The Clinical Application of MathematicalPathology

Mathematical pathology is a research branchof pathology in which mathematics andphysical principles are applied to the study ofdiseases. In the field of cancer research, theobjective of mathematical pathology is to modeland explain the structural and functionalmechanisms that control cancer. By under -standing these mechanisms, we can providequantitative predictions of how the tumor willbehave. Here, highlighted by our recent workon ductal carcinoma in situ (DCIS), we assessand provide the justification for mathematicalpathology as a useful way to characterise cancer.

Current clinical practice of DCIS treatment

DCIS is an early form of breast cancer in whichthe tumor originates within and is confined to

the breast duct. While DCIS is not a fataldisease, it oen progresses to invasive ductalcarcinoma, which accounts for 80% of all breastcancer diagnoses. In many cases, women withDCIS will elect to have breast conservationsurgery, in which the tumor and a portion ofthe surrounding breast are removed. Whenpathologists examine patient biopsies of breasttissue, they are looking to see whether theneoplastic cells have stayed within the duct(in situ disease) or invaded through it. epathologists are also looking at how aggressivethe cells look (i.e., the tumor grade), amongother features. Before planning surgery, a teamof physicians (radiologists, pathologists, andsurgeons) will determine the boundary of thebreast tissue affected by DCIS and decidehow much tissue needs to be removed. Oen,immediately before surgery, a radiologist placesa needle or other marker to localise the centerof the tumor with the help of X-rays. In theoperating room, the surgeon will then measurethe amount of tissue to remove around theneedle-localised tumor.

Determining the optimal surgical volume forresection thus becomes essential for thecontrol of the tumor and the cosmetic resultof the surgery. If too much tissue is removed,the patient could suffer the physical andpsychological effects of a large defect thatleaves the breast looking uneven; if too littleof the affected breast is removed, the patientmay need to undergo one or more surgeries tocompletely remove the affected area ofmalignant cells. e imperfection of breastconservation surgery is exemplified by the factthat the surgery fails to remove all of the tumorin 38-72% of all cases.

The Clinical Application of Mathematical Pathology

1 Department of NanoMedicine and BiomedicalEngineering, University of Texas Health ScienceCenter at Houston (UTHealth) McGovernMedical School, Houston, TX 77054, USA

2 Brown Foundation Institute of MolecularMedicine, University of Texas Health ScienceCenter at Houston (UTHealth) McGovernMedical School, Houston, TX 77030, USA

3 Department of Radiation Oncology, Universityof Texas MD Anderson Cancer Center, Houston,TX 77030, USA

Vittorio Cristini1,2, Eugene J. Koay3, and Zhihui Wang1,2

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A quantitative approach to breastconservation surgery for DCIS

We sought to address the clinical uncertainty ofremoving DCIS. We developed a mathematicaltheory of tumor growth that assumed that thebalance of cell proliferation and cell death is akey determinant of tumor size. We also tookinto account a number of clinical characteristicsof DCIS in building the model. For example,younger tumors will have a viable rimcharacterised by cells in a state of proliferation.More advanced DCIS will be characterised by arim with more cells in a state of apoptosis(programmed cell death) and a necrotic core ofdead cell material that has built up along thecentral ductal axis as a result of the staticpressure delivered by the proliferating cells atthe rim of the tumor. By the time DCIS showsup on a mammogram, most patients willhave had the disease for at least five months,as this is the time necessary for build-up ofmicro-calcifications in individual ducts, whichare then detectable by mammography. isresult indicates that when DCIS is detected, ithas usually progressed past the initial fast-growth phase.

e goal of our mathematical pathology workin relation to DCIS is to accurately describehow DCIS develops. By doing so, we would beable to estimate the size of the tumor and aid intherapeutic planning to define the volume ofbreast tissue which must be surgically removed.An accurate assessment of surgical volume isinstrumental in providing physicians and alsopatients important information before surgery.rough mathematical modeling of cancer, it ispossible to describe where a tumor is in itsdevelopment, and, for DCIS, where the edgeof the tumor is located. We designed ourapproach so that all data being supplied to themathematical models could be derived from apatient’s biopsy and integrated into the clinicalworkflow.

