A Mathematical Model of Breast Cancer Treatment with CMF ...rtd/sem2013/DICMF.pdfA Mathematical Model of Breast Cancer Treatment with CMF 589 by cellular mutation is disregarded. Henceforth

Bull Math Biol (2011) 73: 585–608DOI 10.1007/s11538-010-9549-9

O R I G I NA L A RT I C L E

A Mathematical Model of Breast Cancer Treatmentwith CMF and Doxorubicin

Rachel Roe-Dale · David Isaacson ·Michael Kupferschmid

Received: 19 June 2009 / Accepted: 6 May 2010 / Published online: 8 June 2010© Society for Mathematical Biology 2010

Abstract A mathematical model is presented to investigate the ordering phenom-enon observed in the comparison of alternating to sequential regimens of CMF (cy-clophosphamid, methotrexate, 5-fluorouracil) and doxorubicin used in breast can-cer chemo-therapy. The ordinary differential equation model incorporates cell cyclespecificity and resistance to study why doses of the same drugs given in different or-ders result in different clinical outcomes. The model employs a pulsing condition tosimulate treatment and induced resistance, and we investigate treatment outcome bysimulating a patient population by varying parameters using uniform distributions.The results of these simulations correspond to those observed in prior clinical studiesand suggest that drug resistance might be a key mechanism in the sequential regi-men’s superiority.

Keywords Drug order · Drug resistance · Mathematical model · Breast cancer · Cellcycle

R. Roe-Dale (�)Skidmore College, 815 North Broadway, Saratoga Springs, NY 12866, USAe-mail: [email protected]

D. IsaacsonDepartment of Mathematical Sciences, Rensselaer Polytechnic Institute, 110 Eighth St, Troy,NY 12180, USAe-mail: [email protected]

M. KupferschmidAcademic and Research Computing, Rensselaer Polytechnic Institute, 110 Eighth St, Troy,NY 12180, USAe-mail: [email protected]

mailto:[email protected]



586 R. Roe-Dale et al.

1 Introduction

For most types of cancer, a wide range of chemotherapeutic drug treatments are avail-able, and in many clinical cases no treatment regimen is clearly superior. Recentstudies (Bonadonna et al. 1991; Motwani et al. 2003, 1999; Silvestrini et al. 2000)have documented the benefits of using multiple drugs having synergistic or antag-onistic effects. There is a need for qualitative and quantitative information derivedfrom mathematical models and simulations to guide in vivo and in vitro studies ofthis strategy.

In this paper, we discuss the development of a mathematical model to explore thedrug order dependence in a well-known breast cancer treatment regimen developedby the Italian oncologist Gianni Bonadonna, which consists of doxorubicin and thedrug cocktail CMF (cyclophosphamid, methotrexate, and 5-fluorouracil). We investi-gate the role of cell cycle specificity and drug resistance as possible mechanisms forthis order dependence.

We first present background information on the clinical work of Bonadonna andhis colleagues and discuss cancer cell dynamics in Sect. 2. In Sect. 3, we discussour treatment, cell cycle, and resistance models. We use these models and describeparametric studies and results in Sects. 4 and 5. Our conclusions and discussion arepresented in Sect. 6. Finally, a table summarizing the various parameters and variablesused in our models is given in the Appendix at the conclusion of the paper.

2 Background

2.1 Drug Regimens

In the mid 1980s, Bonadonna and his colleagues at the Instituto Nazionale Tumoriof Milan investigated CMF, the standard regimen for adjuvant breast cancer treat-ment. Bonadonna attempted to improve CMF therapy by coupling it with doxoru-bicin, which is also known by its tradename Adriamycin (A). The group conducted astudy of 405 breast cancer patients having more than three positive nodes. After surgi-cal removal of the tumor, the patients were randomized into two treatment regimens.In the sequential treatment regimen, patients were given four doses of A followed byeight doses of CMF, A4C8. Patients given the alternating regimen received two dosesof CMF followed by one dose of A, repeated 4 times for a total of 12 doses, (CCA)4.Both regimens consisted of the same drugs, doses, dose intensities, and treatment du-ration of 33 weeks. The specific details of the study are outlined in Bonadonna et al.(1991).

The 10-year results of this clinical investigation indicate that the sequential regi-men is superior to the alternating regimen across all patient categories such as age,tumor size, and number of positive nodes. In fact, Bonadonna notes that relapse freesurvival (RFS) for those women with the worst prognosis (more than 10 positivenodes and tumor size greater than 2 cm) given the sequential regimen was similar tothe RFS for women with the mildest form of disease who were given the alternatingregimen. Furthermore, Bonadonna notes that the results for the alternating regimen

A Mathematical Model of Breast Cancer Treatment with CMF 587

are similar to those for patients receiving CMF treatment alone. These results clearlyindicate the importance of drug scheduling in treatment outcome and design, andBonadonna’s group advised changing the clinical standard for breast cancer treat-ment to sequential therapy with CMF and doxorubicin.

In a later study, Silvestrini et al. (2000) explored the pharmacological and bi-ological explanations for the enhanced response to sequential therapy and foundthat drug combinations have cell cycle interactions, metabolic targets, and cross-resistance effects that can contribute to treatment success. Norton and Simon havealso investigated the ordering phenomenon and suggest that clinical outcome is im-proved with dose-dense therapy (Norton and Simon 1986, 1977). Chaudhary’s andRoninson’s investigation (Chaudhary and Roninson 1993) into multidrug resistanceprovides further insight into the sequencing phenomenon. They found that doxoru-bicin, methotrexate, and 5-fluorouracil, (methotrexate and 5-fluorouracil are presentin CMF), increase MDR1 expression from its basal level. The MDR1 gene then acti-vates the P-glycoprotein pump. Doxorubicin is subsequently pumped out of the cellwhile methotrexate and 5-fluorouracil are not transported by the P-glycoprotein pump(Chaudhary and Roninson 1993). In this paper, we explore the influence of cell cyclespecificity and drug resistance via the P-glycoprotein pump expression as a possi-ble mechanism in the comparison of Bonadonna’s alternating and sequential regi-mens.

