the uncertainty quantification of pk parameters from preclinical experiments to clinical has been extensively investigated in the literature .
prediction methods for clearance is by far the most studied case because of its importance for systemic drug elimination and oral bioavailability .
clearance is defined as the volume of blood , from which the drug is removed , per unit of time .
reported evaluations of in vivo clearance prediction - methods are often based on relatively small datasets ( up to 50 compounds ) , although a few investigations are based on larger datasets ( up to 400 compounds).9,12,17,20 the main reason for evaluation - data scarcity is that many compounds are only administered by the oral route to humans , making unambiguous distinction of in vivo clearance and bioavailability impossible .
although some studies show that simple allometric prediction from a monkey performs best,12,17,21 there are typically cost and ethical reasons against using higher species in preclinical research .
it has been suggested , albeit using smaller datasets , that the rat and dog data may perform equally well.6,13 allometry may indeed work well for compounds excreted renally , or whose clearance is limited by liver blood flow , but may not be applicable for low - clearance compounds limited by cytochrome p450 enzyme activity.13,29 in some studies , methods using correction factors , such as the rule of exponents method or fraction ( unbound ) corrected intercept method have been shown to predict better than allometry or at least decrease the risk of overprediction.6,8,12 including a correction for species differences in intrinsic clearance could also possibly decrease the prediction error of allometric methods.30 however , as reported by the publication using the largest dataset,12 the best allometric methods only predict approximately 60% of compounds within twofold of the human clearance . in vitro - in vivo extrapolation methods ( hepatocytes or microsomes ) for hepatic clearance predictions have varying success rates reported in the literature , ranging from 2090% of compounds predicted within twofold.6,13,14,22,3134 the variation may be due to the differences in experimental setups and in experimental / empirical correction factors ( see fagerholm14 for an extensive review ) .
it has been reported that in vitro - in vivo extrapolation , using a regression - line correction established in the utilized hepatocyte assays , may be the method of choice because it will correct for systematic errors in the experimental setup.31 overall , it seems reasonable to suggest that the uncertainty in clearance predictions falls in the order of a factor three for high - performance methods ( this can , for example , be approximated by a lognormal distribution with 95% chance of falling within threefold from the predicted point estimate ) .
volume of distribution at steady state ( vss ) is defined as the proportionality factor between the total amount of drugs in the body at steady state and plasma ( or blood ) concentration .
similar to clearance , vss can be predicted using several different methods with allometry historically being most used .
indeed , allometry may be a reasonable method to predict vss because of the physiological aspects governing this parameter , such as organ weights and volumes .
there is little consensus of the best method for predicting vss.7,11,13,15,18 two publications7,11 suggest that the semimechanistic oie - tozer equation predicts relatively well if nonhuman in vivo data are available .
to confidently predict vss , especially when in vivo data are scarce , it is important to consider the physicochemical properties of the compound and that they conform to the assumptions made in the model .
one way to investigate the predictive power of a method is to apply the method between preclinical species.7 similar to clearance , it seems reasonable to suggest that the vss predictions will fall within threefold of the true value .
the absorption of drugs from the gut into the systemic circulation is a complex dynamic process.35,36 the main determinants for the extent and kinetics of this process are dose , solubility / dissolution rate , and permeability .
however , gut metabolism , stability of the drug in the intestine , fasted / fed state , and mass transfer may also affect the dynamics .
bioavailability ( f ) is defined as : 
 1where fa denotes fraction absorbed , fg denotes fraction that escapes gut metabolism , and fh denotes fraction that escapes hepatic metabolism .
absorption is often predicted using the biopharmaceutics classification system , which categorizes the candidate drugs on the basis of their solubility and permeability .
compounds with high solubility and high permeability ( biopharmaceutics classification system i ) have an almost complete and rapid absorption .
however , for compounds with low permeability or low solubility , or both ( biopharmaceutics classification system ii 
iv ) , it is more difficult to accurately predict the absorption process , and compounds in these classes may display a large variation in both extent and kinetics between patients and administrations.37,38 
 in vitro methods are routinely used to get a qualitative understanding of absorption properties.3941 translating fa and fg from animal to human is difficult because of interspecies differences in intestinal physiology as well as enzymatic activity.42,43 for example , bioavailability in dogs is often higher than in humans , possibly because of a more leaky intestine paracellular pathway,44,45 and monkeys often have lower bioavailability because of a higher enzymatic activity in the gut.46 physiologically based pharmacokinetic models are reported to underpredict bioavailability , but the number of investigated compounds is limited ( 8 per biopharmaceutics classification system class).19 most drug - discovery programs actively strive to develop compounds with sufficiently high solubility and permeability in order to reduce the uncertainty and risk of variation in exposure . for such compounds ,
it is reasonable to assume that absorption can be controlled by formulation . in the three case studies presented here
, the working hypothesis has been that the compounds will have good absorption properties in humans .
