Publications of Thomas Hillen

  Math Bio Cover       PDE Book Cover      Book cover       Book cover       Art book cover      Book cover


G. de Vries, T. Hillen, M. Lewis, J. Müller, B. Schönfisch

A Course in Mathematical Biology; Quantitative Modelling with Mathematical and Computational Methods. SIAM, 2006. ISBN: 0-89871-612-8

Math-Bio  book-page
1st edition
T. Hillen, E. Leonard, H. van Roessel
Partial Differential Equations: Theory and Completely solved problems.
Wiley, 2012-2018. ISBN: 978-1-118-06330-9
 2nd edition
T. Hillen, E. Leonard, H. van Roessel
Partial Differential Equations: Theory and Completely solved problems.
2nd edition, 2019,
Friesen Press
PDE book page

Instructional Videos

P. Ciarletta, T. Hillen, H. Othmer, L. Preziosi, D. Trucu
Mathematical Models and Methods for Living Systems,
Springer Basel, 2016,
Lecture Notes in Mathematics 2167 (eds: L. Preziosi, M. Chaplain, A. Pugliese)

Springer book page
A. Bianchi, T. Hillen, M.A. Lewis, Y. Yi.
The Dynamics of Biological Systems. Springer, Plant-Earth Series, 2018
Springer book page
F. Matthaeus, S. Matthaeus, S. Harris, T. Hillen
The Art of Theoretical Biology. Springer 2019
Springer book page
A. Buttenschoen, T. Hillen.
Non-Local Cell Adhesion Models: Symmetries and Bifurcations in 1-D
CMS/CAIMS Book Series No1, Springer, 2021
Springer book page

Editorships for:

  • Previous associate editorships: Journal of Mathematical Biology (2011 -  2021); SIAM J. Applied Mathematics (2011 - 2017); DCDS-B (Discrete and Continuous Dynamical Systems B, (2011 - 2017); Mathematical Medicine and Biology (2006 - 2018;  Acta Applicandae Mathematicae (2014 - 2018);
    Journal of Biological Dynamics (2006 - 2010); Networks and Heterogeneous Media (2005 - 2010);
    Canadian Applied Mathematics Quarterly (2005 - 2010).

(updated: February 2023)
V. Giunta, T. Hillen, MA. Lewis, JR. Potts,
Bifurcation Analysis of Non-local Multi-species Advection-diffusion Models, 2023
in preparation

T. Hillen, A Shyntar
Modelling of Cancer Stem Cell Driven Solid Tumors
book chapter, 2023

T. Hillen, N. Loy, K.J. Painter, R. Thiessen, Modelling Microtube Driven Invasion of Glioma, 2023 finishing touches

99 K. Deutscher, T. Hillen, J. Newby
A Computational Model for the Cancer Field Effect.
book chapter, 2022.

T. Hillen
A Classification of Musical Scales Using Binary Sequences
Journal of Humanistic Mathematics, 13(1), 118-130, 2023.

open access

97 V. Giunta, T. Hillen, M.A. Lewis, J. Potts,
Detecting minimum energy states and multi-stability in nonlocal advection-diffusion models for interacting species
, Journal of Math. Biol., 85:(56) 1-44,

open access
96 A. Shyntar, A. Patel, M. Rhodes, H. Enderling, T. Hillen
The Tumor Invasion Paradox in Cancer STem Cell-Driven Solid Tumors, Bulletin of Math. Biol. 84(12) 139-163, 2022.

open access
95 M. Rhodes, T. Hillen, V. Putkaradze
Comparing the effects of linear and one-term Ogden elasticity in a model of glioblastoma invasion, Brain Multiphysics, 3:100050,  2022.

open access
94 V. Giunta, T. Hillen, M.A. Lewis, J.R. Potts
Local and Global Existence for Nonlocal Multispecies Advection-Diffusion Models, SIAM Applied Dynamical Systems, 21(3):1686-1708,  2022.

open access
93 R. Thiessen, T. Hillen
Anisotropic Network Patterns in Kinetic and Diffusive Chemotaxis Models, MDPI Mathematics, 9(13), 1-22, 202.
open access: doi 10.3390/math9131561

online access
92 C. Contreras, J. Newby, T. Hillen
Personalized Virus-Load Curves of SARS-CoV-2 Infection,
Viruses 13(9), 1815, 2021.