2.2 Tumor Growth and the Cell Cycle

A first approximation of cancer growth assumes that the growth rate is proportionalto the number of cells in the growing population. If a population of cells N(t) growswith rate r , the change in the population is described by the differential equation,

dN(t)

dt= rN(t) (1)

which yields the exponential solution

N(t) = N(0)ert . (2)

Many studies have noted that the exponential growth model accurately describesin vitro and in vivo cell dynamics during the first stages of growth, before the can-cer population is limited by competition for space or nutrients (Bast et al. 2005;Panetta and Adam 1995). Solid tumors during their initial stages of growth also ex-press exponential dynamics (Edelstein-Keshet 1988). For this reason and because weare modeling treatment in an adjuvant setting, we assume exponential growth dynam-ics rather than using a logistic or Gompertzian growth model.

Cancer cells, like most eukaryotic cells, have a life cycle consisting of five phasesof development: G0, G1, S, G2, and M as in Fig. 1. Cells in G0 have not commit-ted themselves to division; cells in the remaining 4 phases are proliferating. Somecells, especially rapidly reproducing cancer cells, never enter G0. In G1, cells be-gin synthesizing RNA and proteins. These cells progress to S phase where DNA issynthesized. In G2, cells continue to synthesize proteins and RNA in preparation for


Fig. 1 Cell Cycle Diagram:This diagram depicts the 5phases of a cell’s development

mitosis, which occurs in M phase when the parent cell divides to make two daughtercells. A cell’s progression through this cycle can be monitored by the DNA count.Cells in G0, G1, and S are diploid cells, with two copies of DNA. Cells in G2 and Mhave four copies of DNA. Each daughter cell produced at the conclusion of mitosishas two copies of DNA.

Many anticancer drugs specifically target cells in a certain stage of development.Doxorubicin binds to DNA and is considered to affect cells in all phases of the cellcycle but to have an enhanced effect in S phase (Bast et al. 2005; Gringauz 1997; Tan-nock and Hill 1998). The drugs methotrexate and 5-fluorouracil, present in CMF, areboth antimetabolites that interfere with DNA synthesis in S phase (Bast et al. 2005;Gringauz 1997; Tannock and Hill 1998). The other component of CMF, cyclophos-phamid, is an alkylating agent that attacks DNA and is active in all phases of the cellcycle (Bast et al. 2005). Based on these properties, we assume that drug A will affectcells in the first half of their cycle preferentially but also have a significant effect inthe later phases. Drug C, containing one cycle-nonspecific drug and two drugs activein S phase, will also have a greater effect on cells in the earlier phases and a lessereffect in the later phases.

2.3 Drug Resistance

Resistance occurs when a drug that is initially effective becomes ineffective, allowingcontinual cancer growth, emergence of metastasis, or the reappearance of a tumorthat has disappeared (Weldon 1988). Cells exhibit resistance to chemotherapy formany reasons that are dependent on environmental and chemical properties of thecell and the drug. For example, the cellular concentration of drug may decrease ifthe population increases, but the same quantity of drug is administered. In addition,the manner in which the cell metabolizes the drug can be altered by mutations orthe presence of other metabolites. The drug target properties of the cell may also bealtered. For example, if the target of the treatment is the HER2 receptor on a breastcancer cell and the cell increases the number of receptors expressed, the efficacy ofthe drug will decrease (Michelson and Slate 1992).

One resistance mechanism is the MDR1 P-glycoprotein drug efflux pump. Thispump is a hexadimer protein embedded in the cell membrane and is activatedby the expression of the MDR1 gene. The pump is ATP (adenosine triphosphate)dependent and requires the binding of two of these energetic molecules in or-der to function. Michelson and Slate (1992) note that if a cell can utilize theefflux pump to keep the internal drug concentration below a critical threshold,the cell is more likely to survive. Since we are primarily interested in the re-sistance imparted by the activation of the drug efflux pump, resistance acquired


by cellular mutation is disregarded. Henceforth in this discussion, the term re-sistant refers to a cell expressing MDR1 and having an active P-glycoproteinpump.

Chaudhary and Roninson (1993) found that chemotherapy drugs affect the P-glycoprotein pump in two ways. First, the drug treatment has the potential to increaseexpression of the MDR1 gene, thereby activating the pump. Secondly, dependentupon chemical considerations, the drug molecules may or may not be eliminatedfrom the cell through this pump. It is possible for certain drugs to activate the MDR1gene by causing cellular damage but to not be transported out of the cell by the P-glycoprotein pump. Conversely, drugs that do not turn on the gene can be exportedfrom the cell by the efflux pump if it is activated for some other reason. Because ofthis function, the activated P-glycoprotein pump is responsible for classic multidrugresistance (MDR) (Michelson and Slate 1992). The activation of the drug efflux pumpis a stable effect. In Chaudhary and Roninson (1993), in vitro leukemia cells treatedwith drugs for 3–5 days and then allowed to grow in the absence of treatment werefound to still express MDR1 several weeks after treatment. Based on the work ofChaudhary and Roninson, we suppose initial treatment with A will be effective. How-ever, after the initial effective dose, the efflux pump will transport drug A out of thecell. On the other hand, subsequent treatment with CMF will continue to be effectivesince CMF is not pumped out of the cells.

3 Mathematical Models

3.1 Modeling Drug Treatment

Drug treatment can be numerically simulated using a variety of methods. We utilizea bolus injection treatment model where the entire concentration of drug is deliveredinstantly to the tumor site. This method is also referred to as the pulsing condition(Panetta and Adam 1995). We assume this treatment results in instantaneous killingof a certain fraction μ of cells. If the population of cells prior to treatment is N(t−),the number of cells surviving treatment is

N(t+

) = (1 − μ)N(t−

) = αN(t−

)(3)

where α = (1 − μ) is the surviving fraction of cells.Consider a model in which the cells grow exponentially. The drugs are cycle in-

dependent, and the cells do not develop drug resistance. The number of cancer cellsN(t) is given by (2). We consider two drugs, A and C, and simulate treatment bybolus injection with instantaneous kill as in (3) where

μi = kill fraction of drug i,αi = 1 − μi = survival fraction of cells treated with drug i.

