Model verification tools: a computational framework for verification assessment of mechanistic agent-based models

Abstract

Background
Nowadays, the inception of computer modeling and simulation in life science is a matter of fact. This is one of the reasons why regulatory authorities are open in considering in silico trials evidence for the assessment of safeness and efficacy of medicinal products. In this context, mechanistic Agent-Based Models are increasingly used. Unfortunately, there is still a lack of consensus in the verification assessment of Agent-Based Models for regulatory approval needs. VV&UQ is an ASME standard specifically suited for the verification, validation, and uncertainty quantification of medical devices. However, it can also be adapted for the verification assessment of in silico trials for medicinal products.

Results
Here, we propose a set of automatic tools for the mechanistic Agent-Based Model verification assessment. As a working example, we applied the verification framework to an Agent-Based Model in silico trial used in the COVID-19 context.

Conclusions
Using the described verification computational workflow allows researchers and practitioners to easily perform verification steps to prove Agent-Based Models robustness and correctness that provide strong evidence for further regulatory requirements.

Full text

Model verication tools: acomputational
framework forverication assessment
ofmechanistic agent‑based models
Giulia Russo1†, Giuseppe Alessandro Parasiliti Palumbo2†, Marzio Pennisi3 and Francesco Pappalardo1*
From 4th International Workshop on Computational Methods for the Immune System Function (CM-
ISF 2020)
Virtual. 16-19 December 2020
Background
e recent openness from both European and USA Regulatory Agencies [1, 2] to the
possibility of using computer modeling and simulation for providing some of the regula-
tory evidence needed for the assessment of safeness (i.e., when it does not worsen the
health of the recipient) and efficacy (when it does improve the recipient’s health) of novel
medical compounds has paved the way to the application of the so-called in-silico tri-
als. Computational simulations can be used to strengthen, or to possibly substitute, the
Abstract
Background: Nowadays, the inception of computer modeling and simulation in life
science is a matter of fact. This is one of the reasons why regulatory authorities are
open in considering in silico trials evidence for the assessment of safeness and efficacy
of medicinal products. In this context, mechanistic Agent-Based Models are increas-
ingly used. Unfortunately, there is still a lack of consensus in the verification assessment
of Agent-Based Models for regulatory approval needs. VV&UQ is an ASME standard spe-
cifically suited for the verification, validation, and uncertainty quantification of medical
devices. However, it can also be adapted for the verification assessment of in silico trials
for medicinal products.
Results: Here, we propose a set of automatic tools for the mechanistic Agent-Based
Model verification assessment. As a working example, we applied the verification
framework to an Agent-Based Model in silico trial used in the COVID-19 context.
Conclusions: Using the described verification computational workflow allows
researchers and practitioners to easily perform verification steps to prove Agent-Based
Models robustness and correctness that provide strong evidence for further regulatory
requirements.
Keywords: Agent-based models, Verification assessment, In silico trials, Medicinal
product, Regulatory context, COVID-19
Open Access
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SOFTWARE
Russoetal. BMC Bioinformatics 2021, 22(Suppl 14):626
https://doi.org/10.1186/s12859‑022‑04684‑0
BMCBioinformatics
†Giulia Russo and Giuseppe
Alessandro Parasiliti Palumbo
were considered as joint first
authors
*Correspondence:
francesco.pappalardo@unict.
it
1 Department of Drug
and Health Sciences,
University of Catania,
95125 Catania, Italy
Full list of author information
is available at the end of the
article
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results coming from experiments involving cell cultures and animals (i.e., pre-clinical
trials) before and human volunteers (i.e., phase I, II, III, and IV clinical trials) then.
While the regulatory protocol for the assessment of safeness and efficacy (i.e., quali-
fication) of a medical product is well established when classical clinical trials are con-
sidered [3], there is still a lack of a common consensus [4–7] on how to assess the
“credibility” of a computational model. Even if verification and validation techniques can
be in general borrowed from other research fields (i.e., statistics, engineering, mathe-
matics, and physics), it has been mandatory to establish which steps must be carried on
and which methodologies must be used on each of them to qualify, through verification,
validation, and uncertainty quantification (VV&UQ) procedures, any computational
model to be used for In Silico Trials (ISTs).
