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Thread: The summary of pharmacokinetics core equations

  1. #1
    PharmD Year 1 TomHsiung's Avatar
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    Lightbulb The summary of pharmacokinetics core equations

    I summarized several pharmacokinetics important equations, as shown in the screenshot below. However, they are only a part of the core equations. I would update more equations in the future.

    The summary of pharmacokinetics core equations-pharmacokinetics-equations-png

    All the equations are in the word document I have attached.
    Attached Files Attached Files

  2. #2
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    Summary of Pharmacokinetics Core Equations
    Tom Hsiung, B.S. Pharm

    Amount of Drug Absorbed or Raching the Systemic Circulation =S*F*(Dose)
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug.

    Administration Rate RA =S*F*(Dose)/τ
    RA: The administration rate is the average rate at which absorbed drug reaches the systemic circulation.
    τ: Dosing interval

    The Fraction of Free Drug Concentration or Fraction of Unbound fu=C free/(C free+C bound)
    C free: Serum concentration of free or unbound drug
    C bound: Serum concentration of bound drug

    Low Plasma Protein Concentrations



    CNormal Binding: The plasma drug concentration that would be expected if the patient’s plasma protein concentration were normal.
    PNL: The normal plasma protein. An average normal albumin concentration is 4.4 g/dL (range: 3.5 to 5.5 g/dL).
    C’: The patient’s plasma drug concentration
    P’: The patient’s plasma protein concentration
    fu: Fraction of free drug concentration or fraction of unbound

    Loading Dose =V*C/(S*F)
    V: The volume of distribution
    C: The desired plasma level

    Incremental Loading Dose =V*(C desired-C initial)/(S*F)
    Incremental Loading Dose: The additional loading dose that will be required to achieve a higher plasma concentration (Cdesired) than the present concentration (Cinitial).

    Clearance Cl =S*F*(Dose/τ)/Css ave
    Cl: Clearance represents the theoretical volume of blood or plasma which is completely cleard of drug in a given period.
    Css ave: The average steady-state drug concentration
    τ: Dosing interval

    Maintence Dose =Cl*Css ave*τ/(S*F)
    Cl: Clearance represents the theoretical volume of blood or plasma which is completely cleard of drug in a given period.
    Css ave: The average steady-state drug concentration
    τ: Dosing interval
    Last edited by admin; Mon 1st December '14 at 6:17pm.
    B.S. Pharm, West China School of Pharmacy, Class of 2007, Health System Pharmacist, RPh. Hematology, Infectious Disease. Chengdu, Sichuan, China.

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  3. #3
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    Default Selecting the Appropriate Equation

    Loading Dose or Bolus Dose

    Initial Plasma Concentration =(S)(F)(Loading Dose)/V
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug.
    V: The volume of distribution

    Subsequent Plasma Concentration C1=[(S)(F)(Loading Dose)/V]*(e-kt1)
    C1 & t1: Subsequent plasma concentration any time (t1) after the dose has been administered
    k: k is the fraction or percentage of the total amount of drug in the body removed per unit of time and is a function of clearance and volume of distribution (k=Cl/V).
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug.
    V: The volume of distribution

    Continuous Infusion to Steady State (Steady-State Attained)

    Average Steady-State Plasma Concentration Css ave =(S)(F)(Dose/τ)/Cl

    τ(tau): Dose interval, a period
    Cl: Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug

    Subsequent Plasma Concentration C1=(Css ave)(e-kt1)
    C1: The plasma concentration after discontinuation of infusion from the steady-state
    t1: The time from the infusion stops to the time of C1 reaches
    Css ave: The average steady-state plasma concentration

    Continuous Infusion (Steady-State Un-attained)

    The Concentration That Occurs at any time (t1) before steady-state C1=[(S)(F)(Dose/τ)/Cl]*(1-e-kt1)
    τ(tau)
    : Dose interval, a period
    C1 & t1: The concentration that occurs at any time (t1), within infusion and before steady-state obtained
    k: k is the fraction or percentage of the total amount of drug in the body removed per unit of time and is a function of clearance and volume of distribution (k=Cl/V).
    (1-e-kt1): (1-e-kt1) is the fraction of steady-state achieved at t1
    Cl: Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug

    The concentration that Occurs at any time (t2) after the
    discontinuation of infusion C2==[(S)(F)(Dose/τ)/Cl]*(1-e-kt1)(e-kt2)=(C1)(e-kt2)
    C2 & t2
    : The concentration that occurs at any time (t2) from infusion stops (steady-state un-obtained)
    τ(tau): Dose interval, a periodC1 & t1: The concentration that occurs at any time (t1), before steady-state
    k: k is the fraction or percentage of the total amount of drug in the body removed per unit of time and is a function of clearance and volume of distribution (k=Cl/V).
    (1-e-kt1): (1-e-kt1) is the fraction of steady-state achieved at t1
    Cl: Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug

    As a clinical guideline, Mike uses one-sixth of a drug half-life as an arbitrary break point. That is, for those drugs that are absorbed over a period equal to one-sixth of a half-life or less, the bolus model is used; for those drugs absorbed over a period that is greater than one-sixth of the half-life, the short infusion model is used. Whereas the one-sixth of a half-life "rule" is arbitrary, it was selected because the difference in the calculated plasma concentrations when using the bolus or short infusion model is <10%.

