Delay Differential Equations inBiology and Medicine

Yang Kuang

Department of Mathematics and Statistics

Arizona State University

http://math.asu.edu/~kuang

Feb. 11, 2005. Penn State – p.1/63

Delay Differential Equations (DDE) inBiology and Medicine

...Recent theoretical and computational advancements in

DDEs reveal that DDEs are capable of generating rich

and intriguing dynamics in realistic parameter regions. In

this talk, through several examples in ecology (staged

predator-prey interaction and marine bacteriophage in-

fection dynamics) and medicine (glucose-insulin interaction

and tumor growth), we show that naturally occurring com-

plex dynamics are often naturally embodied in DDEs.

Feb. 11, 2005. Penn State – p.2/63

A Stage Structured Predator-PreyModel

Importance of Resource Dynamics and MaturationDelay

Stephen A. Gourley and Yang Kuang: Astage structured predator-prey modeland its dependence on maturation delayand death rate. J. Math. Biol., 49,188-200 (2004).

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Introduction and preliminaries

Many consumer species go through two or more life

stages as they proceed from birth to death. In order to

capture the oscillatory behavior often observed in na-

ture, various mechanisms are proposed. Such mecha-

nisms include delay differential models. However, The

discrete delay logistic equation is ill formulated and pro-

duces unrealistic complex dynamics.

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Unrealistic dynamics

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8Solutions of delayed logistic equation x′=rx(1−x(t−τ))

time t

x(t)

τ=1τ=2τ=3

Solutions of x′ = rx(1 − x(t − τ)) for r = 1, τ = 1 (stable),τ = 2 and τ = 3 (peak to valley ratio is well over 2000).

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Realistic dynamics?

Solutions of Nicholson’s BLOWFLIES (Nature,287(1980), 17-21) model

x′ = px(t − τ)e−ax(t−τ)− dx(t)

exhibit plausible and rich dynamics.

So, are BLOWFLIES birth regulated???

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Aiello and Freedman Model

The work of Aiello and Freedman on a single speciesstage-structured model has received much attention inthe more mathematically oriented literature. Itassumes the population, like the standard logisticequation, is death regulated.

y′ = b e−djτy(t − τ) − my2,

y′

j = by(t)p(x(t)) − b e−djτy(t − τ)p(x(t − τ)) − dj yj,

y(θ) ≥ 0 is continuous on − τ ≤ θ < 0, x(0), y(0), yj(0) > 0.

(1)

Feb. 11, 2005. Penn State – p.7/63

More...

Their model predicts a positive steady state as the glo-

bal attractor, thereby suggesting that stage structure

does not generate the sustained oscillations observed

in nature in single species population growths. Subse-

quent works by other authors suggest that the time de-

lay to adulthood should be state dependent and careful

formulation of such state dependent time delays can

lead to models that produce periodic solutions .

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Our Assumptions

We assume that1): The prey or the renewable resource, denoted by x,can be modeled by a logistic equation when theconsumer is absent.2): The predators or consumers is divided into twostage groups: juveniles and adults and they aredenoted by yj and y respectively below.3): Only adult predators are capable of preying on theprey species and that the juvenile predators live onother resource.

Feb. 11, 2005. Penn State – p.9/63

Our model

We have the following two-stage predator-preyinteraction model:

x′ = rx(1 − x/K) − yp(x),

y′ = b e−djτy(t − τ)p(x(t − τ)) − dy − my2,∫ 0

τp(x(s))ds = M,

y(θ) ≥ 0 is continuous on − τ ≤ θ < 0, and x(0), y(0) > 0.

(2)

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Scaled Model

Both r and K can be easily scaled off.

x′ = x(1 − x) − yp(x),

y′ = b e−djτy(t − τ)p(x(t − τ)) − dy − my2,∫ 0

τp(x(s))ds = M.

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p(x)

The function p(x) is assumed to be differentiable andsatisfy p(0) = 0,p(x) is strictly increasing, p(x)/x isbounded for all x ≥ 0.Reasonable choices for p(x) include the casesp(x) = px (p > 0 constant), and p(x) = px/(1 + ax)(p, a > 0).