online access
91 A. Baabdulla,  H. Now, JA Park, WJ. Kim, S. Jung, JY. Yoo, T. Hillen
Homogenization of a Reaction-Diffusion Equation can Explain Influenza A Virus Load Data, Journal of Theoretical Biology, 527(21) 1-10, 2020. doi: 10.1016/j.jtbi.2021.110816

90 N. Loy, T. Hillen, KJ. Painter,
Direction-Dependent Turning Leads to Anisotropic Diffusion and Persistence. European Journal of Applied Mathematics,  2021.
doi: 10.1017/S0956792521000206

F. Lutscher, T. Hillen
Correlated Random Walks in Heterogeneous Landscapes
: Derivation, Homogenization, and Invasion Fronts.
AIMS Mathematics, 6(8), 8920-8948, 2021

online access

M. Getz Y. Wang G. AnM. Asthana A. Becker C. Cockrell N. Collier M. Craig C.L. Davis R. Faeder A.N. Ford Versypt T. Mapder J.F. Gianlupi J.A. Glazier S. Hamis R. Heiland T. HillenD. Hou M.A. Islam A.L. Jenner F. Kurtoglu C. I. Larkin B. Liu F. Macfarlane P. Maygrundter P.A. Morel A. Narayanan J. Ozik E. Pienaar P. Rangamani A.S. Saglam J. E. Shoemaker A.M. Smith J.J.A. Weaver, P. Macklin

Iterative community-driven development of a SARS-CoV-2 tissue simulator
bioRxiv 2020.04.02.019075v4, 2021

T. Hillen
A Standard Virus-Load Function,
preprint, 2020.
86 A. Rhodes, T. Hillen.
Implications of immune-mediated metastatic growth on metastatic dormancy, blow-up, early detection, and treatment.
J.  Math. Biol., 81: 799-843, 2020. doi: 10.1007/s00285-020-01521-x

85 A. Rhodes, T. Hillen
A Mathematical Model for the Immune-Mediated Theory of Metastasis. J. Theoretical Biology, Vol 482, 2019
Journal link

84 T. Hillen, M.A. Lewis
Dynamical Systems in Biology - A Short Introduction.
In "Biological Dynamics" A. Bianchi, T. Hillen, M.A. Lewis, Y. Yi, Springer 2018.

83 K. Painer, T. Hillen
From Random Walks to Fully Anisotropic Diffusion Models for Cell and Animal Movement.
In "Cell Movement: Modelling and Applications" M. Stolarska, N. Tarfulea (eds) 2018 (book chapter).

82 C. Frei, T. Hillen, A. Rhodes
A Stochastic Model for Cancer Metastasis: Branching Stochastic Processes with Settlement. Math. Medicine and Biology, 37(2) 153-182, 2020.
Journal link

81 A. Buttenschoen, T. Hillen
Nonlocal Adhesion Models for Microorganisms on Bounded Domains. SIAM J. Appl. Math, 80 (1), 382-401, 2020
Journal link

O. Olobatuyi, G. de Vries, T. HIllen
Effects of G2 checkpoint dynamics on the low dose hyper-radiosensitivity and increased radioresistance.
J. Math. Biol. (Hadeler issue), 77:(6–7), 1969–1997, 2018. 


79 A. Buttenschoen, T. Hillen, A. Gerisch, K.J. Painter
A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis. 
J. Math. Biol., 76, (1–2), 429–456, 2018.
biorxiv 093617
Springer link
78 O. Olobatuyi, G. de Vries, T. Hillen
A Reaction-Diffusion Model for Radiation-Induced Bystander Effects
J. Math. Biol., 75, (2), 341–372, 2017. 

biorxiv 094375
Springer link
77 A. Swan, T. Hillen, J. Bowman, A. Murtha
A Patient-Specific Anisotropic Diffusion Model for Brain Tumour Spread
Bulletin Math. Biol. 80, (5), 1259–1291, 2017.