Suppose that drug A is administered sequentially m times at intervals of τ hours.Administration of drug C immediately follows for a duration of m doses again at


intervals of τ hours. The number of cells after m treatments of drug A followed by m

treatments of drug C is then

N(2mτ) = (αCerτ

)m(αAerτ

)mN(0) = (

αCαAe2rτ)m

N(0). (4)

Now suppose that drugs A and C are administered in an alternating order for onecycle. In that case

N(2τ) = (αCerτ

)(αAerτ

)N(0).

If this alternating therapy is continued for m cycles of treatment, the number of cellssurviving is

N(2mτ) = (αCαAe2rτ )mN(0). (5)

Equation (5) is equivalent to (4). Thus, according to this simple model, the numberof cells after sequential treatment is the same as the number of cells after alternatingtreatment. In fact, any permutation of m treatments of drug A and m treatments ofdrug C will yield the same result, and the order of dosing has no effect on the outcomeof treatment.

3.2 Cell Cycle Model

It is well established, however, that the order of drug administration does in manycases make a difference in the effectiveness of treatment. To mathematically accountfor this fact, we incorporate the cell cycle into the exponential growth model of (1) asin Takahashi (1966, 1968). Rather than assuming that all cells are affected by drugsA and C in the same manner, we consider the possibility that each drug affects thecell at a different stage in its life cycle.

In Fig. 2, a population of cancer cells is divided into two compartments. The firstcompartment contains cells in G1, and S; these cells all have two copies of their DNA.Cells in G2 and M phases, having four copies of their DNA, are in compartmenttwo. Since we are primarily interested in modeling the dynamics and accountingfor the order dependence in the proliferating cancer population, we consider onlyproliferating cancer cells and assume that senescent or non-proliferating cells in G0are negligibly affected by drug therapy. We model the transition of cells from eachcompartment to the next by introducing transition rates λ1 and λ2. Checkpoints existat various locations in the cell cycle where the cell examines itself for damage ormutations. Many mutations are repaired at these checkpoints. However, a severelydamaged cell that cannot be repaired will commit itself to programmed cell death atthis stage, a process known as apoptosis. In our models, we assume that the rate ofnatural cell death is insignificant compared to the effects of chemotherapy treatmenton the cancer cells. Thus, in Fig. 2, we set ν1 = ν2 = 0, and the change in eachsubpopulation is described by (6)

dN1

dt= −λ1N1 + 2λ2N2,

dN2

dt= λ1N1 − λ2N2.

(6)


Fig. 2 Two Compartment Cell Cycle Model: In this model, cells are divided into 2 compartments based oncell cycle state. Cells in compartment 1 are in the first two phases of the cell cycle, and cells in compartment2 are in the later phases. Cells transition from compartment 1 and 2 according to rate λi . Cells are alsoremoved from each compartment according to rate νi

Here, N1 and N2 are the numbers of cells in compartments 1 and 2. The factor of 2in the first equation of (6) accounts for mitosis, in which one cell divides into two.

Rewriting (6) in matrix-vector form, we have

dNdt

=( dN1

dt

dN2dt

)=

(−λ1 2λ2λ1 −λ2

)N = AN (7)

with solution

N(t) = eAtN(0) = MN(0) (8)

where M = eAt is a growth matrix.In treatment with drugs A and C, we assume each drug can kill cells in compart-

ments 1 and 2 differently. If

μji = kill fraction in compartment j by drug i,

αji = 1 − μ

ji = survival fraction in compartment j treated with drug i

then we can construct treatment matrices, Ti for i = A, C to describe the fraction ofcells that survive treatment with drug A or C

TA =(

α1A 00 α2

A

), TC =

(α1

C 00 α2

C

). (9)

As we did previously, let us assume we administer m treatments with drug A at inter-vals of τ hours. Following treatment with A, m doses of drug C are administered atintervals of τ hours. The resulting number of cells is

N(2mτ) = (TCM)m(TAM)mN(0). (10)

If the doses of drug A and C are administered m times in an alternating schedulerather than sequentially, the number of cells is given by

N(2mτ) = (TCMTAM)mN(0). (11)

Unlike the one-dimensional case, the cell counts given by (10) and (11) are not nec-essarily equal because the matrix product is not necessarily commutative. That is, in


Fig. 3 Two Compartment Resistance Model With Treatments: In this model, cells are separated into com-partments based on drug sensitivity. Cells sensitive to treatment are in Ns, and cells resistant to treatmentare in Nr. The solid arrows show growth rate, λ. The dashed arrows show fixed proportions κi of cells

transferred by each drug i from the sensitive to the resistant compartment and the fixed percentage μji

ofcells that leave compartment j as a result of treatment i

general,

(TCMTAM)m �= (TCM)m(TAM)m.

Therefore, according to this model, using drugs in different orders yields differentoutcomes.

3.3 Resistance Model

As documented throughout the literature a primary concern in the design of a treat-ment regimen is the emergence of drug resistance. In many instances, as describedin Goldie and Coldman (1983), the presence of even one resistant cell can ultimatelycause treatment to fail.

The simplest resistance model is a two compartment model. One compartmentcontains Ns sensitive cells, and the other compartment contains Nr resistant cells. Asin Weldon (1988) and Goldie and Coldman (1983), we assume that resistant cellsgrow at the same rate as sensitive cells (λ ≡ λs = λr) and that resistant cells do nottransition back to a sensitive state. This model is illustrated in Fig. 3, and the changein populations is described by (12)

dNs

dt= λNs,

dNr

dt= λNr.

(12)

In modeling treatment of resistant cells, we assume an “all or nothing” drug effectas in Weldon (1988) and Westman et al. (2001). Therefore, cells resistant to a spe-cific drug will not die or leave the proliferating resistance compartment as a result oftreatment with this drug. As before, we assume treatment by bolus injection with in-stantaneous delivery of the drug and constant kill fraction μ

ji and that the rate of cells

dying or leaving the proliferating compartments in the absence of drug treatment isnegligible.

In our models, our primary concern is with the presence of the MDR1 gene and ef-flux pump and its role in the outcome of treatment. We assume that this gene has little


activity in cells until treatment with a drug, such as Adriamycin, increases its expres-sion. Resistance induced by MDR1 expression is modeled by converting a specifiedfraction κ of cells from the sensitive to resistant compartments instantaneously fol-lowing the drug’s administration. To a first approximation, this transition simulatesthe activation of the drug efflux pump. The transferred cells are then resistant to sub-sequent treatments with drugs that are removed from the cell by the pump. We furtherassume that resistant cells remain sensitive to drugs that are not transported out of thecell by this mechanism.

