The Location Problem of Medical Drone Vertiports for Emergency Cardiac Arrest Needs

Abstract

The implementation of medical drones can quickly and efficiently expand the coverage range of an area, allowing for a faster response to incidences of out-of-hospital cardiac arrest and improving the subsequent survival rate of such incidences, while promoting sustainable health development goals under the configuration standards for automatic external defibrillators in China. In response to the problem of the selection of locations for medical drone vertiports (for take-off and landing) that are equipped with automatic external defibrillation facilities, a survival function was introduced to establish a model for site selection, with the primary optimization objective of maximizing the average survival rate of patients and taking the operating costs of a system into account. At the same time, considering the constraints of drone phase operation time, energy consumption, coverage range, etc., a medical drone vertiport site selection model was established for emergency cardiac arrest needs. An improved immune algorithm was applied to the model’s calculations and the analysis of the results, using the Jinnan District in Tianjin as an example. The results show that the proposed model and algorithm are feasible and applicable. The Jinnan District in the city of Tianjin requires a total of 24 medical drone vertiports in order to achieve full coverage of an area under the “golden 4-minute” rescue time. When the average survival rate of patients is 0.9, the operation results are deemed optimal, and the average survival rate of patients is 64.06%. Compared to ground ambulances currently used in hospitals, the implementation of medical drones could significantly shorten response time, improve the average survival rate of patients by 41.96%, and effectively improve the existing low survival rate and the accessibility of medical services. The results of this study can provide decision-making support for the planning of automatic external defibrillators in public places and the construction of sustainable and efficient emergency medical service systems.

1. Introduction

The rate of population aging is rapidly advancing, and contemporary life is beset with mounting pressures as a result of this. Notably, the incidence of risk factors associated with cardiovascular diseases is evident. Cardiac arrest (CA) has emerged as a prominent cardiovascular emergency, with grave effects on life and well-being due to its abrupt onset symptoms and its dismal survival rate [1]. When contrasted with instances of in-hospital cardiac arrest (IHCA), the majority of cardiac arrests (ranging from 70% to 80%) take place in non-hospital environments, exhibiting a stochastic and abrupt spatiotemporal distribution. This circumstance exacerbates the already diminished survival prospects for individuals experiencing out-of-hospital cardiac arrest (OHCA). In the event of a patient undergoing cardiac arrest, the timing of rescue becomes paramount. With each minute of delay in intervention, the likelihood of survival diminishes from 7% to 10%. Timely interventions, including the application of automated external defibrillation (AED), prove indispensable in augmenting the survival prospects of OHCA patients [2]. Empirical research has demonstrated that administering AED defibrillation within a 3 to 5 min timeframe can yield a survival rate ranging from 50% to 70% for individuals experiencing OHCA. Consequently, the timely application of defibrillation to OHCA patients is crucial in increasing their likelihood of survival.

At present, automated external defibrillator (AED) installations predominantly comprise conventional, static fixtures located within public spaces, such as subway stations and shopping centers. These installations exhibit deficiencies encompassing unequal spatial dispersion, challenges in detection, restricted maneuverability, and incongruence in sites where out-of-hospital cardiac arrest (OHCA) emergencies occur [3]. Consequently, there are difficulties in providing swift responses to medical emergencies, leading to a suboptimal utilization rate of AEDs [4]. To increase patient survival rates and achieve the judicious refinement of AED distribution, scholars have harnessed mathematical modeling approaches, such as maximum coverage site selection models and two-step mobile search techniques, to optimize deployment strategies. Nevertheless, in China, investigations regarding the use of AEDs are relatively infrequent and primarily consist of qualitative examinations [5]. These examinations undertake a comparative analysis of AED deployment standards and research advancements in various nations, the identification of shortcomings in China’s current AED distribution, and the formulation of recommendations for enhancing AED implementation [6]. Unmanned aerial vehicles (UAVs), commonly referred to as drones, present advantages such as exceptional mobility, convenience, efficiency, delivery speeds akin to those of ground-based emergency medical service (EMS) vehicles, and immunity to ground traffic impediments. Consequently, scholars have proposed the integration of AED-equipped drones to establish a comprehensive specialized medical drone system, enabling swift and effective responses to a wide spectrum of emergency medical demands [7]. Nonetheless, the establishment of suitable take-off and landing sites (vertiports) is of paramount importance for drone usage. Selecting optimal locations for medical drone vertiports could expand their coverage range, improve accessibility to healthcare, reduce environmental impacts and health inequalities, and promote health system reforms, which could then help to guide policy and healthcare resource allocation, promote sustainable urban development, and improve people’s quality of life, thus contributing to the realization of sustainable health development goals (SHDGs). Hence, the location of these medical drone vertiports has become a pressing concern that demands immediate attention. These efforts are crucial for the construction of contemporary, efficient emergency medical service infrastructure, in keeping with the United Nations Sustainable Development Goals and the realization of sustainable healthcare development.

Currently, research on medical drones equipped with AED facilities is in its initial stages. Therefore, when researching the location of vertiports for medical drones, it is advisable to draw insights from the prior development of logistics drones as well as relevant studies in emergency rescue operations. Scholars have conducted a number of studies on decision-making models for the selection of emergency medical facilities, aiming to determine the number and location of emergency facilities and the scope of their service under different qualifying conditions. While the four basic models of facility selection have been applied in varying degrees to the emergency medical facility location problem, in an emergency out-of-hospital cardiac arrest scenario, out of the spirit of humanitarian rescue, emergency medical facilities need to cover as much as possible of an area in demand (or its entirety) at the fastest speed possible; therefore, the coverage model [8] is highly suitable for addressing the location problem of medical drone vertiports.