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Equilibria and their feasibility

Assume FIRST that τ is a positive constant.Apart from the zero solution, system (11) always has(x, y) = (1, 0) as an equilibrium. The components ofany interior equilibrium must satisfy

y =x(1 − x)

p(x), y =

be−djτp(x)

m−

d

m.

Fig. 2 shows the x- and y- isoclines and it is clear thatan interior equilibrium will exist if and only if

b e−djτp(1) > d.

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Positive Equilibrium

x

y

0

x−isocline

the y−isocline,as τ is increased

(x*,y*)

1

The x and y-isoclines of system (2) for various values of τ

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Global stability of the equilibrium(1, 0)

We will need the following result, which is a directapplication of Theorem 4.9.1 in Kuang (1993, p159).The solution of the equation

u′(t) = au(t − τ) − bu(t) − c u2(t), (1)

where a, b, c, τ > 0, and u(t) > 0 for −τ ≤ t ≤ 0, satis£esthe following:

(a) if a > b then limt→∞ u(t) = a−bc

;

(b) if a < b then limt→∞ u(t) = 0.

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Global stability of the equilibrium(1, 0)

Theorem 1 Assume that be−djτp(1) ≤ d. Thensolutions of (2) satisfy x(t) → 1, y(t) → 0 as t → ∞.

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Density-independent death rate

Assume now m = 0. We shall carry out the analysis forgeneral p(x) as far as possible, but will concentrate onthe case p(x) = px later to enable further analyticprogress and for the purposes of numerical simulation.We begin by examining the linear stability of theequilibrium E = (x∗, y∗), assuming of course that

be−djτp(1) > d

so that the equilibrium is feasible. If dj > 0 (i.e. there ismortality among the juveniles) then, as one increasesthe delay τ , the equilibrium loses feasibility at aFINITE value of τ .

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Linearization

Let us linearize (2) at the interior equilibrium (x∗, y∗).Setting x = x∗ + u, y = y∗ + v where u and v are small,and linearising, gives

u′(t) = (1 − 2x∗ − y∗p′(x∗))u(t) − p(x∗)v(t),

v′(t) = be−djτy∗p′(x∗)u(t − τ) + be−djτp(x∗)v(t − τ) − dv(t).

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Linearization

Non-trivial solutions of the form (u, v) = (c1, c2) exp(λt)exist if and only if G(λ, τ) = 0, where

G(λ, τ) = λ2 + (2x∗ + y∗p′(x∗) − 1 + d − be−djτ p(x∗)e−λτ )λ

+ (2x∗ + y∗p′(x∗) − 1)(d − be−djτ p(x∗)e−λτ ) + be−djτ y∗p(x∗)p′(x∗)e−λτ .

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Stability Switch

Our immediate interest is the possibility of stabilityswitches as τ is increased from zero. We shall assumethat the interior equilibrium is stable when τ = 0. Whenτ = 0, the characteristic equation is

λ2 + (2x∗ + y∗p′(x∗) − 1)λ + by∗p(x∗)p′(x∗) = 0

(we have used the fact that when τ = 0 theequilibrium’s components are related by bp(x∗) = d).

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DIFFICULTY

In our situation there are two main complications.1): When one searches for purely imaginary rootsλ = ±iω of the characteristic equation, the polynomialequation that one obtains for ω still has τ in itsCOEFFICIENTS.2): The equilibrium components x∗ and y∗ involve τ . Itis important to know the dependence of thesecomponents on τ , yet often the components cannot becomputed explicitly.

Feb. 11, 2005. Penn State – p.21/63

Method of Beretta and Kuang

Implementation of the technique of Beretta andKuang (2002) (Geometric stability switch criteria in delay differential systems

with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.)requires the explicit expressions for x∗ and y∗. Weassume p(x) = px, p > 0 constant.