76 T. Hillen, K.J. Painter, A. Swan, A. Murtha
Moments of von Mises and Fisher Distributions and Applications
Math. Biosciences and Engineering, 14(3), 673-694, 2017.

75 J. Martin, T. Hillen
The Spotting Distribution of Wildfires.
Applied Sciences, 6(6) 177-211, 2016. doi:10.3390/app6060177

74 A. Rhodes, T. Hillen
Mathematical Modeling of the Role of Survivin on Dedifferentiation and Radioresistance in Cancer.
Bulletin Math. Biol., 78(6), 1162-1188, 2016.

73 A. Konstorum, J. Lowengrub, T. Hillen
Feedback regulation in a cancer cell model can cause an Allee effect.
Bulletin Math. Biology, 78(4), 754-785, 2016.
DOI 10.1007/s11538-016-0161-5

72 K.J. Painter, T. Hillen
Navigating the flow, individual and continuum models for homing behavior in flowing environments.
Royal Society Interface, 12: 20150647, 2015


 biorxiv 029389
I. Bica, T. Hillen, K.J. Painter
Aggregation of biological particles under radial directional guidance.
J. Theor. Biol. 427:77-89, 2017.

biorxiv 096602
J.R. Potts, T. Hillen, M.A. Lewis
Edge effects and the spatio-temporal scale of animal movement decisions.
Theoretical Ecology, 9920 233-247, 2015, free online.

69 T. Hillen, A. Swan
The diffusion limit of transport equations in biology
book chapter, CIME lectures, Springer

68 M. Winkler, T. Hillen, K.J. Painter
Global solvability and explicit bounds for a non-local adhesion model.
European J. Appl. Math., 2017, online first

67 T. Stocks, T. Hillen, J. Gong, M. Burger
A stochastic model for the normal tissue complication probability (NTCP) in radiation treatment of cancer.
Math. Med. and Biol. 2016, 1-24, doi: 10.1093/imammb/dqw013

 arXiv: 1412.3441
66 I. Borsi, A. Fasano, M. Primicerio, T. Hillen
Mathematical properties of a non-local integro-PDE model for cancer stem cells.  Mathematical Medicine and Biology,  2015, doi:10.1093/imammb/dqv037

biorxiv 019604
65 T. Hillen, D. White, A. Dawes, G. de Vries
Existence and uniqueness for a coupled PDE model for motor induced microtubule organization.  J. Biol. Dynamics, 2017, DOI 10.1080/17513758.2017.1310939

64 T. Hillen, B. Greese, J. Martin, G. de Vries
Birth-jump processes and Application to Forest Fire Spotting.
Journal of Biological Dynamics 9, 104-127, 2015.

  63 C. Engwer, T. Hillen, M.P. Knappitsch, C. Surulescu
A DTI-Based multiscale model for glioma growth including cell-ECM interactions.
J. Math. Biol. 71(3), 551--582, 2015.

T. Hillen, J. Zielinski, K.J. Painter
Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model
Discrete and Continuous Dynamical Systems - B (DCDS-B), 18 (10), 2013 Pages 2513-2536.

(big file, 15 MB)
  61 Jeff W.N. Bachman, T. Hillen
Mathematical optimization of the combination of radiation and differentiation therapies for cancer
Frontiers in Molecular and Cellular Oncology (Front. Oncol.) 2013,
doi: 10.3389/fonc.2013.00052 free online at:

K.J. Painter, T. Hillen
Mathematical modelling of glioma growth: the use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion.
J. Theoretical Biol., 323, 25-39, 2013.

  59 M. Delitala, T. Hillen
The Language of Systems Biology: bridging the scales.
book chapter in: J.A. Marsan and M. Delitala et al. "Managing complexity, reducing perplexity. Modeling biological systems", Springer 2013.pages 131-132.

58 T. Hillen, M.A. Lewis,
Mathematical Ecology of Cancer
book chapter in: J.A. Marsan and M. Delitala et al. "Managing complexity, reducing perplexity. Modeling biological systems", Springer 2013, pages 1-14.

