Structural Identifiability of Biological Models Nikki Meshkat - PowerPoint PPT Presentation

Structural Identifiability of Biological Models Nikki Meshkat Santa Clara University Joint work with Zvi Rosen and Seth Sullivant Symbolic/Numeric Seminar at CUNY August 31, 2017

Motivation: Unidentifiable models • Model 1: • Model 2:

Motivation: Unidentifiable models • Model 1: No ID scaling reparametrization! • Model 2: ID scaling reparametrization:

Outline • Identifiable reparametrizations of linear compartment models – Scaling reparametrizations – Necessary and sufficient conditions • Finding identifiable functions, in general – Using Gröbner Bases – Linear algebra techniques

Structural Identifiability Analysis • Model: – 𝑦 state variable – 𝑣 input – 𝑧 output – 𝑞 parameter • Finding which unknown parameters of a model can be quantified from given input- output data

Motivation: biological models Measured drug Drug concentration input Drug exchange Loss from blood Loss from organ

u 1 Example: Linear 2- y Compartment Model 𝑏 21 x 1 x 2 Can we 𝑏 12 determine parameters 𝑏 01 𝑏 02 𝑏 01 , 𝑏 02 , 𝑏 12 , 𝑏 21 from input- output data? 𝑦 1 = − 𝑏 01 + 𝑏 21 𝑦 1 + 𝑏 12 𝑦 2 + 𝑣 1 𝑦 2 = 𝑏 21 𝑦 1 − (𝑏 02 + 𝑏 12 )𝑦 2 𝑧 = 𝑦 1

Linear Compartment Models • Let 𝐻 = (𝑊, 𝐹) be a directed graph with 𝑛 edges and 𝑜 vertices • Model 𝑦(𝑢) = 𝐵(𝐻)𝑦(𝑢) + 𝑣(𝑢) 𝑧(𝑢) = 𝑦 1 (𝑢) • where 𝑦 ∈ ℝ 𝑜 is the state variable 𝑣 is the input where 𝑧 is the output Assumption #1: Single input/output in first compartment

Linear Compartment Models • Let 𝐻 = (𝑊, 𝐹) be a directed graph with 𝑛 edges and 𝑜 vertices • Model 𝑦(𝑢) = 𝐵(𝐻)𝑦(𝑢) + 𝑣(𝑢) 𝑧(𝑢) = 𝑦 1 (𝑢) • where 𝐵(𝐻) has the form: −𝑏 0𝑗 − 𝑏 𝑙𝑗 𝑗𝑔 𝑗 = 𝑘 𝐵(𝐻) 𝑗𝑘 = 𝑙: 𝑗→𝑙 ∈𝐹 𝑏 𝑗𝑘 𝑗𝑔 𝑘 → 𝑗 𝑗𝑡 𝑏𝑜 𝑓𝑒𝑕𝑓 𝑝𝑔 𝐻 0 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓 Assumption #2: Leaks from every compartment, so 𝑏 0𝑗 ≠ 0 for all 𝑗

Linear Compartment Models • Let 𝐻 = (𝑊, 𝐹) be a directed graph with 𝑛 edges and 𝑜 vertices • Model 𝑦(𝑢) = 𝐵(𝐻)𝑦(𝑢) + 𝑣(𝑢) 𝑧(𝑢) = 𝑦 1 (𝑢) • where 𝐵(𝐻) has the form: 𝑏 𝑗𝑗 𝑗𝑔 𝑗 = 𝑘 𝐵(𝐻) 𝑗𝑘 = 𝑏 𝑗𝑘 𝑗𝑔 𝑘 → 𝑗 𝑗𝑡 𝑏𝑜 𝑓𝑒𝑕𝑓 𝑝𝑔 𝐻 0 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓 Assumption #3: Strongly connected graph, with 𝑛 ≤ 2𝑜 − 2

Our class of models • Assumptions: – I/O in first compartment – Leaks from every compartment – 𝐻 strongly connected with at most 2𝑜 − 2 edges 𝑦 = 𝐵𝑦 + 𝑣 𝑧 = 𝑦 1 • Identifiability: Which parameters of model can be determined from given input-output data? – Must first determine input-output equation

