# Mathematical Models for FDG Kinetics in Cancer: A Review

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## Abstract

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## 1. Introduction

- Image reconstruction inverse problem [6]: to reconstruct the spatio-temporal distribution of FDG inside the tissue by solving the integral equation that connects the FDG densito to the measured radiation by means of the Radon transform.
- Compartmental inverse problem [7]: to model the tracer kinetics by solving the non-linear time-dependent equation that connects the tracer coefficients to the reconstructed FDG concentration.

- Only one compartment is allowed to exchange tracer with the environment.
- The input function, i.e., the tracer concentration introduced into the tissue by the blood, is known by means of either experimental measurements or mathematical modeling.
- The overall tracer concentration associated to the organ of interest (typically, the tumor) is known as a function of time.
- Both the linearity of fluxes between compartments and vanishing initial conditions hold.
- The kinetic coefficients are constant and homogeneous in the tissue.

## 2. The Experimental Data in the Compartmental Game

#### 2.1. Standardized Uptake Value

#### 2.2. Input Function

#### 2.3. Activity Concentration of Target Tissue

## 3. The Construction of Compartmental Models

#### 3.1. Generalities

#### 3.2. Basic Applicability Conditions

- Tracer is administered in trace amounts. The number of injected molecules is supposed to be sufficiently high so that diffusion may described by application of a continuous model. However, such a number is not so high as to influence physiological processes and molecular interactions. In particular, tracer does not affect glucose metabolism.
- During an experiment, physiologic conditions are in a steady state which is not affected by measurement devices of tracer concentration. This holds true, in particular, for glucose metabolism.
- The well-mixing condition holds for each compartment. In practice, this means that equilibrium is reached in a time interval, which is rather short with respect to the time of data acquisition. As a consequence, the spatial homogeneity condition follows, which implies that the tracer concentration in each compartment depends only on time.
- Transport of tracer molecules and related composites between compartments follows a first order kinetics, which ultimately leads to linear ODEs.
- Bound tracer in blood is disregarded, and the arterial concentration of tracer available for tissue uptake is regarded as a valuable approximation of capillary concentration.

#### 3.3. Examples of Standard CMs

#### 3.3.1. 1-Compartment Model

#### 3.3.2. 2-Compartment Model

#### 3.3.3. 3-Compartment Model

**Remark**

**1.**

#### 3.4. Compact Formulation and General Formal Solution of the Direct Problem

**Remark**

**2.**

## 4. Patlak and Logan Graphical Approaches

#### 4.1. PGA

**First step.**The vector solution $\mathit{C}$ of the irreversible system of ODE (29) is substituted into the IPE (33), which is then divided by ${C}_{b}$. The resulting equation takes the form$$\frac{{C}_{T}}{{C}_{b}}={\alpha}_{P}\phantom{\rule{0.166667em}{0ex}}{\displaystyle \frac{{\int}_{0}^{t}{C}_{b}}{{C}_{b}}}+{\beta}_{P}\left(t\right),$$(see the Appendix A) where ${\alpha}_{P}$ [min${}^{-1}$] is a constant macroparameter [25], and ${\beta}_{P}$ depends on ${C}_{b}$ and the components of $\mathit{C}$.**Second step.**In a number of relevant cases it may be shown that ${\beta}_{P}$ is asymptotically constant. Then, in the plane referred to Cartesian coordinates $(x,y)$, define the functions $x\left(t\right)$ and $y\left(t\right)$ by$$x\left(t\right)={\displaystyle \frac{{\int}_{0}^{t}{C}_{b}}{{C}_{b}}},\phantom{\rule{2.em}{0ex}}y\left(t\right)={\displaystyle \frac{{C}_{T}}{{C}_{b}}},$$$t\in (0,\infty )$. The points $\left(x\right(t),y(t\left)\right)$ give the parametric representation of a curve which is known as the standard Patlak plot [38]. Comparison with Equation (34) and the condition on ${\beta}_{P}$ show that the curve is asymptotically linear. Thus the slope ${\alpha}_{P}$ and the adimensional constant intercept ${\beta}_{P}$ are estimated in terms of the data by linear regression [38]. The procedure may be applied pixel-wise.**Third step.**The interpretation of ${\alpha}_{P}$ is achieved by comparison with the stationary solution of the system of ODEs (29), corresponding to a constant IF. It is shown that ${\alpha}_{P}$ measures the rate of tracer uptake by the tissue at stationary conditions.