57 T. Hillen, H. Enderling, P. Hahnfeld
The Tumor Growth Paradox and Immune System-Mediated Selection for Cancer Stem Cells
Bulletin for Math Biology, 75(1), 161-184, 2013. 

56 T. Hillen, K.J. Painter, M. Winkler
Anisotropic Diffusion in Oriented Environments can lead to Singularity Formation
European J. Applied Math., 2012. First View online:


55 T. Hillen, K.J. Painter, M. Winkler
Convergence of a cancer invasion model to a logistic chemotaxis model.
Math.  Models and Methods in Applied Sci. (M3AS), 23(1), 165-198,  2013.

54 T. Hillen, K. Painter
Transport Models for Movement in Oriented Habitats and Anisotropic Diffusion.
In: Dispersal, individual movement and spatial ecology: A mathematical perspective. Eds: M.A. Lewis, P. Maini, S. Petrovskii,  Heidelberg, Springer, 2012, 46 pages.

53 J. Gong, M. dos Santos, C. Finlay, T. Hillen
Are More Complicated Tumor Control Probability Models Better?
Math. Medicine and Biol. 2011, 30(1):1-19.
online October 17, 2011 doi:10.1093/imammb/dqr023

52 K. Painter, T. Hillen
Spatio-Temporal Chaos in a Chemotaxis Model
Physica D, 240, 363-375, 2011.

  51 Barber, J., Bose, C., Bourlioux, A., Braun, J., Brunelle, E., Garcia, T., Hillen, T. Ong, B., Burning Issues with PROMETHEUS, the Canada's Wildfire Growth Simulator. 
CAMQ (Canad. Appl. Math Quart) 16(4), 337-378, 2008 (published in 2010).

  50 Hillen, T.
Existence Theory for Correlated Random Walks on Bounded Domains
CAMQ, (Canad. Appl. Math. Quart.) 18(1), 1-40, 2010.

  49 Hillen, T., de Vries, G., Gong, J., Finlay, C.
From Cell Population Models to Tumour Control Probability: Including Cell Cycle Effects.
Acta Oncologica,  49, 1315-1323, 2010. 

 48 Hillen, T. and Hinow, P. and Wang, Z.A.
Mathematical Analysis of a Kinetic Models for Cell Movement in Network Tissues.
Discrete and Continuous Dyn. Syst. - B, 14(3), 1055-1080, 2010

 47 Khassehkan, H. Eberl, H.J. and Hillen, T. 
A Nonlinear Master Equation for a Degenerate Diffusion Model of Biofilm Growth.
Lecture Notes Computer Science, Vol 5544, p 735-744,2009
 46 Lee, J.M., Hillen, T. and Lewis, M.
Pattern Formation in Prey-Taxis Systems
J. Biol. Dynamics, 3(6), 551 – 573, 2009.

 45 K.P. Hadeler, T. Hillen, M.A. Lewis
Mathematical Modeling with Quiescent Phases.
in "Spatial Ecology", S. Cantrell, C. Cosner, S. Ruan (eds), 2009
 44 Babak, P. and Bourlioux, A. and Hillen, T.
The Effect of Wind on the Propagation of a Forest Fire.
SIAM J Appl Math, 70(4), 1364-1388, 2009.
 43 O'Rourke, F. and McAneney, H. and Hillen, T.
Linear Quadratic and Tumour Control Probability Modelling in External Beam Radiotherapy.
J. Math. Biology, 58, 799-817, 2009. 
42 Lee, J.M., Hillen, T. and Lewis, M.
Continuous Travelling Waves for Prey Taxis
Bull. Math. Bio., 70(3) 654-676, 2008.
41 Hillen, T. and Painter, K.
A User's Guide to PDE Models for Chemotaxis.
J. Math. Biol., 58(1),   183-217, 2009.


Chauviere, A. and Hillen, T. and Preziosi, L.

Modeling cell movement in anisotropic and heterogeneous network tissues

NHM (Networks and Heterogeneous Media), 2, 333-357, 2007.


Chauviere, A. and Hillen, T. and Preziosi, L.

Modeling the motion of a cell population in the extracellular matrix

DCDS-B (Discrete and Continuous Dynamical Systems, Series B), Special Issue September  2007, pages 250-259.


