Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

A mathematical model for the quantification of a patient’s sensitivity to checkpoint inhibitors and long-term tumour burden

Nature Biomedical Engineering volume 5, pages 297–308 (2021)Cite this article

Subjects

Abstract

A large proportion of patients with cancer are unresponsive to treatment with immune checkpoint blockade and other immunotherapies. Here, we report a mathematical model of the time course of tumour responses to immune checkpoint inhibitors. The model takes into account intrinsic tumour growth rates, the rates of immune activation and of tumour–immune cell interactions, and the efficacy of immune-mediated tumour killing. For 124 patients, four cancer types and two immunotherapy agents, the model reliably described the immune responses and final tumour burden across all different cancers and drug combinations examined. In validation cohorts from four clinical trials of checkpoint inhibitors (with a total of 177 patients), the model accurately stratified the patients according to reduced or increased long-term tumour burden. We also provide model-derived quantitative measures of treatment sensitivity for specific drug–cancer combinations. The model can be used to predict responses to therapy and to quantify specific drug–cancer sensitivities in individual patients.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Example model fits to clinical immunotherapy response.
Fig. 2: Distribution of model parameters against immune response.
Fig. 3: Patient response classification by ROC analysis.
Fig. 4: Correlation analysis of the literature-derived calibration cohort revealed significant correlation between measured values and model-predicted variables.
Fig. 5: Immune response strength is associated with predicted and measured long-term tumour burden.

Similar content being viewed by others

Data availability

All of the data supporting the results in this study are available within the paper and its Supplementary Information. The raw, de-identified patient data are available from the corresponding authors upon reasonable request and subject to Institutional Review Board approval.

References

  1. Disis, M. L. Mechanism of action of immunotherapy. Semin Oncol. 41, S3–S13 (2014).

    CAS  PubMed  Google Scholar 

  2. Farkona, S., Diamandis, E. P. & Blasutig, I. M. Cancer immunotherapy: the beginning of the end of cancer? BMC Med. 14, 73 (2016).

    PubMed  PubMed Central  Google Scholar 

  3. Topalian, S. L. et al. Safety, activity, and immune correlates of anti-PD-1 antibody in cancer. N. Engl. J. Med. 366, 2443–2454 (2012).

    CAS  PubMed  PubMed Central  Google Scholar 

  4. Rosenberg, J. E. et al. Atezolizumab in patients with locally advanced and metastatic urothelial carcinoma who have progressed following treatment with platinum-based chemotherapy: a single-arm, multicentre, phase 2 trial. Lancet 387, 1909–1920 (2016).

    CAS  PubMed  PubMed Central  Google Scholar 

  5. Schumacher, T. N., Kesmir, C. & van Buuren, M. M. Biomarkers in cancer immunotherapy. Cancer Cell 27, 12–14 (2015).

    CAS  PubMed  Google Scholar 

  6. Chiou, V. L. & Burotto, M. Pseudoprogression and immune-related response in solid tumors. J. Clin. Oncol. 33, 3541–3543 (2015).

    CAS  PubMed  PubMed Central  Google Scholar 

  7. Carthon, B. C. et al. Preoperative CTLA-4 blockade: tolerability and immune monitoring in the setting of a presurgical clinical trial. Clin. Cancer Res. 16, 2861–2871 (2010).

    CAS  PubMed  PubMed Central  Google Scholar 

  8. Ahmadzadeh, M. et al. Tumor antigen-specific CD8 T cells infiltrating the tumor express high levels of PD-1 and are functionally impaired. Blood 114, 1537–1544 (2009).

    CAS  PubMed  PubMed Central  Google Scholar 

  9. Schmidt, H. et al. Pretreatment levels of peripheral neutrophils and leukocytes as independent predictors of overall survival in patients with American Joint Committee on Cancer Stage IV Melanoma: results of the EORTC 18951 Biochemotherapy Trial. J. Clin. Oncol. 25, 1562–1569 (2007).

    CAS  PubMed  Google Scholar 

  10. Phan, G. Q., Attia, P., Steinberg, S. M., White, D. E. & Rosenberg, S. A. Factors associated with response to high-dose interleukin-2 in patients with metastatic melanoma. J. Clin. Oncol. 19, 3477–3482 (2001).

