Mathematical Model of ingested glucose in Glucose-Insulin Regulation
Sourav Chowdhury, Sourabh Kumar Manna, Suparna Roychowdhury, Indranath Chaudhuri
MMathematical Model of ingested glucose in Glucose-Insulin Regulation
Sourav Chowdhury ∗ , Sourabh Kumar Manna † , Suparna Roychowdhury, Indranath ChaudhuriMarch 6, 2020 Department of Physics, St. Xavier’s College(Autonomous), 30 Mother Teresa Sarani, Kolkata-700016.
Abstract
Here, we develop a mathematical model for glucose-insulin regulatory system. The model includes a new parameter whichis the amount of ingested glucose. Ingested glucose is an external glucose source coming from digested food. We assumethat the external glucose or ingested glucose decays exponentially with time. We establish a system of three linear ordinarydifferential equations with this new parameter, derive stability analysis and the solution of this model.
Keywords
Mathematical model, Diabetes mellitus, Linear system, Ingested glucose, Glucose tolerance test, Natural time period, Stabilityanalysis.
Glucose which we get from food is very important for the human body because it is like fuel and energy source for cells as well asthe human body. However diabetes is a condition when blood glucose level exceeds the normal range (75 -110 mg/dl) for a longperiod of time. In 2017, 4 million people died due to diabetes and approximately 425 million adults in the world had diabetes.Statistics says that by 2045 the number of people with diabetes will rise to 629 million. More than 1,106,500 children have type1 diabetes and more than 21 million births are affected by diabetes during pregnancy (2017). Around 352 million people haverisk to develop type 2 diabetes [1]. India was ranked up from 11th (2005) to 7th (2016) due to number of deaths by diabetesand there are 70 million people who suffered from diabetes and statistics says these numbers will grow more than double in thenext decade [2]. For normal person blood glucose level reaches to a homeostasis with the help of two types of hormones, firstwhich reduces blood glucose level like Insulin, Amylin, Somatostatin and second which raises blood glucose level like Glucagon,Epinephrine (adrenaline), Growth Hormone, Thyroxine [3]. However when there is a disorder in secretion of Insulin or cellsbecomes resistant of Insulin or both, then lower amount of glucose reaches to the cells and blood glucose level is very high thanfasting blood glucose level for a long period of time. Since the lower amount of glucose reaches the cells, body weakens andperson become diabetic [4].There are mainly three types of diabetes : • Type 1 diabetes mellitus - Type 1 diabetes mellitus previously known as “insulin-dependent diabetes mellitus” (IDDM)or “juvenile diabetes”. Almost 10% of worldwide diabetics have type 1 diabetes and most of them are children. This typeof diabetes caused due to deficiency of insulin in blood. Feeling very thirsty, urinating frequently, feeling very tired, weightloss, constant hunger are the main symptoms of type 1 diabetes. [5, 6, 7] • Type 2 diabetes mellitus -
Type 2 diabetes mellitus previously known as “non-insulin-dependent diabetes mellitus”(NIDDM) or “adult onset diabetes”. Almost 90% of worldwide diabetics have type 2 diabetes and most of them are adults.This type of diabetes is caused due to the insulin resistance in the cells of the body (may be due to problem in insulinreceptors in the cells). Symptoms are almost similar as type 1 diabetes. However people who suffers from type 2 diabetesare often obese. [5, 6, 7] ∗ Short term project student working under Dr. Suparna Roychowdhury and Dr. Indranath Chaudhuri, Department of Physics, St. Xavier’s College(Autonomous), Kolkata † Short term project student working under Dr. Suparna Roychowdhury and Dr. Indranath Chaudhuri, Department of Physics, St. Xavier’s College(Autonomous), Kolkata a r X i v : . [ q - b i o . T O ] M a r Gestational diabetes -
Gestational diabetes is a type of diabetes which develops in some women when they are pregnant.Most of the time, this type of diabetes goes away after the baby is born. However they and their children have increased riskof type 2 diabetes. Gestational diabetes is mainly diagnosed through parental screening rather than through symptoms.[5, 6, 7]Mathematical models are very important to understand the dynamic behavior of complex biological systems. There are variousmathematical models, statistical methods and algorithms to understand different aspects of diabetes. Models can be classifiedbetween two categories : • Clinical models - Clinical models are developed to understand a disease more accurately so that we can find a bettercure of it. There are various existing models for diagnostics like GTT (glucose tolerance test), IVGTT (intravanous glucosetolerance test), OGTT (oral glucose tolerance test), FSIVGTT (frequently sampled intravenous glucose tolerance test).[4, 8, 9] • Non-clinical models -
Non-clinical models are developed from the partial knowledge of the system. There are variousnon-clinical models to understand insulin-glucose dynamics and also there are many different types of non-clinical models.[4, 8, 9]Since diabetes is very complex in nature, these models needs to be upgraded with respect to the experimental knowledge. Here wepresent a realistic model by considering a new parameter which represents ingested glucose (External glucose which is acquiredfrom intake food).The paper is arranged in the following way : importance of glucose and insulin and their role in the human body, discussionof the previous model (briefly) and the new model, analysis of stability and calculation for model, model fitting of a data-setand finally concluding remarks.
