K-PD Model and Its Application in Research of Veterinary Antimicrobials

Bing LI　Jiyu ZHANG　Jianyong LI　Yajun YANG　Xuzheng ZHOU　Jianrong NIU　Xiaojuan WEI　Jinshan LI

Abstract With the development of the PKPD model， its application in veterinary antimicrobials has also received great attention. This paper expounded the theory of PKPD model， and made a brief analysis of the dosing regimens of various antibiotics according to the parameters of antimicrobial drugs， in order to optimize the clinical drug delivery plan and promote the rational use of clinical antibiotics.

Key words PKPD model; Antimicrobial; Dosing regimen

In pharmaceutical research， pharmacokinetics （PK） and pharmacodynamics （PD） have been considered as two separate disciplines for a long time. Pharmacokinetics is to study the time （T） process of the drug concentration （C） in body fluids at a givendose， i.e.， CT relationship， while pharmacodynamics studies the relationship between drug concentration and pharmacological action intensity （E）， i.e.， CE relationship. Pharmacokinetics is the effect of drugs on the body. However， with the further development of pharmaceutical research， the relationship between the two is getting closer and closer. Combining PK and PD to study the PKPD relationship can help to resolve the individual differences in the clinical manifestations of drugs， and can be used to explore the mechanism of action， the influence of environmental factors inside and outside the body on the invivo process of drugs and the evaluation of the development of new drugs and new formulations. In the past 10 years， the practical application of PKPD joint modeling has developed rapidly and has become a new subject area and discipline growth point. Traditional antimicrobial treatment regimens are based on minimum inhibitory concentration （MIC）， minimal bacteriocidal concentration （MBC）， killing curves （KCs） and post antibiotic effect （PAE） of invitro pharmacodynamics as a guide. Although the above parameters can reflect the antimicrobial activity of antimicrobial drugs to a certain extent， the measurement method is to measure the concentration of the antimicrobial drugs in fixed antimicrobial drug concentrations， and the concentration of the antimicrobial drugs in the body is in a continuous changing state and cannot reflect the dynamic killing process of antimicrobial dugs. Antimicrobial studies combine pharmacokinetics with invitro pharmacodynamic parameters， aiming at investigating the timeeffect process corresponding to a given dose.

This paper expounded the theory of PKPD model， and made a brief analysis of the dosing regimens of various antibiotics according to the parameters of antimicrobial drugs， in order to optimize the clinical drug delivery plan and promote the rational use of clinical antibiotics.

PKPD model

The PKPD model generally consists of two parts： the PK model and the PD model. The two dynamic processes are synchronized in time. The PD model describes the process of the drug in the body， and the pharmacodynamic model describes the relationship between drug concentration and effect at the receptor site. The current pharmacokinetic models of the PK model mainly include the atrioventricular model， the statistical moment model， and the physiological model. The current pharmacokinetic models of the PK model mainly include the compartment model， the statistical moment model and the physiological model. Although the noncompartment model can also be used in the PKPD model， since the compartment model can provide continuous concentrationtime curves and continuous effecttime curves， and can establish an effect compartment easily， the compartment model is usually used to establish the PKPD model. In the steadystate PD model， the drug concentration in the effect site is directly in equilibrium with the detectable body fluid concentration under the steady state of pharmacokinetics， and there is a direct relation， so the steadystate plasma concentration is the only determinant of the observed effect. The most commonly used PD models under steady state conditions are fixed effect model， linear model， loglinear model， maximum effect model， and Shao maximum effect model. Which model to choose depends mainly on the specific drug， the effect that can be observed after administration， the linearity of the effectconcentration curve， and the potential to achieve the maximum possible effect. Under the unsteady state conditions of the unsteadystate PD model， the concentration of the body fluid varies with time， the balance between the plasma concentration and the drug concentration in the effect site does not necessarily exist， and the acting of the drug on the acting site such as the receptor to produce the effect of the drug also takes time， so plasma concentration and effect time process are separated. In order to show the time course characteristics of drugs under unsteady state conditions， it is necessary to combine PK and PD， first to predict the drug doseblood concentration relationship， and secondly to predict the blood drug concentrationeffect relationship. This relationship can be obtained by PKPD model. Lobo et al.[1]used the PKPD model to simulate and analyze the PK and toxicity of the antitumor drug methotrexate after intraperitoneal administration in mice. The results showed that the PK of methotrexate was not related to the dose and method of administration， but the toxicity was closely related to the dosing regimen and the relationship between toxicity and drugtime process， which can guide clinical safe and effective drug use to improve efficacy and reduce toxic side effects. Sun et al.[2]used the PKPD model to compare the differences in Kdifference and therapeutic effect between human recombinant growth hormone （rhGH） microsphere sustainedrelease preparation and osmotic pump delivery system， and common solutions， in order to provide a basis for optimization and screening of the dosage form and the dosing regimen. The common models are as follows.

