Vet Pharmacokinetics Print File #2

Steady State Concept

[Steady State Concept general, 08/30/96]

Steady State

Definition

• Steady State = Condition in which amount of drug eliminated each dose interval equals the dose for that interval.
• [full] [icon] Figure: Intravenous infusion rising to steady state

Questions about why?

• [Ans] Why does the concentration "plateau"?

Questions about time required to plateau

• [Ans] For which of the following conditions would the "plateau" occur sooner in the face of the same infusion rate? (hint: think bathtub!)
  a) very rapid elimination?
  b) very slow elimination?
  
• [Ans] Will the concentration of a drug plateau if it is eliminated at a constant mass per minute, i.e., its elimination rate could be described as 10 mcg/min?

• [Ans] What could one do if the time-to-plateau were too long for the therapeutic need?

Answers/Comments regarding steady state

• Time to plateau: The rate at which steady state is approached when repeated equal doses or an intravenous infusion is given depends in the simplest case on the elimination rate. Steady state is approached faster when elimination is faster. Drug "accumulates" until the concentration gets high enough to make elimination equal input.
  [full] [icon] Figure: Intravenous infusion an an example of "accumulation"
  
• A fact you should remember is that steady state is approached within 3 to 5 elmination half-lives (88% to 97%)
  [full] [icon] Figure: Time to steady state on intravenous infusion
  
• First/zero order: Drug must follow "First-Order Kinetics" These principles do not apply to drugs following "Zero-Order Kinetics." These concepts apply only when elimination is "first order", that is, dependent on the plasma drug concentration. See accompanying Michaelis-Menton plot of drug elimination rate versus concentration.
  [full] [icon] Figure: Michaelis-Menten plot
  
• Note: the final curve after infusion is stopped is the inverted-image" of the initial rise. Both shapes depend on the same processes -- those involved in eliminating the drug.
  [full] [icon] Figure: Rise on start and Fall on cessation are inverted images
  
• Loading Dose: One can give a first dose much larger than the maintenance dose or, in the case of an infusion, one could give a rapid, injection followed by the planned rate of infusion. This concept is similar to induction and maintenance in inhalation anesthesia. More on this later.

Major Pharmacokinetic Parameters

• Volume of Distribution (& Loading dose)
• Clearance
• Ke (Elimination Rate) -- Half-Life
• Bioavailability

Volume of Distribution
and Loading Dose

Calculating an Initial Dose

• Assume you decide to treat 5 expensive fish in a 50 L tank with miraclemycin and that the desired final concentration is 10 mg/L. 
  • How much drug (in mg) should be added?
  Enter (mg)>
  
  

• How much drug should be given to a patient that weighs 75 kg and has a body water content of 50 L (assume uniform distribution of drug). Desired concentrationis 10 mg/L.
  • How much drug (in mg) should be added?
  Enter (mg)>
  
  

• Can you see the similarity in these examples? You have a known volume of fluid in which you want to have a specific concentration of drug.

• In this case,
  • Amount of drug = dose
  • Desired concentration = target concentration (mean or average) in sampled fluid
  • Volume = the volume of sampled fluid in which the drug is distributed
  • These cases assume that no drug has left the tank or body, i.e.,they are values taken at the instant drug was administered. This time is called zero-time.
  • They further assume that no drug was lost during that instant
• How REALISTIC are these assumptions? We will see that at the first level of approximation and given a few qualifiers that you are to know, they are not bad.

Loading Dose

• Loading Dose (DL) = a dose of drug sufficient to produce a plasma conentration that would fall within the therapeutic window after only one or very few doses over a very short interval. It is larger than the dose rate needed to maintain the concentration within the window and would produce toxic concentrations if given in repeated doses.
  