In this section we will discuss briefly, how glucose-insulin plays an important role in the human body. We start with a simpleblock diagram which represents importance of glucose in the human body.Figure 1: Block diagram to show how glucose is converted to energy.Glucose is converted to energy (ATP) in the cells by glycolysis and other processes. In some cells glucose turns into energywith the help of oxygen and in some cells it turns into energy without oxygen [10]. Also insulin is very important, because withthe help of insulin glucose can enter into cells. If there is less insulin or the body cells are insulin resistant then glucose cannotenter into the cells, and it remains in the blood creating different complications [3, 6].2igure 2: Block diagram to show how homeostasis is maintained for a normal person.The human body needs to maintain homeostasis (dynamic equilibrium in human body) of blood glucose level. To maintainthis homeostasis, mainly insulin and glucagon work together. Insulin is released from the β -cells of pancreas, and glucagonis released from the α -cells of pancreas. Figure 2 shows how human body maintains homeostasis with the help of these twohormones. When blood glucose level of the human body is higher than the normal level, the pancreas secretes more insulin.Insulin helps glucose to enter the cells. This also helps the extra glucose to get stored into the liver as glycogen, thus reducingthe blood glucose level to maintain the homeostasis. However, when blood glucose level is low, the pancreas secretes glucagonwhich breaks stored glycogen into glucose and increases blood glucose level to normal level [4, 7, 11].Figure 3: Block diagram to show how glucose level varies for diabetic person.However for a diabetic person, the human body cannot maintain homeostasis of blood glucose level [Figure 3]. In type-1diabetes, β -cells of pancreas are destroyed by ones own immune system (T-cells). Thus less insulin is produced and for this3nsulin deficiency blood glucose level is high for a very long period.In type-2 diabetes, cells of the human body becomes insulin resistant. So, insufficient amount of blood glucose enters intothe cells. Also low amount of blood glucose gets stored into liver in the form of glycogen. To overcome this situation pancreassecretes excess insulin. If this excess insulin is enough to bring back blood glucose level to the normal level then the personis pre-diabetic. However if this excess insulin is not enough then the person is type-2 diabetic. Other reason for which bloodglucose level can be high is due to malfunctioning α -cells of pancreas. In this case they produce excess amount of glucagon[4, 6, 7]. In this section we first discuss a model suggested by previous authors. Next we go on to discuss the relevant factors that wehave introduced into our model.In 1964, E. Ackerman, J. W. Rosevear and W. F. McGuckin developed a mathematical model of Glucose-tolerance test [12].In this model they have considered G ( t ) as blood-glucose concentration and H ( t ) as blood-insulin concentration. Here G b and H b were defined as fasting blood glucose level and fasting hormone (insulin) level respectively. D ( t ) and h ( t ) were defined asdifference of the blood-glucose level and blood-hormone (insulin) level from the fasting levels respectively. So, D ( t ) = G ( t ) − G b and h ( t ) = H ( t ) − H b . The main equations used in that model are, dD dt = − a D − a h + I,dhdt = b D − b h. Where, a = Rate constant of glucose removal independent of insulin. a = Rate constant of glucose removal dependent of insulin. b = Rate constant of insulin release due to glucose. b = Rate constant of insulin removal independent of glucose. I ( t ) = Rate of increase of blood glucose due to absorption of glucose from intestines.We have modified the above model taking into account the contribution of the ingested glucose in the form of an additionaldifferential equation. We explain this model using a block diagram where various parts of the human body represented bycompartments. [Figure 4] Figure 4: Diagram of simplified model of blood glucose regulatory system.Let at time