GLUCOSE REGULATION IN TYPE 1 DIABETIC PATIENTS BY A MULTI-DOSES REGIMEN

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1 GLUCOSE REGULATION IN TYPE 1 DIABETIC PATIENTS BY A MULTI-DOSES REGIMEN Campos-Delgado D.U. Hernández-Ordoñez M. Femat R. Facultad de Cencas, Av. Salvador Nava s/n, Zona Unverstara, C.P , UASLP, Méxco Facultad de Ingenería, CIEP, UASLP, Méxco IPICYT, Depto. de Matemátcas Aplcadas y Sstemas Computaconales, Méxco Abstract: In ths paper several control regmes are suggested for type 1 dabetc patents. The suggested regmes are proposed based on dfferent nsuln formulatons. The nsuln doses are assumed to be nfused by a subcutaneous njecton n a three daly regmen pror to each meal. Mxng two types of nsuln: rapd or short, and ntermedate or long acton, the basal and postprandal nsuln productons of the pancreas are reproduced. The performance n the glucose regulaton s evaluated durng a 10-day tral by open-loop and closed-loop smulaton wth a compartmental patent model. Keywords: Dabetes, Bomedcne, Glucose Regulaton. 1. INTRODUCTION The nsuln s a hormone n charge to promote the processng of glucose (energy) by the body cells. As a result, ths hormone has a regulatory effect n the blood glucose n order to avod hgh (hyperglycema) glucose concentratons from ts euglycemc (normal) level mg/dl (Sorensen 1985), (Puckett 1992). The type 1 dabetes s a desease characterzed by the destructon of the β- cells n the pancreatc slets of Langerhans. Snce the β-cells produce the nsuln n the pancreas, external nsuln nfusons are needed by the patent n order to mantan regulated hs/her blood glucose. Due to contnuous varatons n the blood glucose concentraton (BGC), the dabetes can produce short and long term llnesses (dabetes coma, nephropathy, retnopathy, and other tssue damage). In a healthy pancreas, a constant basal rate of nsuln s produced 22 mu/dl (Sorensen 1985), but n order to assmlate the glucose absorbed by the gut through meals, the basal rate s ncreased (postprandal peaks) temporary. Therefore, ths nsuln release pattern should be mtated externally n order to reduce the rsk of future deseases. As a frst step n the treatment of ths llness, t s necessary to understand the nsuln-glucose dynamcs n dabetc patents. For ths reason, several research efforts have focused on the mathematcal modelng of these nteractons (Puckett 1992), (Sorensen 1985). These models can also be used as educatonal smulators for demonstraton and self-learnng (Lehmann and Deutsch 1998). There are two overall approaches for glucose control, and they depend on the locaton of the nsuln nfusons: (a) subcutaneous (Belazz et al. 2001) and (b) ntravenous (Parker et al. 2001). For the ntravenous approach, a contnuous pump

2 s used to delver a varable nsuln nfuson rate to the patent, accordng wth a control algorthm that processes the glucose measurements. Several control methodologes have been suggested: H robust control (Ruz-Velazquez et al. 2004), optmal control and model predctve (Lynch and Bequette 2002). However, due to the sze of mechancal pumps, ths approach s now lmted to patents under a hosptal treatment. On the other hand, the subcutaneous approach reles on several therapeutc regmes based on combnatons of dfferent types of nsuln (Amercan Dabetes Assocaton 2002), (APhA Specal Report 2001), (Dckerson 1999), (Hrsch 1999); delvered to the patent through a subcutaneous route on multple daly dosng regmes. The doses are programmed accordng wth the nformaton gathered by mplanted glucose sensors (MnMed R), pcks of blood glucose concentraton (Accu-Chek R) or non-nvasve