The treatment of cells and the conversion of sensitive cells to resistant cells cantherefore be modeled by modifying the pulsing condition used in the previous sec-tions. Now we expect that the treatment will instantaneously kill sensitive cells andthen convert a fraction κ of the remaining sensitive cells to resistant cells. Drugs notremoved by the efflux pump will also kill resistant cells. If drug i is exported fromcellular compartment j by the drug efflux pump, the drug will not kill any cells andμ

ji = 0.

Using N(mτ−) to describe the number of cells immediately prior to treatment, thenumber of cells in each compartment after treatment i is

Ns(mτ+) = (

1 − μsi

)Ns

(mτ−) − κ

(1 − μs

i

)Ns

(mτ−)

,

Nr(mτ+) = (

1 − μri

)Nr

(mτ−) + κ

(1 − μs

i

)Ns

(mτ−)

.(13)

In matrix-vector notation, we can write

N(mτ+) =

(Ns(mτ+)

Nr(mτ+)

)=

((1 − κ)(1 − μs

i ) 0κ(1 − μs

i ) (1 − μri )

)(Ns(mτ−)

Nr(mτ−)

)

= TiN(mτ−)

(14)

where Ti is the treatment matrix for drug i. A similar argument to that presentedin Sect. 3.2 indicates that treatment order is also mathematically significant in thisresistance model.

In our simulations, we investigate treatment with two drugs, A and C. While Bastet al. (2005) describe the activation of MDR1 by Adriamycin, Chaudhary and Ronin-son (1993) found that methotrexate and 5-fluorouracil also activate the gene. BothChaudhary and Roninson (1993) and Bast et al. (2005) state that Adriamycin ispumped out of the cell by the efflux pump, and Chaudhary and Roninson (1993) notethat methotrexate and 5-fluorouracil are not transported by this mechanism. There-fore, we assume drug A kills sensitive cells but is transported by the drug efflux pumpif the MDR1 gene is activated. In addition, drug A turns on this gene and convertssensitive cells to resistant cells. Drug C also activates the MDR1 gene and inducesresistance, but drug C is not transported by the efflux pump and therefore is notcross-resistant. We assume that drug C kills cells in both the resistant and sensitivecompartments, with an equivalent effect for both types of cells. Therefore, we abbre-viate μ

j

C as μC since the kill fraction is the same in both compartments. Similarly,since we are not considering cell cycle effects and assume drug A kills only sensitivecells, μr

A = 0, and the parameter μsA is simplified to μA in the treatment matrices


Fig. 4 Four Compartment Cell Cycle and Resistance Model: In this model, cells are separated by cellcycle state as well as drug sensitivity. The solid arrows show constant transition rates between compart-ments. The dashed arrows show the fixed proportions κi of cells transformed from sensitive to resistant and

the constant kill fractions μji

for drug i that are removed from the proliferating compartment at discreteintervals of simulated treatment

below

TA =(

(1 − κ)(1 − μA) 0κ(1 − μA) 1

), TC =

(1 − μC 0

0 1 − μC

). (15)

3.4 Combined Cell Cycle and Resistance Model

We next combine the two previous models to account for both cell cycle specificityand the development of resistance via activation of the P-glycoprotein pump. Cellsare separated by cell cycle state into two categories: G1 or S and G2 or M. Thesetwo categories are then further separated into cells that are expressing MDR1 andthose that are not, resulting in a four-compartment model, shown in Fig. 4 with theresulting ODE system (16). Here, compartments N1 and N3 contain cells in phasesG1 or S that are respectively sensitive or resistant to drug therapy, and compartmentsN2 and N4 contain cells in phases G2 or M that are either sensitive or resistant totherapy. As before, we assume that if a cell is expressing MDR1 it will be resistantto drugs that are transported by the P-glycoprotein pump. We also assume that cellsexpressing MDR1 will transition and grow at the same rates as the cells that do notexpress the gene

dN1

dt= −λ1N1 + 2λ2N2,

dN2

dt= λ1N1 − λ2N2,

dN3

dt= −λ1N3 + 2λ2N4,

dN4

dt= λ1N3 − λ2N4.

(16)

As in the previous sections, drug treatment is modeled using the pulsing condi-tion which instantaneously delivers a bolus injection of drug. The effects are also


immediate, either killing a fraction of the cells, moving a fraction to the resistantcompartment where the drug efflux pump is functioning, or both killing a fractionand moving a fraction of the surviving cells. The treatment equation for this systemis therefore again

N(mτ+) = TiN

(mτ−)

.

Treatment is simulated using drugs A and C, with treatment matrices TA and TC,given in (17). Drug A is assumed to kill only sensitive cells (μ3

A = μ4A = 0) with a cell

cycle dependence, having an enhanced effect in the first half of the cycle (μ1A > μ2

A).Drug A also increases expression of the MDR1 gene and thus transfers a specifiedfraction κA of the surviving sensitive cells to the resistant compartments. We assumethis induction is not cell cycle dependent, and A will induce the same fraction ofcells to transition from N1 → N3 and N2 → N4. Since it is not transported by thepump, drug C is assumed to kill sensitive and resistant cells equally (μ1

C = μ3C and

μ2C = μ4

C). Drug C also is cell cycle dependent, having a preferential effect on cellsin the first phase of the cell cycle, therefore, (μ1

C > μ2C). C will increase expression

of the MDR1 gene, and thus transfer a specified fraction κC of the surviving cells tothe resistant compartments. Again this induction is not cell cycle dependent, and Cwill induce the same fraction of cells to transition from N1 → N3 and N2 → N4. Forthe four compartment model, the treatment matrices have the following form:

TA =

⎛

⎜⎜⎝

(1 − κA)(1 − μ1A) 0 0 0

0 (1 − κA)(1 − μ2A) 0 0

κA(1 − μ1A) 0 1 0

0 κA(1 − μ2A) 0 1

⎞

⎟⎟⎠ ,

TC =

⎛

⎜⎜⎝

(1 − κC)(1 − μ1C) 0 0 0

0 (1 − κC)(1 − μ2C) 0 0

κC(1 − μ1C) 0 (1 − μ1

C) 00 κC(1 − μ2

C) 0 (1 − μ2C)

⎞

⎟⎟⎠ .

