The mathematical modelling of adjuvant chemotherapy scheduling: Incorporating the effects of protocol rest phases and pharmacokinetics

In this paper the modelling objective is to determine the drug alternation time which minimises the formation of resistant tumour cells when delivering two non-cross resistant chemotherapeutics given such drugs cannot be delivered simultaneously and constraints due to pharmacokinetics and protocol rest phases. We initially consider cell cycle phase non-specific models, as investigated by Goldie and Coldman. By extending previous work, these models are generalised to consider chemotherapeutic S-phase specificity. We find with the cell cycle phase non-specific models that once the alternation time of the drugs is reduced below a critical threshold, a substantial improvement in protocol outcome is predicted. Extensive improvements are also observed for the S-phase specific investigation if the drugs can be alternated extremely rapidly. However, this is typically impossible due to pharmacokinetic constraints. Under such circumstances, the most appropriate choice of the alternation time can depend sensitively on the median and variance of the tumour cell cycle time in a complicated manner. For schedulings motivated by Capecitabine protocols, we find that switching the drugs only once, or at most twice, between rest phases gives the most reliable alternation time. The main and novel conclusion of this paper is the modelling prediction that one must be much more specific in the choice of the protocol alternation time if attempting to observe the improvements promised by Goldie and Coldman’s alternation hypothesis for the rest phases, pharmacokinetics and delivery mechanisms typically encountered in cell cycle phase specific chemotherapy protocols.     

Optimal policies of non-cross-resistant chemotherapy on Goldie and Coldman’s cancer model

Mathematical models can be used to study the chemotherapy on tumor cells. Especially, in 1979, Goldie and Coldman proposed the first mathematical model to relate the drug sensitivity of tumors to their mutation rates. Many scientists have since referred to this pioneering work because of its simplicity and elegance. Its original idea has also been extended and further investigated in massive follow-up studies of cancer modeling and optimal treatment.     

ON THE EXISTENCE OF A Go-PHASE IN THE CELL CYCLE

The model is based on the assumption that the cell cycle contains a G_o-phase which cells leave randomly with a constant probability per unit time, γ. After leaving the G_o-phase, the cells enter the C-phase which ends with cell division. The C-phase and its constituent phases, the‘true’G₁-phase, the S-phase, the G₂-phase and mitosis are assumed to have constant durations of T, T₁T_s, T₂ and T_m, respectively. For renewal tissue it is assumed that the probability per unit time of being lost from the population is a constant for all cells irrespective of their position in the cycle. The labelled mitosis curve and labelling index for continuous labelling are derived in terms of γ, T, and T_s. The model generates labelled mitosis curves which damp quickly and reach a constant value of twice the initial labelling index, if the mean duration of the G_o-phase is sufficiently long. It is shown that the predicted labelled mitosis and continuous labelling curves agree reasonably well with the experimental curves for the hamster cheek pouch if T has a value of about 60 hr. Data are presented for the rat dorsal epidermis which support the assumption that there is a constant probability per unit time of a cell being released from the Go-phase.     

Bioluminescence Imaging of DNA Synthetic Phase of Cell Cycle in Living Animals

Bioluminescence reporter proteins have been widely used in the development of tools for monitoring biological events in living cells. Currently, some assays like flow cytometry analysis are available for studying DNA synthetic phase (S-phase) targeted anti-cancer drug activity in vitro; however, techniques for imaging of in vivo models remain limited. Cyclin A2 is known to promote S-phase entry in mammals. Its expression levels are low during G1-phase, but they increase at the onset of S-phase. Cyclin A2 is degraded during prometaphase by ubiquitin-dependent, proteasome-mediated proteolysis. In this study, we have developed a cyclin A2-luciferase (CYCA-Luc) fusion protein targeted for ubiquitin-proteasome dependent degradation, and have evaluated its utility in screening S-phase targeted anti-cancer drugs. Similar to endogenous cyclin A2, CYCA-Luc accumulates during S-phase and is degraded during G2/M-phase. Using Cdc20 siRNA we have demonstrated that Cdc20 can mediate CYCA-Luc degradation. Moreover, using noninvasive bioluminescent imaging, we demonstrated accumulation of CYCA-Luc in response to 10-hydroxycamptothecin (HCPT), an S-phase targeted anti-cancer drug, in human tumor cells in vivo and in vitro. Our results indicate that a CYCA-Luc fusion reporter system can be used to monitor S-phase of cell cycle, and evaluate pharmacological activity of anti-cancer drug HCPT in real time in vitro and in vivo, and is likely to provide an important tool for screening such drugs.     

