« back to all changes in this revision
Viewing changes to chapters/numerical-simulations-2.tex
- browse files at revision 128
- compare with another revision
- download diff
- download tarball
- view history from revision 128
- Committer: Harish Narayanan
- Date: 2007-10-26 06:26:24 UTC
- Revision ID: hnarayan@umich.edu-20071026062624-fxk074gq1dvvezog
Polished the document some more, and better indented the source
added
removed
chapters/numerical-simulations-2.tex
54
54
imposed by treating the tissue as a whole and solving a summation of
55
55
Equation~(\ref{localbalanceofmomentum}) over all species. This
56
56
simplification necessitated additional assumptions on the underlying
57
micro-mechanics, and these were discussed in
58
Section~\ref{constriction-1}. In contrast, the implementation used in
59
this chapter solves the {\em detailed} momentum balance equations;
60
enforcing the balance of momentum for each species separately. The
61
coupling between the mechanics equations of the individual species is
62
introduced by specifying momentum transfer terms, $\bq^{\iota}$,
63
arising from frictional interaction as discussed in
57
micro-mechanics, and these were discussed in Section~\ref{constrict%
58
ion-1}. In contrast, the implementation used in this chapter solves
59
the {\em detailed} momentum balance equations; enforcing the balance
60
of momentum for each species separately. The coupling between the
61
mechanics equations of the individual species is introduced by
62
specifying momentum transfer terms, $\bq^{\iota}$, arising from
63
frictional interaction as discussed in
64
64
Section~\ref{eu-interaction-forces}.
65
65
66
The balance of mass~(\ref{localbalanceofmass}) and
67
momentum~(\ref{localbalanceofmomentum}) equations for the solid
68
collagen are solved in the reference configuration of the tissue,
69
$\Omega_{0}$, and the balance of mass~(\ref{eu-localbalanceofmass})
70
and momentum~(\ref{eu-localbalanceofmomentum}) equations for the fluid
66
The balance of mass~(\ref{localbalanceofmass}) and momentum~(\ref%
67
{localbalanceofmomentum}) equations for the solid collagen are solved
68
in the reference configuration of the tissue, $\Omega_{0}$, and the
69
balance of mass~(\ref{eu-localbalanceofmass}) and
70
momentum~(\ref{eu-localbalanceofmomentum}) equations for the fluid
71
71
phase are solved in the current configuration, $\Omega_{t}$. Recall
72
72
that this choice is justified because we know the reference
73
73
configuration of the solid phase of the tissue. These equations, along
74
74
with the saturation constraint discussed below, are solved
75
simultaneously for the for the solid
76
concentration,~$\rho_{0}^{\mathrm{c}}$, and
77
displacement,~$\bu^{\mathrm{c}}$; and the fluid
75
simultaneously for the for the solid concentration,~$\rho_{0} ^
76
{\mathrm{c}}$, and displacement,~$\bu^{\mathrm{c}}$; and the fluid
78
77
concentration,~$\rho^{\mathrm{f}}$, velocity,~$\bv^{\mathrm{f}}$, and
79
78
pressure,~$p^{\mathrm{f}}$. Variable-order backward difference
80
79
formulae \citep{leveque2007} are used for forwarding the equations
83
82
In the interest of generality, the formulation and corresponding
84
83
implementation presented in Chapters~\ref{lagrangian-perspective} and
85
84
\ref{numerical-simulations-1} allowed for the possibility of
86
cavitation in tissues under certain ex~vivo/in~vitro
87
conditions. However, since it is well established that under normal
88
physiological conditions soft tissues are fully saturated by the
89
fluid, this condition will be imposed in the following calculations.
85
cavitation in tissues under certain ex~vivo/in~vitro conditions.
86
However, since it is well established that under normal physiological
87
conditions soft tissues are fully saturated by the fluid, this
88
condition will be imposed in the following calculations.
90
89
91
90
The concentration of each species~$\iota$ can be expressed as the
92
91
product of two non-negative scalar fields: $\rho^\iota = \phi^\iota
351
350
these cases, it is observed that the dynamic case decays gradually and
352
351
has oscillations (Figure~\ref{velocity-evolution-dynamic}) while the
353
352
quasistatic case reaches a higher magnitude, and decays nearly
354
instantaneously without manifesting oscillations
355
(Figure~\ref{velocity-evolution-quasistatic}).
353
instantaneously without manifesting oscillations (Figure~\ref%
354
{velocity-evolution-quasistatic}).
