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Antimicrobial Agents and Chemotherapy, July 1998, p. 1731-1737, Vol. 42, No. 7
0066-4804/98/$04.00+0
Copyright © 1998, American Society for Microbiology. All rights reserved.
Comparisons between Antimicrobial Pharmacodynamic
Indices and Bacterial Killing as Described by Using the Zhi
Model
S.
Corvaisier,1
P. H.
Maire,1,2,*
M. Y.
Bouvier
d'yvoire,3
X.
Barbaut,1,2
N.
Bleyzac,1 and
R.
W.
Jelliffe2
ADCAPT, Department of Pharmacy, Hospital
Antoine Charial, Francheville,1 and
Hoechst-Marion-Roussel Co., Paris,3
France, and
Laboratory of Applied Pharmacokinetics, University
of Southern California School of Medicine, Los Angeles,
California2
Received 24 January 1997/Returned for modification 20 June
1997/Accepted 3 May 1998
 |
ABSTRACT |
Various suggestions have been made for empirical pharmacodynamic
indices of antibiotic effectiveness, such as areas under the drug
concentration-time curve in serum (AUC), AUC>MIC, AUC/MIC, area under
the inhibitory curve (AUIC), AUC above MIC, and time above MIC
(T>MIC). In addition, bacterial growth and killing models, such as the Zhi model, have been developed. The goal of the present study was to compare the empirical behavior of the Zhi model of bacterial growth and killing with the other empirical pharmacodynamic indices described above by using simulated clinical data analyzed with
the USC*PACK PC clinical programs for adaptive control of drug therapy,
with one model describing a concentration-dependent antibiotic
(tobramycin) and another describing a concentration-independent antibiotic (ticarcillin). The computed relative number of CFU was
plotted against each pharmacodynamic index, with each axis parameterized over time. We assumed that a good pharmacodynamic index
should present a clear and continuous relationship between the time
course of its values and the time course of the bacterial killing as
seen with the Zhi model. Preliminary work showed that some
pharmacodynamic indices were very similar. A good sensitivity to the
change in the values of the MIC was shown for AUC/MIC and also for
T>MIC. In addition, the time courses of some other
pharmacodynamic indices were very similar. Since AUC/MIC is easily
calculated and shows more sensitivity, it appeared to be the best of
the indices mentioned above for the concentration-dependent drug, because it incorporated and used the MIC the best. T>MIC
appeared to be the best index for a concentration-independent drug. We also propose a new composite index, weighted AUC (WAUC), which appears
to be useful for both concentration-dependent and
concentration-independent drugs.
| |
INTRODUCTION |
Success in antibiotic therapy is
defined by bacterial killing and by improvement in the patient's
clinical status. It is a retrospective measure. However, before
antibiotic therapy is begun, the choice of the drug and the dosage
regimen must be planned in advance to achieve maximum predicted
efficacy with a tolerable risk of toxicity. Adaptive control of
antibiotic dosage regimens, by using pharmacokinetic models, can
predict (and therefore control) plasma antibiotic concentrations
(2). However, drug levels at the site of the infection
itself may be somewhat different from the plasma drug concentrations,
except during septicemia when the infection is in the bloodstream. Time
delays exist between antibiotic administration and the achievement of
antibiotic efficacy. Because of this, plasma antibiotic concentrations
at time t cannot be automatically correlated with
pharmacodynamic effect at the site of the infection at the same time
t, even though general improvement in clinical status can be
empirically correlated with such plasma antibiotic concentrations.
Various empirical pharmacodynamic indices for a 24-h period of therapy
using plasma or serum antibiotic levels have been proposed to predict
antimicrobial effectiveness at steady state and at the site of
infection (17-20). At the steady state, their values at the
end of a typical 24-h period appear to be related to the success or
failure of therapy
some of them for a concentration-dependent drug,
some of them for a concentration-independent drug (8, 11, 18,
20). However, the initial exposure to effective serum antibiotic
levels is also most important and may be necessary to minimize the
emergence of adaptive resistance (4) and as well as the
emergence of resistant bacterial subpopulations (5). Indeed,
the time course of pharmacodynamic indices during the initial 24 h
of therapy appears to be especially important in achieving early
success in therapy. With the Zhi model, as the number of CFU is
calculated and plotted over time, a dynamic view of the other empirical
pharmacodynamic indices (the evolution of each of the other indices
over time) has been studied here. The value of each index was computed
throughout the initial 24 h of therapy, and the index value was
also found at the end of the initial 24-h period, rather than at some
subsequent steady state. This was done to emphasize the
importance of rapid and effective bacterial killing at the start of
therapy, especially for life-threatening infections.