Cell cycle analysis by the relative movement approach: effect of variability across S-phase of DNA synthesis rate

Cell populations pulse-labelled with BrdUrd, and sampled at increasing times after the pulse, yield DNA-BrdUrd distributions from which the relative movement (RM) and the depletion function (DF) of labelled, undivided cells can be calculated. In this paper we present an extension of the equation for the time course of RM, given by White and Meistrich (Cytometry 1986, 7 , 486–490), to the case in which the rate of DNA synthesis changes across S-phase. Some modalities of cell loss were also considered. Computer simulations showed that different patterns of DNA synthesis rate across S-phase can result in appreciably different RM curves. An analytical expression of the RM curve, in which the variability across S-phase of the rate of DNA synthesis is accounted for by only one parameter, was proposed. This expression was used for the simultaneous fitting of time sequences of RM and DF data of U937 cells, in order to estimate the phase transit times T_S and T_G2+M, and the potential doubling time T_pot. The use of the extended model gave better results than those obtained under the assumption of constant rate of DNA synthesis across S-phase.     

Non-Circadian Periodicity in the Duration of the Cell Cycle and the G1 Phase in the Hindlimb Epidermis of the Anuran Tadpole,Rana Pipiens

Using the percentage labeled mitoses method, seven cell cycle determinations were initiated at 6-hr intervals over a 36-hr span in order to see if the cell cycle in the tadpole hindlimb epidermis varied with time or showed rhythmicity. There was a pattern of two long cell cycles followed by a shorter one. Total cell cycle length (Tc) and the length of the G₁ phase plus one-half of the mitotic time (TG₁+½M) fluctuated the most, although only TG₁+½M varied significantly with the Chi-square test. The proportion of TC spent in each phase was also calculated. Only TG₁+½M/Tc had statistically significant fluctuations with time.

Rhythmicity was analyzed by a computer program using the method of least squares for cosine curve fitting. Statistically significant ultradian rhythms of 18.4 hr in TC, 18.5 hr in TG₁+½M and 18.6 hr in TG₁+½M/TC and the length of the DNA synthetic phase/total cell cycle length (TS/TC) were found. Circadian rhythmicity was not observed. The acrophases of the ultradian rhythms of TC and TG₁+½M coincided, suggesting that the rhythm of TC was due mainly to variation in TG₁+½M. In the absence of significant variation in TS, the longest phase of the cell cycle, whenever G₁+½M was short, TS/TC increased, so that the 18.6 hr rhythm in TS/TC was also a result of the periodicity in TG₁+½M.     

Low-intensity combination chemotherapy maximizes host survival time for tumors containing drug-resistant cells

Recent clinical trials have shown that for some cancers, high-intensity alternating chemotherapy does not significantly improve either survival times or response rates compared with nonalternating therapy. The current study uses optimal control to determine the best way to treat a tumor that contains drug-resistant cells that cannot be destroyed. The delivery of two non-cross-resistant chemotherapeutic agents is limited by bounds on the drug concentration and the dose intensity. This ensures that the drug toxicity stays within a tolerable range. The aim of the therapy is to maximize the host survival time, defined as the time over which the tumor burden can be kept below a fixed bound. The model is posed as a free terminal time, optimal parameter selection problem in which the constraints are continuously parametrized by time and the number of courses of therapy is free to vary. New theory is developed so that the optimal parameter selection problem can be solved as a sequence of fixed terminal time problems using existing optimal control software. Numerical simulations of Gompertz tumor growth showed that a treatment maintaining a high tumor burden doubled and sometimes tripled with survival time under aggressive therapy. When these simulations were repeated using exponential and logistic tumor growth models, the tumor burden during treatment had little influence upon survival time. In all simulations, survival time was not extended by delivering the anticancer drugs concurrently instead of staggering the treatment arms.     

