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Inadequacy of the "Minimum Inhibitory Concentration" > THE
INADEQUACY OF MIC: ILLUSTRATIONS
The Inadequacy of the Minimum Inhibitory Concentration
by Joachim Gruber
HERMAN MATTIE,
LICHEN ZHANG, ELISABETH VAN STRIJEN, BEATE RAZAB SEKH,
AND ANNA E. A. DOUWESIDEMA
Department of Infectious Diseases,
University Hospital Leiden, 2300 RC Leiden, The Netherlands
ANTIMICROBIAL AGENTS AND CHEMOTHERAPY,
Oct. 1997, p. 2083?2088 Vol.
41, No. 10
Abstract
Problem
When a bacteria population can be
exposed to an antibiotic invitro in the same way as it is in vivo, the
invitro experiments quantify the efficacy of the antibiotic.
When the antibiotic has strong side
effects, one has to optimize between those effects and the risk of losing
control over the bacteria population. In such critical cases an invitro
model with only one parameter, the Minimum Inhibitory Concentration, may
not be able to describe the development of an invitro bacteria population
accurately enough. The Emax model presented here provides more accurate
predictions, thus helping with the optimization. It can do so only at the
expense of needing more information about the effect of the antibiotic,
i.e. more model parameters. The pharmaceutic industry is willing to provide
that information. It seems that the ball is now in the field of the treating
physician: S/He needs to voice interest in those data.
Methods
The development of the number N[C,
t] of the bacteria over time t under the influence of an antibiotic at
concentration C is described by a nonlinear first order differential equation.
Four approximations of the growth or decay rate R[C, t] are used.

In the traditional MIC model, the rate
R is constant in time and independent of C, except at two threshold concentrations
C = MIC (Minimum Inhibitory Concentration) and C = MBC (Minimum Bactericidal
Concentration, MBC > MIC). When C increases across these thresholds, the
rate jumps:

when starting from C = 0 C passes
through MIC, the rate jumps from a positive value, R, to 0, and

when C passes through MBC the rate jumps
from 0 to  R.

In three socalled Emax models the rate
R may depend on the antibiotic concentration C and the time t over which
the bacteria have been exposed to the antibiotic.

In the "adaptation" model, the rate
R is a smoothed version of the MIC model jump at C = MIC, and thus also
independent of t, i.e. R[C]. The time dependency in R describes the process
of adaptation of the bacteria population to the antibiotic with a simple
exponential law.

In the two other Emax models the time
dependency of R[C, t] is represented by an asymmetrical bellshaped curve
with a rate maximum E_{R}at time t_{max}.

In the complete version of this model,
both the rate maximum E_{R }and its time t_{max }depend
on C: The functions E_{R}[C] and t_{max}[C] are socalled
"Hill expressions", which are basically smoothed step functions.

In a simple version, t_{max}
does not change with C. Only the rate maximum E_{R} is concentration
dependent as given by a "Hill expression" E_{R}[C].
Results

When the antibiotic (meropenem) dose
is chosen so that the bacteria population neither grows nor decays, i.e.
when the meropenem concentration C is by definition near MIC, the MIC model gives a number of surviving bacteria typically deviating from the experimental values by less than plus or minus one order of magnitude.

At concentrations C >> MIC, the MIC model gives too large bacteria populations (typically by two or more orders of magnitudes), thus being a conservative model.

Independent from the meropenem concentration C the (compared to the MIC model more sophisticated) Emax model predicts bacteria population sizes that deviate from the data by typically a factor of three.
The results have been calculated with
a program written in Mathematica (version 3.0). The program
is a useful tool for conveying a feeling for the quantitiative influence
of the system parameters. The program and my examples can be read with
the free MathReader:
The Mathematica Notebook Reader.
Introduction:
Pharmacokinetics, Pharmacodynamics and
the Influence of the Immune System

Pharmacokinetics calculates the distribution
of the pharmaceutical within the body, i.e. in a system of interconnected
compartments.