356
355
357
356
\begin{figure}[!hptb]
358
357
\centering
518
517
holding one of the longitudinal edges fixed while subjecting the other
519
518
to the suitable displacement load. The lateral edges of the solid
520
519
remain traction free. For the fluid, there is no flow relative to the
521
solid at the longitudinal edges, i.e., $\bv^{\mathrm{f}} =
522
\bv^{\mathrm{c}}$. Since we are simulating the tissue being held by
523
grips at the longitudinal edges, this boundary condition ensures that
524
there is no outflow or inflow along those edges. The lateral edges
525
expose the fluid to the bath, and therefore the fluid pressure is
526
equated to that of the bath, 0~MPa, along those edges.
520
solid at the longitudinal edges, i.e., $\bv^{\mathrm{f}} = \bv ^
521
{\mathrm{c}}$. Since we are simulating the tissue being held by grips
522
at the longitudinal edges, this boundary condition ensures that there
523
is no outflow or inflow along those edges. The lateral edges expose
524
the fluid to the bath, and therefore the fluid pressure is equated to
525
that of the bath, 0~MPa, along those edges.
527
526
528
527
Figure~\ref{poro-stress-relax-0p01} shows the stress relaxation in a
529
528
quasistatic calculation\footnote{Since the displacement condition
545
544
response, the next test doubles the strain rate to $\dot{\epsilon} =
546
545
0.02$ Hz. The tissue is subjected to the same maximum strain of 0.085,
547
546
now in 4.25~s, and is held fixed for the remainder of the test. The
548
stress relaxation resulting from this test is shown in
549
Figure~\ref{poro-stress-relax-0p02}. The initial peak stress is now
550
increased; an observation which is in agreement with classical results
551
in viscoelasticity theory. This is because the increased strain rate
547
stress relaxation resulting from this test is shown in Figure~\ref{%
548
poro-stress-relax-0p02}. The initial peak stress is now increased;
549
an observation which is in agreement with classical results in
550
viscoelasticity theory. This is because the increased strain rate
552
551
results in an increased relative velocity between the phases
553
552
initially, which correspondingly increases the frictional interaction
554
553
between the phases.
654
653
When this average strain rate is decreased to $\bar{\dot{\epsilon}} =
655
654
0.001$ Hz, (1/10$^\mathrm{th}$ the rate of the preceding calculation),
656
655
the relative velocity between the phases correspondingly decreases,
657
resulting in reduced dissipation and area of the hysteresis loop
658
(see Figure~\ref{medium-hysteresis-dynamic-0p001-d1p037}). In other
659
words, this slower process proceeds closer to thermodynamic
660
equilibrium. In an analogous comparison, when the average strain rate
661
is maintained at $\bar{\dot{\epsilon}} = 0.01$ Hz, but the magnitude
662
of the frictional coefficient tensor is increased by a factor of 10 to
663
$D=10.37$ MPa.s.mm$^{-2}$, it is observed that the dynamic effects of
664
the fluid flow are much more prominent (as observed in Figure~\ref%
656
resulting in reduced dissipation and area of the hysteresis loop (see
657
Figure~\ref{medium-hysteresis-dynamic-0p001-d1p037}). In other words,
658
this slower process proceeds closer to thermodynamic equilibrium. In
659
an analogous comparison, when the average strain rate is maintained at
660
$\bar{\dot{\epsilon}} = 0.01$ Hz, but the magnitude of the frictional
661
coefficient tensor is increased by a factor of 10 to $D=10.37$
662
MPa.s.mm$^{-2}$, it is observed that the dynamic effects of the fluid
663
flow are much more prominent (as observed in Figure~\ref%
665
664
{medium-hysteresis-dynamic-0p01-d10p37}). This is because the
666
665
comparable strain rates ensure similar relative velocities between the
667
666
phases (in the calculations corresponding to Figures~\ref{medium%
685
684
any of the dynamic effects arising from the fluid flow.
