Effects of drug resistance in the tumour-immune system with chemotherapy treatment
José Trobia, Enrique C Gabrick, Evandro G Seifert, Fernando S Borges, Paulo R Protachevicz, José D Szezech Jr, Kelly C Iarosz, Moises S Santos, Iberê L Caldas, Kun Tian, Hai-Peng Ren, Celso Grebogi, Antonio M Batista
aa r X i v : . [ q - b i o . T O ] A ug Effects of drug resistance in the tumour-immune systemwith chemotherapy treatment
Jos´e Trobia , Enrique C Gabrick , Evandro G Seifert , Fernando S Borges , PauloR Protachevicz , Jos´e D Szezech Jr , Kelly C Iarosz , Moises S Santos , Iberˆe LCaldas , Kun Tian , Hai-Peng Ren , Celso Grebogi , Antonio M Batista Graduate Program in Science - Physics, State University of Ponta Grossa, 84030-900, Ponta Grossa, PR,Brazil. Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900, PontaGrossa, PR, Brazil. Department of Physics, State University of Ponta Grossa, 84030-900, Ponta Grossa,PR, Brazil. Center for Mathematics, Computation, and Cognition, Federal University of ABC, 09606-045, S˜ao Bernardo do Campo, SP, Brazil. Institute of Physics, University of S˜ao Paulo, 05508-900, S˜aoPaulo, SP, Brazil. Faculty of Telˆemaco Borba, FATEB, 84266-010, Telˆemaco Borba, PR, Brazil. GraduateProgram in Chemical Engineering Federal Technological University of Paran´a, Ponta Grossa, 84016-210,Paran´a, Brazil. Shaanxi Key Lab of Complex System Control and Intelligent Information Processing, Xi’anUniversity of Technology, Xi’an 710048, PR China. Xi’an Technological University, Xi’an, 710021, PRChina. Institute for Complex Systems and Mathematical Biology, University of Aberdeen, AB24 3UE,Aberdeen, Scotland, United Kingdom.Corresponding author: [email protected]
Abstract
Cancer is a term used to refer to a large set of diseases. The cancerous cells growand divide and, as a result, they form tumours that grow in size. The immune sys-tem recognise the cancerous cells and attack them, though, it can be weakened bythe cancer. One type of cancer treatment is chemotherapy, which uses drugs to killcancer cells. Clinical, experimental, and theoretical research has been developedto understand the dynamics of cancerous cells with chemotherapy treatment, aswell as the interaction between tumour growth and immune system. We study amathematical model that describes the cancer growth, immune system response,and chemotherapeutic agents. The immune system is composed of resting cellsthat are converted to hunting cells to combat the cancer. In this work, we considerdrug sensitive and resistant cancer cells. We show that the tumour growth can becontrolled not only by means of different chemotherapy protocols, but also by theimmune system that attacks both sensitive and resistant cancer cells. Furthermore,for all considered protocols, we demonstrate that the time delay from resting tohunting cells plays a crucial role in the combat against cancer cells.