this has been based on in vitro ( permeability , crystalline solubility ) combined with in vivo animal pk studies . naturally ,
if solubility and permeability properties are less favorable , or gut metabolism is suspected , one should account for uncertainty in absorption parameters ( using the same principles as outlined for cl and vss ) . as mentioned earlier
, differences in pd effect can be attributed to differences in metabolism between species ( i.e. , active metabolites ) , leading to different pharmacological / toxicological responses.1,47 another example is species differences in receptor distribution , which have been reported for the opioid receptors in the brain48 and in histamine receptor subtypes in bronchi.49 differences in relative receptor density between species can be found both for serotonergic and dopaminergic receptors.50,51 cardiac glycosides ( e.g. , digoxin and related compounds ) are an example of compounds with a large difference in affinity between species , in the order of 100-fold lower affinity in rodents compared with humans.52 further , translational modeling of disease progression is not straightforward , but can sometimes be scaled with the expected lifespan of the species.53 taking all these factors into account , uncertainty in pd can influence the dose prediction significantly , especially for a first - in - class compound targeting an unprecedented class of target or pathway . on the other hand ,
confidence increases when clinical data on the same target / pathway have been generated , the preclinical disease model has been shown to translate to human , or the system pharmacology of the target is understood , or a combination of those.5457 the human dose is often predicted using relatively simple models , which can be represented by closed - form expressions . as a base case , the dose prediction for a compound with assumed one - compartment kinetics , systemic clearance cl , a desired average steady - state concentration ( css ) , bioavailability f , and dosing interval  is calculated as : 
 2 alternatively , for a compound with a desired coverage above a minimum effective concentration ( mec ) , the predicted dose is calculated as : 
 3where ka denotes the absorption rate - constant . for nonlinear models with no closed - form expression for dose ,
simulation - based methods are needed . to investigate the uncertainty of the dose prediction , pk and
pd parameters can be represented by empirical distributions based on available information on translational performance . specifically , for a pk parameter , the distribution around the point estimate reflects the uncertainty of each method , as well as potential variability in input data .
the shape of the distribution can be chosen based on reasonable assumptions or inferred empirically from compounds tested in man .
a log - normal distribution is useful to avoid negative values ( exemplified in figure 
 1d ) .
alternatives to log - normal distributions include fuzzy numbers or other interval analyses that are common in engineering applications58 and that have also been used in physiologically based pharmacokinetic applications2,23 ( exemplified in figure 
 1e ) . in those probabilistic frameworks , it is straightforward to include correlations between parameters .
however , there is presently not much information in the literature on potential correlations of pk / pd parameter uncertainties . in the case studies ,
no correlations were incorporated in the monte - carlo simulations . in the next step ,
the model and the parameter distributions are fed into a monte - carlo simulation to calculate the distribution of the dose prediction .
basically , random samples for each parameter are drawn from the chosen distribution ( the number of samples depends on the problem , and is typically greater than 1,000 ) .
the dose is calculated for each set of parameters , resulting in an empirical dose distribution .
monte - carlo simulations using closed - form dose expressions , like eqs . 2 and 3 , are computationally feasible and can typically be generated within milliseconds on a standard computer . for more complex pk / pd models
, the calculation may be significantly slower , because each dose calculation involves an optimization ( e.g. , to assure that plasma concentration is above a certain threshold at a specific timepoint ) , and each evaluation step in the optimization requires numerical integration of the ordinary differential equations of the model .
it is often expedient to present the resulting dose distribution as a cumulative probability plot ( which describes on the y - axis the probability that the dose is less than or equal to the value on the x - axis , cf .
science communication in general and uncertainty communication in particular are actively discussed in the environmental science,5961 as well as in other areas , like radiotherapy.62,63 we mention the following main challenges in order to effectively communicate uncertainty in human dose - predictions : communication of results based on modeling and simulation is hard and requires careful preparation .
such preparation is often performed iteratively and involves complete listing of modeling assumptions , translation of model equations into words or overview figures , choice of simulations and interpretation of results , and reflection on consequences of the modeling predictions .
note in particular that formulas and technical terms are easy to list but difficult to interpret for scientists without adequate training in mathematics.60 
 human dose - predictions tend to focus on pk properties of the drug , and not so much on pd properties .
however , translating the pd model from animal pharmacology to the human situation is based on assumptions and associated with uncertainties.1,23,24 therefore , no or low emphasis on pd uncertainty may underestimate overall uncertainty / risk and , hence , bias decision - makers . 

it is easy to mistake variability for uncertainty , and a consequent language is essential to avoid this problem . 
 
such preparation is often performed iteratively and involves complete listing of modeling assumptions , translation of model equations into words or overview figures , choice of simulations and interpretation of results , and reflection on consequences of the modeling predictions .
note in particular that formulas and technical terms are easy to list but difficult to interpret for scientists without adequate training in mathematics.60 
  human dose - predictions tend to focus on pk properties of the drug , and not so much on pd properties .
however , translating the pd model from animal pharmacology to the human situation is based on assumptions and associated with uncertainties.1,23,24 therefore , no or low emphasis on pd uncertainty may underestimate overall uncertainty / risk and , hence , bias decision - makers . 
 
it is easy to mistake variability for uncertainty , and a consequent language is essential to avoid this problem . 
 
fundamental communication elements are words , numbers , and visualizations . concerning words , it is important to be aware of variation in interpretation of phrases of likelihood ( e.g. , likely and probable).64,65 to elucidate the meaning of such words , it is recommended to define a consistent language according to numerical levels of confidence as exemplified by the verbal quantifiers in table 
 1 .
the number of levels of such a table depends on the amount and quality of data , the type of research question , and the expected level of granularity of the response .
it is also wise to strive for a simple language and avoid technical jargon.61 concerning numbers , they can be directly used to communicate , for example , confidence levels , and there is empirical evidence that including both verbal and numerical characterization improves communication.59 communication by visualization is central in science to convey causal relationships , temporal trends , data , etc.27 probability words as quantitative subjective probability judgments mapping of probability words into quantitative subjective probability judgments , used by the intergovernmental panel on climate change third assessment59,65 based on recommendations developed by moss & schneider.69 successful communication depends on several factors : identify the science relevant to decision - making ; separate need - to - know science from nice - to - know science . determine the decision - maker 's background knowledge in the subject area . 