686
685
687
686
\begin{figure}[!hptb]
688
\centering
689
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
690
eulerian/pulling/plots/poro-elastic/medium-hysteresis-static-0p01-d1p037}
691
\caption{Quasistatic poroelastic model, $\dot{\epsilon}=0.01$ Hz, $D=1.037$
692
MPa.s.mm$^{-2}$.}
693
\label{medium-hysteresis-static-0p01-d1p037}
694
\end{figure}
695
696
\begin{figure}[!hptb]
697
\centering
698
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
699
eulerian/pulling/plots/poro-elastic/medium-hysteresis-dynamic-0p01-d1p037}
700
\caption{Dynamic poroelastic model, $\bar{\dot{\epsilon}}=0.01$ Hz,
701
$D=1.037$ MPa.s.mm$^{-2}$.}
702
\label{medium-hysteresis-dynamic-0p01-d1p037}
703
\end{figure}
704
705
\begin{figure}[!hptb]
706
\centering
707
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
708
eulerian/pulling/plots/poro-elastic/medium-hysteresis-dynamic-0p001-d1p037}
709
\caption{Dynamic poroelastic model, $\bar{\dot{\epsilon}}=0.001$ Hz,
710
$D=1.037$ MPa.s.mm$^{-2}$.}
711
\label{medium-hysteresis-dynamic-0p001-d1p037}
712
\end{figure}
713
714
\begin{figure}[!hptb]
715
\centering
716
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
717
eulerian/pulling/plots/poro-elastic/medium-hysteresis-dynamic-0p01-d10p37}
718
\caption{Dynamic poroelastic model, $\bar{\dot{\epsilon}}=0.01$ Hz,
719
$D=10.37$ MPa.s.mm$^{-2}$.}
720
\label{medium-hysteresis-dynamic-0p01-d10p37}
721
\end{figure}
722
723
\begin{figure}[!hptb]
724
\centering
725
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
726
eulerian/pulling/plots/visco-elastic/medium-hysteresis-visco-0p01-t0p3}
727
\caption{Dynamic viscoelastic model, $\dot{\epsilon}=0.01$ Hz,
728
$\tau=0.3$ s.}
729
\label{medium-hysteresis-visco-0p01-t0p3}
687
\centering
688
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
689
eulerian/pulling/plots/poro-elastic/medium-hysteresis-static%
690
-0p01-d1p037}
691
\caption{Quasistatic poroelastic model, $\dot{\epsilon}=0.01$ Hz,
692
$D=1.037$ MPa.s.mm$^{-2}$.}
693
\label{medium-hysteresis-static-0p01-d1p037}
694
\end{figure}
695
696
\begin{figure}[!hptb]
697
\centering
698
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
699
eulerian/pulling/plots/poro-elastic/medium-hysteresis-dynamic%
700
-0p01-d1p037}
701
\caption{Dynamic poroelastic model, $\bar{\dot{\epsilon}}=0.01$ Hz,
702
$D=1.037$ MPa.s.mm$^{-2}$.}
703
\label{medium-hysteresis-dynamic-0p01-d1p037}
704
\end{figure}
705
706
\begin{figure}[!hptb]
707
\centering
708
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
709
eulerian/pulling/plots/poro-elastic/medium-hysteresis-dynamic%
710
-0p001-d1p037}
711
\caption{Dynamic poroelastic model, $\bar{\dot{\epsilon}}=0.001$ Hz,
712
$D=1.037$ MPa.s.mm$^{-2}$.}
713
\label{medium-hysteresis-dynamic-0p001-d1p037}
714
\end{figure}
715
716
\begin{figure}[!hptb]
717
\centering
718
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
719
eulerian/pulling/plots/poro-elastic/medium-hysteresis-dynamic%
720
-0p01-d10p37}
721
\caption{Dynamic poroelastic model, $\bar{\dot{\epsilon}}=0.01$ Hz,
722
$D=10.37$ MPa.s.mm$^{-2}$.}
723
\label{medium-hysteresis-dynamic-0p01-d10p37}
724
\end{figure}
725
726
\begin{figure}[!hptb]
727
\centering
728
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
729
eulerian/pulling/plots/visco-elastic/medium-hysteresis-visco%
730
-0p01-t0p3}
731
\caption{Dynamic viscoelastic model, $\dot{\epsilon}=0.01$ Hz,
732
$\tau=0.3$ s.}
733
\label{medium-hysteresis-visco-0p01-t0p3}
730
734
\end{figure}
731
735
732
736
\section{Mechanics and the growing tumour}
745
749
Similar to the approach followed in Section~\ref{simple-physics}, the
746
750
computations presented below serve only to demonstrate aspects of the
747
751
coupled physics underlying the problem, and the actual constitutive
748
modelling choices (and corresponding numerical parameters) chosen are
749
not intended for direct comparison with experiment. Incorporating more
750
realistic modelling choices (such as the use of more sophisticated
751
biochemistry involving additional species \citep{tjacks2000}), and the
752
ascertainment of corresponding parameters, is a direction for future
753
work.
752
modelling choices made (and the corresponding numerical parameters
753
used) are not intended for direct comparison with experiment.
754
Incorporating more realistic modelling choices (such as the use of
755
more sophisticated biochemistry involving additional species
756
\citep{tjacks2000}), and the ascertainment of corresponding
757
parameters, is a direction for future work.
754
758
755
759
The computations presented in this section are motivated by and aim to
756
760
replicate a fundamental experimental observation: Compressive solid
821
825
1.1~kg.m$^{-3}$/1~kg.m$^{-3}$, as one would expect.