In logistics-focused drone research, scholars have considered three main aspects: meeting customer demands, operational costs, and drone performance. Shavarani et al. considered drone flight endurance and speed, aiming to minimize the overall construction cost and the number of customers out of reach of the drones. They developed models for the number and location of drones and their distribution [9]. Chauhan et al. considered drone energy consumption and range limitations, with the goal of maximizing demand coverage. They established models for the strategic placement of logistics drone facilities [10]. Ren et al. integrated time-sensitive requirements to create a fully automated vertiport location model with the shortest pickup time as the objective, using queuing theory to determine the optimal number of vertiports [11]. Zhang et al. considered various levels of logistics drones operating in urban areas and constructed a vertiport location planning model with the objectives of minimizing total costs and maximizing customer satisfaction [12]. In the field of emergency rescue research, primary objectives include adapting to the temporal variations in emergency needs and service areas, minimizing dispatch time or tardiness, and maximizing service coverage. Golabi et al. assessed the survival probability of individual streets based on the distance from the earthquake epicenter and the risk level of the corresponding area; they established a disaster relief siting model with the objective of minimizing the total number of trips [13]. Wang et al. combined the characteristics of suddenness and urgency in civil aviation emergency rescue work and used the Analytic Hierarchy Process (AHP) and p-median models to analyze the factors influencing the layout of civil aviation emergency regional support centers. They selected appropriate layout points from existing alternative airports, contributing to a scientifically planned network layout for regional support centers [14].

Reviewing the studies about the integration of AED with drones, Chan et al. employed a maximum coverage location model to improve the coverage of AED for OHCA accidents. Through comparative analysis, they demonstrated that the maximum coverage location model is superior to population-based AED location methods [15]. Baumgarten et al. assessed the feasibility of drone systems equipped with AED facilities to improve patient survival rates in rural areas of northeastern Germany [16]. Schierbeck et al. identified areas with high incidences of OHCA nationwide for the strategic placement of drone-equipped AED facilities. They quantified the number of drones required to reach 50%, 80%, 90%, and 100% of the target population within 8 minutes, proving that a minimal number of drone EMS systems significantly enhances the national OHCA coverage [17]. Wankmüller et al. optimized the allocation of defibrillator drone stations within a designated geographic area by developing an integer linear programming model that minimizes the number of drones used and the average travel time for drones responding to patients [18]. Claesson et al. considered the differences in response times of emergency medical facilities in urban and rural areas. They used GIS to design an appropriate layout for drone systems, significantly reducing the response time of drone-equipped AED facilities [19]. Pulver et al. utilized a maximum coverage location model to optimize the positioning of drone-equipped AED facilities, aiming to reduce emergency response times for OHCA patients. The results showed that positions determined by the maximum coverage location model could meet 90.3% of emergency demands and save costs [20].

Upon reviewing the relevant literature, it is evident that there is limited research on drone-equipped AED facilities; previous models for locating emergency medical facilities have typically focused on traditional single-objective functions, such as “minimizing response time”, “maximizing coverage”, or “minimizing cost”, with minimal consideration for the patient. In this paper, we address this gap by introducing the survival rate function and the concept of “golden 4-minute” rescue coverage from the patient’s perspective. Simultaneously, we consider the performance constraints related to the operation of medical drones in various phases. And we construct a multi-objective medical drone vertiport location model that addresses the emergency needs of sudden cardiac arrest, considering both patient survival rates and system operational costs. The immune optimization algorithm is applied to solve this model. Using Tianjin’s Jinnan District as an example, this paper obtains a layout of medical drone vertiports. These results can offer decision-making support for the construction of sustainable and healthy municipalities.

2. Problem Description and Assumptions

2.1. Medical Drones and Modes of Operation

Medical drones, derived from conventional drones through advanced technological development, utilize aerodynamic take-off and operate through wireless remote-control equipment and self-contained program control devices. They are capable of repetitive deployment and can transport various types of portable specialized medical equipment or medical supplies. In this study, the term “medical drone” specifically refers to a dedicated integrated drone designed for OHCA emergency scenarios, equipped with AED facilities. Taking reference from the VIVEST Air series drone with integrated AED, this design incorporates a customized cabin for vertical take-off and landing operations as well as recharging, facilitating efficient and effective adaptation to OHCA emergency rescue scenarios.

The operational mode of specialized integrated medical drones equipped with AED facilities for responding to sudden public health emergencies related to OHCA is depicted in Figure 1. Key stakeholders in the event of an abrupt OHCA accident include OHCA patients, medical drones and their vertiports, first responders and bystanders, emergency response platforms, and higher-level medical institutions. Given the stochastic and sudden nature of OHCA accidents in both temporal and spatial dimensions, this study employs a randomized approach to generate specific OHCA accident locations within the designated area. The potential number of cases per year in the selected area is calculated by multiplying the number of permanent residents in the selected area by the proportion of elderly people aged 60 years old or above and further multiplying the incidence of cardiac arrest in the past. OHCA accident points are then randomly distributed according to the population distribution in the area, effectively emulating their random occurrences. This methodology accurately replicates the characteristics of OHCA accidents within the selected region while upholding the privacy and confidentiality of medical data. Furthermore, it provides viable options for strategic planning and research, demonstrating operational feasibility.

Considering the stakeholders involved in accidents, the operational characteristics of medical drones, and the specificities of OHCA accidents, the operational procedure of medical drones unfolds as follows: OHCA accidents occur randomly within the designated research area. Upon discovering a patient, first responders and bystanders use mobile phones or wearable devices to initiate emergency calls and distress signals. Subsequently, they administer cardiopulmonary resuscitation (CPR) following operational guidelines provided by the emergency platform. Upon receiving the distress signal, the emergency platform ascertains the accident’s location, leveraging bystander descriptions and mobile GPS positioning. It promptly dispatches the most time-efficient medical drone to provide immediate medical assistance to the patient. Simultaneously, relevant data and information are shared with higher-tier medical platforms. Upon receipt of the emergency directives, the medical drone swiftly departs from its designated vertiport (referred to as the drone’s accompanying cabin). Following ascent to an appropriate altitude, the drone transitions into a horizontal cruising phase. As it nears the accident location, it hovers and executes a landing. Bystanders retrieve the AED from the medical drone and use it to perform defibrillation on the OHCA patient, following operational instructions, until the arrival of emergency personnel. This comprehensive approach not only ensures the patient’s well-being but also contributes to an enhanced survival rate.

