A healthy person's body automatically maintains normoglycemia, i.e. the proper level of sugar in the blood. Insulin and glucagon are pancreatic hormones of key importance for the regulation of energy metabolism and blood glucose concentration. They work in opposite ways - the role of insulin is to prevent hyperglycemia, while glucagon plays a diferent role - it prevents hypoglycemia. In a person with diabetes, the above process is impaired (or does not work at all). Therefore, external mechanisms for achieving the proper blood sugar level are necessary, for the most part, administering exogenous insulin. In the paper we present an elementary Petri net model of normoglycemia maintaining in a healthy person, and a basic model of the processes occurring in the body of a person sufering from diabetes. Comparison and analysis of both models allows for a better understanding of the mechanisms operating in the (healthy and sick) human body. Such an exploration also makes it possible to better adapt the treatment to a sick person and constitutes the initial step on the way to our long-time goal to create the whole body model of the glucose regulation in a healthy human and a person with diabetes.
The body of a healthy person strives to maintain normoglycemia, i.e. the appropriate level of sugar in the
blood. According to the World Health Organization [
Diabetes is a chronic disease that occurs either when the pancreas does not produce enough insulin
(a hormone that regulates blood glucose) or when the body cannot efectively use the insulin it produces.
There are three main types of diabetes: type 1, type 2, and gestational diabetes (diabetes while pregnant).
Type 1 diabetes is a condition in which the immune system destroys insulin-making cells (beta cells)
in the human pancreas. That means that the body cannot produce either enough endogenous insulin,
or none at all. Type 2 diabetes is a chronic condition that happens when high blood sugar levels
(hyperglycemia) persist in a body. It happens when pancreas is not able to produce enough insulin,
body does not use insulin properly, or both. For people with diabetes, blood sugar level targets are as
follows [
Diabetes is considered one of the civilization diseases. According to the International Diabetes
Federation data for 2021 [
It is therefore not surprising, that in many fields of science, intensive work has being undertaken not only to cure the disease, but also, and perhaps above all, to prevent or delay its future health complications (such as heart disease, chronic kidney disease, nerve damage, and other problems with feet, oral health, vision, hearing, and mental health) or improving the quality of life of patients and their families.
More and more advanced systems are being developed to continuously measure blood glucose
levels without the need to puncture the skin (CGM - continuous glucose monitoring [
Our long-term goal is to create a simple and intuitive mathematical model representing the glucose regulation mechanisms occurring in the body of a healthy person and a person with diabetes. This model should be easily analysable and clear, but at the same time, capable of representing complex processes consisting of interactions between many components. In our opinion, Petri nets (PNs) constitute a perfect tool for this purpose. Due to PNs intuitive graphical representation and mathematical properties, the model would be useful for people with and without medical background. This could allow for a better understanding of the processes occurring in a human body, predicting new therapeutic targets and designing drug therapies. We are aware, that our goal (modelling the entire process) is ambitious and would not be reached at once. Hence, our preliminary step, presented below, is to model processes of achieving normoglycemia in the case of a healthy or sufering from diabetes person.
In this paper, we present an elementary model of the glucose level regulating processes in a healthy body, as well as very basic model of processes taking place in the body of a sick person, aiming to achieve normoglycemia. We believe that the analysis of both models may be of great importance for understanding the processes taking place in a healthy body and disease of diabetes and shows how complicated the process of maintaining the appropriate sugar level may be. Such knowledge may allow for appropriate adjustment of therapy to a given person.
We use standard Petri net analysis tools, such as the reachability graph and t-invariants, to study out models. In the following section, we recall the basic concepts of Petri nets, and in subsequent parts of the paper we introduce and discuss Petri nets models of glucose levels regulating processes in a healthy and diabetic person. The paper ends with a summary and future plans.
In this section we recall the basic notions concerning Petri Nets and its properties [
A finite labelled transition system with initial state is a tuple = (, →, , 0) with nodes (a finite set of states), edge labels (a finite set of letters), edges → ⊆ ( × × ), and an initial state 0 ∈ . A label is enabled at ∈ , denoted by [⟩, if ∃′ ∈ : (, , ′) ∈ →. A state ′ is reachable from through the execution of ∈ * , denoted by [ ⟩′, if there is a directed path from to ′ whose edges are labelled consecutively by . The set of states reachable from is denoted by [⟩. A sequence ∈ * is allowed, or firable, from a state , denoted by [ ⟩, if there is some state ′ such that [ ⟩′.