    How about the affection of distribution phase? When drugs are administered orally, the primary concern is with the absorption phase because the distribution component associated with two-compartment modelling is usually negligible. This principle is for the calc of Maximum and Minimum Plasma Concentrations at Steady-State.

    Maximum and Minimum Concentration at Steady-State (Here we assumes drug absorption and distribution rates are rapid in relation to the drug elimination half-life and the dosing interval. This assumption is valid as long as drug concentrations are not sampled during the absorption and distribution phases)

    Css max =[(S)(F)(Dose)/V]/(1-e-kτ)
    τ(tau): Dose interval, a period
    k: k is the fraction or percentage of the total amount of drug in the body removed per unit of time and is a function of clearance and volume of distribution (k=Cl/V).
    (1-e-kτ): (1-e-kτ) represents the fraction of drug that is eliminated in the dosing interval
    Cl: Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug
    V: Volume of Distribution

    Css min =[(S)(F)(Dose)/V]/(1-e-kτ)*(e-kτ)
    τ(tau): Dose interval, a period
    k: k is the fraction or percentage of the total amount of drug in the body removed per unit of time and is a function of clearance and volume of distribution (k=Cl/V).
    (1-e-kτ): (1-e-kτ) represents the fraction of drug that is eliminated in the dosing interval
    Cl: Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug
    V: Volume of Distribution

    Css 1 =[(S)(F)(Dose)/V]/(1-e-kτ)*(e-kt1)
    Css 1 is concentration obtained at time (t1) other than the peak or trough, within a dose interval
    τ(tau): Dose interval, a period
    k: k is the fraction or percentage of the total amount of drug in the body removed per unit of time and is a function of clearance and volume of distribution (k=Cl/V).
    (1-e-kτ): (1-e-kτ) represents the fraction of drug that is eliminated in the dosing interval
    Cl: Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug
    V: Volume of Distribution

    Short Infusion Model

    This model more closely approximates the actual absorption and plasma concentration curve during drug absorption and elimination. Therefore if there is any uncertainty about which model is more appropriate, the short infusion model should be used.

    Ctin =[(S)(F)(Dose/tin)/Cl]*(1-e-ktin)

    Ctin & tin: The plasma concentration at the end of a short infusion (Ctin) where the infusion time of period is tin
    Cl: Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug
    V: Volume of Distribution

    C2 =[(S)(F)(Dose/tin)/Cl]*(1-e-ktin)(e-kt2)
    C2 & t2
    : Subsequent drug plasma concentration (C2) at any time interval (t2) since the end of the short infusion
    Cl: Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug
    V: Volume of Distributions

    Css max and Css min for intermittent short infusion

    Css max =[(S)(F)(Dose/tin)/Cl/(1-e-kτ)]*(1-e-ktin)
    Css max
    : Maximal plasma concentration at steady state (Css max) and minimal plasma concentration at any time interval (t2) since the end of the short infusion
    tin: Infusion time of period
    Cl
    : Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug
    V: Volume of Distribution
    τ: Dose interval

    Css min =[(S)(F)(Dose/tin)/Cl/(1-e-kτ)]*(1-e-ktin)[e-k(τ-tin)]
    Css min
    : Minimal plasma concentration at steady state (Css min)
    tin: Infusion time of period
    Cl: Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug
    V: Volume of Distribution
    τ: Dose interval

    For short infusion model, after N doses (not reaching steady state yet).

    C2
    =[(S)(F)(Dose/tin)/Cl/(1-e-kτ)]*(1-e-ktin)(1-e-knτ)(e-kt2)
    C2 & t2
    : Subsequent drug plasma concentration (C2) at any time interval (t2) since the end of the short infusion
    Cl: Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug
    V: Volume of Distribution
    τ: Dose interval
    n: number of doses


    PS: Whether a bolus or infusion model is used to represent the input or absorption of drug into the body depends on the relationship between the duration of drug input relative to the drug's half-life (as a clinical guideline the cut point is 1/6 of half-life). For example, if a drug is administered rapidly as an IV bolus or if an orally administered drug is absorbed rapidly relative to the drug's half-life, very little drug will be cleared or eliminated during the administration or absorption process. Therefore, absorption can be thought of as instantaneous, and the bolus model can be used.