Pharmacodynamics calculates the development
over time of the bacteria population under the effect of a pharmaceutical.
(A) The pharmacokinetic model is potentially
misleading to the unexperienced pharmacologist when

the data and paramters describing a
certain compartment have not been determined (as is often the case for
intracellular locations) and

the fraction of the bacteria population
in that compartment is nonnegligible.
(B) Two similar statements apply to
pharmacodynamic models. The model might not help in directing the therapy
and control the illness when

the dynamics of a bacteria population
in a certain compartment under the influence of the pharmaceutical is unknown,

the reaction of the immune system to
the combined action of the pharmaceutical and the bacteria cannot be quantified.
Therefore, the pharmaceutical has to
be and often is chosen such that neither (A) nor (B) is relevant, meaning
that given over a fixed period valid for all of us the drug kills off
or inhibits the growth a relevant fraction of the bacteria population,
and the immune system does the rest to restore our health.
For short, I will call this "the
one size fits all concept", meaning that all of us will be cured by the
same therapy. When this therapy fails and we are sure that our diagnosis
was correct, we talk of "niches"
into which the bacteria can withdraw and be out of reach of the drug and
immune system. Niches have been demonstrated

in technical systems, e.g. in water
supply systems where one possible niche is a biofilm, as well as

in living systems, e.g. in our sinusmaxillary
floor.
Such "niches" are the subject of research
and therefore not usually known to our treating physicians. We as patients
need to call his attention to the forefront of research  ideally by finding
the relevant medical literature with the help of e.g. Medline.
Treatment failure manifests itself
in a continuing shift of body parameters from normal to borderline or "positive"
for a more or less wide range of illnesses. In many cases we do not know
the mechanism that produces the shift, but from experience we know how
to interpret it. Our interpretation might be called a "global understanding".
Of such global character is a large
part of our practical knowledge of the immune response that we apply in
therapies. Here is an example in the context of a case of neuroborreliosis
[Gruber, 2002]:

Whereas
the antibiotic itself brings about only exponential changes of the number
of bacteria,

the immune
system is shown to effect temporal oscillations of the number of the bacteria,
some of which are "selforganized".
A conservative antibiotic treatment
regime

must consider selforganized immune
system oscillations a sign for an ongoing active infection [Burrascano
2003, Gruber 2003]. and

can have its end only after the cessation
of these oscillations.
A more sophisticated treatment would
be accompanied by mathematical model calculations. When sufficiently quantitative
microscopic models of the immune response are missing, one might conceptually
attempt to describe the development with time t of the number N of bacteria
with the following global nonlinear differential equation
(1)
where the nonlinearity as incorporated
in R_{t} summarizes the combined action of

the antibiotic and

the immune system
Such models
have been presented in the literature and are being used to optimize the
therapy. But they exceed the scope of this paper. So, the pharamcodynamics
discussed here will ignore the influence of the immune system.
I. The MIC Concept
By definition, a population of N bacteria
exposed in vitro to a concentration C of an antibiotic stops growing when
C is equal or higher than the Minimum Inhibitory Concentration (MIC).
When C drops below the MIC, the population grows with a rate R_{0},
where R_{0} is now the fraction of the population that enters the
cell division phase per unit time.
I. 1 NonLinear Differential Equation
(2)
I. 2 Growth and KillRates R_{0}
As soon as the antibiotic concentration
exceeds the Minimum Bactericidal Concentration (MBC), the bacteria population
decays.
In the case of a cell wall
antibiotic, the straighforward model would assume that when exposed to
an antibiotic concentration C > MBC any bacterium dies that tries to cell
divide. Thus, the fraction of bacteria population that dies per unit time
is equal to the fraction of that population that would cell divide in the
absence of the cell wall antibiotic:
R[C > MIC] =  R[C
< MIC] =  R_{0} (3)
With the definitions
R[C <
MIC] = growth rate
 R[C > MIC]
= kill rate,
eq. (3) can be written as
kill rate = growth rate
With this simplification, the equation
describing the population dynamics in the presence or absence of an antibiotic
can be written as
(4)
where
(5)
I. 3 Bacteria Population Dynamics N[C,
t]
The bacteria population dynamics N[C,
t] can be calculated from eq. (4) as
("MIC model")(6)
It is shown in Figs. 1
and 1a.
Fig. 1: Development of a
bacteria population as described by the MIC concept. xaxis is the time
axis (t measured in hours) yaxis gives the concentration C of meropenem
in mg/l. The zaxis gives the logarithm of the number N[C, t] of S. aureus.
N[C, t] as specified in (6), with R_{0}
= 0.33 /h, MIC =
0.029 mg/l. The figure shows the development
of the population when MBC = MIC. (Click on Figure to see the development
for MBC = 3 MIC. )
The population

either grows with a growth rate R_{0}
when the antibiotic concentration C < MIC,

decays with the (kill) rate R_{0}
when C > MIC = MBC.
The publication of
Mattie
et al. quantifies the drawback of the MIC concept and proposes an improved
pharmacodynamic model, which belongs to the class of socalled "Emax models".
II. The Emax Model: Mathematical Formulation
The improved mathematical model presented
by Mattie
et al. is based on 3 assumptions:

The bacteria population dynamics is
given by a nonlinear differential equation.

