Conditional Dosing Pharmacometric Example

Chris Rackauckas

In this example we will show how to model a conditional dosing using the DiscreteCallbacks. The problem is as follows. The patient has a drug A(t) in their system. The concentration of the drug is given as C(t)=A(t)/V for some volume constant V. At t=4, the patient goes to the clinic and is checked. If the concentration of the drug in their body is below 4, then they will receive a new dose.

For our model, we will use the simple decay equation. We will write this in the in-place form to make it easy to extend to more complicated examples:

using DifferentialEquations
function f(du,u,p,t)
    du[1] = -u[1]
end
u0 = [10.0]
const V = 1
prob = ODEProblem(f,u0,(0.0,10.0))

ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
timespan: (0.0, 10.0)
u0: 1-element Vector{Float64}:
 10.0

Let's see what the solution looks like without any events.

sol = solve(prob,Tsit5())
using Plots; gr()
plot(sol)

We see that at time t=4, the patient should receive a dose. Let's code up that event. We need to check at t=4 if the concentration u[1]/4 is <4, and if so, add 10 to u[1]. We do this with the following:

condition(u,t,integrator) = t==4 && u[1]/V<4
affect!(integrator) = integrator.u[1] += 10
cb = DiscreteCallback(condition,affect!)

DiffEqBase.DiscreteCallback{typeof(Main.##WeaveSandBox#4067.condition), typ
eof(Main.##WeaveSandBox#4067.affect!), typeof(DiffEqBase.INITIALIZE_DEFAULT
), typeof(DiffEqBase.FINALIZE_DEFAULT)}(Main.##WeaveSandBox#4067.condition,
 Main.##WeaveSandBox#4067.affect!, DiffEqBase.INITIALIZE_DEFAULT, DiffEqBas
e.FINALIZE_DEFAULT, Bool[1, 1])

Now we will give this callback to the solver, and tell it to stop at t=4 so that way the condition can be checked:

sol = solve(prob,Tsit5(),tstops=[4.0],callback=cb)
using Plots; gr()
plot(sol)

Let's show that it actually added 10 instead of setting the value to 10. We could have set the value using affect!(integrator) = integrator.u[1] = 10

println(sol(4.00000))
println(sol(4.000000000001))

[0.18316389222022983]
[10.183163892210045]

Now let's model a patient whose decay rate for the drug is lower:

function f(du,u,p,t)
    du[1] = -u[1]/6
end
u0 = [10.0]
const V = 1
prob = ODEProblem(f,u0,(0.0,10.0))

ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
timespan: (0.0, 10.0)
u0: 1-element Vector{Float64}:
 10.0

sol = solve(prob,Tsit5())
using Plots; gr()
plot(sol)

Under the same criteria, with the same event, this patient will not receive a second dose:

sol = solve(prob,Tsit5(),tstops=[4.0],callback=cb)
using Plots; gr()
plot(sol)

Appendix

These tutorials are a part of the SciMLTutorials.jl repository, found at: https://github.com/SciML/SciMLTutorials.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this tutorial, do the following commands:

using SciMLTutorials
SciMLTutorials.weave_file("tutorials/models","02-conditional_dosing.jmd")

Computer Information:

Julia Version 1.6.2
Commit 1b93d53fc4 (2021-07-14 15:36 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (ORCJIT, znver2)
Environment:
  JULIA_DEPOT_PATH = /root/.cache/julia-buildkite-plugin/depots/a6029d3a-f78b-41ea-bc97-28aa57c6c6ea
  JULIA_NUM_THREADS = 16

Package Information:

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  [8e850ede] nghttp2_jll
  [3f19e933] p7zip_jll