#### 4.1.1. PGA for 2-CM

**Remark**

**3.**

**Remark**

**4.**

#### 4.1.2. PGA for 3-CM

**Remark**

**5.**

#### 4.2. LGA

- First step. Consider the integral in time of the IPE equation in the compact form (33):$${\int}_{0}^{t}{C}_{T}={\int}_{0}^{t}\mathit{\alpha}\phantom{\rule{0.166667em}{0ex}}\mathit{C}+{V}_{b}\phantom{\rule{0.166667em}{0ex}}{\int}_{0}^{t}{C}_{b}.$$$$\mathit{C}={\mathit{M}}^{-1}\phantom{\rule{0.166667em}{0ex}}\dot{\mathit{C}}-{k}_{1}\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathit{e}\phantom{\rule{0.166667em}{0ex}}{C}_{b}.$$$${\int}_{0}^{t}\mathit{\alpha}\phantom{\rule{0.166667em}{0ex}}\mathit{C}(\tau ;\mathit{k})\phantom{\rule{0.166667em}{0ex}}d\tau =\mathit{\alpha}\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}^{-1}\mathit{C}-{k}_{1}\phantom{\rule{0.166667em}{0ex}}\mathit{\alpha}\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathit{e}\phantom{\rule{0.166667em}{0ex}}{\int}_{0}^{t}{C}_{b}\phantom{\rule{0.166667em}{0ex}}d\tau $$$$\frac{{\int}_{0}^{t}{C}_{T}}{{C}_{T}}={\alpha}_{L}\phantom{\rule{0.166667em}{0ex}}\frac{{\int}_{0}^{t}{C}_{b}}{{C}_{T}}+{\beta}_{L}\left(t\right)$$$${\alpha}_{L}=-{k}_{1}\phantom{\rule{0.166667em}{0ex}}\mathit{\alpha}\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathit{e}+{V}_{b},$$$${\beta}_{L}\left(t\right)=\frac{\mathit{\alpha}\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}^{-1}\mathit{C}}{{C}_{T}}.$$
- Second step. Following the analogy with the GPA approach of Section 4.1, in a number of relevant cases it may be shown that ${\beta}_{L}$ is asymptotically constant. In such a case, consider the functions $x\left(t\right)$ and $y\left(t\right)$ defined by$$x\left(t\right)={\displaystyle \frac{{\int}_{0}^{t}{C}_{b}}{{C}_{T}}},\phantom{\rule{2.em}{0ex}}y\left(t\right)={\displaystyle \frac{{\int}_{0}^{t}{C}_{T}}{{C}_{T}}},$$$t\in (0,\infty )$. In analogy with (34) the points $\left(x\right(t),y(t\left)\right)$ of the Cartesian plane define a parametric representation of the standard Logan plot, which is an asymptotically linear curve. The adimensional slope ${\alpha}_{L}$ and the intercept ${\beta}_{L}$ are macroparameters determined by the data, which are estimated by linear regression.
- Third step. The interpretation of ${\alpha}_{L}$ follows from the equilibrium solution of the system (29) at constant IF ${C}_{b}^{*}$. The equilibrium state ${\mathit{C}}^{*}$ is given by$${\mathit{C}}^{*}=-{k}_{1}\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathit{e}\phantom{\rule{0.166667em}{0ex}}{C}_{b}^{*}.$$According to the IPE (33), the previous equation, and the definition of ${\alpha}_{L}$, it follows that$${C}_{T}^{*}=\mathit{\alpha}\phantom{\rule{0.166667em}{0ex}}{\mathit{C}}^{*}+{V}_{b}\phantom{\rule{0.166667em}{0ex}}{C}_{b}^{*}=-{k}_{1}\phantom{\rule{0.166667em}{0ex}}\mathit{\alpha}\phantom{\rule{0.166667em}{0ex}}{\mathit{M}}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathit{e}\phantom{\rule{0.166667em}{0ex}}{C}_{b}^{*}+{V}_{b}\phantom{\rule{0.166667em}{0ex}}{C}_{b}^{*}={\alpha}_{L}\phantom{\rule{0.166667em}{0ex}}{C}_{b}^{*}$$Thus, the slope ${\alpha}_{L}={C}_{T}^{*}/{C}_{b}^{*}$ is the ratio between the constant equilibrium value of the total tissue concentration and the blood concentration.

**Remark**

**6.**

#### 4.2.1. LGA for 2-CM

#### 4.2.2. LGA for 3-CM

## 5. Issues on the Solvability of the Inverse Problem

#### 5.1. Identifiability of Linear CMs

#### 5.2. Sensitivity Analysis

## 6. Physiology-Driven Compartmental Models

#### 6.1. Reference Tissue Models

#### 6.2. CMs for Liver

#### 6.3. CMs for the Renal System

- An extravascular compartment ${\mathcal{C}}_{f}$ accounting for tracer outside cells, whose exchange with blood is free.
- A compartment ${\mathcal{C}}_{p}$ containing the phosphorylated FDG, the FDG in the cells, and the preurine pool. In particular, following the flow of liquid, tracer is filtered in the preurines and carried towards the proximal tubule. This compartment has been denoted as ${\mathcal{C}}_{p}$ because tracer can also be in phosphorylated form.
- A tubular compartment ${\mathcal{C}}_{t}$, where tracer flows towards bladder. Here, the concentration varies (increases) because of the re-absorption of liquids through the tubular walls.
- The urinary pool ${\mathcal{C}}_{u}$, anatomically identified with the bladder, where the tracer carried by the urine is accumulated. Notice the bladder volume varies with time.