The goal of the present study was to compare the bacterial growth and
killing calculated according to the model proposed by Zhi et al.
(22) with the various empirical pharmacodynamic index values
and their time courses during a typical simulated first day of therapy.
In this analysis, we have assumed that a good pharmacodynamic index
should show a clear and continuous relationship between its values and
the bacterial killing observed with the Zhi model: that is, that during
periods of bacterial killing, the value of the pharmacodynamic index
under consideration should increase, and during periods of bacterial
growth, the index should decrease (or at least increase only very
slowly).
| |
MATERIALS AND METHODS |
Pharmacokinetic model.
We chose to use a one-compartment
pharmacokinetic model with intravenous administration. With maximum a
posteriori probability (MAP) Bayesian fitting to individual patient
data, plasma antibiotic levels can be predicted in clinical situations
with such a linear one-compartment model. Using pharmacokinetic
population data available in the USC*PACK PC clinical programs (9,
10), we simulated typical clinical treatment situations for two
representative antibiotics by using a representative simulated adult
patient (age, 30 years; height, 175 cm [69 in.]; weight, 70 kg; and
creatinine clearance, 120 ml/min). We assumed he was being treated for
Pseudomonas aeruginosa septicemia. Antibiotic concentrations
at the site of the bloodstream infection were assumed to be equal to
serum antibiotic concentrations. Two patient data files were created:
one for a concentration-dependent drug, tobramycin, and one for a
concentration-independent drug, ticarcillin. The dosage regimens used
in these simulations were 200 mg of tobramycin infused over 30 min
every 12 h (slightly less than 6 mg/kg per day) and 6 g of
ticarcillin infused over 30 min every 8 h (slightly more than 260 mg/kg per day). Table 1 shows the
pharmacokinetic parameter values for each drug and the characteristics
for each dosage regimen.
Zhi pharmacodynamic model.
Bacterial growth and killing were
described by computing the relative number of CFU versus time. CFU were
computed with the Zhi model (22), which states that
|
(1)
|
where B is the initial bacterial inoculum (CFU per
milliliter), G is the rate constant for exponential
bacterial growth of a single bacterial population in the absence of
antibiotics (per hour) Kt is the instantaneous
rate constant for bacterial kill in the presence of the antibiotic (per
hour), which depends on the profile of serum antibiotic levels,
Ct (micrograms per milliliter) present at any
time t.
Kt follows a sigmoid
Emax
model:
|
(2)
|
where
Kmax is the maximum possible rate
constant for bacterial killing (per hour)
C50 is
the antibiotic concentration which
produces a kill rate equal to
Kmax/2 (micrograms per milliliter),
and
is
the Hill sigmoidicity coefficient.
Killing was computed by using values for
G,
Kmax, and
previously obtained (
1,
3) for
P. aeruginosa in the presence
of tobramycin or
ticarcillin (Table
2).
The USC*PACK PC clinical programs for adaptive control of tobramycin
and ticarcillin therapy compute serum antibiotic concentrations
every 6 min. The pharmacodynamic effect of bacterial killing was
computed by
using these concentration profiles as input to the
effect model
described in equation 1 above.
When the serum antibiotic level is equal to the pharmacodynamic MIC,
the bacterial killing rate equals the bacterial growth
rate, the
bacterial population remains unchanged, and equation
1 becomes
|
(3)
|
The bacterial apparent growth rate (
G 
Kt) is then equal to zero. By combining
equations 2 and 3, an expression relating
C50 to
the pharmacodynamic MIC (zMIC) can be obtained (
13):
|
(4)
|
By combining equations 1, 2, and 4, the time course of the
bacterial population can be computed by using a model for which
the
relevant variables (for a single antibiotic and for a single
bacterial
population with an unchanging growth rate constant and
for which the
MIC of the antibiotic is unchanging) are the serum
antibiotic levels at
time
t (
Ct),
B,
G,
Kmax, the zMIC, and
(
14).
Computation of pharmacodynamic indices.