blood glucometers (GlucoWatch R) (Tamada et al. 2002), and physcan advce. Thus, algorthms for the optmal tme and amount of nsuln have been suggested n (Doyle et al. 2001), (Shmauch et al. 1988). The subcutaneous approach s ndeed a challengng control problem snce the nsuln absorpton has to be consdered, and consequently a tme-lag s present n the plasma nsuln concentraton. Nevertheless, ths s the most common therapeutc regme for Type 1 dabetc patents n a chroncal stage. The paper s organzed as follows. Secton 2 descrbes the types and characterstcs of dfferent comercal nsulns, and the control problem framework s defned n Secton 3. The mult-doses regmens are llustrated n Secton 4. Secton 5 outlnes the mathematcal model for a type 1 dabetc patent. The mplementaton of the control strateges by smulaton s shown n Secton 6, and some conclusons and fnal remarks are ntroduced n Secton INSULIN TYPES AND CHARACTERISTICS If the nsuln s njected subcutaneously to the patent, there s an absorpton process from the perphery toward the blood stream. As a result, there s an nherent delay tme n the nsuln acton. Now, n some cases to reproduce the basal nsuln rate, t s desred to reduce the absorpton rate of the njected nsuln. For ths purpose, to the nsuln formulatons s added protamne or znc to delay the absorpton and the bologcal actvty of the nsuln (Dckerson 1999). In general, the nsuln s classfed accordng wth ts orgn: bovne, porcne, and human; and wth ts acton: rapd (Aspart and Lspro), short (Regular), ntermedate (NPH and Lente), and long (Ultralente and Glargne) (APhA Specal Report 2001). Human nsuln s syntheszed by chemcal modfcaton of pork nsuln, or through a recombnant DNA technology. Snce the human nsuln s less antgenc than anmal nsuln, and t also has a more rapd onset of acton and shorter absorpton process, human nsuln s preferred n therapeutcal regmes. Table 1 llustrates the dynamc characterstcs of the dfferent types of human nsuln. For some types of nsuln, Berger y Rodbard (1989) proposed a mathematcal model to reproduce the assmlaton pattern after a subcutaneous njecton. The tme evoluton of Lspro, Regular, NPH, Lente and Ultralente nsuln after a 10 U nfuson s shown n Fgure 1. Plasma Insuln Concentraton ( µ U/ ml ) Table 1. Insuln Characterstcs After Subcutaneous Infuson. Type Acton (hours) Onset Peak Duraton Rapd Aspart Lspro Short Regular Intermedate NPH Lente Long Ultralente Glargne Lspro Regular NPH Lente Ultralente tme (hours) Fg. 1. Tme Evoluton of Plasma Insuln Concentraton after a Subcutaneous Insuln Infuson of 10 U. 3. CONTROL PROBLEM DESCRIPTION Accordng to Mexcan customs, three major meals are taken per day: breakfast (desayuno) (7:00-10:00 hrs), lunch (comda) (13:00-15:00 hrs) and dnner (cena) (20:00-22:00 hrs); where the comda meal s the major one of the day. Roughly, there s a tme nterval of 6 hrs among each meal of the day. The approach presented n the paper reles n a three daly njectons usng rapd or short, and ntermedate or long acton types of

3 nsuln. These doses are programmed mnutes before takng a meal for rapd nsuln, and 30 mnutes for short acton nsuln. In Fgure 1, the plasma nsuln concentraton for dfferent types of nsuln are shown for a subcutaneous njecton of 10 U. Due to the delayed acton of the ntermedate or long acton nsuln, the doses for lunch-tme are omtted, and only rapd or short acton nsuln s njected. In order to prevent log-term llnesses, the control objectve s defned as to regulate the BGC around an euglycemc concentraton (EC), defned as EC = mg/dl (1) usng three daly doses of a preparaton of two types of nsuln. In