• Maintenance Dose (DM) = The dose needed to maintain the concentration within the therapeutic window when given repeatedly at a constant interval
  

Information needed to calculate a DL

• Volume of "tank" (Volume of distribution, usually in L or L/Kg)
  • from literature
• Desired concentration
  • from literature
• Knowledge of assumptions
  • absorption very rapid relative to elimination
  • volume of tank is in terms of the sampled fluid (i.e., plasma or serum) because this is the only concentration you know by direct measurement.
  • one must use the same terms throughout calculations
    • mg / L; mcg/L; mcg/mL, etc.
      Note that mg/L and mcg/mL are the same!
    • total drug vs unbound drug
      Usually use total drug concentration due to cost-effectiveness

Finding Vd

• Look it up!

• Simple experiments

  • Conceptual approach given here. Actual determinations are much more complex and by different methods to get more accurate values
    

• Vd Experiment #1 (closed system)

  • MEASURE DILUTION!
  • Add 500 mg of drug to ____ L of water in a beaker
  • Measured drug concentration after mixing is 5 mg/L
  • What is the volume of distribution?
  Enter (L)>

• Vd Experiment #1 (open [real] system)

  • 70 kg patient given dose of 2,800 mg of miraclemycin
  • Plasma drug concentrations --

  Times (hrs)	2	4	6	8
  Conc (mg/L)	10	5	2.5	1.25

  • Plot the data on arithmetic graph paper with labels
    • X- axis = 0 hours to 20 hours
    • Y-axis = 0 mg/L to 25 mg/L
    • What is the shape of the plot?
    • Draw light horizontal lines from each data point to the 0 h axis (parallel to bottom of graph paper!)
    • Is the vertical distance between each of the horizontal lines constant? Why or why not?
    • Can you make a meaningful extrapolation of the data to "zero hours"?
    • See Excel Spreadsheet [plot]

    ---

  • Plot the data on semi-logarithmic graph paper with labels
    (Keep this graph for later use!)
    • Note that on a 2 or 3-cycle sheet of paper, each "1' could represent 0.1, 1.0, 10.0, and 100.0! For this experiment, make the top-most "1" = 100 mg/L.
    • Plot the data and connect the data points.
    • Draw light horizontal lines to each data point as you did previously
    • What difference do you note? The vertical distances between each of the data points are now the same!
    • Semi-log paper represents a "geometric progression" where each multiple is distributed evenly, i.e., 1, 2, 4, 8, 16, 32, 64, 128, ..., n*2 are distributed evenly. OR 1, 10, 100, 1,000, ..., n*10 are distributed evenly on one of the axes.
    • By the same token, if we start with 100 ug/ml and cut the concentration by half to 50 and then again to 25, the vertical distances are the same. Each FRACTIONAL change of 0.5 gets the same vertical distance.
    • Can you make a meaningful extrapolation of these data to "zero hours"? YES! [Excel plot]
    • What is the concentration at the "zero hours" intercept?
  Enter (mg/L)>

  • What is the volume into which the drug APPEARS to have distributed? That is, how much plasma-equivalent is required to explain the dilution?

  Enter (L)>

  • How does this compare to the weight of the patient?

  • If body water is approximately 70% of body weight, how can a drug appear to have distributed into 2 L/Kg of body weight?
  • Hint: Think about what fluid we are measuring and our assumption of uniform distribution? Are these data consistent with the notion that miraclemycin distributes in such a way that the concentration outside the plasma is higher than that in the plasma?
  • How can one use an estimate of Vd to calculate a loading dose?
  • What is the "pharmacokinetic usefulness" of the Vd term?

Summarizing statements for Vd

• Definition: volume of plasma/serum (Vd) required to explain the plasma/serum concentration (C) when there is a known amount (A) of drug in the body. The amount (A) is equivalent to the dose (D) at zero time after administration of a single dose. For many of our purposes, this would equate to an initial dose, or a loading dose (DL)
• Note the dimensions of the terms below. Vd can be in L or in L/Kg. The former would be the Vd for the entire patient, whereas the term "L/Kg" is a way of normalizing the Vd to account for differences in body mass.