Keywords: tumour-immune, chemotherapy, drug resistance
An abnormal growth of cells can cause a malignant or cancerous tumour to invadenearby tissues and possibly to spread to other organs [1]. Cancer is a group of diseases,being a public health problem in all countries of the world [2]. Many types of treatmenthave been developed to eliminate cancer cells, such as surgery [3], chemotherapy [4],and radiation [5]. One of the chemotherapeutic treatments is the immunotherapy [6].1athematical models have been used to study different types of cancer and stagesof tumour progression [7, 8]. In 1972, Greenspan [9] constructed a mathematicalmodel of tumour growth to analyse the evolution of carcinoma. A model of tumourinduced capillary growth was proposed by Balding and McElwain [10] in 1985. Inthe 1990s, Tracqui et al. [11] and Panetta [12] added chemotherapy to study the ef-fects of chemotherapeutic agents on spatio-temporal growth and tumour recurrence,respectively. Recently, L´opez et al. [13, 14] formulated a model of tumour growth withcytotoxic chemotherapeutic agents to analyse the role of dose-dense protocols.The immune system has as its main function to protect the body against infectionand illness. It can recognise the cancerous cells and eliminate them, though, the cancercan weaken the immunity [15]. The cancer treatment that takes advantage of the im-mune system is known as immunotherapy [16]. Some therapies based on the immunesystem consists of monoclonal antibodies, vaccines, and T-cell transfer [17]. Mathe-matical and computational studies of cancer immunotherapy have been performed tounderstand the interactions between immunity and tumour growth [18, 19]. Borges etal. [20] presented a tumour-immune model with chemotherapy treatment. They con-sidered a time-delay between the conversion from resting to hunting cells, the mainimmune system reaction. Ren et al. [21] demonstrated analytical result for impulsechemotherapy parameter to eliminate the cancer cells.Cancers can develop resistance to chemotherapeutic agents [22]. Drug resistanceis a phenomenon that occurs when cancer cells are unaffected by chemotherapy. Ex-periments have yielded information about the mechanisms of cancer drug resistance[23]. Sun et al. [24] modelled drug sensitive and resistant cancer cells in response tochemotherapeutic treatment. Trobia et al. [25] created a model of brain tumour growthwith drug resistance. They demonstrated that the time interval of the drug applicationplays an important role in the treatment to eliminate the cancerous cells.In this work, we include drug resistance in the tumour-immune model proposed byBorges et al. [20] and analyse its effect on the system. In our mathematical model, theimmune system is composed of resting and hunting cells, while the cancer is separatedinto drug sensitive and drug resistant cells. We consider chemotherapy to combat thetumour growth. However, the chemotherapeutic agents also attack the immune system.We show that the tumour growth can be controlled by means of different chemotherapyprotocols.The paper is organised as follows. In Section 2, we introduce the tumour-immunesystem with drug resistance. Section 3 presents our results about the effects of the drugresistance. In the last Section, we draw our conclusions.
Cancer drug resistance has been a difficulty in chemotherapy cancer treatment [26],and the challenge is how to identify and avoid the resistance [27]. Many researchershave carried out tests to find new strategies in the treatment of tumours associated withdrug resistance [28].In this work, we proposed a mathematical model that describes cancerous cellgrowth, where we include cancer drug resistance. The cancer cells are separated into2ensitive and resistant cells, as they are attacked by the immune system. In the immunesystem, the resting cells are converted to hunting cells. We consider that the cancerousand resting cells have a logistic growth, while the hunting cells have a form of pro-grammed cell death, known as apoptosis. The chemotherapeutic agent is applied to killthe cancer, it affects all cells, except the drug resistant cancer cells, as shown in Fig. 1.Figure 1: (Colour online) Schematic representation of the model.The model is given by dC S dt = q C S (cid:18) − C S + C R K (cid:19) − α C S H − uF [ Z ] C S − p C S Za + C S , (1) dC R dt = q C R (cid:20) − C S + C R K (cid:21) − α C R H + uF [ Z ] C S , (2) dHdt = β HR ( t − τ ) − d H − α H [ C S + C R ] − p HZa + H , (3) dRdt = q R (cid:18) − RK (cid:19) − β HR ( t − τ ) − p RZa + R , (4) dZdt = Φ − (cid:18) ζ + g C S a + C S + g Ha + H + g Ra + R (cid:19) Z , (5)where C S and C R are the concentration of drug sensitive and resistant cells (kg.m − ),respectively, H is the concentration of hunting cells (kg.m − ), R is the concentration ofresting cells (kg.m − ), Z is the concentration of the chemotherapeutic agent (mg.m − ), t is the time (day), τ is the delay time from resting to hunting cells, and F ( Z ) is afunction defined as F ( Z ) = (cid:26) , Z = , Z > . (6)3esides that, p i represents the predation coefficient of the chemotherapeutic agent, a i corresponds to the rate at which the cells achieve the carrying capacity when there is nocompetition and predation, and g i represents the combination rates of the chemothera-peutic agent with the cells [29]. The parameters p i and g i are related with the strengthof the Holling type 2 interaction functions. Holling [30] proposed types of functionalresponses to different types of interactions. The type 2 function describes the responseof many interacting predators and has the characteristics of decelerating the intake rate.The parameter values that we use in our simulations are given in Table 1.Table 1: Parameters values according to the literature.Parameter Values Description q .
