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Exact Parameter Identification in PET Pharmacokinetic Modeling: Extension to the Reversible Two Tissue Compartment Model

Martin Holler, Erion Morina, Georg Schramm

TL;DR

This work establishes analytic identifiability for the reversible two-tissue PET pharmacokinetic model under a multi-region framework. The authors show that most tissue kinetic parameters $(K_1^i,k_2^i,k_3^i,k_4^i)$ can be uniquely recovered from image-based tissue concentrations $C_{ ext{T}}^i(t)$ across regions when the number of time points $T$ satisfies $T\ge 2(p+4)$ and the arterial input is modeled via a polyexponential $C_{ ext{P}}(t)=\sum_{j=1}^p \lambda_j e^{\mu_j t}$; full identifiability including $K_1^i$ also requires measurements of the arterial input $C_{ ext{WB}}(t)$ (or $C_{ ext{P}}(t)$) with a mild parameterization of the attenuation $f(t)$. The analysis extends prior results for the irreversible model to the practically important reversible case, employing interpolation-based arguments and region-diversity conditions to guarantee uniqueness of the exponents $\mu_j$ and the rate constants across regions. The results imply that, up to a global scale for the $K_1^i$ and $\lambda_j$, image-derived data alone can determine the kinetic parameters, with full recovery achievable when arterial input information is available. This provides a theoretical foundation for SIME-like multi-region approaches in dynamic PET without requiring invasive blood sampling, while also outlining pathways to incorporate realistic blood input and mixture voxel models.

Abstract

This paper addresses the problem of recovering tracer kinetic parameters from multi-region measurement data in quantitative PET imaging using the reversible two tissue compartment model. Its main result is an extension of our previous work on the irreversible two tissue compartment model. In analogy to our previous work, we show that also in the (practically highly relevant) reversible case, most tracer kinetic parameters can be uniquely identified from standard PET measurements (without additional full blood sample analysis that is usually performed in practice) and under reasonable assumptions. In addition, unique identifiability of all parameters is shown provided that additional measurements from the (uncorrected) total arterial blood tracer concentration (which can be obtained from standard PET measurements or from a simple blood sample analysis) are available.

Exact Parameter Identification in PET Pharmacokinetic Modeling: Extension to the Reversible Two Tissue Compartment Model

TL;DR

This work establishes analytic identifiability for the reversible two-tissue PET pharmacokinetic model under a multi-region framework. The authors show that most tissue kinetic parameters can be uniquely recovered from image-based tissue concentrations across regions when the number of time points satisfies and the arterial input is modeled via a polyexponential ; full identifiability including also requires measurements of the arterial input (or ) with a mild parameterization of the attenuation . The analysis extends prior results for the irreversible model to the practically important reversible case, employing interpolation-based arguments and region-diversity conditions to guarantee uniqueness of the exponents and the rate constants across regions. The results imply that, up to a global scale for the and , image-derived data alone can determine the kinetic parameters, with full recovery achievable when arterial input information is available. This provides a theoretical foundation for SIME-like multi-region approaches in dynamic PET without requiring invasive blood sampling, while also outlining pathways to incorporate realistic blood input and mixture voxel models.

Abstract

This paper addresses the problem of recovering tracer kinetic parameters from multi-region measurement data in quantitative PET imaging using the reversible two tissue compartment model. Its main result is an extension of our previous work on the irreversible two tissue compartment model. In analogy to our previous work, we show that also in the (practically highly relevant) reversible case, most tracer kinetic parameters can be uniquely identified from standard PET measurements (without additional full blood sample analysis that is usually performed in practice) and under reasonable assumptions. In addition, unique identifiability of all parameters is shown provided that additional measurements from the (uncorrected) total arterial blood tracer concentration (which can be obtained from standard PET measurements or from a simple blood sample analysis) are available.

Paper Structure

This paper contains 15 sections, 9 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model
  3. Unique identifiability
  4. Uniqueness of at least $p-3$ exponents $(\mu_j)_{j=1}^p$.
  5. Uniqueness of the remaining exponents $(\mu_j)_{j=1}^p$.
  6. Case I.
  7. Case II.
  8. Uniqueness of $\alpha_1^i, \alpha_2^i$ for at least three regions.
  9. Uniqueness of $k_2^i,k_3^i,k_4^i$ for at least three regions.
  10. Case I.
  11. Uniqueness of the remaining $k_2^i,k_3^i, k_4^i$ and of the $K_1^i$ and $(\lambda_j)_j$ up to a constant factor.
  12. Case I.
  13. Case II.A.
  14. Case II.B.
  15. Conclusion

Key Result

Theorem 1

Let $(K_1^i,k_2^i,k_3^i,k_4^i)$ be the kinetic parameters of different tissues $i=1,\ldots,n$ of the reversible two tissue compartment model, let $T$ be the number of time-points where PET measurements of $C_{\text{T}}^i(t)$ that are available, and let $p$ be the degree of a polyexponential parametr

Figures (1)

  • Figure 1: Reversible two tissue compartment model. The (sub-)compartments are illustrated by boxes around the concentrations $C_{\text{P}}, C_{\text{B}}, C_{\text{F}}$ and $C_{\text{T}}$. The (directional) exchange rates $K_1, k_2, k_3, k_4$ between the (sub-)compartments are represented by arrows.

Theorems & Definitions (22)

  • Theorem 1: Main result - informal version
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5: Unique interpolation
  • proof
  • Definition 6: Parameter configuration
  • ...and 12 more