Convert From Slope to Exponent Parameterization for Drug i in the MuSyC Model

This can be used for setting priors and interpreting parameter estimates

Usage

MuSyC_si_to_hi(si, Ci, E0, Ei)

Arguments

si

numeric value of slope of drug i at its AC50 and doses of all other drugs are zero

Ci

numeric value of the AC50 of drug i

E0

numeric value of the response with no treatments

Ei

numeric value of the response of infinite drug i and no other treatments

Value

hi numeric value of the exponent in the MuSyC equation for drug i

Details

Claim: When d1=0 and d2=C2 then the gradient of the response with respect to d2 is the s2, symbolically d(Ed)/d(d2) = s2 where s2 = h2 * (E0 + E2) / (4 * C2) then

d(Ed)/d(d2)
  = d/d(d2)
    (C1^h1 * C2^h2 * E0 + C1^h1 * d2^h2 * E2) /
    (C1^h1 * C2^h2      + C1^h1 * d2^h2)

Cancel the C1^h1 terms:

  =  d/d(d2)
     (C2^h2 * E0 + d2^h2 * E2) /
     (C2^h2      + d2^h2)

Distribute the derivative across the terms in the numerator

  =  E0 * C2^h2 * (d/d(d2) 1     / (C2^h2 + d2^h2)) +
     E2         * (d/d(d2) d2^h2 / (C2^h2 + d2^h2))

  =  E0 * C2^h2 * (h2 * d2^(h2-1) / (C2^h2 + d2^h2)^2) +
     E2 * (C2^h2 * h2 * d2^(h2-1) / (C2^h2 + d2^h2)^2)

  =  (E0 + E2) * C2^h2 * h2 * d2^(h2-1)/(C2^h2 + d2^h2)^2

Evaluate at d2 = C2:

  =  (E0 + E2) * h2 * C2^(2*h2-1) / (4*C2^(2*h2)))
  =  h2 * (E0 + E2) / (4 * C2)

See also

MuSyC_hi_to_si