Modeling of Microbial Population Responses to Time-Periodic Concentrations of Antimicrobial Agents

Modeling of Microbial Population Responses to Time-Periodic Concentrations of

Antimicrobial Agents

Michael Nikolaou,1* Amy N. Schilling,2 Giao Vo,2 Kai-tai Chang,2 Vincent H. Tam,2*

1Department of Chemical Engineering; 2College of Pharmacy

University of Houston, Houston, TX

*Corresponding authors and mailing addresses:

Michael Nikolaou

Chemical Engineering Department

University of Houston

Houston, TX 77204-4004

Phone: (713) 743 4309; Fax: (713) 743 4323

Email: nikolaou@uh.edu

Vincent H. Tam

University of Houston College of Pharmacy

1441 Moursund Street

Houston, TX 77030

Phone: (713) 795-8316; Fax: (713) 795-8383

Email: vtam@uh.edu

Abstract

We present the development and first experimental validation of a mathematical

modeling framework for predicting the eventual response of heterogeneous (distributed-

resistance) microbial populations to antimicrobial agents at time-periodic (hence

pharmacokinetically realistic) concentrations. Our mathematical model predictions are

validated in a hollow-fiber in vitro experimental infection model. They are in agreement

with the threshold levofloxacin exposure necessary to suppress resistance development of

P. aeruginosa in a murine thigh infection model. Predictions made by the proposed

mathematical modeling framework can offer guidance for targeted testing of promising

regimens. This can aid the development and clinical use of antimicrobial agents that

combat microbial resistance.

Keywords: Antimicrobial agents, Antimicrobial resistance, Mathematical modeling,

Pharmacodynamics, Pharmacokinetics.

1 Introduction

Resistance of microbial populations to antimicrobial agents is a problem that has reached

alarming proportions, as has been repeatedly emphasized by a number of investigators

and clinicians4; 16; 8; 10; 7; 23; 12. Combating microbial resistance involves both preservation

of the efficacy of existing agents as well as development of new ones. The design of

effective dosing regimens (dose and dosing interval) for either task is crucial. Indeed,

even when developing new agents, for which identification of a new molecule with

potentially therapeutic properties is usually thought to be the limiting step, using the right

dosing regimen may make a dramatic difference. This is exemplified by the case of

daptomycin (Cubicin®, Cubist Pharmaceuticals)22, whose development was abandoned

in the 1970s because of muscle toxicity concerns only to be rekindled by its new

developer in the 1990s and finally reach FDA approval for clinical use in 2003 using only

a different dosing regimen19 (once daily and weight-based vs. the original three-times

daily). Therefore, in addition to their obvious use in the design of clinically useful dosing

regimens, predictive methods guiding the experimental dosing regimen testing can

greatly affect the development of antimicrobial agents. Yet such predictive methods are

currently lacking, leaving the use of empirical indices as the main tool that guides dosing

regimen testing. Given the combinatorially large number of alternative dosing regimens

that need to be tested before successful ones are identified, the use of empirical indices,

which may leave out promising tests or suggest unnecessary ones, may seriously hamper

the development or use of antimicrobial agents. To remedy this situation, we propose in

this paper an approach to the identification of promising dosing regimens that is based on

the mathematical modeling framework introduced by Nikolaou and Tam17. This

framework, originally developed to model the effect of antimicrobial agents on microbial

populations of heterogeneous resistance, is here extended to account for the effect of

antimicrobial agents at pharmacokinetically realistic time-periodic concentrations.

In the remainder of the paper we provide a review of the basic mathematical

modeling framework and preliminary results derived from it. Next, we discuss our main

theoretical results along with experimental validation. Finally, we draw conclusions and

provide directions for further research.

2 Background Review and Preliminary Results

2.1 Homogeneous microbial population under time-invariant agent

concentration

A homogeneous microbial population in an environment of an antimicrobial agent at

concentration ( )C t satisfies the mass balance

kill rate physiological

due to agentgrowth rate

( ) ( ( )) ( )= −gdN K N t r C t N tdt

, 0(0) =N N (1)

where ( )N t is the total number of microbial units at time t; gK is the physiological

microbial growth rate per microbial unit (net effect of natural microbial growth and

death); and ( )r C is the microbial kill rate coefficient induced by the antimicrobial agent

per microbial unit, which is a non-decreasing function of the antimicrobial agent

concentration C. A commonly used expression for ( )r C is25

Cr C KC C

where kK is the maximum kill rate coefficient; 50C is the concentration at which 50% of

the maximum kill rate coefficient is attained; and H is the Hill factor, indicating whether

( )r C is heavily inflected ( 1)H >> or not.

For a time-invariant concentration C the standard solution of eqn. (1) is

( )ln ( )gN t K r C tN

= − , (3)

indicating an exponentially growing, declining or constant microbial population,

depending on whether ( )ˆ gK r Cα = − is positive, negative or zero. Therefore, the value

of ( )r C in comparison with gK represents the microbial resistance to a specific

antimicrobial agent at concentration C. The critical concentration of the antimicrobial

agent, crC , for which 0α = , i.e.

cr( ) 0gK r C− = , (4)

is related to the standard and widely used concept of minimal inhibitory concentration

(MIC), defined as the agent concentration for which the size of a microbial population at

24 hours shows no visible growth.5; 14

Remark 1 – Limits of applicability of eqn. (1)

In reality, a growing microbial population will eventually reach a saturation value maxN

as described by the logistic growth equation 20

max kill rate

due to agentphysiological growth rate

( )( ) 1 ( ( )) ( )gdN N tK N t r C t N tdt N

⎛ ⎞= − −⎜ ⎟

⎝ ⎠, 0(0) =N N (5)