Direction link model and nondirect link model

The direct link model assumes that the concentrations of the drug in the central compartment （blood compartment） and the effect compartment can be quickly balanced， so the measured blood concentration can be directly used as the input value Ce in the PD model to directly link the blood concentration and the effect compartmentcan. In this case， the concentration and effect maximums will occur simultaneously， and the effect concentration curve will not show hysteresis. For example， Auler et al.[3]used the SigmoidEmax model to directly link the analgesic effect of diclofenac with plasma concentration. It was found that the protein binding level of diclofenac in plasma directly affected the analgesic effect produced by it， and the higher the concentration of free diclofenac， the greater the maximum analgesic effect expressed by Emax. On the contrary， if the distribution of the drug from the central compartment （blood compartment） to the effect compartment takes a certain time， the peak of the drug effect often lags behind the peak of the peak plasma concentration， and an indirect link model is required which uses the drug concentration curve of a modeled peripheral compartment （equivalent to the effect compartment） as the input value Ce in the PD model. The direct link and nondirect link， although different， both reflect that the effect of the drug is directly related to the drug concentration in the acting site， only the correlation between the plasma concentration and the drug concentration in the acting site is different in that there is a direct relation in the former， but an indirect relation in the latter.

Direct response model and nondirect response model

The socalled direct response and indirect response are distinguished by the correlation between the effect produced by the drug and the drug concentration in the acting site. Direct response means that the effect of the drug is directly related to the concentration in the acting site， that is， the drug can immediately produce an effect after reaching the acting site， and there is no time lag. The abovementioned direct link model and indirect link model both can be considered as direct response models， because the effect of the drug is directly related to the drug concentration in acting site. For this type of drugs， the corresponding PKPD model can be established by the direct or indirect link manner. Indirect response means that the effect of the drug is not directly related to the concentration in the acting site， that is， the drug does not immediately produce an effect at the acting site， and the effect of the drug is significantly delayed. This lag is not caused by the transport of the drug from the plasma to the acting site， but by the action mechanism of the drug itself. These drugs often exert their effects by changing certain endogenous substances in the body. For this type of drugs， the corresponding models should be established according to the action mechanism of the drugs， which is more practical and widely used， such as the PKPD model analysis of the anticoagulant， warfarin， the antipyretic analgesic， naproxen， the hypoglycemic drug， insulin， and the antianemia drug， recombinant human erythropoietin.

Soft link model and hard link model

If the mechanism of action is not considered， only the effect compartment is used to explain the hysteresis in the concentrationeffect relationship， then the commonly used PKPD model belongs to the soft link model. In addition to concentration and effect data， the hard link model can also use other data such as receptor binding affinity， minimum inhibitory concentration of antibiotics or other variables related to the mechanism of action[4]. Therefore， the hard link model is clearly predictive， and the PK activity of a new compound can be predicted only with the need for the PK data of the compound and its invitro test.

Timeinvariant model and timevariant model

Most drugs are adaptive to the timeinvariant model， the effect changes are only related to changes in the effect compartment concentration， and the PD model parameters remain unchanged. However， there are also some drugs whose PD parameters such as Emaxand EC50values change in a timedependent manner. Although the drug concentration in the effect compartment does not change， but the effect intensity changes， and the model at this time is a timevariant model[4].

The reduction and increase in sensitivity to stimuli are manifested as tolerance and susceptibility， respectively. Tolerance is caused by a decrease in the number of receptors or affinity for the receptor， both of which produce a clockwise hysteresis loop in the concentration effect relationship， while susceptibility can cause a counterclockwise hysteresis loop. In summary， the cause of the hysteresis loop may be timevariant PK， or it may be caused by a configuration delay or an indirect response mechanism.

Common PKPD Application Programs Abroad

NONLIN series PKPD programs （NONLIN， PCNONLIN and WINNONLIN）

PCNONLIN and WINNONLIN are currently recognized as PKPD generalpurpose programs， all of which originated from the early NONLIN programs. NONLIN uses three different parameter calculation methods such as Marquardt method， which can automatically calculate the initial value， perform compartment model judgment， and obtain the pharmacokinetic parameters. Based on NONLIN， Statistical Consortants （lnc， SCl） developed PCNONLIN， which is the most widely used PKPD program in the world and can be used for the calculation of all nonlinear mathematical models. In the late 1990s， Pharsight Company of the United States developed the WINNONLIN program based on the calculation method and model library of PCNONLIN. At present， WINNONLIN Standard， WINONLIN Pro and other versions have been released. WINNONLIN has increased the type of noncompartment model and the PKPD binding model， and also added noncompartmental analysis function to derive PK parameters from urine drug data; and WINNONLIN Pro can also analyze the variance of the data， which can be used for the calculation of bioavailability and bioequivalence[5].

