Computational simulation including interpretation of the importance of the immune system in tumor
growth
Nidhi S Vaishnaw1 & S. Shankar Narayan2
1,2 Department of Mathematics, CMR Institute of Technology , Bengaluru, India Shankar.n@cmrit.ac.in1 & Nidhi.v@cmrit.ac.in2
Article History:Received:11 January 2021; Accepted: 27 February 2021; Published online: 5 April 2021
Abstract: The secret to tumor growth and survival is proliferation. Therefore, to plan and formulate a required therapy process, the study of the propagation rate is very important. The objective of the immune system is to protect against disease or other potentially harmful foreign bodies, a set of processes and mechanisms within the body. In regulating the growth of a tumor, different cells of our immune system perform their assigned function. Natural Killer cells, Dendritic cells, and CD8+ cells are among the many cells. In our scientific literature, in the presence of numerous family components of the human immune system, we have developed a mathematical model to evaluate the dynamics involved in tumor formation. We separated the population of tumor cells into proliferating and inactive subsets and felt that the cells of the immune system had little effect on the dormant cells. This partition of the universal tumor set into the subsets as mentioned has never been made in any of the previous research works. Also, to the best of authors’ knowledge, the compartment combination considered for the present work adds novelty to the article.
Keywords: Proliferation, Quiescent, Natural killer cells, Dendritic cells, CD8+ cells
1. INTRODUCTION
The role of the immune system in controlling cancer development, both mathematically and clinically, means that mathematical models that integrate tumor growth and treatment would do well to include a portion of the immune system. It is also possible to model how various immune therapies, once the variable is in operation, will influence the system in combination with each other or one by one. The latest clinical evidence presented and applied by Wheeler et al.[1] showed that paired with traditional cancer therapy, the capacity of the immune response has important benefits. In tandem with chemotherapy, it has been found that vaccine therapy makes outpatient survival periods more successful than either chemotherapy or vaccine therapy alone. As a result, the resulting statistical oncology continues to be a fundamental area for developments in medical therapy. It assures the inclusion of tools by precisely structured statistical formulas for both quantifying success indicators from patient-specific scientific data and tailoring cancer care. In the past few years, mathematical oncology has progressed into a complex project, advancing at a phenomenal pace.
One of the key facts of oncology statistical modeling is the simulation of the various oncological treatments used by the different forms of cancer. Model for surgery way of treatment for various cancers has been made by Hanin et al. [2] and Enderling et al. [3] Farrar et al.[4] developed a mathematical model to study the significance of the immune system in fighting cancer. The chemotherapy model was developed by several teams. Powathil et al. [5] investigated a model for chemotherapy and radiation therapy for brain tumors. The team considered a simple Spatio-temporal model to include the effect of both radiation and chemotherapeutic treatments.
It is known that persons with weakened immune systems are more likely to develop certain rare forms of cancer, particularly those with AIDS. This event can be viewed as providing more proof that the role of the immune response in the battle against cancer is crucial. See, for example, Dalgleish and O'Byrne [6]. A two-compartment tumor cell model consisting only of proliferating and quiescent cells were proposed by Gyllenberg and Webb [7]. They believed that dead cells will be automatically eliminated from the tumor, which is in reality, incorrect. Their major contribution is their theory that in a normal multicellular avascular tumor spheroid, proliferating cells may travel into and out of sleep.
Many mathematical models have suggested that the absence of the immune system leads to the growth of the tumor [8].Studies have figured out that the Natural Killer cells and cytotoxic T cells are responsible for killing the tumor cells.Natural killer cells and cells CD8+T are responded to tumor cells with high cytolytic behavior as per Osadaet al.[9]. Natural killer cells are part of the innate immune system, although there are no tumor cells. Roitt et al.,[10] noticed enabled tumor-specific CD8+T cells are found only in substantial numbers while tumor cells are present as part of the immediate immune response; natural killer and CD8+T cells are transformed into inactive cells after multiple tumors encounter[11].