18 day − Proliferation q .
18 day − rate [31, 32] q . − d . − Death rate [33] β . × − (cells · day) − Conversion rate [33] Φ −
200 mg(m .day) − Chemotherapy [34, 35] ζ . − Absorption rate [20] u − day − Mutation rate [25] α , α . × − (cells · day) − Competition α . × − (cells · day) − coefficients [33] K × cells Carrying K × cells capacity [31, 32] τ . c s = C S / K T , c r = C R / K T , h = H / K T , r = R / K T , and z = ζ Z , where K T = K + K and t ∗ = t / day . We consider K ∗ = K / K T , K ∗ = K / K T , u ∗ = u day, d ∗ = d day, β ∗ = β K T day, Φ ∗ = Φ day, ζ ∗ = ζ day, q ∗ i = q i day, α ∗ i = α i K T day, p ∗ i = p i / ( ζ K T ) day, g ∗ i = g i day, and a ∗ i = a i / K T ( i = , , dc s dt = q c s (cid:18) − c s + c r K (cid:19) − α c s h − uF [ z ] c s − p c s za + c s , (7) dc r dt = q c r (cid:20) − c s + c r K (cid:21) − α c r h + uF [ z ] c s , (8) dhdt = β hr ( t − τ ) − d h − α h [ c s + c r ] − p hza + h , (9) drdt = q r (cid:18) − rK (cid:19) − β hr ( t − τ ) − p rza + r , (10) dzdt = Φζ − (cid:18) ζ + g c s a + c s + g ha + h + g ra + r (cid:19) z , (11)The dimensionless parameter values are given in Table 2.4able 2: Dimensionless parameters.Parameter Values q . q . q . d . β . × − p × − p × − p × − a × − a × − a × − g , g , g . α , α . α . × − K / K / Many different powerful chemicals and clinical protocols have been used to elimi-nate a wide variety of cancers. In this work, we consider both continuous and pulsedchemotherapy treatments. In our simulations, the initial conditions are given by c s ( ) = . c r ( ) = . h ( ) = . r ( ) = .
48, and z ( ) = . Figure 2 displays the behaviour of the time evolution of c s (red line), c r (blue line), h (black line), and r (green line) when there is no cancer drug resistance ( u =
0) fora continuous chemotherapy treatment. Increasing the value of chemotherapy dose Φ from 0 .
02 (Fig. 2(a)) to 0 .
025 (Fig. 2(b)), we observe that the cancer (red line) is killedwhile the cells of the immune system (black and green lines) remain alive.Drug resistance is one of the many problems in the cancer therapy. This phe-nomenon is considered in our model when the mutation rate u >
0. In Fig. 3, we seethe appearance of drug resistant cancer cells (blue line) due to u = . Φ from 0 .
02 (Fig. 3(a)) to 0 .
035 (Fig. 3(b)), we verify a temporary cancer remission( c s ( t ) < . c r ( t ) < . t equal to 434 and 557 days, respectively. Thesensitive cancer cells are suppressed by the chemotherapy and the immune system.However, the immune system by itself is not sufficient to suppress the resistant cancercells.We compute the parameter space p × Φ to identify the regions in which the cancerremission occurs. Figure 4(a) displays the situation without drug resistance, namely5 t (day) (a)(b) Figure 2: (Colour online) Time evolution of c s (red line), c r (blue line), h (black line),and r (green line) for (a) Φ = .
02 and (b) Φ = . t (day) (a)(b) Figure 3: (Colour online) Time evolution of c s (red line), c r (blue line), h (black line),and r (green line) for u = . Φ = .
