Solving & Simulation of a ODE/solved system (and solving options) equation

This uses RxODE family of objects, file, or model specification to solve a ODE system. There are many options for a solved RxODE model, the first are the required object, and events with the some-times optional params and inits.

rxSolve(
  object,
  params = NULL,
  events = NULL,
  inits = NULL,
  scale = NULL,
  method = c("liblsoda", "lsoda", "dop853", "indLin"),
  transitAbs = NULL,
  atol = 1e-08,
  rtol = 1e-06,
  maxsteps = 70000L,
  hmin = 0,
  hmax = NA_real_,
  hmaxSd = 0,
  hini = 0,
  maxordn = 12L,
  maxords = 5L,
  ...,
  cores,
  covsInterpolation = c("locf", "linear", "nocb", "midpoint"),
  addCov = FALSE,
  matrix = FALSE,
  sigma = NULL,
  sigmaDf = NULL,
  sigmaLower = -Inf,
  sigmaUpper = Inf,
  nCoresRV = 1L,
  sigmaIsChol = FALSE,
  sigmaSeparation = c("auto", "lkj", "separation"),
  sigmaXform = c("identity", "variance", "log", "nlmixrSqrt", "nlmixrLog",
    "nlmixrIdentity"),
  nDisplayProgress = 10000L,
  amountUnits = NA_character_,
  timeUnits = "hours",
  stiff,
  theta = NULL,
  thetaLower = -Inf,
  thetaUpper = Inf,
  eta = NULL,
  addDosing = FALSE,
  stateTrim = Inf,
  updateObject = FALSE,
  omega = NULL,
  omegaDf = NULL,
  omegaIsChol = FALSE,
  omegaSeparation = c("auto", "lkj", "separation"),
  omegaXform = c("variance", "identity", "log", "nlmixrSqrt", "nlmixrLog",
    "nlmixrIdentity"),
  omegaLower = -Inf,
  omegaUpper = Inf,
  nSub = 1L,
  thetaMat = NULL,
  thetaDf = NULL,
  thetaIsChol = FALSE,
  nStud = 1L,
  dfSub = 0,
  dfObs = 0,
  returnType = c("rxSolve", "matrix", "data.frame", "data.frame.TBS", "data.table",
    "tbl", "tibble"),
  seed = NULL,
  nsim = NULL,
  minSS = 10L,
  maxSS = 1000L,
  infSSstep = 12,
  strictSS = TRUE,
  istateReset = TRUE,
  subsetNonmem = TRUE,
  maxAtolRtolFactor = 0.1,
  from = NULL,
  to = NULL,
  by = NULL,
  length.out = NULL,
  iCov = NULL,
  keep = NULL,
  indLinPhiTol = 1e-07,
  indLinPhiM = 0L,
  indLinMatExpType = c("expokit", "Al-Mohy", "arma"),
  indLinMatExpOrder = 6L,
  drop = NULL,
  idFactor = TRUE,
  mxhnil = 0,
  hmxi = 0,
  warnIdSort = TRUE,
  warnDrop = TRUE,
  ssAtol = 1e-08,
  ssRtol = 1e-06,
  safeZero = TRUE,
  sumType = c("pairwise", "fsum", "kahan", "neumaier", "c"),
  prodType = c("long double", "double", "logify"),
  sensType = c("advan", "autodiff", "forward", "central"),
  linDiff = c(tlag = 1.5e-05, f = 1.5e-05, rate = 1.5e-05, dur = 1.5e-05, tlag2 =
    1.5e-05, f2 = 1.5e-05, rate2 = 1.5e-05, dur2 = 1.5e-05),
  linDiffCentral = c(tlag = TRUE, f = TRUE, rate = TRUE, dur = TRUE, tlag2 = TRUE, f2 =
    TRUE, rate2 = TRUE, dur2 = TRUE),
  resample = NULL,
  resampleID = TRUE,
  maxwhile = 1e+05
)

