The basis of clinical pharmacology
(calculation of drugs concentrations in blood)

Before reading this chapter, look through a chapter of General Pharmacology.
Here, several examples of the calculation of blood concentrations of administered drugs are shown by
utilizing the various parameters and figures of the pharmacokinetics which are described in the attached drug sheets.
Since drugs are administered repetitively rather than single dosing and also continuous infusion is carried out
by intravenous drip in many cases. These examples are also described.
Calculate by yourself and confirm them. It is good for calculation to use Excel.

When calculation is wrong, please let me know.

Exercise 1: From the attached sheets of Lithium carbonate

Lithium carbonate (Li2CO3, MW=74) is a drug for mania, and 100mg and 200mg tablets are marketed.
The parameters after an oral single dose of the 200mg are described in the attached sheets
(Taisho Pharmaceutical Co.) as follows.  

      Cmax: the maximal concentration in serum, Tma: time reached to the maximal concentration in serum,
      F: bioavailability

　Now, when you administer a dose (D) of 800mg orally to a manic patient every 12 hours, answer the next questions.

1) Calculate Vd of lithium carbonate.

　　Apparent volume of distribution (Vd) and elimination rate constant (Ke) obey the following formulae.

　　　Since AUC is expressed with mEq/L, it is changed to mg/L.

　　　　(Ans) Vd=1*200/(0.693/14.7)/(2.26/2*74)=50.9 L

2) How many hours are required to reach a steady state of the blood concentration?
　 By repetitive administration, the steady state is achieved after four to 5 times administration at a time interval
    of the drug half-life. 　

　　　　(Ans)  14.7* 5= 73.5 hours = three days

3) When the drug reaches a steady state after administered repetitively at a time interval of half-life (tau= 12 hours),
   calculate the maximum concentration (a peak value, Cpk), minimum concentration (a trough value, Ctr) and
  average concentration (Cav) in blood of lithium carbonate.

　　The following formulae are used.

　　　　(Ans)

　　　　　　Cpk=36.5 mug/ml =0.99 mEq/L

　　　　　　Cav=27.9 mug/ml =0.75 mEq/L

　　　　　　Ctr=20.7 mug/ml = 0.56 mEq/L

4) The effective blood concentrations for mania are 0.3 - 1.2 mEq/L. Is the clinical effect obtained by this dose?

　　　(Ans) As two molecules of Li are included in LiCO3, the average blood concentration is 27.9/74*2=0.75 mEq/L
               The highest concentration is calculated to be 0.99 mEq/L and this concentration is an effective dose.

5) Monitoring of blood concentration (TDM)
　Since there is a correlation between the concentration in blood of drugs, and the clinical effects or
  adverse reactions, monitoring of blood concentrations of specific drugs (TDM, therapeutic drug monitoring) is
  performed to use the drugs most properly.　　Moreover, a patient's compliance can also be detected.

　　The conditions of drugs for which TDM is required are as follows.
　　　a) A drug which has a clear correlation between the blood concentration and the clinical effects,
          or adverse reactions.
　　　b) The drugs whose effective dose and toxic dose are close and the drugs having narrow therapeutic index.
  　　c) The drugs which have large individual difference, large changes and nonlinearity of blood concentrations
          in pharmacodynamics.
　　　d) Drugs which has drug interaction

　　The drugs for which TDM is required are as follows.
         Antimanic drugs: lithium carbonate, Antiepileptics: phenytoin, phenobarbital, Cardiac glycosides:digoxin,
         Antiarrhythmic drugs: lidocaine, procainamide, Bronchodilators: theophylline,
         Antibiotics: gentamicin, vancomycin, amikacin, Immunosuppressants: cyclosporine, etc.

Exercise 2:  From the attached sheets of Amlodipine

　Amlodipine is a long-acting calcium antagonist, and 2.5mg and 5.0mg tablets are marketed.
　The following Fig. 1, Fig. 2, and Table are described in the attached sheets (Sumitomo Pharmaceuticals).