(x∗, y∗) =

(

dedjτ

bp,

bpe−djτ − d

bp2e−djτ

)

.

Typically, the interior equilibrium loses stability andthen regains stability at a larger τ , before disappearingat τ = (1/dj) ln(bp/d) ≈ 3.

Feb. 11, 2005. Penn State – p.22/63

Stability Switch Figure

Graph of stability switch for S_0(T) and S_1(T), here T=tau

–14

–12

–10

–8

–6

–4

–2

0

2

y

0.5 1 1.5 2 2.5 3T

p = 1, b = 10, dj = 1, d = 0.5 and m = 0. The equilibrium isfeasible for 0 ≤ τ < (1/dj) ln(bp/d) ≡ τe ≈ 3.

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Simulation Results

Thanks to DDE23.

0 100 200 3000

0.5

1

1.5

2

2.5

3

3.5τ=0.1

x, y

preypredator

0 100 200 3000

1

2

3

4

5τ=0.6

x, y

preypredator

0 100 200 3000

0.5

1

1.5

2

2.5

3τ=1.2

time t

x, y

preypredator

0 100 200 3000

0.5

1

1.5

2τ=1.8

time t

x, y

preypredator

A solution of (2) with p(x) = px, x(θ) = 0.3, y(θ) = 1, θ ∈ [−τ, 0] where p = 1.0, b = 10,

dj = 1.0, d = 0.5, and τ varies from 0.4 to 1.4.

Feb. 11, 2005. Penn State – p.24/63

Simulation Results for dj = 0

Graph of stability switch by S_0(T), S_1(T) and S_2(T), here T=tau

–20

–10

0

10

20

y

5 10 15 20 25 30

T

Each intersection point of these lines with the T = τ axisprovides the threshold values at which the number of roots withpositive real parts of the characteristic equation about thepositive steady state jumps by 2.

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Simulation Results for dj = 0

0 100 200 300 4000

1

2

3

4τ=0.1

x, ypreypredator

0 100 200 300 4000

2

4

6

8τ=5

x, y

preypredator

0 100 200 300 4000

2

4

6

8τ=15

time t

x, y

preypredator

0 100 200 300 4000

2

4

6

8τ=20

time t

x, y

preypredator

A solution of model (2) with p(x) = px, x(θ) = 0.3, y(θ) = 1,θ ∈ [−τ, 0] where p = 1.0, b = 10, dj = 0 d = 0.5 and τ variesfrom 0.25 to 25.

Feb. 11, 2005. Penn State – p.26/63

Model implications: constant τ

Our linear stability work shows that if the resource isdynamic, then there is a window in time delayparameter values that provides sustainable oscillatorydynamics. This suggests that resource dynamics mustbe explicitly modelled.

Feb. 11, 2005. Penn State – p.27/63

Model implications: constant τ

However, in the unlikely case that juveniles do not suf-

fer any mortality (dj = 0), the oscillatory dynamics will

persist and gain complexity (in the sense that the num-

ber of characteristic roots with positive real parts incre-

ases) when we increase the delay τ . Such distinct dyna-

mical outcomes highlight the importance of incorporating

the juvenile death rate in stage structured population mo-

dels.

Feb. 11, 2005. Penn State – p.28/63

Model implications:state dependent τ

Thanks to a program written by Zdzislaw on May 15.Observation 1: Longer maturation time delay (largervalues of M) stabilizes the interaction.

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

t

x,y

r=4 K=1 b=3 d=0.3 dj=0.6 m=0.8

xy

0 10 20 30 40 50 60 70 80 90 1001

1.5

2

2.5

3

3.5

t

τ

Feb. 11, 2005. Penn State – p.29/63

Model implications:state dependent τ

Observation 2: Maturation time delay does notgenerate complex dynamics.

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

t

x,y

r=4 K=1 b=3 d=0.3 dj=0.6 m=0.6

xy

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

t

τ

Feb. 11, 2005. Penn State – p.30/63

Model implications:state dependent τ

Observation 3: Maturation time delay MAY not workwhen the dynamics is far away from the steady state.Death rate shall be resource dependent.