To date, few model credibility standards and approaches have been discussed [8–10].
In the field of In Silico Trials, Viceconti etal. [11] proposed a theoretical framing for
the problem of assessing the credibility of a predictive model for ISTs that considers the
epistemic specificity of the research field and is general enough to be used for different
types of models, including simulators based on Agent-Based Models (ABMs), that have
become increasingly popular in this scenario.
anks to ABMs ability to readily describe complicated biological behaviors, laws, and
interactions involving cells and molecules without the need for complex mathemati-
cal formulas, these models are increasingly applied to simulate human pathophysiol-
ogy. Specifically, AMBs are useful to predict disease progression and related response
to various treatment administrations, or in specific conditions where the immune sys-
tem involvement is considered, and also to assist in discovering and developing novel
vaccines.
In ABMs, entities (also called agents) are tracked individually, and interactions are
recorded one by one, allowing for the inference of the system global emergent behavior
as the sum of the agents individual behaviors (bottom-up approach). As ABMs lack a
strong mathematical formalization and a standard verification process, some verifica-
tion steps must be refined and customized better to meet their characteristics [12, 13].
In this scenario, Curreli etal. [14] adapted the theoretical framework mentioned above
for assessing the credibility of an ABM simulator of the immune system in the presence
of tuberculosis disease. However, the numerical and statistical procedures carried on the
verification procedure have been executed manually, using different tools and software.
To facilitate the work of researchers employed in the development of computational
models for ISTs and to speed up the verification procedure, we developed “Model Verifi-
cation Tools” (MVT), a suite of tools based on the same theoretical framework described
above and with a user-friendly interface for the evaluation of the deterministic verifica-
tion of discrete-time models, with a particular focus on agent-based approaches. e
toolkit makes it simple for researchers to check many parts of models for possible flaws
and inconsistencies that could influence their outcomes.
Implementation
The verication workow andits application toABMs
Curreli et al. proposed a theoretical framework that aims at defining the steps for
assessing the credibility of mechanistic models used in the context of in-silico trials for
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medicinal products [14]. Here, we recall such a verification workflow, as it represents the
starting point for MVT development. e workflow considers two verification proce-
dures that can be carried out independently, i.e., deterministic and stochastic model ver-
ification. ABMs usually make use of pseudo-random number generators initialized with
different random seeds for reproducing different stochastic behaviors. Keeping constant
or varying the random seed over a set of simulations, it is then possible to analyze the
model behavior from a deterministic or stochastic point of view. Hence, stand-alone
procedures for the deterministic and stochastic verification procedures can be provided.
For the deterministic model verification, the workflow takes into consideration the fol-
lowing steps:
1. Existence and uniqueness analysis
2. Time Step Convergence Analysis
3. Smoothness Analysis
4. Parameter sweep analysis
Moreover, for stochastic model verification, the following steps are also considered:
5. Consistency
6. Sample Size.
At present, MVT includes the analysis tools for steps 1–4, as these represent the most
important ones. Steps 5 and 6 will be implemented in the next release of the tool.
e existence procedure checks for solution existence in the acceptable range of the
input parameters. Uniqueness focuses on checking for possible numerical and discre-
tizations (i.e., round-off errors) due to the limited numerical precision of computing
platforms that may influence solution results over different runs executed with the same
seed. While existence can be checked by assuring that the computational model returns
an output value for a given reasonable input range, uniqueness can be verified by check-
ing that identical input sets always entitle the same outputs with, at most, a minimal
tolerated variation determined by the used numerical rounding algorithm.
