- 1 - 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* 1 Department of Chemical Engineering; 2 College 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: [email protected]Vincent H. Tam University of Houston College of Pharmacy 1441 Moursund Street Houston, TX 77030 Phone: (713) 795-8316; Fax: (713) 795-8383 Email: [email protected]
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Modeling of Microbial Population Responses to Time-Periodic Concentrations of Antimicrobial Agents
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- 1 -
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
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,
2007.
- 27 -
Appendix 1 – Proof of Theorem 1
Eqn. (1) implies
( )0 0
( )
( ) exp ( )⎛ ⎞⎡ ⎤= −⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠∫
t
gN t N K r C dρ θ
θ θ (29)
The integral in the above eqn. (29) can be written as
( ) ( )
1 ( 1)
00
1 ( 1)
0
1
0 00
0
( ) ( ) ( )
( ) ( )
( ) ( )
( )
nt j T t
jT nTj
n j T t
jT nTj
n T t nT
j
t nT
g g
d d d
jT d kT d
d 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
η η
− +
=
− +
=
− −
=
−
= + =
= − + − =
= + =
⎡ ⎤= − + − +⎣ ⎦
∑∫ ∫ ∫
∑∫ ∫
∑∫ ∫
∫
(30)
which implies eqn. (15). Substitution of t by nT in eqn. (30) immediately yields the
result. ■
Appendix 2 – Proof of Theorem 2
Eqn. (16) implies
- 28 -
( )
( ) ( )( )
( )( )
( ) ( )
( )
0
cr cr cr0
cr avg cr
cravg cr
crcr
cr
1 ( )
1 1 ( )
1
1
AUC /1 1
g
T
g g
T
g K
gg
g
g
D r C t dtK K T
r C r C C t C dtK T
K r C C CK
r CC C
K
r C TCK C
=
⎡ ⎤⎢ ⎥′≈ + −⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤′= + −⎣ ⎦
′= + −
′ ⎛ ⎞= + −⎜ ⎟
⎝ ⎠
∫
∫
Appendix 3 – Proof of Theorem 3
Substitution of eqn. (23) into eqn. (16) and integration yields
[ ]
50 max
50 max
lnexp
H Hk
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
50 max
lnexp
H Hk
H Hg g
K C CDK K HT C C kTH
+=
+ − (33)
To get eqn. (25)b, we have
1/
cr( ) 0H
k gg
g
K KK r C z y
K⎛ ⎞−
− = ⇒ = ⎜ ⎟⎜ ⎟⎝ ⎠
(34)
which yields the result by substitution. ■
- 29 -
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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
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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.
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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.
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
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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
AH 10
bH 24
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
DK
as a function of kT and avg
cr
CC
. ........................ 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
g
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
Cmax
Ccr
CHtL
Figure 3 – Potential eradication of a microbial population during the first dosing interval.
- 36 -
- 37 -
Figure 4 – A library of behaviors of g
DK
as a function of kT and avg
cr
CC
.
- 38 -
0
2
4
6
8
10
0 6 12 18 24
time (hr)
logN
(CFU
/ml)
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
g
DK > , eqn. (18).
- 42 -
(a)
(b)
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 -
(a)
(b)
0
2
4
6
8
10
0 1 2 3 4 5 Time (d)
Log
CFU
/ml
TotalResistant
(b)
0
2
4
6
8
10
0 1 2 3 4 5 Time (d)
Log
CFU
/ml
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.