ADAPT II

The ADAPT II program was developed in 1988 by a researcher at the University of Southern Californias Biomedical Simulations Resource （BMSR）. The data processing function of ADAPTII is extremely powerful. It provides multiple curve fitting methods such as least squares method， weighted least squares method， maximum likelihood method and Bayesian method. The ADAPTII model library has 37 different types of models， including 16 PK models， 8 PKPD binding models， and 13 special models[6].

Other PKPD programs

There are many programs developed abroad for PKPD operations， most of which use nonlinear least squares for curve fitting， such as： RSTRIP， DIFFEQ， STELLA， MKMODEL， NPEM2 and SAS[7]. These programs have their own characteristics. RSTRIP and STELLA are suitable for students teaching; SAS can not only be used for nonlinear fitting and various statistical analysis， but also has the most powerful drawing function; MKMODEL provides PD model; and NPEM2 is more suitable for clinical pharmacokinetic research.

Classification of Antimicrobials Based on PKPD

Antimicrobial therapy has traditionally been guided by invitro pharmacodynamic data， MIC， MBC， KCs， PAE， etc. However， although the above parameters can reflect the antimicrobial activity of antimicrobial drugs to some extent， their determination methods are to place the bacteria in a fixed state for determination， while the concentrations of antimicrobial drugs in the body are in a continuously changing state， so the dynamic microbekilling process of the antimicrobials cannot be reflected. The drugtime curve of different antimicrobials and the antibacterial action mode are shown in Fig. 1. The antimicrobial PKPD study combines pharmacokinetics with invitro pharmacodynamic parameters， and based on the correlation between antimicrobial effect and plasma concentration or action time， the drugs can be roughly divided into such three types as the concentrationdependent， timedependent and timerelated longPAE types. This classification provides important theoretical significance for the optimal design of different dosing regimens.

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Concentrationdependent antimicrobials

The concentrationdependent antimicrobial drugs have a good rapid antimicrobial effect， and the concentration is a factor determining the clinical efficacy. The antimicrobial effect on the pathogens depends on the peak concentration， and is not closely related to the action time. Most of the antimicrobial concentrations have a maximum limit. When the drug concentration is lower than the highest antimicrobial concentration， the antimicrobial activity increases with the increase of the drug concentration. When the highest antimicrobial concentration is reached， the antimicrobial activity is the strongest， and the clinical efficacy can be improved by Cmax. However， the concentration should not exceed the minimumtoxic dose， and special attention should be paid to aminoglycosides with narrow therapeutic windows. Such drugs include aminoglycosides， fluoroquinolones， ketolides and amphotericin， which are characterized by a first dose effect and a longer PAE. PKPD parameters for the evaluation of antimicrobial effects of concentrationdependentdrugs are mainly AUC/MIC and Cmax/MIC.

Timedependent antimicrobials with a short half life

When the concentration of the drug is maintained above the MIC of pathogens， it is critical for the removal of pathogens. The antimicrobial effect of such drugs is saturated when concentration is 4-5 times of that of corresponding MIC. Blindly increasing the dose is meaningless. When the concentration in tissues or serum is lower than the MIC， the microbes quickly begin to grow. Such drugs include most βlactams， lincomycins， and the like. The efficacy evaluation parameter is T>MIC. The key to rational and scientific use of timedependent antimicrobial drugs is to optimize the exposure time of the microbes to the drugs. Within 24 h after the drug is used， the antimicrobial effect is best when the blood concentration in the body exceeds the MIC of pathogens by 40%-60%. Clinically， multiple daily dosing or continuous intravenous infusions are often required[8]. If the administration method is improper， the drug concentration can be maintained at a sublethal dose， and instead of killing the microbes， the microbes can be selected， resulting in the growth of the resistant mutant microbes， which gradually occupy the dominant position of the flora. Therefore， in order to prevent the occurrence of drug resistance， the sublethal time of drugs should be minimized to achieve the desired therapeutic effect.

Timedependent antimicrobials with a long PAE

Such drugs exhibit a small concentrationdependent antimicrobial effect and show a certain PAE， as well as a timedependent antimicrobial effect. The goal of the dosing regimen is to prolong the exposure time of drugs and allow the drug concentration to be lower than its MIC over a significant time interval of the dosing interval. Such antimicrobial drugs include macrolides， streptomycin， carbapenems， glycopeptides， antifungal azoles， etc.[9]， and the main evaluation index is AUC/MIC.