ths control scheme, several glucose measurements are avalable daly whch could be derved from blood samples, n-vvo sensors or non-nvasve means (Tamada et al. 2002). The control problem posed s very demandng snce the doses gven by a physcan can vary abruptly from patent to patent. Moreover, the nsuln-glucose dynamcs for a type 1 dabetc patent are hghly non-lnear and can be modfed by dfferent parameters lke det, exercse, etc. (Puckett 1992), (Sorensen 1985). Note that a det s assgned by the physcan accordng wth age and weght, however n most of the cases, the patent cannot follow tghtly the amount of carbohydrates per meal assgned. So, the nsuln regme should be robust enough to mantan the BGC regulated despte these ssues. In a systems pont of vew, a type 1 dabetc patent can be vewed a SISO (sngle-nput sngleoutput) system, where the control output s the subcutaneous BGC and the control nput s the external nsuln. It s mportant to pont out that n the absence of a control nput (nsuln), the system s unstable snce the BGC rses contnuously. On the other hand, the control objectve s dffcult to tackle wth classcal control theory, snce the strategy wth multple daly nfusons can be thought as dscrete mpulses wth varable samplng tme gven by the meal tmes. So, control strateges that rely on alternatve technques as fuzzy-logc (Campos-Delgado et al. 2003), selftunng algorthms (Campos-Delgado et al. 2004), neural networks, and adaptve control (Belazz et al. 2001) have been suggested. 4. MULTI-DOSES THERAPEUTICAL REGIMENS Accordng wth the pharmacologcal effect of each nsuln, several combnatons of fast and slow acton nsuln can be suggested (APhA Specal Report 2001), (Amercan Dabetes Assocaton 2002). Consequently, Lspro o Regular nsuln are Table 2. Three Daly Doses Control Regmens. Breakfast Lunch Dner Lspro+NPH Lspro Lspro+NPH Lspro+Lente Lspro Lspro+Lente Lspro+Ultralente Lspro Lspro+Ultralente Regular+NPH Regular Regular+NPH Regular+Ultralente Regular Regular+Ultralente combned wth NPH, Lente or Ultralente nsuln. Fve therapeutc regmes are llustrated n Table 2. These regmens are also known as flexble nsuln regmens or basal-bolus nsuln therapy (Hrsch 1999), snce they allow the patent to adjust the tmng and amount of nsuln n accordance wth changes n meal carbohydrate content or exercse. Note that the mxng of short-actng (Regular) and lente nsuln s not recommended snce the absorpton dynamcs can be serously delayed (Amercan Dabetes Assocaton 2002), so ths strategy s not consdered n the paper. At the tme of ths study, there was not accurate data and models to dentfy the absorpton dynamcs of the (rapd-actng) Aspart and (long-actng) Glargne nsuln (see Table 1), hence ther performance was not nvestgated and wll be objectve of future research. Intally, n type 1 dabetc patents, the amount of nsuln s calculated based on the patent weght, as 0.3 to 0.8 U per klogram. Ths amount s contnuously updated by the physcan n collaboraton wth the patent n order to reach an euglycemc control, and t could change accordng wth food consumpton, exercse, llness, stress, hormonal changes, travelng and any change of routne (APhA Specal Report 2001), (Amercan Dabetes Assocaton 2002). Hence t looks promsng and rewardng the dea of an automated nsuln adjustment algorthm for dabetc patents. 5. TYPE 1 DIABETIC MATHEMATICAL MODEL In ths secton, the mathematcal model of a type 1 dabetc patent s descrbed. For smplcty, ths model s presented n three parts: Insuln-Glucose Compartmental Model : the nsuln-glucose model used n ths work has a physologcal structure based on a compartmental technque (Sorensen 1985). Ths model departs from expermental evdence to formulate and valdate metabolc processes of the compartmental model on the whole organ and tssue level ncludng counter-regulatory effects. Thus, the nsuln-glucose model s governed by 19 nonlnear ordnary dfferental equatons, and s dvded nto three subsystems () Glucose,

