the arithmetic , geometric and harmonic means , as well as the standard deviation ( sd ) , as the measures of variability , are the ones most frequently used descriptive statistics in the calculation of pharmacokinetic ( pk ) ( cocchetto et al .
1980 ; lam et al . 1985 ; roe and karol 1997 ; griffin et al . 1999
a common problem in the analysis and interpretation of comparative pharmacokinetic parameters is the high value of sd ( davit et al .
2008 ; haidar et al . 2008 ; van peer 2010 ) . in extreme cases ,
the high value of sd makes it impossible to make the right decision as to the fate of the study .
this problem concerns many types of studies , from preclinical studies , pilot pharmacokinetic studies to bioequivalence ( be ) ( riley 2001 ; chien et al .
one way to solve the problem of comparative analysis of data burdened with high values of sd is to optimize the sample size of the group in the study , which consists of a precise determination of the number of subjects or animals on the basis of intrasubject variability ( fda 2006 ; ramirez et al .
however , even in be studies , in some cases , to determine the correct number of subjects , a pilot study is needed or even a two - stage study model ( ema 2010 ) .
a high value of variability ( cv  % ) of observed concentrations or pk parameters , especially often makes it impossible to properly interpret the outcome of the study in case of research concerning new drugs , for which their value of variability of key pk parameters is unknown .
this situation is further complicated by the fact that research in the early stages of drug discovery and pilot studies is usually conducted on a small number of subjects ( for example first in man ) .
the scatter of results described by mean values and sd is the result of many factors . to a large extent
it depends on the fate of the drug in the organism , such as absorption , distribution , re - distribution , metabolism and elimination .
one of the factors standing  outside  of a living organism is the scatter of results , which comes from the precision of the used analytical method .
currently , the limit value for the cv  % of precision for the calibration curve excluding the lower limit of quantitation point ( lloq ) is 15  % , while in the point equal to lloq , this value can be 20  % ( fda 2001 ) . in relation to incurred sample analysis for classic drugs , as the acceptable range of differences for repeated analysis , the range 20  % is proposed ( ema 2009 ; rozet et al .
this means that every bioanalytical result introduces an error to the pharmacokinetic calculations as well as the chosen research model  human , laboratory animal or cells in the in vitro studies ( jansen et al .
the problem of analysis of pk after per os administration of the drug is high variability in the c  t values from first sampling to the phase of distribution .
the common factor influencing maximum concentration ( cmax ) and last concentration ( clast ) values is the spread of analysis result , which determines the precision of the analytical method .
all three , absorption , distribution and elimination , processes which in point of time corresponding to cmax occur simultaneously . in case of a single administration of the drug in the elimination phase , the values of the concentration can be observed , which illustrate almost exclusively elimination processes ( excluding the redistribution phenomenon ) , until the interval between the end of the distribution phase and the value equal to \documentclass[12pt]{minimal } 
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				 \begin{document}$$ t_{\hbox{max } } + t_{1/2kel } \times 3 $ $ \end{document } ( emea 2001 ; veng - pedersen 2001 ; fda 2006 ; hc 2010 ) .
it can be assumed that the factors that affect the cmax relative standard deviation ( cmax , cv  % ) , resulting from simultaneously occurring elimination , are to some extend dependent on the value of last concentration relative standard deviation ( clast , cv  % ) minus the error resulting from the precision of the analytical method ( cv  % , an ) .
pk studies are usually conducted in the conditions of good laboratory practice and good clinical practice , or in accordance with the principles of these quality systems .
it can be therefore assumed that the sum of errors connected to the subject , experimental animal , used formulation or bioanalytical method is constant , while keeping the experiment conditions , controlled by the quality system .
it can also be assumed that in each pk study , a minimum range of sd for c  t is possible to achieve . in relation to a single administration , the closest value in many cases could be the lowest value of cv  % for the last points of sampling in the elimination phase . in this phase of the study ,
the deviations from the mean are usually the lowest in the whole series , as the elimination phase is the dominant one and no other process , which is characterized by high variability ( for example absorption ) influences the sd of the analyzed concentrations .
taking the above into account , the aim of this study was to propose a method of transformation of c  t , which would allow significantly reducing the sd value of observed concentrations , without the statistically significant influence on the value of the mean and median for each sampling point .
in the presented study , the lowest value of relative standard deviation ( rsd  % ) of concentrations observed in the elimination phase and the value of precision of the used analytical method were used to optimize the arithmetic and geometric mean and the value of sd obtained after single oral administration of itraconazole , which is characterized by high variability of pharmacokinetic parameters .
a single dose of 100  mg of itraconazole was administered orally ( sporanox 100  mg tab .
