Mathematical Modeling Predicts Response to Chemotherapy and Drug Combinations in Ovarian Cancer

Platinum-based chemotherapy constitutes the backbone of clinical care in advanced solid cancers. Cisplatin, and its less toxic analogues, carboplatin, oxaliplatin, and nedaplatin, have been used for more than 40 years in most solid cancers, including metastatic colorectal (1), triple-negative breast (2), and prostate (3) cancers. Platinum is curative in testicular cancers (4); in epithelial ovarian cancer, it has improved the 10-year survival rate by over 10% and more than doubled the number of complete responses (5, 6). Although platinum-based chemotherapy has prolonged the survival of millions of patients, majority of them still develop resistance, which usually leads to a refractory disease with very short life expectancy.

We focus herein on high-grade serous ovarian cancer (HGSOC), which is the most common epithelial ovarian cancer subtype and accounts for 70% to 80% of ovarian cancer–related deaths (7, 8). The first-line standard-of-care (SOC) therapy for HGSOC consists of a cytoreductive surgery, followed by platinum-based chemotherapy (8). In a recent guideline, the Society of Gynecologic Oncology and the American Society of Clinical Oncology recommends that patients with a high perioperative risk profile or a low likelihood to obtain optimal cytoreduction should receive neoadjuvant chemotherapy (NACT) before surgery and adjuvant chemotherapy (8). In a majority of patients, NACT reduces tumor burden significantly, as shown in Fig. 1. However, even though 90% of the patients with HGSOC show no clinically detectable signs of cancer after surgery and chemotherapy, the 5-year survival rate in HGSOC is still less than 50%, making epithelial ovarian cancer the fifth leading cause of female cancer–related deaths worldwide (9, 10).

HGSOC tumors are characterized by high intratumoral heterogeneity (11), which leads to cancer cell subpopulations with various platinum resistance mechanisms. Resistance mechanisms can exist in a subpopulation already at diagnosis (intrinsic resistance) or develop in a previously sensitive cancer cell due to chemotherapy (acquired resistance). Here we focus on intrinsic resistance, which is suggested to be the dominant resistance model in HGSOC by evolutionary studies (12, 13). Several platinum resistance–related processes have been identified, including reduced intake or increased efflux or platinum, DNA repair mechanisms, xenobiotics pathways, and dysfunctional apoptosis machinery (7, 14, 15). However, it is still unclear how many of them are active already at diagnosis even though the number of active resistance mechanisms at diagnosis and after chemotherapy has direct implications to planning treatment regimens.

Several drugs that are approved for cancer care could be used in combination with the current SOC to increase survival rates in HGSOC. However, establishing the benefit of a drug or a combination therapy over SOC in clinical trials is expensive and slow. We introduce here a stochastic mathematical model for chemotherapy resistance in HGSOC and use it to create “virtual HGSOC patients.” These virtual patients can be used to estimate the number of active resistance mechanisms at diagnosis or after treatment and to suggest guidelines for developing effective combination therapy regimens.

Our approach takes advantage of rich clinical data, including ¹⁸F-fluorodeoxyglucose PET/CT (¹⁸F-FDG-PET/CT) imaging. Cancer cells with low proliferation rate have low ¹⁸F-FDG uptake and are associated with a less aggressive disease, whereas high ¹⁸F-FDG uptake is associated with a more aggressive disease. Thus, ¹⁸F-FDG-PET/CT allows accurate estimates of total metabolic tumor burden, collected before and after chemotherapy (Fig. 1A). Our results indicate that the virtual cancer patients have indistinguishable treatment responses from real-life patients. Thus, the benefit of a targeted drug or combination therapy over SOC can be evaluated rapidly and rigorously with our approach. This facilitates translating preclinical findings efficiently into clinical trials and, eventually, patient care.

Clinical data were collected prospectively from 62 patients treated for ovarian or primary peritoneal HGSOC at Turku University Hospital from October 2009 to July 2016 (Table 1; Supplementary Table S1). All the calibration cohort patients had been evaluated as inoperable in the diagnostic laparoscopy and were referred to NACT treatment before surgery. NACT consisted of median three (range 1–5) cycles of carboplatin and paclitaxel chemotherapy. For patients who responded to NACT, an interval debulking surgery (IDS) was performed, aiming for optimal cytoreduction, followed by a median of three (range 0–9) cycles of adjuvant chemotherapy.

