MODELING GLUCOSE-INSULIN METABOLIC SYSTEM AND INSULIN SECRETORY ULTRADIAN OSCILLATIONS WITH EXPLICIT TIME DELAYS. Yang Kuang

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1 MODELING GLUCOSE-INSULIN METABOLIC SYSTEM AND INSULIN SECRETORY ULTRADIAN OSCILLATIONS WITH EXPLICIT TIME DELAYS Yang Kuang (joint work with Jiaxu Li and Clinton C. Mason) Department of Mathematics and Statistics Arizona State University Tempe, AZ ISU 5 p. 1/36

2 Large Diabetics Population Large population of diabetics millions, 6.4% in 22 Direct and indirect cost - $132 billions in 22 Better understand how the mechanism functions Causes of the dysfunctions of the system Prevent and detect onset of diabetes More reasonable, effective, efficient and economic treatment ISU 5 p. 2/36

3 Glucose-Insulin Regulatory System Low Plasma Glucose Level High Plasma Glucose Level Pancreas α-cells release glucagon ß-cells release insulin Glucagon Insulin Exercises, fasting and others Liver Glucose Infusion, meal, enteral, oral intake and others Liver converts partial glucagon released from α-cells and partial glycogen stored in liver to glucose Insulin helps to consume plasma glucose Normal Plasma Glucose Level ISU 5 p. 3/36

4 Langerhans Islet, β-cell, α-cell β-cells (65-8%); α-cells (15-2%); δ-cells (3-1%); polypeptide and D 1 cells (1%) ISU 5 p. 4/36

5 Insulin Secretion β-cells secrete insulin in oscillatory manner Rapid oscillations triggered by internal peacemaker (5-15 min.) Pòrksen et al. (22), Pulsatile insulin secretion: detection, regulation, and role in diabetes, Diabetes, 51 (Suppl. 1), S245 S254. Slow oscillations stimulated by elevated glucose (5-12 min.) Simon and Brandenberger (22), Ultradian oscillations of insulin secretion in humans, Diabetes, 51 (Suppl. 1), S258 S261. ISU 5 p. 5/36

6 Insulin Secretion - Oscillatory Manner Glucose (gm/dl) Insulin (µu/ml) Glucose (gm/dl) Insulin (µu/ml) A C min min. Insulin (µu/ml) Glucose (gm/dl) Insulin (µu/ml) Glucose (gm/dl) B D min min. A. meal ingestion; B. oral glucose intake; C. continuous enteral nutrition; D. constant glucose infusion. (Sturis et al, 1991) ISU 5 p. 6/36

7 Insulin Degradation And Clearance Mainly liver and kidney, but most other tissues IDE - major enzyme in the proteolysis of insulin in addition to several peptide. Insulin is degraded by enzymes in the subcutaneous tissue and interstitial fluid Insulin receptors ISU 5 p. 7/36

8 Glucose Productions Exogenous ingestion - e.g., meal ingestion; oral glucose intake; continuous enteral nutrition; and constant glucose infusion Endogenous production - complex - Insulin concentration dominates the hepatic glucose production (conversion from glucagon) and release rate by the liver - substantial time delay (25-5 min.) ISU 5 p. 8/36

9 Glucose Utilization Insulin-independent utilization: mainly by brain and nerve cells. Insulin-dependent utilization: by muscle, fat cells and other tissues ISU 5 p. 9/36

10 Glucose-Insulin System Model with Delay Delay (25-5 min.) Liver Glucose Infusion: meal ingenstion, oral intake, enteral nutrition, constant infusion Liver converts glucagon and glycogen to glucose Glucagon secretion Insulin controls hepatic glucose production Pancreas α cells ß-cells Delay (5-15 min.) Glucose production Glucose Glucose controls glucagon secretion Glucose controls insulin secretion Insulin secretion Insulin production Glucose utilization Insulin helps cells consume glucose Insulin Insulin clearance Insulin independent: brain cells, and others Insulin dependent: fat cells, and others Insulin degradation: receptor, enzyme, and others ISU 5 p. 1/36