# S3 method for default
rxSolve(
  object,
  params = NULL,
  events = NULL,
  inits = NULL,
  ...,
  theta = NULL,
  eta = NULL
)

# S3 method for rxSolve
update(object, ...)

# S3 method for RxODE
predict(object, ...)

# S3 method for rxSolve
predict(object, ...)

# S3 method for rxEt
predict(object, ...)

# S3 method for rxParams
predict(object, ...)

# S3 method for RxODE
simulate(object, nsim = 1L, seed = NULL, ...)

# S3 method for rxSolve
simulate(object, nsim = 1L, seed = NULL, ...)

# S3 method for rxParams
simulate(object, nsim = 1L, seed = NULL, ...)

# S3 method for rxSolve
solve(a, b, ...)

# S3 method for RxODE
solve(a, b, ...)

# S3 method for rxParams
solve(a, b, ...)

# S3 method for rxEt
solve(a, b, ...)

rxControl(..., params = NULL, events = NULL, inits = NULL)

Arguments

object	is a either a RxODE family of objects, or a file-name with a RxODE model specification, or a string with a RxODE model specification.
params	a numeric named vector with values for every parameter in the ODE system; the names must correspond to the parameter identifiers used in the ODE specification;
events	an eventTable object describing the input (e.g., doses) to the dynamic system and observation sampling time points (see eventTable());
inits	a vector of initial values of the state variables (e.g., amounts in each compartment), and the order in this vector must be the same as the state variables (e.g., PK/PD compartments);
scale	a numeric named vector with scaling for ode parameters of the system. The names must correspond to the parameter identifiers in the ODE specification. Each of the ODE variables will be divided by the scaling factor. For example scale=c(center=2) will divide the center ODE variable by 2.
method	The method for solving ODEs. Currently this supports: "liblsoda" thread safe lsoda. This supports parallel thread-based solving, and ignores user Jacobian specification. "lsoda" -- LSODA solver. Does not support parallel thread-based solving, but allows user Jacobian specification. "dop853" -- DOP853 solver. Does not support parallel thread-based solving nor user Jacobain specification "indLin" -- Solving through inductive linearization. The RxODE dll must be setup specially to use this solving routine.
transitAbs	boolean indicating if this is a transit compartment absorption
atol	a numeric absolute tolerance (1e-8 by default) used by the ODE solver to determine if a good solution has been achieved; This is also used in the solved linear model to check if prior doses do not add anything to the solution.
rtol	a numeric relative tolerance (1e-6 by default) used by the ODE solver to determine if a good solution has been achieved. This is also used in the solved linear model to check if prior doses do not add anything to the solution.
maxsteps	maximum number of (internally defined) steps allowed during one call to the solver. (5000 by default)
hmin	The minimum absolute step size allowed. The default value is 0.
hmax	The maximum absolute step size allowed. When hmax=NA (default), uses the average difference + hmaxSd*sd in times and sampling events. The hmaxSd is a user specified parameter and which defaults to zero. When hmax=NULL RxODE uses the maximum difference in times in your sampling and events. The value 0 is equivalent to infinite maximum absolute step size.
hmaxSd	The number of standard deviations of the time difference to add to hmax. The default is 0
hini	The step size to be attempted on the first step. The default value is determined by the solver (when hini = 0)
maxordn	The maximum order to be allowed for the nonstiff (Adams) method. The default is 12. It can be between 1 and 12.
maxords	The maximum order to be allowed for the stiff (BDF) method. The default value is 5. This can be between 1 and 5.
...	Other arguments including scaling factors for each compartment. This includes S# = numeric will scale a compartment # by a dividing the compartment amount by the scale factor, like NONMEM.
cores	Number of cores used in parallel ODE solving. This is equivalent to calling setRxThreads()
covsInterpolation	specifies the interpolation method for time-varying covariates. When solving ODEs it often samples times outside the sampling time specified in events. When this happens, the time varying covariates are interpolated. Currently this can be: "linear" interpolation, which interpolates the covariate by solving the line between the observed covariates and extrapolating the new covariate value. "constant" -- Last observation carried forward (the default). "NOCB" -- Next Observation Carried Backward. This is the same method that NONMEM uses. "midpoint" Last observation carried forward to midpoint; Next observation carried backward to midpoint.
addCov	A boolean indicating if covariates should be added to the output matrix or data frame. By default this is disabled.
matrix	A boolean indicating if a matrix should be returned instead of the RxODE's solved object.
sigma	Named sigma covariance or Cholesky decomposition of a covariance matrix. The names of the columns indicate parameters that are simulated. These are simulated for every observation in the solved system.
sigmaDf	Degrees of freedom of the sigma t-distribution. By default it is equivalent to Inf, or a normal distribution.