4 () Insuln, and () Glucagon. The frst two subsystems were modeled for the bran, arteral system (heart/lungs), lver, gut, kdney, and perphery (muscle and adpose tssue) compartments. The glucagon was modeled as a sngle blood pool compartment. The system output s the perpheral ntersttal glucose, that permts to obtan accurate glucose levels. Glucose Input va Gastrc Emptyng : The amount of glucose n the gut followng the ngeston of a meal contanng Ch mlmoles of glucose equvalent carbohydrate s modeled as a frst order dfferental equaton (Lehmann and Deutsch 1992). In ths model, the rate of gastrc emptyng due to a meal s a functon of the amount of carbohydrates ntake Ch. Fnally, the glucose nput for a meal ntake s gven by a proporton of the glucose n the gut. Subcutaneous Insuln Injecton : Assume that an nsuln dose s njected subcutaneously. Hence the velocty of absorpton can be descrbed by a frst order non-lnear dfferental equaton that depends on of the dfferent types of nsuln: Lspro, Regular, NPH, Lente or Ultralente (Berger and Rodbard 1989). Fnally, the plasma nsuln concentraton due to the subcutaneous njecton s proportonal to the absorbed nsuln. It s assumed that the nsuln effect of prevous njectons s addtve,.e. the nsuln plasma concentraton depends on the combned effect of the actual and prevous dosages. Ths consderaton s not sgnfcant for rapd and short acton nsuln snce ts duraton s approxmately from 3 to 4 hours, and the doses are programmed n perods of 6 hours durng the day and 12 hours at nght. However, t s mportant for ntermedate and long acton nsuln snce ther duraton are from 10 to 18 hours. 6. PERFORMANCE ANALYSIS THROUGH SIMULATION Frst the performance of an open-loop strategy for nsuln nfusons was tested. The fve therapeutc regmens n Table 2 were analyzed based on a performance ndex that measured the error n mantanng an euglycemc control (BGC [70, 120] mg/dl). Next, the best regmens were smulated wth a closed-loop strategy usng a selftunng algorthm for dose adjustment (Campos- Delgado et al. 2004). The numercal smulaton were mplemented n MATLAB/Smulnk c. A total of 12 days (284 hrs.) were smulated wth three meals per day: Breakfast: 8:00 hrs., Lunch: 14:00 hrs., Dnner: 20:00 hrs. The meals carbohydrate ntakes were calculated accordng wth the followng profle: male, 30 years old, kg, 1.75 m, number of hours of sleep per day: 7, number of hours of very lght actvty: 4, number of hours of lght actvty: 9, number of hours of ntense actvty: 4, amount of calores per day: 3734 Cal/day. It s consdered that 50 % of the calores are comng from carbohydrates, and take that 4 calores are equvalent to 1 gr. of carbohydrates (CH). Consequently, t s needed 467 gr. of carbohydrates per day. Assumng a dstrbuton of ths amount of carbohydrates n three meals: 30 % breakfast, 45 % lunch and 25 % dnner, results n the next meal dstrbuton of carbohydrates: Breakfast: 1 gr. CH, Lunch: 210 gr. CH, and Dnner: 117 gr. CH. Therefore, the lunch s the heavest meal of the day accordng to Mexcan customs. Durng the smulaton tme, the amount of carbohydrate ntake per meal was vared around the nomnal values calculated prevously ±15%, but lookng to add up to 3734 Cal/day n average durng the smulaton nterval. Consequently, a varable meal carbohydrate ntake was tested durng smulaton. 