(17)

4 Parametric Studies and Model Simulations

4.1 Simulation Assumptions

We conducted a series of simulations to investigate the question of drug order speci-ficity in Bonadonna’s sequential and alternating regimens. In the simulations, weassume cells grow with exponential growth kinetics, and treatment is simulated bybolus injection with instantaneous kill and by converting a constant fraction of cellsto the resistant compartments. We disregard toxicity and the effects of the treatmenton normal cells. However, we implicitly consider normal cell preservation by restrict-ing τ , the treatment interval. According to Goldie and Coldman (1983), Norton andSimon (1986, 1977), and Weldon (1988), the optimal therapy to eradicate a tumoris through continuous infusion or in a dose-dense therapy by alternating drugs asrapidly as possible. These forms of therapy are sometimes achievable with supple-mentary measures such as concomitant growth factor administration or bone marrow


transplant. We do not examine these cases of small τ in this paper. We study the dos-ing interval τ = 21 days following the protocol described in Bonadonna et al. (1991).A higher concentration of drug is presumed to result in a higher kill fraction, butconcentration dependence is not explicitly considered in these models.

As is conventional in the literature, we assume a tumor reaches clinical detection at109 cells, and death results at 1012 cells (Bast et al. 2005; Weldon 1988). To qualifythe results of the numerical simulations of treatment where Nt represents the totalnumber of cancer cells, we define a cure when Nt < 1, remission when Nt < 109,regression when Nt < Nt(0), steady-state when Nt = Nt(0), and growth when Nt >

Nt(0). These classifications allow direct comparisons of the qualitative outcomes oftherapy.

To determine appropriate rate parameters, λ1 and λ2 in (6) and (16), we assumethat an in vivo breast cancer tumor doubles in size every 29 days. The tumor wouldtherefore grow from 1 cell to the clinically detectable size, 109 cells, in 2.4 yearsand to terminal size in 3.2 years. These values are consistent with observed diseaseprogression. Flow cytometry data presented in Motwani et al. (1999) for the MKN-74gastric cell line prior to treatment indicates that 78.7% of the cells were in G1/S and21.3% in G2/M, and cytometry data for the breast cancer line MCF-7 was found to beconsistent with these values (Motwani et al. 1999). Using this cell cycle distributionand doubling time, we used the ellipsoid algorithm (Ecker and Kupferschmid 2004,§9.7) to estimate λ1 = 0.0368 per day and λ2 = 0.112 per day for (6). For the two-compartment resistance model (12), we use the growth rate λ = 0.024 per day whichcorresponds to a 29 day doubling time.

4.2 Cell Cycle Dependent Simulations

To explore the importance of the cell cycle in the treatment-order effect, we inves-tigated a multiple drug treatment regimen with model (6) with growth parametersλ1 = 0.0368 per day and λ2 = 0.112 per day and initial values N1(0) = 7.87 × 108

and N2(0) = 2.13 × 108. Drug A (Adriamycin) and drug C (CMF cocktail) werechosen for this simulation as in Bonadonna’s clinical trails. A total of 4 doses of Aand 8 doses of C were simulated either in an alternating (CCA)4 or sequential orderA4C8 with τ = 21 days. Each drug was assumed to kill a constant fraction of the cellsin each compartment. The general treatment matrices of survival fractions are givenby (9).

To illustrate a specific choice of parameters, thus simulating a single patient’streatment, the following treatment matrices were used:

TA =(

0.01 00 0.35

), TC =

(0.25 0

0 0.90

). (18)

These values indicate that drug A will affect cells in the first half of their cycle pref-erentially but also have a significant effect in the later phases. Drug C, containingone cycle-nonspecific drug and two drugs active in S phase, will also have a greatereffect on cells in the first phase and a lesser effect in the later phases. A controlsimulation was also conducted in the absence of treatment. The results for the twotreatment regimens and the control tumor population are displayed in Fig. 5; for


Fig. 5 Cell Population Trajectories for Cell Cycle Model with Treatments of A and C: This figure il-lustrates treatment outcome for the simulation of one patient’s treatment with four drug regimens. Theparameters used are defined by (18). The results indicate that the alternating and sequential regimens arenot significantly different

comparison single drug treatments for 12 doses of A and 12 doses of C are alsoshown on this figure. In practice, however, the regimens we are comparing limitthe number of doses of A to 4 and the doses of C to 8 (Bonadonna et al. 1991;Silvestrini et al. 2000).

This data clearly illustrates that both treatment regimens reduce the initial popu-lation of cells while the control simulation (without treatment) illustrates continuedexponential growth. Treatment A12 produces the best results. However, this therapyis not clinically realistic since A is extremely cardiotoxic, and the number of dosesof A that can be administered is limited (Gringauz 1997). These results do indicate,though, that the multiple drug regimens are superior to treatment with C alone.

While the number of cells differs at various points in the treatment course, theresulting number of cells for both regimens at the conclusion of therapy is nearlyidentical for the two multiple drug regimens. For this reason, we cannot establish thateither regimen is better than the other.

To investigate the effects of the alternating and sequential regimens when appliedto a patient population, we conducted a series of simulations using different valuesfor each of the kill fractions. Each drug was assumed to kill a constant fraction ofcells in each compartment. Since the general patient population will exhibit a varietyof responses to chemotherapy, we simulate a population in this study by varying killfractions using uniform distributions. A total of 14,641 combinations were run foreach treatment sequence with

μ1A,μ2

A,μ1C,μ2

C ∈ [0.01,0.10,0.20,0.30,0.40,0.50,0.60,0.70,

0.80,0.90,0.99]. (19)

These parameter values represent a wide range of drug responses from weak (0.01)to strong (0.99).


