The World Health Organization (WHO) has "access to oxygen" on its model list as one of the essentials required by an individual, especially when they are in a critical condition health-wise and they are unable to breathe on their own. Access to oxygen is the only medicine listed that does not have a substitute. Access to oxygen even after the worldwide pandemic is still a major dilemma in middle and low-income countries [1]. A mechanical ventilator is a medical device that is employed in assisting patients in cases where their respiratory system is not functioning well, thus the patient has challenges in breathing or has shortness of breath. The mechanical ventilator can also be used in cases where a patient has been sedated and is undergoing surgery.

The basic operation of a mechanical ventilator is to control a high-pressure region during the inspiration stage. In the inspiration stage, the mechanical ventilator is paused. During expiration, air flows out due to the lung’s natural recoil which creates a higher pressure in the alveoli. Due to the high demand for mechanical ventilators both in the past and at present, it is imperative that efficient and affordable mechanical ventilators should be researched, modelled, designed and implemented. To ensure that robust mechanical ventilators are designed, it is important to formulate new models that can be used in the research and testing stages of the mechanical ventilators.

Tran et al, [2], conducted research whose main goal was to design, control, model, and simulate a mechanical ventilator that is light in weight, portable, and suitable for use at home. [2], modelled the mechanical ventilator as a voltage source. El-Hadj et al [3], applied the fluid-structure Interaction (FSI) to a couple of computational fluid dynamics used for fluid flow with finite element analysis for the solid domain. This facilitated the investigation of fluid behaviour, structural behavior, and interactions. Two designs were proposed for the mechanical ventilator. The first design was reported to be uncontrollable and the second design considers Computational Fluid Dynamics (CDF) with a moving boundary which is applied to the piston cylinder based ventilator. Tharton et al [4], designed and developed a ventilator prototype to be used by professionals in medical emergencies and in any other context where the regular ventilator is not available. The mechanical ventilator was modelled using the crank rocker mechanism in order to meet specific requirements for mechanical ventilator design. Pivk et al, [5], developed an empirical model for a low-cost mechanical ventilator by observing the response of each of the ventilator subsystems. The current progress made globally in the development of affordable and effective designs is important. However, research in mechanical ventilator development lacks model development. The Port-Hamiltonian approach applied in this research work acts as a modelling template for future energy-based mechanical ventilator modelling and design. It is important to develop designs that rest on a solid understanding of a significant aspect of the design. Naggar, [6] developed a piecewise model of a mechanical ventilator which described the artificial behaviour of a mechanical ventilator. A pressure-controlled ventilator is created and simulated. The mechanical ventilator is modelled using a periodic function with inequalities to control the beginning of inspiration and expiration periods. Shi et al, [7] developed a mathematical model of volume-controlled mechanical ventilation. The model is viewed as a pneumatic system where the ventilator is regarded as an air compressor. The exhalation valve is considered as throttle. The compressor and the container represent the ventilator. Naggar et al, [8] proposed an integrated mathematical model of the mechanical ventilator and the lung. Linear quadratic and exponential equations were used to model the system. The integrated model was used to simulate volume-controlled artificial ventilation. Giri et al, [9], proposed a simplified design of a mechanical ventilator, to reduce cost and automate the mechanical ventilation process. The proposed design was simulated on the MATLAB platform. Hannon et al, [10] presented a review on the advancement in the modelling of human anatomy, physiology and pathophysiology via mathematical modelling and computer simulation. Clinical applications in various disease states were emphasized. The research work discussed the current limitations and potential of in-silico modelling. There are currently no existing Port Hamiltonian models of mechanical ventilators intergrated with a human repiratory system. The main contribution in this research work is the development of an intergrated Port-Hamiltonian model representation of a mechanical ventilator-human respiratory system. The model consists of electromechanical and electromagnetic components modelled in the finite-dimensional representation, interconnected with fluid components in the infinite-dimensional representation. As a result, this Port-Hamiltonian model is of a mixed finite/infinite dimensional nature.

This section presents the mathematical preliminaries of the Port Hamiltonian approach that have been followed in later sections. These definitions are standard definitions from literature and are mostly derived from the following references: [11,12] and [13]

Definition 1 (Dirac Structure)

A Dirac Structure (DS) is a pair of elements, $f\in {\mathbb{R}}^n$ and $e\in {\mathbb{R}}^n$, that satisfies the set:

The DS is a subset $\mathcal{D}\subset \mathcal{F}\times \mathcal{E}$, where $\mathcal{F}$ and $\mathcal{E}$ represent the flows and efforts, respectively.