While eqn. (5) is more accurate than eqn. (1) for values of N near maxN , eqn. (1) is

acceptable for max<<N N , which is the focus of this work. ■

2.2 Heterogeneous microbial population under time-invariant agent

concentration

Unlike what eqn. (3) indicates, ( )log ( )N t hardly ever depends linearly on t for real

microbial populations. Most notably, initial decline of ( )N t may be followed by later

regrowth. For time-invariant agent concentration, such behavior is due to the fact that

microbial populations are heterogeneous, consisting of subpopulations with differing

degrees of antimicrobial resistance corresponding to different kill rate coefficients ( )ir C

at a given antimicrobial concentration C . It is customary to lump subpopulations with

( )i gr C K< into one resistant subpopulation and the remaining subpopulations, with

( )i gr C K≥ , into one susceptible subpopulation. Eqn. (1) is then applied to each of the

two subpopulations. While this approach is conceptually appealing and computationally

simple, it is only a rough approximation of the real system and may fail to predict

phenomena such as regrowth17. To account for this, Nikolaou and Tam17 developed a

mathematical modeling framework to capture the effect of antimicrobial agents on

heterogeneous microbial populations. This framework considers the distribution of the

kill rate coefficient ( )r C over a microbial population. A virtually continuous distribution

approximation is reasonable, because the size of a microbial population in an infection is

of the order of 8 910 10− . In this framework, the dynamics of the entire microbial

population N are

( )1( ) ( )gdN K t N tdt

κ= − (6)

1( )nn

d tdtκ κ += − , 1n ≥ (7)

where ( )n tκ , 1n ≥ , are the cumulants26 of the kill rate coefficient distribution function

( , )f r t at time t. Note that the first four cumulants are directly related to the average, µ ,

variance, 2σ , skewness, 33

, and kurtosis excess, 44 3µ

σ− , of the kill rate coefficient

distribution, as 1κ µ= , 22κ σ= , 3 3κ µ= , and 4

4 4 3κ µ σ= − . Therefore, eqns. (6) and

(7) can be written as

( )( ) ( )gdN K t N tdt

µ= − (8)

2( )d tdtµ σ= − (9)

3( )d tdtσ κ= − (10)

Eqn. (8) indicates that the entire population grows or declines according to the average

kill rate coefficient ( )tµ . Eqn. (9) suggests that the decline rate of the average kill rate

coefficient is proportional to the spread of the kill rate over the microbial population.

Finally, eqn. (10) suggests that the change rate of the spread of the kill rate depends on

the initial distribution of the kill rate over the microbial population. For example, if the

initial kill rate is almost normally distributed, then 0nκ = for 3n ≥ , which implies that

the kill rate will remain almost normally distributed with the same spread for a period of

time. Certainly, it would be interesting to determine the evolution of the kill rate

distribution with time for various initial distributions, particularly bimodal. This is a

subject of current investigation. However, under fairly mild assumptions, the above eqns.

(6) and (7) can be proven17 to yield explicit results for the population size

( )2 2

( ) (0) (0)ln (0) 1

N t K t eN A A

σ σµ −

⎛ ⎞ ⎛ ⎞≈ − + + −⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

, (11)

average kill rate coefficient

2 2(0) (0)( ) (0) At At

t e Re bA A

σ σµ µ − −≈ − + ≡ + , (12)

and variance of the kill rate coefficient

2 2( ) (0) Att eσ σ −≈ , (13)

where (0)µ and 2(0)σ are the average and variance, respectively, of ( )r C for the initial

population, and 0A > is the decline rate for the average and variance of ( )r C at a given

Eqn. (11) indicates that for the entire population to be eradicated by the agent, it is

not enough that the initial average kill rate coefficient, (0)µ , be larger than the growth

rate constant, gK , but it must be 2(0)(0)ˆ gb K

Aσµ= − > , namely the eventual kill rate

coefficient lim ( )t

b tµ→∞

= (by eqn. (12)) must exceed gK . The eventual kill rate

coefficient, b, corresponds to the most resistant subpopulation of the original population.

The most resistant subpopulation will eventually dominate, thus rendering the eventual

population homogeneous, in agreement with the limit of eqn. (13) as t →∞ .

The essence of eqns. (6) and (7), particularly when they can be simplified as in

eqns. (11) and (12), is that the entire population can be characterized without explicit

description of each particular subpopulation corresponding to a certain level of resistance.

Remark 2 – Accounting for biofitness

In the preceding analysis the growth rate constant, gK , has been assumed to be the same

for all subpopulations, because of common growth physiology. However, it could well

be that resistant strains of a species adapt their biofitness by lowering their growth rate

constant from gK to g gK Kδ− , where gKδ refers to the biofitness cost. In that case, the

above equations (6) – (13) remain intact, the only notational difference being that all

cumulants refer to the kill rate coefficient ( )r C plus the biofitness cost term gKδ . ■

3 Main Results – Theory

3.1 Homogeneous microbial population under time-periodic

antimicrobial agent concentration

Assume now that the antimicrobial agent concentration does not remain time-invariant

but fluctuates periodically (due to constant elimination and periodic injections) as shown,

for example, in Figure 1. If the fluctuations have period corresponding to a dosing

interval T, then

( )( )C t C t T= + , 0t ≥ (14)

The kill rate coefficient ( ( ))r C t corresponding to the above agent concentration will

satisfy a similar periodic relationship as in eqn. (14).

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Under these conditions it can be shown that the total population ( )N t exhibits

periodic patterns with period T, and the values of ( )log ( )N nT , 0,1,2,...,n = lie on a

straight line, similarly to eqn. (3), as made explicit by the following result:

Theorem 1 – Homogeneous population dynamics under time-periodic agent

concentration

Assume that a homogeneous population of 0N bacteria satisfying eqn. (1) is subjected to

periodically fluctuating antimicrobial agent concentration ( )C t satisfying eqn. (14).

a. The bacterial population is

( )ln ( )tt TT

gN t tK t DT r C dN T

η η−

= − − ∫ (15)

where tT

indicates the integer part of the real number tT

1 ( )ˆT

D r C dT

η η= ∫ (16)

is the time-averaged kill rate coefficient.

b. At times t nT= , 0,1,2,...=n the total population satisfies the equation

( )ln gN nT K D nT

N= − , 0,1, 2,...=n (17)

Proof: See Appendix 1. ■

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Remark 3 – Interpretation of eqn. (17)

Similarly to eqn. (3) (for time-invariant agent concentration C), eqn. (17) (for time-

periodic agent concentration ( )C t ) indicates that the points 0

( )ln N nTN , 0,1,2,...=n will lie

on a straight line corresponding to the time-averaged kill rate coefficient D, eqn. (16). In

other words, the points 0

( )N nTN appear as if they were generated by a system under time-

invariant agent concentration D, a fact that significantly simplifies the ensuing analysis.

