# Antifragile Control Systems: The Case of an Anti-Symmetric Network Model of the Tumor-Immune-Drug Interactions

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## Abstract

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## 1. Introduction

#### The Need for Control in Cancer Therapy

## 2. Materials and Methods

#### 2.1. Tumor-Immune-Drug Network Model

#### Drug-Free Evolution

#### 2.2. Antifragile Control

#### 2.2.1. Preliminaries

#### 2.2.2. Formalism

#### Dynamical Systems on Manifolds

#### Control Error Function

#### Parallel Transport Map

#### 2.2.3. Control Synthesis

#### Redundant Overcompensation

#### Stucture Variability

- the motion equation of the sliding mode, as nicely framed in Slotine et al. [32], can be designed linear and homogeneous, despite that the tumor-immune-drug model is governed by nonlinear equations,
- the sliding surface does not depend on the process dynamics, but it is determined by parameters selected by the designer, as suggested in deCarlo et al. [33], i.e., desired trajectory of the system in the antifragile region of the drug dose–response curve (e.g., Hill function),
- once the sliding motion occurs (i.e., the system dynamics are on the surface), the system has invariant properties which make the motion independent of certain system parameter variations, uncertainty, and disturbances, as described in the famous work of Utkin [34]. Hence, the system performance can be completely determined by the dynamics of the sliding manifold.

#### Bounded High-Frequency Activity

#### 2.3. State-of-the-Art Control Algorithms in Cancer Therapy

#### 2.3.1. Optimal Control

#### 2.3.2. Robust Control

#### 2.3.3. Pulsed Control

## 3. Experiments

Model Parameter | Parameter (Initial) Value | Impact on Dynamics |
---|---|---|

Number of normal host cells N | 1.0 | Interacts with I and T w/wo u |

Number of tumor cells T | 0.25 | Interacts with N and I w/wo u |

Number of host immune cells I | 0.1 | Interacts with N and T w/wo u |

Drug concentration at the tumor location u | 0.01 | induces cell kill by toxicity (normal+tumor) |

Drug administration v | 0.0 | modulates the drug concentration at the tumor |

Fraction normal cell kill ${a}_{1}$ | 0.2 | dose-related weight of normal cell kill (toxicity) |

Fraction tumor cell kill ${a}_{2}$ | 0.3 | dose-related weight of tumor cell kill |

Fraction immune cell kill ${a}_{3}$ | 0.1 | dose-related weight of immune cell kill |

Tumor growth rate ${r}_{1}$ | 1.5 | typically ${c}_{2}+{c}_{3}$ |

Normal cells growth rate ${r}_{2}$ | 1.0 | per capita growth |

Carrying capacity of tumor cells ${b}_{1}$ | 1.0 | weights tumor cells self-excitation |

Carrying capacity of normal cells ${b}_{2}$ | 1.0 | weights normal cells self-excitation |

Tumor-immune competition factor ${c}_{1}$ | 1.0 | competition |

Immune-tumor competition factor ${c}_{2}$ | 0.5 | modulation |

Tumor-normal competition factor ${c}_{3}$ | 1.0 | competition |

Normal-tumor competition factor ${c}_{4}$ | 1.0 | competition |

Immune cells death rate ${d}_{1}$ | 0.2 | regulation through $\frac{s}{{d}_{1}}$ |

Drug influx modulation ${d}_{2}$ | 1.0 | rate of change/decay of u |

Immune threshold rate $\alpha $ | 0.3 | related to the immune response curve |

Immune response rate $\rho $ | 0.01–1.0 | immune-compromised to healthy |

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Therapy closed–loop control system: the main elements of our contribution in a control-theoretic feedback loop diagram. The purpose of the controller is to minimize the deviation from the desired response of the tumor by computing a suitable drug dose. The suitable dose can span from the fragile region (“red”) to the antifragile region (“green”) of the response curve under the effect of the controller output. The evolution of the tumor happens under the impact of the drug and also the immune system, which perturbs its evolution.

**Figure 2.**Network model of tumor-immune-drug interactions. Left panel: populations cells interactions; Right panel: differential equations parameters mapped on the network populations interactions. Each of the circles represents the respective cell population, the connecting lines three the types of interaction (i.e., cooperation—enhancement, competition—mutually induced decrease, and killing—extermination), and the impact of the administered drug.