Table 1 Simulation Results forthe Cell Cycle Model: Resultsfrom these simulations indicatethat the alternating andsequential regimens result insimilar qualitative outcomes

Treatment Cure Regression Steady-state Growth

(CCA)4 351 9,976 9 4,305

A4C8 347 9,957 5 4,332

The specific number of combinations resulting in each type of response for the twosequences is displayed in Table 1. The sequential and alternating regimen results arealmost equivalent. The alternating regimen results in 19 more cases of regression and27 fewer cases of growth than the sequential regimen. Within each specific parametercombination, the regimens can further be classified by direct comparison. A regimenis considered superior at the conclusion of therapy if it results in greater regression orlower progression of the initial population. The regimens are considered equivalentif they both affect the cell population in the same way. Of the 14,641 simulations,the alternating regimen was superior for 11,322 cases, and the sequential regimenwas superior for 47 cases. The two regimens were equivalent for 3,272 parametercombinations. To investigate the nature of the difference between the two regimens,we consider the percent difference between the final cell counts at the conclusionof therapy for each combination of treatment parameters. Rather than using the ab-solute value of the difference between cell counts in this calculation, we calculate thepercent difference using the following method:

% Difference = Nseq − Nalt

Nseq + Nalt× 200%. (20)

Using this form, we are able to determine which regimen is actually reducingcell count further. For example, if the number of cells surviving treatment with thesequential regimen is larger than the number surviving treatment with the alternatingregimen, the percent difference calculation in (20) will result in a positive value, asis most often the case for the cell cycle model. The histogram in Fig. 6 illustrates thedistribution of the percent difference between the alternating and sequential regimensfor this model. Most of the percent differences (79.4% of the trials) are between0–5% in the positive direction. Therefore, while we can conclude that the alternatingregimen reduces cell count further, this difference is not extreme. Nonetheless, theseresults are not consistent with Bonadonna’s clinical observations, indicating that thiscell cycle model does not appropriately capture the biology of the situation. We nowmove forward to consider the effects of drug resistance.

4.3 Resistance Dependent Simulations

We use the two-compartment resistance model depicted in Fig. 3 and described by(12–14) to model the activation of the P-glycoprotein pump. Our simulations inves-tigate the treatment regimens A4C8 and (CCA)4 for specific parameter values. Wesimulate the interval of treatment as τ = 21 days, the growth rate λ = 0.024 per day,and assume that the initial population consists entirely of sensitive cells, Ns(0) = 109

and Nr(0) = 0.


Fig. 6 Percent Difference Between Treatment Outcomes for the Cell Cycle Model: The percent differencebetween the final cell counts at the conclusion of therapy for each combination of treatment parameters isdetermined. The positive values indicate that more cells survived treatment with the sequential regimen

Fig. 7 Cell Population Trajectories for Resistance Model with Treatments A and C: This figure illustratestreatment with four drug regimens and is based on the resistance model where κA = 0.1 and κC = 0. Notethat the alternating and sequential regimens result in similar outcomes

We first present an individual case of specific parameter values. For this individual,drug A was assumed to increase expression of MDR1 with κA = 0.10 and to killsensitive cells with μA = 0.92. In the first simulation, drug C was assumed not toaffect MDR1 expression (κC = 0) and to kill sensitive and resistant cells equally withμC = 0.61.

The results for these simulations are shown in Fig. 7. The alternating and sequen-tial regimens give the same outcome because the parameter values in this example aresuch that the products of the growth and treatment matrices are commutative. That


is, given any ordering of 4 doses of A and 8 doses of C, the ending cell count is thesame.

The treatment matrix here for drug C is TC = (1 − μC)I, where I is the identitymatrix. Since the growth rate of the sensitive and resistant cells is assumed to be thesame, the population at time t is

N(t) =(

eλt 00 eλt

)N(0). (21)

The growth matrix in (21) is a multiple of I so the products of two such matrices(TC and the growth matrix) with TA are commutative, and all regimens result in thesame outcome.

Similar to the cell cycle model, the simple resistance model suggests that multi-drug treatment is better than single drug administration. Unlike the case presentedearlier for the cell cycle model, though, treatment with only drug A yields the worstoutcome after no treatment. While drug A successfully eradicates the sensitive cellpopulation, it creates a tumor that is entirely composed of resistant cells. Therefore,in order for treatment to be successful, a non-cross resistant drug must also be ad-ministered to destroy the resistant population. As shown in Fig. 7, cells receiving notreatment continue to grow exponentially, and no resistant cells are created since weare assuming here that resistance is induced only by drug treatment.

While Bast et al. (2005) describe the activation of MDR1 by Adriamycin, the geneis also activated by methotrexate and 5-fluorouracil (Chaudhary and Roninson 1993).Both Bast et al. (2005) and Chaudhary and Roninson (1993) state that Adriamycin ispumped out of the cell by the efflux pump, and Chaudhary and Roninson (1993) notethat methotrexate and 5-fluorouracil are not transported by this mechanism. There-fore, in the next trial, we used κC = κA = 0.10 assuming both drugs turn on the genein the same way. The various other parameter values are kept the same as before. Thegraphical results for this simulation are given in Fig. 8.

Here note, as before, that treatment with A alone results in the worst outcome, withresistant cells comprising the entire tumor population. In this simulation, however, thesequential regimen is more successful than the alternating regimen. Also note that thesequential regimen is superior to treatment with C alone, whereas treatment with onlyC is more successful than the multi-drug alternating administration. These resultsagree with Bonadonna’s observation that sequential therapy is superior to alternatingtherapy.

To investigate the effects of drug resistance on a population, we again simulatea patient population by expanding the comparison of the two Bonadonna regimensfor varying parameter values, allowing for the following kill fractions and resistanceparameters:

μA,μC, κA, κC ∈ [0.01,0.10,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,0.99].All parameter combinations were tested for a total of 14,641 simulations for eachsequence.

