasma
cells (
), averaged around 7 months, which is much longer than the 1 month antibody lifespan.
The estimated lifespan of long-lived plasma cells, averaged around 60 years (i.e.
roughly similar to the average human lifespan). For the second dataset (Havrix™ 720),
the estimated lifespan of short-lived plasma cells was close to 1 months and the estimated
lifespan of long-lived plasma cells was only 10 years. However, due to the additional
measurement at 6 months after the (final) booster dose the adiabatic assumption is
no longer valid (ignoring the antibody lifespan compared to the plasma cell lifespan).
Indeed, at the 6 months post booster point, the observed antibody kinetics are principally
driven by the antibody decay rate, implying that we can no longer assume that its
effect is negligible relative to that of the plasma-cell kinetics. In both cases,
the estimates of
are the result of a combination of antibody and short-lived plasma cell decays. However,
the lifespan of long-lived plasma cells, contributing to long-term persistence of
the humoral response, was found to be 6-fold longer with the more recent and more
potent vaccine formulation (Havrix™ 1440) than with the older formulation (Havrix™
720). The conventional power-law model assumes that the antibody level declines continuously
with time but the data suggest the existence of at least two phases of decline: a
short-term component with a high decay rate in the first 2 years of observation, followed
by a long-term component which could be thought as a “plateau” phase. The results
obtained for the two datasets using the conventional power-law model are similar with
a low decay rate (a = 0.63) reflecting both phases using only one parameter (Table
2). The inclusion of an asymptotic phase in the modified power-law model allows for
a focus on the short term dynamics. For both datasets, the decay rate estimates were
drastically increased compared with conventional power-law approaches. The decay rate
obtained with the second dataset was slightly lower than for the first dataset, but
combined with a lower peak of the antibody titer, the immunity provided by the Havrix™
720 vaccine remains weakest compared to the more recent Havrix™ 1440 vaccine. Finally,
the introduction of the second time scale, governing the long-term behaviour, referred
as “full power-law model” supports the results obtained in our study: the presence
of a supplementary point (6 months post-boosting) in the second dataset allow for
a better estimation of the long-term component. The results obtained with the first
dataset are close to the ones obtained with the asymptotic model with a decay rate
close to 0 (b = 0.07) whereas the second data set permitted to estimate a decay rate
of 0.37 for long-lived plasma cells resulting in a slow but continuous decay of the
antibody population.
10.1371/journal.pcbi.1002418.t002
Table 2
Parameter estimates using power-law model (95% confidence intervals determined using
bootstrap percentile intervals).
Population parameter estimates (CI)
Havrix™ 1440 dataset
Havrix™ 720 dataset
Parameters
Conventional power-law
Asymptotic power-law
Full power-law
Conventional power-law
Asymptotic power-law
Full power-law
k
4.13 (4.04, 4.18)
5.87 (5.67, 6.12)
6.21 (5.65, 6.97)
4.00 (3.89, 4.10)
5.29 (4.48, 5.74)
6.37 (6.12, 6.55)
a
0.63 (0.59, 0.67)
2.26 (2.07, 2.50)
2.79 (2.09, 3.40)
0.60 (0.54, 0.66)
2.01 (0.93,2.48)
3.67 (3.28, 3.88)
π
-
8.1e−4 (4.3e−4, 1.2e−3)
0.0008 (1.8e−4, 1.4e−3)
-
3.2e−3 (1.3e−3, 5.1e−3)
1.7e-3 (9.7e−4, 2.8e−3)
b
-
-
0.08 (1.8e−3, 0.16)
-
-
0.37 (0.29, 0.43)
AIC
−572.83
−1226.77
−1255.01
−128.36
−204.26
−297.35
All models showed a good consistency between individual predictions and observations
with
-shrinkage estimated between 13 and 18%. Additional data points in the early phase
of the kinetics might decrease the
-shrinkage as they provide more information on high-level antibodies. Among the six
models considered throughout this study, the lowest AIC was obtained with the asymptotic
model assuming exponential decays for antibodies and plasma cells. This model is a
derivation of the complete model by constraining the decay rate of long-lived plasma
cells to 0. Figure 1 displays the observation/prediction plot (log10 scale) for the
asymptotic model (
= 0.97).
10.1371/journal.pcbi.1002418.g001
Figure 1
Observations Vs. model predictions (left) and residuals Vs Time (right) plots using
individual parameters (Havrix™ 720 dataset, Asymptotic model, log10 scale).