Therefore, according to eqn. (17),

DK > (18)

implies eradication of the entire population, whereas 1g

DK < implies eventual proliferation

of the microbial population (Figure 2) except for the case where eradication can occur

during the first dosing interval (Figure 3). The latter case can occur if the minimum of

ln ( )N t , 0 t T≤ ≤ , is at or below 0. As Figure 3 indicates, the minimum of ln ( )N t

occurs at time mint , at which

min minmin

ln 1( ) ( ) 0( )

d N dNt tdt N t dt

= = (19)

min cr( )C t C= (20)

by eqn. (4). ■

We can now ask under what conditions Eqn. (18) is satisfied. We provide a

qualitative approximate answer in the subsequent Theorem 2, and a quantitative answer

for representative pharmacokinetics/pharmacodynamics (PK/PD) in Theorem 3.

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Theorem 2 – Effect of AUC/MIC on agent bactericidal activity under realistic PK

For time-periodic agent concentration corresponding to dosing interval T as in eqn. (14),

the value of g

DK is given, to first-order approximation, by

( )cr cravg cr cr

( ) ( ) AUC /1 1 1g g g

r C r CD TC C CK K K C

⎛ ⎞′ ′≈ + − = + −⎜ ⎟

⎝ ⎠ (21)

where the area under the curve (AUC) is defined as

AUC ( )ˆT

C t dt= ∫ (22)

Remark 4 – Significance of Theorem 2

Eqn. (21) indicates that in order to design a dosing regimen that results in bactericidal

effect, the average concentration of the agent, avgC , must be above crC , so that eqn. (18)

hold. It is inferred from eqn. (21) that the effectiveness of an agent is approximately

related to the well known PK/PD parameter crAUC/MIC AUC/C≈ . However, it should

be stressed that the dependence of an agent’s effectiveness on AUC/MIC is only

approximate; if ( )r C is fairly linear in the neighborhood of crC the approximate is

reasonable. A more accurate index would have to be used to account for strong effects of

higher-order derivatives in the series expansion of eqn. (21). This motivates the results

presented in section 3. ■

We will show in the next Theorem 3 that if the agent concentration follows the

realistic pharmacokinetic pattern

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max( ) ktC t C e−= , 0 t T≤ < , (23)

(Figure 1) where 1/ 2

ln 2tk = is the agent elimination rate constant (reciprocally proportional

to the half-time 1/ 2t ) and T is the dosing interval, then the value of g

DK can be influenced

by selecting two dimensionless variables associated with the dose and dosing interval of a

dosing regimen, namely avg

crˆ C

Cz = (or, equivalently, avg

50ˆ C

Cy = ) and ˆx kT= , where

1 ( )ˆT

C C t dtT

= ∫ (24)

is proportional to the administered daily dose (mass of agent over 24 hour period). We

will also show that the functional dependence of g

DK on x , z depends on two

pharmacodynamic parameters: H and k

Theorem 3 – Bactericidal effect as function of avg

crˆ C

Cz = and ˆx kT= for different values of

H and k

Under the assumptions in eqns. (1), (2), (23), and with D defined as in eqn. (16), we have

( )( )

( ) ( )( )

11 ln1

111 ln1

HK e xzK ek

HK xzg g K e

He xyeH

KDK K Hx

− +=

cr50 1ˆ k

HC KHC Kχ = = − ,

- 14 -

daily dose [mg]cr24 [hours] clearance [L/hour]ˆ C

Cz C= = ,

daily dose [mg]5024 [hours] clearance [L/hour]ˆ C

Cy C= = ,

ln 2ˆ Ttx kT= = ,

T is the dosing interval, and crC is defined through eqn. (4).

Remark 5 – Significance of Theorem 3

Eqn. (25) makes it clear that for a given agent, for which the values of H and k

KK have

been estimated from preliminary experimental data, one can visualize the agent

effectiveness, i.e. value of g

DK , as a function of two variables that characterize a dosing

regimen, namely avg

crˆ C

Cz = and ˆx kT= . Figure 4 shows a small library of such patterns for

different values of values of H and ( )50

HK CK C= + , along with associated plots of the

scaled kill rate coefficient ( )k

r CK as a function of

CC . A careful examination of these

patterns for g

DK reveals qualitatively different behaviors for different values of H and

HK CK C= + . Figure 4 quantifies Vogelman and Graig’s well known classification of

antimicrobial agents into two broad categories, characterized according to whether agent

activity is gradually or steeply concentration-dependent24. ■

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3.2 Heterogeneous microbial population under time-periodic

antimicrobial agent concentration

As discussed in section 2.2, realistic microbial populations are heterogeneous, in that not

all parts of a population are killed by an agent at the same rate. Therefore, to characterize

the entire population as a function of time one should follow each subpopulation using an

equation analogous to eqn. (1) with time-periodic ( )C t . This task would be

straightforward to accomplish numerically, if the initial distribution of kill rates and

corresponding parameters were known. Unfortunately, such knowledge would be

unavailable for all practical purposes. In addition, a lumping analytical description, akin

to eqns. (6) and (7), would be challenging as well. While such an analytical description

would be worth pursuing on its own merit from the viewpoint of a mathematical biology,

what is of interest from a therapeutic viewpoint is not the exact behavior of the

population at all times, but rather its eventual behavior, namely whether the population

will eventually be eradicated or survive. It is clear that to rigorously characterize the

eventual behavior of the entire population it is necessary and sufficient to track its most

resistant subpopulation. This task can be accomplished as follows:

For a heterogeneous population subjected to a number of time-invariant agent

concentrations C, eqn. (12) indicates that the population-average kill rate coefficient will

eventually reach a value b for each C. This C-dependent kill rate coefficient, ( )b C ,

corresponds to the most resistant subpopulation, which will eventually dominate the

entire population, and which is homogeneous, as suggested by eqn. (13) when t →∞ .