The specific results for each sequence are given in Table 2, which provides a qual-itative comparison of the two regimens. These results differ from the previous case in


Fig. 8 Cell Population Trajectories for Resistance Model with Treatments A and C: For this simulation,drugs A and C were assumed to induce resistance; therefore, κA = κC = 0.1. In this case, the populationafter treatment with the sequential regimen is 1.5 orders of magnitude less than the population remainingafter alternating therapy

Table 2 Simulation Results for the Resistance Model: Results from these simulations indicate that thesequential regimen while resulting in approximately the same number of cures as the alternating regimen,results in more cases of regression and less cases of growth


(CCA)4 1331 5470 3 7837

A4C8 1342 6743 0 6556

Table 1 where the two regimens have similar qualitative outcomes. Note in Table 2that the sequential regimen results in more successful treatments (cures and regres-sions) than the alternating administration. Among the 14,641 parameter combina-tions, the sequential regimen is better for 13,213 cases. Both regimens are equivalentfor 1,420 combinations (cases where the treatment matrix products are commutative),but the alternating regimen is not superior for any parameter combination. Using (20),we calculate the percent difference in final cell count for both regimens. The distri-bution of percent differences is shown in Fig. 9. Unlike the distribution presented inFig. 6, this distribution contains only values less than or equal to zero. As previouslymentioned, a negative percent difference indicates that the alternating regimen is notas effective at killing cancer cells as the sequential regimen. This result further sup-ports our qualitative assertion that the sequential treatment regimen results in a bettertreatment outcome when the resistance model is used.

For almost all parameter combinations simulated, the treatment order does makea difference. By considering only a simple two-compartment resistance model,we have captured the result Bonadonna and colleagues (Bonadonna et al. 1991;Silvestrini et al. 2000) found in their clinical studies; the sequential regimen is su-perior. When only the cell cycle is considered, the alternating regimen is superior.


Fig. 9 Percent Difference Between Treatment Outcomes for the Resistance Model: The percent differencebetween the final cell counts at the conclusion of therapy for each combination of treatment parameters isdetermined. Negative values, as shown here, indicate that more cancer cells survive after treatment withthe alternating regimen than with the sequential regimen

This result suggests that the emergence of resistance in A and CMF therapy plays asignificant role in treatment outcome.

4.4 Cell Cycle with Resistance Simulations

Results presented in the previous section indicate the role of resistance is importantin drug ordering. We now examine the combined effects of cell cycle specificity anddrug resistance on Bonadonna’s regimens using the growth and treatment model de-picted in Fig. 4 and described by (16) and (17). For all simulations, drug A wasassumed to kill only sensitive cells (μ3

A = μ4A = 0) and to increase expression of

MDR1 in 10% of the remaining cells (κA = 0.10), thus moving them to one of theresistant compartments. Drug C, which is not removed by the pump, was assumedto affect sensitive and resistant cells equally (μ1

C = μ3C and μ2

C = μ4C). Drug C was

also assumed to induce resistance with κC = 0.10. Treatment was simulated everyτ = 21 days, as in Bonadonna’s trials. The growth parameters λ1 = 0.0368 per dayand λ2 = 0.112 per day were used, and the starting population consisted of only sen-sitive cells: N(0) = [7.87 × 108,2.13 × 108,0,0]�.

The results for the simulation of an individual case (μ1A = 0.99, μ2

A = 0.65,μ1

C = 0.75, and μ2C = 0.1) are given in Fig. 10 with a comparison of the alternat-

ing and sequential regimen shown in Fig. 11. The results indicate that sequentialtherapy is the superior regimen for eradicating tumor cells. In this simulation, we areassuming that both drugs A and C activate MDR1 and therefore invoke resistanceand that A is pumped out of the cell by the P-glycoprotein pump, and hence doesnot affect resistant cells. C, on the other hand, is not removed by this mechanism


Fig. 10 Cell Population Trajectories for the Cell Cycle and Resistance Model with Treatments A andC: This figure illustrates the treatment outcome for one individual case where the two drugs are cell cyclespecific and induce resistance in the cancer cells. The cell count after treatment with the sequential regimenis about 1 order of magnitude less than the cell count after treatment with the alternating regimen

Fig. 11 Comparison ofAlternating and SequentialRegimens for the Cell Cycle andResistance Model: Thesepopulation trajectories indicatethe sequential regimen almostalways keeps the totalpopulation below the survivingpopulation level for thealternating case

and is therefore active in both sensitive and resistant cells. The kill fractions wereselected such that A and C have preferential effects on cells in the first phases of thecell cycle, and drug A has higher kill fractions than drug C. The results illustrate thatwhile both regimens result in regression, the sequential regimen decreases cell countmore than the alternating regimen. Similar to the resistance model, Fig. 10 illustratesthat treatment with A alone results in the worst outcome except for no treatment. Asmentioned before, therapy with A alone converts the tumor to only resistant cells.Treatment with C alone is preferable to alternating therapy. These results are similarto the findings for the two-compartment resistant model, suggesting that the conceptof resistance might be more important than the cell cycle in influencing the orderingof drugs.


Table 3 Simulation Results forthe Cell Cycle and ResistanceModel: The sequential regimenresults in more cases ofregression than the alternatingregimen


(CCA)4 242 7047 0 7352

A4C8 245 8731 0 5665

To investigate the behavior of the model on a larger patient population, the studywas expanded as before with 14,641 trials using all possible combinations of

μ1A,μ2

A,μ1C,μ2

C ∈ [0.01,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,0.99]. (22)

The qualitative results for each regimen are given in Table 3. This data indicates thatthe sequential regimen results in more cases of regression and less cases of growththan the alternating regimen. The sequential regimen is also superior for most para-meter combinations. Of the 14,641 combinations tested, the sequential regimen wasthe most successful for 14,342 combinations. The regimens were equivalent for 242combinations, and the alternating regimen was better for 57 combinations. Again weassessed the strength of the comparison by computing the percent difference of can-cer cell count following treatment with the sequential and alternating therapies. Thehistogram in Fig. 12 shows the distribution of values. For most cases, the sequentialregimen elicits the better response. Comparison of Figs. 12 and 9 indicates that whilethe sequential regimen resulted in a better outcome for both models, the percent dif-ference between the sequential and alternating regimens is larger for the cell cycleresistance model, signifying that both cell cycle and resistance effect are importantproperties to consider when modeling treatment results.

The combined model therefore indicates that the sequential regimen results in abetter treatment outcome. This conclusion agrees with the findings of Bonadonna,that the sequential combination A4C8 is superior to the alternating regimen (CCA)4.Several other observations can be made about this set of simulations. Even in thecases where both regimens failed, the population exhibited less growth in the sequen-tial regimen. In the 245 cases where cure (N < 1) was achieved, the parameter valueswere μ1

C,μ2C ∈ {0.80,0.90,0.99} for varying values of μ1

A and μ2A. These values sug-

gest a greater dependence on the efficacy of drug C, which makes sense because C isable to kill both sensitive and resistant cells.