The pharmacodynamic
indices generally proposed to correlate with antimicrobial
effectiveness at the steady state have been the area under the drug
concentration-time curve in serum (AUC); the AUC when C > MIC (AUC>MIC) (the AUC above the MIC; the AUC divided by the MIC,
or AUC/MIC; the area under the inhibitory curve (AUIC), or the AUC when
C > MIC divided by the MIC; and the time above the MIC
(T>MIC) (Table 3)
(18). All indices were computed versus time by employing an
approximate integration based on the trapezoidal method (7).
The accuracy of this method depends on a factor of
1/n2, for which n is the number of
trapezoids which are used (here, 10 per h, with serum drug
concentrations computed every 6 min).
A dynamic view of the various empirical pharmacodynamic indices was
introduced here in order to relate them to the Zhi model.
Each
simulation was performed throughout the initial 24 h (first
doses)
of therapy, and the final index value was obtained at the
end of the
initial 24-h period, rather than for a typical 24-h
period at steady
state, as others have done. Continuing this dynamic
view, it is
noteworthy that the term
T>MIC has had several different
meanings. Some have used this name to refer instead to the percent
of
the time during the dose interval in the steady state when
the serum
drug concentrations are at least the MIC. Another somewhat
similar
definition has been the percent of the time during a typical
day in the
steady state that the serum drug concentrations are
at least the MIC.
However, the name itself specially refers to
the total time, during
some defined time period, that the concentrations
are at least the MIC.
In the present paper, we have used the term
to mean the total time
during the first 24 h of therapy that the
serum drug
concentrations are at least the MIC. This index will
progress from the
minimum value of zero toward the maximum value
(here 24 h) as the
duration of therapy progresses from the very
beginning (time zero) to
the completion of the first 24 h of therapy.
Because the pharmacodynamic indices are many and varied, preliminary
work was first done to compare their dimensional equations,
their
dynamic evolution over time, and the index value at the
end of the
initial 24-h period. They were then evaluated to select
the most useful
indices for comparison with the Zhi model.
Simulation.
Tobramycin and ticarcillin profiles in serum
were simulated by using the USC*PACK clinical programs (10).
The time courses of the computed CFU were then plotted against the time
courses of each pharmacodynamic index (MATLAB software). Each axis was parameterized versus time (as a phase-space plot, like a hysteresis loop), for different MICs of each drug. The MICs of tobramycin studied
were 16, 8, 4, 2, 1, 0.5, 0.25, and 0.125 µg/ml, and those of
ticarcillin were 128, 64, 32, 16, 8, 4, 2, and 1 µg/ml
(21). The initial inoculum was always assumed to be
106 CFU/ml.
| |
RESULTS |
The resulting simulated tobramycin and ticarcillin serum drug
level profiles obtained with their respective population models are
shown in Fig. 1.
Study, comparison, and selection of pharmacodynamic indices.
Table 3 shows the different pharmacodynamic indices, their mathematical
expressions, and their dimensional equations. Comparison of
pharmacodynamic indices was first performed for those indices having
the same dimensional units. Several indices were found to be very
similar. For example, AUIC and AUC/MIC were indistinguishable. For each
MIC and for both antibiotic agents, a linear relationship between AUIC
and AUC/MIC was found. This is probably because the area not included
in the computation of the AUIC was extremely small, because the serum
drug levels were above the MIC the great majority of the time. The
method of calculating the index value is simpler for AUC/MIC than for
AUIC. In addition, regarding calculation of the AUC>MIC and AUC above
the MIC during the first 24-h period (Table 3),
Ct MIC was nearly equal to
Ct for low MICs, and for each
antibiotic-bacterium pair. The comparison of the mathematical expressions of AUC>MIC and AUC above MIC shows that AUC>MIC is nearly
equivalent to AUC above the MIC for low MICs.
Because of this, only four pharmacodynamic indices were finally
compared to the computed CFU by using the Zhi model. They
were AUC,
AUC>MIC, AUC/MIC, and
T>MIC.
Comparison between pharmacodynamic indices and the Zhi model of
bacterial growth and killing.
Plots of CFU versus AUC, AUC>MIC,
AUC/MIC, and T>MIC, are shown in Fig.
2 for tobramycin (panel A for AUC, B for
AUC>MIC, C for AUC/MIC, and D for T>MIC) and for
ticarcillin (panel A' for AUC, B' for AUC>MIC, C' for AUC/MIC, and D'
for T>MIC). Figure 3 shows
the relationship between final pharmacodynamic index values found at
the end of the first 24-h period (on a logarithmic scale) and the MICs.