822
826
823
827
\begin{figure}[!hptb]
824
\centering
825
\includegraphics[width=0.9\textwidth]{images/examples/%
826
eulerian/cancer/isotropic-swelling-0}
827
\caption{A semicircular tumour at time $t=0$ days.}
828
\label{tumour-isotropic-swelling-0}
829
\end{figure}
830
831
\begin{figure}[!hptb]
832
\centering
833
\includegraphics[width=0.9\textwidth]{images/examples/%
834
eulerian/cancer/isotropic-swelling-100}
835
\caption{A semicircular tumour at time $t=100$ days.}
836
\label{tumour-isotropic-swelling-100}
837
\end{figure}
838
839
\begin{figure}[!hptb]
840
\centering
841
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
842
eulerian/cancer/isotropic-swelling-area-evolution}
843
\caption{The area of the tumour evolving over 100 days.}
844
\label{tumour-isotropic-area-evolution}
828
\centering
829
\includegraphics[width=0.9\textwidth]{images/examples/%
830
eulerian/cancer/isotropic-swelling-0}
831
\caption{A semicircular tumour at time $t=0$ days.}
832
\label{tumour-isotropic-swelling-0}
833
\end{figure}
834
835
\begin{figure}[!hptb]
836
\centering
837
\includegraphics[width=0.9\textwidth]{images/examples/%
838
eulerian/cancer/isotropic-swelling-100}
839
\caption{A semicircular tumour at time $t=100$ days.}
840
\label{tumour-isotropic-swelling-100}
841
\end{figure}
842
843
\begin{figure}[!hptb]
844
\centering
845
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
846
eulerian/cancer/isotropic-swelling-area-evolution}
847
\caption{The area of the tumour evolving over 100 days.}
848
\label{tumour-isotropic-area-evolution}
845
849
\end{figure}
846
850
847
851
\subsection{A constraining wall and soft contact mechanics}
876
880
smooth in time.
877
881
878
882
Starting with the same initial conditions as the previous test
879
(Figure~\ref{tumour-isotropic-swelling-0}),
880
Figure~\ref{tumour-constrained-swelling-120} depicts the compressive
881
horizontal stress built-up in the solid after 120~days due to the
882
presence of the wall (not visible in the figure). Notice that the
883
velocity vectors are much smaller in the constrained
884
direction. Figure~\ref{tumour-constrained-stress-evolution} shows the
885
time evolution of the compressive horizontal stress at a point near
886
the extreme right of the domain. The stress increases sharply as the
887
point gets close to the wall, but remains smooth in time.
883
(Figure~\ref{tumour-isotropic-swelling-0}), Figure~\ref{tumour%
884
-constrained-swelling-120} depicts the compressive horizontal stress
885
built-up in the solid after 120~days due to the presence of the wall
886
(not visible in the figure). Notice that the velocity vectors are much
887
smaller in the constrained direction. Figure~\ref{tumour-constrain%
888
ed-stress-evolution} shows the time evolution of the compressive
889
horizontal stress at a point near the extreme right of the domain. The
890
stress increases sharply as the point gets close to the wall, but
891
remains smooth in time.
888
892
889
893
\begin{figure}[!hptb]
890
\centering
891
\includegraphics[width=0.9\textwidth]{images/examples/%
892
eulerian/cancer/constrained-swelling-120}
893
\caption{The growing tumour constrained by a wall at time $t=120$ days.}
894
\label{tumour-constrained-swelling-120}
894
\centering
895
\includegraphics[width=0.9\textwidth]{images/examples/%
896
eulerian/cancer/constrained-swelling-120}
897
\caption{The growing tumour constrained by a wall at time $t=120$
898
days.}
899
\label{tumour-constrained-swelling-120}
895
900
\end{figure}
896
901
897
902
\begin{figure}[!hptb]
898
\centering
899
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
900
eulerian/cancer/constrained-stress-evolution}
901
\caption{The horizontal stress in the tumour evolving over 120 days.}
902
\label{tumour-constrained-stress-evolution}
903
\centering
904
\includegraphics[width=0.6\textwidth,angle=270]{images/examples/%
905
eulerian/cancer/constrained-stress-evolution}
906
\caption{The horizontal stress in the tumour evolving over 120
907
days.}
908
\label{tumour-constrained-stress-evolution}
903
909
\end{figure}
904
910
905
911
\clearpage
919
925
\begin{equation}
920
926
\Bsigma^{\mathrm{c}} =
921
927
\underbrace{\frac{1}{J}\ \frac{\rho_{0}^{\mathrm{c}}}
922
{\tilde{\rho_{0}^{\mathrm{c}}}}\
923
\frac{\partial \hat{\psi^{\mathrm{c}}}}{\partial
924
\bF} \bF^{\mathrm{T}} }_{\text{Passive}}
925
+ \underbrace{\tau\ \rho^{\mathrm{c}} \rho^{\mathrm{cell}}
926
(N - \rho^{\mathrm{cell}})\ \bone.