#### 6.4. The Role of the Endoplasmic Reticulum

#### 6.5. Comparison among Different Models

## 7. Some Numerics: Optimization Schemes

- They typically suffer numerical instabilities related to the non-uniqueness and sensitivity limitations discussed in Section 5.
- Since the operator $\mathcal{F}$ is clearly non linear and, further, the space where possible minimizers can be searched for is typically big, they may suffer local minima.
- Particularly, in the case of three-compartment models, the number of kinetic parameters to determine is high, which implies that they are computationally demanding.

#### 7.1. Deterministic Approaches

#### 7.2. Statistical Approaches

#### 7.3. Biology-Inspired Approaches

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 3.**Illustrative example of the Patlak graphical approach. (

**A**) Experimental input function ${C}_{b}\left(t\right)$ as a function of time. (

**B**) Simulated total concentration ${C}_{T}\left(t\right)$ as a function of time. (

**C**) Patlack plot (black circles). The red dotted line is the results of a linear regression model fitted to the points of the plot. The value of the corresponding slope ${\alpha}_{P}$ and of the coefficient of determination ${R}^{2}$ are also reported.

**Figure 4.**Illustrative example of the Logan graphical approach. (

**A**) Experimental input function ${C}_{b}\left(t\right)$ as a function of time. (

**B**) Simulated total concentration ${C}_{T}\left(t\right)$ as a function of time. (

**C**) Logan plot (black circles). The red dotted line is the results of a linear regression model fitted to the points of the plot. The value of the corresponding slope ${\alpha}_{L}$ and of the coefficient of determination ${R}^{2}$ are also reported.

**Figure 8.**(

**A**) Kinetic parameters estimated for the reference tissue CM with a deterministic approach (

**B**) Kinetic parameters estimated for the liver CM with a deterministic approach (

**C**) Kinetic parameters for a simplified CM for the renal system estimated with ant colony optimization, (

**D**) Kinetic parameters for the CM of the renal system estimated with a statistical approach (

**E**) Kinetic parameters of the CM including the endoplasmic reticulum estimated with a regularized Gauss-Newton approach.

**Table 1.**Comparison among different compartment models. For each of the models we summarize the main characteristic (Scope), the involved tracer compartments (Compartments), and the number of unknow rate constants to be estimated (Size of $\mathit{k}$). The last to columns indicate whether the Patlak graphical approach can be applied (PGA), and whether the identifiability of the considered model has been proved (Identifiability).

Model | Scope | Compartments | Size of k | PGA | Identifiability |
---|---|---|---|---|---|

2-CM | Basic standard model | ${\mathcal{C}}_{f}$, ${\mathcal{C}}_{p}$ | 4 | ✓ | ✓ |

3-CM | Focus on endoplasmic reticulum | ${\mathcal{C}}_{f}$, ${\mathcal{C}}_{p}$, ${\mathcal{C}}_{r}$ | 5 | ✓ | ✓ |

RTM | Avoids use of IF | ${\mathcal{C}}_{f}$, ${\mathcal{C}}_{p}$, ${\mathcal{C}}_{R}$ | 6 | ✕ | ✓ |

Liver | Role of gut for the dual input (HA and PV) | ${\mathcal{C}}_{f}$, ${\mathcal{C}}_{p}$, ${\mathcal{C}}_{f}^{\prime}$, ${\mathcal{C}}_{p}^{\prime}$, ${\mathcal{C}}_{v}$ | 8 | ✕ | ✕ |

Kidney | Focus on tubules and bladder | ${\mathcal{C}}_{f}$, ${\mathcal{C}}_{p}$, ${\mathcal{C}}_{t}$, ${\mathcal{C}}_{u}$ | 7 | ✕ | ✓ |

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**MDPI and ACS Style**

Sommariva, S.; Caviglia, G.; Sambuceti, G.; Piana, M.
Mathematical Models for FDG Kinetics in Cancer: A Review. *Metabolites* **2021**, *11*, 519.
https://doi.org/10.3390/metabo11080519

**AMA Style**

Sommariva S, Caviglia G, Sambuceti G, Piana M.
Mathematical Models for FDG Kinetics in Cancer: A Review. *Metabolites*. 2021; 11(8):519.
https://doi.org/10.3390/metabo11080519

**Chicago/Turabian Style**

Sommariva, Sara, Giacomo Caviglia, Gianmario Sambuceti, and Michele Piana.
2021. "Mathematical Models for FDG Kinetics in Cancer: A Review" *Metabolites* 11, no. 8: 519.
https://doi.org/10.3390/metabo11080519