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FIG. 2.
CFU time course versus each pharmacodynamic index for a
concentration-dependent drug, tobramycin, and for a
concentration-independent drug, ticarcillin, during the first 24-h
period of therapy, determined with the previous plasma drug level
profiles.
|
|
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FIG. 3.
Course of pharmacodynamic index values found at the end
of the first 24-h period versus the different tested MICs of a
concentration-dependent antibiotic and a concentration-independent
antibiotic determined with the plasma drug level profiles shown in Fig.
1.
|
|
| |
DISCUSSION |
Many pharmacodynamic indices have been developed to predict
antimicrobial effectiveness. Some of them were not studied
herespecifically the peak serum antibiotic level/MIC ratio (6,
15, 16), because no dynamic or cumulative evolution of this index
was possible for comparison with the Zhi model.
For the other indices shown in Table 3, two types of index could be
distinguished, depending on their dimensional units. The indices of
AUC, AUC>MIC, and AUC above MIC are all in micrograms times hours per
milliliter. The indices of AUC/MIC, AUIC, and T>MIC are in
hours. Contrary to Schentag et al. (20), none of these
pharmacodynamic indices was found to be dimensionless.
The indices AUC/MIC, AUC>MIC, AUIC, and AUC above MIC were defined
from the AUC by including the MIC in their calculations in different
ways, as shown in Table 3. T>MIC is clearly distinguishable from the others. Indeed, it is the only index for which the value is
not computed with the AUC. As mentioned above, some indices were very
similar and could not be distinguished from each other, by using our
dynamic view, especially AUC/MIC versus AUIC and AUC>MIC versus AUC
above the MIC. For these reasons, only four pharmacodynamic indices
were finally compared with the Zhi model: AUC, AUC/MIC, AUC>MIC (two
different ways of including MIC in the calculation), and the
T>MIC (independent of the AUC).
Figure 2 shows the CFU versus each of the pharmacodynamic indices (AUC,
AUC/MIC, AUC>MIC, and T>MIC) and their evolution during the initial 24-h period. In the MIC ranges tested, and for the first
24 h, the relationships of CFU to each evolving pharmacodynamic index were somewhat similar for both the concentration-dependent antibiotic and the concentration-independent one when the AUC/MIC was
less than 250 h. However, for the concentration-independent drug,
there was little relationship between AUC/MIC and killing beyond this
point. This illustrates the saturation of the effect relationship for
the concentration-independent drug. In fact, the relationship of any of
the pharmacodynamic indices studied to bacterial killing depended
essentially on the AUC and the MIC.
It was evident that AUC alone does not take into account the
sensitivity of the bacterium, since MIC is not included in its calculation. For different bacterial sensitivities represented by the
different MICs, the values of the AUC were always the same at the end
of the initial 24-h period (Fig. 2A and A'). Thus, AUC alone was not a
good index.
For the AUC>MIC index, derived from AUC, but which takes into account
the relationship of the plasma drug concentration with at least the MIC
in the calculation, discontinuities were seen as CFU increased while
the index value did not (Fig. 2B and B'). This is because this method
does not include concentrations below the MIC, which means that the
bacterial killing rate constant at those times is zero. However, this
does not seem realistic. Even if the bacterial population increases,
the apparent growth rate constant is reduced, and the killing rate is
not zero. Because of this assumption, a continuous relationship does
not exist between the values of the AUC>MIC and CFU. In addition,
there was only a minimal relationship between CFU and AUC>MIC. The
relationship between AUC>MIC and CFU was not very different from that
found for AUC versus CFU.
In contrast, for the AUC/MIC, the more its values increased, the
greater was the killing seen with the Zhi model (Fig. 2C and C').
Conversely, treatment was not effective when AUC/MIC was small. A
continuous relationship exists between AUC/MIC values and the resulting
CFU. Even if serum antibiotic levels are below the MIC with the
apparent growth rate only slightly reduced, AUC/MIC values increased
very slowly, showing the small to modest effect of the sub-MIC serum
antimicrobial level upon the bacterial growth rate. The difference
between concentration-dependent and concentration-independent killing
is also shown in Fig. 2C and C'. For those concentrations relatively
near the MIC, killing was almost linear with concentration. However,
deviations from linearity were seen above this, especially in Fig. 2C',
showing the lack of concentration dependence at the higher AUC/MICs, in
the saturable region of the relationship. It is for this reason that
when concentrations are reasonably above the MIC, the T>MIC
becomes the most significant empirical index for killing. All such
relationships were well shown with the Zhi model.