Lewis, M.A., Lutscher, F., Hillen T.

Spatial dynamics in ecology 

In M.A. Lewis, J. Keener, P. Maini and M. Chaplain (Eds.), Park City Mathematics Institute Volume in Mathematical Biology, (pp 25-45). Insitute for Advanced Study, Princeton.

Wang, Zhi An, Hillen, T. and Li, M.
Mesenchymal Motion Models in One Dimension

SIAM J. Appl. Math. 69 (2) 375-397, 2008
  36 Wang, Zhi An, and Hillen, T.
Pattern Formation for a Chemotaxis Model with Volume Filling Effects
Chaos, 17(3), 037108 (13 pages), 2007
  35 Wang, Zhi An and Hillen, T.
Shock Formation in a Chemotaxis Model
Math. Methods in the Appl. Sciences, 31(1), 45-70, 2008  

  34 Liu, W. Hillen, T. and Freedman, H.I.
A Mathematical Model for M-phase Specific Chemotherapy Including the
G0-Phase and Immuno-Response

Mathematical Biosciences and Eng., 4(2), 239-259, 2007.

  33 de Vries, Gerda and Hillen, Thomas
Teaching Mathematical Biology in a Summer School for Undergraduates
in Mathematical Modeling of Biological Systems, Volume II: Epidemiology, Evolution and Ecology, Immunology, Neural Systems and the Brain, and Innovative Mathematical Methods, A. Deutsch, R. Bravo de la Parra, R. de Boer, O. Diekmann, P. Jagers, E. Kisdi, M. Kretzschmar, P. Lansky, and H. Metz, editors, Birkhauser, Boston, 2008, pp. 369-377.
  32 Hillen, Thomas
A Classification of Spikes and Plateaus
SIAM REVIEWS,  49(1) 35-51, 2007.

Hillen, Thomas
M^5, Mesoscopic and Macroscopic Models for Mesenchymal Motion
2006,  J. Math. Biol. 53(4), 585-616, 2006.
(electroinc: DOI 10.1007/s00285-006-0017-y)
The original publication is available a

  30 Dawson Andria and Thomas Hillen,
Derivation of the Tumour Control Probability (TCP) from a Cell Cycle Model
Comput. and Math. Meth. in Medicine, 2006, 7:121-142.

  29 Hillen, Thomas and Painter, Kevin and Schmeiser, Christian
Global Existence for Chemotaxis with Finite Sampling Radius
Discr. Cont. Dyn. Syst. B (DCDS-B), 7(1) 125-144, (2007).

  28 Hillen, Thomas and Renclawowicz, Joanna and Wang, Zhian 
Analysis of an Attraction-Repulsion Chemotaxis Model
2007, unpublished manuscript.


Hadeler, K.P. and Hillen, T.
Coupled Dynamics and Quiescent States
Deterministic and Stochastic Modelling in Biomedicine, Economics and Industry.
Dedicated to the 60th birthday of Vincenzo Capasso.
G. Aletti, M. Burger, A. Micheletti, D. Morale Editors,
Springer, Heidelberg, 2006.


Hillen, Thomas, Hadeler, K.P.
Hyperbolic Systems and Transport Equations in Mathematical Biology
p 257-279 in G. Warnecke (ed), Analysis and Numerics for Conservation Laws, Springer, Berlin, Heidelberg, 2005.


Hillen, Thomas
On the L^2-Moment Closure of Transport Equations:
The General Case

Discr. Cont. Dyn. Systems, Series B, 5(2) 299-318, 2005


Hillen, Thomas
On the L^2-Moment Closure of Transport Equations:
The Cattaneo Approximation

Discr. Cont. Dyn. Systems, Series B, 4(4), 961-982, 2004


Hillen, Thomas and Potapov, Alex
The One-Dimensional Chemotaxis Model: Global Existence and Asymptotic Profile
Math. Meth. Appl. Sci., 27:1783-1801, 2004.