    If, however, a drug is absorbed over a long time relative to its half-life, a significant amount of drug will be eliminated during the input or absorption period and the plasma level concentrations resulting from oral administration would resemble those resulting from an infusion model.

    As a general rule, if the drug input time (tin) is less than 1/10 its half-life, then it can be successfully modelled as a bolus dose; however, if the drug input time is greater than one-half its half-life, it is more appropriate to use an infusion model.

    Sustained-Release Dosage Forms

    Sustained-release dosage forms could be thought as continuous infusion because the tin (tin equals to the time required for overall absorption of all dosage) is long which is approximately as long as dose interval (tin approximates tau/tin = tau).

    So we can get a Css max whose value is approximately as same as Css min for sustained-release dose forms, that is,

    Css max
    = Css min =(S)(F)(Dose/tin)/Cl
    Css max: Maximal plasma concentration at steady state (Css max)
    Css min
    : Minimal plasma concentration at steady state (Css min)
    tin: Infusion time of period (absorption time of the sustained-release dosage forms)
    Cl: Clearance
    F: Bioavailability factor
    S: Fraction of the administered dose that is the active drug
    V: Volume of Distribution
    τ: Dose interval

    Notice: If the tin is exactly equal to tau, the input from one dose stops at the same time the next dose begins its infusion process. As a result, an average steady-state concentration with on rise or fall within the dosing interval is achieved. This would be exactly the same as changing an IV bag for a constant infusion without interrupting the infusion process.

    In practice, absorption times are not exactly equal to the dosing interval, but for most sustained-release drug products, they are reasonably close and, therefore, plasma concentrations can be considered as an average steady-state value. As a general rule, absorption times that exceed the dosing interval are not a problem. However, if the duration of absorption (tin) is substantially less than the dosing interval, then there will be some fluctuation of the plasma concentration. A useful approach is to consider the duration over which the plasma concentrations will decay following the end of absorption.

    As a clinical guideline, if tau - tin is <= 1/3 t1/2, the average steady-state equation can be used and the plasma concentrations can be considered as with little fluctuation.
    Last edited by admin; Sun 1st January '17 at 2:49pm.
    B.S. Pharm, West China School of Pharmacy, Class of 2007, Health System Pharmacist, RPh. Hematology, Infectious Disease. Chengdu, Sichuan, China.

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    Default Series of Individual Doses

    When a series of individual doses is administered and a concentration before steady state must be calculated, there are several approaches that can be taken. One approach is to sum the contributions of each individual dose. This is done by decaying the peak concentration of each dose to the time at which the plasma concentration needs to be predicted.

    Csum = [(S)(F)(D1)/V*(e-kt1)] + [(S)(F)(D2)/V*(e-kt2)] + [(S)(F)(D3)/V*(e-kt3)]

    The t1, t2, and t3 represent the time from each administered dose to the time at which the plasma concentration (Csum) is to be calculated.
    Last edited by admin; Fri 26th June '15 at 12:40pm.
    B.S. Pharm, West China School of Pharmacy, Class of 2007, Health System Pharmacist, RPh. Hematology, Infectious Disease. Chengdu, Sichuan, China.

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  5. #5

    Default Pharmacokinetic-Pharmacodynamic/PK-PD Equations

    Relationship of Dose to Pharmacologic Effect

    The onset, intensity, and duration of the pharmacologic effect depend on the dose and the pharmacokinetics of the drug. As dose increases, the drug concentration at the receptor site increases, and the pharmacologic response (effect) increases up to a maximum effect. A plot of the pharmacologic effect to dose on a linear scale generally results in a hyperbolic curve with maximum effect at the plateau (Fig. 19-1). The same data may be compressed and plotted on a log-linear scale and results in a sigmoid curve (Fig. 19-2).

    The summary of pharmacokinetics core equations-screen-shot-2016-04-08-at-8-37-37-pm-png

    The summary of pharmacokinetics core equations-screen-shot-2016-04-08-at-8-37-47-pm-png

    For many drugs, the graph of the log dose-response curve shows a linear relationship at a dose range between 20% and 80% of the maximum response, which typically includes the therapeutic dose range for many drugs. For a drug that follows one-compartment pharmacokinetics, the volume of distribution is constant; therefore, the pharmacologic response is also proportional to the log plasma drug concentration within a therapeutic range, as shown in Fig. 19-3.

