Bayesian theory

Bayesian theory in population pharmacokinetics--
1) INTRODUCTION TO BAYESIAN THEORY
2)BAYESIAN PROBABILITY TO DOSING OF DRUGS
3)APPLICATIONS AND USES OF BAYESIAN THEORY IN APPLIED PHARMACOKINETICS:
therapeutic drug monitoring and clinical pharmacokinetics-fifth pharm d notes

1. THERAPEUTIC DRUG MONITORING





ASSIGNMENT ON BAYESIAN
THEORY









SUBMITTED BY
PAVITHRA .V
V- PHARMD

2. POPULATION PHARMACOKINETICS
Population pharmacokinetics (PopPK) is the study of variability in
plasma drug concentrations between and within patient populations
receiving therapeutic doses of a drug. Traditional pharmacokinetic
studies are usually performed on healthy volunteers or highly
selected patients, and the average behavior of a group (ie, the mean
plasma concentration–time profile) is the main focus of interest.
PopPK examines the relationship of the demographic, genetic,
pathophysiological, environmental, and other drug- related factors
that contribute to the variability observed in safety and efficacy of
the drug.
The PopPK approach encompasses some of the following features
 The collection of relevant pharmacokinetic infor- mation in
patients who are representative of the target population to be
treated with the drug
 The identification and measurement of variability during drug
development and evaluation
 The explanation of variability by identifying fac- tors of
demographic, pathophysiological, environ- mental, or
concomitant drug-related origin that may influence the
pharmacokinetic behavior of a drug
 The quantitative estimation of the magnitude of the unexplained
variability in the patient population
 The resolution of the issues causing variability in patients
allows for the development of an optimum dosing strategy
for a population, subgroup, or indi- vidual patient.
 The importance of developing opti- mum dosing
strategies has led to an increase in the use of PopPK
approaches in new drug development.

3. INTRODUCTION TO BAYESIAN THEORY

Bayesian theory was originally developed to
improve forecast accuracy by combining subjective prediction with
improvement from newly collected data. In the diagnosis of disease,
the physician may make a pre- liminary diagnosis based on symptoms
and physical examination.
Later, the results of laboratory tests are received. The
clinician then makes a new diagnostic forecast based on both sets of
information. Bayesian theory provides a method to weigh the prior
informa- tion (eg, physical diagnosis) and new information (eg,
results from laboratory tests) to estimate a new probability for
predicting the disease.
 The Bayesian view of probability is related to degree of belief. It is
a measure of the plausibility of an event given incomplete
knowledge.
 It was originally developed to improve the forecast accuracy by
combining subjective prediction with improvement from newly
collected data.
 Bayesian theory of probability evaluates the probability of a
hypothesis by specifying some prior probability, which is then
updated in the light of new relevant data.

 For e.g. In the diagnosis of a disease, a physician may make a
preliminary or differential diagnosis based on the patient’s
symptoms and physical examination (subjective evidence/
priori). The results from relevant laboratory data when
combined with the prior evidence a new diagnostic forecast is
made.

4.  Thus Bayesian method of probability provides a method to weigh
the prior information (subjective evidence) and the new information
(objective evidence) to estimate a new probability for predicting the
disease.
 This approach has been used in drug dosing for patients. Based on
the patient’s medical history, an average or population
pharmacokinetic parameters appropriate for the patient’s condition
(priori) are used to calculate the initial dose.
 After the initial dose, the patient’s pharmacokinetic parameters
(like Clearance and volume of distribution)and plasma / serum drug
concentrations are obtained from the patient that provide new
information to assess the adequacy of the dosage. Based on the new
parameters, adjustments are made.
 Bayesian method gives better prediction to PK parameters of the
patient and is particularly useful when only a few blood samples
are available.
 This approach is beneficial when data is scarce, noisy (erroneous),
or biased. Bayesian methods can be used to combine results from
different experiments
 Because of inter and intra subject variability, PK parameters of an
individual patient must be estimated from limited data in the
presence of unknown random errors (assays etc ), known covariates
and variables such as clearance, weight, disease factors etc.
 In the presence of large amount of variation in data, Bayesian
approach employs weighted least-squares (WLS) to allow for
improved estimation of patient’s pharmacokinetic parameters by
taking into consideration the mean of population PK parameters
and their variability.

5. EXAMPLE:

After diagnosing a patient, the probability of having a particular
disease was 0.4 according to the physician. Based on the preliminary
assessment, the physician ordered a lab test. Positive value has a 0.8
probability of positively identifying the disease in patient s with the
disease (true positive) and a 0.1 probability of positive result in
patients without the disease (false positive). From the prior
information (physician’s assessment)and the current patient specific
data (lab tests), what is the posterior probability of the patient having
the disease using the Bayesian method?