Residual disease was reported based on the surgeon's visual estimation of the amount of macroscopic tumor left after IDS. The primary therapy response was defined according to RECIST 1.1 criteria (16). Twenty-eight (45%) patients had complete response, 13 (21%) had partial response, 20 (32%) patients had progressive disease, and one patient died during primary therapy and thus the response could not be evaluated. All patients participating in the study gave written informed consent. The study and the use of all clinical material have been approved by (i) The Ethics Committee of the Hospital District of Southwest Finland (ETMK): ETMK 53/180/2009 § 238 and (ii) National Supervisory Authority for Welfare and Health (Valvira): DNRO 6550/05.01.00.06/2010 and STH507A.

We used The Cancer Genome Atlas (TCGA) data as an independent validation cohort (clinical data downloaded from the TCGA data portal in December 2016; ref. 17). The TCGA cohort consists of 489 patients with clinical data. Patient selection criteria for the validation cohort were as follows: (i) advanced HGSOC (grade >1, stage: IIIb–IV), (ii) an indication of attempted surgical debulking, that is, information available on the residual disease, and (iii) treatment by platinum-based chemotherapy alone or in combination in the first-line setting. To conform to current therapy standards, we did not exclude patients who received bevacizumab, an angiogenesis inhibitor, which has been shown to improve progression-free survival in HGSOC (18), as a maintenance therapy. After applying these selection criteria, the validation cohort consisted of 170 patients with HGSOC.

The patient characteristics of the calibration and validation cohorts are summarized in Table 1. The patients in the calibration cohort were older, had more often advanced stage (IV) disease, and received more often single platinum therapy and bevacizumab maintenance after the first-line treatment. The patients in the calibration cohort also had less residual disease after debulking surgery compared with the validation cohort. There were more patients with progressive disease in the calibration cohort, and progression-free survival was shorter in the validation cohort. However, there were no significant differences in therapy responses, that is, platinum-free interval (PFI) or progression-free/overall survival, between the cohorts.

For 28 patients in our calibration cohort, ¹⁸F-FDG-PET/CT imaging was performed at the time of diagnosis and after NACT, as described in ref. 19. The ¹⁸F-FDG-PET/CT images were used to evaluate the total metabolic tumor volume (MTV) before and after NACT. Two patients did not have visible residual tumor in PET/CT images after NACT and were excluded from further analysis as their tumor burden could not be evaluated.

The total tumor burden was estimated using the MTV, which is defined as the sum volume of all the tumor lesions with increased ¹⁸F-FDG uptake. The MTV calculations were done by expert nuclear medicine physician as follows. The PET images were reconstructed in 3D mode and 128 × 128 matrix size using the ML-OSEM reconstruction algorithm. Imaging analysis was performed using ADW 4.6 workstation (General Electric Medical Systems). The metabolically active lesions were identified semiautomatically with PET Volume Computer Assisted Reading (VCAR) program by comparing the pre- and post-NACT ¹⁸F-FDG-PET/CT images. All the suggested lesions with increased ¹⁸F-FDG uptake were either bookmarked as a tumor lesion or removed in cases of physiologic uptake, such as ureters and bladder.

We applied a stochastic multitype branching process (20) to model tumor growth and evolution. Briefly, our model describes clonal expansion of cancer cells starting from a single platinum-sensitive cell that can divide at rate b and die at rate d, giving the net growth rate λ = b − d. A cell can acquire resistance mechanisms at rate u per cell division independent of chemotherapy. Resistance accumulates sequentially, that is, a cell with n active resistance mechanism can accumulate n+ 1 resistance mechanisms that are irreversible. The schematic of the model is given in Fig. 2A and in more detail in Supplementary Text S1 to S3.

The integration of clinical data and modeling is outlined in Fig. 2B. We estimated the key model parameters using our calibration cohort, and the values for other parameters were collected from previous studies (21–23). All model parameters together with their values are listed in Table 2. Details of the parameter calibration are provided in Supplementary Text S4.

