Overview of mathematical biological models and their application in CML research

Transcription

1 Overview of mathematical biological models and their application in CML research March 27 th 2015 Christina Fitzmaurice, MD, MPH Faculty discussants: Vivian Oehler, MD FHCRC/UW William Hazelton, PhD FHCRC Computational Biology Program

2 Outline Definitions and history of mathematical biological models Application in CML Michor model Roeder model Conclusions Future perspectives

3 Definition mathematical biological models Explaining the pathophysiology of a disease using mathematical formulas, which are based on mechanistic understanding of the underlying process NOT purely based on statistical inference Model components have meaningful biological interpretation In contrast STATISTICAL MODELS are not required to have a meaningful biological interpretation Emmers Streib: Frontiers in Genetics 2014

4 Differences between statistical and mathematical models Waller et al: Ecology 2010

5 Historic evolution of mathematical models for cancer 1954 Armitage and Doll Multistage theory of carcinogenesis Incidence curves predict number of mutations required 1971 Knudson Two hit hypothesis for retinoblastoma Two mutations in tumor suppressor gene are necessary for cancer to form 1979 Moolgavkar Two stage clonal expansion model Tumor initiation, clonal expansion, malignant transformation 1991 Loeb Mutator phenotype Mutations in genes that stabilize genome lead to large number of mutations in cancer cells

6 CML mathematical models Despite being a rare disease compared to solid tumors CML has been studied extensively for mathematical models Simple etiology with a known mutation Availability of molecular data (ease of obtaining samples) Availability of successful treatment (TKI)

7 Examples of CML mathematical models Carcinogenesis CML as a model for carcinogenesis (Melo et al: Nat Rev Cancer 2007) Predicting stem cell numbers (Radivoyevitch et al: Math Biosci 1999) Interplay between malignant cells and surrounding environment (Roeder et al: Bull Math Biol 2009) Risk estimation Risk of radiation on CML incidence (Radivoyevitch et al: Radiat Environ Biophys 2001) Gender differences in risk (Radivoyevitch et al: Radiat Environ Biophys 2014) Treatment Cyclic drug treatment/combination of TKIs (Komarova: Math Biosci Eng 2011) Risk for relapse after SCT (Vincent et al: Stem Cells 1999) Risk for relapse after TKI discontinuation (Michor et al: Nature 2005; Foo et al: Plos Comput Biol 2009; Stein et al: Clin Cancer Res 2011; Tang et al: Blood 2011; Tang et al: Haematologica 2012; Horn et al: Blood 2013)

8 MICHOR CML model 2005

9 Mathematical model structure Normal cells Leukemic cells Differentiation rate without imatinib Differentiation rate with imatinib Death rate Stem cells x 0 y 0 d a 0 a' Progenitors x 1 y 1 d 1 b b' Differentiated cells x 2 y 2 d 2 c c Terminally differentiated cells x 3 y 3 d 3 Michor et al, Nature 2005

10 Model equations Michor et al, Nature 2005

11 Mathematical model assumptions BCR/ABL is present in all leukemic cells and leads to slow clonal expansion of leukemic stem cells BCR/ABL leads to increased rate of leukemic cell proliferation compared to normal cells Imatinib reduced the proliferation rate of leukemic stem cells Michor et al, Nature 2005

12 Model fitting and statistical model Measurement of BCR/ABL over BCR ratio in 169 CML patients treated with first line imatinib therapy Biphasic decline in BCR/ABL levels 5% per day 0.8% per day Michor et al, Nature 2005

13 Translation of statistical model results to mathematical model First slope represents turnover of differentiated cells Once differentiated cells have reached steady state with progenitor cells, second slope is decline of progenitor cells 5% per day 0.8% per day Michor et al, Nature 2005

14 Mathematical model results First slope leads to ~1000 fold decline in leukemic cell burden Translates to a 1000 fold reduction of differentiation rate from progenitor cells to differentiated leukemic cells 5% decline per day translates to lifespan of 20 days After ~200 days differentiated cells reach steady state with the progenitor cells and they decrease at the same rate Second slope leads to >7 fold decline in progenitor cell burden 0.8% decline per day this translates to lifespan of 125 days Overall 5000 fold decline in leukemic cell burden Michor et al, Nature 2005