sIn sour sprevious sstudy, swe semphasized sthe simportance sof sdead scell sdensity sin sthe scase sof sthe sdormant sstage sof sthe stumor. sAs sa scontinuation sof sour sprevious sstudy, swe smodel sthe simmune ssystem swith sa sgrowing spopulation sof stumor scells. sWe sdescribe sthe sinteraction sbetween sfour smain scell spopulations namely s sTumor scells ,Natural
skilling scells , sDendritic scells , sand sCytotoxic s sCD8+ scells .This sarticle scomprises sfour ssections. sIn ssection s2, swe sexplore sthe srole sof simmune scells sin santi-tumor sactivity stheoretically, sand sthen sa ssuitable smathematical smodel sis sconstructed sto sportray sthe stheoretical slearning. sSection s3 spresents sthe smathematical sanalysis sof sthe smodel sconstructed- sstability shas sbeen sdiscussed. sIn ssection s4, swe sanalyze sthe sresults sobtained sby ssolving sthe ssystem sformulated sin sthe sprevious ssection susing snumerical stechniques, sthe sRunge sKutta smethod. sIn sthe sfinal ssection, swe spresent sthe sadvantages sand sdisadvantages sof sthe smodel sformed swith sfuture sdevelopments spossible. A. MATHEMATICAL MODEL
Specific sprelude slymphocytes sin sthe ssame sfamily sas sT scells sand sB scells sare snatural skilled s(NK) scells. sNK scells showever sreact srapidly sto sa swide svariety sof spathological scomplications sas sendogenous simmune ssystem scells. sNK scells sare sbetter spositioned sto scope swith scontaminated scells sand sto sdetect scancer searly sindications. sFirst, sit swas snoticed sthat sNK scells scould sdestroy sor sactivate sleading stumor scells. sNK scells soften scontain scytokines, swhich sare sused sto senhance sthe simmune sresponse sagainst scertain simmune scells, sfor sexample, smacrophages sand sdendritic scells. sAs sthe sNK scells sare sin sdirect scommunication swith sother scells, swhether sor snot sthe sNK scells sdestroy sthem sdepends son sthe scompatibility sof sthe ssignals sof sactively sreceiving scells sand sthe sinhibitory sreceptors sof sthe sNK scell ssurface. sReceptor sreceptors srecognize sand sdisable smolecules sreleased son sthe ssurface sof scancer sand scontaminated scells.
Dendritic scells s(DC) sare sa sdiverse, sheterogeneous sgrouping sof sthe stumor sinvading sinnate simmune scells. sDC splays san sessential spart sin sthe simmunity srecovery sof santi-tumor sT scells sand sthus sis sone sof scancer simmunotherapy’s smain stherapy sstrategies.
CD8 s+ sT scells sare sanother sgroup sof scells sinvolved sin stumor scell scombat. sFor simmune sreaction sto sand scontrol sof sintracellular sdiseases, sincluding sviruses sand sbacteria, sthe scytotoxic sT slymphocytes sor sCD8 s+ scells sare sextremely simportant. sA sCD8+T scell sdevelops sthree ssignificant spathways sfor sthe sdestruction sof sdamaged sor sdiseased scells sin sappreciation sof sthe santigen. sIn saddition sto sthe smajor simmune sprotection sroles sof sCD8+ sT scells sagainst spathogens, sintracellular sbacteria, sand stumors, sdefective simmune sresponses sthat smay slead sto simmune sdisturbances sand simmune-mediated sharm sare salso spossible.
Figure s1. sBlock sdiagram srepresenting sthe smodel sconstructed
Many sof sthe sabove-mentioned scells sare sthe sactivity sin svitro seffectors sfor svarious stumors. sHowever, stumor sor scell-associated sinhibitory ssignals simpede stheirs sin svivo sefficacy. sIf sdormancy scells sare snot saffected sby sNK, sDC, sand
sCD8+, swe sdevise sthe smodel sbelow sto sdetermine show stime-dependent sthe scell sdensities sare. sUsing sall sof sthe saforementioned spremises, swe sare sformulating sthe sfollowing snonlinear sordinary sdifferential sequation sscheme. sFigure s1 srepresents sthe scompartmental smodel sfor sthe smodel sconstructed.
s s s s s s s s s s s(1) Where
Here sP(t) sand sR(t) srepresents sthe sproliferation sand sdormant scell sdensities srespectively, sr(t) splays sthe srole sof sproliferating scells’ slimiting snutrient suptake srate sfunction, sf(r) sand sg(r) sare sthe srates sof sproliferation sand srate sof stransition sfrom sproliferation sto sdormancy sstate swhich sis sassumed sto sbe sfixed, sα sis sthe srate sof stransition sof sdormant scells sto sproliferating sphase, sµ sis sthe srate sof sdeath sof squiescent spopulation, stransmission sbetween simmune scells snamely s sTumor scells s,Natural skilling scells , sDendritic scells , sand sCytotoxic s sCD8+ scells sis spresented sfurther. sThe svalues sof sthe svarious sparameters sinvolved sin sthe smodel sare smentioned sin sthe supcoming stable s1.