Fig. 1

Dose	Tmax	Cmax	AUC
5.0 mg	7.7 hr	3.39 g/mL	178.2 ng*hr/mL

1) Fig. 1 shows the time course of blood concentrations of amlodipine (long-acting calcium antagonist),
    when 5.0 mg dosage is orally administered.

   Calculate the half-life of amlodipine and its apparent volume of distribution (Vd).
   Moreover, calculate body clearance (Clb).   F (bioavailability) is assumed to be 0.7.

    Use Clb=Ke*Vd.  

　　　　　(Ans)

　　　　　　　　From the elimination phase of graph, half-life is calculated about 40 hours.

　　　　　　　　Ke=0.693 / 40= 0.0173,

　　　　　　　　Vd=F*D/Ke/AUC=0.7*5.0/0.0173/178.2=1135 L

　　　　　　　　Clb=0.0173*1135=19.6 L/hr

　　　　　　　　Since Vd is very large, amlodipine is considered to bind with the components in the body.

2) What days are required to reach the steady state, if amlodipine is administered to the patient every 24 hours?
    Calculate the peak value (Cpk) at that time, and compare with the blood concentration at first time administration.

　　　　(Ans)

　　　　A steady state is reached in 24* 5= 120 hours (4-5 days).

　　　　Cpk=0.7*5.0/1135/(1-exp(-0.0173*24))=9.1 ng/mL

　　　　It is about 3 times compared with the peak value of initial dose (Fig. 2).

Fig. 2

3) Fig. 2 shows the blood concentrations of a young healthy person and an old age hypertensive patient,
   when 5mg amlodipine is administered at single time (A) or continuous eight days (B).

　Fill up the blank spaces of the following Table; Cmax,  Tmax and T1/2.
   Moreover, calculate Vd and Clb.

   What parameters of the old age hypertensive patient are different from those of the young person?
   And explain the reasons.

	Old age hypertensive patient		Young healthy person
	Single dose administration	Continuous administration	Single dose administration	Continuous administration
Cmax (ng/mL)
Tmax (hr)
T1/2 (hr)
AUC (ng*hr/mL)	116.9	-	63.2	-
Vd(L)		-		-
Clb (L/hr)		-		-

　　(Ans)

　　Ke-old=0.693/37.5=0.0185, Ke-young=0.693/27.7=0.025

　　Vd-old=0.65*5.0/0.0185/116.9=1502 L,

　　Clb=Ke-old*Vd-old=1502*0.0185=27.8 L/hr

　　Vd-young=0.65*5.0/0.025/63.2=2056 L,

　　Clb=Ke-young*Vd-young=2056*0.025=51.4 L/hr

　Since the metabolism (body clearance) of amlodipine is slower in the aged person, the concentrations in serum
  become higher and you should begin from a low dose of the drug.

4) Pharmacokinetics of elderly people
　　As shown in 3),  the parameters of pharmacokinetics such as half-life (T1/2), body clearance (Clb) and
    volume of distribution (Vd) are changed.

　　　a) The decrease of renal function : the kidney function falls with aging and decreases 30-40% at the age of 70.
            Therefore, the half-life of drugs is extended and the clearance is decreased.
  　　b) The increase in relative amounts of fat: lipophilic drugs become more accumulated in the body.
　　　c) Reduction of body water: since the total amount of water decrease 15-20% at the age of 70, the Vd decreases.
  　　d) Since liver capacity and blood stream decrease about 45%, it is thought that drug metabolic activity falls similarly.

    As mentioned above, since the blood concentrations of drugs in old aged patients become higher and the risk
    of toxic symptoms is higher, dosage used for the elderly persons should begin from 1/2 to 1/3 of normal adults.

Exercise 3: From the attached sheets of cefotiam

　cefotiam is a cephem antibiotic and the drugs for i.v. injection (1g, 0.5g, and 0.25g) are marketed.
  Attached sheets of cefotiam (Takada Chemical Industries) show the measured parameters as shown in Fig. 3 and Fig. 4.

Fig. 3  

1) Fig. 3 shows the changes of the blood concentration after the intravenous injection of 1g of cefotiam.
  　Calculate a half-life (T1/2) and an elimination rate constant (Ke), and also Vd.