, janssen pharmaceuticals ) for male subjects 20 to 40  years old , with a body mass index 20 to 25  kg / m .
blood samples were collected just prior to administration and at 0.5 , 1.0 , 1.5 , 2.0 , 2.5 , 3.0 , 3.5 , 4.0 , 4.5 , 5.0 , 6.0 , 8.0 , 12.0 , 24.0 , 36.0 , 48.0 , and 72.0  h after the administration .
the concentration analyses were performed using tandem mass spectrometry , using the method described previously ( grabowski et al .
the study was approved by independent ethics committee of district council of physicians , baniowa 3 , warsaw ( resolution no .
itraconazole is a drug with high intrasubject variability , and the formulation belongs to the group of high variability drug product ( hvdp ) .
therefore , the majority of pharmacokinetic profiles began and ended at different time points ( different time of absorption delay and concentration with values < lloq in last sampling points ) .
t profiles that were chosen were those which originated from different subjects and those having identical number of indicated concentrations in the same interval .
t profiles were obtained this way between 1.5 and 48  h after the drug administration ( table  1).table  1concentrations of itraconazole administered orally at a single dose of 100  mg ( sporanox 100  mg tab . , janssen pharmaceuticals ) for 10 male subjects .
data before ( 148  h sampling points ) and after transformation ( 136  h sampling points)time ( h)12345678910 
 drug concentration before transformation ( ng    ml 
 ) 
 1.58.778.7611.843.446.544.443.764.703.0217.752.036.0315.4217.898.3324.536.2111.0148.305.5031.862.570.6775.3631.7613.6729.508.4726.3585.5810.9968.103.064.6380.3822.5820.9647.8115.3730.0176.8913.1266.203.555.6069.6822.6736.4838.2520.9429.8060.4118.6557.944.057.9861.1931.0040.2955.2833.0032.1777.4724.2653.524.557.0662.7742.7415.3661.9733.9143.9477.4724.1348.265.047.8256.8635.5045.2547.7437.7064.7280.0732.2144.126.038.9639.4928.5029.7932.5437.1254.6871.1333.4924.798.031.7125.1416.3122.5530.3127.7532.5442.8626.5618.4212.024.0419.7811.5412.6316.1815.3421.7231.6918.8313.0224.012.2611.925.115.867.177.4010.3611.184.626.1536.06.234.545.053.413.974.025.627.415.184.1048.03.135.263.393.143.173.643.224.593.182.92 
 drug concentration after transformation ( ng    ml 
 ) 
 1.56.756.759.124.238.045.464.625.783.7113.672.027.7518.9622.0010.2418.897.6413.5437.206.7624.542.554.4358.0439.0616.8136.2810.4232.4165.9113.5252.453.049.7861.9027.7725.7836.8218.9036.9159.2216.1450.983.542.8253.6627.8844.8647.0425.7536.6546.5222.9444.624.044.6547.1338.1349.5542.5740.5839.5659.6629.8441.224.543.9448.3452.5618.8947.7341.7054.0459.6629.6837.175.058.8143.7943.6655.6558.7146.3749.8461.6739.6154.266.047.9130.4135.0536.6440.0245.6542.1154.7841.1930.498.024.4230.9220.0627.7323.3421.3725.0633.0132.6622.6512.018.5115.2314.1915.5319.9018.8716.7324.4114.5016.0124.09.449.186.287.218.829.107.988.615.687.5636.04.805.583.894.194.884.944.335.713.995.0448.0 
 3.135.263.393.143.173.643.224.593.182.92 
 data are not subject to transformation concentrations of itraconazole administered orally at a single dose of 100  mg ( sporanox 100  mg tab . , janssen pharmaceuticals ) for 10 male subjects .
data before ( 148  h sampling points ) and after transformation ( 136  h sampling points ) 
 data are not subject to transformation the source of variability in cmax point and for concentrations illustrating clast are different and are the result of different processes , which are subject to the drug molecule in the two time points .
components that generate the cmax , cv  % value are inter alia : variability resulting from the absorption process ( cv  % , abs ) , variability resulting from the distribution process ( cv  % , dist ) , variability resulting from the elimination process ( cv  % el ) and cv  % an .
cv  % an which in this case is expressed by the precision of the method designated for the value equal to lloq . the main components that generate the clast , cv  % value are cv  % el and cv  % an .
both cv  % el and cv  % an to a large extent influence the value of cmax , cv  % , as the drug elimination process is simultaneous to the processes of distribution and absorption .