02, and (b) Φ = . u =
0. We separate into three regions: cancer growth ( c s > c s < . h < . p and Φ (black region), but it is suppressed for larger values (yellow region).Higher values of these parameters not only lead to the killing cancerous cells, but alsoweaken the immune system with the remission of the hunting cells (red region). Whenthere is drug resistance, the temporary cancer remission for 100 days (0 . ≤ c r < . p and Φ for u = . Pulsed administration of chemotherapeutic drugs, also known as intermittent therapy,is a clinical protocol in which the drug is administered and followed by a rest period.In our simulations, we use periodically pulsed chemotherapy and analyse different restperiods to find cancer remission. 6 -4 -3 -2 p -4 -3 -2 p Φ Cancer Cancer RemissionHunting Cells RemissonCancer Remission Cancer Hunting Cells Remisson (b)(a)
Figure 4: (Colour online) Parameter space p × Φ for (a) u = . u = . c s > . c s < .
001 and 0 . ≤ c r < .
01, yellow), and hunting cells remission ( h < . c s , (b) c r , (c) h , and (d) r , respectively, for Φ = . u = . × time interval). We do not observe a significant difference between theprotocols 2 ×
15 (blue line) and 1 ×
10 (black line). However, both are better than theprotocol 5 ×
23 (red line), due to the fact that the times for suppression and remissionof c s and c r , respectively, are shorter than 5 ×
23. The suppression of c s occurs for t approximately equal to 625 for 5 ×
23, and about 500 for 2 ×
15 and 1 ×
10. Thetemporary remission ( c r < .
1) starts approximately 615 days after the chemotherapytreatment according to the protocol 5 ×
23, and about 450 days for the protocols 2 × × α ),and the delay time from resting to hunting cells ( τ ). Figure 6 exhibits the parameterspace α × τ for the protocol 5 ×
23, where we consider cancer remission when c s < . c r < . c s ≥ . c r ≥ . Φ from 0 . . Φ value, thecancer remission is obtained for smaller τ value.We also compute the parameter space Φ × τ for the protocols 1 ×
10 and 5 ×
23, asshown in Figs. 7(a) and 7(b), respectively. Comparing Fig. 7(a) and Fig. 7(b), we seethat not only Φ and τ are important, but also the type of protocol is relevant to increasethe cancer remission region. The cancer remission region is smaller for 5 ×
23 than1 ×
10. 7 c s ( t ) c r ( t ) h ( t ) t (day) r( t ) (a)(b)(c)(d) Figure 5: (Colour online) Time evolution of (a) c s , (b) c r , (c) h , and (d) r for Φ = . u = . × time interval):5 ×
23 (red line), 2 ×
15 (blue line), and 1 ×
10 (black line). α Cancer Remission Cancer 0.81.21.51.838 41 45 48 α τ Cancer Remission Cancer (a)(b)
Figure 6: (Colour online) Parameter space α × τ for the protocol 5 ×
23, (a) Φ = . Φ = .
25. The yellow and black regions correspond to the cancer remissionand cancer.
Drug resistance is responsible for a vast majority of cancer deaths and it is one of themajor challenges in chemotherapy treatment. Initially some cancers are susceptible tochemotherapeutic agents, however over time they can become resistant. Due to thisfact, strategies have been used to eliminate resistant cancer cells.In this work, we study the effects of the drug resistance in the tumour-immune sys-tem with chemotherapy treatment. The immune system is composed of resting cellsthat can transform into hunting cells. We separate the cancer into drug sensitive and8 .10.40.70.9 30 35 40 45 50 55 60 Φ Φ τ Cancer Remission Cancer Cancer Remission Cancer (a)(b)
Figure 7: (Colour online) Parameter space Φ × τ for the protocols (a) 1 ×
10 and (b)5 ×
23. The yellow and black regions correspond to the cancer remission and cancer.drug resistant cells. In our simulations, we consider continuous and pulsed chemother-apy treatment.In the continuous chemotherapy treatment, we verify that cancer remission is pos-sible for smaller values of the chemotherapy intensity and the coefficient of chemother-apeutic agent on the sensitive cancer cells. The sensitive cells are eliminated, while theresistant cells are responsible for the remission. With regard to the pulsed chemother-apy, we analyse three types of protocols (days of administration × time interval):5 ×