Time step convergence analysis aims at assuring that the time approximation intro-
duced by the Fixed Increment Time Advance (FITA) approach used by most ABM
frameworks and tools does not extensively influence the quality of the solution. e
same model is run with different time-step lengths to calculate the percentage discre-
tization error according to the following equation:
where qi* is an output reference quantity (i.e., the peak value of the simulation, final value
or mean value) obtained by the simulation executed at the smallest reference time-step
that maintains the execution of the model still computationally tractable (i*); qi repre-
sents the same output reference quantity obtained with a time-step i (with i > i*), and eqi
is the percentage discretization error. In their work, Curreli etal. proposed to assume
that the model converges if the error eqi < 5%.
e
i
q=
q
i
∗−q
i
q
i∗
∗
100
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Smoothness analysis was proposed to calculate the smoothness of the solution, bear-
ing in mind that possible errors in the numerical solution may lead to singularities, dis-
continuities, and buckling. e coefficient of variation D is computed as the standard
deviation of the first difference of the time series scaled by the absolute value of their
mean for all the output time series to evaluate the smoothness. To this end, a moving
window is used, and thus for each time observation yt in the output time series, the k
nearest neighbors are considered in the window: ykt = {yt-k, yt-k + 1,…, yt, yt + 1,…, yt + k}.
Currelli etal. used k = 3. e higher D is, the higher is the risk of stiffness, singularities,
and discontinuities.
Finally, parameter sweep analysis is used to assure that the computational model is not
numerically ill-conditioned. In general, the procedure involves sampling the entire input
parameter space to check if for particular input sets, the model fails to produce a valid
solution or if the solution is valid but outside the expected validity range. Furthermore,
by introducing slight variations on the input values, the analysis can be used to verify if
such slight variations entitle significant variations on the output values, suggesting an
abnormal sensitivity to some inputs.
While in their paper Curreli etal. proceed by using a two-step procedure for reducing
the input set size first, and to check the effects of the most relevant selected inputs on
the outputs then, we believe that similar results can be obtained by using well-known
standard stochastic sensitivity analyses, such as variance based (Sobol) sensitivity analy-
sis or Latin Hypercube Sampling-Partial Rank Correlation Coefficient (LHS-PRCC),
which have been then introduced inside MVT. e latter, in particular, uses a Latin
Hypercube Sampling (LHS) over the entire input parameter range to calculate the Partial
Rank Correlation Coefficient (PRCC) values between the input values and the selected
output value. In this way, it is possible to estimate the influence that any input parameter
has on the output value, independently from the variation carried over the other input
parameters from a stochastic point of view. is procedure can also be carried at any
time point to check the influence of the inputs on the output over time. LHS-PRCC is a
robust sensitivity analysis technique for nonlinear but monotonic relationships between
inputs and output.
Model verication tools
Model Verification Tools (MVT) is an open-source tool1 that offers helpful analysis
to verify discrete-time stochastic simulation models. Figure1 shows the architecture,
the software, and the libraries used to develop MVT. e tool is fully developed using
Python 3.9 programming language [15], the Django2 environment to create the web
infrastructure, and Docker.3 anks to this last component, we were able to build up
a stand-alone software platform (a docker container) that can be used on any oper-
ating system. is represents a huge leap ahead in respect to its preliminary web-
based implementation [16]. is version brings several improvements, among which
a considerable reduction of the latency times related to large file uploading and the
1 https:// github. com/ COMBI NE- Group/ docker_ verify
2 https:// www. djang oproj ect. com/
3 Django (Version 1.5) [Computer Software]. (2013). Retrieved from https:// www. djang oproj ect. com/
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possibility of taking full advantage of the system resources for more complex analyses.
Among the libraries used for Uncertainty and Sensitivity Analysis, “Pingouin” [17],
“Scikit’’ [18] and “Scipy” [19] were used to perform the LHS-PRCC analysis, while the
library “SALib” [20] was chosen to perform the Sobol sensitivity analysis.
Regarding the libraries used for the deterministic model verification techniques,
we used “Numpy” [21], the fundamental python package for scientific computing.
e Graphical User Interface (GUI) of MVT (Fig.2) consists of two main menus: 1)
Time Step Convergence Analysis
Smoothness Analysis
Uniqueness Analysis
Sobol Analysis
LHS-PRCC Analysis
Model Verification Tools
Graphical User Interface
Deterministic Model Verification
Fig. 1 Components and libraries of model verification tools
Fig. 2 The documentation page of MVT
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Documentation and 2) Model Verification. e documentation menu gives a brief
description of each technique and explains in detail each input parameter.
e second menu consists of five sub-menu: i) Smoothness Analysis, ii) Time step
Convergence Analysis, iii) Uniqueness Analysis, iv) Sobol Analysis, and v) LHS-PRCC
Analysis.
