Extracorporeal Shock Wave Therapy as a Non-Linear Physical System
P. C. Encina1, 2, A. Albeeshi1, M. R. Brañes3,5, and E. Tirapegui4
1
Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON, Canada,
2
Facultad de Ciencias, Departamento de Física, Universidad de Chile, Casilla 653, Santiago, Chile.
3
Facultad de Ciencias, Departamento de Química, Universidad de Chile, Casilla 653, Santiago. Chile.
4
Facultad de Ciencias Fis. & Mat., Dept. de Física, Universidad de Chile, Casilla 487-3, Santiago.
Chile.
5
Arauco Salud Clinic, Santiago. Chile.
Abstract
A theoretical non-linear physical model of the healing process observed in extracorporeal shock wave
therapy (ESWT) is developed. The model helps to understand the dynamics of the healing process as a
transition from a passive state to an active state under the effect of a shock wave consistent with a well
defined set of experimental parameters.
Keywords: Extracorporeal Shock Wave Therapy; Nonlinear System
1. Introduction
The use of shock waves (SW) for kidney stone disintegration is well established in urology.
However, the most thriving approach of SW in medicine, as a new emerging health technology, is in
the treatment of orthopaedic diseases (Wild et al. 2000) and recently in the healing process in general
(Shrivastava and Kailash 2005); including wound healing(Dumfarth et al. 2008), rotator cuff tendon
healing responses (Brañes et al. 2012), bone-to-tendon junction healing(Wang et al. 2008), non-union
fracture healing (Schaden et al. 2015) and, in addition, it has been found that low-energy extracorporeal
shock wave therapy (ESWT) may be bactericidal (Novak et al.
2008). ESWT uses pressure waves that generate highly
localized positive pressures and cavitations. The reports point to
the ability to accelerate healing by modifying the local
intracellular and extracellular biological environment. For
instance, as illustrated in Figure 1 (a and b, rotator cuff enthesis
to bone; and c, rotator cuff tendon samples; all of them
representing biopsies taken at 8 weeks after shock wave
application), the microscopic changes observed after ESWT are
characterized by revascularization (blue arrow-head) and
hypercellularity (white arrows in c), which later underwent a
cellular differentiation into normal tissues producing new bone
(black arrows, a-b) or fibroblastic repair in tendon tissue (c). It
is interesting to note similar response of two different living
tissues to shockwaves, considering that their impedances are
quite different.(Gabriel et al. 1996; Kalvoy et al. 2009).
The shock waves currently used for medical applications are
pulse shaped waves generated and propagated in water. Shock
waves are characterized by a surge-type pressure distribution.
At least three different types of shock waves generators are used in medical field today (electrohydraulic, electro-magnetic and piezo-electric). Generally a shock wave can be described as a single
pulse with a wide frequency range (up to 20 MHz), high-pressure amplitude (up to 120 MPa), low
tensile wave (up to 10 MPa), small pulse width at – 6 dB and a short rise time. The rise time of an
electro-hydraulic generated shock wave measured with an optical hydrophone is below 10 ns. The
positive pressure amplitude is followed by a diffraction-induced tensile wave with a few μs duration.
The flux energy density (FED, up
Figure 1. ( a & b ), rotator cuff enthesis to bone; (c), rotator cuff tendon sample, biopsies taken at 8
weeks after shockwave application).
to 3 mJ/mm2) and the pulse energy (up to 100 mJ) are determined from the temporal and spatial
distribution of the pressure profile.(Wess 2006)
2. Nonlinear physical model
It is believed that “the whole spectrum of medicine consists of complex non-linear systems that are
balanced and interact with each other”.(Petros 2003) Further, it has been proposed that “ the ‘art of
medicine’ consists of an intuitive ‘tuning in’ to these complex systems and as such is not so much an art
as an expression of non-linear science.”(Petros 2003) This approach have additional applications that
may help to elucidate and support understanding of complex nonlinear systems.(Bell Iris et al. 2002;
Meyer-Hermann et al. 2009) In this work, to model the ESWT, let’s start with the simplest wave
(oscillating system), that of the harmonic oscillator;
ẋ= y
ẏ= − x
[1]
The first degree of complexity can be added using an external oscillating force with zero averange.
When the frequency of the oscillating force is greater than the natural frequency of the harmonic
oscillator(Landau and Lifshitz 1960), the total action on the system (including the harmonic oscillator)
is different from zero, and a net force shows up. This happens in the following example of a sinusoidal
external force;
=
x
y
=
2
y