We used the stochastic mathematical model and simulation approach to create a virtual HGSOC patient cohort that has similar responses to chemotherapy as observed in the real-life HGSOC cohort. Briefly, for each virtual patient, the values of tumor burden at diagnosis (M) and chemotherapy effect (d_chemotherapy) were sampled from log-normal distributions while the other parameters were kept constant, as shown in Table 2. The log-normal distribution was the best match to describe the data among six statistical distributions using Bayesian Information Criterion (Supplementary Material S4). All virtual patients were simulated until tumor relapse, that is, to the point where the total number of cancer cells reaches M_relapse cells.

In the simulations, the effect of a drug is measured by PFI, which is the interval from the last date of chemotherapy until progression (24). PFI is used in clinical practice to measure a patient's sensitivity to platinum and to group them into platinum-sensitive, -resistant, and refractory, and to guide treatment decisions (14, 25, 26).

We have constructed a stochastic multitype branching process mathematical model that describes chemotherapy sensitive and resistant cell growth in HGSOC (Fig. 2A). A stochastic model allows modeling an arbitrary number of resistant mechanisms and thus our analyses are not limited by the number of resistance mechanisms.

Figure 2B shows how clinical data were integrated into the model, and the SOC simulation is described in Fig. 2C. Briefly, during the prediagnosis phase, the tumor burden increases without intervention. After the diagnosis, the treatment phase starts with NACT, followed by cytoreductive surgery and adjuvant chemotherapy. The posttreatment phase continues until a relapse is detected. Detailed descriptions of the model structure and major assumptions are given in Materials and Methods and Supplementary Text S1.

To quantify the effect of chemotherapy, we measured both tumor burden and PFI and used them to create a virtual HGSOC cohort of 1,000 patients (see Materials and Methods). The ratio of the SD to the mean tumor burden was 0.46 before NACT and 1.15 after NACT in the calibration cohort, which indicates high individual variation in response to chemotherapy. Response variation in the virtual HGSOC cohort is indistinguishable from the real-life calibration cohort (Kolmogorov–Smirnov test P > 0.05; Supplementary Fig. S1). Median PFI of patients in the virtual HGSOC cohort is 7 months, which is in line with the median PFI in the calibration cohort (5 months) and is identical to the validation cohort, as shown in Fig. 3A. These results show that the clinically important characteristics of treatment responses, that is, individual variation and PFI, of real-life patients with HGSOC are faithfully reproduced by the mathematical model.

To identify the most influential parameters in the model, we conducted a sensitivity analysis. Briefly, sensitivity analysis quantitatively ranks the strength of the relationship between all model parameters (Table 2) and platinum-free survival (Supplementary Text S5). We considered a parameter sensitive if the sensitivity coefficient (SC), which measures the ratio of the change in the output (area under survival curve) to the change in input, is at least 50%.

The parameters that had high sensitivity, and therefore a high impact on the PFI, were the fraction of cells removed in surgery (β), death rate (d), and division rate (b), as illustrated in Fig. 3B. The death and division rates reflect the inherent growth pattern and aggressiveness of a tumor. Thus, their ranking as sensitive parameters was expected. The impact of IDS on PFI is supported by several clinical trials that have demonstrated that optimal cytoreduction yields a significant survival benefit (27, 28).

Interestingly, tumor burden at diagnosis (M) did not have a large effect on PFI (SC, 12%). This is in accordance with the training cohort, in which the tumor burden at diagnosis did not correlate well with response to NACT (Pearson correlation coefficient = 0.17). The modest effect of tumor burden at diagnosis on PFI suggests that earlier detection of HGSOC will only lead to a relatively modest survival benefits. Our results thus further corroborate previous modeling efforts, which concluded that detection of HGSOC with small tumor volume does not significantly improve response to therapy (29, 30).

Given the significant effect of tumor aggressiveness on PFI, we next explored the number of active resistance mechanisms present in HGSOC tumors at the time of diagnosis by using the virtual cohort of 1,000 patients. In addition to the number of active resistance mechanisms, we also investigated the proportion of resistance mechanisms in tumors before and after NACT.