Chronomodulated chemotherapy and irradiation: an idea whose time has come?

The chronomodulated delivery of systemic chemotherapy given with irradiation (chemoradiation) is driven by an understanding of: the chronobiology of normal tissue response to cytotoxic insult, chronopharmacology, and by technologic advances in vascular access and in the availability of portable programmable pumps. Since circadian variation exists in the proliferative activity of acute-reacting normal tissues like the gut and bone marrow, a potential therapeutic gain can be realized by the chronomodulated administration of S-phase chemotherapeutic agents at biological times when these normal tissues are in a different cell phase and thus relatively spared (chronotolerance). The reasons for this are complex and possibly include newly described time-keeping genes that may influence the cell cycle. Another important aspect of chronotolerance is based on chronopharmacologic behavior of S-phase chemotherapeutic radiation sensitizing agents, especially 5-fluorouracil (5-FU). In this review laboratory and clinical evidence is presented for using chronomodulated 5-FU or the topoisomerase-I inhibitor, camptothecin, when best tolerated biologically. Although the main body of this work has been accomplished with pure chemotherapy schedules, there is emerging clinical evidence this approach to treatment also applies to the application of chemoradiation. This knowledge has been exploited only recently in the clinic. These data should be viewed as a call for additional studies to investigate the precise timing of systemic chemotherapeutic radio sensitizers to ameliorate toxicity and maximize treatment effect, especially with newer and potentially more toxic chemoradiation programs.     

The Proliferative State of Clonogenic Cells In the Lewis Lung Tumour After Treatment With Cytotoxic Agents

The proportion of clonogenic cells from the Lewis lung carcinoma which are in S-phase of the cell cycle has been measured as the fraction killed by a short exposure to hydroxyurea in vitro. Estimates of the proportions of Sphase cells before and 30 min after doses of γ-radiation of 1000–2000 rad suggest no alternation in the cell cycle age distribution due to these doses of radiation. As the survivors of these high doses of radiation are predominantly hypoxic, the results imply that hypoxic cells have the same cell cycle age distribution as oxygenated cells in Lewis lung tumours. After treatment with cyclophosphamide or CCNU, the proportion of S-phase cells among the survivors exceeds the faction of S-phase cells in untreated populations. This increase is consistent with a relative resistance of S-phase cells to alkylating agents and nitrosoureas.     

INTERACTIONS OF LIGHT/DARK CYCLES AND NITROGEN PULSES ON THE TIMING OF CELL DIVISION IN THE NITROGEN-LIMITED MARINE DIATOM CYLINDROTHECA FUSIFORMIS (BACILLARIOPHYCEAE)1

Cell division in most eukaryotic algae grown on alternating periods of light and dark (LD) is synchronized or phased so that cell division occurs only during a restricted portion of the LD cycle. However, the phase angle of the cell division gate, the time of division relative to the beginning of the light period, is known to be affected by growth conditions such as nutrient status and temperature. In this study, it is shown that the phase angle of cell division in a diatom, Cylindrotheca fusiformis Reimann and Lewin, is affected by the N-limited growth rate; cell division occurred later in the dark period (12:12 h LD cycle) when the growth rate was infradian (D = 0.42 d^?1) than when it was ultradian (D = 1.0 d^?1). Nitrogen-pulses did not affect the phase angle of the division gate, but could shift the time of peak cell division activity within the division gate. The effects, if any, of N-pulses were dependent upon the growth rate and the time of day that the pulses were administered. These responses indicate that the timing of cell division in this diatom is not determined solely by the zeitgeber from the LD cycle, but rather that a LD cycle control mechanism and a N-mediated control mechanism are both involved and are somewhat interdependent. In addition, an increase in protein was observed immediately after administering a N-pulse to C. fusiformis in the ultradian growth mode indicating that the accumulation of protein can be uncoupled from the cell division cycle.     