Mathematical modeling of DCIS

In order to describe the nature and developmentof specific tumors within a patient, specificvalues are required for parameters inmathematical equations. In our modeling workon DCIS (Edgerton et al., Anal Cell Pathol2011, PMC3613121), three key parametersare required, and they can all be assessed inpathological analysis of patient biopsiesperformed at a single point in time as a step inbetween detection through mammography andsurgical planning. ey are: the ProliferativeIndex (PI), the Apoptotic Index (AI), and thediffusion-penetration length. e fractions ofcells in the proliferative and apoptotic states aredefined as the proliferative index and apoptoticindex, respectively; the ratio of these twoindices is a key parameter in a cell-scale modelof tumor growth that forms the starting pointfor modeling the area of the entire tumor.e diffusion-penetration length (or depth) is a measure of how far molecules (e.g.,chemotherapy drug, nutrients) diffuse througha medium (e.g., the tumor tissue). Note thatthe three key parameters are related to tumordevelopment, involve cell proliferation anddeath, as well as the physical ability of cells toreceive nutrients depending on the diffusionproperties of the microenvironment, and willprovide output useful in determining the tumormargin. Our major purpose with the modelwas to find out if a specific diagnosis of tumorsize is achievable through measurements thatare taken from a single biopsy (for quantifyingapoptotic/proliferative indices and diffusionpenetration length).

We assume that the tumors (of our primaryinterest here) have passed their initial fast-growth phase and reached a size relatively closeto their final volume due to the static pressurebalance between tumor cell proliferation andtumor cell death. e tumors in DCIS grow to apoint where their growth is arrested due to the

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lack of viable cells capable of proliferating; atthis point nearly all the cells in the core of theduct have undergone lysis (death), with theirdecayed leover material contributing to thenecrotic core of the duct and accompanyingmicro-calcifications. Hence, there is a balancebetween tumor cell proliferation and tumorcell death, and the corresponding ratio oftumor cell proliferation-to-apoptosis, along withthe diffusion penetration length. is balancereveals the mechanism behind which themathematics can model the tumor’s growth andpredict – when the models are scaled up to thetissue scale – the final geographic area of thetumor, based on one measurement at a singletime point.

From a previously developed model (Cristini etal., J Math Biol 2003, 46(3):191-224), we obtainthe following analytic solution:

A=3._. , (1)

where A is the patient-specific ratio of cellapoptosis to proliferation rates averaged overthe multitude of ducts within the surgicalvolume, L is the diffusion penetration distanceof nutrients, and R is the geometric-meantumor surgical radius. is equation can becalibrated from the results of immunohisto -chemistry (IHC), which determines cellproliferation and death. IHC involves stainingthe cells in the proliferative and apoptotic state,and provides the measurements that drive theproliferative and apoptotic index values.Specifically, in Eq. 1, A and L can be derivedfrom pathology measurements taken onspecific patients’ tissue (see the original article).When A and L are known, determining thevalue of R is simply a matter of mathematics.In this way, this equation uses the cell-scalevalues for calibrating the tissue-scalecontinuum model predicting surgical volume.

How is mathematical pathology more useful indetermining the surgical volume for DCIS

Figure 1. Correlation of tumor size with the death-to-proliferation ratio parameter. Tumor geometric-meandiameters 2R (dashed) vs. L/A predicted by the model compared to the corresponding pathology measurements.In contrast, grades based on histopathology are clearly poor predictors of tumor size. Data were obtained fromthe 17 excised tumors (symbols, with de-identified case numbers). Reproduced with permission from Edgerton etal., Anal Cell Pathol 34(5):247-63.

LR

LR(_________–_)1

tanh(R/L)

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patients than current standard-of-careimaging? Our data show that surgical volumesdetermined by radiology and tumor grade arelargely inaccurate when compared to surgicalresults. Our study involved examining 17excised DCIS tumors, and we found thatmammography overestimated the tumor sizein ten cases and underestimated the tumorsize in seven cases. e correlation betweenmammography, nuclear grade, and the finalobserved tumor size is poor. However, thecorrelation between the sizes observed aersurgery and those predicted by the model wereclose (see Fig. 1 for an example). Briefly, modelresults were very encouraging, with predictionsof tumor size and surgical volume being farmore accurate when based on patient-specificbiomarkers than were the predictions of thesame tumor’s surgical volume based onmammography.

e implication of the accuracy of our modelpredictions is that with standard mammogramand pathologic specimens, physicians should beable to accurately predict the surgical volumeof DCIS tumors mathematically. is willresult in a lower chance of additional requiredsurgeries. e study also confirms that tumorgrade or tumor dimensions from mammographyare inconsistent with actual tumor volume.Together, this study represents a proof ofprinciple that it is possible to incorporate amathematical modeling step within currentclinical practice to aid in and improve surgicalplanning by estimating the surgical volumeand the outcome of surgery before treatment.Since IHC and morphometric measurementscan be performed on patient-specific breastbiopsies, the clinical value of this mathematicalpathology approach is that the prediction andresulting surgical planning can be tailored forthat particular patient. e practice will lead toless subjective analysis of tumors and improvedsurgical treatment efficacy through individualisedtreatment design.

