BENG 221: Poblem Solving Poject Multiscale Model of Oxygen tanspot in Diabetes Decembe 1, 2016 Austin Budick Nafeesa Khan Sihita Rudaaju
Motivation Diabetes emains a significant health condition today, and affects not only key egulatoy systems within the body, but cetain bodily tissues as well. This disease is pehaps best associated with a distubance in the pocessing of suga consumed though food, due to eithe a lack of insulin o to a deceased efficiency of this homone within an affected peson [1]. Howeve, it is equally impotant to note that diabetes has demonstated effects within the composition of the micovasculatue, shown by thickening of the basement membane in human capillaies [2]. This alteation of the vessel stuctue can be linked to a deficiency in the amount of oxygen that is ultimately tanspoted to the suounding body tissue [3]. By modelling the physiological effects of diabetes on the capillaies, the effects of this disease can be bette undestood in ode to potentially mitigate its accompanying health isks. The scope of diabetes within the United States alone has been well established. The Centes fo Disease Contol and Pevention (CDC) have estimated that 29.1 million, o 9.3%, of the population have developed some fom of diabetes, with appoximately 8 million of this numbe emaining undiagnosed [4]. This condition has been elated to othe health isks as well, as death ates fom cadiovascula disease have been shown to be almost twice as high in affected individuals ove eighteen yeas old. Additionally, hospitalization ates fo instances of both heat attack and stoke ae almost twice as high in affected individuals ove twenty yeas old. The effects of diabetes also include significant medical costs, with $176 billion, ove two times the costs fo those without this disease, estimated fo aveage expenditues [4]. Poblem Statement Basement membane thickening is a significant change affected by the development of diabetes in both type I and type II vaieties. Because it is composed of poteins that ae closely elated to collagen, this membane expeiences an incease in potein synthesis when subjected to vaious types of damage caused by diabetes and its elated health conditions [2]. This state of the vessel bounday, as pictued in Figue 1, has been linked to seveal conditions esulting fom diabetes, such as neuopathy [3]. Additionally, 1
eseach [5] has yielded distibutions of membane thickness fo both diabetic patients as well as unaffected individuals. The esults, shown in Figue 2, display a significant incease in membane width that is enhanced with age 5. Futhe studies have implied that this incease in thickness is due to a lage tunove accompanied by a significant decease in the eventual degadation of the membane [2]. Figue 1: Images of capillaies with both nomal and inceased thickness (left) along with distibutions of membane aea in nomal and diabetic patients. [3] Figue 2: Effects of diabetes and aging on membane thickness compaed to unaffected individuals. [5] The effects of diabetes on the capillay basement membane width esult in multiple disuptions in the final steps of oxygen delivey. Afte passing the vessel wall following elease, diffusion is the pimay diving foce of oxygen to each the suounding tissue. Howeve, the distance that can be coveed by this pocess is extemely limited, equiing 2