The DS in electric circuits can be represented as:

Proposition 1:

The subspace is called a DS iff, there exist $A,B\in {\mathbb{R}}^{n\times n}$ such that satisfying the condition

Definition 2 (Resistive Relation)

Any relation $\mathcal{R}\subset {\mathbb{R}}^n\times {\mathbb{R}}^n$ is said to be resistive if $\forall (f_{\mathcal{R}},e_{\mathcal{R}})\in \mathcal{R}$ the is satisfied.

Definition 3 (Port-Hamiltonian (pH) System).

The set $(\mathcal{D},\mathcal{L},\mathcal{R})$, defines a pH system where:

The elements of the sets:

where $n_{\mathcal{L}},n_{\mathcal{R}}$ and $n_{\mathcal{P}}\in {\mathbb{N}}_0$.

The dynamics of the pH system are given by the differential inclusion where $(x\left(t\right),e_{\mathcal{L}}\left(t\right))\in \mathcal{L}$, $(f_{\mathcal{R}}\left(t\right),e_{\mathcal{R}}\left(t\right))\in \mathcal{R}$ and $(f_{\mathcal{P}}\left(t\right),e_{\mathcal{P}}\left(t\right))\in \mathcal{P}$

Definition 4 (Interconnection of n pH Systems)

Let (${\mathcal{D}}_i,{\mathcal{L}}_i,{\mathcal{R}}_i$) denote the space of the $i^{\mathrm{th}}$ pH system in a set of n pH systems that are to be interconnected. The space of flows is divided into an external part and a part to be linked and is given by

Similarly, the space of efforts is divided into an external part and a part to be linked and is expressed as

The interconnection of two pH systems $({\mathcal{D}}_1,{\mathcal{L}}_1,{\mathcal{R}}_1)$ and $({\mathcal{D}}_2,{\mathcal{L}}_2,{\mathcal{R}}_2)$ with respect to the link $(F_{\mathcal{P}\mathrm{link}},E_{\mathcal{P}\mathrm{link}})$ results in a new interconnected pH system, $(\mathcal{D},\mathcal{L},\mathcal{R})$. This interconnected system is given by the expression

Definition 5 (Directed Graph).

A directed graph is a quadruple $\mathcal{G}(V,E,\mathrm{l},\mathrm{r})$ where

Definition 5.1 (Loop-free directed Graph).

If G is a directed graph, then G is said to be loop-free if for all $e\in E$,

Definition 5.2 (Subgraphs).

Given the graphs $\mathcal{G}(V,E,\mathrm{l},\mathrm{r})$ and ${\mathcal{G}}^{\prime }(V^{\prime },E^{\prime },\mathrm{l},\mathrm{r})$, then ${\mathcal{G}}^{\prime }$ is said to be a subgraph of ${\mathcal{G}}^{\prime }$ if $E^{\prime }\subset E$ and $V^{\prime }\subset V$. Furthermore,

Definition 6 (Paths, Connectivity and Cycles).

Let $\mathcal{G}=(V,E,\mathrm{l},\mathrm{r})$ be a directed finite graph.

Definition 7 (Incidence Matrix).

Let $\mathcal{G}=(V,E,\mathrm{l},\mathrm{r})$ be a directed graph that is finite and loop-free such that $E=e_1,\dots ,e_m$ and $V={\upsilon }_1,\dots ,{\upsilon }_n$. Then the $j^{th}$ row and $k^{th}$ columns of the incidence matrix, $A_0\in {\mathbb{R}}^{n\times m}$ of $\mathcal{G}$ is given by

If the $\mathrm{rank}\left(A_0\right)=n-k$, then the graph $\mathcal{G}$ has $k\in \mathbb{N}$ components, such that k rows can be removed from $A_0$ resulting in matrix with same rank.

Definition 8 (Kirchhoff-Dirac Structure, Kirchhoff-Lagrange Submanifold).

The Kirchhoff-Dirac structure of $\mathcal{G}$ can thus be defined by the set where i and u are the currents and voltages at the edges of the graph, respectively, whereas q and $\varphi$ are the charges and potentials at the vertices of the graph, respectively.

Assuming that $S=\left\{v_1,\dots ,v_{\left|S\right|}\right\}$, the Kirchhoff-Lagrange submanifold of $\mathcal{G}$ with respect to S is defined as where $\mathcal{G}=(V,E,\mathrm{l},\mathrm{r})$ is a directed graph that

If ${\mathcal{G}}_1,\dots ,{\mathcal{G}}_k$ are the components of $\mathcal{G}$ with corresponding vertices $V_1,\dots ,V_k\subset V$ so that there exists a subset $\mathcal{S}\subset V$ such that $\mathcal{S}$ containing at most one vertex from each component, i.e. $\forall s,s^{\prime },\in S,i\le k:v,v^{\prime }\in V_i\Rightarrow v=v^{\prime }$, then $A\in {\mathcal{R}}^{(n-k)\times m}$ can be constructed by deleting the rows corresponding to the vertices $\mathcal{S}$ from $A_0\in {\mathbb{R}}^{n\times m}$.