Smoothness analysis
e model may suffer from singularities, discontinuities, and buckling errors. e
Smoothness Analysis allows detecting these errors. To perform this analysis, a setting
up of the following parameters is required (Fig.3): i) “Skip rows” panel allows the ignore
specific rows from the analysis, for example, header lines of the input/output files that
have to be removed; ii) “Column to analyze” panel allows the selection of the output col-
umn to be analyzed from the input/output file uploaded by the user; iii) “Window Size”
panel allows to define the size of the window, i.e., the choice of the k nearest neighbors
for the analysis; iv) “Separator character” panel allows defining the correct separator
character of the input files (i.e., comma, space, tab); v) “File to analyze” panel allows the
user to upload a CSV or ASCII file defining the output file on which to perform the anal-
ysis. After clicking on the submit button, the analysis applies the procedure described
above on the column selected by the user. en, by choosing the listed results on the
box “your analysis” reported on the right side of Fig.3, the user can look at the analysis
results and the related produced plots.
Time step convergence analysis
Time step Convergence Analysis allows one to determine if the model behavior
converges as the time-step length becomes narrower. In this version of MVT, the
numerical measures considered to evaluate the global convergence of the model
are the maximum value achieved throughout the simulation (Peak Value—PV), the
time-to-peak-value, the final value (FV), the Pearson Correlation Coefficient (PCC),
and the root-mean-square error (RMSE). To perform the study of the model con-
vergence, it is essentially setting up the parameters from the interface (Fig.4). is
Fig. 3 The smoothness analysis GUI. The box on the left side represents the list of parameters to perform the
analysis; on the right side, the “Your Analysis” box contains the list of the completed results analysis
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analysis essentially takes into account the same parameters described for the smooth-
ness analysis. After clicking on the submit button, the user can retrieve the plots of
the measures mentioned above produced by the algorithm in the box “your analysis”
shown on the right side of Fig.4.
Uniqueness analysis
As described in the previous version of the tool [16], the user can set up the analysis
parameter in the GUI (Fig.5). In particular, the analysis needs as an input: i) “Skip
rows” parameter that allows ignoring specific rows from the analysis; ii) “Separator
Character” panel allows defining the correct separator character of the input files, and
iii) “Files to analyze” panel allows to select the files to analyze. After clicking on the
submit button, the tool calculates the mean and Standard Deviation (SD) of all the
rows among all the files. If the maximum value of the previously calculated SD is not
equal to 0, the files are different. In this case, the tool returns a warning message in
a pop-up window, showing the row and the column of the first occurrence where the
SD is different from 0 (see Fig.6); on the other hand, MVT returns a success message
in a pop-up window.
Fig. 4 The Time step convergence analysis GUI. The box on the left side represents the list of parameters
to perform the analysis; on the right side, the “Your Analysis” box contains the list of the completed results
analysis
Fig. 5 The uniqueness analysis GUI
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Uncertainty analysis
In this version of MVT, we used the scikit-optimize library to integrate the Latin
Hypercube Sampling (LHS) methodology and the SALib python library for imple-
menting the Sobol Sensitivity Analysis methodology [22–24].
e user can set up the analysis parameters in the GUI for LHS presented in Fig.7.
In particular, the box allows to set: i) the “Number of samples” to define the number
of samples to generate; ii) the “Seed” parameter that is used to define the random
seed of the pseudo-random generator, in order of creating sample input parameter
sets that can be reproduced; iv) “Iteration” field defines the number of iterations for
optimizing LHS; v) “Separator Character” panel allows defining the proper separator
character of the input file; vi) “Input parameter file” is the field that allows uploading
a CSV file, defining the model inputs on which to perform the simulations. is file
must have a header and three columns defined as follows: i) param_name: is the first
column and represents the name of the parameter; ii) min: is the second column and
represents the minimum value of the parameters; iii) max: is the third column and
represents the maximum value of the parameters. In this version of MVT, the LHS
tool can generate a sample set drawn from a uniform distribution of the parameters.