Figue 3: Oxygen tanspot unde nomal and diabetic conditions. Distance between the individual capillaies is inceased, educing the effects of nomal diffusion. [7] close aangement of the capillay aay. In patients with diabetes, the space between individual capillaies is geatly inceased, while the limited ange of diffusion ceates egions in the tissue whee ischemia can develop (Figue 3). In this case, theapies might be necessay in ode to incease the patial pessue of oxygen within the vessels, estoing delivey [6]. Pio to diffusion outside the vessels, oxygen deficiency may also occu due to an incease in blood velocity due to diabetes. It has been epoted that duing an incease in membane width, a shunt may occu within the vessel netwok, petubing flow within the capillaies and peventing delivey to the suounding tissue due to less time fo oxygen extaction [3]. While this pompts an incease in blood flow to meet the needs of the tissue, hypoxia can aise in the adjacent cells [3]. This incease in fluid velocity has also been obseved in othe studies of diabetes, listing an intequatile ange fom 15.9 x 10 3 cm/s to 89.0 x 10 3 cm/s in those unaffected by diabetes, and a ange fom 68.4 x 10 3 cm/s to 21.8 x 10 2 cm/s in those with the condition [7]. Both the incease in distance between capillaies and the distubed flow within the vasculatue povide the basis fo a model that can epesent the extent of diabetes with espect to oxygen delivey. In each case, the final tansfe of oxygen to the body is inhibited, due to inceases in the flow of the blood itself as well as to inceased distances coveed by the basement membane. By taking the paametes found unde each of these conditions into account, oxygen movement within the capillaies can be chaacteized at each step. This is modelled fom the elease of molecules fom hemoglobin into an individual vessel, followed by passage though the suounding wall and basement membane, and finally though the tissue itself. In this way, the extent of change 3
to this pocess unde diabetic conditions can be bette undestood by way of a compehensive model. Model The model deived is based on oxygen tanspot at multiple stages within a vessel and its suounding tissues. In this instance, movement though the vessel itself, the vessel wall, and a potion of tissue beyond the wall has been accounted fo. The blood vessel is modeled as a single capillay whee hemoglobin eleases oxygen, which exits the cell as it entes into the plasma. Fom hee, oxygen tavels to the vessel wall, nomally expeiencing both convection and diffusion while moving towads a laye of endothelial cells, and subsequently diffusing futhe into the absobing tissue (Figue 4). Figue 4: Model of oxygen tanspot within a single capillay. This pathway depicted indicates oxygen elease fom hemoglobin and exit fom the cell, followed by flow down the vessel and diffusion to the vessel wall and suounding tissue. Within the developed model, cetain assumptions ae made to poduce a mathematical analysis of oxygen tanspot. The intestitial space between the vessel wall and the tissue has been disegaded. Assumptions ae detemined fo each stage of tanspot, and ae made based on easonable estimates as well as a focus on diffusion in a single dimension within the model. The following sections explain each stage in futhe detail. Blood vessel The assumptions consideed fo the blood vessel ae listed below. 1) Homogenous solution of hemoglobin 2) No diffusion in the axial diection 4
Figue 5: Model of oxygen tanspot within a blood capillay. 3) No flow in the adial diection 4) Hb, HbO 2 and O 2 ae in equilibium 5) Constant velocity of oxygen along the axial (z) diection 6) Constant diffusion coefficient D c Convection cuents ae not impotant fo ou model and tanspot of oxygen occus pimaily in the adial diection. Consideing these simplifications, the blood vessel is modeled as a small capillay [10]. Diffusion occus exclusively in the adial diection at a constant diffusivity D c and axial diffusion is ignoed. Deoxygenated hemoglobin and oxygen ae assumed to be in equilibium with oxygenated hemoglobin, and the concentation of hemoglobin [Hb] is held constant. Finally, the velocity of oxygen tanspoted is only consideed in the z-diection at a constant ate v. Fom these paametes, the following balance equations ae utilized fo hemoglobin and oxygen. +v. [O 2 ] = D O2 2 [O 2 ] R O2 (1) [HbO 2 ] +v. [HbO 2 ] = D HbO2 2 [HbO 2 ] R O2 (2) These equations ae simplified fo the case of steady state flow and oxygen tanspot in a cylindical conduit. Consideing neglected diffusion in axial (z), o flow diection and no flow in the adial diection (), the equations ae tanslated to as below. v [O 2] z = D O2 ( 2 [O 2 ] + 1 2 ) k [Hb][O 2 ]+k[hbo 2 ] (3) 5
