This example shows how to perform noncompartmental analysis to calculate NCA parameters and
estimate the tumor growth model [1] parameters from experimental
data using nonlinear regression in the **SimBiology Model Analyzer**
app.

The model used in this example is a SimBiology implementation of the
pharmacokinetic/pharmacodynamic (PK/PD) model by Simeoni et al. It quantifies the effect of
anticancer drugs on tumor growth kinetics from *in vivo* animal studies.
The drug pharmacokinetics are described by a two-compartment model with IV bolus dosing and
linear elimination (*ke*) from the *Central* compartment.
Tumor growth is described as a biphasic process with an initial exponential growth followed
by linear growth. The growth rate of the proliferating tumor cells is described by

$$\frac{{L}_{0}*{x}_{1}}{{\left[1+{\left(\frac{{L}_{0}}{{L}_{1}}*w\right)}^{\psi}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\psi $}\right.}}$$

.

Here, *L _{0}*,

*w* =
*x*_{1}

. In the presence of an anticancer
agent, it is assumed that a fraction of the proliferating cells is transformed into
non-proliferating cells. The rate of this transformation is assumed to be a function of the
drug concentration in the plasma and an efficacy factor
Some adjustments were made to the pharmacodynamics of tumor growth.

Instead of defining the tumor weight as the sum of

*x1*,*x2*,*x3*, and*x4*, the tumor weight is determined by the reaction named*Increase*,`null → tumor_weight`

, with the reaction rate $$\left(\raisebox{1ex}{$2*{L}_{1}*{L}_{0}*{x}_{1}{}^{2}$}\!\left/ \!\raisebox{-1ex}{${L}_{1}+2*{L}_{0}*{x}_{1}$}\right.\right)*tumor\_weight$$.*tumor_weight*is the total tumor weight,*x*is the weight of the proliferating tumor cells, and_{1}*L*, and_{0}*L*are tumor growth parameters._{1}Similarly, the decrease in tumor weight is defined by the reaction named

*Decay*,`tumor_weight → null`

, with the reaction rate

.*k**_{1}*x*_{4}*k*is the forward rate parameter and_{1}*x*is the last species in the series of transit reductions in tumor weight._{4}*ke*is defined as a function of the clearance and the volume of the central compartment:`ke = Cl_Central/Central`

.

The experimental (synthetic) data contains measurements from eight patients for three responses: measured drug concentrations in the central compartment, in the peripheral compartment, and measured tumor weight. The data also contains the dosing information, and each patient receives an IV dose at day 7.

The data set contains the following columns.

*ID*— Patient IDs*Time*— Times when measurements are taken*CentralConc*— Drug concentration in the central compartment*PeripheralConc*— Drug concentration in the peripheral compartment*Dose*— Dosing information for each patient

`NaN`

values are used whenever there is no measurement or no dose is
given.

Open the

**SimBiology Model Analyzer**app by typing`simBiologyModelAnalyzer`

at the command line or by clicking the app icon on the**Apps**tab.On the

**Home**tab of the app, select**Open**.Navigate to the folder

.*matlabroot*\examples\simbio\*matlabroot*is the folder where you have installed MATLAB. Select the project file named`tumor_growth_fitPKPD.sbproj`

. In the**Browser**pane, the**Models**folder contains the`Tumor Growth Model`

and the**Data1**folder contains the experimental data.Classify the measured responses of the data as the dependent variables. In the

**Browser**pane, click**Documents**and double-click**Datasheet1**.In the

**Data1**table, double-click**Classification**under*CentralConc*. Select`dependent`

. Repeat the same process for*PeripheralConc*and*TumorWeight*. Now all the data columns have proper classifications and the data is ready for use.

After you load the data, you can visualize the measured responses.