sigmaLower	Lower bounds for simulated unexplained variability (by default -Inf)
sigmaUpper	Upper bounds for simulated unexplained variability (by default Inf)
nCoresRV	Number of cores used for the simulation of the sigma variables. By default this is 1. To reproduce the results you need to run on the same platform with the same number of cores. This is the reason this is set to be one, regardless of what the number of cores are used in threaded ODE solving.
sigmaIsChol	Boolean indicating if the sigma is in the Cholesky decomposition instead of a symmetric covariance
sigmaSeparation	separation strategy for sigma; Tells the type of separation strategy when simulating covariance with parameter uncertainty with standard deviations modeled in the thetaMat matrix. "lkj" simulates the correlation matrix from the rLKJ1 matrix with the distribution parameter eta equal to the degrees of freedom nu by (nu-1)/2 "separation" simulates from the identity inverse Wishart covariance matrix with nu degrees of freedom. This is then converted to a covariance matrix and augmented with the modeled standard deviations. While computationally more complex than the "lkj" prior, it performs better when the covariance matrix size is greater or equal to 10 "auto" chooses "lkj" when the dimension of the matrix is less than 10 and "separation" when greater than equal to 10.
sigmaXform	When taking sigma values from the thetaMat simulations (using the separation strategy for covariance simulation), how should the thetaMat values be turned int standard deviation values: identity This is when standard deviation values are directly modeled by the params and thetaMat matrix variance This is when the params and thetaMat simulates the variance that are directly modeled by the thetaMat matrix log This is when the params and thetaMat simulates log(sd) nlmixrSqrt This is when the params and thetaMat simulates the inverse cholesky decomposed matrix with the x^2 modeled along the diagonal. This only works with a diagonal matrix. nlmixrLog This is when the params and thetaMat simulates the inverse cholesky decomposed matrix with the exp(x^2) along the diagonal. This only works with a diagonal matrix. nlmixrIdentity This is when the params and thetaMat simulates the inverse cholesky decomposed matrix. This only works with a diagonal matrix.
nDisplayProgress	An integer indicating the minimum number of c-based solves before a progress bar is shown. By default this is 10,000.
amountUnits	This supplies the dose units of a data frame supplied instead of an event table. This is for importing the data as an RxODE event table.
timeUnits	This supplies the time units of a data frame supplied instead of an event table. This is for importing the data as an RxODE event table.
stiff	a logical (TRUE by default) indicating whether the ODE system is stiff or not. For stiff ODE systems (`stiff = TRUE`), `RxODE` uses the LSODA (Livermore Solver for Ordinary Differential Equations) Fortran package, which implements an automatic method switching for stiff and non-stiff problems along the integration interval, authored by Hindmarsh and Petzold (2003). For non-stiff systems (`stiff = FALSE`), `RxODE` uses DOP853, an explicit Runge-Kutta method of order 8(5, 3) of Dormand and Prince as implemented in C by Hairer and Wanner (1993). If stiff is not specified, the `method` argument is used instead.
theta	A vector of parameters that will be named THETA\[#\] and added to parameters
thetaLower	Lower bounds for simulated population parameter variability (by default -Inf)
thetaUpper	Upper bounds for simulated population unexplained variability (by default Inf)
eta	A vector of parameters that will be named ETA\[#\] and added to parameters
addDosing	Boolean indicating if the solve should add RxODE EVID and related columns. This will also include dosing information and estimates at the doses. Be default, RxODE only includes estimates at the observations. (default FALSE). When addDosing is NULL, only include EVID=0 on solve and exclude any model-times or EVID=2. If addDosing is NA the classic RxODE EVID events are returned. When addDosing is TRUE add the event information in NONMEM-style format; If subsetNonmem=FALSE RxODE will also include extra event types (EVID) for ending infusion and modeled times: EVID=-1 when the modeled rate infusions are turned off (matches rate=-1) EVID=-2 When the modeled duration infusions are turned off (matches rate=-2) EVID=-10 When the specified rate infusions are turned off (matches rate>0) EVID=-20 When the specified dur infusions are turned off (matches dur>0) EVID=101,102,103,... Modeled time where 101 is the first model time, 102 is the second etc.
stateTrim	When amounts/concentrations in one of the states are above this value, trim them to be this value. By default Inf. Also trims to -stateTrim for large negative amounts/concentrations. If you want to trim between a range say c(0, 2000000) you may specify 2 values with a lower and upper range to make sure all state values are in the reasonable range.
updateObject	This is an internally used flag to update the RxODE solved object (when supplying an RxODE solved object) as well as returning a new object. You probably should not modify it's FALSE default unless you are willing to have unexpected results.