v [HbO 2] z = D HbO2 ( 2 [HbO 2 ] 2 + 1 [HbO 2 ] )+k [Hb][O 2 ] k[hbo 2 ] (4) In ode to obtain the analytical solution, the poblem is futhe simplified. Fluid velocity is only consideed in the adial diection, oxyhemoglobin and oxygen ae in local chemical equilibium (o the ate of eaction is zeo), and diffusion only in the adial diection. No net oxygen exchange was assumed acoss the bounday. Thus, zeo flux bounday conditions ae consideed at both boundaies of capillay. The initial concentation of oxygen in the capillay is modeled on the basis of fluid velocity inside a tube. Consequently, the concentation is highest at the cente of the capillay C 0 and is zeo at the bounday, vaying adially in a linea fashion. The above assumptions lead to the following equation. = D c ) (5) Bounday conditions: (0,t) = 0 ( cw,t) = 0 (6) Initial condition: [O 2 ](,0) = C 0 (1 cw ) (7) Vessel wall Figue 6: Model of oxygen tanspot within the vessel wall. The assumptions fo this stage ae as follows: 6
- Diffusion only occus in the adial diection - Unifom consumption of oxygen by mitochondia in endothelial cells - Constant diffusivity D w Simila assumptions as the capillay ae made fo the vessel wall. Additionally, a constant consumption tem of oxygen by mitochondia is consideed. This consumption tem is epesented as M w. A simple way of connecting the thee stages of oxygen tanspot is devised. The steady state value of oxygen concentation is used as a bounday condition fo the consequent stage. Thus, steady state oxygen concentation deived fom the capillay stage is used as a bounday condition hee. Zeo flux condition is consideed at the othe bounday. It is also assumed that thee is no oxygen pesent in the vessel wall initially. These assumptions lead to the following equations. = D w ) M w (8) Bounday conditions: [O 2 ]( cw,t) = C 0 0.5 ( wt,t) = 0 (9) Initial condition: [O 2 ](,0) = 0 (10) Tissue Figue 7: Model of oxygen tanspot within the suounding tissue. The assumptions consideed fo the suounding tissue ae as given: - Diffusion only occus in the adial diection - Unifom consumption of oxygen by mitochondia in endothelial cells 7
- Constant diffusivity D t Assumptions fo the suounding tissue ae simila to those consideed in the vessel wall. Steady state oxygen concentation in the vessel wall is used as a bounday condition hee. It is assumed that thee is no oxygen pesent in the tissue initially. The equations ae as follows. = D t ) M t (11) Bounday conditions: [O 2 ]( wt,t) = M w 4D w 2 wt +C 0 0.5 ( t,t) = 0 (12) Initial condition: [O 2 ](,0) = 0 (13) Analytical Solution Blood vessel = D c ) (14) Bounday conditions: (0,t) = 0 ( cw,t) = 0 (15) Initial condition: [O 2 ](,0) = C 0 (1 cw ) (16) This simplified model esembles a homogenous patial diffusion equation in the cylindical coodinates. It can be solved using sepaation of vaiables. u(,t) = X()T(t) (17) Substituting this new definition into the homogenous PDE and eaanging esults in one time-dependent equation and one space-dependent equation. Bessel functions ae 8
used fo the solution. T(t) = Ce Dλ2 t (18) 2 2 X 2 + X +2 λ 2 x = 0 (19) X() = AJ 0 (λ) (20) Applying the bounday conditions and using local extemes of Bessel function J n (0) = 0 J n (0) = 0 (21) X = λ n0 (22) Using the pinciple of supeposition, the final solution to the homogenous equation is the weighted sum of all possible solutions: u(,t) = A n J 0 (λ n )e Dλ2 t n=1 (23) To solve fo the constant A n, the initial condition is applied to the above solution. The esult is integated ove the adius of the capillay. u(,0) = u 0 () = A n J 0 (λ n ) (24) n=1 cw 0 u 0 ()J 0 (λ m )d = cw A n J 0 (λ n )J 0 (λ m )d (25) n=1 0 cw 0 u 0 ()J 0 (λ m )d = A m 2 cw 2 J 1(λ m cw ) 2 (26) A m = 2C cw 0 (1 0 cw )J 0 (λ m )d (27) cwj 2 1 (λ m cw ) 2 Solving fo integals of the above Bessel functions becomes complicated. Hence, the 9