t glucose disturbance of the digestive system (1st compartment) is D ( t ) . In this system we assume that thefasting glucose level is zero, thus glucose level and glucose disturbance is same. Let at time t the blood (2nd compartment)glucose level is G ( t ) and the effective hormone level is H ( t ) . By the term effective hormone level means the net effect of allhormones (like insulin, glucagon) which can increase or decrease blood glucose level. The fasting blood glucose and the effectivehormone levels G b and H b respectively. Let, D ( t ) and h ( t ) are the disturbances of the blood glucose level and the effectivehormone level at time t respectively. Thus we can write D ( t ) = G ( t ) − G b and h ( t ) = H ( t ) − H b . [12, 13]In this model we make following assumptions : • Rate of decrease in glucose disturbance in digestive system is proportional to its glucose disturbance ( D ( t ) ) at time t .4 Rate of increase in glucose disturbance in blood is proportional to the glucose which enters blood from digestive system( D ( t ) ) at time t . • Rate of hormone independent decrease of glucose disturbance in blood is proportional to its glucose disturbance ( D ( t ) )at time t . • Rate of hormone dependent decrease of glucose disturbance in blood is proportional to the effective hormone disturbance( h ( t ) ) at time t . • Rate of increase in effective hormone disturbance is proportional to the glucose disturbance in blood ( D ( t ) ) at time t . • Rate of decrease in effective hormone disturbance is proportional to the effective hormone disturbance ( h ( t ) ) at time t .Using these assumptions the model equations can be written as, dD dt = − D τ , D (0) = A G D, (1) dD dt = D τ − a D − a h, D (0) = 0 , (2) dhdt = b D − b h, h (0) = 0 . (3) τ = Time constant of decreasing glucose level in the digestive system, which is the total time to decrease glucose level to /e of the maximum value. a = Rate constant of the hormone independent decrease of glucose level in the blood. a = Rate constant of the hormone dependent decrease of glucose level in the blood. b = Rate constant of release of the hormone due to blood glucose disturbance. b = Rate constant for the removal of the hormone due to disturbance of the blood hormone level. D = Amount of food that has been taken. A G = Percentage of glucose obtained in the body from the food that has been taken.
To determine fixed points we can write, dD dt = dD dt = dhdt = 0 . Substituting dD dt = 0 in equation (1) we get, D = 0 = D . (4)Now making dD dt = 0 and dhdt = 0 and putting D in equation (2) we get, D τ − a D − a h = 0 ,a D + a h = 0 . (5)From equation (3) we get, b D − b h = 0 . (6)Solving equation (5), (6) we get, D =0 (7)and h =0 . (8)5o the fixed point is (D , D , h ) = (0 , , . Thus equations (1) , (2) and (3) can be written in the matrix form as, dD dtdD dtdhdt = − τ τ − a − a b − b D D h Let, − τ τ − a − a b − b = A Now characteristic equation is given by, det ( A − λI ) = 0 Here λ is the eigen values of matrix A . So, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( − τ − λ ) 0 0 τ ( − a − λ ) − a b ( − b − λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 From this determinant we can write, (cid:18) − τ − λ (cid:19) [( a + λ ) ( b + λ ) + a b ] = 0 . The above equation can write into two algebraic equations as, (cid:0) − τ − λ (cid:1) = 0 and λ + ( a + b ) λ + ( a b + a b ) = 0 .Hence the first eigenvalue is λ = − τ where τ > so λ < and the other two eigenvalues are λ = − ( a + b ) + (cid:112) ( a − b ) − a b ,λ = − ( a + b ) − (cid:112) ( a − b ) − a b . If ( a − b ) > a b then (cid:112) ( a − b ) − a b < ( a + b ) . So, λ and λ are both negative.If ( a − b ) < a b then (cid:112) ( a − b ) − a b is a complex quantity. Hence λ and λ are both complex with negative realpart.Hence the model is stable around (0 , , .Figure 5: Phase portraits of these model shows it is globally stable. (0,0,0) point is represented in the plots as red ( × ).Figure 5 represents a three dimensional phase portrait with different initial conditions. From this plots we can see that allthe trajectories converge to the point (0,0,0). Hence (0,0,0) is a stable equilibrium point and the model is globally stable.6 Calculation of glucose level in blood