11 Why Are There Delays? Insulin response time delay and glucose utilization delay (5-15 min.): GLUT2 transport glucose into β-cells;...; K + channel close; Ca 2+ channel open and influx release insulin; Insulin receptors activate the signaling cascade for GLUT4 translocation. GLUT4 transporters lead glucose molecules into muscle cells then consume the glucose and convert into energy. (Sturis-Tolic Model splits insulin in two compartments.) Hepatic glucose time delay (25-5 min.): appearance of insulin in plasma and its inhibitory effect. Exact pathway is unknown. (Sturis-Tolic Model simulates the delay by introducing additional variable into the system.) ISU 5 p. 11/36

12 Modeling Insulin Secretion Ultradian Oscillation Glucose change rate = glucose production - glucose utilization Insulin change rate = insulin production - insulin clearance Introduce explicit time delays and modify original Sturis Model G = G in f 2 (G(t)) f 3 (G(t))f 4 (I(t)) + f 5 (I(t τ 2 )), I = f 1 (G(t τ 1 )) d i I(t), ISU 5 p. 12/36

13 f i, i = 1, 2, 4, 5, (J. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, 1998.) G I 3 4 f 2 (G) f 4 (I) x G f 5 (x) f 1 (G) ISU 5 p. 13/36

14 Modeling Ultradian Oscillations - existing V. W. Bolie (1961) J. Sturis et al (1991) I. M. Tolic et al (2) K. Engelborghs et al (21) D. L. Bennett and S. A. Gourley (24) A. Mari (22), Mathematical modeling in glucose metabolism and insulin secretion, Curr.Opin.Clin.N utr.m etab.care, 5 : A. Makroglou et al.(25), Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: an overview, Appl. Numerical Math., in press, available on line ISU 5 p. 14/36

15 Sturis-Tolic ODE Model G (t) = G in f 2 (G(t)) f 3 (G(t))f 4 (I i (t)) + f 5 (x 3 ), I p(t) = f 1 (G(t)) E( I p(t) I i(t) ) I p(t), V p V i t p I i(t) = E( I p(t) I i(t) V p x 1(t) = 3(I p x 1 )/t d, x 2(t) = 3(x 1 x 2 )/t d, x 3(t) = 3(x 2 x 3 )/t d ) I i(t), V i t i ISU 5 p. 15/36

16 Bennett-Gourley Single Time Delay Model D. L. Bennett and S. A. Gourley (24) G (t) = G in f 2 (G(t)) f 3 (G(t))f 4 (I i (t)) + f 5 (I p (t τ)), I p(t) = f 1 (G(t)) E( I p(t) V p I i(t) V i ) I p(t) t p, I i(t) = E( I p(t) V p I i(t) V i ) I i(t) t i, ISU 5 p. 16/36

17 Observations of Sturis-Tolic Model and Bennett-Gourley Model ST1 The oscillation is critically dependent on hepatic glucose production time delay. ST2 The period ω of the oscillation (95, 14) (min). ST3 It is critical to break insulin into two compartments for sustained oscillation. ST4 The oscillation is sensitive to the max s of f 1 and f 5. BG1 Same as [ST4]. Also, hepatic glucose production rate R g can not be too small. BG2 Hepatic glucose production time delay τ and insulin interstitial transfer time t i and t p can not be too small. ISU 5 p. 17/36

18 Preliminaries of Two Time Delay Model Proposition 1 (i) If lim f 3(x) > (G in M 2 + M 5 )/m 4, x then the system has unique positive steady state (G,I ) and I = d 1 i f 1 (G ). Further, all solutions of the system exist for all t > and are positive and bounded. (ii) If lim f 3(x) < (G in M 2 )/m 4, x then lim sup t G(t) =. ISU 5 p. 18/36

19 Global Analysis of Two Time Delay Model Theorem 1 Let If F(x,y) = f 3 (x)f 4 (d 1 i f 1 (y)) + f 5 (d 1 i f 1 (x)), x,y. F(x,y) F(y,x), x y, then the steady state (G,I ) is globally asymptotically stable. Remark 1 Higher hepatic glucose production helps oscillations occur. Remark 2 If no hepatic production, then (G,I ) is globally stable and thus no oscillation. ISU 5 p. 19/36