Once the analysis is complete, it produces the LHS matrix with N rows and M col-
umns, where N represents the number of samples and M the number of parameters.
After that, the user can download the matrix and run the model on the parameter set
generated.
MVT allows the user to use another algorithm to generate samples based on Saltelli
methodology [20, 23] using the SALib library. Specifically, the user can use the appro-
priate GUI (Fig.8) to choose the parameters for the generation samples: i) "Number
of combinations" panel allows to set the number of samples to be generated; ii) "skip
values," according to the SALib library, is the number of points in Sobol’ sequence to
skip for getting different samples. Furthermore, it is worth mentioning that this value
Fig. 6 An example of the output of the uniqueness analysis, where the files are not the same. Information
about the row and the column in which SD is not equal to zero is also provided
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must be an exponent to the power of 2; iii) "Separator Character" allows to define the
proper separator character and, iv) "Input Parameter File" allows to choose a CSV file
with the model parameters, its range values and the type of distribution to be used
for the generation of each specific parameter. e CSV file must have a header with
the following structure: i) param_name: this field represents the name of the param-
eters; ii) first_value: this value depends on the type of distribution and represents the
minimum value if the value itself of the distribution field is "uniform"; otherwise, the
value represents the mean; iii) second_value: this field also depends on the type of
Fig. 7 The Latin hypercube sampling analysis GUI
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distribution chosen. is value represents the maximum if the value of the distribu-
tion field is "uniform"; otherwise, the value represents the standard deviation; iv) dis-
tribution: this field represents the type of distribution used to generate the samples
for each parameter. e allowed values are: "unif", "norm" and "lognorm", which spe-
cifically represent the "uniform", "normal" and "lognormal" distribution. After click-
ing on the submit button, the algorithm produces a matrix having N * (2D + 2) rows.
Where D is the number of parameters in input and N is the number of combinations.
e number of rows in the matrix is equal to the number of samples.
Sensitivity analysis: Sobol analysis
is analysis is used to evaluate the sensitivity of the model to the input parameters
using the matrix obtained from the Sobol sample generation procedure. e user can
perform this analysis using the appropriate GUI (Fig.9), which allows to set up the fol-
lowing parameters: i) “Separator Character for Input parameter file” and “Separator
Character for Output file from the model” allows to define the proper separator char-
acter for parameter file and for Output file which derive from the model; ii) “Column to
Fig. 8 The SOBOL sample generation analysis GUI
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analyze” represents the column on which to perform the analysis; iii) “Input parameter
file” represents the same file used in Sobol sample generation; iv) “Output files from the
model” panel is used for uploading the model outputs files, in ASCII or CSV format,
without header. To perform the analysis, it is mandatory to rename the files according
to a predefined scheme. e naming of the output model files should follow the follow-
ing nomenclature: "0_1.csv", "0_2.csv", and so on, with the second index representing
the model output obtained by the respective input row value of the Sample Generation
output matrix. For example, "0_1.csv" represents the model output from the first input
row, while "0_2.csv" represents the model output obtained by the second row from the
Sample Generation output matrix.