 design communication to fill the critical gaps between what people know and need to know . 
 collect systematic feedback provided by empirical evaluation , and use the information to evaluate the adequacy of the communication and to revise it accordingly.60 
  identify the science relevant to decision - making ; separate need - to - know science from nice - to - know science . determine the decision - maker 's background knowledge in the subject area . 
  design communication to fill the critical gaps between what people know and need to know . 
  collect systematic feedback provided by empirical evaluation , and use the information to evaluate the adequacy of the communication and to revise it accordingly.60 
 
the case studies that we show in the next section represent the final outcome of this iterative approach ( points 13 above ) .
the feedback ( step 3 ) was continuously collected from various industrial and academic stakeholders during approximately one year and is summarized in the discussion section .
we have chosen to communicate by visual means complemented with verbal quantifiers ( exemplified in table 
 1 ) .
specifically , we have condensed complex information from multiple sources into a single uncomplicated plot that captures the uncertainty in human dose - prediction , and , in that way , pinpoints the need - to - know science .
we present three case studies of how uncertainty in preclinical human dose - prediction can be quantified and communicated . in these case studies ,
clearance was estimated from in vitro and in vivo data using two different scaling methods .
first , clearance was predicted from human hepatocyte data using a regression correction method based on 36 marketed drugs reflecting the same principles as in sohlenius - sternbeck et al.31 this study reports that 66% of compounds have a predicted intrinsic clearance within twofold .
second , clearance was also scaled from in vivo data obtained from rats and dogs using the liver blood flow method.17 using a clinical dataset of 18 compounds , this method predicts in vivo clearance within twofold for 67% of the compounds and within threefold for 94% of the compounds.6 the uncertainty distribution for each clearance method was represented by a log - normal distribution with 95% of samples falling within threefold of the predicted point estimate .
variability in input data to the scaling methods was considered negligible in comparison to the translational uncertainty .
if two predictions are available for the same parameter , it is common to take the average value of the predictions as the point estimate . for each iteration of the monte - carlo simulation , one sample
is drawn from each of the two distributions , and the average value is used . in this way , two ( similar ) predictions of cl ,
for example , result in a consensus prediction with less uncertainty than each individual prediction .
volumes of distribution for all three case studies were predicted using the semi - mechanistic oie - tozer equation.66 for this method , 78% of the predictions were within twofold and 89% within threefold of the in vivo result using a dataset of 18 compounds.7 for a larger dataset of 400 compounds , 67% of the predictions were within twofold.11 unbound vss for each series in the case studies was consistent across species ( rat and dog ; i.e. , the slope of a simple unbound allometric plot fell between 0.8 and 1.2 ) , which further adds confidence in prediction .
vss was assumed log - normally distributed with 95% of samples falling within threefold of the predicted point estimate .
similar to clearance predictions , variability in input data to the scaling method was considered negligible in comparison to the translational uncertainty . for all case studies , it was assumed that rate of absorption was rapid ( ka  =  1h ) , unless a scenario with an absorption - modifying formulation was simulated .
in addition , the fraction of the dose absorbed was assumed high ( fa  =  1 ) and gut metabolism negligible ( fg  =  1 ) .
hence , the major contributor to f was first - pass metabolism in the liver and f was calculated as : 
 4where lbf is the liver blood flow and cl is the systemic blood clearance .
the target audiences for communication of the three case studies have been pk / pd - modelers , medicinal chemists , bioscientists , and governance bodies .
the first case study considers five compounds ( cmp 1a1e ) , that were evaluated in the lead - optimization phase of a drug - discovery project . the aim was to support ranking of the compounds for subsequent progression into toxicological studies . for each compound ,
3 . however , because of an estimated short half - life ( 34 hours ) of the compounds , small changes in pk parameters had a large impact on the dose prediction , thereby increasing overall uncertainty .
the project team was considering an extended release ( er ) formulation in order to generate a flatter concentration - time profile and to decrease dependence on elimination half - life and thereby obtain a more clinically realistic dose level . during these discussions
, it became apparent that understanding and efficiently communicating uncertainty in the dose predictions were pivotal to progress the project .
parameters and distributions were generated as described in the  uncertainty quantification  section ( table 
 2 ; example in figure 
 1d ) .
predictions for cl using methods based on in vitro data ( in vitro - in vivo extrapolation from hepatocytes ) and in vivo data ( rat / dog liver blood flow17 ) were in a similar range and did not diverge more than twofold for any of the compounds .
the point estimate of mec was derived from potency measures in a whole blood in vitro assay ( table 
 2 ) .
the in vitro assay potency results fell within twofold of the potency seen ex vivo in human whole blood , for two reference compounds with reported clinical data , targeting the same pathway .
hence , the uncertainty in pd was set to let 95% of the samples fall within twofold of the point estimate .
the absorption constant , ka , was fixed because it is a parameter that can be largely controlled by formulation choices as long as the compounds are judged to show properties suitable for er formulation .
specifically , ka was kept fixed at 1 h and 0.1 h when simulating immediate release ( ir ) and er formulations .
parameters for case study 1 cl , clearance ; lbf , liver blood flow ; mec , minimum effective concentration ; vss , volume of distribution at steady state ; heps , hepatocytes . point estimates and 95% uncertainty ranges .
ranking of the compounds was communicated by a cumulative probability distribution of the daily dose for each compound , all summarized into a single plot combined with verbal quantifiers ( figure 
 2a ) .