For all parameter combinations investigated, the tumor load at the conclusion oftherapy contained a greater percentage of resistant cells. This result supports the priorobservation that in investigating sequence order for a particular regimen, the conceptof resistance is more important than the effects of the cell cycle. Also, this is con-sistent with the observation that the success of the regimen is more sensitive to thestrength of drug C. The greater the percentage of resistant cells in the population, theharder it is to attain a cure.

An additional simulation was developed assuming that C does not induce MDR1,i.e. κC = 0. The results, not shown here, were similar to those presented for the simpletwo compartment resistance model where κC = 0. The results from alternating andsequential treatment were equivalent, again suggesting that induction of resistance isa predominant feature to consider in the modeling of these biological systems.


Fig. 12 Percent Difference Between Treatment Outcomes for the Cell Cycle and Resistance Model: Thenegative values indicate that more cells survived treatment with the alternating regimen. The sequentialregimen is more effective for 98% of the parameter combinations

Control simulations were also run to simulate treatment with C only (C12) andwith A only (A12) where the kill parameters for each drug were chosen from theset previously described. In all cases, treatment with A resulted in a worse outcomethan either sequential or alternating therapy. This result makes sense since A killsonly sensitive cells in addition to inducing resistance in cells. In some cases aftercompleting 12 cycles of treatment, the sensitive population is driven to zero, andthe remaining tumor population is comprised entirely of resistant cells. Treatmentwith C alone resulted in a better outcome than either of the combination regimes forsome parameter choices. As previously discussed, the response of the combinationtherapy is sensitive to the kill fraction values for drug C, since C is able to affectboth sensitive and resistant cells. The sequential regimen is superior when drug Chas lower kill fractions and drug A has higher kill fractions. This result is consistentwith clinical findings that Adriamycin is one of the most effective drugs for treatinga variety of cancers when used in combination with other treatments.

5 Simulation and Clinical Results

In the 10-year study (Bonadonna et al. 1991), Bonadonna notes that the sequentialtreatment regimen is superior; he also provides a qualitative summary of patient re-sponse to each regimen. If we combine these results with the results of our cell cy-cle and resistance parametric study, we can compare positive response (either sur-vival or regression/cure) between the two treatments regimens, outlined in Table 4.Bonadonna’s results indicate the percentage of patients in each treatment arm that are


Table 4 Comparison of Model Results and Bonadonna’s Clinical Results: Simulation results generatedfrom the cell cycle and resistance model and from (Bonadonna et al. 1991) both indicate that the sequentialregimen results in a superior treatment outcome

A4C8 (CCA)4 Difference p-value

Bonadonna 10 yr Survival 62% 48% 14% 0.004

Model Regression/Cure 62% 50% 12% <0.0001

still alive ten years post-therapy. The model results report the percentage of paramet-ric study cases that result in patient cure or disease regression, as previously defined.The results indicate a 14% difference in the survival of patients from each treatmentregimen in the comparison of Bonadonna’s patient population, which yields a p-value of 0.004. The model simulations result in a 12% difference, yielding a p-valueof less than 0.0001. Therefore, it is reasonable to conclude at any statistical levelthat the two regimens result in different outcomes and that the outcome observed fol-lowing sequential treatment of doxorubicin and CMF is preferred to the alternatingregimen.

While the models presented in this section do not prove that the MDR1 mechanismof resistance and the cell cycle dependence of drug therapy are the only mechanismsresponsible for the outcome that Bonadonna observed, the results are consistent withhis findings and suggest a possible mechanism. In addition as indicated in Table 4, oursimulation results also appear to appropriately model the patient response observedby Bonadonna in terms of the percentage of survival for each treatment arm.

6 Discussion

If a different initial population is assumed or if different transition rates for growth arechosen, the outcomes for any of these parametric studies might be different. Futureinvestigation into the parameter space pertaining to the initial conditions on the sys-tem and the rate constant values is warranted to determine this relationship. However,the above investigation indicates that for various kill fractions and growth constants,different drug orders result in different treatment outcomes.

We have shown that without any assumptions regarding dose density, it is resis-tance rather than cell cycle specificity that is responsible for the superiority of thesequential regimen. Both the dose dense and the drug resistance theories may becorrect. However, more careful definitions, additional experiments, and theoreticalstudies need to be done for this combination of drugs.

A significant amount of laboratory and clinical research has been conducted toexplore the benefit of multi-drug sequencing. Perhaps mathematical modeling ofchemotherapy treatment can help guide future studies and provide insight into thequestion of why the same drugs given in a different order result in different outcomes.

Acknowledgements The authors wish to thank Dr. Gary Schwartz from Memorial Sloan-Kettering Re-search Hospital in New York City and Dr. Igor Roninson from the Molecular Oncology Laboratory atthe Ordway Research Institute in Albany, NY, for valuable discussions about this material and for sharingrecent papers on this topic. The authors also wish to thank the reviewers for their helpful recommendations.


Appendix: Variable List

Table 5 An index of the variables used in this manuscript

Symbol Meaning and value or range

α Survival fraction; α ∈ [0,1]α

ji

Survival fraction of cells treated with drug i in compartment j ; αji

∈ [0,1] and αji

= 1 − μji

λi Rate constant; λ1 = 0.0368 per day, λ2 = 0.112 per day, and λ = 0.024 per day; See Sect. 4.1for discussion

κ Fraction of acquired resistance; κ ∈ [0,1]κi Fraction of acquired resistance after treatment with drug i; κi ∈ [0,1]μ Kill fraction; μ ∈ [0,1] and μ = 1 − α

μji

Kill fraction from drug i in compartment j ; μji

∈ [0,1] and μji

= 1 − αji

νi Kill rate of cells from compartment i; νi = 0 in this study

τ Period between treatments; As per (Bonadonna et al. 1991;Silvestrini et al. 2000), τ = 21 days

A Coefficient matrix in ODE system

m Number of treatments in a regimen

M Growth matrix

N(t) Total population of cells at time t (also written as N or Nt )

Ni Population of cells in compartment i

r Growth rate

Ti Treatment matrix with drug i

t Time

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