# Trials

In Pumas QSP, the different variations of the model to be ran are called the *trials*. For example, one trial may specify that the model should be solved with a loading dose of 150mg, while the next trial specifies that the loading dose is 250mg. Each trial is then optionally tied to a dataset which, when defined in an inverse problem, specifies a multi-simulation optimization problem that the further functions (`calibrate`

, `vpop`

, etc.) generate solutions for. The type of trial is used to signify what the data corresponds to measuring, i.e. whether the trials is used to match data of time series, or steady states, etc.

## Trial Types

The following describes the types of trials which can be generated.

`PumasQSP.Trial`

— Type`Trial(data, model; kwargs...)`

The `Trial`

describes an experiment in which `data`

was obtained. The dynamics of the investigated system are represented in `model`

as an ODESystem. The trial is used within the optimization problem, as part of `InverseProblem`

to fit the unknown `model`

parameters and initial conditions to `data`

. In the case of Global Sensitivity Analysis (GSA), the `data`

is not needed and can be assigned to `nothing`

.

**Required Keyword Arguments**

`tspan`

, which indicates the timespan for which the model equations are solved.

**Optional Keywords Arguments**

`u0`

modifies the default values of the initial conditions of some or all of the states, e.g.`u0 = [state_name => custom_initial_value]`

`params`

modifies the default values of some or all the parameters, e.g.`params = [p1 => specific_value, p2 => other_value]`

`err`

specifies the trial's contribution to the cost function. Defaults to the`l2loss`

function. The function requires 2 arguments, the solution of the trial and the data and is expected to return a scalar value corresponding to the cost of the trial, i.e.`err = (sol, data) -> compute_error`

.`likelihood`

is used for`MCMCOpt`

methods. This likelihood describes the distribution that generated the data. By default, a multivariate Normal distribution is used, centered at the solution of the trial at each saved timepoint`u`

with a standard deviation`s`

around it, but different distributions can be used, e.g.,`likelihood = (u, s) -> MvNormal(u, s)`

.`noise_priors`

is used for`MCMCOpt`

methods. Describes the prior for`s`

. The`s`

parameter represents any observation noise around the solution of the trial at each timepoint,`u`

. The default value for this prior distribution is an`InverseGamma(2,3)`

, but this can be changed, e.g.`noise_prior = [s1 => Exponential(1), s2 => Gamma(3,2)]`

). If only one prior is given, then it is assumed that it applies to all saved states (`save_names`

). Otherwise, a vector containing pairs of states and prior distributions with length equal to`saved_names`

needs to be provided, to set different priors for each state. In the example above, the model had two states`s1`

and`s2`

so two pairs were provided. The noise parameters`s`

are assumed to be unique for each trial and to be constant across timepoints for each state of each trial.`reduction`

is used for GSA methods. The output of`reduction`

is expected to be the quantity whose sensitivity is being investigated.`doses`

specifies the doses that occur in a trial. This argument can be equal to an instance of`Bolus`

,`PeriodicBolus`

,`Infusion`

,`PeriodicInfusion`

or a`Vector`

of multiple instances of any of these dose types.`save_names`

is used to specify which`model`

states are saved. The same states are extracted from`data`

.`trial_name`

, an identifying name. The default name is "Trial".`forward_u0=true`

, if the trial is part of a collection of`SteadyStateTrials`

, then`forward_u0=true`

signals that the trial should use the outcome of the`SteadyStateTrial`

of the same collection as its initial condition.

If additional keywords are passed, they will be forwarded to the `solve`

call. For example, one can pass `alg=Tsit5()`

to specify what solver will be used. More information about supported arguments can be found here.

`PumasQSP.SteadyStateTrial`

— Type`SteadyStateTrial(data, model; kwargs...)`

Describes a trial that is ran until a steady state is reached. This object can be initialized in the same way as a `Trial`

object, with the only difference being that `data`

needs to be a `Vector`

here. The `data`

in this case represents the values of the saved states when the system has reached its steady state.

See the SciML documentation for background information on steady state problems.

`PumasQSP.IndependentTrials`

— Type`IndependentTrials(trials...)`

This trial collection type indicates that each trial can be solved individualy and that there is no interaction between them. This trial type is automatically created it the trials are passed as a `Vector`

(i.e. [trial1, trial2])

`PumasQSP.SteadyStateTrials`

— Type`SteadyStateTrials(ss_trial, trials...; postprocess=last)`

`SteadyStateTrials`

are a trial collection that describes a steady state trial (see `SteadyStateTrial`

) (specified as the first argument) followed by subsequent trials that can continue using the steady state by setting `forward_u0=true`

. The steady state solution that is passed on can be modified using the `postprocess`

keyword argument, which accepts a function with a single argument that represents the solution of the first trial and returns the state to be further passed on.

## Simulation and Analysis Functions

To better understand and debug trials, the trials come with associated analysis functions to allow for easy investigation of the results in a trial-by-trial form. The following functions help the introspection of such trials.

`JuliaSimModelOptimizer.simulate`

— Function`simulate(experiment::AbstractExperiment, prob::InverseProblem, x)`

Simulate the given `experiment`

using optimization-state point `x`

, which contains values for each parameter and initial condition that is optimized in `InverseProblem`

`prob`

.

## Loss Functions

By default, the loss function associated with a trial against its data is the standard Euclidian distance, also known as the L2 loss. However, PumasQSP provides alternative loss definitions to allow for customizing the fitting strategy.

`JuliaSimModelOptimizer.l2loss`

— Function`l2loss(sol, data)`

Squared error loss :

$\sum_{i=1}^{M} \sum_{j=1}^{N} \left( \text{sol}_{i,j} - \text{data}_{i,j} \right)^2$

where N is the number of saved timepoints and M the number of measured states in the solution

`JuliaSimModelOptimizer.ARMLoss`

— Function`ARMLoss(sol, bounds)`

Allen-Rieger-Musante (ARM) loss :

$\sum_{i=1}^{M} \sum_{j=1}^{N} \text{max} \left[ \left( \text{sol}_{i,j} - \frac{\text{u}_{i,j} + \text{l}_{i,j}}{2} \right)^2 - \left( \frac{\text{u}_{i,j} - \text{l}_{i,j}}{2} \right)^2, 0 \right]$

where N is the number of saved timepoints, M the number of measured states in the solution and `l, u`

are the lower and upper bounds of each measured state respectively.

**Reference**

Allen RJ, Rieger TR, Musante CJ. Efficient Generation and Selection of Virtual Populations in Quantitative Systems Pharmacology Models. CPT Pharmacometrics Syst Pharmacol. 2016 Mar;5(3):140-6. doi: 10.1002/psp4.12063. Epub 2016 Mar 17. PMID: 27069777; PMCID: PMC4809626.