Remark. According to Proposition 1, Equations 1 and 2 indicate that ${\mathcal{D}}_K^S\left(\mathcal{G}\right)$ is a DS and ${\mathcal{L}}_K^S\left(\mathcal{G}\right)$ is a LS in the space ${\mathbb{R}}^{n-\left|S\right|}\times {\mathbb{R}}^{n-\left|S\right|}$.

Definition 8 caters for an introduction of a pH system $({\mathcal{D}}_K^S\left(\mathcal{G}\right),{\mathcal{L}}_K^S\left(\mathcal{G}\right),\left\{0\right\})$ with dynamics where $(q\left(t\right),\varphi \left(t\right))\in {\mathcal{L}}_K^S\left(\mathcal{G}\right)$.

Using the equivalence of $(q\left(t\right),\varphi \left(t\right))\in {\mathcal{L}}_K^S\left(\mathcal{G}\right)$ to $q\left(t\right)=0$ and $\varphi \left(t\right)\in {\mathcal{R}}^{n-\left|S\right|}$, we see that equation 3 holds, iff, the condition holds.

On this basis, one can define a DS with dynamics and an LS with dynamics

Together, they form a the pH system defined by the set $(D_K^{\prime }\left(\mathcal{G}\right),{\mathcal{L}}_K^{\prime }\left(\mathcal{G}\right),\left\{0\right\})$.

Figure 3 shows a solenoid valve in the open and closed position. It is assumed that the air gaps are sufficiently small such that the effect of fringing of the magnetic flux is negligible. Consider a solenoid in which the permeability of the core and the length of the part of the magnetic circuit inside the core are denoted by ${\mu }_c\in {\mathbb{R}}_{+}^1$ and $l_c\in {\mathbb{R}}_{+}^1$, respectively. The equivalent length of the solenoid’s magnetic circuit, $l_{eq}\left[·\right]:\mathbb{R}\to \mathbb{R}$ is dependent on the displacement of the spool, $q_s\in \mathbb{R}$, and can be written as where ${\mu }_0\in {\mathbb{R}}_{+}^1$ are the permeability of air and $q_{{\mathrm{s}}_{\mathrm{tot}}}\in {\mathbb{R}}_{+}^1$ is the total air-gap. The solenoid coordinate systems is represented in Figure 4.

Thus, the inductance of the solenoid varies with displacement of the spool and hence can be expressed by the function $L_s\left[·\right]:\mathbb{R}\to \mathbb{R}$ given by where $N\in {\mathbb{R}}_{+}^1$ is the number of turns in the coil of the solenoid and $A_e\in {\mathbb{R}}_{+}^1$ is the effective cross-sectional area of the path of the magnetic flux. The magnetic and mechanical subsystems in the solenoid valve are therefore coupled magnetically due to the dependence of the inductance on the displacement of the spool.

The total energy of the solenoid is given by the Hamiltonian, ${\mathcal{H}}_s\left[·\right]:{\mathbb{R}}^3\to \mathbb{R}$, which is a function of the state vector ${\mathcal{X}}_s={\left[\begin{array}{ccc}{\varphi }_s,& p_s,& q_s\end{array}\right]}^{\top }\in {\mathbb{R}}^3$, expressed as the sum of the magnetic, kinetic and potential energies denoted by ${\mathcal{H}}_{\mathrm{magnetic}}\left[{\varphi }_s,q_s\right]:{\mathbb{R}}^2\to \mathbb{R}$, ${\mathcal{H}}_{\mathrm{kinetic}}\left[p_s\right]:\mathbb{R}\to \mathbb{R}$ and ${\mathcal{H}}_{\mathrm{potential}}\left[q_s\right]:\mathbb{R}\to \mathbb{R}$, respectively. Thus given a magnetic flux ${\varphi }_s\in \mathbb{R}$.

Assuming that the pretension of the spring is set to $q_0\in \mathbb{R}$, the Hamiltonian is where $p_s\in \mathbb{R}$ is the momentum of the spool, $m_s\in {\mathbb{R}}_{+}^1$ is the mass of the spool, $k_s\in {\mathbb{R}}_{+}^1$ is the spring stiffness and $g\in {\mathbb{R}}_{+}^1$ is the acceleration due to gravity.