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

t

x,y

r=4 K=1 b=3 d=0.3 dj=0.6 m=0.3

xy

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

t

τ

Feb. 11, 2005. Penn State – p.31/63

Model implications:state dependent τ

Observation 4: Maturation time delay cut transitiontime.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.5

1

1.5

2

2.5

3

3.5

x

y

Feb. 11, 2005. Penn State – p.32/63

Stoichiometry Of Tumor Dynamics:Models And Analysis

Y. Kuang, J. Nagy and J. Elser: Disc. Cont. Dyn. Sys.,series B, 4, 221-240. 2004.

Feb. 11, 2005. Penn State – p.33/63

Summary

We integrate natural selection driven by competitionfor resources, especially phosphorus P , intomathematical models consisting of delay differentialequations.

We show that tumor population growth and sizes are

more sensitive to total phosphorous amount than their

growth rates. If an mechanism (treatment) can half the

tumor cells’ P uptake, then it may lead to a three quar-

ter reduction in ultimate tumor size, indicating the po-

tential of such a treatment.

Feb. 11, 2005. Penn State – p.34/63

Tumor growth with nutrient limitation

The phosphorus is distributed into two compartments:

extracellular and intracellular, which contains fraction

f (approximately 2/3) of the total FLUID within a ty-

pical organ (Ganong 1999). Let n represent mean

phosphorus content (millimoles/g) in 1 gram of healthy

cells, within the tumor stroma. Similarly, let m be mean

phosphorus content of 1 gram of tumor tissue. Then

P −nx−my−nz ≡ Pe is extracellular phosphorus within

the organ.

Feb. 11, 2005. Penn State – p.35/63

Homogeneous Tumor

For healthy cells, if the extracellular phosphorus con-

centrationPe

fkh

(millimoles/kg) =Pe

fkh

(millimoles/g) is

less than n, the mean phosphorus content (millimo-

les/g) in healthy cells, then the per capita growth rate

becomes aPe/fnkh. The tumor growth rate decreases

to bPe

fmkh

when concentration of extracellular phospho-

rus drops below m.

Feb. 11, 2005. Penn State – p.36/63

Homogeneous Tumor

The tumor proliferation rate begins to decreasewhenever vascularization drops below a certainthreshold. In particular, whenever g(z − αy) < 1, thenthe maximum proliferation rate of the tumor becomesg(z − αy), where α represents the mass of tumor tissuethat one unit of blood vessels can just barely maintain,and g measures sensitivity of tumor tissue to lack ofblood.The tumor’s growth rate decelerates as it approachesits carrying capacity kt.

Feb. 11, 2005. Penn State – p.37/63

Homogeneous Tumor

We assume that new microvessels arise from activated

vascular endothelial cell (VEC) precursor cells within

the tumor stroma at per capita rate c. Furthermore, we

assume there is a delay between activation of vascular

precursor cells and construction of functional vessels

(Ji et al. 1998). The vascular network within tumors is

constantly being remodeled (Columbo et al. 1996).

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Homogeneous Tumor

We introduce a parameter β to measure the power ofa possible tumor cell P uptake regulation mechanismin the tumor growth equation. These considerationslead to the following model:

dx

dt= x

(

a min

(

1,Pe)

fnkh

)

− dx − (a − dx)x + y + z

kh

)

,

dy

dt= y

(

b min

(

1, βPe

fmkh

)

min(1, L) − dy − (b − dy)y + z

kt

)

,

dz

dt= c min

(

1,Pe

fnkh

)

y(t − τ) − dzz,

L =g(z − αy)

y.