therefore , it was assumed that1\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c _ { { { \hbox{max } } , { \text{cv \% } } } } \approx { \text{cv \% } } _ { \text{abs } } { \text { + cv \% } } _ { \text{dist } } { \text { + cv \% } } _ { \text{el } } { \text { + cv \% } } _ { \text{an } } $ $ \end{document}while in the case of a point in the elimination phase:2\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c _ { { { \hbox{max } } , { \text{cv}}\,\% } } \approx { \text{cv}}\,\%_{\text{el } } + { \text{cv}}\,\%_{\text{an } } $ $ \end{document}subtracting eq .  2 from eq .  1 the value 3\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{{{\text{max , cv\,\% } } } } = { \text{cv\,\% } } _ { \text{abs } } { \text { + cv\,\% } } _ { \text{dist } } $ $ \end{document}is obtained , which allows to observe the cmax value , without the factors responsible for the variability of the qualifying process and the analytical method . adopting the above assumptions , a scheme of data transformation was proposed , which is illustrated by the following example : the arithmetic mean ( ma ) for the sampling point equal to 1.5  h is 7.30  ng  ml , concentration value ( cn ) for one of the subjects in the analyzed series is before the transformation 8.77  ng  ml ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } > m_{a } $ $ \end{document } ) ; the lowest value of variability in the elimination phase is the value obtained for the sampling point in the 48th hour and equals to 21.26  % ( clast , cv  % ) ; cv  % an for the lloq value is 7.06  % ; cmax , cv  % in the analyzed group is 30.82  % therefore4\documentclass[12pt]{minimal } 
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				 \begin{document}$$ x = c_{\text{last , cv\,\%}}-(c_{\text{last , cv\,\%}})\times { \text{cv\,\%}}_{\rm an}/100$$\end{document}which in the case of the analyzed point is 19.76  % , 5\documentclass[12pt]{minimal } 
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				 \begin{document}$$ y = c_{\text{max , cv\,\%}}-(c_{\text{max , cv\,\%}})\times { \text{cv\,\%}}_{\rm an}/100$$\end{document}which in the case of the analyzed point is 28.64  % , 6\documentclass[12pt]{minimal } 
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				 \begin{document}$$ y_{1 } = ( y \times c_{n } ) /100 $ $ \end{document}represents the percentage of concentration value before the transformation ( cn ) calculated with the value of the variability cmax , cv  % reduced with cv  % an , which in the case of the analyzed point gives the value of 2.51  ng    ml.7\documentclass[12pt]{minimal } 
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				 \begin{document}$$ x_{1 } = ( y_{1 } \times x)/100 , $ $ \end{document}represents the percentage of value y1 calculated with the value clast , cv  % reduced with cv  % , an , which in the case of the analyzed point gives the value of 0.496  ng    ml . the value of cn after the transformation of ( cnt ) is8\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{{n{\text{t } } } } = c_{n } + ( y_{1 } - x_{1 } ) , $ $ \end{document}if9\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } < m_{a}\ ; { \text{and}}\;c_{{n{\text{t } } } } = c_{n } - ( y_{1 } - x_{1 } ) $ $ \end{document}if \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } > m_{a } $ $ \end{document}. in the case of the analyzed concentration point \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } > m_{a } $ $ \end{document } therefore\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{{n{\text{t } } } } = 8.77 - ( 2.51 - 0.496 ) \ , $ $ \end{document } , which after transformation gives the concentration equal to \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{{n{\text{t } } } } = 6.75 $ $ \end{document }  ng    ml . the transformation of all of the concentration points was made in an analogous way , excluding the series of concentration , which were the source for clast , cv  % . in developed form , used formulas take the form : if \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } < m_{a } $ $ \end{document } , then cnt takes the value:10\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \begin{gathered } c_{{n{\text{t } } } } = c_{\text{n } } + [ ( ( ( c_{{{\text{max , cv\,\% } } } } - ( c_{{{\text{max , cv\,\% } } } } \times { \text{cv\,\% } } _ { \text{an } } ) /100 ) \times c_{n } ) /100 ) - ( ( ( ( c_{{{\text{max , cv\,\% } } } } - ( c _ { { { \hbox{max } } , { \text{cv\,\% } } } } \times \\ { \text{cv\,\% } } _ { \text{an } } ) /100 \times c_{n } ) /100 ) \times ( c_{{{\text{last , cv\,\% } } } } - ( c_{{{\text{last , cv\,\% } } } } \times { \text{cv}}\,\%_{\text{an } } ) /100))/100 ) ] \\ \end{gathered } $ $ \end{document}if \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } > m_{a } $ $ \end{document } , then cnt takes the value:11\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \begin{gathered } c_{{n{\text{t } } } } = c_{n } - [ ( ( ( c_{{{\text{max , cv \% } } } } - ( c _ { { { \hbox{max } } , { \text{cv \% } } } } \times { \text{cv \% } } _ { \text{an } } ) /100 ) \times c_{n } ) /100 ) - ( ( ( ( c_{{{\text{max , cv \% } } } } - ( c_{{{\text{max , cv \% } } } } \times \\ { \text{cv \% } } _ { \text{an } } ) /100 \times c_{n } ) /100 ) \times ( c_{{{\text{last , cv \% } } } } - ( c_{{{\text{last , cv \% } } } } \times { \text{cv } } \%_{\text{an } } ) /100))/100 ) ] \\ \end{gathered } $ $ \end{document } non - compartmental modeling was used to estimate pharmacokinetic parameters of itraconazole .
pharmacokinetic calculations were performed with the use of phoenix winnonlin 6.3 ( certara l.p . ) .
the area under the c  t curve ( auc ) from time 0 to the last concentration time point and for infinity ( auc0-tlast ; auc0-inf ) as well as area under first moment of concentration time curve ( aumc ) from time 0 to the last concentration time point ( aumc0-tlast ) , were determined by the trapezoidal method .
mean residence time ( mrt0-tlast ) from time 0 to the last concentration time point was calculated using the standard formula \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \text{mrt } } _ { { 0 { \text { - tlast } } } } { \text { = aumc } } _ { { 0 { \text { - tlast } } } } / { \text{auc } } _ { { 0 { \text { - tlast } } } } .