　　　(Ans)

　 　　　　T1/2=0.8 hr, Ke=0.693 / 0.8= 0.866

　　　　　C₀ is calculated by using the C=C₀*exp (-Ke*t), from logarithm of the elimination phase of Fig. 3.
           It is calculated to be 60microgr/mL. Since the dose (D) is 1g,

            Vd=D/C₀=1/60=16.7 L is calculated. 　　

2) Calculate the blood concentration when the intravenous drip infusion of 2g of cefotiam for 1 hour. Moreover,
   find the blood concentration when the intravenous drip infusion of the same amount for 2 hours.    

   The average blood concentration is expressed as a following formula. Here, Rinf is drip infusion speed (g/hr).

　　　(Ans)

　　　　　　In the case of 1 hour: infusion speed (Rinf) is 2 g/hr.

　　　　　　Cav=2* (1-exp (-0.866*2)) /16.7/0.866= 114microgr/mL

　　　　　　In the case of 2 hours: Rinf=2g/2 hr, 　

　　　　　　Cav=1* (1-exp (-0.866*2)) /16.7/0.866= 56.9microgr/mL

　　　　　　This blood concentration is obtained at the end of the intravenous drip infusion.

　　　　　　A steady state will be achieved if the intravenous drip is infused for 4 hours which correspond to
             5 times of the half-life.

　　　　　　Css=Rinf/Vd/Kel=2/4/16.7/0.866= 34.6microgr/mL is obtained.

Fig. 4

3) It is clear that the renal dysfunction patients of Mr. A and Mr. B show higher blood concentrations,
    compared to a healthy person.

Calculate T1/2 and Clb of Mr. A and Mr. B.

4) Pharmacokinetics in renal dysfunction
  　In renal dysfunction, Vd increases by hypoproteinemia and decrease of the binding rate to serum proteins.
    As it is considered as Drug body clearance (Clb) = Hepatic clearance (Clh) + Renal clearance (Clr),
    Clr decreases and Clb falls in renal dysfunction.
 
    Since it is Clb=Vd*Ke, the fall of Clb will be based on the fall of Ke, if supposed that Vd does not change,
　　Moreover, since it is considered Ke=renal elimination constants (Kr) + extra-renal elimination constants (Knr),
     Ke can be rectified by the method of  Giusti which utilizes creatinine clearance.

　　When creatinine clearance (Clcr) is not known, Clcr can be estimated from serum creatinine
     from the following formulae.

　　　　Clcr(male) =(140-age) * weight/72/serum creatinine concentration [Kg/mg/dl]

　　　　Clcr(female) =0.85*Clcr (male)

　　The compensation coefficient (G) of the dose to renal dysfunction is expressed with the following formula.

     
　　　Here, ‘100’ means a healthy person's Clcr [mL/min] and fu is a rate of  the non-metabolites in urine.

5) Compensation of the dosage for renal dysfunction  

　　The compensation of dosage and intervals are required in renal dysfunction.
    　Creatinine clearance (Clcr) of Mr. A in Fig. 4 is <5mL/min.

　　Mr. B is 50.6 mL/min.

　　Calculate the rectified dose for Mr. B, in the case of 3) ( 2g cefotiam and 1-hour intravenous drip infusion)
     and assuming that fu of cefotiam is about 70%.

     Moreover, calculate the infusion time in renal dysfunction when 2g of cefotiam is infused.

　　(Ans)   　　

　　　　G=1-0.7*(1-50.6/100)=0.654

　　　　　Therefore, a dose for Mr. B is calculated to be D=2*0.654=1.3 g.
    　 　　When you want to change the infusion time in stead, it is extended to tau=1hr/G = 1/0.654= 1.5 hours.

Exercise 4 : From attached sheets of Theophylline

　Theophylline is a drug for bronchial asthma Theophylline sustained-release tablets of 100mg and 200mg
  and the syrup of 20 mg/mL are marketed.

　The parameterｓ after an oral single dose of 200mg are shown in the attached sheets
   (Mitsubishi Tokyo Pharmaceuticals) as follows.