Table s1. sThe sparameter svalues sof sthe smodel B. sStability sAnalysis s& sEquilibrium spoints
In sthis ssection, swe sapply smathematical sinterpretation sto sdetermine stumor-free sequilibrium swhich swill sidentify sthe sconditions sto seliminate stumor scells.
Let be sa stumor-free sequilibrium sof sthe sgoverned ssystem(1). sFor s san sequilibrium spoint, swe
scan sput s s .
Where s sand sby s ssetting sleads sto sa squadratic sequation sgiven s
s Both stumor-free sequilibrium
spoints sare s
biologically smeaningful sif
Where s , s s , , , , , , , , s, , , , ,
The sTumor-free sequilibrium spoints sgive sthe sfollowing s
Then sthe sCharacteristic sEquation sof sthe sabove smatrix sis s given sby
On sexpanding sthe scharacteristic sequation,we sget
By sthe sabove sequation, sit sis sclear sthat smost sof sthe seigenvalues sof sthe sdeterminant sare sreal sand snegative sthat simplies sthe ssystem sis sstable sat sthe stumor-free sequilibrium spoint.
C. Results and Discussion
The sabove-formulated ssystem sis ssolved susing snumerical smethods s(Fourth sorder sRunge-Kutta sMethod) sand sthe sbelow-attached sgraphs sare sobtained.
Figure s2 srepresents sthe sgraphical srepresentation sof sthe spopulation sdensities sof stumor scells s(Proliferating sand sdormant), simmune ssystem scells s(NK scells, sDC, sand sCD8+ sT scells). sSubsequent sgraphs sdenote sthe spopulation sof scalls staken sindividually.
Figure s2. sPopulation sdensity sprofile sof sdifferent s cells sinvolved sin sthe sstudy.
Figure s3.Population sdensity sprofile sof s proliferative scells.
Figure s5.Population sdensity sprofile sof sNatural sKiller sCells
Figure s6.Population sdensity sprofile sof sDendritic scells spopulation.
Figure s7. sPopulation sdensity sprofile sof scytotoxic s cell spopulation.
The srate sat swhich sdormancy scatches sproliferation sis shigh, sas sreported sin sour sprevious swork, sand sit sis svery sclear sfrom sFigures s3 sand s4. sIt scan salso sbe sshown sthat sin sthe scurrent sresearch, sthe srate sat swhich sthe sdensity sof sproliferation sdeclines sis sexceptional sbecause simmune scells sare sknown sto splay san santi-tumor srole sresponsible sfor
seliminating sproliferating scells sfrom sthe senvironment. sThe smain scause sof scancer-related sdeaths[13] sin sthe stumor scell sprocess sis sthe screation sof sdangerous smetastatic sconditions sthat sdestroy snon-primary sorgans. sThe soccurrence sand smetastasis sof scancer sare sassociated swith sthe shibernation sof scancer[14], sreferring sto sthe speriod sof sprolonged sidiopathy safter sthe sprimary streatment sof scancer. sDormancy sof scancer smay sbe sconsidered sone sof sthe scauses sof sadjuvant scancer swhich smay sbe sattributed sto sfailure sto scure soriginal streatment sof sthe sprimary stumor sfor smalignancies[15]. sAs ssuch, sa sbasic sgrasp sof scancer sdormancy swill sbe sneeded simmediately sto shelp simprove scancer simmunotherapy, swhich scan sbe san seffective salternative sto smetastatic scancer.