Therefore, it is reasonable to assume that the functional dependence of b on C follows

eqn. (2), namely

- 16 -

b H Hb

Cb C KC C

Similarly, eqn. (12) suggests that the average kill rate coefficient at time 0t = is R b+ .

Nikolaou and Tam17 showed that it is reasonable to assume that the initial average kill

rate coefficient depends on C in a way similar to eqn. (2), namely

50 50 50

( ) ( ) ( )b

k k b H HH H H Hb

C C CR C b C K R C K KC C C C C C

+ = ⇒ = −+ + +

Finally, it is reasonable to assume that the constant A, corresponding to the rate of decline

of the kill rate coefficient with respect to time depends on C as

A H HA

CA C KC C

Therefore, if experimental data are available from time-kill studies (measurements

of population size at various sampling points in time, for a number of time-invariant

concentrations C), then the parameters involved in eqn. (11) after the expressions for b,

R, and A are substituted from eqns. (26), (27), and (28), respectively, can be identified.

Then, predictions can be made for the effectiveness of a dosing regimen according to eqn.

4 Main Results – Experiments

4.1 Materials and Methods

Antimicrobial agent. Levofloxacin powder was obtained from Johnson &

Johnson Pharmaceutical Research and Development, LLC (Raritan, NJ). A stock

solution at 1024 µg/ml in sterile water was prepared, aliquoted, and stored at -70°C.

Prior to each susceptibility testing, an aliquot of the drug was thawed and diluted to the

- 17 -

desired concentrations with cation-adjusted Mueller-Hinton II broth (Ca-MHB) (BBL,

Sparks, MD).

Microorganism. A standard wild type P. aeruginosa strain ATCC 27853

(American Type Culture Collection, Rockville, MD) was used in the study. The bacteria

were stored at -70°C in Protect® (Key scientific products, Round Rock, TX) storage

vials. Fresh isolates were sub-cultured twice on 5% blood agar plates (Hardy

Diagnostics, Santa Maria, CA) for 24 hours at 35°C prior to each experiment.

Susceptibility studies. Levofloxacin minimum inhibitory concentration (MIC) /

minimum bactericidal concentration (MBC) were determined for the bacterial strain in

Ca-MHB using a macrobroth dilution method as previously described 15. The final

concentration of bacteria in each macrobroth dilution tube was approximately 5×105

colony forming units (CFU)/ml of Ca-MHB. Serial twofold dilutions of drug were used.

The MIC was defined as the lowest concentration of drug that resulted in no visible

growth after 24 hours of incubation at 35°C in ambient air. Samples (50 µl) from clear

tubes and the cloudy tube with the highest drug concentration were plated on cation-

adjusted Mueller-Hinton agar (Ca-MHA) plates (Hardy Diagnostics, Santa Maria, CA).

The MBC was defined as the lowest concentration of drug that resulted in ≥ 99.9% kill of

the initial inoculum. Drug carry-over effect was assessed by visual inspection of the

distribution of colonies on media plates. In addition, MIC to a screening panel of

antimicrobial agents (consisting of ciprofloxacin, cefepime, imipenem, meropenem and

tobramycin) was also determined by E-test (AB Biodisk, Piscataway, NJ) according to

the manufacturer’s instructions. The studies were conducted in duplicate and repeated at

least once on a separate day.

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Time-kill studies. Time-kill studies were conducted with different and escalating

concentrations of levofloxacin. Nine concentrations of levofloxacin were used: 0

(control) to 64 µg/ml. An overnight culture of the isolate was diluted 30-fold with pre-

warmed Ca-MHB and incubated further at 35°C until reaching late log phase growth.

The bacterial suspension was diluted with Ca-MHB accordingly based on absorbance at

630 nm; 15 ml of the suspension was transferred to 50 ml sterile conical flasks each

containing 1 ml of a drug solution at 16x the target concentration. The final

concentration of the bacterial suspension in each flask at baseline was approximately 1 x

108 CFU/ml. The high inoculum was used to simulate the bacterial load in a severe

infection (e.g., nosocomial pneumonia). Furthermore, the high inoculum used would

allow resistant sub-population(s) to be likely present at baseline. The experiment was

conducted for 24 hours in a shaker water bath set at 35°C. Serial samples (baseline, 0.5,

1, 2, 4, 8, 12, and 24 hours) were obtained from each flask over 24 hours to characterize

the effect of various drug exposures on the total bacterial population. Prior to culturing

the bacteria quantitatively, the bacterial samples (0.5 ml) were centrifuged at 10,000G for

15 minutes, and reconstituted with sterile normal saline to their original volumes in order

to minimize drug carry-over effect. Total bacterial populations were quantified by spiral

plating 10x serial dilutions of the samples (50 µl) onto Ca-MHA plates. The media plates

were incubated in a humidified incubator (35°C) for 18-24 hours, and the bacterial

density from each sample was determined by CASBA-4 colony scanner / software (Spiral

Biotech, Bethesda, MD). The reliable limit of detection was 400 CFU/ml. The

experiment was repeated once on a separate day.