15 Cessation of TKI therapy Imatinib leads to 5000 fold decrease in leukemic cell burden, therefore cessation of therapy should lead to lower levels of BCR/ABL if stem cells had been eradicated by prior therapy HOWEVER, after discontinuation of treatment BCR/ABL increased to pretreatment levels or higher in 3 patients We conclude that leukemic stem cells, which drive CML disease, are not depleted by imatinib therapy Michor et al, Nature 2005

16 Criticism If imatinib does not affect leukemic stem cells this compartment will continue to expand. Clinical long term data is not consistent with this result Model predicted BCR/ABL response under continued imatinib therapy and without resistance Glauche et al, British Journal of Cancer 2007

17 MICHOR CML Model 2011

18 Mathematical model structure Normal cells Leukemic cells Differentiation rate without imatinib Differentiation rate with imatinib Death rate Division rate Stem cells x 0 y 0 d 0 r a a' y,z Progenitors x 1 y 1 d b 1 b' Differentiated cells x 2 y 2 d 2 c c Terminally differentiated cells x 3 y 3 d 3

19 Mathematical model assumptions BCR/ABL is present in all leukemic cells and leads to slow clonal expansion of leukemic stem cells BCR/ABL leads to increased rate of leukemic cell proliferation compared to normal cells Imatinib leads to reduction of differentiation rates in leukemic cells Imatinib possibly reduces the growth rate of cyclic stem cells Michor et al, Blood 2011

20 MODEL FITTING IRIS Trial Newly diagnosed patients treated with imatinib 400 mg qd Inclusion of 22 patients in the model No BCR/ABL measurements for the first 3 months so unable to determine first slope kinetics with IRIS data TIDEL trial Newly diagnosed patients treated with imatinib mg qd Inclusion of 44 patients in the model Initial slope kinetics (1, 2, 3, 6, 9, 12 months data) Michor et al, Blood 2011

21 Biphasic exponential model IRIS data TIDEL data Second slope First slope Second slope Third slope Michor et al, Blood 2011

22 Results for slope and turning point TIDEL data IRIS data Michor et al, Blood 2011

23 Purpose of using biological mathematical model Provision of biological processes explaining observed slope kinetics Able to predict treatment response for different cell compartments

24 Results for mathematical model TIDEL data IRIS data Decline of differentiated cells Decline of progenitor cells Decline of stem cells Michor et al, Blood 2011

25 Results for cell compartments Leukemic cells Normal cells Michor et al, Blood 2011

26 Conclusion Initial slope in IRIS patients is due to decline of progenitor cells, second slope is due to decline in stem cells Analysis was done on highly selected patients who responded to imatinib Michor et al, Blood 2011

27 ROEDER CML Model

28 Mathematical model structure Stem cells exist in 2 different environments (A and Ω) Compartment A: stem cell supporting niche promoting cellular quiescence and regeneration. Compartment Ω: active proliferation a: marker for stemness Leukemic cells have increased proliferation activity Roeder et al, Blood 2013

29 Mathematical model type Stochastic microsimulation of all cells at the single cell level Roeder et al, Blood 2013

30 Mathematical model assumptions TKIs induce an apoptotic effect TKIs inhibit proliferation of leukemic cells Roeder et al, Blood 2013

31 MODEL FITTING German cohort of IRIS Trial Newly diagnosed patients treated with imatinib 400 mg qd Inclusion of 51 patients in the model CML IV trial 280 patients in imatinib 400 mg arm Inclusion of 31 patients (11%) in the model Roeder et al, Blood 2013

32 Statistical model First slope (α) Statistical regression results for 51 IRIS trial patients Second slope (β) Roeder et al, Blood 2013

33 Correlation between first and second slope Roeder et al, Blood 2013

34 Mathematical model fitting Use of results for slope kinetics from statistical model to fit to mathematical model Iterative process to find degradation rate and probability to move to active stem cell compartment for each patient Assumption TKIs induce apoptosis TKIs inhibit leukemic cell proliferation Model estimation Change degradation rate of leukemic cells Change leukemic cell probability to move to active stem cell compartment Roeder et al, Blood 2013

