## Abstract

In healthy subjects some tissues in the human body display metabolic flexibility, by this we mean the ability for the tissue to switch its fuel source between predominantly carbohydrates in the postprandial state and predominantly fats in the fasted state. Many of the pathways involved with human metabolism are controlled by insulin and insulin-resistant states such as obesity and type-2 diabetes are characterised by a loss or impairment of metabolic flexibility. In this paper we derive a system of 12 first-order coupled differential equations that describe the transport between and storage in different tissues of the human body. We find steady state solutions to these equations and use these results to nondimensionalise the model. We then solve the model numerically to simulate a healthy balanced meal and a high fat meal and we discuss and compare these results. Our numerical results show good agreement with experimental data where we have data available to us and the results show behaviour that agrees with intuition where we currently have no data with which to compare.

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## References

Acheson KJ, Schutz Y, Bessard T, Anantharaman K, Flatt J, Jéquier E (1988) Glycogen storage capacity and de novo lipogenesis during massive overfeeding in man. Am J Clin Nutr 48:240–247

Adiels M, Westerbacka J, Soro-Paavonen A, Häkkinen AM, Vehkavaara S, Caslake MJ, Packard C, Olofsson SO, Yki-Järvinen H, Taskinen MR, Borén J (2007) Acute suppression of VLDL1 secretion rate by insulin is associated with hepatic fat content and insulin resistance. Diabetologia 50:2356–2365

Andersen V, Sonne J, Sletting S, Prip A (2000) The volume of the liver in patients correlates to body weight and alcohol consumption. Alcohol Alcohol 35:531–532

August E, Parker KH, Barahona M (2007) A dynamical model of lipoprotein metabolism. Bull Math Biol 69:1233–1254

Bennet DL, Gourley SA (2004) Asymptotic properties of a delay differential equation model for the interaction of glucose with plasma and interstitial insulin. Appl Math Comput 151:189–207

Bickerton AST, Roberts R, Fielding BA, Tornqvist H, Blaak AA, Wagenmakers AGM, Gilbert M, Humphreys SM, Karpe F, Frayn KN (2008) Adipose tissue fatty acid metabolism in insulin-resistant men. Diabetologia 51:1466–1474

Blakemore C, Jennett S (2001) The oxford companion to the body: glycogen. OUP, Oxford

Bock G, Chittilapilly E, Basu R, Toffolo G, Cobelli C, Chandramouli V, Landau BR, Rizza RA (2007) Contribution of hepatic and extrahepatic insulin resistance to the pathogenesis of impaired fasting glucose. Diabetes 56:1703–1711

Chew YH, Shia YL, Lee CT, Majid FAA, Chua LS, Sarmidi MR, Aziz RA (2009) Modelling of glucose regulation and insulin-signaling pathways. Mole Cell Endocrinol 303:13–24

Cobelli C, Federspil G, Pacini G, Salvan A, Scandellari C (1982) An intergrated mathematical model of the dynamics of blood glucose and its hormonal control. Math Biosci 58:27–60

den Boer MAM, Berbée JFP, Reiss P, van der Valk M, Voshol PJ, Kuipers F, Havekes LM, Rensen PCN, Romijn JA (2006) Ritover impairs lipoprotein lipase-mediated lipolysis and decreases uptake of fatty acids in adipose tissue. Arterioscler Thromb Vasc Biol 26:124–129

De Gaetano A, Arino O (2000) Mathematical modelling of the intravenous glucose tolerance test. J Math Biol 40:136–168

Engelborghs K, Lemaire V, Bélair J, Roose D (2001) Numerical bifurcation analysis of delay differential equations arising from physiological modelling. J Math Biol 42:361–385

Flatt JP (2004) Carbohydrate-fat interactions and obesity examined by a two-compartment computer model. Obes Res 12:2013–2022

Frayn KN (2003) Metabolic regulation: a human perspective. Blackwell Publishing, NJ

Hall KD (2006) Computational model of in vivo human energy metabolism during semistarvation and re-feeding. Am J Physiol 291:E23–37

Hall KD, Bain HL, Chow CC (2007) How adaptations of substrate utilization regulate body composition. Int J Obes 31:1378–1383

