The following steps show one of the workflows you can use at the command line to fit a PK model and estimate parameters:
Import data as shown in Importing Data.
Specify the structural model by creating a PK model as shown in Create a Pharmacokinetic Model Using the Command Line. Alternatively, if you have a SimBiology^{®} model that you want to use in fitting, see Prerequisites for Using Custom SimBiology Models in Data Fitting.
Classify the data set to use in fitting. See Specify and Classify the Data to Fit.
Specify the initial guesses for the parameters to be estimated, as shown in Set Initial Estimates.
Perform individual or population fits:
For individual fits:
(Optional) Specify an error model or weights. See Specify an Error Model.
(Optional) Set tolerances.
(Optional) Specify maximum iterations.
(Optional) Specify to pool the data, which simultaneously fits data from multiple dose levels using the same model parameters for each dose.
For population fits:
Specify the statistical model:
Specify the covariate model and the covariance matrix. See Specify a Covariate Model and Specify the Covariance Pattern of Random Effects.
(Optional) Specify the error model. See Specify an Error Model.
(Optional) Set tolerances.
(Optional) Specify maximum iterations.
Obtain and visualize results.
In order to use the imported data in fitting, you must identify required columns in the data set that was previously imported as shown in Importing Data.
Use the PKData
object to specify the data
set containing the observed data to use in fitting. The properties
of the PKData
object specify what each column in
the data represents.
To create the PKData
object:
Create the PKData
object for the
data set data
.
pkDataObject = PKData(data);
PKData
assigns
the data set data
to the readonly DataSet
property.Use the column headers in the data set to specify the following properties for the column in the data set.
Column in Data Set Represents  PKData Object Property 

Group identification labels  GroupLabel 
Independent variable (For example, time)  IndependentVarLabel 
Dependent variable (For example, measured response)  DependentVarLabel 
Amount of dose given  DoseLabel 
Rate of infusion (when applicable). Data must contain rate
(amount/time ) and not infusion time.  RateLabel 
Covariates (For example, age, gender, weight)  CovariateLabels 
For example, for the tobramycin data set [1]:
pkDataObject.GroupLabel = 'ID'; pkDataObject.IndependentVarLabel = 'Time'; pkDataObject.DependentVarLabel = 'Response'; pkDataObject.DoseLabel = 'Dose'; pkDataObject.CovariateLabels = {'WT','HT','AGE', 'SEX', 'CLCR'};
Note:
For the subset of data belonging to a single group (as defined
by the column in your data set that represents group identification
labels, which you map to the

Tip:
If dosing applies to more than one compartment in the model,
specify the pkDataObject.DoseLabel = {'Dose1', 'Dose2'}; Dose1 and Dose2 are
names of columns containing dose information for compartments. A onetoone
relationship must exist between the number and order of elements in
the DoseLabel property and the Dosed property
of the corresponding PKModelMap object. 
Tip:
If your model measures multiple responses, specify the pkDataObject.DependentVarLabel = {'Response1', 'Response2'}; Response1 and Response2 are
names of columns containing response measurements. A onetoone relationship
must exist between the number and order of elements in the DependentVarLabel property
and the Observed property of the corresponding PKModelMap object. 
When you assign a column containing group identification labels
to the GroupLabel
property, PKData
sets
these readonly properties as follows:
The GroupNames
property is set
to the unique names found in the group column.
The GroupID
property is set to
an integer corresponding to the unique names found in the group column.
If you specify a stochastic solver and options in the Configset object
associated with your
model, be aware that during fitting SimBiology temporarily changes:
SolverType
property to the default
solver of ode15s
SolverOptions
property to the options
last configured for a deterministic solver
Caution: If your model includes active variants that specify alternate values for the parameters to estimate, the variants are ignored for those parameters during fitting. 
To set the initial estimates (or initial guesses) for the parameters
with fixed effects to estimate, first identify the sequence of the
parameters in the model by querying the PKModelMap
object.
Next, construct a vector, beta0
, containing the
initial conditions. For information about PKModelMap
objects,
see step 4 in Create a Pharmacokinetic Model Using the Command Line.
Query the Estimated
property of
the PKModelMap
object:
PKModelMapObj.Estimated
MATLAB^{®} returns the sequence of the parameters to be estimated. For example:
ans = 'Central' 'Cl_Central'
Set the initial estimates for the parameters. For example:
beta0 = [10.0, 1.0];
For information on ...  See ... 