35 Example patient 1 (IRIS trial) Statistical model Mathematical model Roeder et al, Blood 2013

36 Example patient 1 (IRIS trial) End of follow up End of follow up Undetectable BCR/ABL Undetectable BCR/ABL Leukemic stem cell eradication Roeder et al, Blood 2013

37 Results from mathematical model for IRIS trial and CML IV trial Roeder et al, Blood 2013

38 Required follow up time 7 year f/u data form IRIS trial was used Can therapeutic responses be predicted after shorter time period? Need for frequent measurements (6 8 BCR/ABL measurements for each slope) 3 years 4 years 5 years 6 years Full f/u time Roeder et al, Blood 2013

39 Simulation of treatment cessation Simulation of imatinib discontinuation in eligible IRIS Trial patients (MR 5.0 for >2 years) 47 of 51 patients are eligible Of those 47 patients 12 are predicted not to relapse Mathematical model TWISTER trial STIM trial Roeder et al, Blood 2013

40 Cessation simulation 12 patients predicted NOT to relapse 35 patients predicted to relapse Roeder et al, Blood 2013

41 Relationship between slopes and leukemic stem cells Are slope kinetics predictive of stem cell eradication? α/β<16 (=log1.2) predicts no relapse Time on therapy needed until discontinuation is safe: time α/(8 x β) Roeder et al, Blood 2013

42 Prediction of relapse with mathematical model Predicted response Recommended treatment time Individualized approach Conventional approach (2 years in MR 5.0 ) Roeder et al, Blood 2013

43 Conclusion from Roeder Model Confirmation of biphasic BCR/ABL decline Initial slope is due to decline in proliferating leukemic stem cells, second slope is decline in quiescent leukemic stem cells Individualized approach to determine best time for discontinuation of therapy improves probability for long time remission Only applies to responders, does not take into account resistance Roeder et al, Blood 2013

44 Conclusion Models are always wrong and a simplification of biological processes They allow testing of hypotheses that are impossible or difficult to test in vivo/vitro Dependence on assumptions (e.g. treatment effect of TKIs on stem cells, stem cell number, etc.) Models that are clinically relevant need fitting to clinical data Often required advanced mathematical skills need for close collaboration between mathematicians, disease experts (system biology)

45 FUTURE PERSPECTIVES Data from high throughput technology are impossible to understand with conventional methods Rather than focusing on individual genes and proteins mathematical models allow us to evaluate the system Increase in computational power will allow models to become more complex Data mining of electronic health records can add vast amounts of data outside of clinical trials Mathematical models should inform clinical trial design