In the

**Browser**pane, click**Workspace**.Click

**Data1**.On the

**Home**tab, in the**Plot**section, click the**time**plot. The app generates a time plot of all three responses, namely:*CentralConc*,*PeripheralConc*, and*TumorWeight*.In the default time plot,

**Responses**correspond to the measured responses and are plotted using different line styles.**Scenarios**refers to different groups (eight patients) in the data and are plotted using different colors.### Tip

Plots are backed by data that are currently present in the app workspace. Plots are not snapshots. When the data (either experimental data or simulation results) is removed or changed, the plots are also updated according to the changes in the underlying data.

You can customize the plot to make it clearer. For example, you can plot the PD data
(*TumorWeight*) on a different axis than the PK data
(*CentralConc* and *PeripheralConc*). To do so, create two
diffferent groups (*sets*) of responses, where the first set contains only
*TumorWeight* and the second set contains *CentralConc* and
*PeripheralConc*.

Right-click

**TumorWeight (gram)**in the**Responses**table (the middle table in the**Property Editor**) and select`Create New Set`

. The app creates**Set 1**and**Set 2**.**Set 1**contains only*TumorWeight*, which is now plotted on a different axis than**Set 2**, which contains*CentralConc*and*PeripheralConc*.### Note

You can create a set only if there are

*two*or more responses.The

**Slice**table (the top table in the**Property Editor**) now contains**Sets**. This table is a summary table of all slicing variables that are currently present in the plot and their corresponding plot styles. In this current plot, the slicing variables are**Sets**,**Responses**,**Scenarios**, and**Dose**.### Tip

You can slice your data by using different slicing variables. Each slicing variable is shown in the plot using a different visual

*style*(or channel) such as color, line style, and axes position. Slicing variables can represent attributes of data, such as responses or scenarios (that is, groups or simulation runs). Slicing variables can also be covariates or parameter values associated with a scenario or group. By default, the slicing variables are provided for different response variables and different scenarios in the plotted data. Additional visual styles (or channels) can be added for sets of responses and associated parameter or covariate variables.

You can also group the responses based on different dose amounts that the patients receive. There are three different dose groups: 30, 75, and 150 mg.

In the

**Slice**table, at the**Dose**row, double-click the empty cell and select`Color`

. A red indicator appears because another slicing variable (**Scenarios**) has the same plot style. Clear the style (visual channel) for**Scenarios**by selecting empty.In the

**Dose**table (the bottom table), the app has automatically binned the dose amounts. Set**Number of Bins**to`3`

. You can now see that the dose amount has an impact on the tumor size. The higher the dose is, the smaller the tumor becomes.You can also query the corresponding dose group from each line by showing its data tip. Press

**Ctrl**and click a blue line to display its data tip. To remove it,**Ctrl****+****Click**again anywhere on the same line.

Using the drug pharmacokinetic data, you can estimate NCA parameters. NCA is model agnostic and can give insights into the drug pharmacokinetics without any underlying assumptions. You can use some of the NCA results as initial estimates when calibrating the model to the data, as discussed later in this example. For details on the list of available NCA parameters and their formulas, see Noncompartmental Analysis.

On the

**Home**tab, select**Program****>****Non-Compartmental Analysis**. A new program (**Program1**) appears.The

**Data***setup*step of the program defines the data set to use for the NCA analysis. In this example, the program automatically selects**Data1**.The

**NCA***execution*step defines the data column associations and algorithm details. In the**Definition**table, set**Concentration**to`CentralConc`

. Leave the other settings unchanged.On the

**Home**tab, click**Run**.

Once the NCA analysis is complete, the app saves the results in the
**LastRun** folder of the program by default.

On the

**Home**tab, select**New Datasheet**. A new datasheet (**Datasheet2**) opens on a new tab.In the

**Browser**pane, expand the**Program1**folder. Then expand**LastRun**folder. The NCA results are stored in the table named**results**.Drag

**results**onto**Datasheet2**. The calculated NCA parameter values are now shown in the datasheet. Use the scroll bar at the bottom of the table to scroll and view the NCA data columns. For details about calculated NCA parameters, see Noncompartmental Analysis.

You can export the NCA results to the MATLAB^{®} workspace and perform further data analyses at the command line.