Hall KD (2010a) Mechanisms of metabolic fuel selection: modelling human metabolism and body weight change. IEEE Eng Med Biol Mag 29(1):36–41

Hall KD (2010b) Predicting metabolic adaptation, body weight change and energy intake in humans. Am J Physiol 298(3):E449–466

Hallgreen CE (2009) The interplay between glucose and fat metabolism: a biosimulation approach. PhD Thesis, Technical University of Denmark, Copenhagen

Harrison DE, Christie MR, Gray DWR (1985) Properties of isolated human islets of Langerhans: insulin secretion, glucose oxidation and protein phosphorylation. Diabetologia 28:99–103

Iozzo P, Hallsten K, Oikonen V, Virtunen KA, Kemppainen J, Solin O, Ferrannini E, Knuuti J, Nuutila P (2003) Insulin-mediated hepatic glucose uptake is impaired in type 2 diabetes: evidence for a relationship with glycemic control. J Clin Endocrinol Metab 88:2055–2060

Jordan PN, Hall KD (2008) Dynamic coordination of macronutrient balance during infant growth: insights from a mathematical model. Am J Clin Nutr 87:692–703

Kelley DE, Mandarino LJ (2000) Fuel selection in human skeletal muscle in insulin resistance. Diabetes 49:677–683

Kelley DE (2005) Skeletal muscle fat oxidation: timing and flexibility are everything. J Clin Investig 115:1699–1702

Kim J, Saidel GM, Cabrera ME (2007) Multi-scale computational model of fuel homeostasis during exercise: effect of hormonal control. Ann Biomed Eng 35:68–90

Klinke DJ (2008) Integrating epidemiological data into a mechanistic model of type 2 diabetes: validating the prevalence of virtual patients. Ann Biomed Eng 36:321–334

Kotronen A, Juurinen L, Tiikainen M, Vehkavaara S, Yki-Jrvinen H (2008) Increased liver fat, impaired insulin clearance, and hepatic and adipose tissue insulin resistance in type 2 diabetes. Gastroenterology 135:122–130

Lambeth MJ, Kushmerick MJ (2002) A computational model for glycogenolysis in skeletal muscle. Ann Biomed Eng 30:808–827

Li J, Kuang Y, Mason CC (2006) Modelling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays. J Theor Biol 242:722–735

Li Y, Soloman TP, Haus JM, Saidel GM, Cabrera ME, Kirwan JP (2010) Computational model of cellular metabolic dynamics: effect of insulin on glucose disposal in human skeletal muscle. Am J Physiol Endocrinol Metab 298:E1198–1209

Meyer C, Dostou JM, Welle SL, Gerich JE (2002) Role of human liver, kidney and skeletal muscle in the postprandial glucose homeostasis. J Physiol Endocrinal Metab 282:E419–E427

Mizuno A, Arai H, Fukaya M, Sato M, Hisami YO, Takeda E (2007) T Early-phase insulin secretion is disturbed in obese subjects with glucose intolerance. Metab Clin Exp 56:856–862

Olefsky JM (1981) Insulin resistance and insulin action: an in vitro and in vivo perspective. Diabetes 30:148–162

Pearson T, Wattis JAD, O’Malley BJ, Pickersgill L, Blackburn H, Jackson KG, Byrne HM (2009) Mathematical modelling of competitive LDL/VLDL binding and uptake by hepatocytes. J Math Biol 58:845–880

Perival V, Chow CC, Bergman RN, Ricks M, Vega GL, Summer AE (2008) Evaluation of quantitative models of the effect of insulin on lipolysis and glucose disposal. Am J Physiol Regul Integr Comp Physiol 295:R1089–1096

Peterson KF, Dufour S, Savage DB, Bilz S, Solomon G, Yonemitsu S, Cline GW, Befroy D, Zemany L, Kahn BB, Papademetris X, Rothman DL, Shulman GI (2007) The role of skeletal muscle insulin resistance in the pathogenesis of the metabolic syndrome. Proc Natl Acad Sci 104:12587–12594

Phielix E, Mensink M (2008) Type 2 diabetes mellitus and skeletal muscle metabolic function. Physiol Behav 94:252–258

Prager R, Wallace P, Olefsky JM (1986) In vivo kinetics of insulin action on peripheral glucose disposal and hepatic glucose output in normal and obese subjects. J Clin Investig 78:472–481

Pratt A, Wattis JAD, Salter AM (2014) Mathematical modelling of hepatic lipid metabolism.