The parameters added to the model  
Default units for the above parameters  Unit Conversion for Imported Data 
Suppose data for a nonlinear regression model falls into one of m distinct groups i = 1, ..., m. (Specifically, suppose that the groups are not nested.) To specify a general, nonlinear, mixedeffects (NLME) model for this data:
Define groupspecific model parameters φ_{i} as linear combinations of fixed effects β and random effects b_{i}.
Define response values y_{i} as a nonlinear function f of the parameters and groupspecific covariate variables X_{i}.
The model is:
$$\begin{array}{l}{\phi}_{i}={A}_{i}\theta +{B}_{i}{\eta}_{i},where\text{}{\eta}_{i}\sim N(0,\Psi )\\ {y}_{i}=f({\phi}_{i},{X}_{i})+{\epsilon}_{i}\\ Alternatively,\mathrm{log}{y}_{i}=\mathrm{log}f({\phi}_{i},{X}_{i})+{\epsilon}_{i}\end{array}$$
This formulation of the nonlinear, mixedeffects model uses the following notation:
φ_{i}  A vector of groupspecific model parameters 
θ  A vector of fixed effects, modeling population parameters 
η_{i}  A vector of multivariate, normally distributed, groupspecific, random effects 
A_{i}  A groupspecific design matrix for combining fixed effects 
B_{i}  A groupspecific design matrix for combining random effects 
X_{i}  A data matrix of groupspecific covariate values 
y_{i}  A data vector of groupspecific response values 
f  A general, realvalued function of φ_{i} and X_{i} 
ε_{i} 

Ψ  A covariance matrix for the random effects 
σ^{2}  The error variance, assumed to be constant across observations 
For example, consider a onecompartment model with firstorder dosing and linear clearance. The groupspecific parameters (φ) in the model are clearance (Cl), compartment volume (V), and absorption rate constant (k_{a}). From the model:
$$\left(\begin{array}{c}Cl\\ V\\ ka\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}\theta Cl\\ \theta V\\ {\theta}_{ka}\end{array}\right)+\text{\hspace{0.17em}}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}\eta Cl\\ \eta V\\ {\eta}_{ka}\end{array}\right)$$
In SimBiology, B_{i} is
an identity matrix. That is, sbionlmefit
does not
support the specification of a different randomeffects design matrix.
You can alter the design matrices, as necessary, to introduce weighting
of individual effects.
The Statistics and Machine Learning Toolbox™ function nlmefit
fits
the general, nonlinear, mixedeffects model to data, estimating the
fixed and random effects. The function also estimates the covariance
matrix Ψ for the random effects.
Additional diagnostic outputs allow you to assess tradeoffs between
the number of model parameters and the goodness of fit. See MixedEffects Models (Statistics and Machine Learning Toolbox) for
more information.
Construct a CovariateModel
object
to define the relationship between parameters and covariates. After
constructing the object, modify the FixedEffectValues
property
of the object before using the object as an input argument to sbionlmefit
or sbionlmefitsa
,
to estimate nonlinear mixed effects.
If the NLME model assumes a groupdependent covariate such as weight (w), the model becomes:
$$\left(\begin{array}{c}Cl\\ V\\ ka\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& wi\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right)\left(\begin{array}{c}\theta Cl\\ \theta V\\ \begin{array}{l}{\theta}_{ka}\\ {\theta}_{Cl/w}\end{array}\end{array}\right)+\text{\hspace{0.17em}}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}\eta Cl\\ \eta V\\ {\eta}_{ka}\end{array}\right)$$
Thus, the parameter for clearance (Cl) for an individual is Cl_{i} = θ_{Cl} + θ_{Cl/w} * w_{i} + η_{Cli}.
Use the following procedure to specify a covariate model. If you are using the tobramycin data set, make sure you first complete the following procedures:
Use the CovariateModel
constructor
function to construct an empty CovariateModel
object:
covModel = CovariateModel;
Set the Expression
property
of the object to define the relationship between parameters and covariates
in the CovariateModel
object, where Cl
, v
,
and ka
are parameters, weight
is
a covariate, theta1
, theta2
, theta3
,
and theta4
are fixed effects, and eta1
, eta2
,
and eta3
are random effects.
covModel.Expression = {'Cl = exp(theta1 + theta4*weight + eta1)',... 'v = exp(theta2 + eta2)',... 'ka = exp(theta3 + eta3)'};
Display a list of the descriptions of the fixed effects
(theta1
and theta2
) in the CovariateModel
object:
disp('Fixed Effects Descriptions:'); disp(covModel.FixedEffectDescription)
Your output appears as follows, where each character vector describes the role of a fixed effect in the expression equation:
Fixed Effects Descriptions: 'Cl' 'v' 'ka' 'Cl/weight'
Use the constructDefaultFixedEffectValues
method
of the CovariateModel
object to create a structure
containing the initial estimates for the fixed effects in the object.