Right-click

**results**. Select**Export Data to MATLAB Workspace**.The

**SimBiology Data Export**dialog box opens. Change the name of the variable to**ncatable**. Click**OK**.

After you export the data to the MATLAB workspace, you can analyze the data at the command-line. For instance, you can calculate the average drug clearance from the NCA data and use it as the model parameter value.

SimBiology provides different regression techniques to estimate model parameters based on
experimental data. This example details the steps for using the nonlinear regression method
`lsqnonlin`

(requires Optimization
Toolbox™) to fit the model to the data. If you do not have Optimization
Toolbox, the app uses `fminsearch`

instead. For the purposes of the
example, only some of the PK/PD model parameters are estimated, namely: *k1*,
*L0*, *L1*, *Cl_Central*,
*k12*, and *k21*.

From the

**Home**tab, select**Program****>****Fit Data**. A new program (**Program2**) appears on a new tab. The**Data**and**Model**steps have been prepopulated with**Data1**and**Tumor Growth Model**, respectively.By default, the

**Fit**step autogenerates plots after the fitting is complete. Disable the plot generation by clicking the plot icon at the top of the**Fit**program step for now. The plots will be explored later in the example.Define the mapping between the model components and the data columns from the experimental data. Specifically, map

*CentralConc*to the model species*Drug*in the central compartment (*Central.Drug*). Similarly, map*PeripheralConc*to*Peripheral.Drug*.Map

*TumorWeight*to*[Tumor Growth Model].tumor_weight*.Map the

*Dose*column to*Central.Drug*to indicate that the*Drug*species in the*Central*compartment is being dosed.Define the model parameters to estimate in the

**Estimated Parameters**table. Double-click the empty cell in the**Estimated Parameters**column and type*k1*. The app shows model components with matching names. Select`k1`

from the list.By default, the parameter is log-transformed as indicated by the expression

**exp(k1)**. You can change the expression to no transformation,`probit`

, or`logit`

transformation. For this example, keep the default log transform because it often improves the convergence. The**Initial Untransformed Value**is automatically set to the model value which is 0.5.Enforce the biological parameters to stay positive by specifying the

**Untransformed Lower Bound**and**Untransformed Upper Bound**as`1e-5`

and`10`

, respectively.Similarly, add the following parameters:

*Cl_Central*,*L0*,*L1*,*k12*, and*k21*.Select

**Pooled fit**to estimate one set of parameters for all patients (population fit). If you do not select**Pooled fit**, the app estimates one set of parameters for each patient (individual fit).The default error model is the constant error model. SimBiology supports constant, proportional, exponential, and combined error models. For details, see Error Models. For now, use the constant error model.

Keep the rest of the fitting settings unchanged. These settings are

Estimation Method — The default method is

`lsqnonlin`

if you have Optimization Toolbox. If you do not, the app uses`fminsearch`

.For more information, see Supported Methods for Parameter Estimation.

Algorithm Settings — Most common options for the estimation method. Click to expand the section and see the options. To see the description of each option, click the info icon to the right of the header.

Advanced Algorithm Settings — Advanced settings for the estimation method. The table is empty by default.

After you set the fitting options, you can run the **Fit** step.

At the top of the

**Fit**step, click the**Run this program step**button.By default, the

**Fit**step shows the progress of parameter estimation in a separate figure. The progress plot shows the live status of parameter estimation and fitting quality measures such as log likelihood. For details, see Progress Plot.The progress plot shows that the fit converged. You can close the progress plot.

If you are using

`fminsearch`

, the fit can fail to converge due to reaching the maximum number of iterations. You can increase**MaxIter**in the**Algorithm Settings**, but for the purposes of this example, you can continue completing the steps without doing so.

Once the parameter estimation is complete, the fit results are stored in the
**LastRun** folder of the program.