(2.1)

Feb. 11, 2005. Penn State – p.39/63

Heterogeneous tumor

In actual malignant tumors, one often £nds a varietycell types coexist. We simplify the system to only twocompeting varieties, the masses of which arerepresented by y1 and y2.

dx

dt= x

(

a min

(

1,Pe

fnkh

)

− dx − (a − dx)x + y1 + y2 + z

kh

)

,

dy1

dt= y1

(

b1 min

(

1,β1Pe

fm1kh

)

min(1, L) − d1 − (b1 − d1)y1 + y2 + z

kt

)

,

dy2

dt= y2

(

b2 min

(

1,β2Pe

fm2kh

)

min(1, L) − d2 − (b2 − d2)y1 + y2 + z

kt

)

,

dz

dt= c min

(

1,Pe

fnkh

)

(y1(t − τ) + y2(t − τ)) − dzz,

L = gz − α(y1 + y2)

y1 + y2

,

(2.2)Feb. 11, 2005. Penn State – p.40/63

Simulation analysis of the models

For realistic parameter values and initial conditions, the

ultimate outcome of all three models is the same: solu-

tions tend to a positive steady state where phosphorus

limits both healthy and tumor cell growth. One nonin-

tuitive phenomenon is that at this steady state, tumor

growth is no longer limited by its blood vessel infra-

structure.

Feb. 11, 2005. Penn State – p.41/63

Simulation analysis of the models

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Homogeneous tumor growth

time t (days)

x/2,

y, z

, s, g

shalf healthy cells masstumor cells masstumor microvessels massP limitation indicator (0≤ s≤ 1)blood supply limitation indicator (0≤ gs≤ 2)

Feb. 11, 2005. Penn State – p.42/63

Simulation analysis of the models

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Homogeneous tumor growth with 20% reduction of P, β=1

time t (days)

x/2,

y, z

, s, g

shalf healthy cells masstumor cells masstumor microvessels massP limitation indicator (0≤ s≤ 1)blood supply limitation indicator (0≤ gs≤ 2)

Feb. 11, 2005. Penn State – p.43/63

Simulation analysis of the models

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

Heterogeneous tumor growth, β1=β

2=1

time t (days)

x/4,

y1, y

2, z, y

1+y2, s

,gs

1/4 healthy cells mass (kg)tumor cells type 1 masstumor cells type 2 masstumor microvessels masstotal tumor cellsP limitation indicator (0≤ s≤ 1)blood supply limitation indicator (0≤ gs≤ 2)

Feb. 11, 2005. Penn State – p.44/63

Simulation analysis of the models

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

Homogeneous tumor growth by model (4.1) with β=0.36

time t (days)

x/2,

y, z

, s, g

s

half healthy cells masstumor cells masstumor microvessels massP limitation indicator (0≤ s≤ 1)blood supply limitation indicator (0≤ gs≤ 2)

Here we assume a treatment yielding a 64% reduction in phosphorus uptake by tumorcells, and the construction of blood vessel is NOT phosphorus limited.

Feb. 11, 2005. Penn State – p.45/63

Mathematical Results

We assume that(A1): The construction of blood vessels is NOT limitedby phosphorus supply.With (A1), model (2.1) becomes the following:

dx

dt= x

(

a min

(

1,Pe

fnkh

)

− dx − (a − dx)x + y + z

kh

)

,

dy

dt= y

(

b min

(

1, βPe

fmkh

)

min(1, L) − dy − (b − dy)y + z

kt

)

,

dz

dt= cy(t − τ) − dzz,

L =g(z − αy)

y.

(4.1)

Feb. 11, 2005. Penn State – p.46/63

Mathematical Results

Our main concern is the stability of E∗ of (4.1). Weobserve: in both models (2.1) and (4,1), at this steady

state E∗,Pe

fnkh

< 1 and L > 1. Hence we also assume

(A2): For model (4.1),Pe

fnkh

< 1 and L > 1 at E∗.

Clearly (A2) implies thatβPe

fmkh

< 1. We obtain

x∗ =kh

a − dx

[

an

βbm

(

dy + (b − dy)y∗ + z∗

kt

)

− dx

]

− y∗− z∗,

y∗ = dzktN/D,

z∗ = cktN/D.