Rijkelijkhuizen JM, Doesburg T, Girman CJ, Mari A, Rhodes T, Gastaldelli A, Nijpels G, Dekker JM (2009) Hepatic fat is not associated with \(\beta \)-cell function or postprandial free fatty acid response. Metab Clin Exp 58:196–203

Sedaghat AR, Sherman A, Quon MJ (2002) A mathematical model of metabolic insulin signaling pathways. Am J Physiol Endocrinal Metab 283:E1084–E1101

Song B, Thomas DM (2007) Dynamics of starvation in humans. J Math Biol 54:27–43

Taylor R, Magnusson I, Rothman DL, Cline GW, Caumo A, Cobelli C (1996) Direct assessment of liver glycogen storage by 13C nuclear magnetic resonance spectroscopy and regulation of glucose homeostasis after a mixed meal in normal subjects. J Clin Invest 97:126–132

Tindall MJ, Wattis JAD, O’Malley BJ, Pickersgill L, Jackson KG (2009) A continuum receptor model of hepatic lipoprotien metabolism. J Theor Biol 257:371–384

Tolic IM, Mosekilde E, Sturis J (2000) Modelling the insulin-glucose feedback system: the significance of pulsatile insulin secretion. J Theor Biol 207:361–375

Vicini P, Kushmerick MJ (2000) Cellular energetics analysis by a mathematical model of energy balance: estimation of parameters in human skeletal muscle. Am J Physiol Cell Physiol 279:213–224

Wattis JAD, O’Malley BJ, Blackburn H, Pickersgill L, Panovska J, Byrne HM, Jackson KG (2008) Mathematical model for low density lipoprotein (LDL) endocytosis by hepatocytes. Bull Math Biol 70:2303–2333

## Acknowledgments

TP acknowledges support from BBSRC and Unilever Corporate Research. The authors are grateful to the referees for their helpful comments on earlier versions of the manuscript.

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T. Pearson: (sadly died in March 2013, following a long illness).

## Appendices

### Appendix 1: Properties of Solutions of the Governing Equations

### 1.1 Uniqueness of Steady State

In this appendix we will prove the uniqueness of our steady states. We begin by writing down which variables each steady state is a function of (except \(I^*\)),

The expressions for \(A_b^ =f_4\) and \(A_l^*=f_3\) form a linear system for \(A_b^*\) and \(A_l^*\) which has a unique solution for each quantity in terms of \(I^*\) and \(G_b^*\), and hence the solutions can be written explicitly in terms of \(I^*\) using \(G_b^*=f_1(I^*)\). Now we substitute the expression for \(P^*=f_9\) into the equations for \(G_m^*=f_5\), \(A_m^*=f_6\), \(Y_m^*=f_7\), etc., to obtain a simpler system given by

Thus, all quantities can now be written as a function of \(I^*\), only \(T_m^*\) still depends on other quantities, and these can be eliminated using other expressions in (33), for example, \(T_m^*=\tilde{f}_8(I^*,\tilde{f}_6(I^*))\).

Considering the steady-state equation for insulin, we write down an expression for \(I^*\),

We now examine the steady-state values for \(G_b^*\) and \(A_b^*\) given in Table 2 and note that \(G_b\) terms contribute more significantly to the steady state than the \(A_b^*\) terms. Hence we temporarily neglect the \(A_b^*\) term. Under this assumption we substitute in the expression for \(G_b^*\) to find

The left-hand side of Eq. 35 is an increasing function of \(I^*\) for \(I^* > 0\), satisfying rhs \(=0\) when \(I^*=0\), and the right-hand side is a decreasing function of \(I^*\) for \(I^* > 0\), with rhs \(>0\) at \(I^*=0\); therefore, there is a unique solution for \(I^*\).

Given that \(I^*\) has a unique solution, we can now follow the chain of reasoning

Hence the system as whole has a unique steady-state solution.

Reinstating the \(A_b^*\) term theoretically could make the rhs of (35) *non*monotone, and so there could be multiple steady-states, however, for the physically realistic parameter values of interest to us, this does not occur.