At diagnosis, majority of patients (55%) had tumors that contained cancer cells with five active resistant mechanisms, and their average PFI was 6 months. Forty percent of the patients had four active resistance mechanisms and an average PFI of 10 months, whereas 5% of the patients with HGSOC had six active mechanisms and an average PFI of 6 months. The fraction of resistant cells at diagnosis had a high correlation with PFI (Pearson correlation coefficient = 0.60). Notably, 83% of the cancer cells were platinum sensitive at diagnosis, and only 1% of the cells had three or more active resistance mechanisms (Fig. 4A), which is consistent with the clinical efficacy of platinum on reducing tumor burden in HGSOC.

An average virtual patient with HGSOC had approximately 10⁷ cancer cells after NACT. Of all the cancer cells that survived NACT, the fraction of platinum-sensitive cells was dramatically reduced to 11% (Fig. 4B), which suggests limited efficiency of the subsequent platinum-based chemotherapy cycles. However, 99% of the cancer cells that survived NACT had at most two active resistant mechanisms (Fig. 4B). This implies that the vast majority of the cancer cells removed at IDS are platinum resistant, but combining targeted therapies to the adjuvant chemotherapy regimens after NACT and IDS could provide a significant survival benefit.

We next investigated the benefit of combining targeted therapies to SOC. High proportion of cancer cells with less than four active resistance mechanisms even after chemotherapy suggests that combining SOC with only a few additional drugs should improve to patient survival significantly. We tested this hypothesis by simulating up to six targeted therapies in combination with SOC in the virtual patient cohort.

We calculated PFI estimates for SOC alone and for regimens where SOC was combined with at most six drugs targeting resistance mechanisms. The PFIs for the combination therapies are shown in Fig. 5. The median PFI for SOC was 7 months, and adding only one effective targeted therapy to SOC increased the median PFI to 25 months. The addition of more targeted drugs to SOC improved the PFI, but relatively less than the first drug. For example, addition of a second effective targeted therapy to the regimen improved the median PFI to 45 months and a third targeted therapy improved the median PFI to 60 months. Adding a fourth, fifth, or sixth drug did not bring any significant benefit over combining three targeted therapies with SOC.

Resistance to chemotherapies is the major contributor to cancer-related mortality, and a deeper understanding of the prevalence and dynamics of active resistance mechanisms is needed to improve patient survival. HGSOC is typically diagnosed at an advanced stage with significant innate heterogeneity and, likely, several active resistance mechanisms already at diagnosis. Indeed, only few targeted agents with encouraging preclinical results in HGSOC have converged into significant clinical responses, and the survival rate of patients with HGSOC has not markedly improved over the past 30 years. We propose here that mathematical modeling and simulation using rich clinical data offers a means to reliably evaluate the value of preclinical agents in virtual cohorts. This approach allows testing the added value of combination therapies, which may be difficult or expensive to test in vivo or in patients, in an efficient and inexpensive manner, as previously suggested by us and others (31–33).

Our approach takes advantage of the accumulated knowledge of rich longitudinal clinical data from patients with HGSOC. In particular, the values of key parameters in the model, such as tumor burden and the effect of chemotherapy, were quantified with ¹⁸F-FDG-PET/CT, which is an order of magnitude times more sensitive than a contrast-enhanced CT scan. This allowed creating virtual HGSOC patients whose responses to SOC are indistinguishable from real-life patients with HGSOC.

Approximately 30% of patients with HGSOC are platinum refractory already at diagnosis, and most patients initially responding to platinum will develop a platinum-resistant relapse. These patients with advanced platinum-resistant disease have currently very limited treatment options (8). Therefore, we focused on this patient population to uncover the active resistance mechanisms and the value of targeted therapies. All patients in the calibration cohort had been clinically assigned to receive NACT due to advanced and disseminated disease, whereas majority of patients in the validation cohort were treated with primary debulking surgery followed by adjuvant chemotherapy, indicating a less advanced disease at diagnosis. Therefore, the observed good fit to patient responses in both cohorts provides strong validation that the model is able to accurately simulate clinical outcome after first-line SOC in a clinically heterogeneous HGSOC patient population treated with or without NACT.