Optimized Design and Application of PKPD and Dosing Regiment

Fluoroquinolone antimicrobials

Fluoroquinolone antimicrobials are concentrationdependent drugs. AUC024 h/MIC is most relevant to bacteriological efficacy， and is not closely related to the action time. Fluoroquinolone antimicrobials can exert good bacteriological effects when AUC024 h/MIC ≥ 100 and/or Cmax/MIC > 8; and the significance of the Cmax/MIC ratio is of the utmost importance[10]. The administration interval can be referred to t1/2， PAE， Cmax/MIC and AUC/MIC， and most of them are administered in one to two doses per day.

Aminoglycoside antibiotics

Aminoglycosides are concentrationdependent antimicrobial drugs， and the main PKPD parameters for evaluating the clinical efficacy of such drugs are Cmax/MIC. Craig believes that the maximum killing rate can be achieved by using aminoglycoside antibiotics to treat Gbacilli infections by maintaining a Cmax/MIC ratio of 8-10 times. In the case of constant daily dose， a single administration can achieve a larger Cmaxthan multiple administrations a day and thus increase the Cmax/MIC ratio， thereby significantly improving the antimicrobial activity and clinical efficacy. Moreover， the PAE of aminoglycosides on pathogens is also concentration dependent. Single administration of daily dose can reduce the incidence of adaptive drug resistance and ototoxicity and renal toxicity. The process of uptake of aminoglycosides by cochlear hair cells and renal tubular epithelial cells is a saturation process. At low concentrations， if the uptake of aminoglycosides by cells has reached saturation， the uptake will not increase with the increase of drug concentration. In the case of multiple dosing a day or continous intravenous drip infusion， although the peak drug concentration in plasma is relatively low， because of the long maintenance time， a higher proportion of the drug is taken up by the renal cortex， which is likely to cause accumulation of poisoning. Single administration of the same daily dose can produce a was relatively high Cmax， but the uptake by the renal cortex is not significantly increaseed[11].

Aminoglycoside qd administration can achieve higher Cmax/MIC， and achieve better clinical and microbial effect， as wells the effects of prolonging PAE， reducing the incidence of adaptiveresistance and preventing the emergence of drugresistant strains， and the ototoxicity and renal toxicity are similar or slightly lower than the traditional dosing regimen.

βlactam antibiotics

βlactam antibiotics including penicillins， cephalosporins， carbapenems and aztreonam are timedependent antimicrobial agents. When the drug concentration reaches a higher level， increasing the concentration does not increase its antimicrobial effect. The invitro killing curve of ceftriaxone shows that the antimicrobial activity and concentration are not concentration dependent. Craig treated a rat model of neutropenia induced by Klebsiella pneumoniae with different doses and dosing intervals of cefotaxime. The number of colony units （cfus） remaining in the lungs after 24 h of treatment was calculated. The number of colonies remaining in the lungs was not correlated with AUC24 h/MIC （area under drugtime curve within 24 h/minimum inhibitory concentration， AUIC） and Cmax/MIC （maximum plasma concentration/minimum inhibitory concentration）， but correlated with time when the plasma concentration was higher than MIC （R2=0.94）. In the cefazolin treatment of Staphylococcus aureus infection， the maximum bacterial clearance was achieved when the time with a plasma concentration higher than above MIC was 55% of the 24 h course of treatment. However， not all βlactam antibiotics need to increase times of administration to improve clinical efficacy. For some βlactam antibiotics with a long halflife， increasing the times of administration does not increase the efficacy. For example， ceftriaxone has a halflife of 8.5 h， and once administered， the plasma drug concentration can be maintained continuously without reducing the efficacy for 12 to 24 h[12]. Imipenem and meropenem in carbapenem antibiotics have strong bactericidal activity against both reproductive and stationary bacteria， and also show a longer PAE. Therefore， for such drugs， the dosing interval can be appropriately extended when clinically applied， and the dosing regimen of 1 to 2 times a day can be adopted.

Macrolide antimicrobial drugs

Macrolides are basically timedependent antimicrobial drugs in classification， but due to differences in the invivo conditions and pharmacodynamic characteristics of each drug， it is difficult to describe with a certain type of parameters. The results of the killing curves of four macrolides against Streptococcus pneumoniae showed that the two macrolides， telithromycin and ABT.773， were concentrationdependent， and the results of the macrolide lactone， ABT.733 showed that it had best correlation between AUC/MIC and microbiological efficacy with R2 of 0.88[13-14]. Clarithromycin and azithromycin show time dependence. When the plasma concentration of clarithromycin and roxithromycin is high， the time with the plasma concentration above the MIC is related to the clinical pharmacodynamic evaluation; and when the blood concentration is low， the AUC condition should be considered， and the expected time with a concentration above MIC is generally should be 50% of the dosing interval[13-14].