same setup is solved in Catesian coodinates. The solution is as follows. O 2 (,t) = (C 0 0.5)+ n=1 2C 0 (nπ) 2(1 cos(nπ))cos(nπ cw )e Dc( 2 nπcw)t (28) Figue 8: Analytical and numeical solution fo blood vessel modeled as capillay in Catesian coodinates. Vessel wall Bounday conditions: = D w ) M w (29) [O 2 ](0,t) = C 0 0.5 (L w,t) = 0 (30) Initial condition: [O 2 ](,0) = 0 (31) Fo the paticula solution, a steady state condition was assumed to aive at the solution. 0 = D w ) M w (32) 10
u p (,t) = M w 4D w 2 +C 0 0.5 (33) The homogenous patial diffusion equation in cylindical coodinates can be solved using sepaation of vaiables. u h (,t) = X()T(t) (34) Substituting this new definition into the homogenous PDE and eaanging esults in one time-dependent equation and one space-dependent equation. Bessel functions ae used fo the solution. T(t) = Ce Dλ2 t (35) 2 2 X 2 + X +2 λ 2 x = 0 (36) X() = AJ 0 (λ) (37) Applying the bounday conditions and using local extemes of Bessel function J n (0) = 0 J n (0) = 0 (38) X = λ n0 (39) Using the pinciple of supeposition, the final solution to the homogenous equation is the weighted sum of all possible solutions: u h (,t) = A n J 0 (λ n )e Dλ2 t n=1 (40) To solve fo the constant A n, the initial condition is applied to the sum of paticula and homogenous solution. The esult is integated ove the width of vessel wall. u(,0) = u 0 () = A n J 0 (λ n ) (41) 11 n=1
Lw u 0 ()J 0 (λ m )d = 0 Lw A n J 0 (λ n )J 0 (λ m )d (42) n=1 0 Lw cw 2 u 0 ()J 0 (λ m )d = A m 0 2 J 1(λ m L w ) 2 (43) u(,t) = n=1 2 ( J 2(λ m L w J 3 (λ m L w )) 1 +(C L 2 0 0.5)L w w λ m J 1 (λ m L w ) λj 1 (λ m L w ) )J 0(λ m L w )e Dwλmt (44) Tissue Bounday conditions: = D t ) M t (45) [O 2 ](0,t) = M w 4D w 2 cw +C 0 0.5 (L t,t) = 0 (46) Initial condition: [O 2 ](,0) = 0 (47) Fo the paticula solution, a steady state condition was assumed to aive at the solution. 0 = D t ) M t (48) The solution is simila to that of vessel wall. u(,t) = n=1 2 L 2 t u p (,t) = M t 4D t 2 +C 0 0.5 (49) ( J 2(λ m L t J 3 (λ m L t )) +( M w 2 1 λ m J 1 (λ m L t ) 4D wt+c 0 0.5)L t w λj 1 (λ m L t ) )J 0(λ m L t )e Dtλmt (50) 12
Table 1: Values of Paametes Constant Desciption V alue v O 2 velocity in capillay 0.1 cm/s [10] v d O 2 velocity in capillay in diabetes 0.12 cm/s [7] C o Initial O 2 concentation in capillay 1.62 x 10 4 g/ml [10] [Hb] Deoxyhemoglobin Concentation in capillay 0.55 g/ml [10] [HbO 2 ] Hemoglobin Concentation in capillay 0.34 g/ml [10] k p Oxygen and Hemoglobin association ate 30 x 10 6 /Ms [11] k Oxygen dissociation ate 20 x 10 6 /Ms [11] D c Diffusivity of O 2 in capillay 1.62 x 10 5 cm 2 /s [10] D w Diffusivity of O 2 in vessel wall 8.73 x 10 6 cm 2 /s [10] D t Diffusivity of O 2 in tissue 2.41 x 10 5 cm 2 /s [10] cw Inne adius of vessel wall 4.0 x 10 4 cm [10] wt Oute adius of vessel wall 6.0 x 10 4 cm [8] wt,d Oute adius of vessel wall in diabetes 9.0 x 10 4 cm [8] t Oute adius of tissue 15.0 x 10 4 cm (abitay) M w O 2 consumption in vessel wall 5.0 x 10 3 mlo 2 /ml/s [12] M t O 2 consumption in tissue 1.58 x 10 4 mlo 2 /ml/s [12] Numeical Model The numeical validation is computed using MATLAB and its built-in PDE solve. Additional factos not included in the analytical solution ae accounted fo including: velocity of blood flow in the z-diection, dissociation ates of hemoglobin and oxygen, and coupling of bounday conditions between the diffeent sections of the model. The equations used fo the numeical validation ae as follows and paametes ae those listed in the table. Bounday and initial conditions mentioned in the model desciption have been used. Blood vessel: v [O 2] z = D c ( 2 [O 2 ] + 1 2 )+R O 2 (51) R O2 = k[hbo 2 ] k p [Hb][O 2 ] (52) Vessel wall: = D w ) M w (53) 13