In this section we obtain a solution for blood glucose level from the model equations by considering all initial conditions.Solving equation (1) we get, D = A e − tτ . Where A = Integrating constant. Substituting initial conditions, D (0) = A G D = A . Hence, D ( t ) = A G De − tτ . (9)Differentiating equation (2), d D dt = − a dD dt − a dhdt + 1 τ dD dt . (10)Substituting the relation of dhdt from equation (3) in equation (10) we get, d D dt = − a dD dt − a ( b D − b h ) + 1 τ dD dt . After substituting relation of h from equation (2) in equation (11) and rearranging we get, d D dt +( a + b ) dD dt + ( a b + a b ) D = b τ D + 1 τ dD dt . (11)Now using, a + b = 2 α , which is sum of the rate of hormone independent glucose removal and the hormone removal and a b + a b = ω we get, d D dt +2 α dD dt + ω D = b τ D + 1 τ dD dt . (12)Substituting D ( t ) from equation (9) we get, d D dt +2 α dD dt + ω D = ( b A G Dτ − A G Dτ ) e − tτ . (13)Using C = b A G D τ − A G D τ d D dt +2 α dD dt + ω D = Ce − tτ . (14)Solving equation (14), D = e − αt [ A cos ( ωt ) + A sin ( ωt )] + C e − tτ . (15)Where ω = ω − α and α < ω and C = A G D ( b τ − ατ − + ω τ .Now applying initial conditions D (0) = 0 and dD dt | t =0 = A G Dτ in equation (15) we get,So finally we get A = − C and A = A G D + C − αC τωτ .So the disturbance in 2nd compartment is, D = e − αt [ − C cos ( ωt ) + A G D + C − αC τωτ sin ( ωt )] + C e − tτ . (16)7ence glucose level in 2nd compartment is, G ( t ) = G b + D = G b + e − αt [ − C cos ( ωt ) + A G D + C − αC τωτ sin ( ωt )] + C e − tτ . (17)From equation (16) we can see that the solution of D ( t ) is like a oscillator with a small damping. h ( t ) can have both positiveand negative values. Positive h ( t ) corresponds to the effective hormone level which decreases the blood glucose level. Similarly,negative h ( t ) corresponds to the effective hormone level which increases the blood glucose level. As we see, α and ω can havemany different values [12, 13]. So the crucial parameter turns out to be ω which in other words is a natural time period T = πω .If T < . hours then the person is normal and if T ≥ . hours then the person is pre-diabetic [12, 13]. In this section, the blood glucose level G ( t ) that is obtained from our model is fitted with a data-set and essential parameters ofour model are estimated. The data-set represents a glucose vs time data for a normal (Subject A) and a diabetic (Subject B)person. This data-set is taken from “Modeling Diabetes” by Joseph M. Mahaffy (Math 636 - Mathematical Modeling) [16].t in hour Subject A Subject B0 70 1000.5 150 1850.75 165 2101 145 2201.5 90 1952 75 1752.5 65 1053 75 1004 80 856 75 90Table 1: Blood glucose level data for normal (Subject A) and diabetic person (Subject B).Figure 6: Fitted curve for normal and diabetic persons with σ interval.Figure 6 shows the best fit curve with σ confidence interval. σ confidence interval confidence interval is a statisticallycalculated interval from a observed data-set. This interval tells us that the true value of the parameter may be lie within it.Here, 1 σ confidence interval is defined as G ( t ) ± σ √ N , σ =standard deviation of data and N = total number of data. If, N increasesthen confidence interval decreases but confidence remains same.8he values of the obtained parameters are,Subject G b α ω C τ in hourA 73.8161 1.1733 2.4128 116.4327 0.6447B 94.4838 0.8685 1.2823 208.3634 0.6242Table 2: Values of the fit parametersSubject α ω ω = √ ω + α T = πω in hourA 1.1733 2.4128 2.6829 2.3418B 0.8685 1.2823 1.5487 4.0569Table 3: Values of main parameters ω and T G b = The fasting blood glucose level. It can vary person to person. α = Decay parameter of the damped oscillator. ω = Angular frequency of damped oscillator.From Table 2 we can see that value of both α and ω for diabetic person is less than the normal person. τ = Time constant of decreasing glucose level in the digestive system, which is the total time to decrease glucose level to /e of the maximum value. From Table 2 we can see that the