we next investigated the effect of an er formulation using the same communication layout ( figure 
 2b ) .
as already pointed out , the uncertainty around the point estimate of the ir dose was large because of the short half - life of the compounds .
for example , the 90% confidence interval for cmp 1e ranges from 10 mg to more than 100,000 mg .
the confidence interval for an er preparation is significantly narrower and ranges from 450 mg .
cmp 1b was predicted to have a somewhat longer half - life compared with the others , and , consequently , this compound exhibits the steepest slope in figure 
 2a . along the same line of reasoning
, the cumulative probability curves for the er formulation generally show much steeper slopes , compared to the corresponding curves for the ir formulation .
the tendency of the compounds to fall into two groups in the er scenario illustrates a substantial difference in potency between the compounds and it was deemed likely that cmp 1c1e would have a daily dose below 100 mg . based on the information in figure 
 2a , b and table 
 2 , the project team concluded that the point estimates for an ir dose were low enough to facilitate once daily dosing for the best compounds .
however , the uncertainty in these dose estimates was relatively large and the lead - optimization program was planned to account for er feasibility .
figure 
 2c efficiently communicated this rationale to decision - makers for one of the top compounds . in summary , the first case study demonstrates how proper uncertainty quantification and communication support compound selection in the lead - optimization phase of a drug - discovery project .
it also shows how different administration schedules and formulations can be analyzed within this framework .
the second case study considers five compounds ( cmp 2a2e ) , that were evaluated in the lead - optimization phase of another drug - discovery project .
the project team explored a first - in - class target , and , in the current phase of the project , it was not known if the css ( eq .
the goal was to select one compound for further toxicological and pharmacological profiling , taking both uncertainty in model and choice of dosing schedule into account .
pk parameters and their uncertainties were quantified as described in the  uncertainty quantification  section . for pd ,
therefore , the mec and css for each compound were predicted from in vitro potency data , and threefold translational uncertainty was assumed around the predicted point estimates ( table 
 3 ) .
dose predictions for the case of css - driven effect show that cmp 2a2d are likely to have a human dose < 500 mg ( figure 
 3a ) .
however , taking the more conservative mec prediction into account reveals a pronounced separation with cmp 2a as the preferred compound because of its relatively long half - life ( 14 hours compared to 36 hours for the other compounds ; figure 
 3b ) .
the plots in figure 
 3a , b illustrated the effect of the long half - life on the predicted human dose , and facilitated decision - making in the project .
cmp 2a is likely to have a human dose < 500 mg , whereas the other compounds have a medium likelihood of a dose > 1,000 mg .
parameter ranges ( table 3 ) , compounds ( cmps ) 2a2d are likely to have a human dose below 500 mg , and cmp 2e is likely to have a dose below 1,000 mg for a model with steady - state concentration ( css)-driven effect ( a ) .
if the effect is driven by the minimal effective concentration , cmp 2a is likely to have a human dose below 500 mg , whereas the others are not ( b ) ; and assuming this model is medium likelihood that cmp 2a can be administered once daily ( c ) .
parameters for case study 2 cl , clearance ; heps , hepatocytes ; mec , minimum effective concentration ; vss , volume of distribution at steady state .
taking also other factors than the human dose - prediction into account , the project team prioritized cmp 2a and b and was interested in understanding the influence of once daily vs. twice daily dosing schedules .
simulations indicate that cmp 2a may be feasible for once daily administration while cmp 2b is not ( figure 
 3c ) .
hence , cmp 2a stands out as superior to cmp 2b based on the human dose - prediction .
this compound was assumed to be least sensitive for different dosing approaches and was selected for further profiling . in summary , the second case study adds to the first by demonstrating how model uncertainty ( css - driven vs. mec - driven effect ) influences dose quantification and uncertainty level for different dosing schedules , and how these aspects can be quantified and efficiently communicated .
the third case study applies a pk / pd - link model containing a one - compartment pk model ( first order absorption , linear elimination ) connected to a receptor - occupancy model with elementary kinetics defined as : 
 5where c denotes drug plasma concentration , rtot denotes the total receptor concentration , rc denotes the drug - receptor complex and kon and koff are kinetic parameters ( the dissociation constant is derived as kd  =  koff / kon ) . for this case study ,
the human dose was predicted by requiring a certain level of receptor occupancy ( e.g. , 50% ) over a certain period of the day ( e.g. , 16 hours ) .
the aim was to predict the required human dose and its uncertainty for a candidate drug , cmp 3a , in order to assure a certain level of receptor occupancy .
human pk parameters were predicted from empirical scaling from both in vitro ( hepatocytes ) and in vivo ( mouse / rat / dog ) data ( table 
 4 ) .
parameters for case study 3 cl , clearance ; cmp , compound ; heps , hepatocytes ; lbf , liver blood flow ; pk , pharmacokinetic ; ro , receptor occupancy ; vss , volume of distribution at steady state ; pd , pharmacodynamic . point estimates and 95% uncertainty ranges for six parameters based on domain knowledge , data , and literature . for pd parameters , the required receptor occupancy level was observed at 24 hours in rodents ( rat 37% , mouse 63% ) and translated to the same average level , 50% , but at 16 hours in humans . the difference in coverage ( 24 hours in rodents compared to 16 hours in humans ) reflects well - established species differences in the particular disease area . the uncertainty for human translation
was represented by a log - normal distribution with 95% chance of falling in the range of 3763% .