**Figure 3.**Drug-free evolution of the network model. The immune system (I) reacts by proliferating T-cells when the tumor (T) population (and the tumor itself) reaches a threshold triggering immune surveillance. The normal cells (N) production decreases as the tumor expands. In this case, without any drug administration, the tumor proliferates uncontrollably, escaping immune surveillance.

**Figure 4.**Dynamical null-space analysis of the drug-free model and the prerequisites for therapy control design. We are interested in having two attractors, one for “healthy” state ($\mathbf{H}$) and one for “disease” (${\mathbf{C}}_{\mathbf{2}}$) so that the control can push the dynamics to reduce tumor size under drug intake.

**Figure 5.**Model equilibria as a function of the immune cell influx without tumor s and the immune response $\rho $. The control design will revolve around the marked region in the landscape of coexisting equilibria with two stable attractors, one for “healthy” state ($\mathbf{H}$) and one for “disease” (${\mathbf{C}}_{\mathbf{2}}$).

**Figure 6.**Variable structure control using SMC. (

**a**) the sliding condition, from initial states, the system is pushed towards the surface S; (

**b**) the dynamics on the sliding surface; (

**c**) drug dose–response curve with fragile and antifragile regimes depending on curvature (adapted with permission from West et al. [5]).

**Figure 7.**Antifragile principle. (

**a**) The drug dose–response basis for the imposed dynamics of the network. The survival function assumes that the system needs to provide a drug dose such that the value of r decreases; (

**b**) The derivative of the survival function. This demonstrates that the inflection point describes the point where the dosage (control law) can drive the tumor-immune-drug system in the antifragile region (adapted with permission from West et al. [5]).

**Figure 8.**Antifragile control therapy. Individual cell populations evolution under the computation of the antifragile control law of drug dose v. The antifragile drug dose timing and duration are computed based on the control law in Equation (27).

**Figure 9.**Optimal control therapy. Individual cell populations evolve under optimal computation of the drug dose v. The drug dosing amplitude, timing, and duration is computed as a constrained optimization problem that slows down and, eventually, disrupts tumor growth. When the tumor (T) decays under the detectable size (≈80 days), the normal cells population (N) restarts the normal proliferation pattern, and immune reaction (I) is enhanced to support the complete tumor extinction.

**Figure 10.**Robust control therapy. Individual cell populations evolution under the computation of the robust control law of drug dose v based on Equations (31). The bang-bang drug dose timing and duration are computed such that the tumor population (T) decreases rapidly after treatment cessation (≈ 50 days). This decrease in tumor size is also determined by the exponentially increasing immune response (I) and marks also the normal cell (N) proliferation/death (i.e., through toxicity) pattern.

**Figure 11.**Pulsed control therapy. Individual cell populations evolution under the computation of the robust control law of drug dose v. The pulsed control computes densely timed, equally sized, and dosed drug administrations which only trigger short decays in tumor (T) proliferation, which at the end of the therapy resumes its uncontrolled growth which suppresses the normal cells (N) production and a weakening immune response (I).

Criteria/Control | Pulsed | Robust | Optimal | Antifragile |
---|---|---|---|---|

Time to tumor elimination (days) | 150+ | 68 | 100 | 65 |

Max drug concentration (mg/L) | 0.455 | 0.987 | 0.659 | 0.736 |

Total drug administered (mg/L) | 15.00 | 12.00 | 15.00 | 14.31 |

Max tumor cells ($\times {10}^{11}$ cells ) | 0.553 | 0.258 | 0.257 | 0.251 |

Min normal cells ($\times {10}^{11}$ cells) | 0.447 | 0.706 | 0.750 | 0.592 |

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**MDPI and ACS Style**

Axenie, C.; Kurz, D.; Saveriano, M. Antifragile Control Systems: The Case of an Anti-Symmetric Network Model of the Tumor-Immune-Drug Interactions. *Symmetry* **2022**, *14*, 2034.
https://doi.org/10.3390/sym14102034

**AMA Style**

Axenie C, Kurz D, Saveriano M. Antifragile Control Systems: The Case of an Anti-Symmetric Network Model of the Tumor-Immune-Drug Interactions. *Symmetry*. 2022; 14(10):2034.
https://doi.org/10.3390/sym14102034

**Chicago/Turabian Style**

Axenie, Cristian, Daria Kurz, and Matteo Saveriano. 2022. "Antifragile Control Systems: The Case of an Anti-Symmetric Network Model of the Tumor-Immune-Drug Interactions" *Symmetry* 14, no. 10: 2034.
https://doi.org/10.3390/sym14102034