Although care has to be taken using these models based on 10 years of data, long-term
individual extrapolations of antibody kinetics were derived from the individual empirical
parameter estimates for each model (complete, adiabatic and asymptotic) and the two
data sets (Figure 2). In accordance with international current practice, the positivity
threshold was fixed to 20 mIU/ml and subjects with antibody levels below this threshold
for the ELISA test were considered seronegative. Immunity was considered as lost when
a subject passed from seropositive to seronegative status [24], [34]. A focus around
the positivity threshold (20 mIU/ml, thick black line) was plotted for each model
and dataset to monitor the population serological response according to time post-boosting.
For the first dataset, including only one point in the first year after vaccination
(1 month), the asymptotic, complete and power-law models gave similar results with
a life-long immunity for all vaccinated patients. Conversely, for the adiabatic PCDK
model a proportion of the population loses humoral immunity, with the first seronegative
patient occuring 20 years after vaccination. However, the proportion of seronegative
patients 100 years after vaccination did not exceed 15% (figure 3), showing a good
long-term efficacy of the vaccine. The mean time to immunity waning was 216 years
(95%confidence interval [143.0, 848.6], table 3). The results for the second data
set differ according to the model assumptions. Although the asymptotic model gave
similar results as for the first dataset predicting lifelong immunity due to the supposed
asymptot, results with the complete and adiabatic approach were divergent. The complete
model was found closer to the adiabatic due to the existence of an additional sample
time in the early phase of the kinetics (6 months). Although the power-law models
predicted lifelong immunity for both vaccines, the estimate of the decay rate of long
lived plasma-cells was found to be higher for the second dataset, confirming that
the “plateau” assumption in the asymptotic model provides crude approximations of
the actual long-term kinetics. Adiabatic model predictions showed that the total population
lost immunity within 100 years after vaccination. Moreover, the mean time to lose
immunity was evaluated to be 43 years (95% confidence interval [34.8, 52.0]; Table
3).
10.1371/journal.pcbi.1002418.g002
Figure 2
Individual prediction plots with a focus around the positivity threshold (20 mIU/ml,
black line).
(a,c,b) Havrix™ 1440 dataset, (d,e,f) Havrix™ 720 dataset; (a,d) complete model, (b,e)
plasma-cell driven kinetics model, (c,f) asymptotic model.
10.1371/journal.pcbi.1002418.g003
Figure 3
Predicted proportion of seropositive patients according to time post vaccination from
the plasma-cell driven kinetics model (full blue line: Havrix™ 1440 dataset , dashed
green line: Havrix™ 720 dataset).
10.1371/journal.pcbi.1002418.t003
Table 3
Long-term prediction of HAV antibody dynamics obtained with complete and plasma cell
driven kinetics (PCDK) models (95% confidence intervals determined using bootstrap
percentile intervals).
Havrix™ 1440 dataset
Havrix™ 720 dataset
Complete Model
PCDK Model
Complete Model
PCDK Model
Mean Time to immunity waning (years)
1.7e5 (4.7e4, 6.7e6)
216.1 (143.0, 848.6)
237.1 (188.5, 1.7e3)
43 (34.8, 52.0)
Time below 95% of immune patients (years)
7.6e4 (1.7e4, 3.4e5)
63 (31.6, 576.9)
147.1 (111.2, 1.1e3)
23.4 (17.7, 25.3)
Time below 90% of immune patients (years)
1.0e5 (2.8e4, 4.3e5)
77.4 (52.6, 681.4)
169.4 (126.6, 1.2e3)
24.4 (22.2, 29.3)
Discussion
A mathematical model, based on the “imprinted plasma cell lifespan model” proposed
by Amanna and Slifka, was developed to study the long-term persistence of antibodies
after vaccination with inactivated HAV vaccines [2]. Previous studies showed that
anti-HAV antibodies can persist for at least 25 years and that a two-phase decay of
antibody levels occurs according to the time since vaccination [35], [36]. However,
the models used for the estimations were solely based on the antibody dynamics and
did not handle the underlying immunological mechanisms. Plasma-cells are the main
antibody-secreting cells and it is currently recognized that some of these cells can
survive for extended periods when located in survival niches, especially in the bone
marrow [12], [13], [14]. The model used in our study assumed that the antibody kinetics
are determined by three time-scales: the antibody, the short-lived plasma cell and
long-lived plasma cell lifespans (complete model). Two other approaches were derived
from the complete model:
assuming a constant long-lived plasma cell population (asymptotic model) close to
the model of Fraser et al. [20].
ignoring the antibody lifespan (assumed to be short compared with plasma-cell lifespans
(plasma cell driven kinetic model)).