### 1.2 Positivity of Solutions

Since all of the governing Eqs. (1)–(4), (7)–(14) have the form \(\mathrm{d}X/\mathrm{d}t = A - B X\) with \(A>0\), we can be sure that if \(X=0\) ever occurs, then \(X\) would increase, and so, provided we start with positive initial data, the concentrations will remain positive for all time.

Since the nonlinearities in the model are all analytic and have at most linear growth, the standard theory of ordinary differential equations implies uniqueness for the initial value problem.

### Appendix 2: NonDimensionalisation

Before we attempt to solve the system numerically, we nondimensionalise it so that we may reduce the number of parameters in the model. This process also allows for simpler numerical simulations. We rescale each variable by its steady-state value, except for \(Y_L\) which we rescale by \(Y_{max}\), and \(T_L\) which we rescale by a typical healthy liver fat concentration, denoted by \(T_L^H\). This means that we are now concerned with the following nondimensional variables

The forcing functions are nondimensionalised by \(F_G(t) = \beta _G \widehat{F}_G(\widehat{t})\) and \(F_T(t)=\beta _T \widehat{F}_T(\widehat{t})\), where the nondimensional forcing functions are given by

In terms of their nondimensional variables, the functions \(f_1\), \(f_2\) and \(f_3\) are given by

Using the above rescalings we obtain the following system of nondimensional equations where we have dropped the hats for convenience

where the new dimensionless parameters are given by

The values for the nondimensional parameters are given in Table 4.

### Appendix 3: Graphs of Fluxes

In this section we plot various combinations of the governing concentration variables to illustrate the evolution of the fluxes in the model. Noting that the three functions \(f_1,f_2,f_3\) are defined to be unity over the vast majority of their ranges, we only need to consider 10 combinations of concentrations of functions, which are plotted in Figs. 10 and 11.

The flux of glucose from plasma to muscle has a component which depends on insulin according to the product \(IG_b\), which we plot in the top left panel of Fig. 10. This quantity also influences the flux from plasma glucose to liver glycogen and liver FFA, see Eqs. (2), (7), (8) and (11). The flux from plasma TAG to adipose tissue as modelled in (3) depends on the product \(IT_b\), which is plotted in the top central panel. This is the only case where the high fat meal induces a higher response than the balanced meal. The product \(IA_L\) describes the rate of conversion of FFA into TAG in the liver (8)–(9).

In the muscle, both glucose and FFA are used to convert \(P\) into ATP, with rates that depend on \(PIG_m\) and \(PA_m\), respectively, as described by Eqs. (10), (11) and (13). These two fluxes are plotted in the lower left and lower centre plots in Fig. 10. The difference between balanced and high fat meals is more pronounced in the glucose term than the FFA term. The product \(P T_m\) determines the rate of conversion of TAG to FFA in muscle as described in Eqs. (13) and (14), (note that in our current parameterisation, \(k_{DI}=0\)).

In the top left panel of Fig. 11, we plot the rate of release of glucose into the plasma from liver glycogen, this is inhibited by insulin, which we have modelled by \(\beta _G/(1+k_{GL}I^2)\) in Eqs. (2) and (7). The top right panel shows the flow of FFA from adipose tissue to plasma, which has a similar form, but with \(k_{AA}\) replacing \(k_{GL}\), see Eq. (4). The flux from liver TAG to plasma TAG is shown in the lower left panel. This has weaker insulin dependence, being of the form \(\beta _T/(1+k_{TL}I)\) in Eqs. (3) and (9). The oxidation of hepatic FFA is shown in the lower right panel of Fig. 11, this has the form \(S_L A_L / (1+k_{AS}I)\), see Eq. (8). All four fluxes show significant reductions for the time that insulin is elevated, and not a great difference between the high fat mean and the balanced meal.

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Pearson, T., Wattis, J.A.D., King, J.R. *et al.* A Mathematical Model of the Human Metabolic System and Metabolic Flexibility.
*Bull Math Biol* **76**, 2091–2121 (2014). https://doi.org/10.1007/s11538-014-0001-4

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DOI: https://doi.org/10.1007/s11538-014-0001-4