Find Input-Output Equation • Rewrite system eqns as 𝜖𝐽 − 𝐵 𝑦 = 𝑣 • Cramer’s Rule: 𝑏 11 ⋯ 𝑏 1𝑜 𝑏 22 ⋯ 𝑏 2𝑜 ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ 𝐵 = 𝐵 1 = 𝑏 𝑜1 ⋯ 𝑏 𝑜𝑜 𝑏 𝑜2 ⋯ 𝑏 𝑜𝑜 𝑦 1 = 𝑒𝑓𝑢 𝜖𝐽 − 𝐵 1 𝑣 1 𝑒𝑓𝑢(𝜖𝐽 − 𝐵) • I/O eqn: 𝑒𝑓𝑢 𝜖𝐽 − 𝐵 𝑧 = 𝑒𝑓𝑢 𝜖𝐽 − 𝐵 1 𝑣 1 𝑧 𝑜 + 𝑑 1 𝑧 𝑜−1 + ⋯ + 𝑑 𝑜 𝑧 𝑜−1 + 𝑑 𝑜+1 𝑣 1 𝑜−2 + ⋯ + 𝑑 2𝑜−1 𝑣 1 = 𝑣 1

Identifiability • Can recover coefficients from data [Soderstrom & Stoica 1998] • Identifiability: is it possible to recover the parameters of the original system, from the coefficients of I/O eqn? [Ljung & Glad 1994] – Two sets of parameter values yield same coefficient values? – Is coeff map 1-to-1? 𝑧 𝑜 + 𝑑 1 𝑧 𝑜−1 + ⋯ + 𝑑 𝑜 𝑧 𝑜−1 + 𝑑 𝑜+1 𝑣 1 𝑜−2 + ⋯ + 𝑑 2𝑜−1 𝑣 1 = 𝑣 1

Identifiability from I/O eqns • Question of injectivity of the coefficient map 𝑑: ℝ 𝑛+𝑜 → ℝ 2𝑜−1 • If c is one-to-one: globally identifiable finite-to-one: locally identifiable infinite-to-one: unidentifiable • Concerned with generic local identifiability – Check dimension of image of coefficient map • If dim im c = m+n, then locally identifiable • If dim im c < m+n , then unidentifiable

Local identifiability vs. Unidentifiability • Ex: 2-comp model = 𝑏 11 𝑏 12 𝑦 1 𝑦 1 𝑦 2 + 𝑣 1 0 , 𝑧 = 𝑦 1 𝑏 21 𝑏 22 𝑦 2 I/O eqn: 𝑧 − 𝑏 11 + 𝑏 22 𝑧 + 𝑏 11 𝑏 22 − 𝑏 12 𝑏 21 𝑧 = 𝑣 1 − 𝑏 22 𝑣 1 Coefficient map 𝑑: ℝ 4 → ℝ 3 (𝑏 11 , 𝑏 12 , 𝑏 21 , 𝑏 22 ) ⟼ (− 𝑏 11 + 𝑏 22 , 𝑏 11 𝑏 22 − 𝑏 12 𝑏 21 , −𝑏 22 ) Jacobian has rank 3: ⇒ Unidentifiable!

Unidentifiable models • Cannot determine all parameters, but can we determine some combination of the parameters? Ex: 𝑏 12 + 𝑏 21 or 𝑏 12 𝑏 21 • A function 𝑔: ℝ 𝑛+𝑜 → ℝ is called identifiable from c if ℝ(𝑔, 𝑑 1 , … , 𝑑 2𝑜−1 )/ℝ(𝑑 1 , … , 𝑑 2𝑜−1 ) is an algebraic field extension

Unidentifiable models • Cannot determine all parameters, but can we determine some combination of the parameters? Ex: 𝑏 12 + 𝑏 21 or 𝑏 12 𝑏 21 • A function 𝑔: ℝ 𝑛+𝑜 → ℝ is called identifiable from c if 𝑔 = Φ(𝑑)

Identifiable functions • Coefficients: • Identifiable functions: • What do we do with identifiable functions? • Any special meaning in original model?

2-compartment model as graph = 𝑏 11 𝑏 12 𝑦 1 𝑦 1 𝑦 2 + 𝑣 1 Model: 𝑏 21 𝑏 22 𝑦 2 0 𝑧 = 𝑦 1 Graph: 1 2 𝑏 12 𝑏 21 • Cycle: 𝑏 11 , 𝑏 22 • “Self” cycles:

Coeff map factors through cycles • Coeff map c in terms of cycles of graph G • Due to cyclic permutations in determinant (Leibniz formula) • Strongly connected? • G is strongly connected: m+1 independent cycles (= n self-cycles + m-n+1 cycles) Thus dim im c ≤ m+1