  Vd = D / Measured Conc {L = mg / (mg/L)}

• One can rearrange the formula for various uses. To find a loading dose (DL),

  DL = Vd * Desired Conc {mg = L * (mg/L)}

• A frequent use is in formulae to replace the "initial concentration" (C0),

  Predicted Conc = DL /Vd { mg/L = mg / L }

• The easiest way to remember these is to use the most obvious form of the equation, the one above where (mg/L) = (mg) / (L), and then do simple rearrangements as needed. ALWAYS CHECK the TERMS to be sure you are correct.
• The Vd is not a specific anatomic space
• Vd changes in various conditions, e.g., with age, pregnancy, disease, levels of protein binding of drug, fat content of the animal, etc.
• The simplest conceptual way to determine Vd is to administer a dose of drug, measure concentration over time and extrapolate the concentrations to "zero-time" to find the initial concentration. Then use the known dose (DL) and concentration (C0) to calculate Vd.
  [full] [icon] Figure: Example of using a simple plot to determine Vd

CLEARANCE

Clearance

• Clearance is a measure of how rapidly the organ(s) of elimination "clear" drug from the plasma.
• Clearance (CL) can be defined as the volume of plasma completely freed of drug per unit of time (e.g., L/h). It is a "virtual" volume that depends on two major factors for any specific organ of clearance:
  • blood (plasma) flow to the organ
  • fraction of drug extracted from the blood (plasma)
• Extraction ratio
  • Extraction ratio (ER) is the fraction of drug removed in a single pass through the organ of elimination. It can be measured in a single organ by sampling drug concentration in the arterial blood into (Cin) and venous blood from (Cout) the organ.

    ER = [Cin - Cout] / Cin

    Thus if the arterial blood contains 15 mg/L of drug "A" and venous blood from the organ contains 9 mg/L, the ER is (15-9)/15 = 6/15 = 0.4.

  • In the preceding example, an ER of 0.4 means that 40% of the drug in the plasma is removed as it passes through the organ.
  • Extraction ratios for drugs in various organs can range from 0 to 1.0 and may vary depending on the functional capacity of the organ and whether or not the drug concentration is in the "first-order" range of concentrations.
  • Disease conditions and other drugs can change the ER of a drug
  • Extraction ratios can also vary with blood flow if it causes the capacity of the organ to be exceeded.
• Blood flow (often symbolized by Q with a dot on top, but not in these notes because of limitations in the formatting language, html) can be expressed as mL/min, L/h, etc.
  • If an organ of elimination is capable of removing all drug in a single pass, i.e., ER = 1.0, then changes in blood flow to the organ can be very important in determining the elimination rate because blood flow is the rate limiting factor.
  • Conversely, if ER is very low, changes in blood flow are less important.
  • Renal blood flow can vary widely. "Renal shut-down" can dramatically slow elimination of a drug as can slowed renal blood flow due to hypovolemic conditions. For this reason, it is common in clinical medicine to administer fluids to keep renal perfusion high to aid in elimination of some drugs.
• Single organ clearance --

  CL = ER * Flow
  [L/h = frcn x L/h]

  Notice that there are no "mass" terms in the final CL formula. It is the "virtual" volume (in liters) of plasma completely cleared of drug per hour in this case. When the ER is 0.5, the concentration in "twice the CL volume" is lowered by 50%.

• Amount eliminated -- So, how does one find the "amount" of drug actually eliminated during a specific time interval from these computations? Awareness of the difference between "amount" and "fraction" is crucial to understanding clinical pharmacokinetics. Be sure to pay attention to which of these is being used in various equations and terms. In general, when a pharmacokineticist speaks of a "rate", one is referring to a fraction/time.
  • One finds the "amount" eliminated at any instant in time (e.g., one minute) by multiplying the average drug concentration in the plasma "during that minute" by the CL.

    Amount eliminated(t) = CL * Cin(t)
    (mg/min = L/min * mg/L)

    where Cin(t) represents the average concentration of drug in the arterial plasma during the time interval denoted by "t".