$ $ \end{document } the elimination rate constant ( kel ) was determined by linear regression of the last three points on the c  t curve . in relation to calculations of t1/2kel in the specified population
it is recommended to conduct the analysis with the harmonic mean and the proper value of pseudo sd ( lam et al .
t data , the data distribution is inclined according to the log - normal model .
thus , the geometric mean ( mg ) and the corresponding coefficient of variation are the factors of descriptive statistics for t1/2kel , which are considered to be more appropriate than the arithmetic mean ( ma ) ( keene 1995 ; senn 2002 ; gad 2009 ) . in relation to t1/2kel , the harmonic mean and the value of pseudo sd were calculated . in relation to the other parameters
as tool for measurement of central tendency , median ( m ) and his standard deviation ( sdm ) were used . a statistical analysis of ma , mg ,
m and their sd ( sda ; sdg ; sdm ) was performed using microsoft office excel software .
the percent of relative standard deviation ( cv  % ) was calculated using formula \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \text { cv\,\%}}\;=\;{\text{sd}}/{\text{m}}\times{100 } $ $ \end{document } raw and transformed data correlations were confirmed by sign test and all pharmacokinetics correlations were confirmed by student - t test . differences with p  <  0.05 were regarded as statistically significant .
in the presented study , the lowest value of relative standard deviation ( rsd  % ) of concentrations observed in the elimination phase and the value of precision of the used analytical method were used to optimize the arithmetic and geometric mean and the value of sd obtained after single oral administration of itraconazole , which is characterized by high variability of pharmacokinetic parameters .
a single dose of 100  mg of itraconazole was administered orally ( sporanox 100  mg tab .
, janssen pharmaceuticals ) for male subjects 20 to 40  years old , with a body mass index 20 to 25  kg / m .
blood samples were collected just prior to administration and at 0.5 , 1.0 , 1.5 , 2.0 , 2.5 , 3.0 , 3.5 , 4.0 , 4.5 , 5.0 , 6.0 , 8.0 , 12.0 , 24.0 , 36.0 , 48.0 , and 72.0  h after the administration .
the concentration analyses were performed using tandem mass spectrometry , using the method described previously ( grabowski et al .
the study was approved by independent ethics committee of district council of physicians , baniowa 3 , warsaw ( resolution no .
itraconazole is a drug with high intrasubject variability , and the formulation belongs to the group of high variability drug product ( hvdp ) .
therefore , the majority of pharmacokinetic profiles began and ended at different time points ( different time of absorption delay and concentration with values < lloq in last sampling points ) .
t profiles that were chosen were those which originated from different subjects and those having identical number of indicated concentrations in the same interval .
t profiles were obtained this way between 1.5 and 48  h after the drug administration ( table  1).table  1concentrations of itraconazole administered orally at a single dose of 100  mg ( sporanox 100  mg tab . , janssen pharmaceuticals ) for 10 male subjects .
data before ( 148  h sampling points ) and after transformation ( 136  h sampling points)time ( h)12345678910 
 drug concentration before transformation ( ng    ml 
 ) 
 1.58.778.7611.843.446.544.443.764.703.0217.752.036.0315.4217.898.3324.536.2111.0148.305.5031.862.570.6775.3631.7613.6729.508.4726.3585.5810.9968.103.064.6380.3822.5820.9647.8115.3730.0176.8913.1266.203.555.6069.6822.6736.4838.2520.9429.8060.4118.6557.944.057.9861.1931.0040.2955.2833.0032.1777.4724.2653.524.557.0662.7742.7415.3661.9733.9143.9477.4724.1348.265.047.8256.8635.5045.2547.7437.7064.7280.0732.2144.126.038.9639.4928.5029.7932.5437.1254.6871.1333.4924.798.031.7125.1416.3122.5530.3127.7532.5442.8626.5618.4212.024.0419.7811.5412.6316.1815.3421.7231.6918.8313.0224.012.2611.925.115.867.177.4010.3611.184.626.1536.06.234.545.053.413.974.025.627.415.184.1048.03.135.263.393.143.173.643.224.593.182.92 
 drug concentration after transformation ( ng    ml 
 ) 
 1.56.756.759.124.238.045.464.625.783.7113.672.027.7518.9622.0010.2418.897.6413.5437.206.7624.542.554.4358.0439.0616.8136.2810.4232.4165.9113.5252.453.049.7861.9027.7725.7836.8218.9036.9159.2216.1450.983.542.8253.6627.8844.8647.0425.7536.6546.5222.9444.624.044.6547.1338.1349.5542.5740.5839.5659.6629.8441.224.543.9448.3452.5618.8947.7341.7054.0459.6629.6837.175.058.8143.7943.6655.6558.7146.3749.8461.6739.6154.266.047.9130.4135.0536.6440.0245.6542.1154.7841.1930.498.024.4230.9220.0627.7323.3421.3725.0633.0132.6622.6512.018.5115.2314.1915.5319.9018.8716.7324.4114.5016.0124.09.449.186.287.218.829.107.988.615.687.5636.04.805.583.894.194.884.944.335.713.995.0448.0 
 3.135.263.393.143.173.643.224.593.182.92 
 data are not subject to transformation concentrations of itraconazole administered orally at a single dose of 100  mg ( sporanox 100  mg tab . , janssen pharmaceuticals ) for 10 male subjects .