The bacteria population reproduces or
dies with a rate R[C, t], for which an empirical
form is chosen. R[C, t] depends on 2 parameters E_{R}[C] and t_{max}[C].

For the parameters E_{R}[C]
and t_{max}[C] empirical Hill expressions
are chosen.
II. 1 NonLinear
Differential Equation
The number of bacteria in the presence
of the antibiotic is calculated with a nonlinear first order differential
equation similar to the one used for the MIC model:
(7)
II. 2 Growth and KillRates
R
For the rate R[C, t] the authors chose
an empirical expression:
(8)
where e is the base of the natural
logarithm (e = 2.718). This empirical expression is visualized in Fig.
2.
A yet simpler approximation would
replace the time dependence of R expressed in equation (8)
with an adaptation term associated with the growth rate R_{0}
[Mouton et al. 1997, Koop et al.
2000, Zhi et al. 1986].
(8m)
where 1/a
is
the time constant for the bacteria population adaptation to the antibiotic.
Note the slightly different definitions
of ER[C] in eqs. (8) and (8m): E_{R}[C]
in eq. (8) has a factor e = 2.718 in front of it, whereas
in eq. (8m) it does not.
Fig. 2: Staphylococcus aureus
strain 1 rate R[C, t] as a function of time t and meropenem concentration
C, calculated with "complete model" and the pharmacodynamic
data eq. (16).
Clickon
upper part of figure to see R[C, t] calculated with "complete",
"adaptation" and MIC model.
Click here
to see a "lake" with surface R[C, t] = 0 filled into the eq.(8) R[C,
t] surface. The shoreline of this lake is the contour line R[C, t]
= 0, and thus by definition the funtion MIC[t].
The lower part of the figure shows
cross sections through the "complete" model surface in the upper part:

left side: cross sections at constant
C (C = 0, 0.05, 0.1, 0.15, 0.2, 0.25, ..., 1 mg/l) and

right side: cross sections at constant
t (t = 0, 1, 2, 3, 4, 5, 6h).
In the absence of meropenem (C = 0),
S. aureus strain 1 growth rate is R_{0 }= 0.33 /h. The minimum
of R[C, t] as a function of t lies at t_{max}[C].
II. 3 Bacteria Population Dynamics N[C,
t]
The differential equation (7)
is then solved by separation of the variables
(9)
and subsequent integration of both
sides of the equation (9):
(10)
("complete model") (11)
A simpler versioin of eq. (11) would
neglect the concentration dependence of t_{max}[C],
replacing t_{max}[C]
with the constant T.
("simplified model") (12)
When the adaptation concept eq. (8m)
is plugged into eq. (9) the integration yields
("adaptation model") (13)
(4) Hill
Expressions for E_{R}[C] and t_{max}[C]
Hill expressions are being used to quantify
the dependencies of E_{R }and t_{max} on the antibiotic
concentration C:
(14) (click on equation to see graphical representation of E_{R}[C]
for meropenem)
(15) (click on equation to see graphical representation of t_{max}[C]
for meropenem)
III. Results
III. 1 Pharmacodynamic Data for Meropenem
Emax Model
The following parameters are the ones
that produced the best fits of the Emax models to the experimental data
(Fig. 5)
"complete" model"
EC50 = 0.047 mg/l;
(16)
S = 2.28 h^{1};
T = 0.79 h;
s_{1} = 2.48;
s_{2} = 1.6.