furthermore , human point - estimates for receptor - occupancy parameters kd and koff were obtained from pk / pd analysis of mouse data ( with kd protein - binding corrected ) . uncertainty in determination of kd and
we assume that , with 95% confidence , kd and koff in humans independently varies by a factor of three to corresponding mouse parameters . in summary , uncertainty in six key
pk and pd parameters influenced the human dose prediction ( table 
 4 ) . using a pk / pd - link model , we algorithmically searched for the minimal required human dose for each combination of receptor - occupancy coverage and dosing schedule .
this calculation was defined as an optimization problem : find the minimal predicted dose that ensures a certain level of receptor occupancy , given a certain number of doses per day , a certain coverage time , and the predicted human pk parameters .
a simple half - interval search was used to find the minimal dose . here , we took advantage of a flexible and efficient matlab library to ensure computational feasibility.67 then , the monte - carlo simulation was used to integrate uncertainty from all specified sources and to predict empirical distributions for dose and cmax ( figure 
 4 ) .
data indicate that it is likely that the human dose is < 200 mg when uncertainty in both pk and pd are integrated ( figure 
 4a ) .
understanding of human pk would significantly reduce the overall uncertainty ; here simulated for the case with pk parameters fixed at the predicted point estimates ( figure 
 4a ) . in order to design preclinical safety studies and interpret in vitro safety screens , it is pivotal to have a good understanding of expected exposure levels in humans .
the monte - carlo simulation outputs such predictions as well , and data indicate that it is likely that cmax is < 0.4 m taking pk / pd uncertainty into account ( figure 
 4b ) .
overall , data indicate that pd uncertainty is important to consider , in particular when key biomarkers are not accessible in at least one of the species , and when change in disease state is not well understood .
parameter ranges ( table 4 ) , it is likely that the human dose of compound ( cmp ) 3a is < 150 mg ( black solid line ) ( a ) ; it is likely that cmax in humans is < 0.5 m ( black solid line ) ( b ) . in both a and b
, the red steeper curve illustrates the information gain if human pharmacokinetic ( pk ) was known ( here assumed known at the predicted point estimate ) and only uncertainty in pharmacodynamic ( pd ) contributed to overall uncertainty . in summary
, the third case study adds to the two others first by demonstrating how arbitrary ( nonlinear ) pk / pd model structures , as well as uncertainty in several pd parameters can be considered .
the analysis can be connected to a dose optimization step and still be computationally feasible when no closed - form dose expression can be analytically derived .
the third case study also shows how other parameters , like cmax , can be communicated in the same way as dose .
the first case study considers five compounds ( cmp 1a1e ) , that were evaluated in the lead - optimization phase of a drug - discovery project . the aim was to support ranking of the compounds for subsequent progression into toxicological studies . for each compound ,
3 . however , because of an estimated short half - life ( 34 hours ) of the compounds , small changes in pk parameters had a large impact on the dose prediction , thereby increasing overall uncertainty .
the project team was considering an extended release ( er ) formulation in order to generate a flatter concentration - time profile and to decrease dependence on elimination half - life and thereby obtain a more clinically realistic dose level . during these discussions
, it became apparent that understanding and efficiently communicating uncertainty in the dose predictions were pivotal to progress the project .
parameters and distributions were generated as described in the  uncertainty quantification  section ( table 
 2 ; example in figure 
 1d ) .
predictions for cl using methods based on in vitro data ( in vitro - in vivo extrapolation from hepatocytes ) and in vivo data ( rat / dog liver blood flow17 ) were in a similar range and did not diverge more than twofold for any of the compounds .
the point estimate of mec was derived from potency measures in a whole blood in vitro assay ( table 
 2 ) .
the in vitro assay potency results fell within twofold of the potency seen ex vivo in human whole blood , for two reference compounds with reported clinical data , targeting the same pathway .
hence , the uncertainty in pd was set to let 95% of the samples fall within twofold of the point estimate .
the absorption constant , ka , was fixed because it is a parameter that can be largely controlled by formulation choices as long as the compounds are judged to show properties suitable for er formulation .
specifically , ka was kept fixed at 1 h and 0.1 h when simulating immediate release ( ir ) and er formulations .
parameters for case study 1 cl , clearance ; lbf , liver blood flow ; mec , minimum effective concentration ; vss , volume of distribution at steady state ; heps , hepatocytes . point estimates and 95% uncertainty ranges . the parameter t1/2 is derived from cl and vss .
ranking of the compounds was communicated by a cumulative probability distribution of the daily dose for each compound , all summarized into a single plot combined with verbal quantifiers ( figure 
 2a ) .
we next investigated the effect of an er formulation using the same communication layout ( figure 
 2b ) .
as already pointed out , the uncertainty around the point estimate of the ir dose was large because of the short half - life of the compounds .
for example , the 90% confidence interval for cmp 1e ranges from 10 mg to more than 100,000 mg .
the confidence interval for an er preparation is significantly narrower and ranges from 450 mg .
cmp 1b was predicted to have a somewhat longer half - life compared with the others , and , consequently , this compound exhibits the steepest slope in figure 
 2a . along the same line of reasoning , the cumulative probability curves for the er formulation generally show much steeper slopes , compared to the corresponding curves for the ir formulation .
the tendency of the compounds to fall into two groups in the er scenario illustrates a substantial difference in potency between the compounds and it was deemed likely that cmp 1c1e would have a daily dose below 100 mg . based on the information in figure 
 2a , b and table 
 2 , the project team concluded that the point estimates for an ir dose were low enough to facilitate once daily dosing for the best compounds .
however , the uncertainty in these dose estimates was relatively large and the lead - optimization program was planned to account for er feasibility .