Rat C-6 Glioma Cells Maintained In an Organ Culture System: A Study of Kinetic Parameters

Rat C-6 glioma cells were grown on a sponge foam matrix in an organ culture system and the cell cycle parameters, including the growth fraction (GF), were assessed after autoradiography. the zones of growth consisted of a compact upper layer (UL) at the gaseous interface, a central necrotic layer and a deeper lower layer (LL) which invaded the matrix. the fraction of continuously labeled mitoses (FCLM) was similar in both the UL and LL cells. the derivatives of the FCLM curves obtained in three experiments gave an average modal T_G2 of 5 hr. A mathematical model relating GF, T_G2, T_C and labeling index as a function of time, LI(t), was devised for cells in a steady state exposed continuously to tritiated thymidine and was applied to data obtained from UL cells. A mean GF of 9% (range: 8–10%) and a mean cell cycle time (T_C') of 27 hr (range: 13–47 hr) were obtained. the mean T_S was calculated to be 11 hr (range: 8–16 hr) by the method of grain counts per mitotic figure or grain index (GI). Knowledge of T_S permitted alternative calculation of the cell cycle time from the equation T_S/T_C= LI(0)/GF: this gave a mean cell cycle time (T_C) of 29 hr (range: 20–45 hr). Except for the GF, the cell kinetics were comparable to those of the same cell line grown in monolayer culture. the GF in the in vitro system described is in the lower range reported in some human malignant gliomas in vivo.     

Reversible release of chick embryo fibroblast cultures from density dependent inhibition of growth

Initiation of proliferation in density-inhibited chick embryo fibroblast cultures induced by insulin or trypsin was partially reversed by replacing the medium with supernatants from parallel non-stimulated cultures. Growth stimulation by neuraminidase, pokeweed mitogen, bacterial lipo polysaccharide or purified tuberculin was less, or not at all, affected by this procedure. Medium change per se caused some proliferation in non-stimulated cultures. Increased rate of sugar uptake in insulin-stimulated cultures returned to the level of that in non-stimulated cultures within a few hours after medium change. This reversion took place apparently irrespective of the phase of the cell cycle. Replacing the medium with supernatants from non-stimulated cultures induced a rapid decline in subsequent thymidine incorporation during the first S-phase, and completely abolished the second peak of DNA synthesis. The fraction of cells irreversibly committed to mitosis increased when the time after stimulation increased. Less than three hours' incubation with insulin or trypsin was needed to initiate proliferation of a significant fraction of the cell population. It is concluded that reversion of the initiated cycle of a given cell is no more possible after the cell has entered the S-phase.     

Cell kinetic studies on the JB-1 ascites tumour of the mouse

Abstract. The FLM method, modified by double labelling with [³H]- and [¹⁴C]-thymidine, has been applied to the 4-day old JB-1 ascites tumour of the mouse. It results in well separated waves of purely [³H]- and purely [¹⁴C]-labelled mitoses, which show a remarkable asymmetry with long tails to the right. The following values for the mean transit times of the cells have been derived from this FLM curve, for a tumour age of 4–6 days: T_C= 32.5 hr, T_S= 16.7 hr, T_G1= 3.7 hr, T_G1= 11.0 hr and T_M= 1.1 hr. A further evaluation of the FLM curve, however, is difficult, due to the non-stationary growth of the tumour. A number of other experimental findings (growth curve, decrease of the labelling and mitotic index with increasing tumour age, two single-labelled FLM curves starting 4 and 6 days after tumour inoculation) indicate that the cell cycle time increases during the experimental period of the double-labelled FLM curve (about 2 days). A lengthening of the cycle time should result in an increasing enlargement of the areas under the waves of the modified FLM curve. However, such an increase in area has not been found; the areas are constant. All the results of the present cell kinetic studies would be consistent if it were postulated that the cell cycle time lengthens with increasing tumour age up to about 4 days after inoculation, then remains relatively constant at between 4 and 6 days and thereafter increases again. Short-term double labelling experiments suggest that this is actually the case. Under the assumption of nearly constant phase durations during the 5th and 6th day of tumour growth further conclusions can be drawn from the modified FLM curve. In particular, it follows that the transit times of the cells through successive cycle phases are uncorrelated and the variances of the transit times through a cycle phase are proportional to the duration of this phase.     