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Mathematical modeling. Instead of using the conventional two-subpopulations

modeling approach for rational design of dosing regimens 9; 11; 3; 21, we used the

mathematical framework described in the previous section to develop a method to predict

whether a microbial population will be eradicated or survive under an antimicrobial

pressure whose concentration over time follows a clinically realistic (time-periodic)

profile. Our approach is summarized as follows: In step one we used short-term (24-

hour) time-kill data at various time-invariant concentrations of the antimicrobial agent,

and fit parameters values of the model that describes the effect of the agent on the

heterogeneous microbial population using eqns. (11), (26), (27), and (28). In step two we

used the results of step one to predict whether the microbial population would be

eradicated or survive by focusing via eqn. (25) on g

DK , for various dosing regimens. A

dosing regimen (combination of daily dose and dosing interval) with ratio 1g

DK > , eqn.

(18), predicted to be associated with high likelihood of suppressing resistance emergence

and a dosing regimen with ratio 1g

DK < , predicted to be associated with resistance

emergence were suggested for testing in the hollow-fiber in vitro infection model.

Hollow fiber infection model. The schematic of the system has been described

previously2. Drug was directly injected into the central reservoir to achieve the peak

concentration desired. Fresh (drug-free) growth media was infused continuously from

the diluent reservoir into the central reservoir to dilute the drug, in order to simulate drug

elimination in humans. An equal volume of drug-containing media was removed from

the central reservoir concurrently to maintain an iso-volumetric system. Bacteria were

inoculated into the extracapillary compartment of the hollow fiber cartridge (Fibercell

Systems, Inc., Frederick, MD); the bacteria were confined in the extracapillary

- 20 -

compartment, but were exposed to the fluctuating drug concentration in the central

reservoir by means of an internal circulatory pump in the bioreactor loop. The optional

absorption compartment of the system was not used.

Experimental setup. The inoculum was prepared as described above. The

bacteria (20 ml) were inoculated into the hollow fiber infection models at a concentration

of approximately 1 x 108 CFU/ml. Dose ranging studies were conducted for 5 days in a

humidified incubator set at 35°C. The bacteria were exposed to placebo and 2 dosing

regimens simulating the steady-state pharmacokinetic profiles of unbound levofloxacin

(terminal half-life = 6 hours) 18 resulting from a daily dose of 750 mg (the highest FDA-

approved clinical dose, predicted to allow for resistance emergence) and 3000 mg (4x

standard clinical dose, predicted to suppress resistance) given every 24 hours,

respectively.

Pharmacokinetic validation. Serial samples were obtained from the infection

models on days 0, 2, and 4. Levofloxacin concentrations in these samples were assayed

by a validated bioassay method as described below. The concentration-time profiles

were modeled by fitting a one compartment linear model to the observations, using the

ADAPT II program 6. A least-square error model structure was used.

Bioassay. Levofloxacin concentrations were determined by a microbioassay

utilizing Klebsiella pneumoniae ATCC 13883 as the reference organism. The bacteria

were incorporated into 30 ml of molten Ca-MHA (at 50°C) to achieve a final

concentration of approximately 1×105 CFU/ml. The agar was allowed to solidify in 150

mm media plates. Size #3 cork-bore was used to create nine wells in the agar per plate.

Standards and samples were tested in duplicate with 40 µl of the appropriate solution in

- 21 -

each well on the same day. The levofloxacin standard solutions ranged from 0.5 to 64

µg/ml in Ca-MHB. The media plates were incubated at 35°C for 24 hours and the zones

of inhibition were measured. The assay was linear (correlation coefficient ≥ 0.99) using

zone diameter versus log of the standard drug concentration. The intraday and interday

coefficients of variation (CV) for all standards were < 2% and < 10%, respectively.

Microbiologic response (prospective model validation). Serial samples (0.5

ml) were also obtained at baseline, 4 hours, 8 hours (on day 0), and daily (before dose) in

duplicate from each hollow fiber system, for quantitative culture to define the effect of

various drug exposures on the total bacterial population and on selection of resistant

bacterial sub-populations. Total bacterial populations were quantified as described

above. Sub-populations with reduced susceptibility (resistant) were quantified by

culturing onto Ca-MHA plates supplemented with levofloxacin at a concentration 3x

MIC of levofloxacin. Since susceptibility testing is performed in twofold dilutions and 1

tube (2× in concentration) difference is commonly accepted as reasonable interday

variation, quantitative cultures on drug supplemented media plates (at 3×MIC) would

allow reliable detection of bacterial sub-populations with reduced susceptibility. The

media plates were incubated at 35°C for up to 24 (total population) and 72 hours (sub-

populations with reduced susceptibility); bacterial density from each sample was

estimated as described.

Studies on levofloxacin resistant isolates. Bacterial isolates were recovered

from levofloxacin supplemented media plates at the end of the experiment and their

susceptibility to levofloxacin was repeated to confirm the presence of resistance (to rule

out degradation of drug supplementation in Ca-MHA resulting in detecting falsely

- 22 -

resistant isolates). To provide insights on the mechanism(s) of levofloxacin resistance

and cross resistance to other agents, susceptibilities of the resistant isolates were also

repeated using a screening panel of antimicrobial agents (consisting of ciprofloxacin,

cefepime, imipenem, meropenem and tobramycin). Based on the resistant phenotypic

profiles, the quinolone resistance determining regions (QRDR) of genes encoding for the

topoisomerases (gyrA and parC) were amplified by polymerase chain reaction (PCR),

using primers and thermocycling conditions as published previously13. The genetic

sequences of the PCR products were subsequently determined with both primers (forward

and reverse sequences), using the ABI 3730 XL DNA analyzer (Applied Biosystems,

Foster City, CA) and compared to the parent strain to detect point mutation(s) resulting in

amino-acid residue substitution(s). The wild type PAO1 sequences [GenBank accession

numbers L29417 (gyrA) and AB003428 (parC)] were also used for comparison.

4.2 Experimental results and validation of model predictions

Susceptibility studies. The values of MIC and MBC for levofloxacin were both

found to be 2 µg/ml.

Time-kill studies. The bactericidal activity of levofloxacin was found to be

concentration-dependent. With increasing concentrations of levofloxacin used, a faster

killing rate and a greater extent of killing were seen, as shown in Figure 5. Regrowth is

evident after the initial reduction in bacterial burden in almost all time-kill studies.