From sfigure s5 sit sis svery sclear sthat sthe sNK scell spopulation sdepletes srapidly sdue sto sits sactions son sthe stumor starget. sNK scell sexhaustion sis san sinteresting saspect sthat sis soften slargely signored sin sNK sresponses sto sdifferent stumors. sSimilar sto swhat shappens swith sT scells, sthe sconstant saccess sto scertain starget santigens sresults sin sNK scells sbeing sdepleted. sStudies salso sindicated sthat sthe sexplanation sof swhy sthe sclinical sresult sof sNK scell sadoption sis sminimal scould sbe sdue sto sthe srapid spenetration sofNK scells sinto stumor santigens s[16]. sNK scells sshowed spotent santi-tumor sproperties sand scytokine sdevelopment searly son safter sadoption; showever, sNK scells sexhibited spoor sdevelopment sand scytotoxicity sat sthe stumor ssites sbeginning son sday s5 spost-transfer, sfigure s5. sTherefore, sdepletion sappears sto sbe sa smajor sconcern sin sthe stherapeutic suse sand stargeting sof sNK scells sduring scancer, sbecause srapid sdepletion sof sNK scells snegatively simpacts stheir santi-tumor sfunctions. sRepeated sadoptive stransfers sor sadditional simmunotherapy smay sbe srequired sto simprove sthe sfunctionality sof sNK scells sto spreserve sthe sNK scells' sstrong santi-tumor srole s[16]. s
The sdendritic scells s(DC) sare sintegral sto simmune ssystem sactivation.DC sis stargeted sto sinduce simmunologic sreactivity sin sthe ssite sof santigen sexpression, staking santigens, sand smoving sinto ssecondary slymphoid sorgans swhile sreceiving spathogens, sdying scells, sand/or scells sactivation ssignals s[17]. sTumors smay sescape sthe simmune sfunction sby sinteracting swith sthe srecruitment sof sDCs sor sby snot scommunicating seffectively. sTumors salso sfacilitate sthe sproduction sof sfactors sinhibiting sDC sdistinction sand sfunctions. sFigure s6 sshows sthe sdendritic scell sdensity swith stime. sThe sdendritic sdensity sis snever sexhausted seven sthough sthey sare sdeactivated sby sthe smisleading ssignals sfrom sthe stumor scells. sIt shas sbeen scalculated sthat sin sdifferent sspecies, sdendritic scells sare srestocked sfrom sthe sblood sat sa srate sof s4000 scells sper shour, sand sexperience sa ssmall snumber sof sspleen sdivisions sover s10 sto s14 sdays sduring stheir sresidence s[18]. sHence sthe sdepletion sis snot sseen sin sthe scase sof sthe sdendritic scells. sBut, stheir sactivity sis sregulated sby sthe stumor scells. sHence simmune stherapy sthat scan sreactivate sthese sdormant sdendritic scells shas sto sbe semployed sto sovercome sthe sfalse ssignals simparted sby sthe snaughty scancer scells. s
Easy sCD8 s+ sT-cell sactivation srequires sthe sconnection sto scells spresenting sa scompetent santigen, smainly swith smature sdendritic scells. sThis swill sincrease sthe snumber sof scells sthat sattack sthe starget sand sthat smay sthen sbe sscanned sfor santigen-positive ssomatic scells sacross sthe swhole sbody s[19].From sfigure s7, sit scan sbe sseen sthat sat sthe sbeginning sof sthe sstudy sthe sincrease sin sthe sCD8+ scell sdensity sis sslow sas sthe sactivation sof sthese scells sneeds sthe srecruitment sof sdendritic scells. sAs stime sincreases ssteadily, sthe snumber sof scytotoxic scells salso sincreases sin snumber sgradually. s
D. Conclusion
In sthe spresent swork, swe sformulated sa smathematical smodel sfor scancer sgrowth sin spresence sof svarious simmune scells sthat sfight sback sthese sfast-dividing scells sof sthe stumor. sWe svalidated sthe sresults sobtained sby scomparing sthem swith svarious sOncologies soriented sresearch spapers. sWe shave sinvestigated sthe srole sof seach simmune scell sin scombating sagainst sthe stumor scell sentities sand smisleading scharacteristic sfeatures sof sthe scancer scells sin sinactivating sthese simmune scells. sThe sstudy sclearly sshows sthe sneed sfor svarious stherapies sto simprove sthe sefficiency sof sthe simmune scells sagainst stheir sinactivating sagents. sHence simmunotherapy sis svery smuch snecessary sfor simproving sthe simmune ssystem sfunctionality sto soppose sproliferation. sNow sin sthe snext supcoming sworks sof sour screw, swe swould sconcentrate son sthe sdifferent stherapies savailable sor snecessary sfor san soncologist sto semploy. s
The smodel sformulated sdoesn't sinvolve sthe sentire screw sof shuman simmune ssystem. sThus, sthe smodel scould sbe sdeveloped sto sanalyze sthe sproliferation-dormancy srate sby sincluding sthe sother simmune scells sof sour simmune ssystem, swhich splay sa sprominent srole sin santi-tumor sactivity. sAlso, swe sassumed sthat sthe srate sof sproliferation sand sthe srate sof stransition sfrom sproliferation sto sdormant sstate swas sassumed sto sbe sfixed sduring sthe smodel sconstruction swhich sis snot svalid sfor sthe spractical ssituations. sHence swe spretend sto sconstruct sa smodel sin swhich sthe srate sis sa svariable squantity. s
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