23, 2 ×
15, and 1 ×
10. The protocols 2 ×
15 and 1 ×
10 exhibit almost the sameresults. In both protocols, the time for the elimination of sensitive cancer cells and thebeginning of the temporary remission are less than the protocol 5 ×
23. Furthermore,for all protocols, we show that the time delay from resting to hunting cells plays acrucial role in the combat against cancer cells.Our results are in agreement with recent experimental findings related to chemo-immunotherapy. In 2020, Roemeling et al. [36] carried out treatments to induce im-mune response against a type of brain tumour. They reported a therapeutic modulationthat is able to generate potent hunting cells. In our model, the hunting cell efficiencyis increased by means of the competition coefficient between hunting cells and cancerin which the hunting cells kill the cancerous cells. Maletzki et al. [37] in 2019 demon-strated that the combination of immune-stimulating vaccination and cytotoxic therapycan improve long-term survival. Depending on the protocol, they observed tumour freein mice from 25 to 65 weeks. In our simulations, the tumour free occurs about 25weeks. Nevertheless, for small time delay from resting to hunting cells in our model, itis possible to use different protocols aiming to maximise the tumour free time.9 cknowledgement
This study was possible by partial financial support from the following Brazilian gov-ernment agencies: Fundac¸˜ao Arauc´aria, National Council for Scientific and Techno-logical Development, Coordination for the Improvement of Higher Education Person-nel, and S˜ao Paulo Research Foundation (2015/07311-7, 2017/18977-1, 2018/03211-6,2020/04624-2).
References [1] M H¨ockel and U. Behn, Frontiers in Oncology
416 (2019)[2] R L Siegel, K D Miller and A Jemal, CA: A Cancer Journal for Cinicians
64 (2005)[4] V T DeVita Jr and E Chu, Cancer Research
21 (2004)[6] J Couzin-Frankel, Science
113 (2015)[8] H N Weerasinghe, P M Burrage, K Burrage and D V Nicolau Jr, Journal of On-cology
317 (1972)[10] D Balding and D L S McElwain, Journal of Theoretical Biology
53 (1985)[11] P Tracqui, G C Cruywagen, D E Woodward, G T Bartoo, J D Murray and E CAlvord Jr, Cell Proliferation
17 (1995)[12] J C Panetta, Bulletin of Mathematical Biology
425 (1996)[13] A G L´opez, K C Iarosz, A M Batista, J M Seoane, R L Viana and M A F Sanju´an,Communications in Nonlinear Science and Numerical Simulations
307 (2019)[15] H Gonzalez, C Hagerling and Z Werb, Genes & Development
73 (2016)[17] S J Oiseth and M S Aziz, Journal of Cancer Metastasis and Treatment
159 (2000)[19] A Konstorum, A T Vella, A J Adler and R C Laubenbacher, Journal of the RoyalSociety Interface
43 (2014)[21] H P Ren, Y Yang, M S Baptista and C Grebogi, Phylosophical Transactions A
615 (2002)[24] X Sun, J Bao and Y Shao, Scientific Reports
299 (2019)[27] J P Godefridus, Cancer Drug Resistance
980 (2019)[29] S T R Pinho, H I Freedman and F Nani, Mathematical and Computer Modelling
773 (2002)[30] C S Holling, Memoirs of the Entomological Society of Canada
268 (2008)[33] V A Kuznetsov, I A Makalkin, M A Taylor and A S Perelson, Bulletin Mathe-matical Biology (2) 295 (1994)[34] R Stupp, W P Mason, M J Van den Bent, M Weller, B Fisher, M J B Taphoorn, KBelanger, A A Brandes, C Marosi, U Bogdahn, J Curschmann, R C Janzer, S KLudwin, T Gorlia, A Allgeier, D Lacombe, J G Cairncross, E Eisenhauer and RO Mirimanoff, The New England Journal of Medicine
987 (2005)[35] H M Strik, C Marosi, B Kaina and B Neyns, Current Neurology and NeuroscienceReports
286 (2012)[36] C A von Roemeling, Y Wang,Y Qie, H Yuan, H Zhao, X Liu, Z Yang, M Yang,W Deng, K A Bruno, C K Chan, A S Lee, S S Rosenfeld, K Yun, A J Johnson, DA Mitche, W Jiang and B Y S Kim, Nature Communications7