Prediction of tumor growth

We then developed an even more detailedmodel to predict tumor progression starting atthe microscopic scale using a lattice-free agent-based modeling (ABM) approach (Macklin etal., J eor Biol 2012, PMC3322268). ismodel is the first to account for how cellularcalcification influences tumor progression, andis fully constrained to patient-specific clinicaldata easily obtained from histopathology.

We coupled the cells (i.e., agents) with themicroenvironment by introducing fieldvariables for key microenvironmentalcomponents (such as oxygen, growth factors,and ECM) that are governed by reaction-diffusion equations. In the model, cells alter theevolution of the environmental variables, andthese variables also affect the cells’ behavior.We obtained most of the model parametervalues from histopathology data analysis usingthe method described above (Edgerton et al.,Anal Cell Pathol 2011, PMC3613121). Withparameter values quantified for each individualpatient, the model can then be used to simulatethe growth dynamics of DCIS. For instance,we used the ABM to predict the quantitativerelationship between the mammographic (xV)and pathologic tumor sizes (xC), and obtain:

xV ≈ 0.4203+1.117xC mm. (2)

We compare the plot of this equation againstour simulated DCIS data (blue points; Fig. 2a)and a set of published clinical data (red squares;Fig. 2b) extracted from (De Roos et al., World JSurg Oncol 2004, 2:4, PMC394346). We findthat our model not only correctly predicts alinear correlation between a DCIS tumor’smammographic and pathologic sizes, but alsodemonstrates an excellent agreement withpublished clinical data two orders of magnitudelarger than our simulation data.

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ese successful quantitative comparisonsshow that this model may serve as a predictivesimulator to create a patient-specific mapbetween the micro-calcification geometry (asobserved in mammography) and the actualtumor morphology. e model allows surgeonsto more precisely plan DCIS surgical marginswhile removing less normal tissue. While themathematics behind the predictions is complex,the model begins with data that is elegant in itssimplicity: every portion of data taken frompatient pathology can be generated in mostcancer centers nationwide – the practice canbe replicated just about anywhere cancer istreated, and is the starting point for what drivesthe models. No extra steps or tests are neededto obtain the necessary parameters that drivethe model, making the model easy to apply andconvenient to use for physicians.

Clinical implications and future directions

e successes in predicting tumor volume andgrowth rates based on measurements takenfrom microscopic pathology illustrate thepromise of mathematical modeling of cancer inthe development of clinical treatment plans,especially when planning for breast conservationsurgery. Integration into clinical trials canhelp establish this quantitative method asstandard practice.

Our mathematical approach also has relevanceto cancer therapeutics like chemotherapy. Asdemonstrated in our other clinically relevantmodeling work (Pascal et al., Proc Natl AcadSci U S A 2013, PMC3761643; Koay et al., JClin Invest 2014, PMC3973100), macroscopicimaging such as computed tomography (CT)

Figure 2. Comparison of mammographic (xV) and pathologic (xC) DCIS sizes. (a) A linear correlationbetween xC and the actual pathology-measured xV is found from our ABM simulation results. (b) A linearleast-squares fit to our simulation data (blue circles) fits a clinical dataset (red squares), furtherdemonstrating the predictivity of the ABM model. Reproduced with permission from Macklin et al., J eorBiol 2012 301:122-40.

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scans can also be used to derive data to drivemodels predicting the fraction of tumor killedby chemotherapy in patients with colorectalcancer that metastasised to the liver and inglioblastoma. ese will be the foci of futurearticles.

In the development of our theories andmathematical models, we use data that can begained and integrated in patient treatmentplans in almost any cancer treatment facility.With further research, we hope to show thatwe can accurately predict clinically relevantoutcomes so that these models can assistphysicians and patients in making treatmentdecisions.

Contact information:

Vittorio Cristini, PhDUTHealth McGovern Medical School Department ofNanomedicine and Biomedical EngineeringCenter for Advanced Biomedical Imaging (CABI)building, Room 3SCR6.46441881 East Road, Houston, TX 77054, USAhttps://med.uth.edu/nbme/faculty/vittorio-cristini/ISI Highly-Cited Researchers in Mathematics:http://highlycited.comGoogle scholar:http://scholar.google.com/citations?user=uwl5tw0AAAAJ&hl=en&oi=aoTel: +1 713 486 2315Fax: +1 713 796 9697Email: Vittorio.Cristini@uth.tmc.edu

Further reading:http://physics.cancer.gov/http://www.nsf.gov/funding/pgm_summ.jsp?pims_id=6673http://www.internationalinnovation.com/taking-cancer-out-of-the-equation/http://www.pnas.org/content/110/35/14266.longhttp://www.jci.org/articles/view/73455

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www.uth.edu