Method to generate a large cohort in-silico for type 1 diabetes

Highlights

• Large in-silico cohort for type 1 diabetes is generated through linear regression.

• Linear regression uses covariance to generate large cohorts through published data.

• The overlapping dynamics is tested by an algorithm of clustering.

• Covariant and random cohorts are compared their qualitative behaviour.

• The methodology to obtain a large cohort can be extended to any science problem.

Abstract

Background and objective: In the last decade, several technological solutions have been proposed as artificial pancreas systems able to treat type 1 diabetes; most often they are built based on a control algorithm that needs to be validated before it is used with real patients. Control algorithms are usually tested with simulation tools that integrate mathematical models related mainly to the glucose-insulin dynamics, but other variables can be considered as well. In general, the simulators have a limited set of subjects. The main goal of this paper is to propose a new computational method to increase the number of virtual subjects, with physiological characteristics, included in the original mathematical models. Methods: A subject is defined by a set of parameters given by a mathematical model. From the available reduced number of subjects in the model, the covariance of each parameter of every subject is obtained to establish a mathematical relationship. Then, new sets of parameters are calculated using linear regression methods; this generates larger cohorts, which allows for testing insulin therapies in open-loop or closed-loop scenarios. The new method proposed here increases the number of subjects in a virtual cohort using two versions of Hovorka’s mathematical model. Results:Two covariant cohorts are obtained with linear regression. Both cohorts are clustered to avoid overlapping in the glucose-insulin dynamics and are compared in terms of their qualitative and quantitative behaviours in the normoglycemic range. As a result, there have been generated two larger cohorts (256 subjects) than the original population, which contributes to improving the variability in in-silico tests. In addition, for analysing the characteristics of the covariant generation method, two random cohorts have been generated, where the parameters are obtained individually and independently from each other, exhibiting only distribution limitations so that these cohorts do not have physiological subjects. Conclusions: The proposed methodology has enabled the generation of a large cohort of 256 subjects, with different characteristics that are plausible in the T1DM population, significantly increasing the number of available subjects in existing mathematical models. The proposed methodology does not limit the number of subjects that can be generated and thus, it can be used to increase the number of cohorts provided by other mathematical models in diabetes, or even other scientific problems.

Introduction

Type 1 diabetes mellitus (T1DM) is a chronic disease in which the patient’s immune system destroys the beta cells of the pancreas. This phenomenon causes an insulin deficiency in patients and leads to high glucose levels in blood. High glucose levels cause long-term health complications.

In 2018, Forouhi and Wareham [1] showed, in a study about the epidemiology of diabetes, that the clinical and public health burden of T1DM is high and rising globally. The epidemiology of T1DM shows distinct patterns of distribution by age, sex, and ethnicity among others. The global prevalence of T1DM was 425 million adults in 2017, but it is expected to rise to 629 million by 2045 [2].

To treat T1DM, the patient has to constantly maintain its blood glucose level within the normal margins. This can be done by delivering insulin through Intensive Conventional Therapy (ICT), which is administered with insulin pens, or Continuous Subcutaneous Insulin Infusion (CSII), which is administered with insulin pumps. Both therapies are known as open-loop control (OL).

T1DM is characterised by high blood glucose levels known as hyperglycemia. Hypoglycemia events occur when: 1) high insulin dosages that are poorly fitted to the patient’s metabolism are administered or 2) the insulin therapy is well fitted but the patient’s daily actions, such as physical activity or consumption of alcohol or others drugs, affect the glucose metabolism [3]. The occurrence of hypoglycemia events has to be minimized to increase the effectiveness of insulin therapy. A mechanism for the effective treatment of glucose is the artificial pancreas (AP), whose goal is to reduce hypoglycemia and hyperglycemia events by keeping changes in blood glucose levels within a narrow normoglycemia range. AP is also known as closed-loop (CL) control.

In the last three decades, research on AP has been increasing and its main goal is to achieve normoglycemia maintenance 24 h a day, independently of meal intakes or physical activity and with minimal patient intervention, using a variety of control algorithms with full or hybrid CL action [4], [5], [6], [7]. CL control algorithms must be safe, and for this reason they are usually complemented with other features such as meal detection modules [8], [9], exercise estimation [10], personalisation to patients [11], glucose prediction [12], [13], and/or prediction of hypoglycemia events [14] among others. AP systems have demonstrated their superiority over CSII through long-term outpatient studies in free-living scenarios [15].