46 Special thanks to: Vivian Oehler, MD Bill Hazelton, PhD Jerry Radich, MD

47 References Armitage, P. & Doll, R. A two stage theory of carcinogenesis in relation to the age distribution of human cancer. Br. J. Cancer 11, (1957). Emmert Streib, F., Dehmer, M. & Haibe Kains, B. Untangling statistical and biological models to understand network inference: the need for a genomics network ontology. Frontiers in Genetics 5, (2014). Foo, J., Drummond, M. W., Clarkson, B., Holyoake, T. & Michor, F. Eradication of chronic myeloid leukemia stem cells: a novel mathematical model predicts no therapeutic benefit of adding G CSF to imatinib. PLoS Comput. Biol. 5, e (2009). Glauche, I., Horn, M. & Roeder, I. Leukaemia stem cells: hit or miss? Br. J. Cancer 96, ; author reply (2007). Horn, M. et al. Model based decision rules reduce the risk of molecular relapse after cessation of tyrosine kinase inhibitor therapy in chronic myeloid leukemia. Blood 121, (2013). Knudson, A. G. Mutation and cancer: statistical study of retinoblastoma. Proc. Natl. Acad. Sci. U.S.A. 68, (1971). Komarova, N. L. Mathematical modeling of cyclic treatments of chronic myeloid leukemia. Math Biosci Eng 8, (2011). Loeb, L. A. A mutator phenotype in cancer. Cancer Res. 61, (2001). Melo, J. V. & Barnes, D. J. Chronic myeloid leukaemia as a model of disease evolution in human cancer. Nat. Rev. Cancer 7, (2007). Michor, F. et al. Dynamics of chronic myeloid leukaemia. Nature 435, (2005). Moolgavkar, S. H. & Venzon, D. J. Two event models for carcinogenesis: incidence curves for childhood and adult tumors. Mathematical Biosciences 47, (1979). Radivoyevitch, T., Kozubek, S. & Sachs, R. K. Biologically based risk estimation for radiation induced CML. Inferences from BCR and ABL geometric distributions. Radiat Environ Biophys 40, 1 9 (2001). Radivoyevitch, T., Ramsey, M. J. & Tucker, J. D. Estimation of the target stem cell population size in chronic myeloid leukemogenesis. Radiat Environ Biophys 38, (1999). Radivoyevitch, T. et al. Sex differences in the incidence of chronic myeloid leukemia. Radiat Environ Biophys 53, (2014). Roeder, I., Herberg, M. & Horn, M. An age structured model of hematopoietic stem cell organization with application to chronic myeloid leukemia. Bull. Math. Biol. 71, (2009). Stein, A. M. et al. BCR ABL Transcript Dynamics Support the Hypothesis That Leukemic Stem Cells Are Reduced during Imatinib Treatment. Clinical Cancer Research 17, (2011). Tang, M. et al. Selection pressure exerted by imatinib therapy leads to disparate outcomes of imatinib discontinuation trials. Haematologica 97, (2012). Tang, M. et al. Dynamics of chronic myeloid leukemia response to long term targeted therapy reveal treatment effects on leukemic stem cells. Blood 118, (2011). Vincent, P. C. et al. Relapse in chronic myeloid leukemia after bone marrow transplantation: biomathematical modeling as a new approach to understanding pathogenesis. Stem Cells 17, 9 17 (1999). Waller, L. A. Bridging gaps between statistical and mathematical modeling in ecology. Ecology 91, ; discussion (2010).

48 Additional slides not used during oral presentation

49 Armitage and Doll model M(0)(t): rate of mutation 0 X(t): number of normal (stem) cells at age t I(1): number of cells with 1 mutation I(2): number of cells with 2 mutations M: Malignant tumor Little 2010

50 Problems Experimental data did not support the number of mutations required by this model Next step was a two stage model proposed by Armitage and Doll, inspired by letter from Robert Platt

51 Hornsby et al: Lancet Oncology 2007

52 Clonal evolution hypothesis CML

53 Results from statistical models Exponential model: Biphasic exponential model:

54 Statistical models 1. Exponential model: assumption that leukemic cells decline exponentially 2. Biphasic exponential model: based on the initial observation of biphasic decline TIDEL DATA: R 2 exponential model TIDEL data = 0.81 R 2 biphasic exponential model TIDEL data = 0.98 IRIS DATA: R 2 exponential model = 0.8 R 2 biphasic exponential model = 0.89 Michor et al, Blood 2011

55 Observation that mortality rate of many solid tumors increase with power of age Hypothesis that the power integer translates into the number of ratelimiting steps required for formation of malignant tumor For most solid tumors this relates to 5 7 mutational steps

56 Observations that hereditary and sporadic cases of retinoblastoma could arise form same genetic mechanism with the same mutation rate two hit hypothesis Led to the acceptance of the somatic mutation hypothesis and discovery of tumor suppressor genes

57 Modeling for TKI cessation strategies Modeling stem cell pool to determine safe time for cessation of TKI Current approach in clinical trials STIM/TWISTER trial eligibility: minimum duration of UMRD (no detectable BCR/ABL mrna in any sample tested during the preceding 2 years, whether in peripheral blood (PB) or bone marrow (BM). Requirement of 2 qrt PCR results per year Minimum of 36 months of IM treatment Mahon et al, Lancet Oncology 2010 Ross et al, Blood 2013