"adaptation model"
EC50 = 0.047
mg/l; (16m)
S = 1.0 h^{1};
a = 0.5
h^{1};
s_{1} = 2.48;

III. 2 Population Dynamics in the Presence
of Meropenem
Fig. 3 is a plot
of the Emax population dynamics models ("complete" and "simplified") of
S. aureus as a function of time t and meropenem concentration C.
Fig. 3: Development of number
N[C, t] of S. aureus as a function of time t (xaxis) and meropenem
concentration C (yaxis). As in Fig. 1, the zaxis
gives the logarithm of the number N[C, t] of S. aureus. N[C, t] is specified

in the "complete" model (11,
lower surface) with empirical socalled Hill expressions (14)
and (15) for E_{R}[C] and t_{max}[C],
respectively, and

in the "simplified" model (12,
lower surface), where an empirical Hill expression is used only for E_{R}[C],
whereas t_{max}[C] = T (a constant).
III. 3 "Concentration" and
"TimeDependent" Antibiotics
The Emax models Fig. 3,
eqs. (11,
12) show that the effect
of an antibiotic can be dose or timedependent, depending on the concentration
of the antibiotic.

At low antibiotic concentrations (e.g.
C = 0.05 mg/l, point A in Fig. 3),

the effect of the antiibiotic is called
"concenrtationdependent", meaning that

increasing the antibiotic C while keeping
the exposure time t constant (e.g. t = 2 h) will reduce the number N more
effectively than increasing the exposure time. Hence, here

Example: moving from initial position
A {t = 2 h, C = 0.05 mg/l} to final position B {t = 2 h, C = 0.1 mg/l}
reduces the population of S. aureus by approximately a factor of 20. A
similar reduction of N cannot be achieved by inreasing the exposure time
t.

At high antibiotic concentrations (e.g.
at C = 0.15 mg/l, point C in Fiig. 3) the situation is reversed
and the effect is called "time dependent":

increasing concentration C effects only
little change, whereas

increasing t is much more effective.
Often, an antibiotic is characterized
as having a concentration or a timedependent effect, when its usually
applied dosage places it in the concentrationdependent region A or in
the timedependent region C, respectively [e.g. Goodman
and Gilman's, Craig 1998]. This habit is misleading,
since it is not the antibiotic that has that property but rather the dosage.
III. 4 Comparison of Population Dynamics
Models
Fig. 4 compares the four population
dynamics models

the "complete" model, eq. (11),

the "simplified" model, eq.(12),

the "adaptation" model, eq. (13).
Note: S is refitted (S = 0.9 h) to improve position of the "adaptation
model" surface N[C, t] relative to experimental data.

the "MIC model", eq. (4),
Fig. 4: Comparison of MIC
model (6, Fig. 1, MBC = MIC) with
Emax models (11, 12, 13).
Click on Figure to see comparison with MIC model in which MBC = 3 MIC.
The MIC model underestimates the
effect of the antibiotic except at concentrations near EC50
= 0.047 mg/l.
Fig. 5 shows experimental data presented
in the paper [Mattie
et al.] and sections through Fig. 4 at constant meropenem concentrations
C. The comparison with the "adaptation model" has been placed into a second
layer Fig. 5a.
Fig. 5: Number N[C, t] of S.
aureus 1 organisms exposed to meropenem in vitro as a function of exposure
time t, calculated with

eq. (11) ("complete"
Emax model, heavy curves),

eq. (12) ("simplified"
Emax model, normal curves), click on figure to see "adaptation" model eq.
(13) instead of "simplified" model eq. (12).

eq. (6) (MIC model
with MIC = MBC = 0.03 mg/l, i.e. the surface N[C,t] in Fig. 1,
dashed curves).

Points are experimentally determined
numbers N_{x}[C,t] of S. aureus 1 (Fig.2 of [Mattie
et al]),
Meropenem concentrations C are 0.016,
0.032, 0.064, 0.128, 2.0 mg/l. Model (11) and
(12) curves coincide at C = 2.0 mg/l. MIC = 0.029 mg/l
is taken from Fig. 6, i.e. MIC is arbitrarily evaluated
at t = 1 h. If bacteria would be killed only when they cell divide, their
number would decrease as shown by lower dashed curve. Experimentally determined
numbers (coinciding more or less with heavy curves) show a faster decay
of population than that.
III. 5 Comparison Summary

The MIC model, eq. (6),
has no explanation for the observed more rapid decay of the bacteria population
at intermediate times:

In the MIC model the cell wall antibiotic
can at most kill all those cells that undergo cell division. So the rate
of growth R_{0} and
the maximum kill rate are conceptually bound to
be identical.