figure 
 2c efficiently communicated this rationale to decision - makers for one of the top compounds . in summary , the first case study demonstrates how proper uncertainty quantification and communication support compound selection in the lead - optimization phase of a drug - discovery project .
it also shows how different administration schedules and formulations can be analyzed within this framework .
the second case study considers five compounds ( cmp 2a2e ) , that were evaluated in the lead - optimization phase of another drug - discovery project .
the project team explored a first - in - class target , and , in the current phase of the project , it was not known if the css ( eq .
the goal was to select one compound for further toxicological and pharmacological profiling , taking both uncertainty in model and choice of dosing schedule into account .
pk parameters and their uncertainties were quantified as described in the  uncertainty quantification  section . for pd ,
therefore , the mec and css for each compound were predicted from in vitro potency data , and threefold translational uncertainty was assumed around the predicted point estimates ( table 
 3 ) .
dose predictions for the case of css - driven effect show that cmp 2a2d are likely to have a human dose < 500 mg ( figure 
 3a ) .
however , taking the more conservative mec prediction into account reveals a pronounced separation with cmp 2a as the preferred compound because of its relatively long half - life ( 14 hours compared to 36 hours for the other compounds ; figure 
 3b ) .
the plots in figure 
 3a , b illustrated the effect of the long half - life on the predicted human dose , and facilitated decision - making in the project .
cmp 2a is likely to have a human dose < 500 mg , whereas the other compounds have a medium likelihood of a dose > 1,000 mg .
parameter ranges ( table 3 ) , compounds ( cmps ) 2a2d are likely to have a human dose below 500 mg , and cmp 2e is likely to have a dose below 1,000 mg for a model with steady - state concentration ( css)-driven effect ( a ) .
if the effect is driven by the minimal effective concentration , cmp 2a is likely to have a human dose below 500 mg , whereas the others are not ( b ) ; and assuming this model is medium likelihood that cmp 2a can be administered once daily ( c ) .
parameters for case study 2 cl , clearance ; heps , hepatocytes ; mec , minimum effective concentration ; vss , volume of distribution at steady state .
taking also other factors than the human dose - prediction into account , the project team prioritized cmp 2a and b and was interested in understanding the influence of once daily vs. twice daily dosing schedules .
simulations indicate that cmp 2a may be feasible for once daily administration while cmp 2b is not ( figure 
 3c ) .
hence , cmp 2a stands out as superior to cmp 2b based on the human dose - prediction .
this compound was assumed to be least sensitive for different dosing approaches and was selected for further profiling . in summary , the second case study adds to the first by demonstrating how model uncertainty ( css - driven vs. mec - driven effect ) influences dose quantification and uncertainty level for different dosing schedules , and how these aspects can be quantified and efficiently communicated .
the third case study applies a pk / pd - link model containing a one - compartment pk model ( first order absorption , linear elimination ) connected to a receptor - occupancy model with elementary kinetics defined as : 
 5where c denotes drug plasma concentration , rtot denotes the total receptor concentration , rc denotes the drug - receptor complex and kon and koff are kinetic parameters ( the dissociation constant is derived as kd  =  koff / kon ) . for this case study ,
the human dose was predicted by requiring a certain level of receptor occupancy ( e.g. , 50% ) over a certain period of the day ( e.g. , 16 hours ) .
the aim was to predict the required human dose and its uncertainty for a candidate drug , cmp 3a , in order to assure a certain level of receptor occupancy .
human pk parameters were predicted from empirical scaling from both in vitro ( hepatocytes ) and in vivo ( mouse / rat / dog ) data ( table 
 4 ) .
parameters for case study 3 cl , clearance ; cmp , compound ; heps , hepatocytes ; lbf , liver blood flow ; pk , pharmacokinetic ; ro , receptor occupancy ; vss , volume of distribution at steady state ; pd , pharmacodynamic .
point estimates and 95% uncertainty ranges for six parameters based on domain knowledge , data , and literature . for pd parameters ,
the required receptor occupancy level was observed at 24 hours in rodents ( rat 37% , mouse 63% ) and translated to the same average level , 50% , but at 16 hours in humans . the difference in coverage
( 24 hours in rodents compared to 16 hours in humans ) reflects well - established species differences in the particular disease area . the uncertainty for human translation
was represented by a log - normal distribution with 95% chance of falling in the range of 3763% .
furthermore , human point - estimates for receptor - occupancy parameters kd and koff were obtained from pk / pd analysis of mouse data ( with kd protein - binding corrected ) . uncertainty in determination of kd and koff from mouse data
we assume that , with 95% confidence , kd and koff in humans independently varies by a factor of three to corresponding mouse parameters . in summary , uncertainty in six key
pk and pd parameters influenced the human dose prediction ( table 
 4 ) . using a pk / pd - link model
, we algorithmically searched for the minimal required human dose for each combination of receptor - occupancy coverage and dosing schedule .
this calculation was defined as an optimization problem : find the minimal predicted dose that ensures a certain level of receptor occupancy , given a certain number of doses per day , a certain coverage time , and the predicted human pk parameters .
a simple half - interval search was used to find the minimal dose . here , we took advantage of a flexible and efficient matlab library to ensure computational feasibility.67 then , the monte - carlo simulation was used to integrate uncertainty from all specified sources and to predict empirical distributions for dose and cmax ( figure 
 4 ) .
data indicate that it is likely that the human dose is < 200 mg when uncertainty in both pk and pd are integrated ( figure 
 4a ) .
understanding of human pk would significantly reduce the overall uncertainty ; here simulated for the case with pk parameters fixed at the predicted point estimates ( figure 
 4a ) . in order to design preclinical safety studies and interpret in vitro safety screens , it is pivotal to have a good understanding of expected exposure levels in humans .
