3 T-cell populations

Question 1

T-cell populations

Answer 1

One of the most important types of cells of the immune system are T cells (T lymphocytes).
They come from the thymus with T-cell receptors on their surface, being released to the “periphery” (i.e. rest of the body, blood, lymph nodes etc). There are more than 107 T cells in an
adult mouse, more than 1011 in an adult human T-cells that die are replaced by new T cells
from the thymus and by division of existing T cells (one cell dividing into two).
We start with a single-equation model, where the number of cells is represented not as an
integer but by a real function of age,

Question 2

T-cell populations
N(t)

Answer 2

N(t) number of cells is represented not as an
integer but by a real function of age,

Question 3

T-cell populations
ODE

Answer 3

dN(t)/ dt
= rate of thymic export − rate of cell death + rate of cell division

dN(t)/dt
= θ − µN(t) + λN(t)
= θ − βN(t),
*ASSUME cells circulate independetly throughout the body
*every cell is equally likely to undergo one round of division or suffer death in any small time interval dt.

θ- rate of export of cells from the thymus

β = µ − λ is the death rate per cell minus the division rate per cell.

[θ] = [β] = T^−1 and N is just a number.
similar to immigration model

Question 4

T-cell populations
ODE
dN(t)/dt
= θ − µN(t) + λN(t)
= θ − βN(t),
if θ does not depend on age and N(0) = 0, solution:

Answer 4

N(t) = (1 − e^{−βt})(θ/β)

with steady state (“homeostatic-pop remains at SS”) value N_ss =θ/β
The time it takes for the T cells of the immune
system to recover from a perturbation, such as radiotherapy, is proportional to 1/β.

(solution from sep of vars)(β>0 o/w sol doesn’t make sense)

as t tends to infinity N tend to N_ss

Time taken for T cells to recover is proportional to 1/β

Question 5

Example If a mouse’s thymus produces 106 CD4+ cells per day, one peripheral na¨ıve cell in 30
dies per day and one in 300 divides per day, then β = 0.03 day−1 and, using (3.1), we conclude
that there ar

Answer 5

N_ss =θ/β

example If a mouse’s thymus produces 106 CD4+ cells per day, one peripheral na¨ıve cell in 30
dies per day and one in 300 divides per day, then β = 0.03 day−1 and, using (3.1), we conclude
that there are 33 million na¨ıve CD4+ T cells at steady state.

Question 6

Ageing of the thymus (MATH5566) Why we consider?

Answer 6

ODE was non-autonomous baby f(N) but f(N,t) for adult

, when we age,
our thymus produces less T cells

Question 7

Ageing of the thymus (MATH5566)Model

Answer 7

the decreasing function
θ(t) = Ae^{−νt} cells per day.

dN(t)/dt=
θ(t)-βN(t)
= Aexp(-vt) - -βN(t)

Non autonomous ODE now as funct of t: solve analytically
or use variation of constants

General solution
N(t) =
[θ(t)/(β − ν)] + Ce^{−βt}

C dep on initial condition

Question 8

Ageing of the thymus (MATH5566)Model

SOLVE

dN(t)/dt=
θ(t)-βN(t)
= Aexp(-vt) - βN(t)

Answer 8

Non autonomous ODE now as funct of t: solve analytically
or use variation of constants

General solution
N(t) =
[θ(t)/(β − ν)] + Ce^{−βt}

IE: Look for solution in the form N(t) = C(t) exp(∫β.dt)

N’(t)= C’(t) exp(-βt)- C(t)βexp(-βt)
comparing
Aexp(-Vt) =C’(t)exp(-βt)

C(t) = Aexp(t(β-v))/(β-v) + Constant of integration

rearrange gives GS

Question 9

General solution
N(t) =
[θ(t)/(β − ν)] + Ce^{−βt} what it tells u?

Answer 9

General solution
N(t) =
[θ(t)/(β − ν)] + Ce^{−βt}

C dep on initial condition

e^{−βt} decreases more rapidly as a function of age than e^-vt

the long-term decrease of the average number of na¨ıve T cells in
a mouse has the same timescale as the waning of thymic export to the periphery.
Plot of N(t) against t in days tells us increase fast then rate of increase slowly starts to decrease then it decreases over long time once hits θ(t)/β for a certain t? slight delay