928
{\tilde{\rho_{0}^{\mathrm{c}}}}\ \frac{\partial
929
\hat{\psi^{\mathrm{c}}}}{\partial \bF} \bF^{\mathrm{T}}
930
}_{\text{Passive}} + \underbrace{\tau\ \rho^{\mathrm{c}}
931
\rho^{\mathrm{cell}} (N - \rho^{\mathrm{cell}})\ \bone.
927
932
}_{\text{Active}}
928
933
\label{activepassivesplit}
929
934
\end{equation}
955
960
neighbourhoods in greater.
956
961
957
962
\begin{figure}[!hptb]
958
\centering
959
\includegraphics[width=0.9\textwidth]{images/examples/%
960
eulerian/cancer/homogeneous-inward-tug}
961
\caption{Homogeneous inward pull due to a uniform distribution of
962
cells.}
963
\label{tumour-homogeneous-inward-tug}
963
\centering
964
\includegraphics[width=0.9\textwidth]{images/examples/%
965
eulerian/cancer/homogeneous-inward-tug}
966
\caption{Homogeneous inward pull due to a uniform distribution of
967
cells.}
968
\label{tumour-homogeneous-inward-tug}
964
969
\end{figure}
965
970
966
971
\begin{figure}[!hptb]
967
\centering
968
\includegraphics[width=0.9\textwidth]{images/examples/%
969
eulerian/cancer/heterogeneous-inward-tug}
970
\caption{Heterogeneous traction due to a non-uniform distribution of
971
cells.}
972
\label{tumour-heterogeneous-inward-tug}
972
\centering
973
\includegraphics[width=0.9\textwidth]{images/examples/%
974
eulerian/cancer/heterogeneous-inward-tug}
975
\caption{Heterogeneous traction due to a non-uniform distribution of
976
cells.}
977
\label{tumour-heterogeneous-inward-tug}
973
978
\end{figure}
974
979
975
980
\clearpage
982
987
we allow for the cells to proliferate and move via diffusion and
983
988
haptotaxis (again, following the work of \citet{namyetal:04}). We
984
989
solve the mass transport equation (\ref{eu-localbalanceofmass}) for
985
the cells to determine their current concentration
986
fields. In order to account for the aforementioned modes of mass
987
transport, we specify the following constitutive form for the cell
988
mass flux:
990
the cells to determine their current concentration fields. In order to
991
account for the aforementioned modes of mass transport, we specify the
992
following constitutive form for the cell mass flux:
989
993
990
994
\begin{equation}
991
\rho^{\mathrm{cell}}\ \bv^{\mathrm{cell}} = \underbrace{h\ \rho^{\mathrm{cell}}\
992
\mathrm{grad}\left(\rho^{\mathrm{c}}\right)}_{\text{Haptotactic flux}}
993
-\underbrace{D^{\mathrm{cell}}\ \mathrm{grad}\left(\rho^{\mathrm{cell}}\right)
994
}_{\text{Cell diffusion}},
995
\rho^{\mathrm{cell}}\ \bv^{\mathrm{cell}} =
996
\underbrace{h\ \rho^{\mathrm{cell}}\ \mathrm{grad}
997
\left(\rho^{\mathrm{c}}\right)}_{\text{Haptotactic flux}}
998
-\underbrace{D^{\mathrm{cell}}\ \mathrm{grad}
999
\left(\rho^{\mathrm{cell}}\right) }_{\text{Cell diffusion}},
995
1000
\end{equation}
996
1001
997
1002
\noindent where $h$ is the haptotactic coefficient and
1018
1023
\clearpage
1019
1024
1020
1025
\begin{figure}[!hptb]
1021
\centering
1022
\includegraphics[width=0.9\textwidth]{images/examples/%
1023
eulerian/cancer/diffusing-proliferating-cells-0}
1024
\caption{The cells diffusing and proliferating at time $t=0$ days.}
1025
\label{tumour-diffusion-proliferation-0}
1026
\end{figure}
1027
1028
\begin{figure}[!hptb]
1029
\centering
1030
\includegraphics[width=0.9\textwidth]{images/examples/%
1031
eulerian/cancer/diffusing-proliferating-cells-33}
1032
\caption{The cells diffusing and proliferating at time $t=33$ days.}
1033
\label{tumour-diffusion-proliferation-33}
1034
\end{figure}
1035
1036
\begin{figure}[!hptb]
1037
\centering
1038
\includegraphics[width=0.9\textwidth]{images/examples/%
1039
eulerian/cancer/diffusing-proliferating-cells-67}
1040
\caption{The cells diffusing and proliferating at time $t=67$ days.}
1041
\label{tumour-diffusion-proliferation-67}
1042
\end{figure}
1043
1044
\begin{figure}[!hptb]
1045
\centering
1046
\includegraphics[width=0.9\textwidth]{images/examples/%
1047
eulerian/cancer/diffusing-proliferating-cells-100}
1048
\caption{The cells diffusing and proliferating at time $t=100$ days.}
1049
\label{tumour-diffusion-proliferation-100}
1026
\centering
1027