  • Amount changes: One should note that this "amount" changes constantly as the concentration of drug in the body changes (increasing while amount absorbed is greater than that eliminated and decreasing when it is less)
  • Fraction stays constant What stays constant when a drug is cleared by first order kinetics is the "fraction" eliminated with each pass of plasma through the organ of clearance.
  • Confused? Try clicking here
    
• Whole body clearance is simply the sum of all organ clearances

  CL(body) = CL(renal) + CL(hepatic) + CL(pulmonary) + CL(etc)

• Relationship between CL(body) and Ke
  • Whole body clearance and the elimination rate constant are related, but emphasize different aspects of the same processes.
  • Elimination Rate constant (Ke) is the FRACTION of drug in the body that is eliminated each unit of time, e.g., "fraction per hour".
  • Calculate Ke from CL & Vd: If one knows the Vd of a drug and the CL(body), one can calculate the FRACTION of the drug in the body that is removed per unit time. This is the same fraction as the fraction of "plasma equivalent" that is completely cleared of drug per unit time. This fraction is the Ke.

    Ke = CL(body) / Vd (fraction/hour)

    CL(body) = Ke x Vd (volume/hour)

  • Diagram of the Relationship among ER, Q, CL, Vd, and Ke.
• Questions / practice

More on "Amounts", Extraction ratios, and Clearance

There may be confusion about how one relates extraction ratio and clearance with how much drug is actually eliminated during any specific time interval. It should be noted that if a drug is cleared by first order kinetics and if blood flow to the organ(s) of elimination is constant, then CL and ER are both constant regardless of how much drug there is in the plasma. However, THE ACTUAL AMOUNT OF DRUG ELIMINATED DURING EACH INTERVAL OF AN HOUR OR A MINUTE IS NOT CONSTANT. If the concentration of drug in the plasma is rising, the amount of drug eliminated (e.g., mg/min) is also rising. Vice versa, if the concentration of drug in the plasma is falling (when drug absorbed per min is less than the amount eliminated), then the amount of drug eliminated each successive minute will also decrease.

To find the actual amount of drug eliminated each minute, one must know the average concentration of drug in the arterial blood to the organ of elimination during that minute. One then multiplies that concentration times the CL to find the amount of drug eliminated during that minute.

Amount of drug eliminated in the minute = Average concentration in incoming plasma x CL

(mg per min (t) = average mg/L in (t) x L/min)

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Questions regarding clearance

• Is it possible to talk about the efficiency of an organ's function in terms of its ability to "clear" a drug from the circulation?
• If one had tests to measure the functional capacity of an organ, e.g., kidney, could one establish relationships that would help one make "predictive dose schedules" for certain drugs?
• Assume that the Clearance of a drug by the kidney is 5 liters/hour and that the Vd is 50 liters. What FRACTION of the Vd is completely "cleared" of drug per hour? Could one point to the specific volume of blood that had been freed of drug in the animal?
• Assume the drug concentration at the beginning was 10 mg/L. How much drug would there be in the animal?
• How much drug would there be in the animal at the end of the first hour? ...the second hour?

Answers to Clearance Questions

• Yes, it IS possible to talk about an organ's function in terms of Clearance
• Yes. In fact it is common to use such measures as Creatinine Clearance as an index of the kidney's filtration capacity. Many formulae have been developed to adjust drug dosage, e.g., of aminoglycosides like gentamicin, based on creatinine clearance.
• If CL = 5 L/h and Vd = 50 L, then 5/50 or 1/10th of the body fluid is cleared of drug per hour (0.1/h).
• Given above parameters and an initial concentration of 10 mg/L, then the animal would contain 10 mg/L * 50 L = 500 mg of drug at the beginning.
• If drug is cleared from 5 L/h, then 5 L * 10 mg/L = 50 mg is eliminated in the first hour. Assuming volume loss was negligible, then the new concentration is 450 mg/50 L = 9 mg/L. Thus 50 /500 or 1/10 of the drug was eliminated. (0.1/h)
• If drug is cleared from 5 L/h, then 5 L * 9 mg/L - 45 mg eliminated in the second hour. This is 45/450 = 1/10 of the drug present was eliminated the second hour (0.1/h). At the same time 1/10th of the Vd was "cleared" of drug in the same hour.
• Note that the FRACTION of drug eliminated in each hour was the same as the FRACTION of the Vd that was "cleared" of drug in each hour. However, the actual amount of drug eliminated decreased as the concentration decreased.
• This leads us to a discussion of elimination rate constants and half-life: Ke & t1/2