Based on the operational mode of medical drones, the issues can be outlined as follows: Multiple OHCA accident sites within a specific region are assumed to exist, and the coordinates of these sites are known. Specialized integrated medical drones equipped with AED facilities are deployed to address out-of-hospital emergency requirements. Upon receiving emergency instructions, the medical drone nearest to the accident site departs from its vertiport, initiating phased flights to reach the OHCA accident site. Following collaborative intervention by first responders and advanced medical facilities, the medical drone equipped with AED facilities returns to its vertiport for recharging and recovery operations to ensure efficient and prompt responses to subsequent emergency calls. This study takes into account various factors, including patient survival rates, the phased operational characteristics of medical drones, and flight coverage. It explores the optimal layout plan for medical drone vertiports within the selected region, as illustrated in Figure 2. The relationship between the drone vertiports and the accident sites is one-to-many, signifying that each OHCA accident site can be serviced by the nearest medical drone vertiport. Conversely, a single medical drone vertiport can respond to the emergency demands of multiple OHCA accident sites.

2.2. Survival Function of OHCA Patient

During the treatment of OHCA patients, survival rates are influenced by various medical factors, including the patient’s physical condition, CPR, and defibrillation. Among these factors, defibrillation plays a significant role in determining patient survival rates. Consequently, this study focuses on evaluating the impact of AED delivery time on the survival rate of OHCA patients. Medical research reveals that, in cases of cardiac arrest, every minute of delay in defibrillation results in a substantial decline in survival rates, ranging from 7% to 10%. When administering defibrillation within the first minute of cardiac arrest, the patient’s survival rate can reach as high as 90%. However, this rate decreases to approximately 50% after 5 min, to around 30% after 7 min, and plunges to a mere 2–5% beyond the 10-minute mark. Furthermore, critical time intervals known as the “golden 4-minute” and the “platinum 10-minute” exist during the treatment of cardiac arrest patients. Within the initial 4 min following cardiac arrest, irreversible brain damage occurs. If this critical timeframe is exceeded, even if the patient is successfully resuscitated, there is a significantly elevated risk of brain death or entering a vegetative state. Considering these research findings and the importance of time intervals in the rescue process, this paper constructs a survival rate function for OHCA patients. Figure 3 provides a schematic representation of the OHCA patient survival rate function.

2.3. Relevant Assumption

In the course of medical drone operations between vertiports and OHCA accident sites, it is crucial to establish foundational assumptions concerning potential scenarios and conditions. Consequently, the relevant assumptions can be delineated as follows:

(1)

All OHCA accident sites hold equivalent priority and are addressed based on their chronological sequence;

(2)

Subsequent to the occurrence of OHCA accidents, first responders and bystanders who discover the accidents will consistently engage in treatment;

(3)

Each OHCA accident location is attended to by a singular medical drone;

(4)

Each medical drone possesses the capacity to serve multiple OHCA accidents;

(5)

Multiple medical drones are stationed at each vertiport;

(6)

Medical drones have the capability to operate safely and expeditiously without constraints within the designated research area.

3. Model Building

3.1. Medical Drone Vertiport Modeling

3.1.1. Objective function

(1)

Survival rate function

In an OHCA emergency scenario, prioritizing the fundamental rescue principle of “life first” is crucial. Therefore, on the basis of the location set covering problem and the maximum coverage location model, this paper adopts a patient-centric approach. It introduces the survival rate function and designates it as the primary optimization goal, specifically focusing on maximizing the average survival rate of OHCA patients:

$$\max SR={\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}SR\left(t_{ij}\right)/n}}$$

(1)

$$SR\left(t_{ij}\right)={\left(e^{0.679+0.131t_{ij}}-0.8\right)}^{-1}$$

(2)

$$t_{ij}=t_0+\frac{H_{\max }-z_i}{v_1}+\frac{\sqrt{{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2}}{v_2}+\frac{H_{\max }-z_j}{v_3}$$

(3)

$SR\left(t_{ij}\right)$ is the survival rate of OHCA patients, and the survival function is a concave function of time with reference to the survival function of OHCA patients proposed in the literature [21] and adjusted in this paper. i is the serial number of the alternative medical drone vertiports. $\left(x_i,y_i,z_i\right)$ are the coordinates of the vertiports; I is the aggregation of vertiports for alternative medical drones; $X_i$ = 1 or 0 indicates whether or not to establish a medical drone vertiport at the point i; j is the serial number for OHCA accident points; $\left(x_j,y_j,z_j\right)$ are the coordinates of the accident points; J is the OHCA accident point set; $Y_{ij}$ = 1 or 0 indicates whether the OHCA accident point is within the coverage area of the medical drone vertiport; $H_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the horizontal cruising altitude for medical drones; $t_{ij}$ is the AED delivery time, i.e., the time it takes to fly from the drone vertiport to the OHCA accident point; $t_0$ is the pre-scheduling and lead time (the AED can initiate external defibrillation once it arrives, with a negligible time to save the patient); and $v_1$,$v_2,v_3$ are the vertical ascent phase speed, horizontal cruise speed, and vertical descent speed of the medical drone.