value of this parameter is nearly equal for both normal and diabeticperson. ω = Natural angular frequency of damped oscillator. T = Natural time period of damped oscillator. From Table 3 we can see that value of T for normal person is small than thediabetic person.Here we plot the data-set and fit the function with 1 σ confidence interval [Figure 6]. Values of T of Subject A and SubjectB are 2.3418 hours and 4.0569 hours respectively [from Table 3], which indicates Subject A is normal and Subject B is milddiabetic. Thus it can be seen that the value of T determines whether a person is pre-diabetic (for which T ≈ . hours) ornot. This result nearly matches with the earlier model result but more accurate than the previous result to predict pre-diabeticcondition. Also we found the value of a important parameter τ which is not included in the previous models. This parametertells us how fast glucose enters in the blood from the digestive system. This parameter should be independent of the diabetesor blood glucose level at any time. From Table 2 we can see that the value of τ is nearly equal for both normal and diabeticperson. So, τ is independent of diabetes and does not depend on blood glucose level.Figure 7: Three dimensional surface plot with different α and ω .9igure 8: Three dimensional simulated scatter plot.Figure 7 represents a three dimensional simulated surface plot which shows how T varies with the possible range of α and ω . We also see as α and ω decreases, T increases sharply after a point. High T means that blood glucose level remains high fora long interval of time, which is the sign of a pre-diabetic case. Figure 8 represents a three dimensional simulated scatter plot.Here, blue dots represent normal cases (for which T < . hours) and red dots represent pre-diabetic or diabetic (for which T ≥ . hours) cases. This plot shows that there exists two completely separate regions for normal and diabetic cases. It alsoshows that there is a large range of α and ω which can vary person to person. In this paper, we have modified an existing model to a more realistic one by considering ingested glucose level, which is describedby an additional ordinary differential equation. Here we have assumed that the externally ingested glucose decreases exponentiallywith time by increasing the blood glucose level, which is modeled by a parameter τ which describes how fast ingested glucosedecreases. We have found that the model is globally stable around the fixed point (0,0,0). The solution imitates the behaviour ofa damped harmonic oscillator and it converges to basal values normally observed in the human body. After fitting this relationto an available data-set, various fit parameters were obtained. Using these, the value of the parameter T , which is the naturaltime period of a damped oscillator, was found. To conclude, we would like to point out that the most important improvementof this model over earlier models is it’s ability to predict the vulnerability of a person to be diabetic in the future. We havededuced that T is less than 4 hours for a normal person and for a diabetic person, this time period is greater than 4 hours,which also matches the earlier established model predictions. Thus if a person has this natural time period T of value around4 hours, then it can be concluded that the person is susceptible to diabetes in future (pre-diabetic). However, in this modelwe consider that pancreas responded instantly with the blood glucose disturbance and hormone disturbance. But a small timedelay of pancreatic response due to these disturbances will be more practical [32]. We can modify our model by considering thistime delay. So, there is a lot of scope for further development of this model, which can enable precise and better control of thispre-diabetic stage and thus modify the quality of life of a human being. Acknowledgment
We would like to thank Dr. Indrani Bose and Dr. Tanaya Bhattacharyya for their useful comments and suggestions. We alsolike to thank the Department of Physics, St. Xavier’s College for providing support during this work.
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