    CAS  PubMed  Google Scholar 

  11. Yuan, J. et al. Integrated NY-ESO-1 antibody and CD8+ T-cell responses correlate with clinical benefit in advanced melanoma patients treated with ipilimumab. Proc. Natl Acad. Sci. USA 108, 16723–16728 (2011).

    CAS  PubMed  Google Scholar 

  12. Yuan, J. et al. Pretreatment serum VEGF is associated with clinical response and overall survival in advanced melanoma patients treated with ipilimumab. Cancer Immunol. Res. 2, 127–132 (2014).

    CAS  PubMed  Google Scholar 

  13. Blay, J. Y. et al. Serum level of interleukin 6 as a prognosis factor in metastatic renal cell carcinoma. Cancer Res. 52, 3317–3322 (1992).

    CAS  PubMed  Google Scholar 

  14. Kelderman, S. et al. Lactate dehydrogenase as a selection criterion for ipilimumab treatment in metastatic melanoma. Cancer Immunol. Immunother. 63, 449–458 (2014).

    CAS  PubMed  Google Scholar 

  15. Rizvi, N. A. et al. Mutational landscape determines sensitivity to PD-1 blockade in non-small cell lung cancer. Science 348, 124–128 (2015).

    CAS  PubMed  PubMed Central  Google Scholar 

  16. Van Allen, E. M. et al. Genomic correlates of response to CTLA-4 blockade in metastatic melanoma. Science 350, 207–211 (2015).

    CAS  PubMed  PubMed Central  Google Scholar 

  17. Meng, X., Huang, Z., Teng, F., Xing, L. & Yu, J. Predictive biomarkers in PD-1/PD-L1 checkpoint blockade immunotherapy. Cancer Treat. Rev. 41, 868–876 (2015).

    CAS  PubMed  Google Scholar 

  18. Madore, J. et al. PD-L1 expression in melanoma shows marked heterogeneity within and between patients: implications for anti-PD-1/PD-L1 clinical trials. Pigment Cell Melanoma Res. 28, 245–253 (2015).

    CAS  PubMed  Google Scholar 

  19. Sabatino, M. et al. Serum vascular endothelial growth factor and fibronectin predict clinical response to high-dose interleukin-2 therapy. J. Clin. Oncol. 27, 2645–2652 (2009).

    CAS  PubMed  PubMed Central  Google Scholar 

  20. Eisenhauer, E. A. et al. New response evaluation criteria in solid tumours: revised RECIST guideline (version 1.1). Eur. J. Cancer 45, 228–247 (2009).

    CAS  PubMed  Google Scholar 

  21. Deisboeck, T. S., Wang, Z., Macklin, P. & Cristini, V. Multiscale cancer modeling. Annu. Rev. Biomed. Eng. 13, 127–155 (2011).

    CAS  PubMed  Google Scholar 

  22. Wang, Z., Butner, J. D., Kerketta, R., Cristini, V. & Deisboeck, T. S. Simulating cancer growth with multiscale agent-based modeling. Semin Cancer Biol. 30, 70–78 (2015).

    PubMed  Google Scholar 

  23. Cristini, V., Koay, E. & Wang, Z. An Introduction to Physical Oncology: How Mechanistic Mathematical Modeling Can Improve Cancer Therapy Outcomes (CRC Press, 2017).

  24. Wang, Z., Butner, J. D., Cristini, V. & Deisboeck, T. S. Integrated PK-PD and agent-based modeling in oncology. J. Pharmacokinet. Pharmacodyn. 42, 179–189 (2015).

    CAS  PubMed  PubMed Central  Google Scholar 

  25. Gillies, R. J., Brown, J. S., Anderson, A. R. A. & Gatenby, R. A. Eco-evolutionary causes and consequences of temporal changes in intratumoural blood flow. Nat. Rev. Cancer 18, 576–585 (2018).

    CAS  PubMed  PubMed Central  Google Scholar 

  26. Wang, Z. & Deisboeck, T. S. Dynamic targeting in cancer treatment. Front. Physiol. 10, 96 (2019).

    PubMed  PubMed Central  Google Scholar 

  27. Dogra, P. et al. Mathematical modeling in cancer nanomedicine: a review. Biomed. Microdevices 21, 40 (2019).

    PubMed  PubMed Central  Google Scholar 

  28. Dogra, P. et al. Image-guided mathematical modeling for pharmacological evaluation of nanomaterials and monoclonal antibodies. Wiley Interdiscip. Rev. Nanomed. Nanobiotechnol. 12, e1628 (2020).