Enhancement of cancer chemotherapy by simultaneously altering cell cycle progression and DNA-damage defenses through global modification of the serine/threonine phospho-proteome

Despite improvements in the therapeutic efficacy of rationally designed cancer treatment regimens, most cancers remain incurable once spread beyond their sites of origin. Failure to achieve sustained control or eradication of cancers arises in large part because a sub-population of quiescent “cancer stem cells” is insensitive to drugs targeting cell growth and replication and because defense mechanisms critical to survival of the normal cell also protect the cancer cell from cytotoxic injury. Global alteration of signal transduction by inhibition of serine/threonine dephosphorylation has recently been shown to markedly potentiate cancer cell killing by the DNA-methylating drug, temozolomide. Inhibition of the multifunctional protein phosphatase 2A appears to drive quiescent cancer cells into cycle and simultaneously inhibits cycle arrest, permitting cancer cell entry into mitosis despite the presence of chemotherapy induced DNA-damage. Absence of toxicity in animal models suggests that multi-site mutations in pathways regulating cell cycle in cancer cells make them more vulnerable than normal cells to large changes in the balance of phosphorylation-regulated signaling. Global modulation of the serine-threonine phospho-proteome may be a general method for enhancing the effectiveness of cytotoxic cancer therapy.     

Cell proliferation kinetics of six xenografted human cervix carcinomas: comparison of autoradiography and bromodeoxyuridine labelling methods

Abstract. Cell kinetic and histologic parameters of six xenografted tumours with volume doubling times ranging from 6 to 43 d were investigated in order to obtain kinetic information on a panel of tumours to be used in radiobiological studies. The six tumours covered a range of histologies and their DNA indices varied from 2–7 to 1–4. The length of the cell cycle (T_c), potential doubling time (T_pot) and labelling index (LI) were determined by continuous labelling with [³H]TdR and autoradiography in three tumours. T_c varied from 30 to 40 h. Determinations of the length of the S phase (T_s) were found to be less reliable by this method. Data on T_s and LI were also determined in all six tumours using bromodeoxyuridine (BrdU) labelling and the single sample method; values of T_pot were slightly longer than those obtained via the autoradiographic method. In addition, multiple samples were taken after BrdU labelling. T_c was determined by fitting the data obtained from mid-S, mid-G₂ and mid-G₁ windows to curves described by a damped oscillator. Data obtained via the mid-S window were found to be most reliable. Generally, cell cycle times obtained by the BrdU method were longer than those observed with the autoradiographic method. Differences between the two methods could be explained by inaccuracies in the determination of T_s, LI and T_c and differences in the experimental approach. We consider the BrdU labelling method to be a suitable alternative for the time-consuming autoradiography, if data on T_s or T_pot are sufficient. Due to difficulties in the reproducibility of the immunofluorescence staining and asynchronization of cells approximately 10 h after labelling, the method of windows analysis was affected by similar problems to those observed in interpretation of percentage labelled mitosis (PLM) curves. However, the method may serve as an alternative to determine cell cycle times in vitro and, if improved technically, in vivo. Careful comparison of the data obtained from mid-S, mid-G₁ and mid-G₂ windows may increase the reliability of the determination of cell kinetic parameters.     