(2)

Total cost function

In addition to the prioritization of the patient survival rate in the rescue, the system operating costs also need to be taken into account. Relevant researchers and scholars have measured the cost-effectiveness of medical drones’ modes of operation, so this paper incorporates the system operating costs as a secondary objective. The goal is to minimize the total cost of the medical drone operation, achieving an optimal equilibrium state where both patient survival rate and system operation costs are appropriately balanced, that is:

$$\min TC=FC+VC$$

(4)

$$FC={\displaystyle \sum _{i\in I}{C}_1{X}_i}$$

(5)

$$VC={\displaystyle \sum _{i\in I}{\displaystyle \sum _{j\in J}2C_2}}{Y}_{ij}{d}_{ij}$$

(6)

TC is the total cost of operating a medical drone, consisting of fixed cost FC and variable cost VC, of which $C_1$ represents the construction and operation of medical drone landing sites measured in CNY, $C_2$ is the flight path costs for AED delivery of medical drones measured in CNY per meter, including maintenance costs per unit distance $C_2^1$, depreciation costs per unit distance $C_2^2$, and battery maintenance costs per unit distance $C_2^3$; $d_{ij}$ is the one-way distance for medical drone flights from a landing site to an OHCA accident site.

(3)

Overall objective function

In this paper, we comprehensively consider two objective functions: the average survival rate of OHCA patients and the total cost of operation. Maximizing the average survival rate of patients serves as the primary optimization objective, while minimizing the cost of medical drone operation acts as the secondary objective. Due to the different magnitudes of these objectives, direct weighting is not feasible. Therefore, this paper normalizes them separately to eliminate discrepancies in magnitudes.

$$MT=\omega \cdot S\tilde{R}+\left(1-\omega \right)\cdot T\tilde{C}$$

(7)

$$S\tilde{R}=\frac{SR-\min SR}{\max SR-\min SR}$$

(8)

$$T\tilde{C}=\frac{\max TC-TC}{\max TC-\min TC}$$

(9)

MT represents the total objective function considering multiple objectives, and $\omega$ is the weighting factor for the average patient survival after normalization. The range of values for $\omega$ is [0, 1]. Considering that maximizing the average survival rate of patients is the primary goal and following the principle of “life is paramount” in relief efforts, the value of $\omega$ should generally be greater than 0.5 and converge to 1. This paper aims to maximize the average patient survival rate, making it a positive objective in Equation (8). The larger the average patient survival rate, the better the result. Conversely, in Equation (9), the total cost of medical drone operation is the objective, and the smaller the total cost, the better the result. This represents an inverse objective.

3.1.2. Constraint

Based on the problem description, assumptions, and the characteristics of medical drone phased operation time, energy consumption, and coverage area, the medical drone vertiport location model constraints are as follows:

(1)

Constraint of the medical drone range

The round-trip distance of a medical drone for a single AED delivery rescue mission cannot exceed its maximum range. The constraint is expressed as follows:

$$2Y_{ij}{d}_{ij}\le L_{\max },\forall i\in I,j\in J$$

(10)

$L_{max}$ is the maximum range for medical drones in Equation (10);

(2)

Constraints of rescue matching relation

For any OHCA accident point j, there will be only one medical drone vertiport responsible for AED delivery, and there is no scenario in which the accident is not responded to or multiple points are responded to. The constraint is expressed as follows:

$${\displaystyle \sum _{i\in I}{Y}_{ij}}=1,\forall j\in J$$

(11)

(3)

Constraint of emergency response relationship

Only when a medical drone vertiport is established at point i can the emergency needs of the OHCA accident site at point j be responded to by a medical drone at point i to provide AED delivery services. The constraint is expressed as follows:

$$Y_{ij}-X_i\le 0,\forall i\in I,j\in J$$

(12)

(4)

Constraints on the number of landing sites

The total number of locations for medical drone vertiports cannot exceed the number of alternative vertiports. The constraint is expressed as follows:

$${\displaystyle \sum _{i\in I}{X}_i\le N}$$

(13)

N is the number of alternative medical drone vertiports;

(5)

Energy consumption constraints for medical drones

Medical drones do not exceed their maximum energy consumption during the phased operation of AED delivery. The constraint is expressed as follows:

$$\frac{\left(W+m\right)g}{370\eta \gamma }(2H_{\max }-z_i-z_j+d_{ij})+e(t_{ij}-t_0)\le E_{\max }$$

(14)

W is the self-weighting for drones, m is the weight of AED facilities carried, g is the acceleration due to gravity (the general value is 9.8 $\mathrm{m}/{\mathrm{s}}^2$),$\eta$ is the drone propeller power transfer efficiency, $\gamma$ is the drone flight lift-to-drag ratio, e is the energy consumption of electronic components of drones, and ${\mathrm{E}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum energy consumption of medical drones;

(6)

Constraint of survival rate threshold

We must ensure that the survival rate of OHCA patients rescued by medical drones does not fall below a set survival rate threshold. The constraint is expressed as follows:

$$SR\left(t_{ij}\right)\ge {\xi }_j,\forall i\in I$$

(15)

${\xi }_j$ is the survival of OHCA patients receiving AED defibrillation within the “golden 4-minute”.