    CAS  PubMed  PubMed Central  Google Scholar 

  29. Kronik, N., Kogan, Y., Vainstein, V. & Agur, Z. Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics. Cancer Immunol. Immunother. 57, 425–439 (2008).

    PubMed  Google Scholar 

  30. Eikenberry, S., Thalhauser, C. & Kuang, Y. Tumor-immune interaction, surgical treatment, and cancer recurrence in a mathematical model of melanoma. PLoS Comput. Biol. 5, e1000362 (2009).

    PubMed  PubMed Central  Google Scholar 

  31. Wells, D. K. et al. Spatial and functional heterogeneities shape collective behavior of tumor-immune networks. PLoS Comput. Biol. 11, e1004181 (2015).

    PubMed  PubMed Central  Google Scholar 

  32. Palladini, A. et al. In silico modeling and in vivo efficacy of cancer-preventive vaccinations. Cancer Res. 70, 7755–7763 (2010).

    CAS  PubMed  Google Scholar 

  33. Pappalardo, F., Lollini, P. L., Castiglione, F. & Motta, S. Modeling and simulation of cancer immunoprevention vaccine. Bioinformatics 21, 2891–2897 (2005).

    CAS  PubMed  Google Scholar 

  34. Walker, R. & Enderling, H. From concept to clinic: mathematically informed immunotherapy. Curr. Prob. Cancer 40, 68–83 (2016).

    Google Scholar 

  35. Konstorum, A., Vella, A. T., Adler, A. J. & Laubenbacher, R. C. Addressing current challenges in cancer immunotherapy with mathematical and computational modelling. J. R. Soc. Interface 14, 20170150 (2017).

    PubMed  PubMed Central  Google Scholar 

  36. Luksza, M. et al. A neoantigen fitness model predicts tumour response to checkpoint blockade immunotherapy. Nature 551, 517–520 (2017).

    CAS  PubMed  PubMed Central  Google Scholar 

  37. Kim, E., Rebecca, V. W., Smalley, K. S. & Anderson, A. R. Phase I trials in melanoma: a framework to translate preclinical findings to the clinic. Eur. J. Cancer 67, 213–222 (2016).

    CAS  PubMed  PubMed Central  Google Scholar 

  38. Cyll, K. et al. Tumour heterogeneity poses a significant challenge to cancer biomarker research. Br. J. Cancer 117, 367–375 (2017).

    CAS  PubMed  PubMed Central  Google Scholar 

  39. Powles, T. et al. MPDL3280A (anti-PD-L1) treatment leads to clinical activity in metastatic bladder cancer. Nature 515, 558–562 (2014).

    CAS  PubMed  Google Scholar 

  40. Motzer, R. J. et al. Nivolumab for metastatic renal cell carcinoma: results of a randomized phase II trial. J. Clin. Oncol. 33, 1430–1437 (2015).

    CAS  PubMed  Google Scholar 

  41. Borghaei, H. et al. Nivolumab versus docetaxel in advanced nonsquamous non-small-cell lung cancer. N. Engl. J. Med. 373, 1627–1639 (2015).

    CAS  PubMed  PubMed Central  Google Scholar 

  42. Plodinec, M. et al. The nanomechanical signature of breast cancer. Nat. Nanotechnol. 7, 757–765 (2012).

    CAS  PubMed  Google Scholar 

  43. Pascal, J. et al. Mechanistic patient-specific predictive correlation of tumor drug response with microenvironment and perfusion measurements. Proc. Natl Acad. Sci. USA 110, 14266–14271 (2013).

    CAS  PubMed  Google Scholar 

  44. Earle, C. C. et al. Aggressiveness of cancer care near the end of life: is it a quality-of-care issue? J. Clin. Oncol. 26, 3860–3866 (2008).

    PubMed  PubMed Central  Google Scholar 

  45. De Larco, J. E., Wuertz, B. R., Manivel, J. C. & Furcht, L. T. Progression and enhancement of metastatic potential after exposure of tumor cells to chemotherapeutic agents. Cancer Res. 61, 2857–2861 (2001).