This is not how the "complete "model
interprets the experimental data: In the fitting process the minimum of
the R[C, t] curves at t_{max} (lower part of Fig. 2)
is not fixed to lie within the range 0 < R[C, t] <  R_{0}.
The "complete" model represents the data best when (again see lower part
of Fig.
2)

the kill rate
exceeds the growth rate R_{0}
initially, i.e.
R[C > EC50, t < t_{max}[C]]
<  R_{0 }(except
for times near t = 0), and

then declines to a value within that
range at later times:
0 > R[C > EC50, t >> t_{max}[C]]
>  R_{0}.
The microbiological process responsible
for the decline has not yet been fully understood. It is assumed that we
see a phenotypical selection during the exposure to the antibiotic [Mouton
et al.].

The simple Emax models, approximations
(12) and (13), might be called
"conservative" in the following sense: These models overestimate the number
of surviving bacteria by 2 orders of magnitude (or less), as one can see
in Figs. 4 and 5.

At large antibiotic concentrations (C
>> EC50) as well as at large exposure times t >> t_{max},
the simplified Emax model eq. (12) differs only insignificantly
from the complete Emax model eq. (11).

The larger the antibiotic concentrations,
the less the complete Emax model and simplified
Emax model curves deviate from each other.

At large exposure times t >> t_{max}
and large antibiotic concentrations C >> EC50, the MIC model eq. (4)
deviates from the full Emax model by less than an order of magnitude (Fig.
5).
IV. REFERENCES
Burrascano
JJ, "Diagnostic hints and treatment guidelines for Lyme and other tick
borne diseases, 14. ed., 2002, COURSE
DURING THERAPY
Craig
WA, Pharmacokinetic/pharmacodynamic
parameters: rationale for antibacterial dosing of mice and men. Clin. Infect.
Dis., 1998, 26:110
Goodman
and Gilman's "The Pharmacological Basis of Therapeutics", Chapter.
43, "Pharmacokinetic Factors", search inside book for "1160", from results
select "on Page 1160".
Gruber
J,
Compartment models
displaying Lyme disease symptom cycles, 2002.
Gruber
J, Burrascano's Guidelines and Immune
Response Modeling, 2003.
Koop
AH, Neef C, van Gils SA.
A mathematical model for the efficacy and toxicity of aminoglycoside (April
2003). Workshop "Predictive Value of PK/PD models of antimicrobial
drugs". Leiden University Medical Center, Leiden, The Netherlands, 4 
5. September 2003.
Mouton
JW, Vinks AA, Punt NC. PharmacokineticPharmacodynamic
Modeling of activity of ceftazidime during continuous and intermittent
infusion. Antimicrobial Agents and Chemotherapy 1997;1(4):733738.
model 3 in Mouton
et al., page 734
Vinks
AA, Punt NC and Mouton JW.
PharmacokineticPharmacodynamic Modeling of Bacterial Growth and Killing
using the Modified Zhi Emax model. Workshop "Predictive Value of
PK/PD models of antimicrobial drugs". Leiden University Medical Center,
Leiden, The Netherlands, 4  5. September 2003.
Zhi J, Nightingale
CH, Quintiliani R. A
pharmacodynamic model for the activity of antibiotics against microorganisms
under nonsaturable conditions. J Pharm Sci 1986;75(11):10637.
Zhi JG, Nightingale CH, Quintiliani
R. Microbial
pharmacodynamics of piperacillin in neutropenic mice of systematic infection
due to Pseudomonas aeruginosa. J Pharmacokinet Biopharm 1988;16(4):35575.
V. APPENDIX
V. 1 Dependence of MIC on Exposure Time
By definition, the Minimum Inhibitory
Concentration is the antibiotic concentration C = MIC at which the bacteria
population neither grows nor decreases, i.e. dN/dt = 0 in eq. (7).
This is the case when R[C, t] = 0.
Solving the "complete" model rate
expresssion eq. (8) for C, one can calculate the dependence
of MIC on the time the bacteria population has been exposed to the antibiotic.
Eq. (17) or (17m) can be solved for
MIC as a function of t. Mathematica can do this graphically. Fig. 6 is
Mathematica's plot of the function MIC[t].
Fig. 6: Dependence of MIC
on the time t during which the bacteria population has been exposed in
vitro to the antibiotic meropenem. The MIC's used in this paper are MIC[t
= 1 h] = 0.029 mg/l, e.g the MIC in Fig.1 and MIC[t
= 3 h] = 0.024 mg/l.
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