Math of two pool system

THE EXPLICIT SOLUTION TO THE TWO POOL SYSTEM

The two pool system is described by the following differential equations *:

d Q₁ / dt = - ( K₁₂ + K_e) Q₁ + K₂₁ Q₂
d Q₂ / dt = - K₂₁ Q₂ + K₁₂ Q₁

where the Qs are the total amounts in the pools, and dQ/dt is the rate of change of Q with respect to time. For each pool, the outward flow (negative terms) is just the amount of drug in the pool times the sum of all the Ks that represent outward flows. The inward flow is the sum of all products of inward Ks times the amount of drug in each pool associated with the K. The solutions are:

Q₁ = a₁ exp( -alpha t) + b₁ exp( -beta t)
Q₂ = a₂ exp( -alpha t) + b₂ exp( -beta t)

It is useful to define these intermediate constants:

K_sum = K₁₂ + K₂₁ + K_e
K₄ = 4 K₂₁ K_e

now we can give alpha and beta as:

alpha = 0.5 (K_sum + sqrt( K_sum² - K₄))
beta = 0.5 (K_sum - sqrt( K_sum² - K₄))

finally we use the dose, D (the value of Q₁ at t=0), to obtain the two a's and two b's:

a₁ = D (K₂₁ - beta) / (alpha - beta)
b₁ = D (beta - K₂₁) / (alpha - beta)
b₂ = - a₂ = K₁₂ D / (alpha - beta)

* On this page I use:

alpha, beta instead of lambda₁, lambda₂
exp(x) instead of e^x

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				The two pool system is described by the following differential equations *: d Q₁ / dt = - ( K₁₂ + K_e) Q₁ + K₂₁ Q₂ d Q₂ / dt = - K₂₁ Q₂ + K₁₂ Q₁ where the Qs are the total amounts in the pools, and dQ/dt is the rate of change of Q with respect to time. For each pool, the outward flow (negative terms) is just the amount of drug in the pool times the sum of all the Ks that represent outward flows. The inward flow is the sum of all products of inward Ks times the amount of drug in each pool associated with the K. The solutions are: Q₁ = a₁ exp( -alpha t) + b₁ exp( -beta t) Q₂ = a₂ exp( -alpha t) + b₂ exp( -beta t) It is useful to define these intermediate constants: K_sum = K₁₂ + K₂₁ + K_e K₄ = 4 K₂₁ K_e now we can give alpha and beta as: alpha = 0.5 (K_sum + sqrt( K_sum² - K₄)) beta = 0.5 (K_sum - sqrt( K_sum² - K₄)) finally we use the dose, D (the value of Q₁ at t=0), to obtain the two a's and two b's: a₁ = D (K₂₁ - beta) / (alpha - beta) b₁ = D (beta - K₂₁) / (alpha - beta) b₂ = - a₂ = K₁₂ D / (alpha - beta)

		* On this page I use: alpha, beta instead of lambda₁, lambda₂ exp(x) instead of e^x