Why are cycles identifiable? • Recall dim im c = m+1 = 2+1 = 3 • Let 𝑕: ℝ 4 → ℝ 3 be the “cycle map” (𝑏 11 , 𝑏 22 , 𝑏 12 , 𝑏 21 ) ⟼ (𝑏 11 , 𝑏 22 , 𝑏 12 𝑏 21 ) • Commutative diagram: 𝑑 ℝ 4 ℝ 3 Φ 𝑕 ℝ 3 𝑕 = Φ(𝑑) Thus

Unidentifiable model = 𝑏 11 𝑏 12 𝑦 1 𝑦 1 𝑦 2 + 𝑣 1 • Model 𝑏 21 𝑏 22 𝑦 2 0 𝑧 = 𝑦 1 𝑏 11 , 𝑏 22 , 𝑏 12 𝑏 21 • Identifiable functions − 𝑏 01 + 𝑏 21 , − 𝑏 02 + 𝑏 12 , 𝑏 12 𝑏 21 i.e. • Reparametrize: 4 independent parameters 3 independent parameters?

Identifiable reparametrization Let 𝑑: ℝ 𝑛+𝑜 → ℝ 2𝑜−1 be our coefficient map An identifiable reparametrization of a model is a map 𝑟: ℝ 𝑙 → ℝ 𝑛+𝑜 such that: • 𝑑 ∘ 𝑟: ℝ 𝑙 → ℝ 2𝑜−1 has the same image as 𝑑 • 𝑑 ∘ 𝑟 is identifiable (finite-to-one)

Scaling reparametrization 𝑜 : ℝ 𝑛+𝑜 → ℝ where we • Choice of functions 𝑔 1 , … , 𝑔 replace 𝑦 1 , … , 𝑦 𝑜 with 𝑌 𝑗 = 𝑔 𝑗 𝐵 𝑦 𝑗 • Set 𝑔 1 = 1 since 𝑧 = 𝑦 1 is observed 𝑦 = 𝐵𝑦 + 𝑣 , each parameter 𝑏 𝑗𝑘 is • Since model is replaced with 𝑏 𝑗𝑘 𝑔 𝑗 𝐵 𝑔 𝑘 𝐵 • Only graphs with at most 2n-2 edges

Identifiable scaling reparametrization • Use scaling: 𝑌 1 = 𝑦 1 , 𝑌 2 = 𝑏 12 𝑦 2 𝑌 1 𝑏 11 1 𝑌 2 + 𝑣 1 𝑌 1 = 𝑏 12 𝑏 21 𝑏 22 0 𝑌 2 𝑧 = 𝑌 1 • Re-write: 𝑌 1 𝑌 2 + 𝑣 1 𝑌 1 = 𝑟 11 1 𝑟 21 𝑟 22 0 𝑌 2 𝑧 = 𝑌 1 • Map 𝑑 ∘ 𝑟 has same image as 𝑑 and is 1-to-1 (𝑟 11 , 𝑟 21 , 𝑟 22 ) ⟼ (− 𝑟 11 + 𝑟 22 , 𝑟 11 𝑟 22 − 𝑟 21 , −𝑟 22 )

Identifiable scaling reparametrization • Use scaling: 𝑌 1 = 𝑦 1 , 𝑌 2 = 𝑏 12 𝑦 2 𝑌 1 𝑏 11 1 𝑌 2 + 𝑣 1 𝑌 1 = 𝑏 12 𝑏 21 𝑏 22 0 𝑌 2 𝑧 = 𝑌 1 • Why useful? – Nondimensionalization of original model – New model: know qualitative behavior of 𝑌 1 , 𝑌 2 • Worked for 2-comp model. Does this always work?

Motivation: Unidentifiable models • Model 1: No ID scaling reparametrization! 2 1 3 • Model 2: ID scaling reparametrization: 2 1 3

Main question: Which models admit an identifiable scaling reparametrization?

Main result: Theorem (M-Sullivant): The following are equivalent: The model has an identifiable scaling reparametrization The model has an identifiable scaling reparametrization by monomial functions of the original parameters All the monomial cycles in G are identifiable functions dim im c = m+1

Non-Example: Model 1 2 1 3 Model: dim im c = 4, so no ID scaling reparametrization!

Example: Model 2 2 1 3 Model: Identifiable cycles:

Algorithm to find identifiable reparametrization 1) Form a spanning tree T 2) Form the directed incidence matrix E(T) : 3) Let E be E(T ) with first row removed 4) Columns of E -1 are exponent vectors of monomials 𝑔 𝑗 (𝐵) in scaling 𝑌 𝑗 = 𝑔 𝑗 𝐵 𝑦 𝑗

Identifiable reparametrization 2 • Spanning tree 1 3 • Rescaling: • Identifiable scaling reparametrization