On the

**Home**tab, select**New Datasheet**. A new datasheet opens in a new tab.In the

**Browser**pane, expand the**Program2**folder. Then expand**LastRun**folder, which contains**results**and**simdataI**.**results**contains estimated parameter values and fit statistics.**simdataI**contains the simulated model responses for each individual (patient or group) using the estimated parameter values.Drag

**results**onto the new datasheet.The

**Statistics**table contains fit quality measures such as AIC, BIC, and log likelihood. These measures can be useful, for example, to compare the performance of different error models.In addition to quality statistics, you can also view various fit plots, such as actual versus predicted plots and residual distribution plots. First, select

**results**in the**Browser**pane. Available fit plots are automatically listed in the**Plot**section on the**Home**tab. Then select**Act vs Pred**from the list.The actual versus predicted plot appears on a separate tab. The predicted responses are plotted on the

*x*-axis and the observed (experimental) responses are plotted on the*y*-axis.You can change the plot to other supported plots by selecting one of the plots from the

**Style**section in the**Property Editor**. If you want the new plot on its own separate tab and you do not want to reuse the existing plot tab, select the plot from the**Plot**section on the**Home**tab.Change the plot to a residual distribution plot by selecting

**Res Dist**in the**Style**section.The plot shows whether the residuals for each response are normally distributed. In an ideal normal probability plot of residuals, the residuals line up along the diagonal line across the plot and the histograms indicate a normal fit. However, from the plot, the residuals for all three responses, especially

*CentralConc*and*PeripheralConc*, do not seem to be normally distributed. It could be an indication that the constant error model assumption is incorrect.

The following steps show how to change the error model to an exponential error model to fit the data again and compare the fit statistics of two different error models.

**Save Fit Results. **Before fitting the data again using the exponential error model, save the constant error
model result in a separate folder. Otherwise, the program, by default, overwrites results from
the **LastRun** folder every time you run the fit.

Right-click the

**LastRun**folder of the fit program in the**Browser**pane.Select

`Save Data`

.In the

**Save Data**dialog, enter`fit_constant`

as the data name.

**Rerun Fit With Exponential Error Model. **After you save the data, you can rerun the fit program with a different error
model.

Return to the fit program by clicking the

**Program2**tab. In the**Error Model**section, select**exponential**.At the top of the

**Fit**step, click the**Run this program step**button.Close the progress plot after the fit completes.

If you closed the datasheet (

**Datasheet3**) that contains the fit statistics from the previous fit, reopen the datasheet. In the**Browser**pane, click**Documents**. Then double-click**Datasheet3**.Click

**Workspace**to go back to the fit program results. Expand**LastRun**folder of the fit program. Then drag**results**onto**Datasheet3**. New columns (**Program2_LastRun**) containing the latest fit results are added next to the previous fit results (**Program2_fit_constant**).The

**Statistics**table compares the fit quality measures. From the comparison, both the AIC and BIC of the fit using the exponential error model are smaller than those of the previous fit. This indicates that the exponential error model fits the data better than the constant error model. The larger log likelihood of the exponential error model also indicates it is a better fit.Next, look at the residual distribution plot. Click

**results**from the**LastRun**folder. Then click**Residual Dist**from the**Plot**section on the**Home**tab.Compared to the residual distributions of the constant error model, the residual distributions from the exponential error model look more normal, indicating that the exponential error model indeed fits the data better.

Another way to assess the quality of fit results is to compute 95% confidence intervals for the estimated parameters and model predictions — that is, model simulation results using the estimated parameters. This step requires Statistics and Machine Learning Toolbox™.

Return to the fit program. Click the (+) icon at the upper left and select

`Confidence Interval`

. A**Confidence Interval**step appears following the**Fit**step.At the top of the

**Confidence Interval**step, disable the autogeneration of plots by clicking the plot icon. For both**Parameter Confidence Intervals**and**Prediction Confidence Intervals**, use the default method`gaussian`

and`95%`

confidence level. Click the**Run this program step**button to compute confidence intervals.For parameter confidence intervals, the supported methods are gaussian, profileLikelihood, and bootstrap.

For prediction confidence intervals, the supported methods are gaussian and bootstrap.

