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          MCSim User' Manual (Derived from ./doc/mcsim.html)

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Table of Contents

   * Software License
        o PREAMBLE
        o TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
   * Overview
        o General procedure
        o Types of simulations
        o Major changes introduced with version 4.2.0
   * Installation
        o System requirements
        o Distribution
        o Machine-Specific Installation
   * Working Through an Example
   * Defining Models
        o Using Mod to process model description files
        o Syntax of the model description file
             + General syntax
             + Global parameter declarations
             + Special functions
             + Input functions
             + Dynamics specifications
             + Parameter scaling
             + Output calculations
             + Comments on style
             + Note about models
   * Specifying Simulations
        o Using the compiled program
        o Syntax of the simulation definition file
        o Global specifications
             + SimType() specification
             + Integrate() specification
             + OutputFile() specification
             + MonteCarlo() specification
             + Distrib() specification
             + MCMC() specification
             + SetPoints() specification
        o Specifying basic conditions to simulate
             + Experiment definition
             + StartTime() specification
             + Print() specification
             + PrintStep() specification
             + Data() specification
        o Specifying a statistical model
             + Level definition
        o Analyzing results
        o Error Handling
   * Bibliographic References
   * Common Pitfalls
   * Using make
   * Examples
        o `linear.model'
        o `1cpt.model': A sample model description file
        o `perc.model': A sample model description file
        o `perc.lsodes.in'
   * Concept Index

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Software License

GNU GENERAL PUBLIC LICENSE





Overview

MCSim consists of two pieces, a model generator and a simulation engine. The
model generator, mod, was created to facilitate the model maintenance and
simulation definition, while keeping execution time fast. Other programs
have been created to the same end, the Matlab family of graphical
interactive programs being some of the more general and easy to use. Still,
many available tools are not optimal for performing time and computer
intensive Monte Carlo analyses. MCSim was created specifically to this end:
to perform Monte Carlo analyses in a highly optimized, and easy to maintain
environment.

General procedure

Model building and simulation proceeds in four stages:

  1. You create with any text editor a model description file. The reference
     section on mod, later in this manual gives you the syntax to use (see
     section Defining Models). This syntax allows you to describe the model
     variables, parameters, equations, inputs and outputs in a C-like
     fashion without having you to actually know how to write a C program.
  2. You instruct the model generator, mod, to preprocess your model
     description file. Mod creates a C file, called `model.c'.
  3. You compile and link the newly created `model.c' file together with the
     other C program files. MCSim C code is standard, so you should be able
     to compile it with any standard C compiler, for example GNU gcc. After
     compiling and linking, an executable simulation program `mcsim' is
     created, specific of the particular model you have designed. These
     preprocessing and compilation steps may seem clumsy but they produce
     the most efficient code for your particular machine.
  4. You design any number of simulation specification files and run them
     with the mcsim program. The simulation specification files describe the
     kind of simulation to run (simple simulations, Monte Carlo etc.),
     various settings for the integration algorithm if needed, and a
     description of one or several experimental conditions or observations
     to simulate (see section Specifying Simulations). The simulation output
     is written to standard ASCII files.

Little or no knowledge of computer programming is required, unless you want
to tailor the program to special needs, beyond what is described in this
manual (in which case you should contact us). You need, however, some
familiarity with program compilation under your operating system (see
section Installation). The software manual for your compiler should be able
to help you.

Types of simulations

Four types of simulations are available:

   * A DefaultSim simulation will simply solve (eventually integrate) the
     equations you specified, using the default parameter values and
     possible overridings imposed in a simulation specification file.
     User-requested outputs are sent to an output file (see section
     SimType() specification).
   * A MonteCarlo simulation will perform repeated (stochastic) simulations
     across a randomly sampled region of the model parameter space (see
     section MonteCarlo() specification).
   * A MCMC simulation performs a series of simulations along a Markov chain
     in the model parameter space (see section MCMC() specification). These
     are Monte Carlo simulations in which the choice of a new parameter
     value is influenced by the current value. They can be used to obtain
     the Bayesian posterior distribution of the model parameters, given
     their prior distributions (that you specify) and data for which a
     likelihood function can be computed. The program handles hierarchical
     (random effect) statistical models, such as population pharmacokinetic
     models (see section Specifying a statistical model).
   * A SetPoints simulation solves the model for a series of specified
     parameter sets, listed in a separate ASCII file (see section
     SetPoints() specification). You can create these parameter sets
     yourself or use the output of a previous Monte Carlo or MCMC
     simulation.

Major changes introduced with version 4.2.0

   * A new input function, Spikes() is available. It simulates instantaneous
     inputs (see section Input functions).
   * Six new distributions are available for Monte Carlo simulations:
     InvGamma (inverse-gamma), Piecewise, Normal_v, LogNormal_v,
     TruncNormal_v, TruncLogNormal_v (see section Distrib() specification).
   * In Monte Carlo simulations, shape parameters of Distrib() statements
     can now reference other sampled parameters (see section MonteCarlo()
     specification).
   * A simTypeFlag option, in the MCMC() specification, allows the printing
     of times, data and predictions for easy checking of the model fit. It
     can also be set to switch MCMC sampling from component by component
     sampling to vector sampling (see section MCMC() specification).

Installation

System requirements

MCSim is written in ANSI-standard C language. We are distributing the source
code and you should be able to compile it for any system, provided you have
an ANSI C compiler. On a Unix system we recommend that you install the GNU
gcc compiler (freeware) and the make utility (which is in fact standard on
most systems). For PC-type computers we recommend the Unix operating system
Linux. For the Macintosh, you can use version 8.0 (or higher) of SymantecÕs
Think C compiler, although other compilers should work.

Distribution

MCSim source code is available on Internet through
`ftp://sparky.berkeley.edu/pub/mcsim',
`http://sparky.berkeley.edu/users/fbois/', and
`http://www.gnu.ai.mit.edu/home.html'.

Machine-Specific Installation

To install on a given machine, download (in binary mode) the distributed
archive file to your machine.directory. Move it to an empty directory of any
name. Decompress the archive with GNU gunzip (gunzip <archive-name.gz>).
Untar the decompressed archive with tar (tar xf <archive-name>; do man tar
for further help). On Macintosh machines the programme "Stuffit Expander"
should be able to both uncompress and untar the archive. This decompression
will create two directories: `mod' and `sim'.

Move to the `mod' directory. Under Unix, compile the mod program using the
`Makefile' in that directory (the command make should just do that). Under
other operating systems, refer to the documentation of your compiler to
create an executable `mod' file from the source code provided in the `mod'
directory. Move the executable `mod' program you just created to the `sim'
directory.

You are then ready to use MCSim. This requires creating a model definition
file, processing it with the mod program, and compiling the resulting
`model.c' file with all the other C files in the `sim' directory. You can
then run simulations files. We recommend that you go to the next section of
this manual (see section Working Through an Example, which walk you through
an example of model building and running.

The makefile `Test_mcsim' can be used to test whether the program output on
your Unix machine is the same as the one on our machines. Just type: make;
make -f Test_mcsim when in your `sim' directory. All input files will be run
and their output compared to the corresponding output files. You will need
to have the `mod' program already compiled and inside the `sim' directory,
or on the command path. In case of differences, don't panic: check the
actual differences between the culprit output file and the file `sim.out'
produced by the makefile. Small differences may occur from different machine
precision. This can happen for random numbers, in which case the Markov
chain simulations (MCMC) can diverge greatly after a while.

Working Through an Example

Pharmacokinetics models describe the transport and transformation of
chemical compounds in the body. These models often include nonlinear
first-order differential equations. The following example is taken from our
own work on the kinetics of tetrachloroethylene (a solvent) in the human
body (Bois et al., 1996; Bois et al., 1990) (see section Bibliographic
References). Go to the `sim' directory (under Unix) or to the `Development'
folder (on a Macintosh). Open the file `perc.model' with any text editor
(e.g., emacs or vi under Unix). This file is distributed as an example of a
model definition file (see section `perc.model': A sample model description
file). You can use it as a template for your own model, but you should leave
it unchanged for now. Notice that it defines:

   * state variables for the model (for which differentials are defined),
     for example:

     States = {Q_fat,        # Quantity of PERC in the fat
               Q_wp,         #   ...   in the well-perfused compartment
               Q_pp,         #   ...   in the poorly-perfused compartment
               Q_liv,        #   ...   in the liver
               Q_exh,        #   ...   exhaled
               Qmet}         # Quantity of metabolite formed

   * output variables (obtainable at any time as analytical functions of the
     states, inputs and parameters), for example:

     Outputs = {C_liv,           # mg/l in the liver
                C_alv,           # ... in the alveolar air
                C_exh,           # ... in the exhaled air
                C_ven,           # ... in the venous blood
                Pct_metabolized, # % of the dose metabolized
                C_exh_ug}        # ug/l in the exhaled air

   * input variables (independent of the others variables, and eventually
     varying with time), for example:

     Inputs = {C_inh} # Concentration inhaled

   * model parameters (independent of time), such as:

     LeanBodyWt = 55; # lean body weight

   * system's dynamics (differential or algebraic equations defining the
     model per se),
   * parameters' scaling (the parameters used in the equations can be made
     functions of other parameters: for example volumes can be computed from
     masses and densities),
   * equations to compute the output variables.