Since the concentration of macrolides in tissues and cells is usually higher than that of the same period， it needs to be considered in the PKPD study. For example， azithromycin can accumulate in macrophages and has a characteristic of slow efflux from cells， and a drug release system can be formed in the infected site with a high concentration of white blood cells， so the drug acts for a long time.

Glycopeptide antimicrobial drugs

Vancomycin is a timedependent antimicrobial drug. Flandroris et al.[15]found that the bactericidal effect of vancomycin against S. aureus was observed to be most obviously in the first 4 h by killing curve experiments at 1， 2.5， 5 and 10 times MIC， and later， the amount of bacteria remained at a constant level and was independent of drug concentration. Further studies found that the optimal bactericidal concentration of vancomycin is 4-5 times MIC. The clearance rate for S. aureus is independent of MIC/AUCand is related to T>MIC. Vancomycin has a longer t1/2and PAE， and whether clinical application of vancomycin requires continuous infusion should be studied in more depth.

Streptomycin antimicrobial drugs

Linezolid[16]（linezolid， LZD） is a timedependent antibiotic with a longer PAE. The PAE of LZDagainst S. pneumoniae is about3-4 h. The study on the relationship between PKPD parameters and bacteriological efficacy of LZDtreated rats with S. pneumoniae infection showed that the correlation coefficient between T>MIC% and bacteriological efficacy of LZDwas the highest， and a good bacteriological effect can be achieved when T>MICwas 40%.

Antifungal drugs

Polyenes， flucytosines and azoles are the most effective antifungal drugs. Polyenes belong to concentrationdependent drugs， and their antifungal effects are related to AUC24 h/MIC and Cmax/MIC.Amphotericin B is a drug that is concentration dependent and has a long PAE， and its Cmax/MIC is associated with antifungal efficacy. Flucytosines are timedependent drugs， the antifungal effects of which are most relevant to T>MIC. Imidazoles are timedependent and have a long PAE. The AUIC of fluconazole is associated with antifungal efficacy， and the effect is best when the AUIC is 20-25 h. When fluconazole is used to treat fungal infections， the AUC/MIC ratio should be greater than 20， and when the MIC against fungi is equal to or lower than 8 mg/L， only 200 mg of fluconazole is required to achieve this ratio. When the MIC for fungi is between 16 and 32 mg/L， a dose of 400 mg and/or 800 mg of fluconazole is required to achieve this ratio[17].

Choice of drug combination

For serious infections and mixed infections， and to prevent bacterial resistance， combined use of antibiotics is often adopted clinically. When a reasonable combination regimen is used， the PAE can be longer than when used alone， i.e.， there is a synergistic or additive effect after the combination of two drugs. In principle， the single dose of the drugs should be reduced accordingly， and the interval of administration should be extended. For example， when combining aminoglycoside drugs with βlactam drugs， the daily dose of aminoglycosides can be appropriately reduced and administered in a single dose[18]. Linmycin and ciprofloxacin have additive effects on the PAEs of the pathogenic bacteria S. aureus and Escherichia coli clinically isolated， especially for S. aureus， suggesting that the dosing interval can be appropriately extended when combining fosfomycin and ciprofloxacin. For drugs with clinically significant side effects and definite curative effects， such as aminoglycosides， the combination with fluoroquinolone drugs can reduce the dose， appropriately extend the dosing interval， and minimize toxic side effects. Proper application of PKPD can help ensure clinical and bacteriological cures while minimizing the potential for bacterial resistance. In summary， the PKPD model theory has been continuously developed in the field of pharmacology research. For veterinary antimicrobial drugs， the pharmacodynamic parameters， pharmacokinetic parameters and the establishment of the PKPD model will provide a better reference for their clinical application. In the future development， the PKPD model will be more and more perfect， and its application will become more and more extensive. Making full use of the PKPD model， researchers will be more confident in the development of new antimicrobial drugs to predict the optimal clinical dosing plan from preclinical experimental data， and optimize the design of clinical trials to achieve rapid and economical goals. It is also expected to use the PKPD model to study the interaction of antimicrobial drugs with other drugs such as nonsteroidal antiinflammatory drugs in the body， and to develop new， highly effective and drugresistant veterinary antiinfective drugs.

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