6.1 Open-Loop Smulaton tme t Doses Assgment I1 I 2 Subcutaneous Infusons Dabetc Patent Fg. 2. Open-Loop Doses Assgnment. Glucose Measurement A total of 29 U/day ( 0.36 U/kg) were assgned for each nsuln formulaton, the dstrbuton was dfferent n every case but the total per day mantaned. In some cases, an ncrease n the total amount of nsuln per day wll mprove the BGC regulaton, however n some others (formulatons wth NPH nsuln), ths could nduce an hypoglycemc scenaro (BGC < mg/dl) due to the dynamcs of the nsuln. So, t was decded to mantan the total nsuln per day to 29 U/day for the ntal open-loop comparson. A total of sx nsuln dosages are defned: (1) I b 1: breakfast dose of rapd or short actng nsuln. (2) I l 1: lunch dose of rapd or short actng nsuln. (3) I d 1 : dnner dose of rapd or short actng nsuln. (4) I b 2: breakfast dose of ntermedate or long actng nsuln. G

5 Table 3. Open-loop Regmens for Performance Analyss. Formulaton I b 1 /Ib 2 I l 1 I d 1 /Id 2 J (U) (U) (U) Lspro/NPH 2.75/ / Lspro/Lente 3.0/ / Lspro/Ultralente 3.0/ / Regular/NPH 2.5/ / Regular/Ultralente 2.75/ / (5) I d 2 : dnner dose of ntermedate or long actng nsuln. The dagram n Fgure 2 was followed. The performance ndex (blood glucose devaton) was measured durng smulaton J = 1 T T 0 φ 2 (t)dt (2) where T represents the total smulaton tme, and φ(t) (pontwse devaton from EC) s defned as { G(t) 120 mg/dl G(t) > 120 mg/dl φ(t) = G(t) 70 mg/dl G(t) < 70 mg/dl (3) 0 70 G(t) 120 mg/dl wth G(t) representng the contnuous glucose concentraton. Durng the smulaton, T was set to 10 days (2 hours) n order to avod the effect of ntal condtons n the performance analyss. The results are presented n Table 3. Ths table shows that the best formulaton uses Ultralente or Lente nsuln as basal nsuln for ether Lspro or Regular nsuln. Consequently, these three combnatons wll be tested durng closed-loop smulaton n the next subsecton. 6.2 Closed-Loop Smulatons The best three formulatons obtaned durng the open-loop test: (a) Lspro-Lente, (b) Lspro- Ultralente, and (c) Regular-Ultralente wll be analyzed n a closed-loop fashon (see Fgure 3). The doses adaptaton s performed by reducng the error n the BGC from euglycemcs (Campos- Delgado et al. 2004). In ths control scheme, several glucose measurements are assumed to be avalable daly whch could be derved from blood samples, n-vvo sensors or non-nvasve means (Tamada et al. 2002), n order to compute a cost functon for the doses adaptaton. Snce the objectve of the paper s not to ntroduce the tunng algorthm, rather to analyze the dfferent nsuln formulatons, the detals of the algorthm are omtted and the reader s referred to (Campos- Delgado et al. 2004). The results are summarzed n Table 4. It s notceable that the the BGC s almost regulated to the EC durng the evaluaton tme wth Ultralente as basal nsuln, wth ether Regular or Lspro as fast-actng nsuln. There s not sgnfcant dfference between the Lspro/Ultralente or Regular/Ultralente formulaton. Note that the best strateges requre more BGC (mg/dl) BGC (mg/dl) Table 4. Closed-Loop Performance Analyss wth Self-Tunng. Formulaton Total Insuln per Day J (U) Lspro/Lente Lspro/Ultralente Regular/Ultralente nsuln per day, but due to the propertes of nsuln mxture, there s no occurrence of hypoglycema. Thus the self-tunng algorthm mproves the prevous performance wth the open-loop strategy, and t accomplshes a better BGC regulaton. BGC (mg/dl) EC days EC days EC days Fg. 4. Closed-Loop Smulatons (TOP) Lspro / Lente, (MIDDLE) Lspro / Ultralente, (BOTTOM) Regular / Ultralente.