3.2. Algorithm Design

The medical drone vertiport location problem is a classical NP problem. Existing research often tackles NP problems using optimized intelligent algorithms, which can reduce the difficulty in solving problems and simplify the solution process. The genetic algorithm is an efficient adaptive evolutionary search algorithm that realizes the improvement of each individual’s adaptability through the mechanisms of natural selection, heredity, and mutation. However, numerous practices and studies have shown that the standard genetic algorithm has defects of poor local search ability and “early maturity”. In response to these challenges, the immune optimization algorithm emerges as a new type of intelligent search algorithm inspired by the biological immune system. This algorithm exhibits strong robustness and finds wide applications in emergency facility site selection. Therefore, this paper applies the immune optimization algorithm to solve the model. The flowchart of the algorithm is shown in Figure 4 and the specific flow of the algorithm is as follows:

Step 1: Recognition of antigens

Define stock information as a structure;

Step 2: Generate initial antibody population

The initial antibody population is randomly generated in the feasible solution space using a simple coding approach. Each medical drone vertiport scheme forms an antibody of a specific length, where each antibody represents the sequence of OHCA accident points selected as medical drone vertiports. For example, the antibody [2 7 14 31 52] represents the accident points 2, 7, 14, 31, 52 selected as medical drone vertiports;

Step 3: Evaluation of the diversity of solutions

After generating the initial antibody population, the diversity of solutions is evaluated using the expected reproduction probability as the criterion. The higher the expected reproduction probability of the antibody, the stronger the adaptive ability of the antibody. Throughout the algorithmic search process, antibodies with strong adaptive abilities are retained in the memory bank. Under the effect of immune balance, both individuals with high adaptive abilities and those with higher concentrations are promoted and suppressed. This mechanism enhances the diversity of individuals in the population.

(1)

Antibody–antigen affinity

The affinity between an antibody and an antigen is used to indicate the degree of recognition of the antigen by the antibody. The affinity function is designed based on the medical drone vertiport siting model:

$$A_v=\frac{1}{F_v}=\frac{1}{\omega \cdot \frac{SR - \min SR}{\max SR - \min SR}+\left(1 - \omega \right)\cdot \frac{\max TC - TC}{\max TC - \min TC}}$$

(16)

$A_v$ is the affinity function and $F_v$ is the overall objective function;

(2)

Antibody–antibody affinity

The affinity between antibodies reflects the degree of similarity between them. In this context, the R-site continuum method proposed by Yang et al. [22] is borrowed to calculate the affinity between antibodies. The formula is as follows:

$$S_{\alpha ,\beta }=\frac{k_{\alpha ,\beta }}{L}$$

(17)

$k_{\alpha ,\beta }$ represents the identical parts in antibody $\alpha$ and antibody $\beta$ and L is the length of the antibody.

(3)

Antibody concentration

The concentration of antibodies is the proportion of similar antibodies in the population:

$$C_v=\frac{1}{n}{\displaystyle \sum S_{\alpha ,\beta }}$$

(18)

$$S_{\alpha ,\beta }=\{\begin{array}{l}1,S_{\alpha ,\beta }>T\\ 0,else\end{array}$$

(19)

n is the total number of antibodies and T is a pre-set threshold.

(4)

Expected probability of reproduction

The expected probability of reproduction is determined by the combination of the affinity between the antibody and antigen and the concentration of the antibody. It is calculated using the following formula:

$$P=\eta \frac{A_v}{{\displaystyle \sum A_v}}+\left(1-\eta \right)\frac{C_v}{{\displaystyle \sum C_v}}$$

(20)

$\eta$ is a constant in mathematics. At the same time, an elite retention strategy is adopted, in which a number of individuals with the highest affinity to the antigen are deposited into the memory bank each time the memory bank is updated. Then, the best individuals from the remaining samples are deposited into the memory bank according to the desired probability of reproduction.

Step 4: Operations of immunization

Selection: Operates according to the roulette wheel selection mechanism, where the probability that an individual is selected is the expected probability of reproduction;

Crossover: This paper employs a single-point crossover method for crossover operations;

Mutations: Mutations are performed using the commonly used mutation method, i.e., randomly selected mutation sites.

4. Case Analysis

4.1. Case Description

In order to validate the model’s effectiveness, this paper chose Jinnan District in Tianjin City, China, as the location for medical drone vertiports. The selected area has a latitude range of 38.83°–39.08°, a longitude range of 117.24°–117.55°, and a total land area of 420.72 square kilometers. Considering the random distribution and suddenness of OHCA patients’ onset in time and space, along with the challenges in obtaining confidential and private patient data, this paper adopts the random generation of a certain number of OHCA accident sites within the study area to simulate the spatial distribution’s randomness.

According to the seventh census data of China in 2020, the number of permanent residents in Jinnan District is 928,066,000. Among them, the proportion of elderly people over 60 years old is 15.9%. OHCA accidents mostly occur in out-of-hospital public places and at home, primarily affecting the elderly, with an incidence rate of 41.8 per 100,000 people. Considering the population size, the proportion of elderly people, and the OHCA incidence rate in the study area, as outlined in Section 2.1, it can be estimated that there are approximately 62 OHCA accidents in Jinnan District per year. For the subsequent distance operations in this paper, Euclidean distance is utilized. Therefore, the latitude and longitude coordinates of the accident points need to be converted to planar coordinates. This conversion facilitates subsequent calculations in the study and is adapted to the algorithm’s model constraints. The MATLAB tool was employed to convert the coordinates using Miller’s projection. The converted planar coordinates of the OHCA accident points are distributed as shown in Figure 5.

4.2. Parameter Settings

To align with the actual situation, this paper establishes specific parameters for medical drone vertiport location. These parameters are set by consulting relevant works in the literature and considering the logistics drone vertiport selection and allocation process. The specific parameters of the model are presented in

Table 1. Model parameter setting.

Parameter	Value	Parameter	Value
$\mathrm{W}/\mathrm{k}\mathrm{g}$	10	${\mathrm{v}}_1^{\mathrm{k}}/\left(\mathrm{m}/\mathrm{s}\right)$	10
$\mathrm{m}/\mathrm{k}\mathrm{g}$	5	${\mathrm{v}}_2^{\mathrm{k}}/\left(\mathrm{m}/\mathrm{s}\right)$	16.67
$\mathsf{\eta }$	0.5	${\mathrm{v}}_3^{\mathrm{k}}/\left(\mathrm{m}/\mathrm{s}\right)$	5
$\mathsf{\gamma }$	3	$\mathrm{N}/\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}$	62
$\mathrm{e}/\mathrm{w}$	100	${\xi }_j$	60%
${\mathrm{H}}_{\mathrm{m}\mathrm{a}\mathrm{x}}/\mathrm{m}$	60	${\mathrm{E}}_{\mathrm{m}\mathrm{a}\mathrm{x}}/\left(\mathrm{k}\mathrm{w}·\mathrm{h}/\mathrm{k}\mathrm{g}\right)$	0.25
${\mathrm{L}}_{\mathrm{m}\mathrm{a}\mathrm{x}}/\mathrm{k}\mathrm{m}$	30	${\mathrm{t}}_0/\mathrm{m}\mathrm{i}\mathrm{n}$	0.1
$C_1$/(CNY)	50,000	$C_2$/(CNY/m)	0.05

.