The initial estimates in this structure are set to a default of zero:
initialEstimates = covModel.constructDefaultFixedEffectValues
Your output appears as:
initialEstimates = theta1: 0 theta2: 0 theta3: 0 theta4: 0
Edit the initialEstimates
structure
to set the initial estimates of the fixed effects:
initialEstimates.theta1 = 1.408; initialEstimates.theta2 = 0.061; initialEstimates.theta3 = 0.31;
Tip: Typically, these initial estimates are values you determine from a previous fit of the data. 
Use the modified initialEstimates
structure
to update the FixedEffectValues
property
of the CovariateModel
object:
covModel.FixedEffectValues = initialEstimates;
Now covModel
, the CovariateModel
object,
is ready to submit as an input argument to sbionlmefit
or sbionlmefitsa
.
By default, the function you use to perform population fits
(nlmefit
or nlmefitsa
) assumes
a diagonal covariance matrix (no covariance among the random effects).
To specify a different covariance pattern of random effects, use the 'CovPattern'
option.
In the previous
example, assuming that each of the parameters has random effects
and that Cl and V exhibit covariance,
the covariance pattern of random effects would be a logical array:
$$\left(\begin{array}{ccc}1& 1& 0\\ 1& 1& 0\\ 0& 0& 1\end{array}\right)$$
Create an options struct
with the
specified covariance pattern:
options.CovPattern = [1, 1, 0; 1, 1, 0; 0, 0, 1];
Specify the arguments for sbionlmefit
or sbionlmefitsa
:
[results, simdataI, simdataP] = sbionlmefit(modelObj,... PKModelMapObj, pkDataObject, beta0, options)
If you are using the tobramycin data set [1], do the following:
Create an options struct
with the
specified covariance pattern:
options.CovPattern = [1, 0; 0, 1];
Specify the arguments for sbionlmefit
:
[results, simdataI, simdataP] = sbionlmefit(modelObj,... PKModelMapObj, pkDataObject, beta0, options)
results = NLMEResults handle Properties: FixedEffects: [2x3 dataset] RandomEffects: [97x2 dataset] IndividualParameterEstimates: [97x2 dataset] PopulationParameterEstimates: [97x2 dataset] RandomEffectCovarianceMatrix: [2x2 dataset] EstimatedParameterNames: {2x1 cell} CovariateNames: {'WT' 'HT' 'AGE' 'SEX' 'CLCR'} FixedEffectsStruct: [1x1 struct] stats: [1x1 struct]
results.FixedEffects
ans = Description Estimate StandardError 'Central' 3.0478 0.064369 'Cl_Central' 1.3054 0.061095
For more information, see nlmefit
or nlmefitsa
.
Fitting the model and estimating the covariance matrix Ψ often leads to further refinements. A relatively small estimate for the variance of a random effect suggests that it can be removed from the model. Similarly, relatively small estimates for covariances among certain random effects suggest that a full covariance matrix is unnecessary. Since random effects are unobserved, Ψ must be estimated indirectly. Specifying a diagonal or blockdiagonal covariance pattern for Ψ can improve convergence and efficiency of the fitting algorithm.
You can specify any of the following error models when using
the sbionlinfit
, sbionlmefit
,
or sbionlmefitsa
function. Each model defines the
error using a standard normal (Gaussian) variable e,
the function value f, and one or two parameters, a and b.
The default error model is 'constant'
.
'constant'
: y = f + a*e (default)
'proportional'
: y = f + b*abs(f)*e
'combined'
: y = f +
(a+b*abs(f))*e
'exponential'
: y = f*exp(a*e),
or equivalently log(y) = log(f)
+ a*e
To define an error model:
Create an optionStruct
input argument
and set the ErrorModel
field to specify one of
the above error models. For example:
optionStruct.ErrorModel = 'proportional';
Specify the optionStruct
input
argument for sbionlinfit
, sbionlmefit
,
or sbionlmefitsa
, as shown in Perform Individual Fitting or Perform Population Fitting.