Cell cycle re-entry sensitizes podocytes to injury induced death

Podocytes are terminally differentiated renal cells, lacking the ability to regenerate by proliferation. However, during renal injury, podocytes re-enter into the cell cycle but fail to divide. Earlier studies suggested that re-entry into cell cycle results in loss of podocytes, but a direct evidence for this is lacking. Therefore, we established an in vitro model to test the consequences of re-entry into the cell cycle on podocyte survival. A mouse immortalized podocyte cell line was differentiated to non-permissive podocytes and stimulated with e.g. growth factors. Stimulated cells were analyzed for mRNA-expression or stained for cell cycle analysis using flow cytometry and immunocytofluorescence microscopy. After stimulation to re-entry into cell cycle, podocytes were stressed with puromycin aminonucleoside (PAN) and analyzed for survival. During permissive stage more than 40% of immortalized podocytes were in the S-phase. In contrast, S-phase in non-permissive differentiated podocytes was reduced to 5%. Treatment with b-FGF dose dependently induced re-entry into cell cycle increasing the number of podocytes in the S-phase to 10.7% at an optimal bFGF dosage of 10 ng/ml. Forty eight hours after stimulation with bFGF the number of bi-nucleated podocytes significantly increased. A secondary injury stimulus significantly reduced podocyte survival preferentially in bi-nucleated podocytes In conclusion, stimulation of podocytes using bFGF was able to induce re-entry of podocytes into the cell cycle and to sensitize the cells for cell death by secondary injuries. Therefore, this model is appropriate for testing new podocyte protective substances that can be used for therapy.     

The cell cycle of spermatogonial colony forming stem cells in the CBA mouse after neutron irradiation

In the CBA mouse testis about 10% of the stem cell population is highly resistant to neutron irradiation (D0, 0.75 Gy). Following a dose of 1.50 Gy these cells rapidly increase their sensitivity towards a second neutron dose and progress fairly synchronously through their first post-irradiation cell cycle. From experiments in which neutron irradiation was combined with hydroxyurea it appeared that in this cycle the S-phase is less radiosensitive (D0, 0.43 Gy) than the other phases of the cell cycle (D0, 0.25 Gy). From experiments in which hydroxyurea was injected twice after irradiation the speed of inflow of cells in S and the duration of S and the cell cycle could be calculated. Between 32 and 36 hr after irradiation cells start to enter the S-phase at a speed of 30% of the population every 12 hr. At 60 hr 50% of the population has already passed the S-phase while 30% is still in S. The data point to a cell cycle time of about 36 hr, while the S-phase lasts 12 hr at the most.     

Synchronization of Anacystis nidulans

A new technique of short alternating lightdark periods was successfully used to synchronize the blue-green alga Anacystis nidulans. Oxygen evolution during the cell cycle is characterized by a maximum in the middle of the cycle and by a minimum at the time of division, a pattern very similar to that found in synchronized green algae.     

Metabolic state and cell cycle as determinants of facilitated uptake of genetic information by yeast <Emphasis Type="Italic">Saccharomyces cerevisiae</Emphasis>

The transformation efficiency of yeast cells during exponential growth might be characterised as undulatory. The aim of the study was to investigate the reason for the fluctuation in transformation efficiency of yeast Saccharomyces cerevisiae p63-DC5 cells during exponential growth. The heightened response to exogenous DNA was observed with the growing yeast culture when budded cells were predominant. To confirm this phenomenon we carried out synchronization of yeast cells with 10 mM hydroxyurea. Results showed that synchronous yeast cells in the S-phase of cell cycle have enhanced transformation efficiency. Furthermore, S. cerevisiae p63-DC5 cells in the S-phase were successfully transformed with plasmid pl13 in the absence of lithium acetate. We indicated that the permeability of yeast cells in the S-phase to tetraphenylphosphonium (TPP) cations was significantly higher than in asynchronous culture. The results of our study showed that the fluctuation in transformation efficiency was strictly dependent on the metabolic state of yeast cells and the capacity of the yeast cells to become competent was related to the S-phase of cell cycle.