data before ( 148  h sampling points ) and after transformation ( 136  h sampling points ) 
 data are not subject to transformation
the source of variability in cmax point and for concentrations illustrating clast are different and are the result of different processes , which are subject to the drug molecule in the two time points .
components that generate the cmax , cv  % value are inter alia : variability resulting from the absorption process ( cv  % , abs ) , variability resulting from the distribution process ( cv  % , dist ) , variability resulting from the elimination process ( cv  % el ) and cv  % an .
cv  % an which in this case is expressed by the precision of the method designated for the value equal to lloq .
the main components that generate the clast , cv  % value are cv  % el and cv  % an . both cv  % el and cv  % an to a large extent influence the value of cmax , cv  % , as the drug elimination process is simultaneous to the processes of distribution and absorption .
therefore , it was assumed that1\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c _ { { { \hbox{max } } , { \text{cv \% } } } } \approx { \text{cv \% } } _ { \text{abs } } { \text { + cv \% } } _ { \text{dist } } { \text { + cv \% } } _ { \text{el } } { \text { + cv \% } } _ { \text{an } } $ $ \end{document}while in the case of a point in the elimination phase:2\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c _ { { { \hbox{max } } , { \text{cv}}\,\% } } \approx { \text{cv}}\,\%_{\text{el } } + { \text{cv}}\,\%_{\text{an } } $ $ \end{document}subtracting eq .  2 from eq .  1 the value 3\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{{{\text{max , cv\,\% } } } } = { \text{cv\,\% } } _ { \text{abs } } { \text { + cv\,\% } } _ { \text{dist } } $ $ \end{document}is obtained , which allows to observe the cmax value , without the factors responsible for the variability of the qualifying process and the analytical method . adopting the above assumptions , a scheme of data transformation was proposed , which is illustrated by the following example : the arithmetic mean ( ma ) for the sampling point equal to 1.5  h is 7.30  ng  ml , concentration value ( cn ) for one of the subjects in the analyzed series is before the transformation 8.77  ng  ml ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } > m_{a } $ $ \end{document } ) ; the lowest value of variability in the elimination phase is the value obtained for the sampling point in the 48th hour and equals to 21.26  % ( clast , cv  % ) ; cv  % an for the lloq value is 7.06  % ; cmax , cv  % in the analyzed group is 30.82  % therefore4\documentclass[12pt]{minimal } 
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				 \begin{document}$$ x = c_{\text{last , cv\,\%}}-(c_{\text{last , cv\,\%}})\times { \text{cv\,\%}}_{\rm an}/100$$\end{document}which in the case of the analyzed point is 19.76  % , 5\documentclass[12pt]{minimal } 
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				 \begin{document}$$ y = c_{\text{max , cv\,\%}}-(c_{\text{max , cv\,\%}})\times { \text{cv\,\%}}_{\rm an}/100$$\end{document}which in the case of the analyzed point is 28.64  % , 6\documentclass[12pt]{minimal } 
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				 \begin{document}$$ y_{1 } = ( y \times c_{n } ) /100 $ $ \end{document}represents the percentage of concentration value before the transformation ( cn ) calculated with the value of the variability cmax , cv  % reduced with cv  % an , which in the case of the analyzed point gives the value of 2.51  ng    ml.7\documentclass[12pt]{minimal } 
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				 \begin{document}$$ x_{1 } = ( y_{1 } \times x)/100 , $ $ \end{document}represents the percentage of value y1 calculated with the value clast , cv  % reduced with cv  % , an , which in the case of the analyzed point gives the value of 0.496  ng    ml . the value of cn after the transformation of ( cnt ) is8\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{{n{\text{t } } } } = c_{n } + ( y_{1 } - x_{1 } ) , $ $ \end{document}if9\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } < m_{a}\ ; { \text{and}}\;c_{{n{\text{t } } } } = c_{n } - ( y_{1 } - x_{1 } ) $ $ \end{document}if \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } > m_{a } $ $ \end{document}. in the case of the analyzed concentration point \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } > m_{a } $ $ \end{document } therefore\documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{{n{\text{t } } } } = 8.77 - ( 2.51 - 0.496 ) \ , $ $ \end{document } , which after transformation gives the concentration equal to \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{{n{\text{t } } } } = 6.75 $ $ \end{document }  ng    ml . the transformation of all of the concentration points was made in an analogous way , excluding the series of concentration , which were the source for clast , cv  % . in developed form , used formulas take the form : if \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } < m_{a } $ $ \end{document } , then cnt takes the value:10\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \begin{gathered } c_{{n{\text{t } } } } = c_{\text{n } } + [ ( ( ( c_{{{\text{max , cv\,\% } } } } - ( c_{{{\text{max , cv\,\% } } } } \times { \text{cv\,\% } } _ { \text{an } } ) /100 ) \times c_{n } ) /100 ) - ( ( ( ( c_{{{\text{max , cv\,\% } } } } - ( c _ { { { \hbox{max } } , { \text{cv\,\% } } } } \times \\ { \text{cv\,\% } } _ { \text{an } } ) /100 \times c_{n } ) /100 ) \times ( c_{{{\text{last , cv\,\% } } } } - ( c_{{{\text{last , cv\,\% } } } } \times { \text{cv}}\,\%_{\text{an } } ) /100))/100 ) ] \\ \end{gathered } $ $ \end{document}if \documentclass[12pt]{minimal } 
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				 \begin{document}$$ c_{n } > m_{a } $ $ \end{document } , then cnt takes the value:11\documentclass[12pt]{minimal } 
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				 \begin{document}$$ \begin{gathered } c_{{n{\text{t } } } } = c_{n } - [ ( ( ( c_{{{\text{max , cv \% } } } } - ( c _ { { { \hbox{max } } , { \text{cv \% } } } } \times { \text{cv \% } } _ { \text{an } } ) /100 ) \times c_{n } ) /100 ) - ( ( ( ( c_{{{\text{max , cv \% } } } } - ( c_{{{\text{max , cv \% } } } } \times \\ { \text{cv \% } } _ { \text{an } } ) /100
\times c_{n } ) /100 ) \times ( c_{{{\text{last , cv \% } } } } - ( c_{{{\text{last , cv \% } } } } \times { \text{cv } } \%_{\text{an } } ) /100))/100 ) ] \\ \end{gathered } $ $ \end{document }
pharmacokinetic calculations were performed with the use of phoenix winnonlin 6.3 ( certara l.p . ) .