To specify parameter transformations, use the ParamTransform
option
in sbionlinfit
, sbionlmefit
and sbionlmefitsa
.
The ParamTransform
option lets you specify either
no transformation, or the log
, probit
,
or logit
transformation.
Note:
Do not use the 
The underlying algorithm in nlmefit
assumes
that parameters follow a normal distribution. This assumption may
not hold for biological parameters that are constrained to be positive,
such as volume and clearance. You may specify a transformation function
for the estimated parameters, so that the transformed parameters follow
a normal distribution.
By default, the SimBiology fitting functions choose a log
transform for all estimated parameters. Parameters that are constrained
between the values 0
and 1
,
like absorption fraction, can be transformed by the probit
or logit
transformations
described below.
The probit
function is the inverse cumulative
distribution function (CDF) associated with the standard normal distribution.
To apply the probit
transform to a variable x
in MATLAB,
use the Statistics and Machine Learning Toolbox function norminv
: t
= norminv(x)
. To untransform a variable t
,
use the function normcdf
: x = normcdf(t)
.
The logit
function is the inverse of the
sigmoid function. To apply the logit
transform
to a variable x
in MATLAB, use the following
expression: t = log(x)  log(1x)
. To untransform
the variable t
, use x = 1/(1+exp(t))
.
For the ParamTransform
option,
specify a vector of values equal to the number of parameters to be
estimated. The values must be one of the integer codes listed in nlmefitsa
or nlmefit
specifying
the transformation for the corresponding value of the parameters to
be estimated. For example
options.ParamTransform = [0 1 2];
Specify the arguments for sbionlmefit
or sbionlmefitsa
,
as shown in Perform Population Fitting.
For individual fitting, see Perform Individual Fitting.
The sbionlmefit
and sbionlmefitsa
functions let you specify
a SimBiology model that you want to use in fitting. These functions
use the nlmefit
and nlmefitsa
functions from the Statistics and Machine Learning Toolbox to
fit data with both fixed and random sources of variation using nonlinear
mixedeffects and return the estimates. Both nlmefit
and nlmefitsa
fit
the model by maximizing an approximation to the marginal likelihood
with random effects integrated out assuming the following:
Random effects are multivariate, normally distributed, and independent between groups.
Observation errors are independent, identically normally distributed, and independent of random effects.
(Optional) Set the tolerance or maximum iteration
options. Use an options structure that is an input argument for sbionlmefit
or sbionlmefitsa
:
optionStruct.Options.TolX = 1.0E4; optionStruct.Options.TolFun = 1.0E4; optionStruct.Options.MaxIter = 200;
Specify the model object, the PKModelMap
object,
the PKData
object, the PKCovariateModel
object,
a vector containing the initial estimates for the fixed effects, and
the options:
[results, simdataI, simdataP] = sbionlmefit(modelObj,... PKModelMapObj, pkDataObject, CovariateModelObject, beta0, optionStruct);
Note:
If your population fit uses multiple doses, make sure each element
in the 
Note:
In your

sbionlmefit
and sbionlmefitsa
return
the following:
results
, an object containing
estimated values and other statistics. For more information, see the sbionlmefit
and sbionlmefitsa
reference
pages.
simdataI
, a SimData object
containing the data from
simulating the model using the estimated parameter values for individuals,
which includes both the fixed and random effects.
simdataP
, SimData object
containing the data from
simulating the model using the estimated parameter values for the
population, which includes only the fixed effects.
Plot the data from the data set. For example, in the
imported data set used for fitting, ds
, ID
, Time
,
and Response
are the column headers for the columns
containing group IDs, time, and the response variable, respectively.
p = sbiotrellis(ds, 'ID', 'Time', 'Response')
Note:
If your data set has multiple responses, with column headers Response = {'Response1', 'Response2'} p = sbiotrellis(ds, 'ID', 'Time', Response) 
Use the plot
method on the trellis
plot object p
, returned by sbiotrellis
to
overlay data, using default values for the second and third input
arguments.
p.plot(simdataP, [], '', PKModelMapObj.Observed);
For a description of the results, see sbionlmefit
.