This model definition file as a simple syntax, easy to master. It needs to
be turned into a C program file before compilation and linking to the other
routines (integration, file management etc.) of MCSim. You will use mod for
that. First, quit the editor and return to the operating system.

To start mod under Unix just type mod perc.model. On a Macintosh, double
click the `Mod' icon; Mod prompts you for the name of your model definition
file; Type `perc.model'. After a few seconds, with no error messages if the
model definition is syntactically correct, Mod announces that the `model.c'
file has been generated. On a Macintosh you need to hit the return key to
exit Mod.

The next step is to compile and link together the various C files that will
constitute the simulation program for your particular model. Note that each
time you want to change an equation in your model you will have to change
the model definition file and repeat the steps above. However, changing just
parameter values or state initial values does not require recompilation
since that can be done through simulation specification files.

   * Under Unix, the simplest is to use the make utility. Just type make and
     compilation will be done automatically (see section Using make). An
     executable `mcsim' is created. You can rename it to better describe the
     fact that it is model specific: rename it `mcsim_perc', for example.
   * On a Macintosh, or PC, the best is to use a Think C project or a
     similar programming environment. You should first use the command make
     or its equivalent to compile the modified `model.c' file and other C
     files. Then create an application (you should give it a name specific
     to the model you are developing, e.g., `MCSim Perc'). Refer to your
     compiler manual for details on how to use your programming environment.
     Your executable `MCSim Perc' program is now ready to perform
     simulations.

To start your MCSim program just type mcsim_perc (if you gave it that name)
under Unix, or double click the `MCSim Perc' Ð or whatever name you
specified Ð icon on your Macintosh. After an introductory banner (telling in
particular which model file the program has been compiled with), you are
prompted for an input file name: type perc.lsodes.in (see section
`perc.lsodes.in', to see this file now). The program then prompts you for
the output file name: type perc.lsodes.out. After a few seconds or less
(depending on your machine) the program announces that it has finished and
that the output file is `perc.lsodes.out' (on a Macintosh you should hit the
return key to exit the program completely). You can open the output file
with any text editor or word processor, you can edit it for input in graphic
programs etc.

Several other models and simulation specification files are provided with
the package as examples (they are in the `sim' directory. Try them and
observe the output you obtain. You can then start programming you own models
and doing simulations. The next sections of this manual reference the syntax
for model definition and simulation specifications.

Defining Models

Three examples of model simulation files: `linear.model', `1cpt.model' and
`perc.model' are included with the program files and appears in Appendix of
this manual (see section Examples).

Using Mod to process model description files

The mod program is a stand-alone facility. It takes a model description file
in the "user-friendly" format described below and creates a C language file
`model.c' which you will compile and link to the simulation program. Mod
allows the user to define equations for the model, assign default values to
parameters or default initial values to model variables, and define scaling
functions for both the input parameters and the outputs variables. Mod lets
the user create and modify models without having to maintain C code.

In Unix, the command line syntax for the mod program is:

mod [input-file [output-file]]

where the brackets indicate that the input and output filenames are
optional. If the input filename is not specified, the program will prompt
for both. If only the input filename is specified, the output is written by
default to the file `model.c'. Unless you feel like doing some makefile
programming, we recommend using this default since the makefile for MCSim
assumes the C language model file to have this name. You have to have
prepared an input file containing a description of the model following the
syntax described in the following (see section Syntax of the model
description file).

On the Macintosh you double-click the `Mod' icon and enter the name of the
model definition file at the prompt (on the Macintosh, names can include
space characters).

Syntax of the model description file

The model description file is a text (ASCII) file that consists of several
sections, including global declarations, dynamics specifications (eventually
with derivative calculations), model scaling, and output computations:

# Model description file (this is a comment)
<Global parameter specifications>
Dynamics {
  <Equations for calculating the dynamics, or state derivatives>
}
Scale {
  <Equations for scaling model parameters>
}
CalcOutputs {
  <Equations for scaling the outputs>
}

where Dynamics, Scale and CalcOutputs are keywords and, if used, must appear
as shown, followed by the curly braces which delimit the section. At least
one of the sections Dynamics or CalcOutputs should be defined. Dynamics must
be used if the model includes differential equations.

General syntax

The general syntax of the file is as follows:

   * Comments begin with a pound sign (#) and continue to the end of the
     line.
   * Blank lines are allowed and ignored.
   * All commands can span several lines and are terminated by a semi-colon
     (;). There should be no blank space before the semi-colon.
   * Variable assignments have the following syntax:

     var-name '=' constant-value-or-expression ';'

     where var-name is any valid C identifier, starting with a letter or
     underscore (_) and followed by any number of alpha-numeric characters
     or underscores, up to a maximum of 80. Variable names are case
     sensitive. Note that the name IFN, in capital letters, is reserved by
     the program and should not be used as parameter or variable name. The
     equal sign is needed. The right-hand side expression can be a valid C
     mathematical expression including already defined variables and ANSI C
     mathematical functions. Additional functions, called special functions,
     can be used, which take variable names, constant values or expressions
     as parameters. Special functions are detailed below (see section
     Special functions). Finally, there should be no space before the
     terminating semi-colon. Colon conditional assignments can also be used.
     Syntax:

     var-name '=' (<condition> ? <value-if-true> : <value-if-false>);

     For example:

     Adj_Parm = (Input > 0.0 ? Parm * Adjust : Parm);

     In this example, if `Input' is greater than 0, the parameter `Adj_Parm'
     is computed as the product of `Parm' by `Adjust'; otherwise `Adj_Parm'
     is equal to `Parm'. The comparison operators allowed are the equality
     operator ==, and non-equality operators !=, <, >, <>, <= and >=.

Global parameter declarations

Commands not specified within the delimiting braces of another section are
considered to be global declarations. In the global section, you first
declare the states, inputs, and outputs variables. There should be at least
one state or output variable in your model.

   * States are variables for which a first-order differential equation is
     defined (higher orders or partial differential equations are not
     allowed).
   * Inputs are variables independent of the others variables, but
     eventually varying with time (for example an exposure concentration to
     a chemical).
   * Outputs are dependent model variables (obtainable at any time as
     analytical functions of the states, inputs or parameters) that do not
     have dynamics.

The format for declaring each of these variables is the same, and consists
of the keyword States, Inputs or Outputs followed by a list of the variable
names enclosed in curly braces as shown here:

States =   {Qb_fat,  # Benzene in the fat
            Qb_bm,   # ...     in the bone marrow
            Qb_liv}; # ...     in the liver and others

Inputs =   {Q_gav,   # Gavage dose
            C_inh};  # Inhalation concentration

Outputs =  {Cb_exp,  # Concentration in expired air
            Cb_ven}; # ...           in venous blood

After being defined, states, inputs and outputs can then be given initial
values (constants or expressions). Note that inputs can also be assigned
input functions, described below. Some examples of initialization are shown
here:

Qb_fat = 0.1; # Default initial value for state variable Qb_fat

# Gavage input assigned a periodic exponential input function
Q_gav = PerExp(1, 60, 0, 1); # Magnitude of 1.0,
                             # period of 60 time units,
                             # T0 in period is 0,
                             # Rate constant is 1.0

If a global state, input, or output variable is not given an initial value,
it will default to zero. Initial values are reset to their specified value
by the simulation program at the start of each simulation of an Experiment
(see section Experiment definition).

All the model parameters you want to be able to change through simulation
files should be declared global. Parameters must be given nominal values,
following the assignment rules given above. For example:

BodyWt = 0.38 + sqrt(0.01); # (kg) Weight of the rat

All parameters and variables are computed in double precision floating-point
numbers. Initialization values should not be such as to cause computation
errors in the model equations; this is likely to cause program crashes (so,
for example, do not assign a default value of zero to a parameter appearing
alone in a denominator). Note that the order of global declarations matters
within the global section itself (i.e., parameters and variables should be
defined and initialized before being used in assignments of others), but not
with respect to other blocks. A parameter defined at the end of the
description file can be used in the dynamics section which may appear at the
beginning of the file. Still, such an inverse order should be avoided. For
this reason, the format above, where global declarations come first, is
strongly suggested to avoid confusing results. Note again that the name IFN,
in capital letters, is reserved by the program and should not be used as
parameter or variable name. Finally, if a parameter is defined several
times, only the first definition is taken into account.

Special functions

The following special functions (whoe name is case-sensitive) are available
to the user for assignment of parameters and variables in the model
definition file:

   * CDFNormal(x): the normal cumulative density function;
   * erfc(x): the error function;
   * lnDFNormal(x, mean, sd): the natural logarithm of the normal density
     function;
   * lnGamma(x): the natural logarithm of the gamma function;
   * UniformRandom(min, max): returns a uniformly distributed random
     variable, sampled between min and max. The algorithm used is that of
     Park and Miller (Barry, 1996; Park and Miller, 1988; Vattulainen et
     al., 1994) (see section Bibliographic References). A default random
     generator seed (314159265.3589793) is used;
   * NormalRandom(mean, sd): returns a normally distributed random variable.
     The default random generator seed is used.