An (initially marked) Petri net (PN) is denoted as = (, , , 0) where is a finite set of places, is a finite set of transitions, is the flow function : (( × ) ∪ ( × )) → N specifying the arc weights, and 0 is the initial marking (where a marking is a mapping : → N, indicating the number of tokens in each place). A transition ∈ is enabled at a marking , denoted by [⟩, if ∀ ∈ : () ≥ (, ). The firing of leads from to ′, denoted by [⟩ ′, if [⟩ and ′() = () − (, ) + (, ). This can be extended, as usual, to [ ⟩ ′ for sequences ∈ * , and [ ⟩ denotes the set of markings reachable from . We call a marking deadlock if it does not enable any transition, i.e. for every ∈ we have ∃ ∈ : () < (, ). The reachability graph ( ) of a bounded (such that the number of tokens in each place does not exceed a certain finite number) Petri net is the labelled transition system with the set of vertices [0⟩, initial state 0, label set , and set of edges {(, , ′) | , ′ ∈ [0⟩ ∧ [⟩ ′}.
0
Note that the reachability graph of a bounded Petri net captures the exact information about the reachable markings of the net, and therefore reflects the entire behaviour of a given net. Figure 2 depicts an exemplary Petri net, together with its reachability graph.
Let ∈ ∪ , ∙ = { ∈ ( ∪ ) | (, ) > 0} and ∙ = { ∈ ( ∪ ) | (, ) > 0}.
A Petri net = (, , , 0) can be represented in the form of matrices with integer coeficients:
an input matrix, an output matrix and an incidence matrix [
An inhibitor net is a quintuple S = (, , , , 0), where: (, , , 0) is a Petri net, as defined above; ⊆ × is the set of inhibitor arcs (depicted by edges ended with a small empty circle). The set of inhibitor entries to is denoted by ∘ = { ∈ | (, ) ∈ }. A transition ∈ (of an inhibitor net) is enabled in a marking whenever ∙ ≤ (all its entries are marked) and (∀ ∈ ∘ ) () = 0, i.e. all inhibitor entries to are empty. The execution of leads to the same marking as in the ordinary Petri nets case.
two hours after eating. A token located in one of places H, N, L informs that the concentration of glucose is in the corresponding level. That level can be afected in a couple of ways.
The body tries to maintain the normal level of glucose. However, glucose is the main source of energy in the body, and every life process consumes it while occurring. It results in a decrease of the amount of glucose. The low concentration of glucose (place L) causes the hunger feeling and the desire to eat. After eating, which is represented by eating transition, the carbohydrates are available for the organism, what is pictured in place Carb. In such a situation, clearly, the level of glucose increases to the normal or high level (transitions t3 and t2, respectively). However, the level of glucose is not only an efect of its delivery and consumption. The organism is not a passive observer of these processes, but actively participates in the glucose regulation, since the abnormal level of glucose is not desirable because it negatively afects the whole organism. The most important organ in the regulation process is the pancreas, represented by Pancreas place. Stimulated by the high level of glucose, the pancreas produces insulin, which is represented by transition t1. The presence of insulin is modelled by place Insulin. Insulin induces processes (transition t4) which result in reduction of the level of glucose and storage of it in various organs - place Storage, mostly liver and fat tissue.
On the other hand, when the level of glucose is low, the pancreas produces glucagon, which is
represented by transition t0 and place Glucagon, visualising the presence of that hormone. Opposite to
insulin, glucagon stimulates processes (transition t5), aiming to release the stored substances from the
liver and fat tissue (place Storage) and increase the level of glucose. Our PN model of processes taking
place in the liver, related to storing and releasing of glucose in the presence of insulin and glucagon, is
presented in [
To compute the reachability graph the PN needs to be bounded. The model presented in Figure 2 is not bounded, because when place L is marked transition eating can produce an unlimited number of tokens. However, such a situation is not possible in the real organism (infinitive eating and at the same time low level of glucose). In order to compute the reachability graph of the model, we removed the eating transition and assumed that only one instance of carbohydrates is available. It would correspond to the situation of the glucose regulation processes after one meal and before the next one. The initial marking used to compute the reachability graph is depicted in Figure 2. The obtained graph is presented in Figure 3.