Hence, the solenoid’s state and output dynamics are expressed in Port-Hamiltonian form in equations 13 and respectively where where $R_s\in {\mathbb{R}}_{+}^1$ is the resistance of the coil, $b_s\in {\mathbb{R}}_{+}^1$ is the viscous damping acting on the spool and ∇ is the gradient operator. $A_{s_1},A_{s_2}\in {\mathbb{R}}_{+}^1$ are the various cross-sectional areas of the lands of the spool and the input vector $u_s\in {\mathbb{R}}^2$ consists of the input voltage $V_s\in {\mathbb{R}}^1$ and the supply pressure $p_s\in {\mathbb{R}}^1$. It can be seen that ${\mathcal{J}}_s=-{\mathcal{J}}_s^{\top }\in {\mathbb{R}}^{3\times 3}$ possesses skew-symmetry, while ${\mathcal{R}}_s={\mathcal{R}}_s^{\top }\in {\mathbb{R}}^{3\times 3}$ is positive semi-definite.

Taking the time derivative of the Hamiltonian

This system has power and resistive ports.

The directed graph of a mechanical ventilator model is composed of the disjoint union of the vertices associated with the patient, as well as the inspiratory and expiratory limbs of the ventilator given by $v_O,{\mathcal{V}}_I$ and ${\mathcal{V}}_E$ respectively. Similarly, the patient, inspiratory and expiratory arcs given by $a_O,{\mathcal{A}}_I$ and ${\mathcal{A}}_E$, respectively. Each arc and vertex can further be subdivided into those associated with blowers, valves, pipes and electric circuits present in the mechanical ventilator. Thus the totality of vertices and arcs are given by and respectively, where the subscripts $i\in \left\{O,p,b,v,p,e\right\}$ are used to indicate the patient, blowers, valves, pipes and electric circuits components, respectively.

Figure 9 shows a detailed graph of the mechanical ventilator given in Figure 2 indicating the patient vertex ${\mathcal{V}}_O$, the arcs and vertices along the inspiratory path given by and respectively, as well as the arcs and vertices along the expiratory path given by respectively.

In this section, the coupling conditions of the port-Hamiltonian network model of the mechanical ventilator are given. The types of interconnections occurring in this model are given below:

The port Hamiltonian model of the pipe is a partial differential equation, continuous in space. As such it is difficult to simulate the dynamics of a pipe section. In order to do this, it is necessary to approximate the model with a discrete model, in this case, a finite difference model which is an approximation of the original system. Within the context of port Hamiltonian systems, an additional requirement is the need to ensure that the discrete approximation maintains the structural properties of the original system e.g. skew symmetry etc.

Each system state can be replaced with a discrete approximation consisting of a total on n elements as can be seen in Figure 10. As such, $\varrho \approx {\varrho }_d=\left[\begin{array}{cccc}{\varrho }_1,& {\varrho }_2,& \cdots ,& {\varrho }_n\end{array}\right]\in {\mathbb{R}}^{n\times 1}$ and $m\approx m_d=\left[\begin{array}{cccc}{m}_1,& m_2,& \cdots ,& m_n\end{array}\right]\in {\mathbb{R}}^{n\times 1}$. As such, state vector ${\chi }_p$ can be replaced by a discrete approximation ${\chi }_d=\left[\begin{array}{cccccc}{\varrho }_1,& \cdots ,& {\varrho }_n,& m_1,& \cdots ,& m_n\end{array}\right]\in {\mathbb{R}}^{2n\times 1}$. The $i^{\mathrm{th}}$ element of ${\varrho }_i,m_i\in {\mathbb{R}}^{1\times 1}$ is located at $z=\left\{\Delta (i-1),\Delta (i-1/2)\right\}$, where $\Delta$ is the fixed discrete step size between points and $i=1,2,\cdots ,n$. In addition, the efforts at the boundaries are given by ${\delta }_{{\chi }_0}{\mathcal{H}}_0$ and ${\delta }_{{\chi }_n}{\mathcal{H}}_n$. Thus the Hamiltonian given in Equation 18 can be approximated by a discrete approximation such that ${\mathcal{H}}_p\left[{\chi }_p\right]\approx \Delta {\mathcal{H}}_d\left[{\chi }_d\right]$ so that the discrete system effort is now ${\delta }_{{\chi }_d}\left({\mathcal{H}}_d\right)$. A finite difference approximation of the spatial derivatives at the $i^{\mathrm{th}}$ point is

The central difference approximation at the $i^{\mathrm{th}}$ point is

In matrix form this is which can be re-written as where