the area under the c  t curve ( auc ) from time 0 to the last concentration time point and for infinity ( auc0-tlast ; auc0-inf ) as well as area under first moment of concentration time curve ( aumc ) from time 0 to the last concentration time point ( aumc0-tlast ) , were determined by the trapezoidal method .
mean residence time ( mrt0-tlast ) from time 0 to the last concentration time point was calculated using the standard formula \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \text{mrt } } _ { { 0 { \text { - tlast } } } } { \text { = aumc } } _ { { 0 { \text { - tlast } } } } / { \text{auc } } _ { { 0 { \text { - tlast } } } } .
$ $ \end{document } the elimination rate constant ( kel ) was determined by linear regression of the last three points on the c  t curve . in relation to calculations of t1/2kel in the specified population
it is recommended to conduct the analysis with the harmonic mean and the proper value of pseudo sd ( lam et al .
t data , the data distribution is inclined according to the log - normal model . thus , the geometric mean ( mg ) and the corresponding coefficient of variation are the factors of descriptive statistics for t1/2kel , which are considered to be more appropriate than the arithmetic mean ( ma ) ( keene 1995 ; senn 2002 ; gad 2009 ) . in relation to t1/2kel ,
the harmonic mean and the value of pseudo sd were calculated . in relation to the other parameters
as tool for measurement of central tendency , median ( m ) and his standard deviation ( sdm ) were used . a statistical analysis of ma , mg ,
m and their sd ( sda ; sdg ; sdm ) was performed using microsoft office excel software .
the percent of relative standard deviation ( cv  % ) was calculated using formula \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \text { cv\,\%}}\;=\;{\text{sd}}/{\text{m}}\times{100 } $ $ \end{document } raw and transformed data correlations were confirmed by sign test and all pharmacokinetics correlations were confirmed by student - t test . differences with p  <  0.05 were regarded as statistically significant .
the lowest values of cv  % for raw data ( rd ) were noted for the sampling point 48  h after the administration of the drug , and the value was 20.61  % ( table  1 ) .
it was used to transform data in the rest of the time points ( 1.535  h ) .
the concentration values after the transformation ( td ) are presented in the table  1 .
image of differences between rd and td for the largest fluctuation of c  t curve is presented in fig .  1 .
on the basis of rd and td , cv  % was calculated for ma , sda , mg , sdg , m and sdm , which is presented in the table  2 . in relation to the value of ma , mg and m between the rd and td data
no statistically significant differences ( p  >  0.05 ) were found between the mean values of individual points of concentration and the rd and td data .
the sda value and sdg were significantly lower ( p  <  0.05 ) in relation to the rd group.fig . 