Input functions

They can be used in a special assignments, valid only for input variables.
Inputs can be initialized as a constant or expression, or assigned one of
the following input functions:

   * PerDose() specifies a periodic input of constant magnitude. The input
     begins at initial-time in the period and lasts for exposure-time time
     units. Syntax:

     PerDose(<magnitude>, <period>, <initial-time>, <exposure-time>);

   * PerExp() specifies a periodic exponential input. At time initial-time
     in the period the input rises instantaneously to magnitude and begins
     to decay exponentially with the constant decay-constant. The input is
     turned off once the magnitude reaches a negligible fraction of its
     original value. Syntax:

     PerExp(<magnitude>, <period>, <initial-time>, <decay-constant>);

   * NDoses() specifies a number of stepwise inputs of variable magnitude
     and their starting times. The first argument, n, is the number of input
     steps and start times. Next come a list of magnitudes and a list of
     initial times. Each list is comma-separated. The duration of input step
     is computed automatically by difference between the specified times.
     Currently this function can only be used in the simulation description
     file, and not in the model description file (which simply implies that
     you cannot use it as a default). Syntax:

     NDoses(<n>, <list-of-magnitudes>, <list-of-initial-times>);

   * Spikes() specifies a number of instantaneous inputs of variable
     magnitude and their exact times of occurence. The first argument, n, is
     the number of inputs and input times. Next come a list of magnitudes
     and a list of times. Each list is comma-separated. Currently this
     function can only be used in the simulation description file, and not
     in the model description file (which simply implies that you cannot use
     it as a default). Syntax:

     Spikes(<n>, <list-of-magnitudes>, <list-of-times>);

The arguments of input functions can either be constants or variables. As an
example, if `GavMag' and `RateConst' are defined model parameters, then the
input variable `Q_gav' can be defined as:

Q_gav = PerExp(GavMag, 60, 0, RateConst);

In this way the parameters of input functions can, for example, be assigned
statistical distributions in Monte Carlo simulations (see section Distrib()
specification).

Variable dependencies are resolved before the simulation is started. For
each of the periodic functions, a single exposure beginning at time
initial-time can be specified by giving an effectively infinite period, The
first period starts at the initial time of the simulation. Magnitudes change
exactly at the times given.

Input functions can be combined to give a lot of flexibility (e.g., an input
can be sum of some others). Separate inputs can be declared in the global
section of the model definition file and combined in the Dynamics and
CalcOutputs sections. The only limitation is that for each input function
used, a separate input must be defined in the model, even though this
function may not be a real element of the model.

Dynamics specifications

The dynamics specification section begins with the keyword Dynamics and is
enclosed in curly braces. The equations given in this section will be called
by the integrator at each integration step.

Additional variables to those declared in the global section may be used for
any calculations within the section. They will be declared as local
temporary variables. (Note, for example, the use of `Cout_fat' and `Cout_wp'
in the `perc.model' sample file). Local variables are not accessible from
the simulation program, or from other sections of the model definition file,
so don't try to output them.

Each state variable declared in the global section must have one
corresponding state equation in the Dynamics section. If a state equation is
missing, mod issues an error message such as:

Error: State variable 'Q_foo' has no dynamics.

If one or more differential equations are missing, no program file will be
created. Most error messages are self-explanatory. Where appropriate, they
also show a line number in the input file where the error occurred. Beware,
however, of cascades of errors generated as a consequence of a first one; so
don't panic: start by fixing the first one and rerun mod.

The derivative of a state variable is defined using the dt() operator, as
shown here:

dt(state-variable) '=' constant-value-or-expression ';'

The right-hand side can be any valid C expression, including standard math
library calls and the special functions mentioned above (see section Special
functions). Note, however, that no syntactic check is performed on the
library function calls. Their correctness is your responsibility.

The dt() operator can also be used in the right-hand side of equations in
the dynamics section to refer to the value of a derivative at that point in
the calculations. For example:

dt(Qm_in) = Qmetabolized - dt(Qm_out);

The integration variable (e.g., time) can be accessed if referred to as t,
as in:

dt(Qm_in) = Qmetabolized - t;

Output variables can also be made a function of t in the Dynamics section.

Note that while state variables, input variables and model parameters can
indeed be used on the right-hand side of equations, they cannot be assigned
values in the Dynamics section. If you need a parameter to change with time,
declare it as output variable in the global section. Assignments to inputs
or parameters in this section causes an error message like the following to
be issued:

Error: line 48: 'YourParm' used in invalid context.
Parameters cannot be defined in Dynamics{} section.

Parameter scaling

The parameter scaling section begins with the keyword Scale and is enclosed
in curly braces. The equations given in this section will define a function
that will be called by the simulation program at the beginning of each
simulation of an Experiment (see section Experiment definition). They can
therefore be used for initialization of the simulations.

All model variables and parameters can be changed in this section.
Modifications to state variables affect initial values only. Modifying an
input is not allowed and state variables can only appear at the left hand
side of equations.

The dt() operator (see section Dynamics specifications) cannot be used in
this section, since derivatives have not yet been computed when the scaling
function is called.

Output calculations

The output calculation section begins with the keyword CalcOutputs and is
enclosed in curly braces. The equations given in this section will be called
by the simulation program at each output time specified by a Print() or
PrintStep() statement (see section Print() specification, and see section
PrintStep() specification). In this way, the output scaling is done
efficiently, only when values are to be saved, and not at each integration
step.

Only variables that have been declared with the keyword Outputs can be
changed in this section.

Assignments to other types of variables cause an error message like the
following to be issued:

Error: line 56: 'Qb_fat' used in invalid context.
Only outputs can be defined in CalcOutputs{} section.

Any reference to an input or state variable will use the last calculated
(current) value of the input. The dt() operator can appear in the right-hand
side of equations, and it refers to values of the derivative as calculated
at the last time step (see section Dynamics specifications). Like in the
Dynamics section, the integration variable can be accessed if referred to as
t, as in:

Qx_out = DQx * t;

Comments on style

For your model file to be readable and understandable, it is useful to use a
consistent style of notation. The example file `perc.model' follows such a
consistent notation (see section `perc.model': A sample model description
file). For example we suggest that:

   * All variable names begin with a capital letter followed by meaningful
     lower case subscripts.
   * Where two subscripts are necessary, they can be separated by an
     underscore, such as in `Qb_fat'.
   * Where there is only one subscript an underscore can still be used to
     increase readability as in `Q_fat'.
   * Where two words are used in combination to name one item, they can be
     separated visually by capitalizing each word, as in `BodyWt'.

These conventions are suggestions only. The key to have a consistent
notation that makes sense to you. Consistency is one of the best ways to:

  1. Increase readability, both for others and for yourself. If you have to
     suspend work for a month or two and then come back to it, the last
     thing you want is to have to decipher your own file.
  2. Decrease the likelihood of mistakes. If all of the equations are coded
     with a consistent, logical convention, mistakes stand out more readily.

Note about models

MCSim can easily deal with algebraic models. You do not need to define state
variables or a Dynamics section for such models. Simply use input and output
variables and paramaters. The model can be specified in the CalcOutputs
section. You can use the time t if that is natural for your model. If you do
not use t in your model, you will still need to specify "output times" in
Print() or PrintStep() statements to obtain outputs: you can use an
arbitrary time, such as 1. If you do not use t you will also need do define
an Experiment (see section Experiment definition) for each combination of
values for the "independent" variables of your model. This may be clumsy if
many values are to be used. In that case, you may want to use the variable t
to represent something else than time.

Ordinary differential models, with algebraic components, can be setup easily
with MCSim. Use state variables and specify a Dynamics section. Time, t is
the integration variable, but here again, t can be used to represent
anything you want. We are not aware of cases in which MCSim has been used
for partial differential equations. Some problems might be solved by
implementing rudimentary line methods...

You can use MCSim for discrete-time dynamic models (or difference models).
It's a bit tricky. Assignments in CalcOutput are volatile (not memorized),
so the model equations have to be in Dynamics. But the model variables
should still be declared as outputs, because they should not be updated by
integration. However, you need at least a true state in the Dynamics
section, so you should declare a dummy one (and give it a constant
derivative of value zero). You also want the calls to Dynamics to be
precisely scheduled, so it is best to use the Euler integration routine (see
section Integrate() specification) which uses a constant step. Since Euler
may call repeatedly Dynamics at any given time, you want to guard against
untimely updating... Altogether, we recommend that you examine the sample
files `discrete.model' and `discrete.in' provided with the source code for
MCSim.

Specifying Simulations

After having your model defined and processed by mod, and the resulting
`model.c' file compiled with the MCSim routines, you are ready to run
simulations. For this you need to write a simulation file.

An example file `perc.lsodes.in', which works with the perchloroethylene
model, appears in Appendix (see section `perc.lsodes.in').

Using the compiled program

The simulation environment MCSim provides several types of simulations for
the models you create. Simulations are specified in a text file of format
similar to that of the model description file.

