# 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. 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, 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 Central1 dynamics. 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.

## Differential Equations DEProblem

DEProblems are types from DifferentialEquations 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.