omega	Estimate of Covariance matrix. When omega is a list, assume it is a block matrix and convert it to a full matrix for simulations.
omegaDf	The degrees of freedom of a t-distribution for simulation. By default this is NULL which is equivalent to Inf degrees, or to simulate from a normal distribution instead of a t-distribution.
omegaIsChol	Indicates if the omega supplied is a Cholesky decomposed matrix instead of the traditional symmetric matrix.
omegaSeparation	Omega separation strategy Tells the type of separation strategy when simulating covariance with parameter uncertainty with standard deviations modeled in the thetaMat matrix. "lkj" simulates the correlation matrix from the rLKJ1 matrix with the distribution parameter eta equal to the degrees of freedom nu by (nu-1)/2 "separation" simulates from the identity inverse Wishart covariance matrix with nu degrees of freedom. This is then converted to a covariance matrix and augmented with the modeled standard deviations. While computationally more complex than the "lkj" prior, it performs better when the covariance matrix size is greater or equal to 10 "auto" chooses "lkj" when the dimension of the matrix is less than 10 and "separation" when greater than equal to 10.
omegaXform	When taking omega values from the thetaMat simulations (using the separation strategy for covariance simulation), how should the thetaMat values be turned int standard deviation values: identity This is when standard deviation values are directly modeled by the params and thetaMat matrix variance This is when the params and thetaMat simulates the variance that are directly modeled by the thetaMat matrix log This is when the params and thetaMat simulates log(sd) nlmixrSqrt This is when the params and thetaMat simulates the inverse cholesky decomposed matrix with the x^2 modeled along the diagonal. This only works with a diagonal matrix. nlmixrLog This is when the params and thetaMat simulates the inverse cholesky decomposed matrix with the exp(x^2) along the diagonal. This only works with a diagonal matrix. nlmixrIdentity This is when the params and thetaMat simulates the inverse cholesky decomposed matrix. This only works with a diagonal matrix.
omegaLower	Lower bounds for simulated ETAs (by default -Inf)
omegaUpper	Upper bounds for simulated ETAs (by default Inf)
nSub	Number between subject variabilities (ETAs) simulated for every realization of the parameters.
thetaMat	Named theta matrix.
thetaDf	The degrees of freedom of a t-distribution for simulation. By default this is NULL which is equivalent to Inf degrees, or to simulate from a normal distribution instead of a t-distribution.
thetaIsChol	Indicates if the theta supplied is a Cholesky decomposed matrix instead of the traditional symmetric matrix.
nStud	Number virtual studies to characterize uncertainty in estimated parameters.
dfSub	Degrees of freedom to sample the between subject variability matrix from the inverse Wishart distribution (scaled) or scaled inverse chi squared distribution.
dfObs	Degrees of freedom to sample the unexplained variability matrix from the inverse Wishart distribution (scaled) or scaled inverse chi squared distribution.
returnType	This tells what type of object is returned. The currently supported types are: "rxSolve" (default) will return a reactive data frame that can change easily change different pieces of the solve and update the data frame. This is the currently standard solving method in RxODE, is used for rxSolve(object, ...), solve(object,...), "data.frame" -- returns a plain, non-reactive data frame; Currently very slightly faster than returnType="matrix" "matrix" -- returns a plain matrix with column names attached to the solved object. This is what is used object$run as well as object$solve "data.table" -- returns a data.table; The data.table is created by reference (ie setDt()), which should be fast. "tbl" or "tibble" returns a tibble format.
seed	an object specifying if and how the random number generator should be initialized
nsim	represents the number of simulations. For RxODE, if you supply single subject event tables (created with [eventTable()])
minSS	Minimum number of iterations for a steady-state dose
maxSS	Maximum number of iterations for a steady-state dose
infSSstep	Step size for determining if a constant infusion has reached steady state. By default this is large value, 420.
strictSS	Boolean indicating if a strict steady-state is required. If a strict steady-state is (TRUE) required then at least minSS doses are administered and the total number of steady states doses will continue until maxSS is reached, or atol and rtol for every compartment have been reached. However, if ODE solving problems occur after the minSS has been reached the whole subject is considered an invalid solve. If strictSS is FALSE then as long as minSS has been reached the last good solve before ODE solving problems occur is considered the steady state, even though either atol, rtol or maxSS have not been achieved.
istateReset	When TRUE, reset the ISTATE variable to 1 for lsoda and liblsoda with doses, like deSolve; When FALSE, do not reset the ISTATE variable with doses.