20 Local Analysis of Two Time Delay Model (λ) = λ 2 + (A + d i )λ + d i A + DBe λτ 1 + DCe λ(τ 1+τ 2 ) =. λ k + k 1 j=1 A := f 2(G ) + f 3(G )f 4 (I ) >, B := f 3 (G )f 4(I ) >, C := f 5(I ) >, D := f 1(G ) >. a j λ j + b + ce λσ 1 + de λσ 2 =, k 2,σ 1,σ 2 >. ISU 5 p. 2/36

21 Local Analysis - Case τ 1 τ 2 Define and Then S 1 = { 2m 2n 1 : m,n Z+,m,n 1} S 2 = { 2m 1 2n : m,n Z +,m,n 1} Q + = S 1 S 2, S 1 = S 2 = R +, S 1 S 2 = ISU 5 p. 21/36

22 Local Analysis - τ 1 τ 2 - General Characteristic Equation Proposition 2 For characteristic equation λ k + k 1 j=1 a j λ j + b + ce λσ 1 + de λσ 2 =, k 2,σ 1,σ 2 >, where b,c,d >,a j R,j = 1, 2, 3,...,k, if b < d c (or b < c d), then σ 1 > and σ 2 > such that the characteristic equation of L.S. has at least one root with positive real part for σ 1 > σ 1 and σ 2 > σ 2 and σ 1 /σ 2 S 1 (or σ 1 /σ 2 S 2 ). ISU 5 p. 22/36

23 Local Analysis - Case τ 1 τ 2 Theorem 2 For the linearized system, If d i A D(B + C), then the steady state is stable. If d i A < D(C B) (or d i A < D(B C)), then τ 1 > and τ 2 > such that the characteristic equation of L.S. has at least one root with positive real part for τ 1 > τ 1 and τ 1 + τ 2 > τ 2 and τ 1 /(τ 1 + τ 2 ) S 1 (or τ 1 /(τ 1 + τ 2 ) S 2 ). ISU 5 p. 23/36

24 Simulation - Function f i,i = 1, 2, 3, 4, 5. f 1 (G) = R m 1 + exp((c 1 G/V g )/a 1 ) f 2 (G) = U b (1 exp( G/(C 2 V g ))) f 3 (G) = G C 3 V g f 4 (I) = U + (U m U ) 1 + (I i /C 4 (1/V i + 1/Et i )) β f 5 (I) = R g 1 + exp(α(i/v p C 5 )) ISU 5 p. 24/36

25 Simulation - Parameters Table Parameters Units Values V g l 1 R m µumin 1 21 a 1 mgl 1 3 C 1 mgl 1 2 U b mgmin 1 72 C 2 mgl C 3 mgl 1 1 Parameters Units Values U mgmin 1 4 U m mgmin 1 94 β 1.77 C 4 µul 1 8 R g mgmin 1 18 α lµu 1.29 a 1 µul 1 26 ISU 5 p. 25/36

26 Simulation - τ 1 = 7,τ = 36,G in = 1.35,d i =.6 Glucose ( mg/dl ) ISR ( µu/ml/min ) time ( min ) Insulin ( µu/ml ) Glucose ( mg/dl ) τ 1 = 7, τ 2 = 36, G in = 1.35, d i =.6 Sturis Tolic ODE Model BG Single Delay Model Two Delay Model Alt Single Delay Model time ( mim. ) ISU 5 p. 26/36

27 Simulation R s = { ( d i, G in ) d i > H 1 (d i, G in ) } R u = { ( d i, G in ) d i < H 2 (d i, G in ) } G in d i H 1 (d i, G in ) := f 1 (G )(f 3 (G )f 4 (I ) f 5 (I )) f 2 (G )+f 3 (G )f 4 (I ) and H 2 (d i, G in ) := f 1 (G ) f 3 (G )f 4 (I )+f 5 (I ) f 2 (G )+f 3 (G )f 4 (I ) ISU 5 p. 27/36