While analysing the reachability graph of the PN model one can notice that the model tends to the state with place N marked. From the initial state, the state N|Pancreas|Storage can be reached by three diferent paths of transitions executions. From that state (and by two side paths) executions of transitions lead to state N|Pancreas. Notice that all those paths are related to maintaining the normal level of glucose (token in place N ) by using available glucose sources, like carbohydrates (place Carb) and substances stored by the body (place Storage). After state N|Pancreas the deadlock marking L|Pancreas is reached, which corresponds to the situation where all available sources of glucose are used and the next meal is necessary.
The dynamic of the model is nicely captured by t-invariants present in the model. The PN model has only two t-invariants. The first contains transitions: eating, life, t3 and is depicted green in Figure 3. The second contains transitions: eating, life, t0, t1, t2, t4, t5 and is depicted blue in Figure 3. It is obvious that the PN model is covered by t-invariants. The first t-invariant corresponds to the process of glucose regulation related only to processes of glucose consumption and providing. Like it was mentioned above, the organism is not a passive observer of these processes, but actively participates in the glucose regulation. Those regulation mechanisms are represented by transitions included in the second t-invariant. By the definition of t-invariant, when the model starts with place N marked like in the initial marking, the execution of all transitions from the t-invariant will lead to the same, initial marking. Hence, firing the transitions corresponding to the glucose level regulation mechanisms of the body, included in the second t-invariant, would result in maintaining the normal level of glucose (token back in place N ).
Now let us look at Figure 4, which depicts how the issue of the glucose level regulation looks like in a
person sufering from diabetes. The model contains 13 places and 27 transitions. For clarity, we have
decided to use inhibitor arcs in this model, which, however, could be eliminated (in accordance with [
In a healthy organism the body produces the suitable amount of insulin in the suitable time. In contrast
to that, in a diabetic case, insulin is administrated from the outside and a diabetic person does not know
their exact level of insulin. Because of that, three possible levels of insulin are present in the model: low
(place LI ), normal (place NI - in a healthy body it is the only one possible) and high (place HI ). Only
the level of glucose can be checked and based on that, in the ideal case, the suitable amount of insulin
is administrated. Moreover, in a diabetic organism the internal mechanisms of glucose regulation do
not work correctly, and even when the person is doing everything correctly by calculating the amount
of eaten carbohydrates and corresponding dose of insulin, low or high glucose levels are still possible
([
Transition PIns corresponds to the administration of insulin, which is represented by place Ins. When insulin is provided, it may (or may not if the amount is not suficient) afect the current level of insulin in the body. This is represented by transitions I1 to I5. When insulin is provided (place InsActive), it can be used by the body to reduce (little or much) the level of glucose - transitions D1 to D6. Still, the efect of insulin supply can be not suficient and the level of glucose could not change. When carbohydrates are eaten (transition PCarb and place Carb), the level of glucose may rise or may not be changed, because of the presence of insulin in the blood. Insulin, during that process, is used by cells, hence its level may be reduced. Transitions from U1 to U7 represent the change of glucose level. Transitions from Ch1 to Ch5, To, From and From2 are responsible for managing the level of insulin.
One can easily notice, that the dynamic of the PN diabetic model is much less certain, and in contrast
to the healthy model, it is not possible to predict the exact result of transitions executions. To calculate
the reachability graph, similarly like for the healthy PN model, transitions PIns and PCarb were removed,
and the initial marking was the one obtained after executions of those transitions once (place Carb
and Ins marked with one token). It represents the situation after the meal and the administration of
insulin. Places NI and N, like in Figure 4, were also marked with one token each. Moreover, inhibitor
arcs were removed according to the procedure described in [
In the paper we have presented the basic Petri net models of normoglycemia maintaining in a healthy person, and a person sufering from diabetes. We believe that the analysis and comparison of both models and their reachability graphs could facilitate the understanding of the entire process, which can be especially useful for a diabetic person. These models can also be used in diabetes education (for instance, with the use a Petri Net simulation displaying the token game), both for sick people and their families, as well as for people involved in medicine and health care.
The presented PN model of the healthy mechanism of glucose regulation seems to be very simple. However, its dynamic is very interesting and could be a source of basic knowledge about glucose regulation. Despite its simplicity, the analysis of the model shows the tendency to maintains normoglycemia. On the other hand, the diabetic PN model shows the much more complex dynamic and the dificulties in obtaining the desired normal level of glucose.
The paper constitutes a preliminary step towards designing a complete model visualizing the regulation of glucose levels in the body, which would aim to better understand the processes occurring in the body of a healthy person, as well as a person sufering from diabetes.