Due to limited existing measurements on the cost of medical drones, this manuscript refers to the relevant cost of logistics drones. The composition of each part of the cost is set at CNY 0.01 per meter for the repair cost per unit distance of the drone ${\mathrm{C}}_2^1$, CNY 0.02 per meter for the depreciation cost per unit distance of the drone ${\mathrm{C}}_2^2$, and CNY 0.02 per meter for the maintenance cost of the battery per unit distance of the drone ${\mathrm{C}}_2^3$. This implies that the variable cost VC is CNY 0.05 per meter.

4.3. Analysis of Results

In our simulation study conducted using MATLAB R2016b (64-bit), the operating environment consisted of a 64-bit Windows 11 operating system. The processor was an AMD Ryzen 7 4800H with Radeon Graphics, operating at a main frequency of 2.90 GHz and with an operating memory of 16.0 GB. The simulation parameters were set as follows: a population size of 50, a memory bank capacity of 10, 1000 iterations, a crossover probability of 0.7, a mutation probability of 0.6, a diversity evaluation parameter of 0.95, and an adjusted elite retention strategy parameter. The convergence comparison of the improved algorithm is shown in Figure 6. It is evident from the figure that the immune algorithm converges at around 500 generations, while the improved immune algorithm converges at around 200 generations, demonstrating a faster convergence. This supports the conclusion that the improved immune algorithm exhibits superior performance in solving the site selection model established in this paper, showcasing better adaptability and efficiency.

4.3.1. Analysis of Changes in the Number of Sites Selected

Analyzing the survival function reveals that, during the rescue of OHCA patients, the shorter the time it takes for the patient to receive external defibrillation and cardiopulmonary resuscitation, the higher the survival rate after rescue. Thus, there exists a critical “golden 4-minute” rescue time in the process of rescuing OHCA patients. If the patient receives operations such as external defibrillation within 4 minutes, the survival rate increases to about 60%. However, if the rescue time exceeds the “golden 4-minute”, the patient’s survival rate sharply declines, adversely affecting survival and recovery prognosis. Given the current challenges in timely ground emergency medical service responses, which are often influenced by ground traffic situations, strategically locating medical drone vertiports can greatly reduce the emergency response time. This has a significant impact on the improvement of survival rates of OHCA patients. Therefore, the number of medical drone vertiports is closely related to the “golden 4-minute” rescue coverage rate, patient survival rate, and total cost. The calculation results are shown in

Table 2. Relationship between coverage, survival rate, cost, and number of facilities.

	Number	Number of Medical Drone Vertiports
Indicator		5	10	15	20	25	30
coverage/%		43.55	64.52	85.49	95.16	100.00	100.00
survival rate/%		39.41	48.87	55.34	60.85	64.75	68.04
cost/CNY		27.61	51.79	76.34	100.99	125.78	150.63
	Number	Number of Medical Drone Vertiports
Indicator		20	21	22	23	24	25
coverage/%		95.16	96.77	96.77	98.39	100.00	100.00
survival rate/%		60.85	61.65	62.55	63.34	64.06	64.75
cost/CNY		100.99	105.95	110.90	115.86	120.82	125.78

, and the relationships between these factors is analyzed below.

(1)

Relationship between “golden 4-minute” coverage and the number of vertiports

Table 2 illustrates that as the number of medical drone vertiports increases, there are varying degrees of increases in the average survival rate of OHCA patients, the “golden 4-minute” rescue coverage rate, and the total cost of the system. For instance, when the number of medical drone vertiports is set as 5, 35 OHCA accident sites remain uncovered by the “golden 4-minute” coverage, resulting in a rescue coverage rate of only 43.55%, as depicted in Figure 7. The coverage rate stabilizes after increasing the number of medical drone vertiports to 20, reaching a 100% “golden 4-minute” rescue coverage rate when the number of sites reaches 24.

(2)

Survival rate of OHCA patients in relation to the number of vertiports

From Table 2, it is evident that as the number of facilities at a medical drone vertiport increases from 5 to 20, the average patient survival rate experiences a notable 21.44% increase. However, as the number of facilities continues to gradually increase to 30, the average patient survival rate sees only a 7.19% increase compared to the rate at 20 facilities. The rate of increase in the average patient survival rate of OHCA patients slows down, indicating that after reaching 20 facilities, the improvement in the patient’s mean survival rate becomes less significant.

(3)

Total system operating costs versus number of vertiports

The relationship between the system operation cost and the number of vertiports is shown in Figure 8. The total cost of system operation includes the fixed cost of construction and operation of medical drone vertiports and the path cost. As the number of selected medical drone vertiports increases, the fixed cost of the system rises. However, with a growing number of vertiports, the distance between the vertiports and the OHCA accident points decreases, leading to a significant reduction in the total flight distance of the system and a gradual decrease in path cost. In Figure 8, it is observed that the total system operation cost increases gradually with the rise in the number of medical drone vertiports. When the number of vertiports is below 20, the total system operation cost rises faster, while when the number of facilities increases from 24 to 25, the total system operation cost rises more slowly, at 4.11%. However, the “golden 4-minute “coverage and the average patient survival rate show little to no change. This suggests that surpassing 24 vertiports will only increase the total cost of operating the system without significantly improving the average patient survival rate and the “golden 4-minute” coverage rate.