the monte - carlo simulation outputs such predictions as well , and data indicate that it is likely that cmax is < 0.4 m taking pk / pd uncertainty into account ( figure 
 4b ) .
overall , data indicate that pd uncertainty is important to consider , in particular when key biomarkers are not accessible in at least one of the species , and when change in disease state is not well understood .
parameter ranges ( table 4 ) , it is likely that the human dose of compound ( cmp ) 3a is < 150 mg ( black solid line ) ( a ) ; it is likely that cmax in humans is < 0.5 m ( black solid line ) ( b ) . in both a and b
, the red steeper curve illustrates the information gain if human pharmacokinetic ( pk ) was known ( here assumed known at the predicted point estimate ) and only uncertainty in pharmacodynamic ( pd ) contributed to overall uncertainty . in summary
, the third case study adds to the two others first by demonstrating how arbitrary ( nonlinear ) pk / pd model structures , as well as uncertainty in several pd parameters can be considered .
the analysis can be connected to a dose optimization step and still be computationally feasible when no closed - form dose expression can be analytically derived .
the third case study also shows how other parameters , like cmax , can be communicated in the same way as dose .
in the case studies , uncertainty in human dose - prediction is quantified by a systematic integration of information from various uncertain sources , and reported as a single uncomplicated plot complemented with verbal quantifiers that facilitates communication and enables informed decisions .
the approach allows drug - project teams to evaluate various scaling scenarios or models , to identify the most influencing parameters , to rank compounds ( e.g. , compounds with similar predicted dose point - estimates but different uncertainty ) , and to properly assess the risk before the first time in humans .
other uncertainty considerations , such as safety aspects , commercial feasibility , cost of goods , and competitive landscape , can be directly connected to the framework as they often use the predicted human dose as one of the input parameters ( cf .
the same reasoning can be applied to the predicted cmax or area under the curve distribution or other derived parameters .
the empirical calculations can be performed efficiently , not only for closed - form expressions , but also for nonlinear ordinary differential equation systems . in this way
, calculations can be instantly revised as soon as new data are generated , or a different scenario is to be tested .
taken together , the approach exemplified by the three case studies represents a useful way of quantifying and communicating uncertainty in human dose - predictions .
however , in our experience , such communications always raise questions ( e.g. , how is ranking influenced by variability / uncertainty , and what is the chance of having a dose less than a certain amount ) . in particular , this is the case when uncertainties differ between compounds ( because of different types , amounts , or quality of input data ) , or a combination of those . a monte - carlo approach can address these and other questions .
there are several ways of communicating the result of a dose prediction based on a probabilistic approach .
a concentration - time plot ( figure 
 1c ) is one obvious alternative to the cumulative probability plot of dose advocated in this tutorial ( figure 
 2a ) .
first , we have experienced that many audiences misinterpret the uncertainty in the concentration - time plot as population variability .
second , the concentration - time plot gets crowded when more than one compound or scenario , or both , are compared ( figure 
 1c shows just one compound and would be hard to interpret if all compounds 1a1e were included , as in figure 
 2a ) .
nevertheless , the concentration - time plot remains a useful complement to the dose distribution . in case study 1 , the need to investigate er feasibility of the compound series was successfully communicated by figure 
 2 .
it was also apparent that an er formulation was likely to generate a dose below 100 mg / day for any of the three compounds 1c  e .
the plots were used to illustrate the risk with the ir once daily approach and gain confidence in the chosen mitigation plan using an er preparation .
in addition , figure 
 2c efficiently communicates that an er formulation would potentially be superior to twice daily administration for one of the compounds .
indeed , the high uncertainty associated with once daily ir formulation ( figure 
 2a ) illustrates the importance of communicating uncertainties ; only communicating point estimates gives a false sense of security . in this and the two other case studies ,
the compounds show good absorption properties and bioavailability was derived from cl only . however , we acknowledge that a more conservative approach with uncertainty in fa can be argued .
generally , for low permeability compounds or compounds predicted to be highly metabolized in the gut , the bioavailability estimate should take fa and fg into consideration .
plausible uncertainty distributions for these parameters must then be proposed in the same way as for other parameters .
the method used for human dose - prediction is drug - project specific and should be chosen based on the pharmacological understanding of the target .
case study 2 shows the impact of model choice on the distribution of the predicted dose ( css vs. mec - driven approach ) .
of course , model averaging can be considered in situations where information is lacking and a single model can not be selected . commonly ,
uncertainty in clearance and bioavailability , and sometimes volume of distribution and rate of absorption , strongly influences overall uncertainty in the prediction .
uncertainty in pd parameters can have a large impact on dose , as seen in case study 3 , but is even more project - specific compared to pk parameters .
uncertainty in pd can be decreased by , for example , understanding the target and its biological pathway , using preclinical animal models with known translational properties , or taking advantage of published clinical data for compounds with the same target or compounds already approved for the indication .
the same type of communication can be used to inform about remaining uncertainty in pd when pk is known from a clinical phase i study ( cf .
furthermore , for early toxicological studies , the uncertainty in predicted exposure levels ( cmax and area under the curve ) , can be used to better assess and communicate uncertainty in safety margins . in order to quantify uncertainty in the dose prediction , it is useful to list the contributing parameters and then assess the uncertainty of each parameter , as well as correlations between those measures if independence can not be assumed .
naturally , it is not uncommon that compounds in a series are supported by various amounts of data ( e.g. , clearance can be predicted from either hepatocyte data , or animal data , or both ) .
