# Collaborative Decision Model for Allocating Intensive Care Units Beds with Scarce Resources in Health Systems: A Portfolio Based Approach under Expected Utility Theory and Bayesian Decision Analysis

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. A Portfolio Selection Approach to Support the Allocation of Patients to ICU Beds

## 4. Materials and Methods

#### 4.1. A Method for Estimating the Patients’ Chances of Survival

#### 4.2. An Information and Decision System to Support the Allocation of ICU Beds

## 5. Results and Discussion

_{1}, A

_{2}, A

_{3}and A

_{4}). The following assumptions were considered:

- a.
- The decision made on a decision round does not affect the parameters of subsequent decision rounds.
- b.
- Once a patient is admitted to an ICU bed, he/she is discharged only if he/she recovers, dies or if it is clear that there is no chance of recovery (in which case, the patient is sent elsewhere for palliative treatment).

Algorithm 1. Simulation algorithm developed to compare the approaches for tackling the ICU allocation problem. |

1: inputs: num_rounds, max_n |

2: for j = 1: 1: num_rounds |

3: generate n from the set {2, 3, …, max_n} by mcm_upd |

4: generate m from the set {1, 2, …, n−1} by mcm_upd |

5: for i = 1: 1: n |

6: generate ${p}_{i}\left({S}_{u}\right)$ from the interval (0,1] by mcm_upd |

7: generate ${p}_{i}\left({S}_{f}\right)$ from the interval [0, ${p}_{i}\left({S}_{u}\right)$) by mcm_upd |

8: end for |

9: define the allocations of ICU beds according to approaches 1, 2, 3 and 4 |

10: for i = 1: 1: n |

11: generate a random number ${x}_{i}$ from the interval [0, 1] by mcm_upd |

12: if ${x}_{i}\le {p}_{i}\left({S}_{f}\right)$ patient i survives regardless of being admitted to the ICU |

13: if ${p}_{i}\left({S}_{f}\right)<{x}_{i}\le {p}_{i}\left({S}_{u}\right)$ patient i survives only if admitted to the ICU |

14: if ${x}_{i}>{p}_{i}\left({S}_{u}\right)$ patient i dies regardless the admission to the ICU |

15: end for |

16: count the number of surviving patients after applying the four allocation rules |

17: end for |

18: output generation: calculate the average number of surviving patients per decision round for the four allocation rules (A_{1}, A_{2}, A_{3} and A_{4}) |

19: mcm_upd–a Monte Carlo method considering a uniform probability distribution. |

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Estimation of the patient’s chance of survival outside the ICU. CS_ICU—chance of survival during the stay in the ICU, CS_OUT—chance of survival outside the ICU. SpO

_{2}—blood oxygen saturation.

**Table 1.**The SOFA score. PaO

_{2}—partial pressure of oxygen in the blood. FiO

_{2}—fraction of inspired oxygen. MAP—mean arterial pressure.

Respiratory System | |
---|---|

PaO_{2}/FiO_{2} (mmHg) | SOFA score |

≥400 | 0 |

<400 | 1 |

<30 | 2 |

<200 with respiratory support | 3 |

<100 with respiratory support | 4 |

Cardiovascular system | |

MAP (mmHg) or administration of vasopressors required (dosage in μg/kg/min) | SOFA score |

MAP > 70 | 0 |

MAP < 70 | 1 |

Dopamine ≤ 5 or dobutamine (any dose) | 2 |

Dopamine > 5 or epinephrine ≤ 0.1 or norepinephrine ≤ 0.1 | 3 |

Dopamine > 15 or epinephrine > 0.1 or norepinephrine > 0.1 | 4 |

Nervous system | |

Glasgow Coma Scale | SOFA score |

15 | 0 |

13–14 | 1 |

10–12 | 2 |

6–9 | 3 |

<6 | 4 |

Kidneys | |

Creatinine (mm/dL) [μmol/L]; urine output (mL/day) | SOFA score |

<1.2 [<110] | 0 |

1.2–1.9 [110–170] | 1 |

2.0–3.4 [171–299] | 2 |

3.5–4.9 [300–440] or urine output < 500 | 3 |

>5.0 [>440] or urine output < 200 | 4 |

Liver | |

Bilirubin (mg/dL) [μmol/L] | SOFA score |

<1.2 [<20] | 0 |

1.2–1.9 [20–32] | 1 |

2.0–5.9 [33–101] | 2 |

6.0–11.9 [102–204] | 3 |

>12.0 [>204] | 4 |

Coagulation | |

Platelets (10^{3}/mL) | SOFA score |

>150 | 0 |

<150 | 1 |

<100 | 2 |

<50 | 3 |

<20 | 4 |

Initial SOFA Score | Chance of Survival Inside the ICU |
---|---|

0–5 | Very high |

6–9 | High |

10–11 | Medium |

12–24 | Very low |

**Table 3.**Results obtained from simulation. num_rounds = 100,000. ${\u2206}_{1i}=\left[\frac{\left({A}_{1}-{A}_{i}\right)}{{A}_{i}}\right].100$, for $i=\left\{2,3,4\right\}$.

Average Numbers of Surviving Patients per Decision Round | Gain Obtained with Allocation Rule 1 Compared to Rules 2, 3 and 4 | ||||||
---|---|---|---|---|---|---|---|

max_n | A_{1} | A_{2} | A_{3} | A_{4} | Δ_{12} | Δ_{13} | Δ_{14} |

2 | 0.869 | 0.832 | 0.758 | 0.750 | 4.49% | 14.70% | 15.81% |

5 | 1.526 | 1.461 | 1.331 | 1.316 | 4.40% | 14.65% | 15.90% |

10 | 2.616 | 2.506 | 2.285 | 2.256 | 4.41% | 14.50% | 15.95% |

15 | 3.712 | 3.552 | 3.239 | 3.194 | 4.50% | 14.62% | 16.23% |

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**MDPI and ACS Style**

Frej, E.A.; Roselli, L.R.P.; Alberti, A.R.; Britto, M.A.; de Barros Campelo Júnior, E.; Ferreira, R.J.P.; de Almeida, A.T.
Collaborative Decision Model for Allocating Intensive Care Units Beds with Scarce Resources in Health Systems: A Portfolio Based Approach under Expected Utility Theory and Bayesian Decision Analysis. *Mathematics* **2023**, *11*, 659.
https://doi.org/10.3390/math11030659

**AMA Style**

Frej EA, Roselli LRP, Alberti AR, Britto MA, de Barros Campelo Júnior E, Ferreira RJP, de Almeida AT.
Collaborative Decision Model for Allocating Intensive Care Units Beds with Scarce Resources in Health Systems: A Portfolio Based Approach under Expected Utility Theory and Bayesian Decision Analysis. *Mathematics*. 2023; 11(3):659.
https://doi.org/10.3390/math11030659

**Chicago/Turabian Style**

Frej, Eduarda Asfora, Lucia Reis Peixoto Roselli, Alexandre Ramalho Alberti, Murilo Amorim Britto, Evônio de Barros Campelo Júnior, Rodrigo José Pires Ferreira, and Adiel Teixeira de Almeida.
2023. "Collaborative Decision Model for Allocating Intensive Care Units Beds with Scarce Resources in Health Systems: A Portfolio Based Approach under Expected Utility Theory and Bayesian Decision Analysis" *Mathematics* 11, no. 3: 659.
https://doi.org/10.3390/math11030659