subsetNonmem	subset to NONMEM compatible EVIDs only. By default TRUE.
maxAtolRtolFactor	The maximum atol/rtol that FOCEi and other routines may adjust to. By default 0.1
from	When there is no observations in the event table, start observations at this value. By default this is zero.
to	When there is no observations in the event table, end observations at this value. By default this is 24 + maximum dose time.
by	When there are no observations in the event table, this is the amount to increment for the observations between from and to.
length.out	The number of observations to create if there isn't any observations in the event table. By default this is 200.
iCov	A data frame of individual non-time varying covariates to combine with the events dataset by merge.
keep	Columns to keep from either the input dataset or the iCov dataset. With the iCov dataset, the column is kept once per line. For the input dataset, if any records are added to the data LOCF (Last Observation Carried forward) imputation is performed.
indLinPhiTol	the requested accuracy tolerance on exponential matrix.
indLinPhiM	the maximum size for the Krylov basis
indLinMatExpType	This is them matrix exponential type that is use for RxODE. Currently the following are supported: Al-Mohy Uses the exponential matrix method of Al-Mohy Higham (2009) arma Use the exponential matrix from RcppArmadillo expokit Use the exponential matrix from Roger B. Sidje (1998)
indLinMatExpOrder	an integer, the order of approximation to be used, for the Al-Mohy and expokit values. The best value for this depends on machine precision (and slightly on the matrix). We use 6 as a default.
drop	Columns to drop from the output
idFactor	This boolean indicates if original ID values should be maintained. This changes the default sequentially ordered ID to a factor with the original ID values in the original dataset. By default this is enabled.
mxhnil	maximum number of messages printed (per problem) warning that T + H = T on a step (H = step size). This must be positive to result in a non-default value. The default value is 0 (or infinite).
hmxi	inverse of the maximum absolute value of H to are used. hmxi = 0.0 is allowed and corresponds to an infinite hmax1 (default). hminandhmximay be changed at any time, but will not take effect until the next change ofHis considered. This option is only considered withmethod="liblsoda"`.
warnIdSort	Warn if the ID is not present and RxODE assumes the order of the parameters/iCov are the same as the order of the parameters in the input dataset.
warnDrop	Warn if column(s) were supposed to be dropped, but were not present.
ssAtol	Steady state atol convergence factor. Can be a vector based on each state.
ssRtol	Steady state rtol convergence factor. Can be a vector based on each state.
safeZero	Use safe zero divide and log routines. By default this is turned on but you may turn it off if you wish.
sumType	Sum type to use for sum() in RxODE code blocks. pairwise uses the pairwise sum (fast, default) fsum uses Python's fsum function (most accurate) kahan uses Kahan correction neumaier uses Neumaier correction c uses no correction: default/native summing
prodType	Product to use for prod() in RxODE blocks long double converts to long double, performs the multiplication and then converts back. double uses the standard double scale for multiplication.
sensType	Sensitivity type for linCmt() model: advan Use the direct advan solutions autodiff Use the autodiff advan solutions forward Use forward difference solutions central Use central differences
linDiff	This gives the linear difference amount for all the types of linear compartment model parameters where sensitivities are not calculated. The named components of this numeric vector are: "lag" Central compartment lag "f" Central compartment bioavailability "rate" Central compartment modeled rate "dur" Central compartment modeled duration "lag2" Depot compartment lag "f2" Depot compartment bioavailability "rate2" Depot compartment modeled rate "dur2" Depot compartment modeled duration
linDiffCentral	This gives the which parameters use central differences for the linear compartment model parameters. The are the same components as linDiff
resample	A character vector of model variables to resample from the input dataset; This sampling is done with replacement. When NULL or FALSE no resampling is done. When TRUE resampling is done on all covariates in the input dataset
resampleID	boolean representing if the resampling should be done on an individual basis TRUE (ie. a whole patient is selected) or each covariate is resampled independent of the subject identifier FALSE. When resampleID=TRUE correlations of parameters are retained, where as when resampleID=FALSE ignores patient covariate correaltions. Hence the default is resampleID=TRUE.
maxwhile	represents the maximum times a while loop is evaluated before exiting. By default this is 100000
a	when using solve(), this is equivalent to the object argument. If you specify object later in the argument list it overwrites this parameter.
b	when using solve(), this is equivalent to the params argument. If you specify params as a named argument, this overwrites the output