however , such differences in input data can be directly incorporated in the uncertainty calculation .
notably , for cases where the predictions of a parameter vary substantially between two or more methods , it is advisable to further investigate the underlying reason because one of the methods may be more suitable for the compound series .
however , in principle , the computational framework also works for this case . in general , it is hard to determine the translational uncertainty in scaling methods and pd relationships , and to assign or infer reasonable distributions representing those uncertainties .
however , there is lack of consensus in ranking these methods , partly because datasets are limited and differ in size .
different physicochemical properties for compounds may also influence which methods are best suited for a certain compound class . as a rule - of - thumb ,
literature data on translation of pk parameters indicate that good methods predict 8090% of the compounds within threefold from the true parameter value.622 however , these evaluations of methods are based on data for compounds that have reached the clinical phase and have been reported in the literature ; potential bias in any direction can not be excluded . in our case studies ,
we have assumed log - normally distributed uncertainty with 95% probability to be within threefold from the point estimate .
it is recommended to increase confidence in prediction by using different types of data ( e.g. , in vitro and in vivo , more than one scaling method , and information from other compounds with similar properties ) . in our experience , decision - makers in drug discovery gain a lot from having a rough indication of the uncertainty levels . integrating information in the way proposed here
quantifies the overall uncertainty ( although not being exact ) in a manner that is easy to understand .
therefore , we believe it is worthwhile doing this analysis when uncertainty levels can be captured in at least the right order of magnitude .
fundamental to science communication is to determine how much and what information the audience needs to make an informed decision ; what is considered good communication for one audience may not suit another . communicating modeling and simulation
can often be difficult , especially when trying to describe the impact of assumptions and uncertainty to people not used to working with prediction models or translational aspects , or both .
also , it is challenging to find the right balance between communication of the point estimate and the uncertainty .
hence , finding the right communication strategy requires iterative trials with intermediate collection and analysis of feedback from the audience .
generally , the reception of the communications has been very positive from various academic and industrial stakeholders from different scientific backgrounds .
for example , medicinal chemists emphasize the value of rough uncertainty quantifications during the design - make - test - analyze process .
negative feedback during the process was mainly related to the three challenges in order to effectively communicate uncertainty in human dose - predictions listed in the  uncertainty communication  section . to overcome these challenges ,
we have strived to avoid technical jargon and minimize the usage of equations in our presentation .
furthermore , as pointed out by pidgeon and fischhoff , scientists from a different field can get an exaggerated sense of scientific uncertainty when uncertainties that are most difficult to understand or model are discussed in depth , whereas uncertainties of commonly accepted methods ( e.g. , a certain pk scaling method ) are never brought up .
the approach used in the case studies is designed to put the same emphasis on all contributing factors to the overall scaling uncertainty in order to avoid this problem .
the hardest communication problem has been related to the distinction between variability for uncertainty ( e.g. , the uncertainty distribution of cl may be incorrectly interpreted as the population variability of cl ) . to overcome this problem ,
we have strived to use a consequent language ( table 
 1 ) , focused on dose distribution plots and not concentration - time plots ( as discussed above ) .
another important factor for successful communication is to allow the audience to take active part in the reasoning behind the prediction . by using efficient software , users can revisit and change the assumptions , or simulate a different scenario , and get an instant reply .
communication should be designed specifically for the target audience ( see point 1 in the numbered list in section  uncertainty communication  ) . in our experience ,
the cumulative probability plots are useful both for modelers and nonmodelers ( e.g. , decision - makers ) . however , the underlying assumptions and calculation steps should be motivated more rigorously for modelers compared to nonmodelers . in this way
, we can balance the right to know and full disclosure with need - to - know .
however , the number of iterations can be reduced by a well chosen starting point .
the approach outlined by three case studies in this tutorial can be such a useful starting point for communicating human dose - predictions in pharmaceutical research organizations . in summary ,
the three case studies have been communicated by three means ( figures 
 4 ; tables24 ) : a single plot showing the cumulative probability of predicted dose levels , allowing several compounds or scenarios , or both , to be simultaneously evaluated . 
 a summary table of underlying assumptions , detailing which parameters were considered uncertain and to what extent . 
 a headline interpreting the presented data and summarizing them into the key finding ( e.g. ,  it is likely that the human dose is < 100 mg  ) . 
  a single plot showing the cumulative probability of predicted dose levels , allowing several compounds or scenarios , or both , to be simultaneously evaluated . 
  a summary table of underlying assumptions , detailing which parameters were considered uncertain and to what extent . 
  a headline interpreting the presented data and summarizing them into the key finding ( e.g. ,  it is likely that the human dose is < 100 mg  ) . 
  together
, these three pieces fit into a single slide in presentation format ( e.g. , powerpoint ) .
simple because the data is easily generated , revised , and summarized into a single slide , efficient because the audience is provided the information that they need in a form they can use , and sound because we have strived to transparently acknowledge main sources of scientific uncertainty .
 human dose - prediction is fundamental for ranking lead - optimization compounds in drug discovery and to inform design of early clinical trials . 
 this tutorial describes how uncertainty in such predictions can be quantified and efficiently communicated to facilitate decision - making . using three drug - discovery case studies , we show how several uncertain pieces of input information can be integrated into one single uncomplicated plot with key predictions , including their uncertainties , for many compounds or for many scenarios , or both . 
Uncertainty quantification
Uncertainty communication
CASE STUDIES
Case study 1
Case study 2
Case study 3
DISCUSSION
CONFLICT OF INTEREST