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Elimination Rate Constant / Half-life

Ke (Elimination Rate) -- Half-Life

• Ke, the elimination rate constant can be defined as the fraction of drug in an animal that is eliminated per unit of time, e.g., fraction/h.
• Elimination half-life is the time required for the amount of drug (or concentration) in the body to decrease by half.
• Although CL can be easily related to the function of a specific organ, it is more difficult to get a "minds-eye-view" of how fast a drug is removed from the whole animal from CL. It is also not easy to conceptualize what a Ke of 0.016 /h means. The elimination "half-life" is better suited for this because after 5 half-lives, approximately 97% of the drug is gone.Therefore, it is useful to be able to interconvert CL, Ke, and half-life.
• CL, Ke, Vd, and Half-life are all inter-related as follows:

  Ke = CL / Vd ---- [frcn/h = (L/h) / L]

  Half-life = 0.693/Ke ---- [h/(half) = 0.693 / (frcn/h)]

• 0.693 is the natural logarithm of 2. Thus, half-life is an arbitrary measure of drug elimination that is useful for humans, but not easy to use in complex formulae. Therefore, many tables contain both, but most formulae require one to use Ke. You should be able to do this as well as to inter-convert CL and Ke when given Vd.
• Discovering Ke and Half-life from a graph

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Graphs -- Ke -- Half-life

• Find the graph of "miraclemycin" that you made earlier.
• What is the "half-life" of the drug in this case?
• Label the slope of the line as being "-Ke"
• From the half-life, calculate the Ke
• Now, calculate the CL
• Answers

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Answers to Graph Questions

• Half-life = 2 h
• Ke = 0.693 / 2 h = 0.3465/h
• Remember that you calculated a Vd in this case of 140 L in the 70 Kg patient

  CL = Ke * Vd

  0.3465/h * 140 L = 48.5 L/h

• One can also express CL relative to weight:

  CL/Kg = Ke * Vd(L/Kg)

  0.693 L/(h * Kg) = 0.3465/h * 2 L/Kg

Bioavailability (F)

• The central circulation is the reference point for all bioavailability calculations. It is the only fluid that has contact with the entire body.
• Not all drug given to an animal per os (or other non-IV routes) is absorbed into the central circulation.
• Not all of an administered dose is "available" to the sites of action.
• Bioavailability is defined as the fraction of drug administered that reaches the central circulation, i.e., the circulating post-portal venous blood.
• By definition, bioavailability for a drug administered intravenously is 100%. The "term" used is "F". An F of 1.00 is equivalent to a bioavailability of 100%.
• One can find the "Area-Under-the-Curve" (AUC) for a drug as a measure of the exposure of the animal to the drug. In its simplest form, one plots the drug concentration versus time and then adds up the areas of successive trapezoids formed under the curve. This is the so-called "trapezoidal rule".
• A fun, simple way of determining the the relative drug exposure would be to cut out the area under the plasma concentration versus time plot and weigh it.
• Find the area (weight!?) for the "test-route", e.g., PO and compare it to the area (weight!?) of an intravenous injection of the same drug dosage.
• Graphic depiction of determining F
• Some representative values of F in animals:
  • Ketamine, cat, IM, F=0.92
  • Ampicillin HCl, cat, SC, F=0.56
  • Sulfadimethoxine suspension, dog, PO, F=0.56
  • Kanamycin Sulfate, sheep, IM, F=1.00
  • Kanamycin Sulfate, sheep, Intratracheally, F=0.38
• Some drugs with low "F" in humans (F < .5) Rowland&Tozer89,p120
  • Alprenolol
  • 5-fluorouracil
  • Lidocaine
  • Morphine
  • Nifedipine
  • Propranolol
  • Salicylamide

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