Considering the relationship between the number of medical drone vertiports and the “golden 4-minute” rescue coverage rate, the average survival rate of patients, and the total cost, selecting 24 or more sites for medical drone vertiports is essential to achieve the “golden 4-minute” rescue coverage for OHCA accident sites. When the number of facilities increases from 24 to 30, the average survival rate of OHCA patients only increases by 3.98%, while the cost increases by 24.67%. However, to fully uphold the humanitarian rescue spirit, even if the survival rate does not increase significantly, every patient should be treated. Therefore, based on the results, with 24 medical drone vertiports, it is possible to achieve 100% coverage of “golden 4-minute” rescue and improve the average survival rate of OHCA patients to 64.06%; the current optimal siting layout scheme is shown in Figure 9.

4.3.2. Analysis of Changes in Objective Function Weights

In this paper, the average survival rate of OHCA patients and the total cost of operation are regarded as two objectives. Changes in the weight of the objective function can influence the results of the case operation. Therefore, the paper conducts experiments based on the current optimal vertiport selection results, that is, 24 medical drone vertiports achieving full “golden 4-minute” coverage. The value of the objective function with varying weights of the average survival rate $\omega$ is shown in Figure 10.

In out-of-hospital cardiac arrest emergencies, prioritizing the maximization of patient survival rates is paramount. Although system operation costs are considered, they are given less weight due to the principle of humanitarian rescue. Consequently, the selection of medical drone vertiports should not solely focus on the system operation cost and discard the patient survival rate. Therefore, in setting up our controlled experiments, we only consider scenarios that aim to maximize the average patient survival rate or achieve a dual objective in balancing survival rates and operational costs.

(1)

Solely consider the maximum mean survival rate objective function

As illustrated in Figure 10, the weight adjustment results in an overall upward fluctuation in the average survival rate of OHCA patients. When exclusively prioritizing the maximum average survival rate, at this time, $\omega$ = 1, and the achieved survival rate reaches 64.02%, accompanied by a total system cost of CNY 1,208,200.

(2)

Consider the bi-objective function

When accounting for both patient survival rate and total system cost, the weight adjustment results in the trend depicted in Figure 10. With $\omega$ = 0.9, the average survival rate of OHCA patients reaches its maximum of 64.06%, accompanied by a minimum total system operation cost of CNY 1,208,170.

The analysis of weights variation indicates that in the context of comprehensive consideration of multiple optimization objectives, prioritizing the maximization of the average survival rate of patients as the primary objective, while also incorporating the total system operation cost with a smaller weight, achieves optimal results. These results effectively adhere to the “life first” principle in patient treatment, emphasizing the importance of considering system operating costs, ultimately achieving a win–win situation for all stakeholders.

4.4. Comparative Analysis with Existing Ground Emergency Ambulances

To assess the impact of medical drones on the survival rate of OHCA patients in the selected area and to compare the efficacy of medical drone-assisted interventions with traditional hospital-based care, this paper, having determined the number and distribution of medical drone vertiports, evaluates the survival outcomes of OHCA patients. The comparative analysis further illustrates the rescue effect of medical drones, emphasizing the potential benefits of increased investment in their infrastructure to improve the patient survival rate.

(1)

Status of ground emergency services’ distribution of life-saving treatment

This paper selected four tertiary hospitals within the Jinnan District in Tianjin as ground emergency service stations, as outlined in

Table 3. Tertiary Hospitals in Jinnan District, Tianjin.

Name	Longitude	Latitude
Tianjin Huanhu Hospital	117.299276	39.07824
Tianjin Haihe Hospital	117.327984	39.048575
Tianjin Chest Hospital	117.298063	39.079867
Tianjin Jinnan Hospital	117.420361	38.987621

, to respond to 62 OHCA accident points within the region. It is assumed that OHCA accidents do not occur simultaneously, and only one ambulance needs to be dispatched for rescue and treatment at a time. The distance between the 62 OHCA accident points and the four hospitals is calculated, and the hospitals with the closest distance to the OHCA accident points are selected and assigned to the corresponding hospitals for treatment. The distribution of hospitals, OHCA accident points in the Jinnan District in Tianjin and the corresponding treatment plan are shown in Figure 11.

When deploying ambulances for OHCA emergency response, the assumed speed of an ambulance in an urban area is 60 km/h. A 10-minute coverage area is delineated, with a coverage rate of 58%, as depicted in Figure 11a. Extending the coverage to 20 minutes results in a 100% coverage rate, as shown in Figure 11b. Examining different time intervals of coverage reveals that the travel distance from an OHCA accident point to a hospital typically ranges from 10 to 20 minutes. For OHCA patients, the survival rate diminishes to less than 10% for those receiving treatment after 10 minutes. However, driven by the principles and purposes of saving lives, rescuing the sick and injured, and prioritizing patient well-being, hospitals remain committed to providing swift treatment.

(2)

Comparative analysis of survival rates in OHCA patients

Upon analyzing the coverage and emergency response capabilities of ground emergency service facilities, namely hospitals and ambulances, to OHCA accident sites, we now compare and analyze the survival rates when ground emergency service facilities respond versus when medical drones are applied for response, the comparison results are shown in Figure 12. The survival rate is compared at 24 medical drone vertiports, revealing that the survival rate is higher when medical drones are applied for response and treatment compared to ground ambulances. The application of medical drones results in an average survival rate of 64.06%, whereas ground ambulances yield an average survival rate of 22.10%. This comparison found that the application of medical drones for OHCA response increases the average survival rate of patients by 41.96%, signifying a crucial advancement in saving lives.