Sensitivity analysis indicated that the most important parameter was the fraction of cells removed in the surgery. Debulking surgery is a very effective strategy to reduce tumor burden and, unlike chemotherapy, it can remove a significant portion of chemotherapy-resistant cells, which reduces the probability of resistance (34). Importance of surgery is supported by clinical trials demonstrating that optimal cytoreduction yields a significant survival benefit (27, 28). Perhaps unexpectedly, platinum efficacy had only a small impact to PFI. This is due to the fact that although chemotherapy effectively kills drug-sensitive cancer cells, it also provides a selective advantage for resistant cells that can grow with less constraints on space and nutrients. A higher fraction of resistant cells rapidly reduces PFI in the subsequent chemotherapy cycles, leading to refractory disease and death. Thus, development of platinum compounds with higher efficacy may have a limited impact to HGSOC patient survival.

Combination therapy has been suggested as an effective means to overcome therapy failures in advanced solid cancers (35). It is, however, not generally known how many drugs should be combined to achieve a significant and long-lasting survival advantage. Our results indicate that at diagnosis there are already up to five resistance mechanisms active in a fraction of the tumor cells. However, before chemotherapy platinum-sensitive cells form 80% of the cancer cell population, which explains the generally observed good response to chemotherapy in HGSOC. Importantly, after NACT, the proportion of chemosensitive cells is reduced to 11%, and 88% of the cancer cells now have only one or two active resistance mechanisms. This provides a window of opportunity to target the dominant-resistant subpopulations with few targeted drugs and potentially gain significant survival benefit. Our results also suggest that combining four or more targeted therapies to SOC has a marginal effect on patient survival over three drugs. This is due to a small proportion of cancer cells having four of more active resistance mechanisms.

Throughout our analyses we have assumed that a targeted therapy kills those cancer cells that contain the respective active resistance mechanism as efficiently as platinum kills platinum-sensitive cancer cells. However, the efficacy of each therapy may vary, and the combination treatments may present synergistic effects (36). We also assumed that targeted therapies are given at tolerated doses in combination with SOC. However, the targeted agents may have significant overlapping toxicities, which need to be modeled for each specific targeted therapy using pharmacological data. Importantly, these biological and clinical factors can be implemented into our model to further improve its clinical value.

Our results provide a theoretical basis for a treatment paradigm where the objective is to maximize platinum's effect on cancer cells by simultaneously overcoming resistance mechanisms with targeted drugs. For this approach, it is crucial to have reliable biomarkers to identify the most dominant resistance mechanisms in tumors before the treatment. Furthermore, our results show that the fraction of the resistance mechanisms significantly changes during the treatment, which suggests that the development and use of serial sampling is needed in planning secondary or maintenance treatments.

The herein presented mathematical model has been developed using clinical data from patients with HGSOC. It is, however, straightforward to modify the model and parameters to accommodate other cancers and therapy regimens. The mathematical model and simulator are available with comprehensive documentation at http://csbi.ltdk.helsinki.fi/pub/home/ekozlows/platinum/.

No potential conflicts of interest were disclosed.

Conception and design: E. Kozłowska, A. Färkkilä, S. Hietanen, S. Hautaniemi

Development of methodology: E. Kozłowska

Acquisition of data (provided animals, acquired and managed patients, provided facilities, etc.): T. Vallius, O. Carpén, J. Kemppainen, S. Grénman, J. Hynninen, S. Hietanen

Analysis and interpretation of data (e.g., statistical analysis, biostatistics, computational analysis): E. Kozłowska, A. Färkkilä, J. Kemppainen, S. Hietanen, S. Hautaniemi

Writing, review, and/or revision of the manuscript: E. Kozłowska, A. Färkkilä, T. Vallius, O. Carpén, J. Kemppainen, S. Grénman, R. Lehtonen, J. Hynninen, S. Hietanen, S. Hautaniemi

Administrative, technical, or material support (i.e., reporting or organizing data, constructing databases): T. Vallius, J. Kemppainen, J. Hynninen

Study supervision: S. Grénman, S. Hautaniemi

We thank Dr. Krzysztof Puszyński, for critical review of the manuscript; MSc Tiia Pelkonen for proofreading; and CSC – IT Center for Science Ltd., for computer resources. This project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 667403 for HERCULES (to S.K. Hautaniemi), the Academy of Finland (project 292402, to S.K. Hautaniemi), the Sigrid Jusélius Foundation (to S.K. Hautaniemi), and Finnish Cancer Associations (to S.K. Hautaniemi). The results published here are in part based upon data generated by TCGA managed by the NCI and NHGRI. Information about TCGA can be found at http://cancergenome.nih.gov.

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.