In Unix the command-line syntax for the MCSim program is:

mcsim [input-file [output-file]]

where the brackets indicate optional arguments. This assume that you have
not renamed the executable file; If you have, substitute the name of your
executable. If no input and output file names are specified, the program
will prompt you for them. If already one file name is given on the
command-line, the program will assume it specifies the input file and will
prompt you for the output file name. You must provide an input file name. If
you just hit the return key when prompted for the output name, the program
will use the name you have specified in the input file, if any, or a default
name. When the program starts up, it announces which model description file
it was created with. The input file describes the simulations to perform and
specifies which outputs should be printed out (see section Syntax of the
simulation definition file).

On the Macintosh you double-click the `MCSim' icon and enter the name of
your simulation definition file at the first prompt and then the name of the
output file (or just hit return if you want the default or the name you have
specified in the input file to be used).

Syntax of the simulation definition file

The file starts with a global declaration section followed by a number of
Experiment (i.e., simulation) definitions (see section Experiment
definition), eventually enclosed in a Level definition if Markov chain Monte
Carlo simulations are to be performed (see section Specifying a statistical
model). Each Experiment defines simulation conditions, from an initial time
(or whatever your dependent variable represents) to a final time. The
initial values of model state variables, parameter values, input variables,
and which outputs are to print at which times can all be changed in a given
Experiment.

The general syntax of the file is the same as that of mod with two
differences:

   * Variables can only be assigned constant values.
   * Input variables' assignments can use any input function (including the
     NDoses() function) or constant values.

Expressions are not allowed (unlike in the model definition file where they
can be used). Similarly, structural change to the model, for instance,
addition of a state, input, output or parameter, cannot be done here and
must be done in the model description file. The simulation specification
file is read until its end is reached, or until an End command is reached.

The general layout of the file is:

# Input file (this a comment)
<Global modifications and analysis specifications>
Experiment {
  <Specifications for first experiment>
}
Experiment {
  <Specifications for second experiment>
}
# Unlimited number of experiment specifications
End # Optional End statement. Everything after this line is ignored

For Markov chain Monte Carlo simulations (see section MCMC() specification),
the general layout of the file includes Level definitions:

# Input file (this is a comment)
SimType(MCMC);
<Global modifications and analysis specifications>
Level {
  # Up to 10 levels of hierarchy
  Experiment {
    Specifications for first experiment
  }
  Experiment {
    Specifications for second experiment
  }
  # Unlimited number of experiment specifications
} # end Level
End # Optional statement. Everything after this line is ignored

Global specifications

The global section is used to give specifications relevant to all
experiments, for example specification of the type of analysis, how the
integrator should work, parameter modifications to be used for all
experiments, etc.

Both the global section and each experiment section can contain
modifications to defined model variables. At the beginning of a simulation,
all model parameters are initialized to the nominal values specified in the
model description file. Next, modifications given in the global section are
applied, and finally any modifications for the current experiment are
applied.

SimType() specification

The type of analysis performed is specified using the SimType()
specification. Example:

SimType(MonteCarlo);

The following keywords can be used:

   * DefaultSim: the list of specified experiments is simulated,
   * MonteCarlo: the specified experiments are simulated several times with
     random input parameters (see section MonteCarlo() specification),
   * MCMC (previously Gibbs): Markov chain Monte Carlo simulations are
     performed to attain the posterior distribution of the model's
     parameters, given their prior distributions and data for which the
     likelihood function can be computed (see section MCMC() specification),
   * SetPoints: the experiments are simulated using several lists of
     user-defined parameters values in input (see section SetPoints()
     specification).

If MonteCarlo, MCMC, or SetPoints simulations are requested, additional
specifications are needed (see below).

Integrate() specification

The integrator settings can be changed with the Integrate specification. Two
integration routines are provided: Lsodes (which originates from the SLAC
Fortran library and is originally based on Gear's routine) (Gear, 1971b;
Gear, 1971a; Press et al., 1989) (see section Bibliographic References) and
Euler (Press et al., 1989).

The syntax for Lsodes is:

Integrate(Lsodes, <rtol>, <atol>, <method>);

Rtol is a scalar specifying the relative error tolerance for each
integration step. Atol is a scalar specifying the absolute error tolerance
parameter. They apply to all integration variables (state variables). The
estimated local error in a state variable y(i) will be controlled so as to
be roughly less (in magnitude) than Thus the local error test passes if, in
each component, either the absolute error is less than atol, or the relative
error is less than rtol. Set rtol toÊzero for pure absolute error control,
and use atol to zero for pure relative error control. Caution: actual
(global) errors may exceed these local tolerances, so choose them
conservatively. The method flag should be 0 (zero) for non-stiff
differential systems and 1 for stiff systems. You should try both and select
the fastest for equal accuracy of output, unless insight from your system
leads you to choose a priori. In our experience, a good starting point for
atol and rtol is about

The syntax for Euler is:

Integrate(Euler, <time-step>, 0, 0);

time-step is a scalar specifying the constant time-step to be taken at each
integration step. The next two scalars are reserved for future use and
should be set to zero.

If the Integrate() specification is not used, the default integration method
is Lsodes with parameters We recommend using Lsodes, since is it highly
accurate and efficient. Euler can be used for special applications (e.g., in
system dynamics) where a constant time step and a simple algorithm are
needed.

OutputFile() specification

The OutputFile() specification allows you to specify a name for the output
file of DefaultSim simulations. If this specification is not given the name
`sim.out' is used if none has been supplied on the command-line or the
initial dialog. The corresponding syntax is:

OutputFile("<OutputFilename>");

MonteCarlo() specification

Monte Carlo simulations (Hammersley and Handscomb, 1964; Manteufel, 1996)
(see section Bibliographic References) require the use of two
specifications, MonteCarlo() and Distrib(), which must appear in the global
section of the file, before the Experiment sections. Such Monte Carlo
specifications are ignored if they appear in an Experiment specification.

The MonteCarlo specification gives general information required for the
runs: the output file name, the number of runs to perform, and a starting
seed for the random number generator. Its syntax is:

MonteCarlo("<OutputFilename>", <nRuns>, <RandomSeed>);

The output filename is a string field and must be enclosed in quotes. If a
null-string "" is given, the default name `simmc.out' will be used. The seed
of the pseudo-random number generator can be any positive real number. Seeds
between 1.0 and 2147483646.0 are used as is, others are rescaled within
those bounds (and a warning is issued). Here is an example of use:

MonteCarlo("percsimmc.out", 5, 9386.630);

The parameters' sampling distributions are specified by a list of Distrib
specifications, as explained in the next section. The format of the output
file of Monte Carlo simulations is discussed later (see section Analyzing
results).

Distrib() specification

This specification indicates which variable to sample, and its sampling
distribution. One Distrib() specification must be included for each variable
to sample. The specification file can include any number of these commands
at the global level, or within any Level section in the case of Markov chain
Monte Carlo sampling (see section Specifying a statistical model). The
syntax is:

Distrib(<identifier>, <iType>, [<shape parms>]);

The iType field specifies the sampling distribution to use and can be one of
following:

   * Uniform,
   * LogUniform,
   * Normal,
   * Normal_v,
   * LogNormal,
   * LogNormal_v,
   * TruncNormal,
   * TruncNormal_v,
   * TruncLogNormal,
   * TruncLogNormal_v,
   * Beta,
   * Gamma,
   * InvGamma,
   * Chi2,
   * Exponential,
   * Binomial,
   * Poisson,
   * Piecewise.

The corresponding shape parameters (Bernardo and Smith, 1994; Gelman et al.,
1995) (see section Bibliographic References) are as follow:

   * Uniform and log-uniform distributions: minimum and maximum of the
     sampling range, real numbers in natural space.
   * Normal distribution (two reals numbers): mean and standard deviation,
     the latter being stricly positive. The variant Normal_v takes the
     variance instead of the standard deviation as second parameter.
   * Truncated normal distribution (four reals numbers): mean, standard
     deviation (stricly positive), minimum and maximum. The variant
     TruncNormal_v takes the variance instead of the standard deviation as
     second parameter.
   * LogNormal distribution (two reals numbers): geometric mean (exponential
     of the mean in log-space) and geometric standard deviation
     (exponential, stricly superior to 1, of the standard deviation in
     log-space). The variant LogNormal_v takes the variance (in log-space!)
     instead of the standard deviation as second parameter.
   * Truncated Lognormal distribution (four reals numbers): geometric mean
     and geometric standard deviation (stricly superior to 1), minimum and
     maximum in natural space. For example:

     Distrib(Var, TruncLogNormal, 1, 2.718, 0.01, 10)

     samples Var such that while Var is truncated to fall between 0.01 to
     10. The variant TruncLogNormal_v takes the variance (in log-space!)
     instead of the standard deviation as second parameter.
   * Beta distribution (at least two strictly positive real numbers): A and
     B. By default the Beta distribution is defined over the interval [0;1].
     If a range is given for the beta distribution, the [0;1] interval is
     mapped onto the specified range.
   * Gamma distribution (two strictly positive real numbers): shape a and
     inverse scale b.
   * Inverse-gamma distribution (two strictly positive real numbers): shape
     a and scale b.
   * Chi-squared distribution (one strictly positive real number): n. This
     distribution is the same as Gamma(n/2, 1/2).
   * Exponential distribution (one strictly positive real number):
     inverse-scale b.
   * Binomial distribution (two strictly positive numbers, a real and an
     integer): p (in the interval [0;1]), and N. If N is not input as an
     integer it will be rounded down during the simulations.
   * Poisson distribution (a strictly positive real): rate l.
   * Piecewise distribution (four reals): minimum, a, b, maximum. The
     distribution has the form of a truncated triangle, with a plateau
     between a and b. If the distribution is the triangular distribution.