Suounding tissue: = D t ) M t (54) In diabetic patients, the width of the vessel wall is lage compaed to healthy condition. The following gaphs illustate the levels of oxygen concentation in diabetic condition in compaison to nomal vessel wall width. In ode to bette visualize the oxygen concentation diffeences between diseased and healthy conditions, 2D gaphs wee plotted epesenting oxygen levels in each stage. Figue 9: Plots of oxygen tanspot within healthy and diabetic capillaies. Figue 10: Plots of oxygen tanspot though the vessel wall unde nomal and diabetic conditions. Figue 11: Plots of oxygen tanspot though the suounding tissue unde nomal and diabetic conditions. 14
Figue 12: Oxygen tanspot cutoff within healthy and diabetic tissue at 0.4 seconds. Figue 13: Oxygen tanspot within the capillay. Plots indicate the tanspot ove time. 15
Conclusion and Futue Wok We constucted a multi-scale model of oxygen diffusion fom a capillay into suounding tissue to show the effects of diabetes on oxygen availability. In the analytical model we simplified the both geomety and diffeent stages involved in the pocess in addition to utilizing steady state solutions as bounday conditions fo the following step in the tanspot of oxygen to the suounding tissue. The effect of velocity incease in the diabetic condition on oxygen availability can be seen in the capillay plots of figue one. Inceased velocity (by appoximately 120%) in diabetes does not decease the amount of oxygen available at the ed blood cell (RBC) site (Z= 0 cm) (seen in figue 13 and plot 3 of figue 9) but does decease the amount of oxygen futhe away (in ou case Z= 0.5cm, figue 9 plot 3). This diffeence in oxygen is caied though to the vessel wall and tissue (figues 10 and 11). The steady state value of the pevious stage was taken as the left bounday condition in these calculations. As can be seen, the oiginal diffeence in oxygen caused by velocity incease in diabetes caies though to the tissue. Howeve, it is notable that the shape of the oxygen availability cuve at the inteface of the vessel wall and tissue (the thid plot of figue 10), changes due to a thicke membane. Figue 12 shows the compounded effect of capillay membane thickening and inceased velocity, whee a cutoff time is set at.04 seconds of diffusion in the vessel wall and is used as the left bounday condition in the tissue stage. This eflects that oxygen elease by RBCs would come in pulses and diffusion would not go to steady state. In that case, the thicke membane in addition to inceased capillay velocity deceases the amount of oxygen available to tissue. Finally, figue 13 shows the isolated effect of wall thickening on oxygen availability. Plots one though thee demonstate that even with velocity being highe in the diabetic condition, oxygen concentation is still the same at Z=0 (site of the RBC. Howeve, even if the same amount of oxygen is available to the vessel wall between the healthy and diabetic case, the inceased thickness still educes oxygen availability (seen by the gap between the ed and blue line in the fouth plot. In the numeical model, we wee able to account fo moe factos involved in the oxygen tanspot pocess and showed a decease in the oxygen available to tissues duing 16