Cmax	Tmax	AUC	T1/2
3.0 microgram/mL	7.2 hr	53.9 microgram*hr/mL	12 hr

1) Calculate Vd and clearance (Clb) in the case of F= 0.9.

　　　　(Ans)

　　　　　　　Ke=0.693/12 =0.058

　　　　　　　Vd=0.9*200/0.058/53.9=5.8 L

　　　　　　　Clb=Vd*ke=0.058*58=3.3 L/hr

2) Urgent saturation by oral administration
　　Effective blood concentration of theophylline is around 8-20microg/mL.
   　Calculate an initial dose (loading dose) and a maintenance dose for obtaining immediately the maintenance-dose
     of 10 mug/mL.

　　A relation between the amount (Dld) of initial dose (loading dose) and the remaining amount (Ab) of drug
    after half-life (Thaf) is expressed as Ab=Dld*exp (-0.693).

　　The eliminating amount (Dm) of drug from the body after half-life is expressed as Dm=Dld*(1-exp (-0.693)) =Dld*0.5.

　  From the above formulae, the initial dose is determined as twice of the maintenance dose and
    after half-life the maintenance dose should be given.

　　　(ANS)

　　　　As we want to obtain a blood concentration (C) of 10 microgr/mL of theophylline at the steady state,
       a formula is expressed as C=F*D/Vd/Ke/Thaf, when defining a maintenance dose as Dm and a half-life as Thaf,

        Therefore, a maintenance dose is expressed as  Dm=10*Vd*Ke*Thaf/F=10*5.8*0.058*12/0.9=448 mg.

       If twice (900 mg) of a maintenance dose is first administered and then 450 mg is given every 12 hours
       after the half-life, a saturated concentration of 10 microgr/mL can be achieved from the first time (rapid saturation).

　　 The prophylaxis of the asthmatic attack by the sustained-release agent of Theophylline is called RTC
      (round the clock: all the day) treatment.

　  This is the method for maintaining the drug effect and preventing the attack by taking the drug twice a day
     and one of the main treatments for child bronchial asthma.

3) Changes of theophylline clearance by age

　　Side effects appear in many cases in proportion to the blood concentration, and TDM. is required for theophylline
    to make suitable plans for patient individuals.

　　As shown in Fig. 5, Clb of theophylline changes markedly with age.

　　Since Clb of theophylline in children becomes twice of adults (especially at the age of 1-2), the cautions are required.

Fig. 5 (modified from China, J. Pedant, and 95, 1735 and 1991)

　　Since maintenance dose is expressed as (medication interval)*(clearance)*(blood concentration),
     it turns out that a maintenance dose is proportional to clearance. Therefore, theophylline tends
     to produce side effects such as convulsion in children.

4) Drug administration to children

　　The dose for children is estimated by assuming that they are small size of  adults　The amounts of drugs are
     determined based on weights shown in next Table and is widely used.

　　　　　　　　　The simple table for determining a children dosage

Age	Newborn infant, premature baby	0.5	1	3	7.5	12	Adult
Dosage	From 1/20 to 1/10	1/5	1/4	1/3	1/2	2/3	1

　　Since a child's pharmacokinetic values are not obtained in many drugs, adult parameters have been used for
     children as an index. In this case, the compensation by water is made.

a) Extracellular fluid  

　　Although Vd is proportional to physiological water distribution and the intracellular fluid of children and
    adults are 35 - 40% of weight, extracellular fluid is about 32% of weight for children and about 20% for adult.
    Inulin clearance is 10 mL/min for children and 120 mL/min for adult.
　　Since many drugs are distributed in extracellular fluid, the formula of Friis-Hansen are used to rectify
    the extracellular fluid.

b) Metabolism and renal function of children 
　　Although glucuronidation is underdeveloped in infant and children, sulfate conjugation is good and
     methylation is also active.
  　Since renal function is also underdeveloped, the parameters of pharmacokinetics are markedly affected.
   
Reference books: Takashi Ishizaki, Clinical pharmacological lecture, Igakushoin  
                         Kanji Takada, Pharmacokinetics study, Jiho Co.

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　                             　 (Miki, Kamisaki)  

 (2002/11/16)