    CAS  PubMed  Google Scholar 

  46. Van Meir, H. et al. Impact of (chemo)radiotherapy on immune cell composition and function in cervical cancer patients. Oncoimmunology 6, e1267095 (2017).

    CAS  PubMed  Google Scholar 

  47. Pascal, J. et al. Mechanistic modeling identifies drug-uptake history as predictor of tumor drug resistance and nano-carrier-mediated response. ACS Nano 7, 11174–11182 (2013).

    CAS  PubMed  PubMed Central  Google Scholar 

  48. Das, H. et al. Impact of diffusion barriers to small cytotoxic molecules on the efficacy of immunotherapy in breast cancer. PLoS ONE 8, e61398 (2013).

    CAS  PubMed  PubMed Central  Google Scholar 

  49. Koay, E. J. et al. Transport properties of pancreatic cancer describe gemcitabine delivery and response. J. Clin. Invest. 124, 1525–1536 (2014).

    CAS  PubMed  PubMed Central  Google Scholar 

  50. Frieboes, H. B., Wu, M., Lowengrub, J., Decuzzi, P. & Cristini, V. A computational model for predicting nanoparticle accumulation in tumor vasculature. PLoS ONE 8, e56876 (2013).

    CAS  PubMed  PubMed Central  Google Scholar 

  51. Wang, Z. et al. Theory and experimental validation of a spatio-temporal model of chemotherapy transport to enhance tumor cell kill. PLoS Comput. Biol. 12, e1004969 (2016).

    PubMed  PubMed Central  Google Scholar 

  52. Brocato, T. A. et al. Understanding the connection between nanoparticle uptake and cancer treatment efficacy using mathematical modeling. Sci. Rep. 8, 7538 (2018).

    PubMed  PubMed Central  Google Scholar 

  53. Koay, E. J. et al. A visually apparent and quantifiable CT imaging feature identifies biophysical subtypes of pancreatic ductal adenocarcinoma. Clin. Cancer Res. 24, 5883–5894 (2018).

    PubMed  PubMed Central  Google Scholar 

  54. Dogra, P. et al. Establishing the effects of mesoporous silica nanoparticle properties on in vivo disposition using imaging-based pharmacokinetics. Nat. Commun. 9, 4551 (2018).

    PubMed  PubMed Central  Google Scholar 

  55. Brocato, T. A. et al. Predicting breast cancer response to neoadjuvant chemotherapy based on tumor vascular features in needle biopsies.JCI Insight 5, e126518 (2019).

    Google Scholar 

  56. Goel, S. et al. Size-optimized ultrasmall porous silica nanoparticles depict vasculature-based differential targeting in triple negative breast cancer. Small 15, e1903747 (2019).

    PubMed  PubMed Central  Google Scholar 

  57. Pichler, M. et al. Therapeutic potential of FLANC, a novel primate-specific long non-coding RNA in colorectal cancer.Gut 69, 1818–1831 (2020).

    CAS  PubMed  PubMed Central  Google Scholar 

  58. Butner, J. D. et al. A multiscale agent-based model of ductal carcinoma in situ. IEEE Trans. Biomed. Eng. 67, 1450–1461 (2020).

    PubMed  Google Scholar 

  59. Butner, J. D. et al. Mathematical prediction of clinical outcomes in advanced cancer patients treated with checkpoint inhibitor immunotherapy.Sci. Adv. 6, eaay6298 (2020).

    CAS  PubMed  PubMed Central  Google Scholar 

  60. Dogra, P. et al. A mathematical model to predict nanomedicine pharmacokinetics and tumor delivery. Comput. Struct. Biotechnol. J. 18, 518–531 (2020).

    CAS  PubMed  PubMed Central  Google Scholar 

  61. Rohatgi, A. WebPlotDigitizer v.4.0 (2017).

  62. Mathematica v.11.2 (Wolfram Research, 2017).

Download references

Acknowledgements

The research reported in this manuscript was supported by the National Science Foundation grant DMS-1930583 (to Z.W. and V.C.), National Institutes of Health grants 1U01CA196403 (to Z.W., E.J.K. and V.C.), 1U01CA213759 (to Z.W. and V.C.), 1R01CA226537 (to Z.W. and V.C.), 1R01CA222007 (to Z.W., G.A.C. and V.C.) and U54CA210181 (to Z.W., E.J.K. and V.C.) and the University of Texas System STARs Award (to V.C.). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