\includegraphics[width=0.9\textwidth]{images/examples/%
1028
eulerian/cancer/diffusing-proliferating-cells-0}
1029
\caption{The cells diffusing and proliferating at time $t=0$ days.}
1030
\label{tumour-diffusion-proliferation-0}
1031
\end{figure}
1032
1033
\begin{figure}[!hptb]
1034
\centering
1035
\includegraphics[width=0.9\textwidth]{images/examples/%
1036
eulerian/cancer/diffusing-proliferating-cells-33}
1037
\caption{The cells diffusing and proliferating at time $t=33$ days.}
1038
\label{tumour-diffusion-proliferation-33}
1039
\end{figure}
1040
1041
\begin{figure}[!hptb]
1042
\centering
1043
\includegraphics[width=0.9\textwidth]{images/examples/%
1044
eulerian/cancer/diffusing-proliferating-cells-67}
1045
\caption{The cells diffusing and proliferating at time $t=67$ days.}
1046
\label{tumour-diffusion-proliferation-67}
1047
\end{figure}
1048
1049
\begin{figure}[!hptb]
1050
\centering
1051
\includegraphics[width=0.9\textwidth]{images/examples/%
1052
eulerian/cancer/diffusing-proliferating-cells-100}
1053
\caption{The cells diffusing and proliferating at time $t=100$ days.}
1054
\label{tumour-diffusion-proliferation-100}
1050
1055
\end{figure}
1051
1056
1052
1057
\clearpage
1057
1062
the same initial conditions for the cells as the previous calculation
1058
1063
(a cell-rich bulb at the centre of the domain), but in order to induce
1059
1064
haptotaxis, we begin with the heterogenous ECM concentration (varying
1060
between 0.5~kg.m$^{-3}$ and 1.5~kg.m$^{-3}$), seen in
1061
Figure~\ref{heterogeneous-ecm-concentration}. In these tests, the
1062
haptotactic coefficient $h$ is 0.1~mm$^2$.day$^{-1}$.mm$^3$.kg$^{-1}$.
1063
Figures~\ref{tumour-haptotaxis-proliferation-0}--%
1064
\ref{tumour-haptotaxis-proliferation-10} show snapshots of the cells
1065
undergoing haptotaxis and proliferating during the course of the
1066
test. The colour contours provide the evolving cell concentration
1067
fields (in~kg.m$^{-3}$) and the arrows provide the deformation
1068
direction of the ECM, induced by the cell traction. We observe that
1069
the cells migrate toward areas of higher ECM while proliferating. Note
1070
that the directionality of the cell traction field changes
1071
correspondingly with the concentration field, as noted by the lengths
1072
and directions of the arrows.
1073
1074
\begin{figure}[!hptb]
1075
\centering
1076
\includegraphics[width=0.9\textwidth]{images/examples/%
1077
eulerian/cancer/heterogeneous-ecm-concentration}
1078
\caption{Heterogeneous extra-cellular matrix concentration
1079
(kg.m$^{-3}$).}
1080
\label{heterogeneous-ecm-concentration}
1081
\end{figure}
1082
1083
\begin{figure}[!hptb]
1084
\centering
1085
\includegraphics[width=0.9\textwidth]{images/examples/%
1086
eulerian/cancer/haptotaxis-proliferating-cells-0}
1087
\caption{Proliferating cells undergoing haptotaxis at time $t=0$
1088
days.}
1089
\label{tumour-haptotaxis-proliferation-0}
1090
\end{figure}
1091
1092
\begin{figure}[!hptb]
1093
\centering
1094
\includegraphics[width=0.9\textwidth]{images/examples/%
1095
eulerian/cancer/haptotaxis-proliferating-cells-3p3}
1096
\caption{Proliferating cells undergoing haptotaxis at time $t=33$
1097
days.}
1098
\label{tumour-haptotaxis-proliferation-3p3}
1099
\end{figure}
1100
1101
\begin{figure}[!hptb]
1102
\centering
1103
\includegraphics[width=0.9\textwidth]{images/examples/%
1104
eulerian/cancer/haptotaxis-proliferating-cells-6p7}
1105
\caption{Proliferating cells undergoing haptotaxis at time $t=67$
1106
days.}
1107
\label{tumour-haptotaxis-proliferation-6p7}
1108
\end{figure}
1109
1110
\begin{figure}[!hptb]
1111
\centering
1112
\includegraphics[width=0.9\textwidth]{images/examples/%
1113
eulerian/cancer/haptotaxis-proliferating-cells-10}
1114
\caption{Proliferating cells undergoing haptotaxis at time $t=100$
1115
days.}
1116
\label{tumour-haptotaxis-proliferation-10}
1065
between 0.5~kg.m$^{-3}$ and 1.5~kg.m$^{-3}$), seen in Figure~\ref{%
1066
heterogeneous-ecm-concentration}. In these tests, the haptotactic
1067
coefficient $h$ is 0.1~mm$^2$.day$^{-1}$.mm$^3$.kg$^{-1}$.