For the index of T>MIC, however, it was also clear, as
shown in Fig. 2D and D', that the dynamic evolution and growth of
T>MIC were strongly related to bacterial killing, with a
significant negative correlation being found between it and CFU.
However, discontinuities were also observed. For all therapy, the
maximum possible value of T>MIC is 24 h. Suppose that
serum drug levels on one regimen are already above the MIC for a
particular dosage regimen of antibiotic and that another regimen of the
same antibiotic generates serum drug levels twice as high. The maximum
value of T>MIC will still be 24 h for either regimen.
Because of this, the effectiveness of antibiotic treatment could not be
distinguished if the second regimen was actually somewhat more
effective than the first one, the smaller but already effective
regimen. On the other hand, the other indices (AUC, AUC>MIC, and
AUC/MIC) do not have such maximum values. Therefore, the entire scale
of therapeutic effectiveness can be explored with the other indices,
whereas that is not the case with T>MIC. Even with
T>MIC, which is expressed as linear percent of the dose
intervals having serum drug concentrations above the MIC, a similar or
the same maximum value (or 100%) is always reached.
The greatest change in the values of the empirical indices found at the
end of the initial 24-h period, for two different MICs, was obtained
with AUC/MIC. For example, with tobramycin MICs of 0.125 and 1.0 µg/ml, there was no relationship between AUC and MIC (58.52 versus
58.52 µg · h · ml
1), a modest one for
AUC>MIC (58.48 versus 57.82 µg · h · ml1), a greater one for T>MIC (23.7 versus
12.1 h), and the greatest one for AUC/MIC (468.16 versus 58.52 h)
(Fig. 3). Somewhat similar results were also seen for ticarcillin,
although saturation of the effect relationship was clearly seen at the
higher concentrations. AUC/MIC was therefore the pharmacodynamic index
which had the greatest sensitivity to differences in the various MICs
of either antibiotic, whether the antibiotic was concentration
dependent or concentration independent (Fig. 3).
A summary of the characteristics of each index (AUC, AUC>MIC,
T>MIC, and AUC/MIC) and of a theoretical ideal index is
presented in Table 4. Of the indices
presented above, AUC/MIC appeared to be the best because it
incorporated and used best the MIC of a concentration-dependent
antibiotic and also, interestingly enough, to some degree, that of a
concentration-independent antibiotic, although saturation of the effect
relationship was clearly seen at higher concentrations. The AUC/MIC for
the initial 24-h period has been previously described as the most
useful index of fluoroquinolone antimicrobial activity against P. aeruginosa (12). In that study, however, it was shown
that AUC/MIC was an index of efficacy only for the
concentration-dependent antibiotic (12) and was not sufficient for the concentration-independent antibiotic. Conversely, T>MIC was the only index to be described as an index of
efficacy for the concentration-independent antibiotic (18).
In addition, a change in antibiotic administration strategy (for
example, more doses at closer intervals) may produce the same AUC/MIC
but two different T>MICs. Equally, a change in the dose
administered might produce a change in AUC/MIC but not in T>MIC.
Because of this, we also propose a new empirical pharmacodynamic index
for which AUC/MIC is weighted by T>MIC, in order to take
into account both the concentration-dependent part of the antibiotic
efficacy and the concentration-independent part. We therefore propose a
new composite pharmacodynamic index, the weighted AUC (WAUC), for the
first 24 h, which is the AUC/MIC weighted by the percentage of the
total time for which the serum drug level is above the MIC:
|
(5)
|
where (
T>MIC)
max equals 24 h (see
above). The units for WAUC are hours. This index considers (i) the
total dose administered
and the clearance of the drug through the AUC,
(ii) the sensitivity
of the bacteria to the MIC, and (iii) the
percentage of time for
which serum drug level is above the MIC through
the ratio
T>MIC/24
h. This index can be used both for a
concentration-dependent drug
and for a concentration-independent drug
with a high sensitivity
to change in MICs (Fig.