Model fit to the data. For absence of agent (placebo, 0 µg/ml)C = we get by

inspection in Figure 5 that 9.4max 10 CFU/mlN = . Using this value for maxN we use

standard nonlinear least squares (differential evolution of Mathematica) to get a growth

- 23 -

rate constant value 10.22 hrgK −= . We assume that gK is the same for all bacterial

subpopulations, owing to common basic physiology (see also Remark 2). For the highest

agent concentration used ( 64 µg/ml)C = there appears to be complete eradication of the

entire microbial population after the third sample in time. Therefore, no experimental

points beyond that are considered in the data fit. Further, we assume that the maximum

kill rate, kK , occurring at an antibiotic concentration high enough to saturate all bacterial

target sites, is the same for all bacteria, namely k bK K= . The value of bK is estimated

from the data at 64 µg/mlC = to be 113.5 hrb kK K −= = . The values of the remaining

parameters are estimated using standard nonlinear least squares for concentrations

0.5,...,32 µg/mlC = (Table 1). Comparison between model fit and measurements is

shown in Figure 6 and Figure 7.

Model simulation / prediction. Substituting the parameter values for gK , bK ,

50bC , and bH of Table 1 into eqn. (25), as discussed in section 3.2, we obtain the 3-

dimensional response surface of Figure 8. It is evident that for a dosing interval of 24

hours, a total of at least 2126 mg of a daily dose is required for eradication of the most

resistant bacterial subpopulation, hence of the entire population as well. It is also evident

that the daily dose of 750 mg recommended in standard literature is going to be

inadequate, according to the mathematical model. These predictions were verified

experimentally in the hollow-fiber experimental infection model system discussed in

section 4.1.

Pharmacokinetic validation in hollow fiber infection models. Both simulated

levofloxacin exposures were satisfactory (r2 > 0.95), as shown in Figure 9.

- 24 -

Microbiologic response (prospective model validation). Placebo control did

not exert a selective pressure on the bacterial population, therefore no resistant sub-

population was detected over the duration of the experiment (Figure 10a). With the

simulated clinical dose (750 mg given once daily), a significant killing of the bacterial

population was observed at 4 and 8 hours. However, regrowth was apparent with

repeated dosing beyond 24 hours, similar to that observed in time-kill studies. Regrowth

observed over time was likely due to selective amplification of pre-existing resistant

mutant resistant subpopulation(s) likely to be present at baseline, as demonstrated in

Figure 10b. This is consistent with our modeling approach. Susceptible bacterial

populations were selectively eradicated, resulting in unopposed growth of resistant sub-

population(s) and consequently the enrichment of the total bacterial population by the

resistant sub-population. As a proof of concept, a supra-clinical dose (3000 mg given

once daily) above the threshold exposure for resistance development was simulated to

verify if resistance in P. aeruginosa could be counter-selected. As predicted, sustained

killing of the total bacterial burden and suppression of resistant sub-population was

achieved over 5 days (Figure 10c).

Studies on levofloxacin resistant isolates. Two random isolates were recovered

from drug supplemented media plates; they were both found to have 16-fold increase in

MIC to levofloxacin, compared to the parent strain. Significant cross resistance (≥ 4 fold

change in MIC) to other antimicrobial agents in the screening panel was not observed

(except for ciprofloxacin), suggesting mutation in QRDR of genes encoding for the

topoisomerases was most likely to be involved (compared to efflux pumps over-

expression). PCR and sequencing studies of the parent strain and PAO1 were identical in

- 25 -

the QRDR of gyrA and parC. Both levofloxacin resistant isolates revealed point

mutations resulting in amino-acid changes in the QRDR of gyrA (T83I), but not in parC,

consistent with a common genotype associated with quinolone resistance in clinical

strains of P. aeruginosa 13; 1.

5 Discussion and Future Work

We have presented the development and first in vitro experimental validation of a

mathematical modeling framework for the response of heterogeneous microbial

populations to antimicrobial agents at time-periodic (pharmacokinetically realistic)

concentrations. Our model predictions were compared to published data regarding the

threshold quinolone exposure necessary to suppress resistance development of P.

aeruginosa in a murine thigh infection model9. Despite the differences between the two

modeling approaches, the estimates of the levofloxacin exposure necessary for resistance

suppression were consistent [approximately 2900 mg daily (total AUC/MIC = 157, free

AUC/MIC = 110) demonstrated previously in the murine thigh infection model versus

2126 mg daily predicted by our model]. The closeness of our mathematical model

predictions to the murine thigh infection model data exemplifies the usefulness of the

proposed approach. Certainly, animal experiments are still likely to be performed for

confirmation of mathematical model predictions. However, such predictions can offer

guidance to targeted testing of promising regimens, thus resulting in significant savings

of time, effort, as well as laboratory animals. While the proposed approach is promising,

further work is needed to identify the limits of the mathematical model’s predictive

- 26 -

ability, sensitivity to available data, choice of experimental conditions, and range of

antimicrobial agents/microbial populations over which it is applicable.

Acknowledgments: Part of this work was presented at the 6th International Symposium

on Antimicrobial Agents and Resistance, Singapore, Republic of Singapore, March 7-9,

- 27 -

Appendix 1 – Proof of Theorem 1

Eqn. (1) implies

( )0 0

( ) exp ( )⎛ ⎞⎡ ⎤= −⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠∫

gN t N K r C dρ θ

θ θ (29)

The integral in the above eqn. (29) can be written as

( ) ( )

1 ( 1)

( ) ( ) ( )

( ) ( )

nt j T t

jT nTj

n j T t

jT nTj

n T t nT

jT d kT d

K D nT K t nT r C d

ρ θ θ ρ θ θ ρ θ θ

ρ θ θ ρ θ θ

ρ η η ρ η η

− −

= − + − =

⎡ ⎤= − + − +⎣ ⎦

∑∫ ∫ ∫

∑∫ ∫

which implies eqn. (15). Substitution of t by nT in eqn. (30) immediately yields the

result. ■

Eqn. (16) implies

- 28 -

( ) ( )( )