Once completed, the results are stored as

*parameterCI*and*predictionCI*in the**LastRun**folder of the program.*parameterCI*contains 95% confidence intervals for the estimated parameters.*predictionCI*contains 95% confidence intervals for the model predictions.Plot 95% confidence intervals for the estimated parameters. Click

*parameterCI*in the**Browser**pane and select**Confidence**in the**Plot**section.The confidence interval for each estimated parameter is shown in a new plot. The plot indicates the successful computation of the confidence intervals for all estimated parameters.

Depending on the outcome (status) of the confidence interval estimation, the app plots the results differently.

If the status of confidence interval estimation is a

*success*(as in the above plot), the app uses the first default color (blue) to plot a line and a centered dot for every parameter estimate. The app also plots a box to indicate the confidence intervals.If the status is

*constrained*or*estimable*, the app uses the second default color (red) and plots a line, centered dot, and box to indicate the confidence intervals.If the status is

*not estimable*, the app plots only a line and a centered cross in red.If there are any transformed parameters with estimated values that are 0 (for the

`log`

transform) and 0 or 1 (for the`probit`

or`logit`

transform), no confidence intervals are plotted for those parameter estimates.

For more details on the definitions of different statuses, see Parameter Confidence Interval Estimation Status.

You can also change the

**Layout**of the plot in**Plot Settings**.The

`'split'`

layout displays the confidence interval for each parameter estimate on a separate axis.The

`'grouped'`

layout displays all confidence intervals on one axis, grouped by parameter estimates. Each estimated parameter is separated by a vertical black line.

In both cases, the parameter bounds defined in the original fit are marked by square brackets. The app uses vertical dotted lines to group confidence intervals of parameter estimates that have been computed in a common fit.

Similarly, plot 95% confidence intervals for the model predictions. Click

*predictionCI*in the**Browser**pane and select**Confidence**in the**Plot**section.In this plot, the confidence interval for each group is plotted in a separate column, and each response is plotted in a separate row. The plot indicates the successful computation of the confidence intervals.

The plotting behavior differs depending on the outcome (status) of the confidence interval calculation.

If the status is

*constrained*or*not estimable*, the app uses the second default color (red) to plot the confidence intervals.Otherwise, the app uses the first default color (blue) and plots the confidence intervals as shaded areas (as in the above plot).

For details, see Gaussian Confidence Interval Calculation for Model Predictions and Bootstrap Confidence Interval Calculation.

Once the parameter estimation is complete, you can plot the simulation results on top of the experimental data. There are generally two ways to get the plot.

The first approach is to use the default **Fit** plot that shows the
fitted results over experimental data for each group.

Click

**results**from the**LastRun**folder of the fit program.Select

**Fit**in the**Plot**section on the**Home**tab.

The second approach is more customized and involved, but you have more control over how the plot looks like. The following steps show how to plot the tumor growth profiles of all groups on the same axes but use different line styles for experimental data and simulation data.

Click

**simdataI**in the**LastRun**folder of the fit program.Click

**time**from the**Plot**section on the**Home**tab.Drag

**Data1**(experimental data) on to the plot.In the

**Property Editor**of the plot, multiselect (**Ctrl****+****Click**) the first three rows (simulation data) of the**Responses**table and click**Create Set**.In the

**Slice**table, use no style for**Scenarios**by selecting empty.Clear the first two rows from each set. Keep only the tumor growth data from each set.

Use

`Line Style`

for**Sets**and`Color`

for**Response**. Also set the**Marker**of the simulation data (Program2) to`none`

. Set the**Style**of the experimental data (Data1) to`none`

. Select`o`

as the**Marker**.

After fitting to data, you can set the model values to the parameter estimates and perform other analyses. For instance, you can find out important model parameters using sensitivity analysis and vary the sensitive parameters to investigate model variability by using virtual patients.

[1] Simeoni, M., P. Magni, C. Cammia,
G. De Nicolao, V. Croci, E. Pesenti, M. Germani, I. Poggesi, and M. Rocchetti. 2004. Predictive
pharmacokinetic-pharmacodynamic modeling of tumor growth kinetics in xenograft models after
administration of anticancer agents. *Cancer Research*.
64:1094-1101.