6 Glucose Devaton Index J tme t Algorthm for Doses Adjustment I1 I 2 Subcutaneous Infusons Dabetc Patent Glucose Measurement G Fg. 3. Closed-Loop Adjustment Algorthm. 7. CONCLUSIONS AND FINAL REMARKS In ths paper several therapeutc regmens based on mult-doses strateges were analyzed. Flexble nsuln schemes were presented, where a combnaton of a rapd or short-actng nsuln s used to cover the postprandal glucose peaks due to meals, and an ntermedate or long-actng nsuln s used to provde a basal nsuln concentraton. Ther performance was evaluated n open-loop and closed-loop schemes. The best formulatons used an Ultralente type as basal nsuln n combnaton wth ether Lspro or Regular nsuln. The results showed n smulaton that a closed-loop scheme can provde an almost perfect BGC regulaton nto the euglycemc concentraton despte varable meal carbohydrate ntake. REFERENCES APhA Specal Report, New Approaches to Insuln Therpy for Dabetes, /pdf/ nsuln-therapy-sr.pdf, Amercan Pharmaceutcal Assocaton, (2001). Amercan Dabetes Assocaton, Insuln Admnstraton, Dabetes Care, 24(2001), Supplement 1 S5-S20. R. Belazz, G. Nucc y G. Cobell, The Subcutaneous Route to Insuln-Dependent Dabetes Therapy, IEEE Eng. n Medcne and Bology, 20(2001), M. Berger and D. Rodbard, Computer Smulaton of Plasma Insuln and Glucose Dynamcs after Subcutaneous Insuln Injecton, Dabetes Care, 12(1989), D.U. Campos-Delgado, R. Femat, E. Ruz-Velázquez and A. Gordllo-Moscoso, Knowledge-Based Controllers for Blood Glucose Regulaton n Type I Dabetc Patents by Subcutaneous Route, Proc. of the Int. Symp. on Intelgent Cont., Houston, 3-5 October (2003). D.U. Campos-Delgado, R. Femat, M. Hernández-Ordoñez and A. Gordllo-Moscoso, Self-tunng Insuln Adjustment Algorthm for Type I Dabetc Patents Based on Mult-Doses Regme., Int. Symp. on Robotcs and Automaton, Queretaro, Mexco, August (2004). E.R. Carson y T. Deutsch, A Spectrum of Approaches for Controllng Dabetes, IEEE Control System Magazne, 12(1992), L.M. Dckerson, Insuln Therapy n the Treatment of Dabetes Melltus, / Med%20Resdent/nsuln.pdf, (1999). F.J. Doyle III, B. Srnvasan y D. Bonvn, Run-to-Run Control Strategy for Dabetes Management, Proc. of the 23rd Annual EMBS Internatonal Conference, Istanbul, Turkey, October (2001). I.B. Hrsch, Type 1 Dabetes Melltus and the Use of Flexble Insuln Regmens, Amercan Famly Physcan, (1999), M. Hu, H. Du y S. Lng, A Dgtal Mnature Pump for Medcal Applcatons, IEEE Trans. on Mechatroncs, 7(2002), E.D. Lehmann and T. Deutsch, A Physologcal Model of Glucose-Insuln n Type I Dabetes Melltus, J. of Bomedcal Engneerng, 14(1992), E.D. Lehmann and T. Deutsch, Compartmental models for glycaemc predcton and decson-support n clncal dabetes care: promse and realty, Computer Methods and Programs n Bomedcne, 56(1998), S.M. Lynch y B.W. Bequette, Model Predctve Control of Blood Glucose n Type I Dabetcs Usng Subcutaneous Glucose Measurements, Proc. of the Amercan Control Conference, Anchorage, AK, May 8-10 (2002). R.S. Parker, F.J. Doyle III and N.A. Peppas, The Intravenous Route to Blood Glucose Control, IEEE Eng. n Medcne and Bology, 20(2001), W.R. Puckett, Dynamc Modellng of Dabetes Melltus, Ph.D. Dssertaton, Chemcal Eng. Department, Unversty of Wsconson-Madson, (1992). E. Ruz-Velazquez, R. Femat and D.U. Campos-Delgado, Blood Glucose Control for Type I Dabetes Melltus: A Robust Trackng H Problem, Control Engneerng Practce, 12(2004), T. Shmauch, N. Kuga, N. Nagata, and O. Takatan, Mcrocomputer-Aded Insuln Dose Determnaton n Intesfed Conventonal Insuln Therapy, IEEE Trans. on Bomedcal Engneerng, 35(1988), J.T. Sorensen, A Physologc Model of Glucose Metabolsm n Man and ts Use to Desgn and Assess Improved Insuln Therapes for Dabetes, Ph.D. Dssertaton, Chemcal Eng. Department, MIT, Cambrdge (1985). J.A. Tamada, M. Lesho y M.J. Terney, Keepng Watch on Glucose, IEEE Spectrum, 39(2002),