1068
Figures~\ref{tumour-haptotaxis-proliferation-0}--\ref{tumour-hapto%
1069
taxis-proliferation-10} show snapshots of the cells undergoing
1070
haptotaxis and proliferating during the course of the test. The colour
1071
contours provide the evolving cell concentration fields
1072
(in~kg.m$^{-3}$) and the arrows provide the deformation direction of
1073
the ECM, induced by the cell traction. We observe that the cells
1074
migrate toward areas of higher ECM while proliferating. Note that the
1075
directionality of the cell traction field changes correspondingly with
1076
the concentration field, as noted by the lengths and directions of the
1077
arrows.
1078
1079
\begin{figure}[!hptb]
1080
\centering
1081
\includegraphics[width=0.9\textwidth]{images/examples/%
1082
eulerian/cancer/heterogeneous-ecm-concentration}
1083
\caption{Heterogeneous extra-cellular matrix concentration
1084
(kg.m$^{-3}$).}
1085
\label{heterogeneous-ecm-concentration}
1086
\end{figure}
1087
1088
\begin{figure}[!hptb]
1089
\centering
1090
\includegraphics[width=0.9\textwidth]{images/examples/%
1091
eulerian/cancer/haptotaxis-proliferating-cells-0}
1092
\caption{Proliferating cells undergoing haptotaxis at time $t=0$
1093
days.}
1094
\label{tumour-haptotaxis-proliferation-0}
1095
\end{figure}
1096
1097
\begin{figure}[!hptb]
1098
\centering
1099
\includegraphics[width=0.9\textwidth]{images/examples/%
1100
eulerian/cancer/haptotaxis-proliferating-cells-3p3}
1101
\caption{Proliferating cells undergoing haptotaxis at time $t=33$
1102
days.}
1103
\label{tumour-haptotaxis-proliferation-3p3}
1104
\end{figure}
1105
1106
\begin{figure}[!hptb]
1107
\centering
1108
\includegraphics[width=0.9\textwidth]{images/examples/%
1109
eulerian/cancer/haptotaxis-proliferating-cells-6p7}
1110
\caption{Proliferating cells undergoing haptotaxis at time $t=67$
1111
days.}
1112
\label{tumour-haptotaxis-proliferation-6p7}
1113
\end{figure}
1114
1115
\begin{figure}[!hptb]
1116
\centering
1117
\includegraphics[width=0.9\textwidth]{images/examples/%
1118
eulerian/cancer/haptotaxis-proliferating-cells-10}
1119
\caption{Proliferating cells undergoing haptotaxis at time $t=100$
1120
days.}
1121
\label{tumour-haptotaxis-proliferation-10}
1117
1122
\end{figure}
1118
1123
1119
1124
While the magnitudes for the cell diffusivity, $D^{\mathrm{cell}}$,
1126
1131
\subsection{Coupling the phenomena}
1127
1132
\label{cacophonous-medley}
1128
1133
1129
With the individual phenomena explored, we are now ready to solve
1130
the coupled problem described initially. The range of physics
1131
incorporated into this problem include proliferating cells undergoing
1132
both diffusion and haptotaxis, a rate law for the production of
1133
additional ECM which scales linearly with the concentration of
1134
cells, the stress within the cells induced by their traction, the
1135
hyperelastic response of the ECM, isotropic kinematic swelling
1136
associated with the increase in tumour mass, and finally, this
1137
swelling constrained by the presence of a wall.
1134
With the individual phenomena explored, we are now ready to solve the
1135
coupled problem described initially. The range of physics incorporated
1136
into this problem include proliferating cells undergoing both
1137
diffusion and haptotaxis, a rate law for the production of additional
1138
ECM which scales linearly with the concentration of cells, the stress
1139
within the cells induced by their traction, the hyperelastic response
1140
of the ECM, isotropic kinematic swelling associated with the increase
1141
in tumour mass, and finally, this swelling constrained by the presence
1142
of a wall.
