Nonlinear Regression

What is Nonlinear Regression?

The purpose of regression models is to describe a response variable as a function of independent variables. Multiple linear regression models describe the response as a linear combination of coefficients and functions of independent variables. Nonlinearities can be modeled using nonlinear functions of independent variables. However, the coefficients always enter the model in a linear fashion.

Nonlinear regression models are more mechanistic models of nonlinear relationships between the response and independent variables. The parameters can enter the model as exponential, trigonometric, power, or any other nonlinear function. The unknown parameters in the model are estimated by minimizing a statistical criterion such as the negative log likelihood or the sum of squared deviations between observed and predicted values.

In the case of pharmacokinetic (PK) studies, the response data usually represent some measured drug concentrations, and independent variables are often dose and time. The nonlinear function often used for such data is an exponential function since many drugs once distributed in a patient are eliminated in an exponential fashion. One PK parameter to estimate in this case is the rate at which the drug is eliminated from the body given the concentration-time data.

For instance, consider drug plasma concentration data from a single individual after an intravenous bolus dose measured at different time points with some errors. Assume the measured drug concentration follows a monoexponential decline: $$C_t=C_0{e}^{-k_{e}t}+a\epsilon$$.

This model describes the time course of drug concentration in the body (C_t), as a function of the drug concentration after an intravenous bolus dose at t = 0 (C₀), time (t), and elimination rate parameter (k_e). ε is the mean-zero and unit-variance variable, that is, $$\epsilon \sim N(0,1)$$ representing the measurement error and a is the error model parameter (here, standard deviation).

More generically, you can write the model as

$$y_i=f(t;p)+g({\epsilon }_i)$$

where y_i is the ith response (such as drug concentration), f is a function of time t and model parameters p (such as k_e), and an error model $$g({\epsilon }_i)$$.

Fitting Options in SimBiology

This table summarizes nonlinear regression options available in SimBiology^®.

Fitting Option	Example
Individual-specific parameter estimation (Unpooled fitting) Fit each individual separately, resulting in one set of parameter estimates for each individual.
Category- or group-specific parameter estimation Fit each category or group separately, resulting in one set of parameter estimates for each category.
Population-wide parameter estimation (Pooled fitting) Fit all of the data pooled together, resulting in just one set of parameter estimates.

In addition, SimBiology supports four kinds of error models for measured or observed responses, namely, constant (default), proportional, combined, and exponential. For details, see Error Models. Depending on the optimization method, you can specify an error model for each response or all responses. For details, see Supported Methods for Parameter Estimation in SimBiology.

Parameter Transformations

SimBiology supports three parameter transformations. These parameter transformations can be useful to improve fitting convergence or to enforce parameter bounds.

The general model explained previously is $$y_i=f(t;p)+g({\epsilon }_i)$$, where p is model parameters that you can transform. Consider the following two equations.

$$\beta =T\left(p\right)$$

$$p=T^{-1}\left(\beta \right)$$

Here, $$\beta$$ represents transformed model parameters, p represents untransformed model parameters, T is the transformation, and T-1 is the inverse transformation.

SimBiology performs parameter estimation using the transformed parameters β, which means that the transformed model $$F\left(t;\beta \right)=f\left(t;T^{-1}\left(\beta \right)\right)$$ is used, where F is the model function using the transformed parameters. Equivalently, the model function can be rewritten as $$f\left(t;p\right)=F\left(t;T\left(p\right)\right)$$.

In other words, the SimBiology optimizer uses the transformed values during maximum likelihood estimation but the reported fit result is reverted back to the model space (untransformed values). For example, if you are estimating a clearance parameter Cl that is log-transformed, you have Cl_β = log(Cl), where Cl_β is what the optimizer uses and Cl is what the model sees.

Specifying parameter transformations imposes implicit bounds on the untransformed parameter values. The log transformation keeps the parameter value to be always positive, and the logit and probit transformations keep the parameter value to between 0 and 1. Alternatively, you can specify further constraints on the parameter values by providing explicit bounds for untransformed parameters p or for the transformed parameters β.

Transformation	Transformed Parameter and Range†	Untransformed Parameter and Range‡	Description
log	$$\beta =\log \left(p\right)\in [-Inf,Inf]$$	$$p=\exp \left(\beta \right)\in [0,Inf]$$	In many cases, the log transformation is useful for parameters, such as rate constants, whose values might span several orders of magnitude and exploring the parameter space for such parameters. The log transformation can be also useful for positive physical quantities, such as clearance or compartment volume. Applying the log transformation imposes an implicit bound on the untransformed parameter so that its value is always positive.
logit	$$\beta =\text{logit(p) = log}\left(\frac{p}{1-p}\right)\in [-Inf,Inf]$$	$$p=1/\left(1+\exp \left(-\beta \right)\right)\in [0,1]$$	The logit transformation imposes an implicit bound (between 0 and 1) on the untransformed parameter value. It can be useful for parameters that have values only between 0 and 1, such as bioavailability. Alternatively, you can specify the parameter bounds (using an EstimatedInfo object) instead of specifying the logit transformation.
probit	$$\beta =\text{norminv}\left(p\right)\in [-Inf,Inf]$$	$$p=\text{normcdf}\left(\beta \right)\in [0,1]$$	Similar to logit, the probit transformation imposes an implicit bound (between 0 and 1) on the untransformed parameter value. SimBiology uses the normal inverse cumulative distribution function norminv (Statistics and Machine Learning Toolbox). The probit transformation requires Statistics and Machine Learning Toolbox™.

† Use the InitialTransformedValue and TransformedBounds properties of an EstimatedInfo object to set the initial transformed value and transformed bounds to the desired subset of the range.

