Tumor Development Under Combination Treatments with Anti-angiogenic Therapies - Mathematical Methods and Models in Biomedicine - page 315

law which makes the tumor volume decrease in D + regardless of the actual control

being used and is true for all growth models F that are strictly increasing and satisfy
F ( 1 0.

If the state

)=

does not lie on the singular arc and inhibitors are still available,
then the optimal policy simply consists in getting to this singular arc in the “best”
possible way as measured by the objective. If p

( p , q )

q , and initially this clearly is the
medically relevant case, then this is done by simply giving a full dose treatment
until the singular arc is reached and then the control switches to the singular regime.
In such a case, since the carrying capacity is high, immediate action needs to be
taken and it would not be optimal to let the tumor grow further. These are the
trajectories in Fig. 1 that are shown by solid green curves. Note that these curves
are almost horizontal and thus show little tumor reduction. Of course, the beneficial
effect of this full dose segment is that it prevents the tumor increase that otherwise
would have occurred. Significant reductions in tumor volume only arise as the
singular arc is reached. Mathematically, it is no problem to include in the solution
initial conditions when p

<

q , but, from the medical side, this case is less interesting.
Here the optimal solution consists of giving no anti-angiogenic agents, but to wait
until the system reaches the singular arc as the carrying capacity increases and then
to start treatment when the singular arc is reached by once more following the
singular regime. Intuitively, inhibitors are put to better use along this curve than
if they would have been applied directly. For small tumor volumes (e.g., those that
lie below the saturation level x u for the singular control,) anti-angiogenic agents are
always given at maximum dose. The minimum tumor volumes are realized when,
after all inhibitors have been exhausted, the trajectory corresponding to u

>

= 0 crosses

the diagonal after termination of treatment.
The interesting feature of this synthesis is the relative simplicity and full
robustness of the resulting optimal controls once the singular curve is known.

Furthermore, not only for the model considered here, but also for various of
its modifications [ 8 , 56 ], it is possible to determine this singular arc and its
corresponding control analytically by means of well-known procedures in geometric
optimal control theory which makes the construction of this synthesis and the
resulting computations of optimal protocols a worthwhile endeavor [ 26 , 30 ].
Below, we give the explicit formulas for the singular arc and control. They
equally apply to the monotherapy problem [A] and to the combination therapy
problem [AC] when no chemotherapeutic drugs are administered.

S

Proposition 8.1 ([ 26 ]). If an optimal control u ∗ is singular on an interval ( α , β )

and v ≡ 0 on ( α , β ) , then the corresponding trajectory ( p ∗ , q ∗ ) follows a uniquely

p
q , can be parameterized in the

defined singular curve S which, defining x =

variables ( p , x ) in the form

dp 3

+ bx ( ln x − 1 )+ μ = 0 (33)

with x in some interval [ x 1 , x 2 ] ⊂ ( 0 , ∞ ) . The singular control u sin that keeps this

curve S invariant is given as a feedback function of p and q in the form