3). It shows a more
direct relationship
between its values and bacterial killing both for
the concentration-dependent
drug and for the concentration-independent
drug (Fig.
4). Obviously,
further
clinical evaluation of this proposed index is needed before
more
conclusions can be drawn.
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FIG. 4.
CFU course versus our new pharmacodynamic index for a
concentration-dependent antibiotic (left panel) and a
concentration-independent antibiotic (right panel) during the first
24-h period of therapy, determined with the previous plasma drug level
profiles. Some saturation of the relationship is still seen at WAUC
values above 400 h.
|
|
Conclusion.
The Zhi model describes a saturable Hill model of
killing, coupled with assumed logarithmic growth. For
concentration-dependent drugs, the concentrations are on the steep
slope of the effect relationship, and saturation of the effect is not
seen. It is for this reason that the model is concentration dependent
when trough concentrations are near or below the MIC; here, AUC/MIC is
usually the best empirical index. However, when concentrations are
significantly above the MIC, saturation of the effect relationship takes place, and killing becomes relatively independent of
concentrations; here, T>MIC is usually the best empirical
index.
A good pharmacodynamic index must present a clear relationship between
its values and bacterial killing. Actually, none of
the empirical
indices generally used brings together all of these
characteristics,
and none approaches the utility of the Zhi model
itself to reflect the
actual dynamic process of bacterial growth
and killing. AUC/MIC appears
the best for the concentration-dependent
antimicrobial agent, and
T>MIC appears best for the concentration-independent
antimicrobial agent. Based on this, a new pharmacodynamic index
for
antimicrobial drugs, WAUC, is also proposed. This new index
appears
useful for both concentration-dependent and concentration-independent
antimicrobial agents. A clear (nearly linear) relationship has
been
found between its values and bacterial growth and killing
reflected by
the Zhi model, as shown in Fig.
4, although some
saturation of the
effect relationship is still seen at higher
concentrations. This index
is dependent not only on time but also
on the concentration and the MIC
throughout the duration of the
therapy.
None of the pharmacodynamic indices, old or new, could be used in order
to estimate the efficacy of a combination of antimicrobial
agents.
Because of this and because it is now easily available
in clinical
software, the Zhi model itself (or any similar dynamic
model of
bacterial growth and killing) probably still represents
the current
optimal clinical index of therapeutic effectiveness.
The Zhi model actually represents a useful "worst-case model" for
the evaluation of the efficacy of a drug dosage regimen.
On the one
hand, while some organisms may have a reduced or slowed
rate of growth
as some substrates for their growth become scarce,
the Zhi model always
makes the worst-case assumption that the
organisms are always in their
logarithmic phase of most rapid
growth. While this may not be entirely
realistic, it nevertheless
furnishes a useful worst-case assumption for
the evaluation of
the potential utility of a proposed drug dosage
regimen.
The Zhi model contains no provision for describing the emergence of
bacterial resistance during therapy. However, if one considers
and uses
the highest MIC the emerging resistant organism is estimated
to attain
during therapy, the Zhi model again provides a useful
worst-case model
for the evaluation of any proposed antibacterial
regimen. If a proposed
dosage regimen is successful in killing
according to the Zhi model and
the highest MIC for the organism
is estimated to be attained, as
described herein, that regimen
is quite likely to be effective
clinically, because the clinical
situation may actually contain a
decreasing rate constant for
growth rather than the fixed one for the
logarithmic phase of
growth, and the organisms are not likely to be
fully resistant
from the very beginning of therapy. Because of this,
the Zhi model
for a single organism, assumed to be at its most
resistant from
the very start of therapy, provides a somewhat more
stringent
and rigorous test of the effectiveness of a dosage regimen
than
more complex models having several subpopulations, if one analyzes
the behavior of the most rapidly growing and most resistant possible
strain of organism. The computations are relatively simple and
have
already been incorporated into clinical software. As more
complex
models become available, they are likely to be less stringent
than the
single Zhi model.
| |
FOOTNOTES |
*
Corresponding author. Mailing address: ADCAPT, Service
Pharmaceutique, Hôpital Antoine Charial, 40 avenue de la Table de Pierre, 69340 Francheville, France. Phone: (33) 04-72-32-34-87. Fax:
(33) 04-72-32-39-08. E-mail:
adcapt{at}cismsum.univ-lyon1.fr.
| |
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Abstract
Introduction