( )( )

( ) ( )

cr cr cr0

cr avg cr

cravg cr

1 1 ( )

AUC /1 1

D r C t dtK K T

r C r C C t C dtK T

K r C C CK

r CC C

r C TCK C

⎡ ⎤⎢ ⎥′≈ + −⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤′= + −⎣ ⎦

′= + −

′ ⎛ ⎞= + −⎜ ⎟

⎝ ⎠

Substitution of eqn. (23) into eqn. (16) and integration yields

50 max

H Hg g

K C CDK K HT C C kTH

+ − (31)

Using eqn. (23) in eqn. we get

[ ]( ) [ ]( )avg max max avg0

1 1( ) 1 expˆ1 exp

T TC C d C kT C CT kT kT

η ητ

= = − − ⇒ =− −∫ (32)

Substitution of eqn. (32) into eqn. (31) yields eqn. (25)a.

50 max

H Hg g

K C CDK K HT C C kTH

+ − (33)

To get eqn. (25)b, we have

cr( ) 0H

K KK r C z y

K⎛ ⎞−

− = ⇒ = ⎜ ⎟⎜ ⎟⎝ ⎠

which yields the result by substitution. ■

- 29 -

References 1Akasaka, T., M. Tanaka, A. Yamaguchi and K. Sato. Type II topoisomerase mutations in

fluoroquinolone-resistant clinical strains of Pseudomonas aeruginosa isolated in 1998 and 1999: role of target enzyme in mechanism of fluoroquinolone resistance. Antimicrob Agents Chemother. 45:2263-8, 2001.

2Bilello, J. A., G. Bauer, M. N. Dudley, G. A. Cole and G. L. Drusano. Effect of 2',3'-didehydro-3'-deoxythymidine in an in vitro hollow-fiber pharmacodynamic model system correlates with results of dose-ranging clinical studies. Antimicrob Agents Chemother. 38:1386-91, 1994.

3Campion, J. J., P. J. McNamara and M. E. Evans. Pharmacodynamic modeling of ciprofloxacin resistance in Staphylococcus aureus. Antimicrob Agents Chemother. 49:209-19, 2005.

4Cohen, M. L. Epidemiology Of Drug-Resistance - Implications For A Postantimicrobial Era. Science. 257:1050-1055, 1992.

5Craig, W. A. Pharmacokinetic/Pharmacodynamic Parameters: Rationale for Antibacterial Dosing of Mice and Men. Clinical Infectious Diseases. 26:1-12, 1998.

6D' Argenio, D. Z. and A. Schumitzky. ADAPT II user's guide: pharmacokinetic / pharmacodynamic systems analysis software. 1997.

7Drlica, K. A. Strategy for Fighting Antibiotic Resistance. ASM News. 67:27-33, 2001. 8Gold, H. S. and R. C. Moellering. Antimicrobial-drug resistance. N. Engl. J. Med.

335:1444-1453, 1996. 9Jumbe, N., A. Louie, R. Leary, W. Liu, M. R. Deziel, V. H. Tam, R. Bachhawat, C.

Freeman, J. B. Kahn, K. Bush, M. N. Dudley, M. H. Miller and G. L. Drusano. Application of a mathematical model to prevent in vivo amplification of antibiotic-resistant bacterial populations during therapy. J Clin Invest. 112:275-85, 2003.

10Levy, S. B. The Challenge of Antibiotic Resistance. Scientific American, 1998. 11Meagher, A. K., A. Forrest, A. Dalhoff, H. Stass and J. J. Schentag. Novel

pharmacokinetic-pharmacodynamic model for prediction of outcomes with an extended-release formulation of ciprofloxacin. Antimicrob Agents Chemother. 48:2061-8, 2004.

12Morens, D. M., G. K. Folkers and A. S. Fauci. The challenge of emerging and re-emerging infectious diseases. Nature. 430:242 - 249, 2004.

13Mouneimne, H., J. Robert, V. Jarlier and E. Cambau. Type II topoisomerase mutations in ciprofloxacin-resistant strains of Pseudomonas aeruginosa. Antimicrob Agents Chemother. 43:62-6, 1999.

14Mueller, M., A. de la Peña and H. Derendorf. Issues in Pharmacokinetics and Pharmacodynamics of Anti-Infective Agents: Kill Curves versus MIC. Antimicrobial Agents and Chemotherapy. 48:369-377, 2004.

15NCCLS. Methods for dilution antimicrobial susceptibility tests for bacteria that grow aerobically; approved standard - sixth edition. 2003.

16Neu, H. C. The crisis in antibiotic resistance. Science. 257:1064-1073, 1992.

- 30 -

17Nikolaou, M. and V. H. Tam. A New Modeling Approach to the Effect of Antimicrobial Agents on Heterogeneous Microbial Populations. Journal of Mathematical Biology. 52:154-182, 2006.

18Pea, F., E. Di Qual, A. Cusenza, L. Brollo, M. Baldassarre and M. Furlanut. Pharmacokinetics and pharmacodynamics of intravenous levofloxacin in patients with early-onset ventilator-associated pneumonia. Clin Pharmacokinet. 42:589-98, 2003.

19Pham, P. A. FDA approves daptomycin, a new cyclic lipopeptide antibiotic, for the treatment of resistant gram positive organisms. FDA approves daptomycin, a new cyclic lipopeptide antibiotic, for the treatment of resistant gram positive organisms, from http://hopkins-abxguide.org/show_pages.cfm?content=F40_100803_content.html

20Robertson, T. B. The chemical basis of growth and senescence. Philadelphia: Lippincott, 1923.

21Tam, V. H., A. Louie, M. R. Deziel, W. Liu, R. Leary and G. L. Drusano. Bacterial-Population Responses to Drug-Selective Pressure: Examination of Garenoxacin's Effect on Pseudomonas aeruginosa. J Infect Dis. 192:420-8, 2005.

