Figure 1.
Cells progress along four phases: G0/G1, S, G2 and M. Transition from one phase to another depends on the circadian time. G1-S phase transition occurs at a rate . Cells in G1 phase can also leave permanently at a rate
. S/G2/M phases have fixed durations
,
,
. At the end of the M phase, cells divide and go back to the G0/G1 phase. Cells in S/G2/M phases die at rates
,
,
. G1-S phase transition
and G2 phase duration
are clock-dependent (24 h periodic).
Figure 2.
Daily evolution of the host, fast growing tumor and slow growing tumor.
(A) S phase fraction. (B) G0/G1 phase fraction. (C) G2/M phase fraction. Dark phases are indicated by black bars (20:00 to 8:00). In panels A–C, time 0 corresponds to 72 h after beginning simulations, to allow for transients to vanish. Initial conditions (at h) are
,
,
, and
. In panels A–C, solid lines denote host, dashed lines fast growing tumor, and dashed-dotted lines slow growing tumor.
Figure 3.
Treatment outcomes as a function of circadian time of administration.
(A) Fast growing tumor treated at optimal time 2:00. (B) Best treatment outcome for fast growing tumors (maximal
) is at 2:00, while the worst is at 17:30 (thick line). (C) Slow growing tumor treated at optimal time
22:00. (D) Best treatment outcome for slow growing tumors is at 22:00, while the worst is at 5:30 (thick line).
Figure 4.
Response of host cells to treatment () as a function of treatment time
and S phase duration
.
(A) TATO, as given by Eq. 1 with sinusoidal. (B) Numerical simulations of the full model. The response is normalized from low tolerance (blue) to high tolerance (red). Daily extrema predicted by TATO are indicated by white lines (dashed: highest toxicity
, solid: lowest toxicity
, same in both panels). TATO predicts well the location of the extrema of the full model (squares: highest toxicity
; circles: lowest toxicity
). At
= 24 h, the location of extrema are shifted by 12 h (thick vs. thin lines).
Table 1.
Best and worst times of treatments.
Figure 5.
Treatment outcomes with different intervals between drug administrations.
(A, C) Outcomes for fast (A) and slow (C) growing tumors, for different first day delivery times () and intervals (
) between administrations. The treatment outcome function used normalized, scaled responses
and
from simulations. Low values (blue) indicate bad treatment outcomes and high values (red), good ones. Outcomes for three intervals with 4.8 h difference (white lines) are compared: 19.2 h, 24 h, and 28.8 h. Eq. 2 predicts the location of the best response, as a function of
(thin white lines). (B, D) Outcomes at 24 h intervals show large amplitudes while small amplitudes occur at 19.2 h and 28.8 h intervals. For fast growing tumors, an interval of 28.8 h is a good alternative to 24 h (B, dotted line), but for slow growing tumors, an interval of 24 h is better (D, solid line).