diabetes by using liteatue-based changes in blood velocity and wall thickness. Consequently, less oxygen is available in the second step duing tanspot though the capillay wall, whee anothe confounding facto in diabetes, basement membane thickening, futhe educes the amount of oxygen that eaches the tissue. The diffeence in oxygen available to tissue at intefaces of the stages and at the end can be seen in figues one though thee. Effectively, in the diabetic condition, oxygen is moving too quickly in the capillay to popely diffuse to the est of the body, which is futhe hindeed by a geate distance within the vessel wall that must be passed befoe eaching the tissue. Chonic eduction in oxygen supply could be eithe coelative o causative with diabetes symptoms and complications. A pimay agument fo this would be a change in cell metabolism in esponse to deceased oxygen. Ostegaad et al. obseved an incease in neuopathy in diabetic patients due to these changes in blood flow, which limited not only oxygen availability to tissue but glucose as well3. While we applied this model to the case of diabetes, it could also be used to analyze othe disease states, such as anemia o sickle cell disease. In addition, modifications could be made to model tanspot of othe nutients o molecules, such as glucose, CO 2 o CO. Multiple complex chemical, geometic and anatomical aspects of oxygen tanspot wee eithe not included o wee simplified fo this model. These included assumptions egading the natue of blood flow, simplifications in oxygen and hemoglobin dissociation kinetics, and simplified anatomy of the capillay (the intestitial space was not included). In ode to build a moe ealistic model we would want to account fo these aspects as well as the discete natue of oxygen diffusion fom individual ed blood cells, instead of a continuum. Ou model shows the decease in oxygen made available to tissue in the diabetic case compaed to the healthy model. This is due to the obseved incease in blood velocity though capillaies in diabetes as well as basement membane thickening. Theefoe oxygen has less time to diffuse though a given section of the capillay wall, and must also tavel a geate distance befoe eaching the tissue. We believe that by taking this step to undestanding the mechanisms of this disease, we povide a basis fo continued eseach into moe effective theapeutics fo those affected. We hope to continue this study to gain futhe insight into inceasingly complex scenaios povided by diabetes. 17
Refeences [1] Kelly, Jane. Diabetes. Centes fo Disease Contol and Pevention, 2016. Accessed 24, Oct. 2016. [2] Williamson, J.R., and C. Kilo. Capillay Basement Membane in Diabetes. Diabetes vol. 32, no. 2, 1983, pp. 96-100. Accessed 26, Oct. 2016. [3] Ostegaad, Leif et al. The effects of capillay dysfunction on oxygen and glucose extaction in diabetic neuopathy. Diabetologia, 2014, pp. 666-77. Accessed 26, Oct. 2016. [4] National Diabetes Repot, 2014. Centes fo Disease Contol and Pevention, 2014. Accessed 27, Oct. 2016. [5] Kilo, Chales, Nancy Vogle, and Joseph R. Williamson. Muscle Capillay Basement Membane Changes Related to Aging and to Diabetes Mellitus. Depatments of Medicine and Pathology, Washington Univesity School of Medicine, 1972, pp. 881-905. Accessed 25, Oct. 2016. [6] Boulton, Andew J. The Diabetic Foot, an Issue of Medical Clinics. Medical Clinics of Noth Ameica, vol. 97, no. 5, 2013, pp. 959-60. Accessed 24, Oct. 2016. [7] Flynn, M.D., M.E. Edmonds, J.E. Tooke, and P.J. Watkins. Diect measuement of capillay blood flow in the diabetic neuopathic foot Diabetologia, 1989. Accessed 27, Oct. 2016. [8] Hellums, J. David, et al. Simulation of intaluminal gas tanspot pocesses in the micociculation. Annals of biomedical engineeing 24.1 (1995): 1-24. [9] Vadapalli, Ajun, Roland N. Pittman, and Aleksande S. Popel. Estimating oxygen tanspot esistance of the micovascula wall. Ameican Jounal of Physiology-Heat and Ciculatoy Physiology 279.2 (2000): H657-H671. [10] Vadapalli, Ajun, Daniel Goldman, and Aleksande S. Popel. Calculations of oxygen tanspot by ed blood cells and hemoglobin solutions in capillaies. Atificial Cells, Blood Substitutes, and Biotechnology 30.3 (2002): 157-188. [11] Vandegiff, K. D., et al. Detemination of the ate and equilibium constants fo oxygen and cabon monoxide binding to R-state human hemoglobin coss-linked between the alpha subunits at lysine 99. Jounal of Biological Chemisty 266.26 (1991): 17049-17059. 18