The shape parameters of the above distribution specifications can reference
other parameters, provided than distributions for these have already been
defined. For example:

Distrib(A, Normal, 0, 1);
Distrib(B, Normal, A, 2);

MCMC() specification

Markov chain Monte Carlo (MCMC) simulations, used in a Bayesian context,
allow the user to specify a statistical model (eventually hierarchical) and
sample parameters from their joint posterior distribution, given a prior
distribution for each parameter, a set of data to simulate, and
corresponding likelihoods. Sampling from the posterior is not immediate: it
requires the simulation chain, which start by sampling purely from the
prior, to reach equilibrium. Checking that equilibrium is obtained is best
achieved, in our opinion, by running multiple independent chains. Hence
these computations are very intensive. For a discussion of Markov chain
Monte Carlo and convergence issues you should consult the appropriate
statistical literature (for example, Bernardo and Smith, 1994; Gelman, 1992;
Gelman et al., in press; Gelman et al., 1995; Gelman and Rubin, 1992; Smith,
1991; Smith and Roberts, 1993) (see section Bibliographic References).
Technically, MCSim uses Metropolis-Hasting sampling and you do not need to
worry about issues of conjugacy or log-concavity of your prior or posterior
distributions. Like simple Monte Carlo simulations, MCMC simulations require
the use of two specifications, MCMC() and Distrib() and of one special
section definition: Level. The syntax for the MCMC() specification is:

MCMC("<OutputFilename>", "<RestartFilename>", "", <nRuns>,
     <simTypeFlag>, <printFrequency>, <itersToPrint>, <RandomSeed>);

The output filename is a string field and must be enclosed in quotes. If a
null-string "" is given, the default name `MCMC.default.out' will be used.

If a restart file name (enclosed in quotes) is given, the first simulations
will be read from that file (which must be a text file). This allows you to
continue a chain where you left it, since an MCMC output file can be used as
a restart file with no change. Note that the first line of the file (which
typically contains column headers) is skipped. Also, the number of lines in
the file must be less than or equal to nRuns. The first column of the file
should be integers, and the following columns (tab- or space-separated)
should give the various parameters, in the same order as specified in the
list of Distrib() specifications in the input file. The third field is
reserved for future use and should just be a pair of empty quotes.

The integer nRuns gives the total number of runs to be performed, including
the runs eventually read in the restart file. The next field, simTypeFlag
should be either 0, 1, or 2. It should be set at zero to start a chain of
MCMC simulations. In that case, parameters are updated by Metropolis steps,
one at a time. If the value of simTypeFlag is set to 1 or 2, a restart file
must also be specified. In the case of 1, the output file will contain codes
for the level sequence, experiment numbers, printing times, data values and
the corresponding model predictions, computed using the last parameter
vector of the restart file. This is useful to quickly check the model fit to
the data. If simTypeFlag is equal to 2, the entire restart file is used to
compute the parameters' covariance matrix. All parameters are then updated
at once using a multivariate normal kernel as proposal distribution of the
Metropolis steps. This results in large improvement in speed. However, we
recommend that this option be used only when convergence is approximately
obtained (therefore, you should run MCMC simulations with simTypeFlag set to
0 first, up to approximate convergence, and then restart the chain with the
flag at 2).

The integer printFrequency should be set to 1 if you want an output at each
iteration, to 2 if you want an output at every other iteration etc.
itersToPrint is the number of final iterations for which output is required
(e.g., 1000 will request output for the last 1000 iterations; to print all
iterations just set this parameter to the value of nRuns). Note that if no
restart file is used, the first iteration is always printed, regardless of
the value of itersToPrint. Finally, the seed of the pseudo-random number
generator can be any positive real number. Seeds between 1.0 and
2147483646.0 are used as is, others are rescaled silently within those
bounds.

To use the MCMC specification, you must define a statistical model precising
each parameter's prior distribution, or conditional distribution (in the
case of a hierarchical model), and the data likelihood (i.e., the
distribution of observation errors). These distributions must be enclosed in
a Level section and are specified with Distrib() statements (see section
Specifying a statistical model). In the context of MCMC sampling, MCSim
provides an extension of the Distrib() specification. First, the first two
shape parameters of distributions may depend on other model parameters. For
example:

Distrib(A, Normal, 0, 1);
Distrib(B, Normal, A, C);

The data distribution is given by a similar statement, which uses the
specification Prediction() to differentiate data from their predicted
counterparts. The Prediction() specification can be used for the first two
shape parameters only (therefore, not for ranges, except in the case of
uniform or loguniform distributions). If Prediction() is used for the first
shape parameter, the variable enclosed in parentheses must be the same as
the variable whose distribution is described. There should be one and only
one distribution specified for a given type of data in the whole input file
(i.e., you cannot redefine a likelihood; this limitation will hopefully be
removed in a future release). Note that only states and outputs can use
Prediction() specifications (but you can always define an output to be equal
to a parameter or an input in your model file). For example:

Distrib(y, TruncNormal, Prediction(y), Prediction(z), -10, 10);

To recapitulate, the extended Distrib() syntax, for use with MCMC
simulations is therefore:

Distrib(<identifier>, <iType>, [<shape parms>]);

where the first two shape parameters can be Prediction(<identifier>), or any
model parameter or numerals, and the last two shape parameters numerals only
(this limitation will also be removed in a future release).

If a statement like:

Distrib(Var, <iType>, Prediction(<Var>), Prediction(<Other_Var>), ...);

is used, the two variables Var and Other_Var must have identical output
times. It is then useful to group them in the same Print() statement.

The other tool MCSim brings you to build a complete statistical model is the
Level keyword. The use of this keyword, is described below (see section
Specifying a statistical model).

Finally, the format of the output file of MCMC simulations is discussed in a
later section (see section Analyzing results).

SetPoints() specification

To impose a series of set points (i.e., already tabulated values for the
parameters), the global section can include a SetPoints() specification. It
allows you to perform additional simulations with previously Monte Carlo
sampled parameter values, eventually filtered. You can also generate
parameters values in a systematic fashion, over a grid for example, with
another program, and use them as input to MCSim. Importance sampling, latin
hypercube sampling, grid sampling, can be accommodated in this way.

This command specifies an output filename, the name of a text file
containing the chosen parameter values, the number of simulations to perform
and a list of model parameters to vary. It has the following syntax:

SetPoints("<OutputFilename>", "<SetPointsFilename>", <nRuns>,
          <identifier>, <identifier>, ...);

If a null string is given for the output filename, the set points output
will be written to the same default output file used for Monte Carlo
analyses, `simmc.out'.

The set points file name is required and must refer to an existing file
containing the parameter values to use. The first line of the set points
file is skipped and can contain column headers, for example. Each of the
other lines should contain an integer (e.g., the line number) followed by
values of the various parameters in the order indicated in the SetPoints()
specification. If extra fields are at the end of each line they are skipped.
The first integer field is needed but not used (this allows you to directly
use Monte Carlo output files for additional SetPoints simulations).

The variable nRuns should be less or equal to the number of lines (minus
one) in the set points file. If a zero is given, all lines of the file are
read. The format of the output file of set points simulations is discussed
below (see section Analyzing results).

Following the SetPoints() specification, Distrib() statements can be given
for parameters not already in the list (see section Distrib()
specification). These parameters will be sampled accordingly before to
performing each simulation. The shape parameters of the distribution
specifications can reference other parameters, including those of the list.

Specifying basic conditions to simulate

Any simulation file must define at least one Experiment to simulate.

Experiment definition

After global simulation specifications, "experiments" must be included in
the input file. They define simulation conditions and specify outputs. An
"experiment" section starts with the keyword Experiment and is enclosed in
curly braces.

An "experiment" can make modifications to any model variable or parameter
that was defined in the global section of the model description file. The
syntax is the same, except that variables or parameters can only take
constant values. So, for example, in an experiment the body weight could be
modified with:

BodyWt = 83.2;

This overrides any previously assigned values, even if randomly sampled, for
the specified parameter.

Inputs can be redefined with the input functions listed in the Mod reference
section above (see section Defining Models). Input functions can reference
other variables (eventually random), as in:

Q_gav = PerExp(GavMag, 60, 0, RateConst);

The maximum number of experiments definable is 200. This can be changed by
changing MAX_INSTANCES and MAX_EXPERIMENTS in the header file `sim.h' and
recompiling. Within an experiment definition, or at the global level (if you
want them to apply to all experiments), several additional specifications
can also be used:

   * StartTime(),
   * Print(),
   * PrintStep(),
   * Data().

StartTime() specification

The origin of time for a simulation, if it needs to be defined, is specified
with the StartTime() specification:

StartTime(<initial-time>);

If this specification is not given, a value of zero is used by default. The
final time is automatically computed to match the largest output time
specified in the Print() or PrintStep() statements (see section Print()
specification; see section PrintStep() specification).

Print() specification

The value of any model variable, input, output or parameter can be requested
for output with Print() specifications. Their arguments are a list of names
of variables (at least one and up to 10), and a list of increasing times at
which to output their value:

Print(<identifier1>, <identifier2>, ..., <time1>, <time2>, ...);

The same output times are used for all the variables specified. The size of
the time list is only limited by the available memory. The limit of 10
variables names can be increased by changing MAX_PRINT_VARS in the header
file `sim.h' and recompiling. The number of Print() statements you can used
in a given Experiment section is only limited by the available memory.

PrintStep() specification

The value of any model variable, input or parameter can be also output with
PrintStep specifications. They allow dense printing, suitable for smooth
plots, for example. The arguments are the name of only one variable, the
first output time, the last one, and a time increment:

PrintStep(<identifier>, <start-time>, <end-time>, <time-step>);

The final time has to be superior to the initial time and the time step has
to be less than the time span between end and start. If the time step is not
an exact divider of the time span the last printing step is shorter and the
last output time is still the end-time specified. The number of outputs
produced is only limited by the memory available at run time. You can use
several PrintStep() in a given Experiment section.

Data() specification

Experimental observations of model variables, inputs, outputs, or
parameters, can be specified with the Data() command. Markov chain Monte
Carlo sampling requires that you specify Data() statements (see section
MCMC() specification; see section Specifying a statistical model). The data
are then used internally to evaluate the likelihood function for the model.
The arguments are the name of the variable for which observations exist, and
a list of data values:

Data(<identifier>, <value1>, <value2>, ...);

This specification can only be used with a matching Print() or PrintStep()
for the same variable (see section Print() specification; see section
PrintStep() specification). You must make sure that there are as many data
values in the Data() specification as output time requested in the
corresponding Print() or PrintStep(). A data value of "-1" is treated as
"missing data" and ignored in likelihood calculations. The convention "-1"
can be changed by changing INPUT_MISSING_VALUE in the header file `mc.h' and
recompiling.

Specifying a statistical model

Statistical models are defined in the simulation specification file, rather
than in the model definition file. It is necessary to define a statistical
model (with parameter dependencies, prior distributions and likelihood) if
you want to use MCMC sampling. MCMC sampling will then give you in output a
sample of parameters drawn from their joint posterior distribution. Take for
example the simple linear regression model:

The first thing to do is to define a model to compute y as a function of x.
Here is such a model (quite similar to the one distributed with MCSim source
code (see section `linear.model'):

# ---------------------------------------------
# Model definition file for a linear model
# ---------------------------------------------
Outputs = {y};

# Model parameters
Alpha = 0;
Beta = 0;
Sigma2 = 1;
x_bar = 0;

CalcOutputs { y = Alpha + Beta * (t - x_bar); }
# ---------------------------------------------

The parameters' initialization values are arbitrary, and could be anything
reasonable. They will be changed or sampled through the input file. Note
that is not used in the model equations, but still needs to be defined here
in order to be part of the statistical model. On the other hand, Finally x
is replaced by the time, t, for convenience. An alternative would be to
define an input `x' and use it instead of t.

We now need to write an input file specifying the distribution of y (i.e.,
the likelihood), and the prior distributions of the various parameters. Here
is what such a file could look like:

# ---------------------------------------------------------------
# Simulation input file for a linear regression
# ---------------------------------------------------------------
SimType (MCMC);
MCMC ("linear.MCMC.out", "", "", 50000, 0, 5, 40000, 63453.1961);
Level {
  Distrib(Alpha,  Normal_v, 0, 10000);
  Distrib(Beta,   Normal_v, 0, 10000);
  Distrib(Sigma2, InvGamma, 0.01, 0.01);
  Distrib(y, Normal_v, Prediction(y), Sigma2);
  Experiment {
    x_bar = 3.0;
    PrintStep (y, 1, 5, 1);
    Data  (y, 1, 3, 3, 3, 5);
  }
} # end Level
End
# ---------------------------------------------------------------

The file begins the obvious SimType() (see section SimType() specification)
and MCMC() (see section MCMC() specification) keywords. The keyword Level
comes next. Level is used to specify the dependence between model parameters
in a hierarchy. There should be at least one Level in every MCMC input file,
even for a non-hierarchical model like the one above (actually,
"non-hierarchical" models can be thought of has having only one level of
hierarchy). See below for further discussion of the Level keyword. You can
also look at the MCMC input files provided as examples with MCSim source
code. The Distrib() statements define the parameter priors. Normal_v
specifications are used since we use variances instead of standard
deviations. The inverse-Gamma distribution is used for the variance
component, since the precision is supposed to be Gamma-distributed. The
likelihood is the distribution of the data, given the model: it is also
specified by a Distrib() statement, valid for every y data point. Again,
note that the variable is not used. Instead the Prediction(y) specification
is used to signify the linear model output. These distributions are in
effect for every sub-level or every Experiment included in the current
level.

The "simulations" to perform, and the corresponding data values, are
specified by the Experiment section. Only one Experiment is needed here, but
several could be specified. In this section, the value of is provided. The
different values of x (time in our model) can be specified via a PrintStep()
specification (see section PrintStep() specification), since they are
equally spaced. More generally, a Print() specification could have been used
(see section Print() specification). The data values are given in a Data()
statement.

Level definition

Markov chain Monte Carlo simulations require the definition of a statistical
model and the use of the Level keyword. At least one level must be defined.
A level section starts with the keyword Level and is enclosed in curly
braces. It can include any number of sub-levels or Experiments. Experiments
(where the data are specified) form the lowest level of the hierarchy (see
section Experiment definition. There must be one and only one top level and
at most 10 sub-levels in the hierarchy. This limit of 10 levels can be
increased (up to 255) by changing MAX_LEVELS in the header file `sim.h' and
recompiling.

A level can make modifications to the sampling distribution of any model
parameter. For example:

Level { # this is the top level

  Distrib(A, Uniform, 0, 1);

  Level { # this is sub-level 1
    Distrib(A, Normal, A, 1);
    Experiment { ... } # experiment 1
    Experiment { ... } # experiment 2
  }
}

These distribution assignments apply to all sub-levels of the level where
they take place. If several assignments are given, their position within the
level section is irrelevant (although a logical order is recommended for
clarity).

A level can also make modifications to any model parameter that was defined
in the global section of the model description file. The syntax is the same,
except that variables can only take constant values. So, for example, in an
experiment, the parameter A could be modified with:

A = 2.0;

This overrides any previously assigned values, even if randomly sampled, for
the specified parameter. This assignment also applies to the sub-levels of
the level where they take place.

An important concept to grasp here is that of "instance". In the code
fragment given above, the parameter A, defined at sub-level 1, is "cloned"
as many times as there are sub-levels or experiments enclosed in sub-level 1
(hence, it will be cloned twice in the example above, once for each
Experiment defined). In that way, the parameters distributions defined at
one level in fact apply to the next lower sub-level, or at the Experiment
level. This convention saves a lot writing and effort in the long run. For
example, the uniform distribution assigned to A, at the top level, applies
to the sub-level 1. There is only one "clone" of A at sub-level 1 since only
one sub-level is included in the top level. In contrast, two
normally-distributed "clones" of A will be defined and sampled. The first
one will apply to experiment 1, and will be conditioned by the data of that
experiment only, and the other will apply to experiment 2. A total of three
variables of "type" A will be sampled and will be printed in the output file
(coded so that the position in the hierarchy is apparent): the "parent"
A(1), a priori uniformly distributed, and two "dependents" A(1.1) and
A(1.2), a priori normally distributed around A(1).

Analyzing results

The output from Monte Carlo or SetPoints simulations is a tab-delimited text
file with one row for each run (i.e., parameter set) and one column for each
parameter and output in the order specified. Thus each line of the output
file is in the following order:

<# of run> <parameters> <outputs for Exp 1> <outputs for Exp2> ...

The parameters are printed in the order they were sampled or set.

The first line gives the column headers. A variable called name requested
for output in an experiment i at a time j is labeled name_i.j.

The output of Markov chain Monte Carlo simulations is also a text file with
one row for each run. It displays a column of iteration labels, and one
column for each parameter sampled. The last three columns contain
respectively, the sum of the logarithms of each parameter's density given
its parents' values (`LnPrior'), the logarithm of the data likelihood
(`LnData'), and the sum of the previous two values (`LnPosterior'). The
first line gives the column headers. On this line, parameters names are
tagged with a code identifying their position in the hierarchy defined by
the Level statements. For example, the second instance of a parameter called
name placed at the fist level of the hierarchy is labeled name(2); the first
instance of the same parameter placed at the second instance of the second
level of the hierarchy is labeled name(2.1), etc.

The tab-delimited file can easily be imported into your favorite
spreadsheet, graphic or statistical package for further analysis.

Error Handling

If integration fails for an Experiment in DefaultSim simulations no output
is generated for that experiment, and the user is warned by an error message
on the screen. In MonteCarlo or SetPoints simulations, the corresponding
simulation line is not printed, but the iteration number is incremented.
Finally, in MCMC simulations, the parameter for which the data likelihood
was computed is simply not updated (which implicitly forbids the
uncomputable region of the parameter space). In all cases an error message
is given on the screen, or wherever the screen output has been redirected.

Bibliographic References

Barry, T. M. (1996). Recommendations on the testing and use of pseudo-random
number generators used in Monte Carlo analysis for risk assessment. Risk
Analysis 16:93-105.

Bernardo, J. M. and Smith, A. F. M. (1994). Bayesian Theory. Wiley, New
York.

Bois, F. Y., Gelman, A., Jiang, J., Maszle, D., Zeise, L. and Alexeef, G.
(1996). Population toxicokinetics of tetrachloroethylene. Archives of
Toxicology 70:347-355.

Bois, F. Y., Zeise, L. and Tozer, T. N. (1990). Precision and sensitivity
analysis of pharmacokinetic models for cancer risk assessment:
tetrachloroethylene in mice, rats and humans. Toxicology and Applied
Pharmacology 102:300-315.

Gear, C. W. (1971a). Algorithm 407 - DIFSUB for solution of ordinary
differential equations [D2]. Communications of the ACM 14:185-190.

Gear, C. W. (1971b). The automatic integration of ordinary differential
equations. Communications of the ACM 14:176-179.

Gelman, A. (1992). Iterative and non-iterative simulation algorithms.
Computing Science and Statistics 24:433-438.

Gelman, A., Bois, F. Y. and Jiang, J. (1996). Physiological pharmacokinetic
analysis using population modeling and informative prior distributions.
Journal of the American Statistical Association 91:1400-1412.

Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (1995). Bayesian
Data Analysis. Chapman & Hall, London.

Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation
using multiple sequences (with discussion). Statistical Science 7:457-511.

Hammersley, J. M. and Handscomb, D. C. (1964). Monte Carlo Methods. Chapman
and Hall, London.

Manteufel, R. D. (1996). Variance-based importance analysis applied to a
complex probabilistic performance assessment. Risk Analysis 16:587-598.

Park, S. K. and Miller, K. W. (1988). Random number generators: good ones
are hard to find. Communications of the ACM 31:1192-1201.

Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T.
(1989). Numerical Recipes (2st ed.). Cambridge University Press, Cambridge.

Smith, A. F. M. (1991). Bayesian computational methods. Philosophical
Transactions of the Royal Society of London, Series A 337:369-386.

Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computation via the
Gibbs sampler and related Markov chain Monte Carlo methods. Journal of the
Royal Statistical Society Series B 55:3-23.

Vattulainen, I., Ala-Nissila, T. and Kankaala, K. (1994). Physical tests for
random numbers in simulations. Physical Review Letters 73:2513-2516.

Common Pitfalls

The following mistakes are particularly easy to make, and sometimes hard to
notice, or understand at first.

   * Putting a space before the end of line ; in the model definition file
     may causes strange error messages.
   * Forgetting about type-related arithmetics in C: `1000/882' gives `1'
     since it is interpreted as an integer division by the compiler. To get
     a floating-point (usual) division use `1000./882.'.

Using make

Make is a utility that facilitates doing repetitive tasks like compilation.
A `makefile' is a text file that contains a description of what make should
do and under what circumstances. For example the compilation `Makefile'
included with the MCSim distribution only compile a C-file if it has changed
since the last compilation. This means that when you change your model and
create a new `model.c' file using mod, only the `model.c' file needs to be
compiled to recreate the simulation engine.

Before you run make for the first time on a machine you must change some
settings in the makefile to specify where your C compiler is on your file
system, and some special settings for that compiler. Refer to the
documentation (or manual pages in Unix) for your compiler to do this. In the
makefile file there are several variables defined which you may need to
change. They are described in the makefile itself.

You run the make program by entering make or make -f Makefile at the prompt
from the directory where the program is that you want to compile.

Examples

You will find here some examples of model description files and simulation
iput files.

`linear.model'

# Linear Model with a random component
# y = A + B * time + N(0,SD_true)
# Setting SD_true to zero gives the deterministic version
#---------------------------------------------------------

# Outputs
Outputs = {y};

# Model Parameters
A = 0;
B = 1;
SD_true = 0;
SD_esti = 0;

CalcOutputs { y = A + B * t + NormalRandom(0,SD_true); }

`1cpt.model': A sample model description file

# One Compartment Model
# First order input and output
#---------------------------------------------------------

# Inputs
Inputs = {Dose};

# Outputs
Outputs = {C_central, AUC, ln_C_central, ln_AUC,
           SD_C_computed, SD_A_computed};

# Model Parameters
ka = 1;
ke = 0.5;
F  = 1;
V  = 2;

# Statistical Parameters
SDb_ka = 0;
SDw_ka = 0;
SDb_ke = 0;
SDw_ke = 0;
SDb_V  = 0;
min_F  = 0;
max_F  = 0;
SD_C_central = 0;
SD_AUC       = 0;
CV_C_cen     = 0;
CV_AUC       = 0;
CV_C_cen_true = 0;
CV_AUC_true   = 0;

# Calculate Outputs
CalcOutputs {

  # algebraic equation for C_central
  C_central = (ka != ke ?
               (exp(-ke * t) - exp(-ka * t)) *
               F * ka * Dose / (V * (ka - ke))):
               exp(-ka * t) * ka * t * F * Dose / V);

  # algebraic equation for AUC
  AUC = (ka != ke ?
         ((1 - exp(-ke * t)) / ke - (1 - exp(-ka * t)) / ka) * F * ka * Dose /
         (V * (ka - ke))):
         F * Dose * (1 - (1 + ka * t) * exp(-ka * t)) / (V * ke));

  C_central = C_central + NormalRandom(0, C_central * CV_C_cen_true);
  AUC       = AUC + NormalRandom(0, AUC * CV_AUC_true);

  ln_C_central = (C_central > 0 ? log (C_central) : -100);
  ln_AUC = (AUC > 0 ? log (AUC) : -100);

  SD_C_computed = (C_central > 0 ? C_central * CV_C_cen : 1e-10);
  SD_A_computed = (AUC > 0 ? AUC * CV_AUC : 1e-10);

} # End of output calculations

`perc.model': A sample model description file

#---------------------------------------------------------
# perc.model
# A four compartment model of Tetrachloroethylene (PERC)
# and total metabolites.
# Copyright (c) 1993.  Don Maszle, Frederic Bois.  All rights reserved.
#---------------------------------------------------------
# States are quantities of PERC and metabolite formed, they can be output

States = {Q_fat,        # Quantity of PERC in the fat
          Q_wp,         #   ...   in the well-perfused compartment
          Q_pp,         #   ...   in the poorly-perfused compartment
          Q_liv,        #   ...   in the liver
          Q_exh,        #   ...   exhaled
          Qmet};        # Quantity of metabolite formed

# Extra outputs are concentrations at various points

Outputs = {C_liv,               # mg/l in the liver
           C_alv,               # ... in the alveolar air
           C_exh,               # ... in the exhaled air
           C_ven,               # ... in the venous blood
           Pct_metabolized,     # % of the dose metabolized
           C_exh_ug};           # ug/l in the exhaled air

Inputs = {C_inh}                # Concentration inhaled

# Constants
# Conversions from/to ppm: 72 ppm = .488 mg/l

PPM_per_mg_per_l = 72.0 / 0.488;
mg_per_l_per_PPM = 1/PPM_per_mg_per_l;

#---------------------------------------------------------
# Nominal values for parameters
# Units:
# Volumes: liter
# Vmax:    mg / minute
# Weights: kg
# Km:      mg / minute
# Time:    minute
# Flows:   liter / minute
#---------------------------------------------------------

InhMag = 0.0;
Period = 0.0;
Exposure = 0.0;

C_inh = PerDose (InhMag, Period, 0.0, Exposure);

LeanBodyWt = 55;    # lean body weight

# Percent mass of tissues with ranges shown

Pct_M_fat  = .16;   # % total body mass
Pct_LM_liv = .03;   # liver, % of lean mass
Pct_LM_wp  = .17;   # well perfused tissue, % of lean mass
Pct_LM_pp  = .70;   # poorly perfused tissue, will be recomputed in scale

# Percent blood flows to tissues

Pct_Flow_fat = .09;
Pct_Flow_liv = .34;
Pct_Flow_wp  = .50; # will be recomputed in scale
Pct_Flow_pp  = .07;

# Tissue/blood partition coeficients

PC_fat = 144;
PC_liv = 4.6;
PC_wp  = 8.7;
PC_pp  = 1.4;
PC_art = 12.0;

Flow_pul   = 8.0;    # Pulmonary ventilation rate (minute volume)
Vent_Perf = 1.14;    # ventilation over perfusion ratio

sc_Vmax = .0026;     # scaling coeficient of body weight for Vmax

Km = 1.0;

# The following parameters are calculated from the above values in
# the Scale section before the start of each simulation.
# They are left uninitialized here.

BodyWt = 0;

V_fat = 0;           # Actual volume of tissues
V_liv = 0;
V_wp  = 0;
V_pp  = 0;

Flow_fat = 0;        # Actual blood flows through tissues
Flow_liv = 0;
Flow_wp  = 0;
Flow_pp  = 0;

Flow_tot = 0;        # Total blood flow
Flow_alv = 0;        # Alveolar ventilation rate

Vmax = 0;            # kg/minute

#---------------------------------------------------------
# Dynamics
# Define the dynamics of the simulation. This section is
# calculated with each integration step. It includes
# specification of differential equations.
#---------------------------------------------------------

Dynamics {

# Venous blood concentrations at the organ exit

Cout_fat = Q_fat / (V_fat * PC_fat);
Cout_wp  = Q_wp  / (V_wp  * PC_wp);
Cout_pp  = Q_pp  / (V_pp  * PC_pp);
Cout_liv = Q_liv / (V_liv * PC_liv);

# Sum of Flow * Concentration for all compartments

dQ_ven = Flow_fat * Cout_fat + Flow_wp * Cout_wp
         + Flow_pp * Cout_pp + Flow_liv * Cout_liv;

# Venous blood concentration

C_ven =  dQ_ven / Flow_tot;

# Arterial blood concentration
# Convert input given in ppm to mg/l to match other units

C_art = (Flow_alv * C_inh / PPM_per_mg_per_l +  dQ_ven) /
        (Flow_tot + Flow_alv / PC_art);

# Alveolar air concentration

C_alv = C_art / PC_art;

# Exhaled air concentration

C_exh = 0.7 * C_alv + 0.3 * C_inh / PPM_per_mg_per_l;

# Differentials

dt (Q_exh) = Flow_alv * C_alv;
dt (Q_fat) = Flow_fat * (C_art - Cout_fat);
dt (Q_wp)  = Flow_wp  * (C_art - Cout_wp);
dt (Q_pp)  = Flow_pp  * (C_art - Cout_pp);

# Quantity metabolized in liver

dQmet_liv = Vmax * Q_liv / (Km + Q_liv);
dt (Q_liv) = Flow_liv * (C_art - Cout_liv) - dQmet_liv;

# Metabolite formation

dt (Qmet)  = dQmet_liv;

} # End of Dynamics

#---------------------------------------------------------
# Scale
# Scale certain model parameters and resolve dependencies
# between parameters. Generally the scaling involves a
# change of units, or conversion from percentage to actual
# units.
#---------------------------------------------------------

Scale {

# Volumes scaled to actual volumes

BodyWt = LeanBodyWt/(1 - Pct_M_fat);

V_fat = Pct_M_fat  * BodyWt/0.92;        # density of fat = 0.92 g/ml
V_liv = Pct_LM_liv * LeanBodyWt;
V_wp  = Pct_LM_wp  * LeanBodyWt;
V_pp  = 0.9 * LeanBodyWt - V_liv - V_wp; # 10% bones

# Calculate Flow_alv from total pulmonary flow

Flow_alv = Flow_pul * 0.7;

# Calculate total blood flow from the alveolar ventilation rate and
# the V/P ratio.

Flow_tot = Flow_alv / Vent_Perf;

# Calculate actual blood flows from total flow and percent flows

Flow_fat = Pct_Flow_fat * Flow_tot;
Flow_liv = Pct_Flow_liv * Flow_tot;
Flow_pp  = Pct_Flow_pp  * Flow_tot;
Flow_wp  = Flow_tot - Flow_fat - Flow_liv - Flow_pp;

# Vmax (mass/time) for Michaelis-Menten metabolism is scaled
# by multiplication of bdw^0.7

Vmax = sc_Vmax * exp (0.7 * log (LeanBodyWt));

} # End of model scaling

#---------------------------------------------------------
# CalcOutputs
# The following outputs are only calculated just before values
# are saved.  They are not calculated with each integration step.
#---------------------------------------------------------

CalcOutputs {

# Fraction of TCE metabolized per day

Pct_metabolized = (InhMag ?
                   Qmet / (1440 * Flow_alv * InhMag * mg_per_l_per_PPM) :
                   0);

C_exh_ug  = C_exh * 1000; # milli to micrograms

} # End of output calculation

`perc.lsodes.in'

#---------------------------------------------------------
# perc.lsodes.in
#
# Copyright (c) 1993.  Don Maszle, Frederic Bois.  All rights reserved.
#
#---------------------------------------------------------

SimType (DefaultSim);

Integrate (Lsodes, 1e-4, 1e-6, 1);

#---------------------------------------------------------
# The following experiment is for a simulation of one of Dr. Monster's
# exposure experiments described in "Kinetics of Tetracholoroethylene
# in Volunteers; Influence of Exposure Concentration and Work Load,"
# A.C. Monster, G. Boersma, and H. Steenweg,
# Int. Arch. Occup. Environ. Health, v42, 1989, pp303-309
#
# The paper documents measurements of levels of TCE in blood and
# exhaled air for a group of 6 subjects exposed to
# different concentrations of PERC in air.
#
# Inhalation is specified as a dose of magnitude InhMag for the
# given Exposure time.
#
# Inhalation is given in ppm
#---------------------------------------------------------

Experiment {

InhMag = 72;            # ppm
Period = 1e10;          # Only one dose
Exposure = 240;         # 4 hour exposure

# measurements before end of exposure and at [5' 30'] 2hr 18 42 67 91 139 163

Print (C_exh_ug, 239.9 245 270 360 1320 2760 4260 5700 8580 10020 );
Print (C_ven, 239.9 360 1320 2760 4260 5700 8580 10020 );

}

END.

Concept Index

'

   * '#' sign
   * ';' sign

a

   * Analyzing results
   * Assignment

b

   * Beta distribution
   * Bibliographic references
   * Binomial distribution
   * Blank lines

c

   * CalcOutputs, output specification
   * CDFNormal() function
   * Chi2 distribution
   * Colon conditional assignment
   * Comments
   * Common pitfalls
   * Comparison operators
   * Conditional assignment
   * Cumulative density function, Normal

d

   * Data() specification
   * DefaultSim
   * Defining models
   * Density function, Normal
   * Derivative specification
   * Distrib() specification
   * Distribution, beta
   * Distribution, binomial
   * Distribution, chi2
   * Distribution, exponential
   * Distribution, gamma
   * Distribution, inverse-gamma
   * Distribution, lognormal
   * Distribution, lognormal_v
   * Distribution, loguniform
   * Distribution, normal
   * Distribution, normal_v
   * Distribution, piecewise
   * Distribution, Poisson
   * Distribution, triangular
   * Distribution, truncated lognormal
   * Distribution, truncated normal
   * Distribution, trunclognormal_v
   * Distribution, truncnormal_v
   * Distribution, uniform
   * Dt() operator
   * Dynamics specifications

e

   * Erfc() function
   * Error function
   * Error handling
   * Euler integrator
   * Examples, Examples
   * Experiment definition
   * Exponential distribution

f

   * Function, CDFNormal()
   * Function, erfc()
   * Function, lnDFNormal()
   * Function, lnGamma()
   * Function, NDoses
   * Function, NormalRandom()
   * Function, PerDose()
   * Function, PerExp()
   * Function, Spikes()
   * Function, UniformRandom()
   * Functions, input
   * Functions, special

g

   * Gamma distribution
   * Gamma function
   * Global specifications

i

   * Input functions
   * Input variables
   * Installation
   * Integrate() specification
   * Integration routine, Euler
   * Integration routine, Lsodes
   * Integration variable
   * InvGamma distribution

l

   * Level definition
   * License
   * LnDFNormal() function
   * LnGamma() function
   * Lognormal distribution
   * LogNormal_v distribution
   * LogUniform distribution
   * Lsodes integrator

m

   * Major changes in versions 4.2.0
   * Make
   * Markov-chain Monte Carlo simulations, Markov-chain Monte Carlo
     simulations
   * MCMC simulations
   * MCMC() specification
   * Mod syntax
   * Mod usage
   * Model definition files
   * Models
   * Monte Carlo
   * MonteCarlo() specification

n

   * NDoses() function
   * Normal cumulative density function
   * Normal density function
   * Normal distribution
   * Normal_v distribution
   * NormalRandom() function

o

   * Output specification
   * Output variables
   * OutputFile() specification
   * Overview

p

   * Parameter declaration
   * Parameter scaling
   * PerDose() function
   * PerExp() function
   * Piecewise distribution
   * Pitfalls
   * Poisson distribution
   * Print() specification
   * PrintStep() specification

r

   * Random number, normal
   * Random number, uniform

s

   * Scale, scaling specification
   * Semi-colon
   * SetPoint() specification
   * SetPoints simulations
   * SimType() specification
   * Simulation definition files
   * Simulation file, syntax
   * Software license
   * Special functions
   * Specification, Data()
   * Specification, Distrib()
   * Specification, Integrate()
   * Specification, MCMC()
   * Specification, MonteCarlo()
   * Specification, OutputFile()
   * Specification, Print()
   * Specification, PrintStep()
   * Specification, SetPoint()
   * Specification, SimType()
   * Specification, StartTime()
   * Specification, statistical model
   * Specifying simulations
   * Spikes() function
   * StartTime() specification
   * State variables
   * Statistical model specification
   * Style
   * Syntax for mod
   * Syntax of simulation files

t

   * Triangular distribution
   * TruncLogNormal distribution
   * TruncLogNormal_v distribution
   * TruncNormal distribution
   * TruncNormal_v distribution

u

   * Uniform distribution
   * UniformRandom() function
   * Unix make utility

w

   * Working Through an Example

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