    The summary of pharmacokinetics core equations-screen-shot-2016-04-08-at-8-37-56-pm-png

    After some mathematical evolution, we get the following equations:

    E = m * log(C) + e

    where E is the drug effect at drug concentration C, m is the slope, and e is an extrapolated intercept.

    log(C) = log(C0) - kt/2.3 (Equation 19.3)

    where C0 is the initial drug concentration after an intravenous dose, k is the elimination constant, and t is the time from the done of administration of the intravenous dose.

    And if we combine the two equations above, we get:

    E = E0 - kmt/2.3 (core equation)

    The core equation predicts that the pharmacologic effect will decline linearly with time for a drug that follows a one-compartment model, with a linear log dose-pharmacologic response. From this equation, the pharmacologic effect declines with a slope of km/2.3. The decrease in pharmacologic effect is affected both the elimination constant k and the slope m. For a drug with a large m, the pharmacologic response declines rapidly and multiple doses must be given at short intervals to maintain the pharmacologic effect.

    Relationship Between Dose and Duration of Activity, single IV Bolus Injection

    The relationship between the duration of the pharmacologic effect and the dose can be inferred from Equation 19.3. After an intravenous dose, assuming a one-compartment model, the time needed for any drug to decline to a concentration C is given by the following equation, assuming the drug takes effect immediately:

    t = 2.3 * [log(C0) - log(C)] / k

    Using Ceff to represent the minimum effective drug concentration, the duration of drug action can be obtained as follows:

    teff = 2.3 * [log(C0) - log(Ceff)] / k = 2.3 * [log(D0/V) - log(Ceff)] / k

    where teff is duration of the pharmacologic effect, D is dose, and V is the volume of distribution.
    Last edited by CheneyHsiung; Fri 8th April '16 at 9:18pm.
    Clinical Pharmacy Specialist - Hematology

  6. #6
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    Default Re: The summary of pharmacokinetics core equations

    Maximum Effect (Emax) Model

    The maximum effect model (Emax) is an empirical model that relates pharmacologic response to drug concentrations. This model incorporates the observation known as the law of diminishing return, which shows that an increase in drug concentration near the maximum pharmacologic response produces a disproportionately smaller increase in the pharmacologic response. The Emax model describes drug action in terms of maximum effect (Emax) and EC50, the drug concentration that produce 50% maximum pharmacologic effect.

    The summary of pharmacokinetics core equations-screen-shot-2016-04-19-at-1-53-58-pm-png

    where C is the plasma drug concentration and E is the pharmacologic effect. This equation is a saturable process resembling Michaelis-Menton enzyme kinetics. As the plasma drug concentration C increases, the pharmacologic effect E approaches Emax asymptotically. Max is the maximum pharmacologic effect that may be obtained by the drug. EC50 is the drug concentration that produce one-half (50%) of the maximum pharmacologic response.

    Examples

    For theophylline, a small gradual increase in FEV1 is obtained as the plasma drug concentrations are increased higher than 10 mg/L. Only a 17% increase in FEV1 is observed when the plasma theophylline concentration is doubled from 10 to 20 mg/L. The EC50 for theophylline is 10 mg/L. The Emax is equivalent to 63% of normal FEV1. A further increase in the plasma theophylline concentration will not yield an improvement in the FEV1 beyond Emax. Either drug saturation of the receptors or other limiting factors prevent further improvement in the pharmacologic response.
    Last edited by admin; Tue 19th April '16 at 3:16pm.
    B.S. Pharm, West China School of Pharmacy, Class of 2007, Health System Pharmacist, RPh. Hematology, Infectious Disease. Chengdu, Sichuan, China.

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  7. #7
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    Default Re: The summary of pharmacokinetics core equations

    Sigmoid Emax Model

    The sigmoid Emax model describes the pharmacologic response-drug concentration curve for many drugs that appear to be S shaped (i.e., sigmoidal) rather than hyperbolic as described by the simpler Emax model. The model was first used by Hill (1910) to describe the association of oxygen with hemoglobin, in which the association with one oxygen molecule influences the association of the hemoglobin with the next oxygen molecule. The equation for the sigmoid Emax model is an extension of the Emax model:

    The summary of pharmacokinetics core equations-screen-shot-2016-04-19-at-3-22-10-pm-png

    where n is an exponent describing the number of drug molecules that combine with each receptor molecule. When n is equal to unity (n = 1), the sigmoidal Emax model reduces to the Emax model. A value of n >1 influences the slop of the curve and the model fit.

    In the sigmoidal Emax model, the slope is influenced by the number of drug molecules bound to the receptor. Moreover, a very large n value may indicate allosteric or cooperative effects in the interaction of drug molecules with the receptor.
    Last edited by admin; Tue 19th April '16 at 3:35pm.
    B.S. Pharm, West China School of Pharmacy, Class of 2007, Health System Pharmacist, RPh. Hematology, Infectious Disease. Chengdu, Sichuan, China.

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