To increase the safety of AP systems, and other treatments for T1DM patients, the use of simulators based on virtual patients’ cohorts is essential as they allow for the evaluation of treatment decisions and their associated risks. A population of virtual patients, which in this study are named “virtual subjects” or simply “subjects”, can be generated with specific mathematical models [16]. These models contain different sets of parameters and several nonlinear equations to describe the behaviour of insulin and glucose dynamics in patients with T1DM. The most frequently used models in the literature for this area are:

Bergman’s minimal model [17], Sorensen’s physiological model [18], Hovorka’s model [19], and Dalla Man’s model [20].

Although the glucose-insulin dynamics can be modified by other variables, such as physical activity, meal intakes, glucagon, etc., the in-silico cohorts represent the main glucose-insulin interaction for the majority of real clinical subjects considered by the authors when the mathematical model was created.

Recently in 2019, new simulators have appeared which consider more complete glucose-insulin dynamics using more complex models. Rashid et al. [21] proposed a mathematical model, which considers physical activity, with a multi-variable Glucose-Insulin-Physiological Variable Simulator that quantifies the effects of meal intakes, insulin administration, and physical activity of the subjects with T1DM. Resalat et al. [22] proposed a simulator that validates single and dual hormones, which represent subjects with T1DM, by adding equations to represent glucagon kinetics and dynamics. Both models are variants of Hovorkas’s model.

In 2018 Fritzen et al. [23] assured that simulation tools defined by mathematical models are good tools to investigate T1DM treatments, including AP systems. To evaluate AP performance, tools based on simulators, such as the Cambridge Simulator (CBSim) or the UVA/Padova Simulator, are available. The common features of the available simulators are: 1) they supply a cohort of subjects, 2) they allow the definition of a diet treatment over 24 h, 3) they allow the delivery of insulin according to CSII therapy or AP, and 4) they provide the glucose dynamics behaviour, which is mainly based on the changes in insulin but other variables can be taken into account.

The UVA/Padova Simulator was introduced by Kovatchev in 2009 [24] and incorporates the Dalla Man’s model with some modifications such as specific models for medical devices (glucose sensors and insulin pumps). In 2014 [25] the simulator was improved by increasing the performance describing hypoglycemic events and by adding new model features such as 1) glucose-kinetics on secretion and action of glucagon models, 2) a new strategy for virtual subject generation, and 3) modification of the insulin-to-carbohydrates ratio and correction factor. In 2018 the simulator provided a scenario modelling the intra-day variability of insulin sensitivity [26]. This update describes several factors that affect glucose metabolism during the day such as subject parameter variations, the composition of meals, physical activity, and illness. The academic version allows for three groups of 10 subjects: adults, teenagers, and children. In addition, the commercial version allows for 100 subjects per group. The simulator was approved by the FDA (Food and Drugs Administration) to replace in vivo animal experimentation by in-silico virtual experimentation in 2010.

The CBSim was introduced by Hovorka et al. in 2002 [19] and since then CBSim has received updates to make the glucose-insulin dynamics more physiological. In 2004 the model started with a cohort of 6 subjects (Hovorka’s Original - HO) [27]. In 2005 Chassin [28] used a cohort of 18 subjects (Chassin’s Original - CO). In 2010 Wilinska et al. [29] published the statistical relations of the Hovorka’s and Chassin’s model parameters and also included an update of Hovorka’s compartmental model related to gut absorption.

In this work, it is used the model proposed by Chassin [28] is used with minimal changes; for this reason, the model’s equations are not mentioned. The equations can be found in the work proposed by Orozco et al. in 2018 [30].

In-silico experiments offer the possibility of reducing the high cost of clinical trials and can aid in evaluating the clinical success of control algorithms in an AP system. Furthermore, for clinical trials, the number of subjects is limited and the procedures to include new individuals from volunteers are invasive and costly. Thus, it is necessary to have a diabetes simulator equipped with a cohort of in-silico subjects that sufficiently spans the observed interpersonal variability of key metabolic parameters in the general population of people, for example, with type 1 diabetes [31].

The main goal of this work is to propose a new computational method to increase existing cohorts of virtual patients often used to simulate patients with T1DM. The method has been applied and tested with the CBSim model, but this method can be extrapolated to other mathematical models or simulators in order to generate larger cohorts.