In the scene of out-of-hospital cardiac arrest, the primary optimization objective is the patient survival rate. This paper focuses on comparing and analyzing the patient survival rate achieved through medical drone vertiport selection versus the existing level of hospital emergency services. The findings demonstrate a significant improvement in patient survival rates with the application of medical drones. Given the diverse fixed and variable costs associated with medical drone systems and hospitals, controlling these variables for effective comparative analyses proves challenging, Additionally, focusing solely on path costs has limited practical significance, as costs play a minor role in influencing vertiport selection outcomes. Therefore, this paper does not carry out the comparative analyses of the costs associated with medical drones and hospitals.

5. Discussion

We focus on out-of-hospital cardiac arrest scenarios, considering the application of medical drones equipped with AED facilities for efficient and rapid emergency response, endeavoring to address diverse emergency medical needs comprehensively. This paper formulates a collaborative and refined operational model for medical drones based on the nature of accidents, operational characteristics of medical drones, and characteristics of out-of-hospital cardiac arrest. Differing from conventional objectives that aim to minimize time [11] or cost [12] in emergency facility location, driven by the humanitarian rescue spirit and from the patient’s perspective, we integrate the patient survival rate into the model as the primary optimization objective while also considering system operating costs. Additionally, informed by real emergency data, we introduce the innovative concept of achieving full “golden 4-minute” coverage in rescue, ensuring a balance between patient survival rates and equitable accessibility to healthcare services.

Through a review and synthesis of existing works in the literature, it is evident that drones hold significant potential and broad prospects in emergency medical rescue [1]. Previous studies have effectively validated the feasibility of medical drones equipped with AED facilities to improve patient survival rates [16]. Building upon this foundation, we integrate the goal of maximizing patient survival rates with the imperative of achieving full “golden 4-minute” coverage. Utilizing the coverage location model, we reduce the rescue time from the previously documented 8–10 minutes [17] to an efficient 4 minutes. Additionally, we elevate the coverage rate of emergency needs from 90.3% [20] to 100%, thereby significantly enhancing both patient survival rates and rescue coverage. The utilization of medical drones for emergency response emerges as a transformative measure, distinctly elevating patient survival rates compared to existing ground-based emergency services.

In our effort to maximize patient survival rates, we take a comprehensive approach by concurrently assessing the overall operating costs of the system. Unlike prior studies on site selection, which predominantly focus on costs associated with the site itself, encompassing construction and maintenance costs [9], we extend this to include the construction and operation costs of medical drone vertiports [12] and incorporate the flight path costs of phased medical drone operation, including repair costs per unit distance, depreciation costs, and battery maintenance costs. The results not only prioritize patient survival but also contribute to a substantial reduction in mortality and long-term healthcare costs, demonstrating the financial feasibility of the model and its results.

The current study primarily addresses the location of medical drone vertiports in normal operation scenarios, without factoring in potential obstacles [23], adverse weather conditions, complex environments, or unforeseen factors. There is a need for further exploration into public acceptance of drones, associated operational risks [24], and considerations of policy and regulations pertinent to medical drone applications. At the same time, economic feasibility aspects, such as the specific operating costs for medical drones, require more detailed examination [25]. The impact of accident first-witness behavior on response times and patient survival rates, along with the skill levels of operators and their emergency response capabilities, introduce a certain level of error between the study’s results and real-world situations. Further studies will strive for increased relevance and reduced errors.

We strategically deploy medical drone infrastructure, including charging stations and vertiports, based on population data and the distribution characteristics of the elderly population, ensuring effective logistical support [20]. Future work can extend to evaluating the layout and operational costs of medical drone vertiports under real demand distributions in specific areas and across various operational scenarios. Additionally, exploring innovative operational modes that integrate seamlessly with ground-based emergency service facilities will be crucial. Furthermore, in the proposed medical drone operational model, there is a focus on enhancing coordination between bystanders and healthcare personnel, ensuring the feasibility of personnel and the efficiency of emergency responses. At the same time, public awareness of emergency care should be strengthened, first aid knowledge popularized, and collaborative efforts with community and neighborhood committees should be fostered to elevate the standard of automated external defibrillator configurations. This approach can contribute to promoting health system reform for sustainable and healthy development. Future advancements in technology and innovation hold the potential to further enhance emergency response by mitigating existing risks and challenges.

6. Conclusions

This paper focuses on scenarios involving out-of-hospital cardiac arrests and explores the application of medical drones equipped with automated external defibrillators for emergency response. Addressing the challenge of locating medical drones vertiports, we introduce the survival rate function and frame the optimization objectives as maximizing the average survival rate of patients while minimizing the total system cost. Simultaneously, we take into account the constraints related to staged operation time, energy consumption, and drone coverage. The model for medical drone vertiport location in cardiac arrest emergencies is solved by applying the improved immune algorithm and verified through numerical examples. The results show that:

The demand for out-of-hospital cardiac arrests in Jinnan District, Tianjin City, based on population, age distribution, and morbidity rate, can be effectively met by deploying 24 medical drone vertiports to achieve full “golden 4-minute” rescue coverage;

By assigning a weight of 0.9 to the average patient survival rate, we optimize the system operation to achieve the highest average patient survival rate while minimizing the total cost. This approach results in an impressive increase in the average patient survival rate to 64.06%;

Comparatively, the application of medical drones for out-of-hospital cardiac arrest treatment surpasses ground ambulances in reducing response time. The average survival rate of patients increases by 41.96%, making a significant improvement over the current situation characterized by low survival rates for out-of-hospital cardiac arrest patients.

The application of medical drones not only significantly improves OHCA emergency services, but it also holds profound implications for various dimensions of sustainable health development. It contributes to improving survival rates, mitigating healthcare disparities, enhancing accessibility to health services, reducing environmental impacts, and facilitating the reform of health systems, thus significantly supporting the fulfilment of sustainable health development goals.