1concentration  time data of itraconazole administered orally at a single dose of 100  mg for 10 male subjects before ( solid line ) and after transformation ( dashed line ) . a represents arithmetic mean and standard deviation , b shows geometric mean and standard deviation and c shows median and standard deviationtable  2mean ( n  =  10 ) , standard deviation , and relative standard deviation of raw data before and after transformationtime ( h)raw data ( rd)transformed data ( td ) 
 m 
 a 
 sda 
 rsd 
 m 
 g 
 sdg 
 rsd 
 m 
 sdm 
 rsd 
 m 
 a 
 sda 
 rsd 
 m 
 g 
 sdg 
 rsd 
 m 
 sdm 
 rsd1.57.304.6663.816.224.5573.235.624.6682.906.812.9443.156.342.8344.645.622.9552.572.020.5114.3770.0916.1114.3388.9616.6614.3786.3018.759.6251.2716.439.4157.3116.339.6659.132.542.0529.6570.5131.5430.0295.1830.6329.6596.7937.9319.7352.0232.1719.5860.8834.3420.1758.743.043.8026.4560.4035.9726.2973.0938.9126.4567.9838.4216.4242.7435.0015.9545.5636.8616.4244.543.541.0418.5345.1537.1418.0148.4937.3718.5349.6039.2810.4326.5537.889.9926.3839.7310.4626.324.046.6216.9836.4243.8316.3537.2946.9116.9836.1943.297.8718.1842.667.4917.5741.907.8718.794.546.7618.9340.4942.5818.4443.3146.1018.9341.0743.3712.1828.0941.4411.7228.2845.8412.1826.585.049.2014.5529.5847.4513.9229.3346.5014.5531.3051.247.6314.9050.717.2614.3145.088.4018.646.039.0513.9535.7237.2113.3635.9035.3113.9539.5140.437.7119.0839.777.3518.4735.848.0222.368.027.427.6427.8726.467.3127.6327.167.6428.1426.124.7018.0025.754.4817.3824.744.7019.0012.018.486.1933.5117.635.9333.6617.516.1935.3717.393.1317.9817.162.9717.3416.373.1319.1024.08.202.9435.807.732.8336.547.292.9440.317.991.2816.097.891.2215.517.771.4018.0436.04.951.2224.534.831.1624.024.801.2225.344.740.6313.284.700.6012.724.840.6312.9948.0 
 3.560.7621.263.500.7220.613.200.7623.68 
  
 p value>0.050.0350.006>0.050.0350.009>0.050.0390.010 
 p - value between raw ( 1.536  h ) and transformed data ( 1.536  h ) 
 m 
 a arithmetic mean ( ng    ml ) ; sd 
 a standard deviation of ma , m 
 g geometric mean ( ng    ml ) , sd 
 g standard deviation of mg , m median ( ng    ml ) , sd 
 m standard deviation of m , rsd relative standard deviation ( % ) 
 value of c 
 last , cv  % 

 data which are not transformed concentration  time data of itraconazole administered orally at a single dose of 100  mg for 10 male subjects before ( solid line ) and after transformation ( dashed line ) . a represents arithmetic mean and standard deviation ,
b shows geometric mean and standard deviation and c shows median and standard deviation mean ( n  =  10 ) , standard deviation , and relative standard deviation of raw data before and after transformation 
 p - value between raw ( 1.536  h ) and transformed data ( 1.536  h ) 
 m 
 a arithmetic mean ( ng    ml ) ; sd 
 a standard deviation of ma , m 
 g geometric mean ( ng    ml ) , sd 
 g standard deviation of mg , m median ( ng    ml ) , sd 
 m standard deviation of m , rsd relative standard deviation ( % ) 
 value of c 
 last , cv  % 
 
 data which are not transformed the results of the key calculations of pharmacokinetic parameters are shown in table  3 .
the analysis of sdg pharmacokinetic parameters in rd and td groups indicated more than twice lower value of sdg in td group .
the cv  % ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \text{sd}}_{a } /m_{a } \times 100 $ $ \end{document } ) of pharmacokinetic parameters , such as kel , t1/2kel , tmax , cmax , auc0-tlast , auc0-inf , aumc0-last and mrtinf calculated on the basis of td was lower by 21.25 , 15.44 , 17.43 , 9.47 , 9.85 , 1.59 , 10.39 and 2.69  % , respectively , than cv  % obtained for the pk parameters in rd group but not statistically significant ( p  >  0.05 ) .
the cv  % ( \documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \text{sd}}_{g } /m_{g } \times 100 $ $ \end{document } ) of pharmacokinetic parameters kel , t1/2kel , tmax , cmax , auc0-tlast , auc0-inf , aumc0-last , and mrtinf calculated on the basis of td was significantly lower ( p  <  0.05 ) by 39.40 , 30.75 , 44.13 , 59.42 , 53.77 , 51.82 and 38.83  % , respectively , from cv  % obtained for the pk parameters in rd group . the ratio of mg in the rd group to mg in the td group ranged between 0.935 and 1.041 and was not statistically significant ( p  >  0.05).table  3pharmacokinetics parameters of itraconazole ( administered orally at a single dose of 100  mg for ten male subjects ) calculated using data before and after transformationstat . 