SOLUTION:

Prior Probability of having the disease (positive) = 0.4
Prior probability of not having the disease = 1-0.4 = 0.6
Ratio of disease positive : disease negative = 0.4/0.6= 2/3, implying
that the physician’s assessment shows that there is a 2/3 chance that
the patient has the disease diagnosed by the physician
The probability of the patient actually having the disease
can be better evaluated by including the lab findings.

For the same patient,
Probability of a true positive = 0.8

6. Probability of a false positive = 0.1 (test wrongly identifying the
disease)
Likelihood Ratio = 0.8/0.1 = 8/1

Combining the prior probability ratio with the likelihood ratio, the
posterior probability may be achieved at:

Posterior probability ratio= (2/3) (8/1) = 16/3

Where 16= number of favorable events and 3 = number of
unfavorable events
Posterior probability = 16/ (16+3) =84.2%
Because total number of events that can occur = 16 +3, and

Probability = total number of favorable events / total number
of events

(Probability of an event occurring or not occurring will always lie
between 0 to 1, with 0 being no probability of the event occurring and
1 indicating a 100% chance of the event occurring, which in this case
will be the disease prediction for the patient.)

7. BAYESIAN PROBABILITY TO DOSING OF DRUGS :

Bayesian probability theory when applied to dosing of
a drug involves a given pharmacokinetic param- eter (P) and
plasma or serum drug concentration (C), as shown in Equation
22.11. The probability of a patient with a given pharmacokinetic
parameter P, taking into account the measured concentration, is
Prob(P/C):

Prob(P/C )=
Prob(P) .Prob(C /P)
Prob(C)


Posterior probability = (Likelihood probability * Prior probability)
Total number of events or occurances



EXAMPLE
Theophylline has a therapeutic window of 10–20 μg/mL. Serum
theophylline concentrations above 20 μg/mL produce mild side effects,
such as nau- sea and insomnia; more serious side effects, such as sinus
tachycardia, may occur at drug concentra- tions above 40 μg/mL; at
serum concentrations above 45 μg/mL, cardiac arrhythmia and seizure
may occur (see Fig. 22-1). However, the probability of some side effect
occurring is by no means certain.

8. Side effects are not determined solely by plasma concentration, as
other known or unknown vari- ables (called covariates) may affect the
side effect outcome. Some patients have initial side effects of nausea
and restlessness (even at very low drug concentrations) that later
disappear when therapy is continued. The clinician should therefore assess
the probability of side effects in the patient, order a blood sample for
serum theophylline determina- tion, and then estimate a combined (or
posterior) probability for side effects in the patient.

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.1
0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

FIGURE Conditional probability curves relating prior probability of
toxicity to posterior probability of toxicity of STC, theophylline serum
concentration

The decision process is illustrated graphically in Fig. The probability
of initial (prior) estimation of side effects is plotted on the x axis, and
the final (posterior) probability of side effects is plotted on the y axis
for various serum theophylline concentrations. For example, a patient
was placed on theophylline and the physician estimated the chance of
side effects to be 40%, but therapeutic drug monitoring showed a
a
b
c
d
e
Posterior probability

9. theophylline level of 27 μg/mL. A vertical line of prior probability at
0.4 intersects curve a at about 0.78 or 78%. Hence, the Bayesian
probability of having side effects is 78% taking both the labora- tory
and physician assessments into consideration. The curves (Fig.) for
various theophyl- line concentrations are called conditional
probability curves.


APPLICATIONS AND USES OF BAYESIAN THEORY
IN APPLIED PHARMACOKINETICS :

 Bayesian theory does not replace clinical judgment, but it
provides a quantitative tool for incorporating subjective
judgment (human) with objective (laboratory assay) in making
risk decisions.
 When complex decisions involving several variables are
involved, this objective tool can be very useful.
 Bayesian probability is used to improve forecasting in medicine.
One example is its use in the diagno- sis of healed myocardial
infarction (HMI) from a 12-lead electrocardiogram (ECG) by
artificial neural networks using the Bayesian concept. Bayesian
results were comparable to those of an experienced
electrocardiographer .
 In pharmacokinetics, Bayesian theory is applied to “feed-
forward neural networks” for gentamicin concentration
predictions
 A brief literature search of Bayesian applications revealed over
400 therapeutic applications between 1992 and 1996.

10.  Bayesian parameter estimations were most frequently used for
drugs with narrow therapeutic ranges, such as the

 aminoglycosides,
 cyclosporin,
 digoxin,
 anti- convulsants –phenytoin, lithium,
 theophylline.


 The technique has now been extended to cytotoxic drugs, factor
VIII, and warfarin.
 Bayesian methods have also been used to limit the number of
samples required in more conventional pharmacokinetic studies
with new drugs

DISADVANTAGE:

 The main disadvantage of Bayesian methods is the subjective
selection of prior probability.
 Therefore, it is not considered to be unbiased by many statisti-
cians for drug approval purposes.