‡ Use the InitialValue and Bounds properties of an EstimatedInfo object to set the initial untransformed value and untransformed bounds to the desired subset of the range.

Maximum Likelihood Estimation

SimBiology estimates parameters by the method of maximum likelihood. Rather than directly maximizing the likelihood function, SimBiology constructs an equivalent minimization problem. Whenever possible, the estimation is formulated as a weighted least squares (WLS) optimization that minimizes the sum of the squares of weighted residuals. Otherwise, the estimation is formulated as the minimization of the negative of the logarithm of the likelihood (NLL). The WLS formulation often converges better than the NLL formulation, and SimBiology can take advantage of specialized WLS algorithms, such as the Levenberg-Marquardt algorithm implemented in lsqnonlin and lsqcurvefit. SimBiology uses WLS when there is a single error model that is constant, proportional, or exponential. SimBiology uses NLL if you have a combined error model or a multiple-error model, that is, a model having an error model for each response.

sbiofit supports different optimization methods, and passes in the formulated WLS or NLL expression to the optimization method that minimizes it. For simplicity, each expression shown below assumes only one error model and one response. If there are multiple responses, SimBiology takes the sum of the expressions that correspond to error models of given responses.

	Expression that is being minimized
Weighted Least Squares (WLS)	For the constant error model, $${\displaystyle \sum _i^N{\left(y_i-f_i\right)}^2}$$
	For the proportional error model, $${\displaystyle \sum _i^N\frac{{\left(y_i-f_i\right)}^2}{f_i^2/f_{gm}^2}}$$
	For the exponential error model, $${\displaystyle \sum _i^N{\left(\ln y_i-\ln f_i\right)}^2}$$
	For numeric weights, $${\displaystyle \sum _i^N\frac{{\left(y_i-f_i\right)}^2}{w_{gm}/w_i}}$$
Negative Log-likelihood (NLL)	For the combined error model and multiple-error model, $${\displaystyle \sum }_i^N\frac{{\left(y_i-f_i\right)}^2}{2{\sigma }_i^2}+{\displaystyle \sum }_i^N\ln \sqrt{2\pi {\sigma }_i^2}$$

The variables are defined as follows.

N	Number of experimental observations
yi	The ith experimental observation
$f_i$	Predicted value of the ith observation
${\sigma }_i$	Standard deviation of the ith observation. For the constant error model, $${\sigma }_i=a$$ For the proportional error model, $${\sigma }_i=b\left|f_i\right|$$ For the combined error model, ${\sigma }_i=a+b\left|f_i\right|$
$$f_{gm}$$	$$f_{gm}={\left({\displaystyle \prod _i^N\left|f_i\right|}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$N$}\right.}$$
$$w_i$$	The weight of the ith predicted value
$$w_{gm}$$	$$w_{gm}={\left({\displaystyle \prod _i^N{w}_i}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$N$}\right.}$$

When you use numeric weights or the weight function, the weights are assumed to be inversely proportional to the variance of the error, that is, $${\sigma }_i^2=\frac{a^2}{w_i}$$ where a is the constant error parameter. If you use weights, you cannot specify an error model except the constant error model.

Various optimization methods have different requirements on the function that is being minimized. For some methods, the estimation of model parameters is performed independently of the estimation of the error model parameters. The following table summarizes the error models and any separate formulas used for the estimation of error model parameters, where a and b are error model parameters and e is the standard mean-zero and unit-variance (Gaussian) variable.

Error Model	Error Parameter Estimation Function
'constant': $$y_i=f_i+ae$$	$$a^2=\frac{1}{N}{\displaystyle \sum _i^N{\left(y_i-f_i\right)}^2}$$
'exponential': $$y_i=f_i\exp (ae)$$	$$a^2=\frac{1}{N}{\displaystyle \sum _i^N{\left(\ln y_i-\ln f_i\right)}^2}$$
'proportional': $$y_i=f_i+b\left|f_i\right|e$$	$b^2=\frac{1}{N}{\displaystyle \sum _i^N{\left(\frac{y_i-f_i}{f_i}\right)}^2}$
'combined': $$y_i=f_i+\left(a+b\left|f_i\right|\right)e$$	Error parameters are included in the minimization.
Weights	$$a^2=\frac{1}{N}{\displaystyle \sum _i^N{\left(y_i-f_i\right)}^2}{w}_i$$

Note

nlinfit only support single error models, not multiple-error models, that is, response-specific error models. For a combined error model, it uses an iterative WLS algorithm. For other error models, it uses the WLS algorithm as described previously. For details, see nlinfit (Statistics and Machine Learning Toolbox).

Fitting Workflow

The following steps show one of the workflows you can use at the command line to fit a PK model.

1. Import data.

2. Convert the data to the groupedData format.

3. Define dosing data. For details, see Doses in SimBiology Models.

4. Create a structural model (one-, two-, or a multicompartment model). For details, see Create Pharmacokinetic Models.

5. Map the response variable from data to the model component. For example, if you have the measured drug concentration data for the central compartment, then map it to the drug species in the central compartment (typically the Drug_Central species).

6. Specify parameters to estimate using an EstimatedInfo object. Optionally, you can specify parameter transformations, initial values, and parameter bounds.

7. Perform parameter estimation using sbiofit or fitproblem.

For illustrated examples, see the following.

• Fit PK Parameters Using SimBiology Problem-Based Workflow

• Fit One-Compartment Model to Individual PK Profile

• Fit Two-Compartment Model to PK Profiles of Multiple Individuals

• Estimate Category-Specific PK Parameters for Multiple Individuals