(4.2)

Feb. 11, 2005. Penn State – p.47/63

Mathematical Results

where

N = aPbβ − dxPbβ − dykhma − afmkhdy + bβndxkh + fdxmkhdy

and

D = −fdxdzmkhb − adzfmkhdy − fdxcmkhb − acfmkhdy − adzmkhdy+

adzmkhb + acfmkhb + adzmktbβ + fdxdzmkhdy − dxdzmktbβ + adzfmkhb

−ktbβdzna + ktbβdzndx + fdxcmkhdy + acmkhb − acmkhdy .

Feb. 11, 2005. Penn State – p.48/63

Mathematical Results

To study stability of E∗ of (4.1), we needLemma 1 (Kuang, p227) Assume that the parametersare positive, and x∗ > 0, y∗ > 0, z∗ > 0:

dx

dt= −x(A1(x − x∗) + A2(y − y∗) + A3(z − z∗)),

dy

dt= −y(B1(x − x∗) + B2(y − y∗) + B3(z − z∗)),

dz

dt= −(−c(y(t − τ) − y∗) + dz(z − z∗)).

(4.3)

If there are positive constants c1, c2 such that1): dz > c/c2,2): B2/c2 > B3 + B1/c1,3): A1/c1 > A1 + A2/c2

then the steady state E∗ = (x∗, y∗, z∗) is G.A.S.Feb. 11, 2005. Penn State – p.49/63

Mathematical Results

For a = 3,m = 20, n = 10, kh = 10, f = 0.6667, P = 150, b =

6, dx = 1, dy = 1, β = 1, we can chose c1 = 0.5 and

c2 = 0.2 to satisfy conditions 1) through 3) in the abo-

ve Lemma. In other words, we have shown that for

this set of parameters, the positive steady state E∗

of model (4.1) is locally asymptotically stable. Ho-

wever, simulation suggests that it is actually globally

asymptotically stable. So, this mathematical question

remains open.

Feb. 11, 2005. Penn State – p.50/63

Mathematical Results

Theorem 1 Assume that in model (4.1) there is aunique positive steady state E∗ = (x∗, y∗, z∗). Assumefurther that there are positive constants c1, c2 such that1): dz > c/c2,2): B2/c2 > B3 + B1/c1,3): A1/c1 > A1 + A2/c2

where A1, A2, B1, B2, B3 are given by (4.4) and (4.5).Then the steady state E∗ = (x∗, y∗, z∗) is locallyasymptotically stable.

Feb. 11, 2005. Penn State – p.51/63

Mathematical Results

Special Case: ma = nb, dx = dy, β = 1.

y∗ =aP − fnkhdx

fn(a − dx)(ρ + 1 + σ) + a(nρ + m + nσ)

=bP − fmkhdy

fm(a − dy)(ρ + 1 + σ) + b(nρ + m + nσ),

where

σ =c

dz

, ρ =

(

b − dy

kt

−a − dx

kh

)

kh(1 + σ)

a − dx

.

The tumor dies out if one can increase the tumor’sdeath rate or the tumor’s P requirement, or lower thetumor’s birth rate to certain threshold levels.

Feb. 11, 2005. Penn State – p.52/63

Discussion

A noteworthy insight gained through this model-

ling exercise is that slower-growing tumor cell ty-

pes, which utilize less phosphorus, will dominate

the tumor over the time. This competitive exclusion

pressure always threatens to push faster-growing cell

types to extinction, which in turn may provide the evo-

lutionary impetus for these more aggressive tumor cells

to metastasize.

Feb. 11, 2005. Penn State – p.53/63

Discussion

We suggest to look for ways to reduce the tumor phos-

phorus uptake rate. This result could be accomplis-

hed by applying drugs capable of selectively blocking

phosphorus uptake by tumor cells. Alternatively, one

could use a nutritionally worthless inhibitor that com-

petes with phosphate for the binding site on membrane

phosphate transporters. This strategy is intriguing sin-

ce it would tend to avoid the toxic effect of excessive

phosphorus liberated from dead tumor cells.