28 Simulation - τ 1 = 7,G in = 1.35,d i = τ 1 = 7, G in = 1.35, d i = In each cycle, G peaks before I does. This curve is the time between these two peaks Glucose concentration amplitude Glucose concentration ( mg/dl ) Glucose concentration limiting value Bifurcation point at τ min. Insulin concentration limiting value Insulin concentration amplitude Insulin concentration ( µu/ml ) Periods of periodic solutions ( min. ) Bifurcation point at τ min. Periods of periodic solutions generated from bifurcation point Time between peaks of G and I in each cycle ( min. ) 75 τ 1 = 7, G in = 1.35, d i = τ ( min. ) τ ( min. ) 2 ISU 5 p. 28/36

29 Simulation - G in = 1.35,d i =.6orτ 1 = 7,τ = τ τ G in.5 1 τ 1 2 ( G in =1.35, d i =.6 ) ( τ 2 = 36, d i =.6 ) ISU 5 p. 29/36

30 Simulation - τ 2 is important Stable region 1.2 Stable region d i.8 Unstable region.8 G in.6 Unstable region G in ( τ 1 =5, τ 2 =6 ) τ 1 ( τ 2 = 1, d i =.6 ) ISU 5 p. 3/36

31 Discussion - 1 Confirmed almost all the results of [STx] and [BGx]. Both delays are critical for sustained oscillations. Effort of insulin compartment split is overcome by introducing an explicit time delay of insulin response and insulin-dependent glucose utilization. Our numerical simulations quantify the time between peaks of glucose and insulin levels in each cycle ( 2 min). ISU 5 p. 31/36

32 Discussion - 2 Neither hepatic glucose production nor its time delay are negligible. Oscillation sustained if d i is relatively moderate (d i < H 2 ) collaborated with appropriate delays. However, if it is too small, insulin level remains high and inhibits hepatic glucose production. Thus keep glucose at low level without oscillation. Oscillatory behavior of glucose-insulin regulation requires the insulin is not utilized too quick (d i < D(C B)/A) and the hepatic glucose production is delayed long enough (τ 2 > τ 2 ). ISU 5 p. 32/36

33 Discussion - 3 Oscillations sustained when glucose infusion rate is not large and insulin response and utilization time is not too soon. modeling IVGTT does not need to consider oscillation due to large bolus of glucose infusion and short dynamics. The system could have complicated behavior. (c.3) ISU 5 p. 33/36

34 Further Research - Regulatory System G = G in f 2 (G(t)) f 3 (G(t))f 4 (I(t τ 3 )) + f 5 (I(t τ 2 )) I = f 1 (G(t)) d i I(t) G = G in f 2 (G(t)) f 3 (G(t))f 4 (I(t τ 3 )) + f 5 (I(t τ 2 )) I = f 1 (G(t τ 1 )) d i I(t) ISU 5 p. 34/36

35 Further Research - Insulin Infusion Model for Type-1 Diabetes G = G in f 2 (G(t)) f 3 (G(t))f 4 (I(t τ 1 )) + f 5 (I(t τ 2 )) I = I in d i I(t) G = G in f 2 (G(t)) f 3 (G(t))f 4 (I(t τ 1 )) + f 5 (I(t τ 2 )) I = I in (t) d i I(t) where I in (t) is a forced term with period ω. Look for subharmonic solutions. ISU 5 p. 35/36

36 Thank You! ISU 5 p. 36/36

ANALYSIS OF A MODEL OF THE GLUCOSE-INSULIN REGULATORY SYSTEM WITH TWO DELAYS

SIAM J. APPL. MATH. Vol. 67, No. 3, pp. 757 776 c 2007 Society for Industrial and Applied Mathematics ANALYSIS OF A MODEL OF THE GLUCOSE-INSULIN REGULATORY SYSTEM WITH TWO DELAYS JIAXU LI AND YANG KUANG