22 Potapov, Alex and Hillen, Thomas
Metastability in Chemotaxis Models
J. Dyn. Diff. Eq. , 17(2),   293-330, 2005.. 

21 Painter, Kevin and Hillen, Thomas
Volume-Filling and Quorum Sensing in Models for Chemosensitive Movement
Canadian Applied Mathematics Quarterly, Vol 10(4), 2002, 501-543.

20 Hadeler, K.P., Hillen, T.,  and Lutscher, Frithjof
The Langevin or Klein-Kramers Approach to Biological Modeling.
M3AS (Math. Models Methods Appl. Sci.), 14(10), 
1561-1583, 2004.,

19 Hillen, Thomas and Levine, Howard
Blow-up in hyperbolic models for chemotaxis.
Zeitschrift fuer Angewandte Mathematik und Physik (ZAMP), vol 54(5), 839-868, 2003.

18 Hillen, Thomas
Transport Equations with Resting Phases.
European J. Appl. Math. 14(5), 613-636, 2003.

17 Hillen, Thomas
Hyperbolic Models for Chemosensitive Movement.
Math. Models Methods Appl. Sci., 12(7), 1007-1034, 2002. 
Review-article  download 
16 Dolak, Yasmin and Hillen, Thomas
Cattaneo Models for Chemotaxis, Numerical Solution and Pattern Formation.
J. Math. Biol. 46 (2003) 2, 153-170.

15 Hillen, Thomas
Transport Equations and Chemosensitive Movement.
Habilitation Thesis, University of Tuebingen, 2001.
14 Othmer, H.G. and Hillen, Thomas
The Diffusion Limit of Transport Equations II: Chemotaxis Equations.
SIAM J. Appl. Math, 62(4), 1222-1250, 2002.
13 Hillen, Thomas and Painter, Kevin
Global Existence far a Parabolic Chemotaxis Model  with Prevention of Overcrowding.
Advances in Applied Mathematics, 26(4), 280-301, 2001.
12 Hillen, Thomas and Rohde, Christian and Lutscher, Frithjof
Existence of weak solutions for a hyperbolic model of chemosensitive movement .
J. Math. Anal. Appl., 260,  173-1999,  2001.
11 Hillen, Thomas and Othmer, H.G.
The Diffusion Limit of Transport Equations Derived From Velocity Jump Processes.
SIAM J. Appl. Math., 61, 751-775, 2000.
10 Hillen, Thomas and Stevens, Angela
Hyperbolic models for Chemotaxis in 1-D.
Nonlinear Analysis: Real World Applications 1(3), 409-433, 2000. 
9 Hillen, Thomas
Orthogonal closure procedure for the first two moments of reaction transport equations.
SFB 382, Report Nr.: 124, 1999.
8 Hillen, Thomas
Qualitative Analysis of semilinear Cattaneo systems.
Math. Models and Methods in Appl. Sci., Vol. 8, No. 3, 507-519, 1998.
   download pdf
7 Müller, Johannes and Hillen, Thomas
Modulation equations and the parabolic limits of reaction random walk systems.
Math. Methods Appl. Sci., 21, 1207-1226, 1998.
  6 Hillen, Thomas 
Invariance Principles for Hyperbolic Random Walk Systems.
J. Math. Analysis Appl., 210, 360-374, 1997.
5 Hillen, Thomas 
Qualitative Analysis of hyperbolic Random Walk Systems.
SFB 382, Report No. 43, April 1996.
4 HIllen, Thomas
Nonlinear Hyperbolic Systems Describing Random Motion and their Application on the Turing Model.
Dissertation Summaries in Math., Vol 1, 121-128, 1996
3 Hillen, Thomas
A Turing model with correlated random walk.
J. of Math. Biology, Vol 35, 49-72, 1996
2 Hillen, Thomas
Nichtlineare hyperbolische Systeme zur Modellierung von Ausbreitungsvorgängen und Anwendung auf das Turing Modell.
Dissertation, Fakultät für Mathematik, Universität Tübingen, 1995.
1 Hadeler, K.P. and Hillen, Thomas
Differential Equations on Branched Manifolds.
in: Evolution Equations, Control Theory and Biomathematics, P. Clement and G. Lumer (eds), 
Han-sur-Lesse, 241-258, Marcel Dekker, 1994.
   download PDF-file