Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 314

18000

S

16000

u=0

14000

u=a

12000

full dosage

10000

beginning of therapy

8000

partial dosages along singular arc

x *

6000

endpoint − (q(T),p(T))

4000

no dose

2000

0

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

carrying capacity of the vasculature, q

Fig. 1 A 2-dimensional projection of the optimal synthesis for the monotherapy model [A] into

( p , q ) -space

optimal solution, no minimization algorithm needs to be invoked, but the procedure
simply consists in evaluating a piecewise defined, albeit somewhat complicated
function.
We briefly describe the structure of optimal controlled trajectories. First of
all, note that we plot the tumor-volume p vertically and the carrying capacity q
horizontally in Fig. 1 . This simply better visualizes a decrease or increase in the
primary cancer volume. The anchor piece of the synthesis is an optimal singular
arc -space along which

the best tumor reductions occur and optimal controls follow this curve whenever
the data allow. That is, if

S shown in blue. This is a unique curve defined in ( p , q )

, and if angiogenic inhibitors
y are still available, then the optimal control consists in giving a specific time-
varying dosage that makes the system stay on this curve, i.e., makes the singular
arc invariant. There is a unique control that has this property, the corresponding
singular control, and as long as its values lie in the admissible range

( p , q ) happens to lie on S

,thisis
the optimal control. For the model by Hahnfeldt et al., there is a unique point on the
singular arc when the control reaches its saturation limit u max which is denoted by
x u in Fig. 1 . Mathematically, the structure of the optimal synthesis near a saturating
singular arc is well understood [ 26 , 55 ], but for simplicity of presentation we limit
our discussions here to cases when saturation does not occur. In this case, once
(

[ 0 , u max ]

, the optimal protocol then simply consists in giving these singular
dosages until all inhibitors are exhausted. At that time, the state of the system lies
in the region

p , q ) lies on S

and, because of aftereffects, the tumor volume
will still be decreasing until it reaches the diagonal

D + = { ( p , q ) : p > q }

of the
system. This behavior is preprogrammed in the properties of the Gompertz growth

D 0 = { ( p , q ) : p = q }