k 
 el ( h ) 
 t 
 1/2kel ( h ) 
 t 
 max ( h ) 
 c 
 max ( ng    ml)auc0-tlast ( ng    h    ml)auc0-inf ( ng    h    ml)aumc0-tlast ( ng    h    ml)mrt0-tlast ( h)pharmacokinetics analysis using raw concentrations ( rd)ma 
 0.0324.33 
 4.0559.06632.48755.788,916.2614.21sda 
 0.0110.511.3018.20184.64171.382,336.140.89rsd38.5643.1932.1130.8229.1922.6826.206.25 mg 
 0.0322.543.8556.37610.57740.338,662.6414.19sdg 
 0.0110.131.2517.48176.53163.322,230.720.84rsd39.9444.9432.4931.0028.9122.0625.755.94m0.0324.334.0559.06632.48755.788,916.2614.21sdm 
 0.019.971.2317.27175.16162.582,216.260.84rsd36.5840.9730.4729.2427.6921.5124.865.93pharmacokinetics analysis using transformed concentrations ( td)ma 
 0.0320.484.5554.94607.90705.078,610.9714.19sda 
 0.018.671.2115.33159.98157.342,021.770.86rsd29.6242.3226.5227.9026.3222.3223.486.08 mg 
 0.0321.074.4254.49602.87697.748,546.8014.18sdg 
 0.015.290.996.8680.57107.761,060.430.52rsd24.2025.1122.5012.5813.3715.4412.413.63m0.0321.694.5554.94607.90705.078,610.9714.19sdm 
 0.015.250.996.8480.42107.511,058.490.52rsd23.3624.2221.6712.4513.2315.2512.293.63 
 h harmonic mean , m 
 a arithmetic mean , sd 
 a standard deviation of ma , m 
 g geometric mean , sd 
 g standard deviation of mg , m median , sd 
 m standard deviation of m ; rsd relative standard deviation ( % ) , k 
 el elimination rate constant ; t 
 1/2kel half life in elimination phase , t 
 max time to reach maximum concentration , c 
 max maximum concentration , auc 
 0-tlast area under the curve between zero and last concentration , auc 
 0-inf area under the curve from zero to infinity , aumc 
 0-last area under the first moment curve between zero and last concentration ,
mrt 
 inf mean residence time ( aumc0-tlast / auc0-tlast ) pharmacokinetics parameters of itraconazole ( administered orally at a single dose of 100  mg for ten male subjects ) calculated using data before and after transformation 
 h harmonic mean , m 
 a arithmetic mean , sd 
 a standard deviation of ma , m 
 g geometric mean , sd 
 g standard deviation of mg , m median , sd 
 m standard deviation of m ; rsd relative standard deviation ( % ) , k 
 el elimination rate constant ; t 
 1/2kel half life in elimination phase , t 
 max time to reach maximum concentration , c 
 max maximum concentration , auc 
 0-tlast area under the curve between zero and last concentration , auc 
 0-inf area under the curve from zero to infinity , aumc 
 0-last area under the first moment curve between zero and last concentration , mrt 
 inf mean residence time ( aumc0-tlast / auc0-tlast )
this manuscript presents the method of data transformation calculated on the basis of pk parameters expressed as ma , mg and m. until now , only a few methods of transforming the values of pk parameters were proposed ( abdallah 1998 ; fujita et al .
frequently , these methods were based on data transformation through the normalization of pharmacokinetic parameters , value of the dose or physiological parameters ( body weight , dose , body surface , normalization etc . ) , ( sathirakul et al . 2003 ; sathyan et al .
normalization and scaling of pharmacokinetic data are also used in the allometric analysis , in scaling either concentrations , time or pharmacokinetic parameters ( mahmood 2005 ) .
these methods , however , do not use the variation values obtained in the study to transform data . in the c 
t data sequences standard deviation in individual time points within the population is different and ranges from low to high .
this is true for both , the analysis of the variability within and between subjects . in the analyzed case ,
the phase of itraconazole absorption is a subject of large fluctuations . after a single administration of the drug , the low values of deviation for individual c 
usually equal to or a bit higher than the lloq value , the variability of the sd value often does not exceed 10  % .
this occurs because these are the concentrations which usually represent only the process of elimination that is not affected by the factors responsible for the variability of the kinetics of absorption and distribution of the drug . in relation to a single administration ,
the exceptions are the cases of redistribution of the drug in the late elimination phase ( davis et al .
2010 ) . also , in the analyzed case the volatility calculated for each sampling point was the lowest at a point closest to the lloq value of the analytical method .
it happens differently in the case of concentrations analyzed in relation to oral administration , while the drug is still present in the stomach or the process of absorption has begun in the intestines .
depending on many factors , the variability in this phase can be very high ( duquesnoy et al . 1998 ; tubic et al .
the proposed method allows to achieve a reliable picture of the pharmacokinetic profile , free of substantial interference in the average values of obtained concentrations and in the values of pharmacokinetic parameters with a simultaneous decrease in the value of sd .
t data at points , in which the sd is particularly high . reducing the value of sd for
studies such as first in man , pilot or the rare ones , due to the difficulty of collecting sufficient number of subjects , can help to make a decision about the further direction of the research .
 for many drugs administered per os , high variability in the concentration  time ( c  t ) values from first sampling to the phase of distribution may cause difficulty in pharmacokinetic analysis . 
 therefore , the aim of this study was to propose a method of transformation of c  
 t data , which would allow significantly reducing the standard deviation ( sd ) value of observed concentrations , without a statistically significant influence on the value of the mean for each sampling point in group . in the presented study , the lowest value of relative standard deviation of concentrations observed in the elimination phase and the value of precision of the used analytical method , were used to optimize the arithmetic , geometric means , median , and the value of sd obtained after single oral administration of itraconazole in human subjects . 
 non - compartmental modeling was used to estimate pharmacokinetic parameters . 
 the analysis of sd pharmacokinetic parameters after c  t value optimization indicated more than twice the lower value of sd . after transforming the itraconazole data , lower variability of concentration data gives more selective pharmacokinetics profile in absorption and early distribution phase . 