Feb. 11, 2005. Penn State – p.54/63

Discussion, end

A second plausible (crazy?) treatment suggested by

this research is to implant into the tumor less malignant

or benign tumor cells, or simply healthy cells if possible,

that require less phosphorus and thus are likely to out-

compete the original tumor cells. Alternatively, one can

try to genetically manipulate existing tumor cells to ge-

nerate less malignant or benign tumor cells that will do

the same thing.

Feb. 11, 2005. Penn State – p.55/63

(Bonus material) glucose-insulindynamics: PRELIMINARIES

The human body wants plasma glucose maintained in

a very narrow range (70-110 mg/dl). Insulin and gluca-

gon are the hormones which make this happen. Both

insulin and glucagon are secreted from the pancreas,

and thus referred to as pancreatic endocrine hormo-

nes. It is the production of insulin and glucagon by the

pancreas that ultimately determines if a person has dia-

betes, hypoglycemia, or some other sugar problem.

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Intimate Relationship of Insulin And Glucose

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INSULIN SECRETION OSCILLATION

Experiments in − vivo and in − vitro have shown thatinsulin secretion oscillates in two different time scales:

1. ultradian oscillation within range of 50-120 minutes2. rapid oscillation with a period of 5-15 minutes

A. meal ingestion; B. oral glucose intake; C. continuous enteral nutrition; D. constant glucose infusionFeb. 11, 2005. Penn State – p.58/63

MODELING ULTRADIAN OSCILLATIONS

R.N. Bergman, D.T. Finegood and S.E. Kahn,Eur. J. Clin. Inv., 32(2002)(suppl.3), 35 − 45

J. Sturis, K. S. Polonsky, E. Mosekilde, E. Van Cauter(1991)

I. M. Tolic, E. Mosekilde and J. Sturis (2000)

K. Engelborghs, V. Lemaire, J. Belair and D. Roose (2001)

D. L. Bennett and S. A. Gourley (2004)

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Bonus material: glucose-insulindynamics

Sturis-Tolic ODE model

G′(t) = Gin − f2(G(t)) − f3(G(t))f4(Ii(t))

+f5(x3)

I ′p(t) = f1(G(t)) − E(p(t)/Vp − Ii(t)/Vi)

−Ip(t)/tp

I ′i(t) = E(Ip(t)/Vp − Ii(t)/Vi) − Ii(t)/ti

x′

1(t) = 3(Ip − x1)/tdx′

2(t) = 3(x1 − x2)/tdx′

3(t) = 3(x2 − x3)/td

(2)

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Bonus material: glucose-insulindynamics

Sturis-Tolic function forms:

f1(G) = Rm/(1 + exp((C1 − G/Vg)/a1)),

f2(G) = Ub(1 − exp(−G/(C2Vg))),

f3(G) = G/(C3Vg),

f4(Ii) = U0 + (Um − U0)/(1+

+ exp(−β ln(Ii/C4(1/Vi + 1/Eti)))),

f5(x) = Rg/(1 + exp(α(x/Vp − C5))),

(3)

Feb. 11, 2005. Penn State – p.61/63

Bonus material: glucose-insulindynamics

Li-Kuang Two Time Delay model:

G(′t) = Gin − f2(G(t)) − f3(G(t))f4(I(t))

+f5(I(t − τ2)),

I ′(t) = f1(G(t − τ1)) − diI(t),

(4)

where the initial condition I(0) = I0 > 0, G(0) = G0 >

0, G(t) ≡ G0 for all t ∈ [−τ1, 0] and I(t) ≡ I0 for t ∈ [−τ2, 0]

with τ1, τ2 > 0. In addition,

Feb. 11, 2005. Penn State – p.62/63

Simulation of the models

0 100 200 300 400 500 600 700 800 90050

60

70

80

90

100

110

120

130

140

150G

luco

se (m

g/dl

)

ODE ModelSingle Delay ModelTwo Delay ModelAlt Single Delay Model

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