The complete model, which should be the best representation of the actual process
including three time-scales (antibody, long- and short-lived cell life-spans), did
not allow for accurate estimates, especially concerning the decay rate of long-lived
plasma cells (RSE>200%). The asymptotic model permits to estimate the antibody decay
rate corresponding to the shortest time scale (around 1 month) [3]. However the hypothesis
of the asymptotic model, assuming a constant antibody production by long-lived plasma
cells residing in niches in the bone marrow and considered as surviving in the host
for life, generates a cost on long-term predictions of the antibody decay which cannot
be studied using this approach. The third approach, called “plasma cell driven kinetic”,
considers the antibody kinetics to immediately reflect the underlying kinetics of
plasma cell populations. Thus, ignoring the antibody decay, which cannot be distinguished
from plasma-cells, allows for fitting the long-term kinetics. However, the interpretation
of the parameters is not straightforward, especially when detailed data are available
in the initial phase of the kinetics, which corresponds to the antibody decay (table
2). Although our model selection criterion (AIC) tends to select the asymptotic model,
all three models have their own interest depending on the research question:
Asymptotic model: Study of the short-term antibody decay and particularly the duration
of antibody lifespan.
Plasma cell driven kinetic model: Study of long-term behavior, permitting to estimate
the mean time to waned immunity.
Complete model: Global approach that could allow dealing with the two previous research
questions. However, this approach would need additional data, especially in the initial
phase of the antibody decay after vaccination, which would permit to identify the
transition between the adiabatic and the asymptotic hypotheses.
Combining the results obtained with each of these models, the average antibody lifespan
was estimated to be around one month that is consistent with the literature whereas
the average plasma cell lifespans varied from 3 to 7 months for short-lived plasma-cells,
and over 60 years for long-lived plasma cell.
Power-law models present a relevant alternative to the modelling framework based on
plasma cells imprinted lifespan, both from a methodological and from a biological
point of view. In absence of emperical evidence for “heterogeneity in the decay rate
of B-cells” given the data at our disposal, exponential decays were assumed for short-lived
and long-lived plasma-cells. The main results of our study rely on the fact that three
time-scales were biologically relevant to explain the antibody decay: the antibody,
the short-lived and long-lived plasma cell lifespans. The power-law models as described
in Fraser et al. [20] included at most two time-scales, which could explain the differences
observed in the fits. This conclusion is supported by the results obtained with the
“Plasma-Cell Driven Kinetics” (PCDK) model, which accounted for two time-scales and
for which the AIC values were close to the one obtained with the full power-law model
(also accounting for two time scales). Thus, whenever relevant data would be available,
the coupling of the two approaches offers an appealing perspective for future immunological
research.
Using individual parameter estimates, the mean time to immunity waning was estimated
to be 43 years for the individuals vaccinated with Havrix™ 720 vaccine. Similar results
were previously obtained by Van Herck et al.
[16] who estimated the individual slow decay rate of antibodies (between months 76
and 128 post boosting) and estimated the mean number of years before an individual
reached the seroconversion level (20mIU/ml) to 45 years. With the same methodology,
less than 15% of individuals vaccinated with the latest vaccine formulation (Havrix™
1440) were estimated to lose their immunity 100 years after boosting, showing possible
life-long vaccine-induced immunity. Although these results are based on long-term
extrapolation and could be influenced by immunosenescence and other distortions of
immunity, they elucidate in a simple way the observed differences between the two
vaccines.
Accounting for correlations between random effects was not found to impact the accuracy
of parameter estimates obtained with the PCDK model (data not shown). Computational
problems, due to convergence failure, avoided the inclusion of such correlations when
analyzing the data with the asymptotic and complete models. However, based on the
results obtained with the PCDK model, the main conclusions of this study are deemed
to be robust to this specific misspecification of the random effects distribution.
The effect of such misspecification would require further research which is beyond
the scope of the present study.
These results have a number of direct implications:
In immunology, it offers a quantitative assessment of the time scales over which plasma
cells and antibodies live and interact. This insight may provide a basis for further
quantitative research on the immunology, with direct consequences for understanding
the epidemiology of infectious diseases.
In vaccinology, it offers an opportunity for clinical trial researchers to collect
relevant information early on, in order to make long term predictions on immunity
conferred by vaccines. We showed in particular that antibody levels measured within
a year after a booster dose provide highly relevant information for long term predictions
of protective immunity over time.
In health policy, it offers more than a purely intuitive basis to make recommendations
on booster vaccinations. Our models for hepatitis A suggest that this would not be
required at least within a 40 year time span after the booster vaccine dose.
A further improvement of our mathematical model could include the explicit interaction
between humoral and cellular immunity. This would involve nonlinear coupling terms.
The validation of such theoretical generalisations would require much more refined
data not only about antibodies but also about B-cell and T-cell subpopulations.