Value

An “rxSolve” solve object that stores the solved value in a special data.frame or other type as determined by returnType. By default this has as many rows as there are sampled time points and as many columns as system variables (as defined by the ODEs and additional assignments in the RxODE model code). It also stores information about the call to allow dynamic updating of the solved object.

The operations for the object are similar to a data-frame, but expand the $ and [[""]] access operators and assignment operators to resolve based on different parameter values, initial conditions, solver parameters, or events (by updating the time variable).

You can call the eventTable() methods on the solved object to update the event table and resolve the system of equations.

Details

The rest of the document focus on the different ODE solving methods, followed by the core solving method's options, RxODE event handling options, RxODE's numerical stability options, RxODE's output options, and finally internal RxODE options or compatibility options.

References

"New Scaling and Squaring Algorithm for the Matrix Exponential", by Awad H. Al-Mohy and Nicholas J. Higham, August 2009

Roger B. Sidje (1998). EXPOKIT: Software package for computing matrix exponentials. ACM - Transactions on Mathematical Software 24(1), 130-156.

Hindmarsh, A. C. ODEPACK, A Systematized Collection of ODE Solvers. Scientific Computing, R. S. Stepleman et al. (Eds.), North-Holland, Amsterdam, 1983, pp. 55-64.

Petzold, L. R. Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations. Siam J. Sci. Stat. Comput. 4 (1983), pp. 136-148.

Hairer, E., Norsett, S. P., and Wanner, G. Solving ordinary differential equations I, nonstiff problems. 2nd edition, Springer Series in Computational Mathematics, Springer-Verlag (1993).

See also

RxODE()

Author

Matthew Fidler, Melissa Hallow and Wenping Wang