Sensitivity analysis: PRCC analysis
Once the input parameters have been generated with the LHS procedure, and the sim-
ulations have been run on such parameters, the Partial Rank Correlation Coefficient
(PRCC) procedure for finalizing the LHS-PRCC analysis [25] can be executed. e user
can then check the PRCC values evolution over time (PRCC_OT) and understand how
the relationship between inputs and outputs evolves over time, and/or visualize the
PRCC results at specific time steps (PRCC_STS); this allows to understand better the
correlation among the input parameters and the output of the model. In the PRCC over-
time analysis, the GUI (the box on the left side in Fig.10) takes into account the same
input parameters described for the other tools, along with the following ones: i) “Time
points interval” allows to pick out the data from the “Column to analyze” for a specific
time point interval selected by the user; ii) “reshold p-value” allows the user to set
up the threshold for the visualization of the level of significance. is analysis produces
a pop-up (Fig.11) that allows the user to download a pdf file containing the temporal
Fig. 9 The SOBOL analysis GUI. The box on the left side represents the list of parameters necessary to
perform the analysis; on the right side, the “Your Analysis” box contains the list of the completed analysis
results
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correlation plots. Furthermore, a JSON file (Fig.12) containing each parameter under
investigation the time points in which the p-value overcomes the threshold set by the
user, meaning that the correlation is significant, is also made available. MVT provides
the PRCC_STS GUI (the central box of Fig.10) to analyze the relationship between
the input and output parameters at specific time points. is analysis takes the same
parameters defined for the PRCC_OT but replaces the “Time point interval” parameter
with “Time step.” At the end, PRCC_STS produces a pop-up window that allows users
to download and visualize a scatter plot for each parameter under study. Such graphi-
cal plots are of significant importance, as they allow to graphically reveal possible non-
monotonic correlations that are not usually detected by the standard PRCC procedure.
Fig. 10 The partial rank correlation coefficients analysis GUI. The box on the left side represents the list of
parameters to perform the PRCC over time analysis; while the box on the center side represents the list of
parameters to perform the PRCC to a specific time step. On the right side, the “Your Analysis” box contains the
list of the completed analysis results
Fig. 11 A pop-up example to download the PRCC over time plots and the time correlation files
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Results anddiscussion
We applied the Deterministic Model Verification and Uncertainty and Sensitivity analy-
sis techniques on UISS for the SARS-CoV-2 scenario (UISS-SARS-CoV-2). UISS-SARS-
CoV-2 is the implementation of COVID-19 disease model in UISS. Hence it owns the
immune system machinery originally developed inside UISS. UISS-SARS-CoV-2 was
further implemented to reflect the dynamics of COVID-19 [26]. Within UISS, it is pos-
sible to change the time-step length for the simulation. In other words, it is possible to
simulate with a time-step length equal to 8h, rather than 20min, or 5min. Within the
Fig. 12 A sketch of time correlation JSON files
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context of UISS-SARS-CoV-2, we run 9 simulations using a time-step length between
8h and 5min and with a total duration of 4months of simulation.
We assumed that the reference trace has a time-step length of 5min. en, we used
the MVT to evaluate the time step convergence analysis on the active TH-1 cells. Panel
A of Fig.13 shows the specific trends of the active TH-1 cells. Panel B shows the PCC,
and the RMSE computed between the reference trace and the other ones. It is worth
mentioning that, at the end of the plot, the value of PCC is about 0.6, and the value of
RMSE remains stable. e last step of time step convergence analysis uses the formula
described above to calculate the convergence of the time-to-peak-value and the final
value; the corresponding plot is shown in Panel C. e x-axis of Panel B and C shows
the number of iterations, that is the number of steps to reach the end of simulation. e
number of iterations depends on time-step length. According to the time step conver-
gence analysis, the obtained results from the time-to-peak value and the final value sug-
gest that convergence is achieved using a time-step length of 15min, that is equal to
11,520 number of iterations.
For this reason, subsequent analyses will be carried out using the outputs obtained
from the simulation with a time-step length of 15min. e next step was to perform
the smoothness analysis of the TH-1 Active (Fig.14). e data presented in the first
part of the plot shows sudden peaks caused by the TH-1 response to specific anti-
gens. en, we perform the Uniqueness analysis to check if repeated executions on
AB
C
Fig. 13 Time step convergence active TH-1 output. Panel A shows the dynamics of the output at different
timestep over time. Panel B shows the Pearson Correlation Coefficient and Root Mean Square Error at
different time step. Panel C shows the convergence of Time to peak value and Final value of the output
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the same input parameter set lead to the identical output. For this reason, we ran
the UISS-SARS-CoV-2 three times with the same set of input parameters obtaining
identical outputs. Sobol is the first technique applied to carry out the Uncertainty
and Sensitivity analysis of the UISS-SARS-CoV-2. We chose to evaluate the sensitiv-
ity of the simulator by varying the values of two parameters: i) Num_Ag, the num-
ber of inoculated antigens within the range; ii) AbMultifact, the number of antibodies
secreted by plasma B cells, within the range. It is important to note that each param-
eter was sampled using a uniform distribution. en, through the GUI of Sobol sam-
ple generation, we chose to set the ’number of combinations’ parameter equal to 16
in order to generate 96 samples. After running the simulations, we chose to analyze
the relationship between the input set and active TH-1 cells and IgG, respectively.