1138
1143
1139
1144
Figures~\ref{tumour-growth-constrained-0}--\ref{tumour-growth%
1140
1145
-constrained-5} show snapshots of the growing tumour constrained by
1170
1175
\vspace{1cm} %Hack
1171
1176
1172
1177
\begin{figure}[!hptb]
1173
\centering
1174
\includegraphics[width=0.9\textwidth]{images/examples/%
1175
eulerian/cancer/growing-tumour-0.eps}
1176
\caption{A constrained growing tumour at $t=0$ days.}
1177
\label{tumour-growth-constrained-0}
1178
\end{figure}
1179
1180
\begin{figure}[!hptb]
1181
\centering
1182
\includegraphics[width=0.9\textwidth]{images/examples/%
1183
eulerian/cancer/growing-tumour-1.eps}
1184
\caption{A constrained growing tumour at $t=20$ days.}
1185
\label{tumour-growth-constrained-1}
1186
\end{figure}
1187
1188
\begin{figure}[!hptb]
1189
\centering
1190
\includegraphics[width=0.9\textwidth]{images/examples/%
1191
eulerian/cancer/growing-tumour-2.eps}
1192
\caption{A constrained growing tumour at $t=40$ days.}
1193
\label{tumour-growth-constrained-2}
1194
\end{figure}
1195
1196
\begin{figure}[!hptb]
1197
\centering
1198
\includegraphics[width=0.9\textwidth]{images/examples/%
1199
eulerian/cancer/growing-tumour-3.eps}
1200
\caption{A constrained growing tumour at $t=60$ days.}
1201
\label{tumour-growth-constrained-3}
1202
\end{figure}
1203
1204
\begin{figure}[!hptb]
1205
\centering
1206
\includegraphics[width=0.9\textwidth]{images/examples/%
1207
eulerian/cancer/growing-tumour-4.eps}
1208
\caption{A constrained growing tumour at $t=80$ days.}
1209
\label{tumour-growth-constrained-4}
1210
\end{figure}
1211
1212
\begin{figure}[!hptb]
1213
\centering
1214
\includegraphics[width=0.9\textwidth]{images/examples/%
1215
eulerian/cancer/growing-tumour-5.eps}
1216
\caption{A constrained growing tumour at $t=100$ days.}
1217
\label{tumour-growth-constrained-5}
1218
\end{figure}
1219
1220
\begin{figure}[!hptb]
1221
\centering
1222
\includegraphics[width=0.9\textwidth]{images/examples/%
1223
eulerian/cancer/growing-tumour-no-wall-4}
1224
\caption{An unconstrained growing tumour at $t=80$ days.}
1225
\label{tumour-growth-no-wall-4}
1178
\centering
1179
\includegraphics[width=0.9\textwidth]{images/examples/%
1180
eulerian/cancer/growing-tumour-0.eps}
1181
\caption{A constrained growing tumour at $t=0$ days.}
1182
\label{tumour-growth-constrained-0}
1183
\end{figure}
1184
1185
\begin{figure}[!hptb]
1186
\centering
1187
\includegraphics[width=0.9\textwidth]{images/examples/%
1188
eulerian/cancer/growing-tumour-1.eps}
1189
\caption{A constrained growing tumour at $t=20$ days.}
1190
\label{tumour-growth-constrained-1}
1191
\end{figure}
1192
1193
\begin{figure}[!hptb]
1194
\centering
1195
\includegraphics[width=0.9\textwidth]{images/examples/%
1196
eulerian/cancer/growing-tumour-2.eps}
1197
\caption{A constrained growing tumour at $t=40$ days.}
1198
\label{tumour-growth-constrained-2}
1199
\end{figure}
1200
1201
\begin{figure}[!hptb]
1202
\centering
1203
\includegraphics[width=0.9\textwidth]{images/examples/%
1204
eulerian/cancer/growing-tumour-3.eps}
1205
\caption{A constrained growing tumour at $t=60$ days.}
1206
\label{tumour-growth-constrained-3}
1207
\end{figure}
1208
1209
\begin{figure}[!hptb]
1210
\centering
1211
\includegraphics[width=0.9\textwidth]{images/examples/%
1212
eulerian/cancer/growing-tumour-4.eps}
1213
\caption{A constrained growing tumour at $t=80$ days.}
1214
\label{tumour-growth-constrained-4}
1215
\end{figure}
1216
1217
\begin{figure}[!hptb]
1218
\centering
1219
\includegraphics[width=0.9\textwidth]{images/examples/%
1220
eulerian/cancer/growing-tumour-5.eps}
1221
\caption{A constrained growing tumour at $t=100$ days.}
1222
\label{tumour-growth-constrained-5}
1223
\end{figure}
1224
1225
\begin{figure}[!hptb]
1226
\centering
1227
\includegraphics[width=0.9\textwidth]{images/examples/%
1228
eulerian/cancer/growing-tumour-no-wall-4}
1229
\caption{An unconstrained growing tumour at $t=80$ days.}
1230
\label{tumour-growth-no-wall-4}
1226
1231
\end{figure}
1227
1232
1228
1233
%
Loggerhead is a web-based interface for Breezy
Version: 2.0.1