Author information

Author notes

  1. These authors contributed equally: Joseph D. Butner, Zhihui Wang.

Authors and Affiliations

  1. Mathematics in Medicine Program, Houston Methodist Research Institute, Houston, TX, USA

    Joseph D. Butner, Zhihui Wang, Prashant Dogra, Sara Nizzero, Javier Ruiz-Ramírez & Vittorio Cristini

  2. Department of Imaging Physics, University of Texas MD Anderson Cancer Center, Houston, TX, USA

    Zhihui Wang & Vittorio Cristini

  3. Department of Radiation Oncology, University of Texas MD Anderson Cancer Center, Houston, TX, USA

    Dalia Elganainy, Karine A. Al Feghali, Geoffrey V. Martin, Caroline Chung, Eugene J. Koay & James W. Welsh

  4. Biozentrum and the Swiss Nanoscience Institute, University of Basel, Basel, Switzerland

    Marija Plodinec

  5. Department of Translational Molecular Pathology, University of Texas MD Anderson Cancer Center, Houston, TX, USA

    George A. Calin

  6. Department of Melanoma Medical Oncology, University of Texas MD Anderson Cancer Center, Houston, TX, USA

    Hussein A. Tawbi

  7. Department of Investigational Cancer Therapeutics, University of Texas MD Anderson Cancer Center, Houston, TX, USA

    David S. Hong

  8. Physiology, Biophysics, and Systems Biology Program, Graduate School of Medical Sciences, Weill Cornell Medicine, New York, NY, USA

    Vittorio Cristini

Authors
  1. Joseph D. Butner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhihui Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dalia Elganainy
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Karine A. Al Feghali
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Marija Plodinec
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. George A. Calin
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Prashant Dogra
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Sara Nizzero
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Javier Ruiz-Ramírez
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. Geoffrey V. Martin
    View author publications

    You can also search for this author in PubMed Google Scholar

  11. Hussein A. Tawbi
    View author publications

    You can also search for this author in PubMed Google Scholar

  12. Caroline Chung
    View author publications

    You can also search for this author in PubMed Google Scholar

  13. Eugene J. Koay
    View author publications

    You can also search for this author in PubMed Google Scholar

  14. James W. Welsh
    View author publications

    You can also search for this author in PubMed Google Scholar

  15. David S. Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

  16. Vittorio Cristini
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

J.D.B. implemented the model and collected the data. Z.W. and V.C. initiated the project, participated in data collection and supervised the model development and analysis. J.D.B., Z.W., M.P., P.D. and V.C. performed the model analysis. D.E., K.A.A.F., G.V.M., C.C., E.J.K., J.W.W. and D.S.H. participated in data collection. J.D.B., Z.W., D.E., K.A.A.F., M.P., G.A.C., P.D., S.N., J.R.-R., G.V.M., H.A.T., C.C., E.J.K., J.W.W., D.S.H. and V.C. interpreted the data and model results. J.D.B., Z.W., M.P., E.J.K. and V.C. wrote the manuscript.

Corresponding authors

Correspondence to Zhihui Wang or Vittorio Cristini.

Ethics declarations

Competing interests

J.W.W. reports research support from GlaxoSmithKline, Bristol Myers Squibb, Merck, Nanobiotix, Mavupharma and Checkmate Pharmaceuticals. J.W.W. serves on the scientific advisory board for RefleXion Medical, MolecularMatch, OncoResponse, Checkmate, Mavupharmaceuticals and Alpine Immune Sciences. J.W.W. is co-founder of Healios Oncology, MolecularMatch and OncoResponse and serves as an advisor to AstraZeneca, Merck, MolecularMatch, Incyte, Aileron and Nanobiotix. J.W.W. has the following patents: MP470 (amuvatinib), MRX34 regulation of PD-L1, and the X-ray tomography technique to overcome immune resistance. The University of Texas MD Anderson Cancer Center has a trademark for RadScopal.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Methods, Discussion, Figs. 1–10, Tables 1–5 and references.

Reporting Summary

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Butner, J.D., Wang, Z., Elganainy, D. et al. A mathematical model for the quantification of a patient’s sensitivity to checkpoint inhibitors and long-term tumour burden. Nat Biomed Eng 5, 297–308 (2021). https://doi.org/10.1038/s41551-020-00662-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41551-020-00662-0

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing