Analytical Solutions and Differential Equations

The dynamical problem types specify the dynamical models that are the nonlinear transformation of the NLME model. There are two major types of dynamical models: analytical models and DEProblems. An analytical model is a small differential equation with an analytical solution. This analytical solution is used by the solvers to greatly enhance the performance. On the other hand, DEProblem is a specification of a differential equation for numerical solution by DifferentialEquations.jl. This is used for specifying dynamical equations which do not have an analytical solution, such as many nonlinear ordinary differential equations (ODEs), or the myriad of differential equation types supported by DifferentialEquations.jl, such as delay differential equations (DDEs) and stochastic differential equations (SDEs).

Analytical Solutions

Analytical problems are a predefined ODE with an analytical solution. While limited in flexibility, the analytical solutions can be much faster for simulation and estimation. In the @model DSL, an analytical solution is declared by name. For example:

@dynamics Central1

declares the use of the Central1. Analytical solutions have preset names which are used in the internal model. These parameters must be given values in the pre block.

Central1

The Central1 model corresponds to the following @dynamics block:

@dynamics begin
  Central' = -(CL/Vc)*Central
end

The variables CL and Vc are required to be defined in the @pre block.

Depots1Central1

The Depots1Central1 model corresponds to the following @dynamics block:

@dynamics begin
  Depot'   = -Ka*Depot
  Central' =  Ka*Depot - (CL/Vc)*Central
end

The variables Ka, CL and Vc are required to be defined in the @pre block.

Depots2Central1

The Depots2Central1 model corresponds to the following @dynamics block:

@dynamics begin
  Depot1'   = -Ka1*Depot1
  Depot2'   = -Ka2*Depot2
  Central'  =  Ka1*Depot1 + Ka2*Depot2 - (CL/Vc)*Central
end

Central1Periph1

The Central1Periph1 model corresponds to the following @dynamics block:

@dynamics begin
  Central'    = -(CL+Q)/Vc*Central + Q/Vp*Peripheral
  Peripheral' =        Q/Vc*Central - Q/Vp*Peripheral
end

The variables CL, Vc, Q, and Vp are required to be defined in the @pre block.

Depots1Central1Periph1

The Depots1Central1Periph1 model corresponds to the following @dynamics block:

@dynamics begin
  Depot'      = -Ka*Depot
  Central'    =  Ka*Depot -(CL+Q)/Vc*Central + Q/Vp*Peripheral
  Peripheral' =                  Q/Vc*Central - Q/Vp*Peripheral
end

The variables Ka, CL, Vc, Q, and Vp are required to be defined in the @pre block.

Central1Periph1Meta1

The Central1Periph1Meta1 model corresponds to the following @dynamics block:

@dynamics begin
  Central'     = -(CL+Q+CLfm)/Vc*Central + Q/Vp*CPeripheral
  CPeripheral' =            Q/Vc*Central - Q/Vp*CPeripheral
  Metabolite'  = -CLm/Vm*Metabolite + CLfm/Vc*Central
end

The variables CL, CLm, Vc, Vp, Vm, Q, and CLfm are required to be defined in the @pre block.

Central1Periph1Meta1Periph1

The Central1Periph1Meta1Periph1 model corresponds to the following @dynamics block:

@dynamics begin
  Central'     = -(CL+Q+CLfm)/Vc*Central + Q/Vp*CPeripheral
  CPeripheral' =            Q/Vc*Central - Q/Vp*CPeripheral
  Metabolite'  = -(CLm+Qm)/Vm*Metabolite + Qm/Vmp*MPeripheral + CLfm/Vc*Central
  MPeripheral' =        Qm/Vm*Metabolite - Qm/Vmp*MPeripheral
end

The variables CL, CLm, Vc, Vp, Vmp, Q, Qm, and CLfm are required to be defined in the @pre block.

LinearODE

The LinearODE dynamic model specification is used for general piece-wise linear ODEs with constant coefficients. These models are solved with a matrix exponential which is faster and more stable than numerical ODE integration. Hence, the LinearODE dynamic model specification is to be preffered over the equation based specification when the method is applicable.

The coefficients of the ODE are specified in the @pre block as a special matrix named A corresponding the $A$ matrix in the linear system of ODEs $u' = Au$. The names of the ODE variables as well as their initial values should be declared in the @init block. Hence, the Central1Periph1 could equivalently be defined as

@pre begin
  ...
  A = [-(CL+Q)/Vc  Q/Vp
             Q/Vc -Q/Vp]
end

@init begin
  Central = 0.0
  Preriph = 0.0
end

@dynamics LinearODE

Differential Equations DEProblem

DEProblems are types from DifferentialEquations.jl which are used to specify differential equations to be solved numerically via the solvers of the package. In the @model interface, the DEProblem is set to be an ODEProblem defining an ODE. The models are defined by writing each of the differential equations in the system, e.g.

@dynamics begin
  Central' = -(CL/Vc)*Central
end

for a simple one compartment model and

@dynamics begin
  Depot'   = -Ka*Depot
  Central' =  Ka*Depot - (CL/Vc)*Central
end

for a one compartment model with first order absorption.

In the function-based interface, any DEProblem can be used, which includes:

  • Discrete equations (function maps, discrete stochastic (Gillespie/Markov) simulations)
  • Ordinary differential equations (ODEs)
  • Split and Partitioned ODEs (Symplectic integrators, IMEX Methods)
  • Stochastic ordinary differential equations (SODEs or SDEs)
  • Random differential equations (RODEs or RDEs)
  • Differential algebraic equations (DAEs)
  • Delay differential equations (DDEs)
  • Mixed discrete and continuous equations (Hybrid Equations, Jump Diffusions)

The problem type that is given can use sentinel values for the initial condition, timespan, and parameters which will be overridden by Pumas during the simulation chain.