Panel A in Fig.15 shows the correlation relationship between the two input param-
eters (Num_Ag and AbMultifact levels) and active TH-1 cells levels, while Panel B
shows the sensitivity result obtained for the IgG values. ese figures make it possible
Fig. 14 Smoothness analysis on the TH-1 Active output. The plot shows the smoothness analysis of the TH-1
Active. The data presented in the first part of the plot shows sudden peaks caused by the TH-1 response to
specific antigens
A B
Fig. 15 The SOBOL analysis output. Panel A and B respectively show the sensitivity of the simulator on the
“Num_Ag ’’ and “AbMultifact ’’ parameters concerning TH1-active cells and IgG
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Russoetal. BMC Bioinformatics 2021, 22(Suppl 14):626
to highlight how, inside our model, the two parameters differently affect the selected
outputs. It is important to remember that the greater the sensitivity indices are, the
more critical parameters are for the model output. erefore, we can observe how
the dynamics of IgG antibodies are more sensitive to the variation of the number of
inoculated antigensdue to the low y value in panel B, while AbMultifact mainly affects
the dynamics of active TH-1.
LHS-PRCC is the second technique applied to perform the Uncertainty and Sen-
sitivity analysis of the model. e parameters and the type of distribution used here
are the same as those taken into account in the previous analysis. In order to gener-
ate the LHS matrix, the parameters of GUI were set as follows: i) ’number of samples’
equals 96; ii) ‘Seed’ parameter equal 2021; iii) ‘Iterations’ equal 1000. After that, the
simulation was run on the LHS matrix. After that, we chose to apply the PRCC_OT
and PRCC_STS analysis to analyze the relationship between parameters set and TH-1
active cells and IgG. e plots of PRCC_OT are shown on panels A and C of Fig.16.
Both panels show a dummy curve (red line), it does not affect the model in any other
way, but it is useful for comparing parameters that have an effect on the model out-
put. Panel A shows a strong correlation at the start of simulations between Num_
Ag and active TH-1 (highlighted in gray), which then turns into a weak correlation
AB
C D
Active TH-1
Active TH-1
Fig. 16 The output of PRCC_OT And PRCC_STS. Panel A and B respectively show the sensitivity of the
simulator on the “Num_Ag” over time and at a specific time step. Panel C and D respectively show the
sensitivity of the simulator on the “AbMultifact” over time and at a specific time step. The scatter plots, Panel
B and D, depict the influence of the Num_Ag and AbMultifact variables (input, x-axis) on the selected output
value (y-axis), respectively
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Russoetal. BMC Bioinformatics 2021, 22(Suppl 14):626
between 0.1*107s and 0.2*107s and then turns again into a strong correlation until
the end of the simulations. Panel C shows a weak correlation at the start of simula-
tions between AbMultifact and the levels of active TH-1 cells, which then turns into a
strong correlation until the end of the simulations. After that, we ran the PRCC_STS
analysis at the specific time step of 5,400,000s, obtaining the plots shown in pan-
els B and D of Fig.16. ese scatter plots, depict the influence of the Num_Ag and
AbMultifact variables (input, x-axis) on the selected output value (y-axis), respec-
tively. Both input and output values are represented as ranked values to remove any
non-linear relationship. Scatter plots may be useful to visually detect the presence of
non-monotonic relationships that are not usually shown by the PRCC value alone. In
this case Fig.16, Panel B, shows a weak positive relationship at time-step 5,400,000
between the Num_Ag variable and the output value (PRCC: 0.3128). is can be seen
by observing that the distribution of the points vaguely approximates an increasing
straight line. Conversely in Fig.16, Panel D, a weak negative relationship between the
AbMultifact variable and the output value (PRCC: 0.3128) holds, with the points dis-
tribution that vaguely approximates a decreasing straight line.