For more information, see the following topics in the Statistics and Machine Learning Toolbox documentation:
Nonlinear Regression (Statistics and Machine Learning Toolbox)
MixedEffects Models (Statistics and Machine Learning Toolbox)
The sbiofitstatusplot
function dynamically
plots the progress of the fitting task. During the task, the function
plots the fixed effects (β), the estimates for the diagonal
elements of the covariance matrix for the random effects (Ψ),
and the loglikelihood. This functionality is useful for large and
complex models when you expect the time to return the results to be
longer than a few minutes. Use the options structure that is an argument
for the sbionlmefit
function:
% Create options structure with 'OutputFcn'. options.Options.OutputFcn = @sbiofitstatusplot; % Pass options structure with OutputFcn to sbionlmefit function. results = sbionlmefit(..., options);
The following figure shows the type of plots obtained.
Tips for interpreting status plots:
The fitting function tries to maximize the loglikelihood. When the plot begins to display a flat line, this might indicate that maximization is complete. Try setting the maximum iterations to a lower number to reduce the number of iterations you need and improve performance. For information on how to set iteration options, see Perform Population Fitting.
Plots for the fixed effects (β) and the estimates for the diagonal elements of the covariance matrix for the random effects (Ψ), should show convergence. If you see oscillations, or jumps without accompanying improvements in the loglikelihood, the model may be overparameterized. Try the following:
Reduce the number of fixed effects.
Reduce the number of random effects.
Simplify the covariance matrix pattern of random effects.
See also sbiofitstatusplot
.
When performing population fitting using nonlinear regression, you can simultaneously fit data from multiple dose levels by either:
Using sbionlmefit
with
a CovariateModel
object input argument and omitting
the random effect (eta
) from the expressions in
the CovariateModel
object.
Using sbionlmefit
with an InitEstimates
input
argument and setting the REParamsSelect
field or
namevalue pair input argument to a 1byn logical
vector, with all entries set to false
, where n equals
the number of fixed effects.
The sbionlinfit
function
lets you specify a SimBiology model to fit using the nlinfit
function
(individual fit). The nlinfit
function uses nonlinear
least squares and returns parameter estimates, residuals, and the
estimated coefficient covariance matrix.
(Optional) Specify an error model, set the tolerance,
set the maximum iteration, or set the data pooling option, which lets
you simultaneously fit data from multiple dose levels using the same
model parameters for each dose. Use an options structure that is an
input argument for sbionlinfit
:
optionStruct.ErrorModel = 'proportional'; optionStruct.TolX = 1.0E8; optionStruct.TolFun = 1.0E8; optionStruct.MaxIter = 100; optionStruct.Pooled = true;
Specify the model object, the PKModelMap
object,
the PKData
object, a vector containing the initial
estimates for the fixed effects, and the options:
[results, simdataI] = sbionlinfit(modelobj,... PKModelMapObj, PKDataObj, beta0, optionStruct);
Note:
If your individual fit uses multiple doses, ensure each element
in the 
Note:
In your

sbionlinfit
returns the following:
A results
array of objects,
with each object containing the following for one group:
ParameterEstimates
— A dataset (Statistics and Machine Learning Toolbox)
array
containing fitted coefficients and their standard errors.
CovarianceMatrix
— Estimated
covariance matrix for the fitted coefficients.
beta
— Vector of scalars
specifying the fitted coefficients in transformed space.
R
— Residuals.
J
— Jacobian of modelObject
.
COVB
— Estimated covariance
matrix for the transformed coefficients.
mse
— Scalar specifying
the estimate of the error of the variance term.
errorparam
— Estimated parameters
of the error model or an empty array if you specified weights using
the 'Weights'
namevalue pair argument.
simdataI
, a SimData
object
containing the data from simulating the model using the estimated
parameter values, for individuals.
Plot the data from the data set. For example, in the
imported data set (ds
), ID
, Time
and Response
are
the column headers for the columns containing group IDs, time, and
the response variable respectively.
p = sbiotrellis(ds, 'ID', 'Time', 'Response')
Note:
If your data set has multiple responses, with column headers Response = {'Response1', 'Response2'} p = sbiotrellis(ds, 'ID', 'Time', Response) 
Use the plot method on the trellis plot object p
,
returned by sbiotrellis
to overlay data, using
default values for the second and third input arguments.
p.plot(simdataI, [], '', PKModelMapObj.Observed);
For more information, see Nonlinear Regression (Statistics and Machine Learning Toolbox) and nlinfit
.