22UCSF. Daptomycin. from http://www.ucsf.edu/idmp/whatsnew/dapto_monograph.htm 23Varaldo, P. E. Antimicrobial resistance and susceptibility testing: an evergreen topic.

Journal of Antimicrobial Chemotherapy. 50:1-4, 2002. 24Vogelman, B. and W. Craig. Kinetics of Antimicrobial Activity. The Journal of

Pediatrics. 108:835-840, 1986. 25Wagner, J. Kinetics of Pharmacologic Response I. Proposed Relationships between

Response and Drug Concentration in the Intact Animal and Man. J. Theoret. Biol. 20:173-201, 1968.

26Weisstein, E. W. Cumulant. MathWorld--A Wolfram Web Resource, from http://mathworld.wolfram.com/Cumulant.html, accessed 2005.

- 31 -

Table 1 – Data fit for experimental time-kill data of Figure 5

Parameter Value gK 0.22 hr–1

maxN 109.4 CFU/ml

gK 0.22 hr–1

k bK K= 13.5 hr–1

50C 0.86 µg/ml

50 AC 0.45 µg/ml

50bC 39.4 µg/ml H 1.7

AK 0.7 hr–1

- 32 -

Figure captions Figure 1 – Example of periodically fluctuating antimicrobial agent profile. Injection points can be seen every 8 hours. Decline of agent concentration is due to agent wash-off according to pharmacokinetics. ........................................................................................ 33 Figure 2 – Eradication (left) or regrowth (right) of a microbial population in an environment of antimicrobial agent concentration following Figure 1. ........................... 34 Figure 3 – Potential eradication of a microbial population during the first dosing interval............................................................................................................................................ 35

Figure 4 – A library of behaviors of g

as a function of kT and avg

. ........................ 37

Figure 5 – Time-kill studies of levofloxacin against P. aeruginosa ATCC 27853 (MIC = 2 µg/ml). For 32 MIC 64 µg/mlC = × = there are no points beyond 1 hour, since all bacteria appear to have been eradicated beyond that point in time. ................................. 38 Figure 6 – Fit of the experimental data shown in Figure 5 by the mathematical model of eqns. (11) and (12). ........................................................................................................... 39 Figure 7 – Correlation between experimental data and model fit for ( )N t according to Figure 6. 2( 0.83)R = ....................................................................................................... 40 Figure 8 – Model prediction of bactericidal effect of levofloxacin for bacterial population of P. aeruginosa, corresponding to the time-kill data of Figure 5. Figure parallels Figure

4 for avgDaily dose(mg)

24(hrs)9.24(L/hr) 0.7C =

× (where the 0.7 term is due to protein binding of 30%)

and 1/ 2 6 hrst = . Dosing regimens (combinations of daily dose and dosing interval) associated with resistance suppression correspond to 1

DK > , eqn. (18)........................... 41

Figure 9 – Simulated levofloxacin pharmacokinetic profiles; observed for daily dose of 750 mg (A), 3000 mg (B) given every 24 hours............................................................... 42 Figure 10 – Prospective validation of the mathematical model in the hollow-fiber infection model for placebo (a), levofloxacin 750mg (b), levofloxacin 3000 mg (c) given every 24 hours. Data presented as mean and standard deviation of duplicate samples... 43

- 33 -

Figure 1 – Example of periodically fluctuating antimicrobial agent profile. Injection points can be seen every 8 hours. Decline of agent concentration is due to agent wash-off according to pharmacokinetics.

- 34 -

Figure 2 – Eradication (left) or regrowth (right) of a microbial population in an environment of antimicrobial agent concentration following Figure 1.

- 35 -

tmintime, t

Figure 3 – Potential eradication of a microbial population during the first dosing interval.

- 36 -

- 37 -

Figure 4 – A library of behaviors of g

as a function of kT and avg

- 38 -

0 6 12 18 24

time (hr)

Placebo0.5 (ug/ml)1248163264

Figure 5 – Time-kill studies of levofloxacin against P. aeruginosa ATCC 27853 (MIC = 2 µg/ml). For 32 MIC 64 µg/mlC = × = there are no points beyond 1 hour, since all bacteria appear to have been eradicated beyond that point in time.

- 39 -

Figure 6 – Fit of the experimental data shown in Figure 5 by the mathematical model of eqns. (11) and (12).

- 40 -

Figure 7 – Correlation between experimental data and model fit for ( )N t according to Figure 6. 2( 0.83)R =

- 41 -

Figure 8 – Model prediction of bactericidal effect of levofloxacin for bacterial population of P. aeruginosa, corresponding to the time-kill data of Figure 5. Figure parallels Figure

4 for avgDaily dose(mg)

24(hrs)9.24(L/hr) 0.7C =

× (where the 0.7 term is due to protein binding of 30%)

and 1/ 2 6 hrst = . Dosing regimens (combinations of daily dose and dosing interval) associated with resistance suppression correspond to 1

DK > , eqn. (18).

- 42 -

Figure 9 – Simulated levofloxacin pharmacokinetic profiles; observed for daily dose of 750 mg (A), 3000 mg (B) given every 24 hours

R2 = 0.958

Peak = 8.8 µg/ml

Terminal t1/2 = 6.8 h

AUC = 86.3 µg.h/ml

R2 = 0.980

Peak = 44.3 µg/ml

Terminal t1/2 = 6.3 h

AUC = 401.1 µg.h/ml

- 43 -

0 1 2 3 4 5 Time (d)

TotalResistant

0 1 2 3 4 5 Time (d)

Figure 10 – Prospective validation of the mathematical model in the hollow-fiber infection model for placebo (a), levofloxacin 750mg (b), levofloxacin 3000 mg (c) given every 24 hours. Data presented as mean and standard deviation of duplicate samples.

0 1 2 3 4 5 Time (d)

Modeling of Microbial Population Responses to Time-Periodic Concentrations of Antimicrobial Agents

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