Conclusions
Mechanistic agent-based models are increasingly employed for developing in silico tri-
als solutions for medicinal products. Consequently, to lower barriers in their regulatory
acceptance, the assessing of their credibility is mandatory. Formal methodologies for
agent-based models verification should be developed and widely adopted. e described
approach proposes a set of automatic tools that help formally verifying the determinis-
tic part of mechanistic agent-based models. is allows researchers and practitioners to
easily perform verification steps to prove ABM robustness and correctness that provide
strong evidences for further regulatory requirements. As ABMs usually own a stochastic
component, statistical consistency and minimum sample size determination need to be
also addressed. We are dealing with this issue and we will expand actual modeling verifi-
cation framework in due course.
Availability andrequirements
Project name: Model Verification Tools
Project home page: https:// github. com/ COMBI NE- Group/ docker_ verify
Operating system(s): Platform independent
Programming language: Python
Other requirements: Docker
Any restrictions to use by non-academics: not applicable.
Abbreviations
VV&UQ: Uncertainty quantification; ISTs: In silico trials; ABMs: Agent-based modeling; MVT: Model verification tools; FITA:
Fixed increment time advance; LHS-PRCC : Latin hypercube sampling-partial rank correlation coefficient (LHS-PRCC);
LHS: Latin hypercube sampling; PRCC : Partial rank correlation coefficient; GUI: Graphic user interface; PV: Peak value; FV:
Final value; PCC: Pearson correlation coefficient; RMSE: Root mean square error; SD: Standard deviation; UISS: Universal
immune system simulator.
Acknowledgements
Not applicable.
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Russoetal. BMC Bioinformatics 2021, 22(Suppl 14):626
About this supplement
This article has been published as part of BMC Bioinformatics Volume 22 Supplement 14 2021: Selected papers from the
4th International Workshop on Computational Methods for the Immune System Function (CMISF 2020). The full contents
of the supplement are available at https:// bmcbi oinfo rmati cs. biome dcent ral. com/ artic les/ suppl ements/ volume- 22-
suppl ement- 14.
Author contributions
GR: provided regulatory context knowledge in modeling and simulation, checked the biological adherence and mean-
ing, evaluated the workflow, drafted the manuscript. GAPP: performed numerical simulations, developed the python
Model Verification Tools, wrote the manuscript. MP: performed numerical verification consistency, analyzed data, and
wrote the manuscript. FP: conceived the model, gave computational immunological knowledge, provided regulatory
knowledge in in silico trials, supervised the project. All authors have read and approved the final manuscript.
Funding
Authors of this paper acknowledge support from the STriTuVaD project. The STriTuVaD project has been funded by
the European Commission and Indian Department of Biotechnology, under the contract H2020-SC1-2017-CNECT-2,
No. 777123. The information and views set out in this article are those of the authors and do not necessarily reflect the
official opinion of the European Commission. Neither the European Commission institutions and bodies nor any person
acting on their behalf may be held responsible for the use which may be made of the information contained therein.
Publication costs are funded by the STriTuVaD project.
Availability of data and materials
The main computational framework is fully described in the paper. The UISS-SARS-CoV-2 framework used for this
research is available at: https:// combi ne. dmi. unict. it/ UISS- COVID 19/. The MVT tool is available at: https:// github. com/
COMBI NE- Group/ docker_ verify.
Declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Author details
1 Department of Drug and Health Sciences, University of Catania, 95125 Catania, Italy. 2 Department of Mathematics
and Computer Science, University of Catania, 95125 Catania, Italy. 3 Computer Science Institute, DiSIT, University of East-
ern Piedmont, 15121 Alessandria, Italy.
Received: 5 April 2022 Accepted: 11 April 2022
Published: 19 May 2022
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