x
+
sin

t
[2]
There is clinical evidence that ESWT may improve the rate of healing, and it can be assumed that the
latter is initiated by energy transfer to the system, where the ESWT can excite complex oscillating
modes in the organism. We can then approach the ESWT using a pulse which promotes a transition of
the tissue from an initial state A (before treatment) to a state B after treatment, and the healing process
would be modelled as a bi-stable system represented by the equation (we have scaled to have variables
without dimensions);
[3]
Here ε is the bifurcation parameter (representing the qualitative change in the dynamic process which
in mathematical terms is the distance to the bifurcation point), where N is the number of pulses applied
and t is the intervals of time between pulses, γ is the energy input(Clerc et al. 2008), a parameter that
controls the amplitude of the white noise.(Risken 1996) η is the white noise generated with Gaussian
noise with zero mean and standard deviation of 0.5, and μ is the energy dissipation term. A typical
ESWT simulation with a total time of 200 units of time (remember we are working with dimensionless
quantities), with N = 1, where the shock wave is applied after a time t0, which is at 100 units of time in
the simulation (with a dissipation of 0.01 and ε = 0.25), is shown in Figure 2, in the phase space.
Figure 2
Fig 2: Effect of the random force in eq. (3). The top figure shows a trajectory in the dimensionless (x,
y) phase space. The bottom figure shows x as a function of the dimensionless time t.
In this model, represented by Eq. 3, one has two equilibrium stable states ±  and one instable
hyperbolic point at zero. The simulation in Figure 2 (in the phase space), is carried out with a SW
intensity equal to 30 (gamma in Eq. 3) with a noise oscillating between ± 1. The Gaussian shape of the
SW, modulated by a white noise, ensures that a wide range of frequencies can be excited. It can be
seen, that there is a certain probability for the system to change from one state to another, and there are
also different pathways for this change to take place. Finding the optimal parameters for the transition,
may lead to a guided medical application for maximum efficiency of the applied ESWT. The two
equilibrium situations represent two distinct states in the tissue; a passive initial state and the new
excited state created by the ESWT excitation. Notably, in-vivo, the excited state is characterized by
higher cell concentration and cell activity, as shown in Figure 3 for the clinical case of tendinopathy,
Figure 3. Hipercellularity illustration for a tendinopathy case, before (left) an after ESWT treatment.
before and after the ESWT treatment. In this particular case (tendinopathy), 4000 pulses were applied
with a flux energy density (FED) of 0.35mJ/mm2 at 36 MPa. In the case of bone non-healing
treatments, between 6000 - 10000 pulses are used with FED of ca. 0.40mJ/mm2 at 40 MPa; the
treatments were carried out using either the electrohydraulic device Orthospec / Medispec or the
electromagnetic device Duolith SD1/ Storz, with similar results (Fig.3a).
Figure 3a.
a
b
c
Figure 3a. In a, a non-healing bone condition is observed clearly (white arrow). After treatment (10
months), it is obtained a complete bone healing (b,c, white arrows).
The epsilon parameter, in Eq. 3, defines an interval where the change of the state can take place.
Therefore, there is no chance for state shifting (no healing) when epsilon approaches zero, or when its
value is too large and the states are too far away for a transition to occur.
It is useful to explore the effect of the ESWT for a system near the equilibrium. For instance, a
particular shock wave perturbs the stable equilibrium located at 0.5 producing an oscillation around the
three equilibria (0.5, 0, -0.5), and moving to the state at -0.5. The latter simulation can also illustrate the
case where the excitation does not lead to the new (healing) state, and the system goes back to the
initial state (a typical case of a false positive); there is enough energy; but there is no transition to the
new state (active state).
The “potential energy” of the bio-system is found by integration of the dynamic equation, given a
potential with two minima,
x2 x4
U
()
x
 
[4]
2 4
The “potential energy” diagram, shown in Figure 4, permits to visualize the most important aspects of
the physical model. The external energy pulse may produce the change in the states leading to a new
equilibrium state, a state of high cell activity responsible for the healing process. In the case of the
ESWT treatment, the physical model is developed to connect the physical parameters to the treatment,
and the objective, is to use them to tune the frequency and the energy of pulses for maximum efficiency
in promoting the change of the state. For instance, the parameter ε may be correlated with the type of
tissue being treated and its magnitude would be associated, within a specific range, with tendon tissue
or bones. Further, the transition energy can be written,
 

U

U
0

U 

4
2
[5]
and the mean first passage time from one stable state to the other can be estimated as (Van-Kampen
1987)
ΔU
ε2
τ e =e ,
γ2
[6]
4γ2
and both states have the same mean first passage time, since the potential well is symmetric. We
assume this approximation is valid for  1 , where   t  is a normalized gaussian white noise, with
t 0 and
Since, 

t
t' 
t
t'.


1 , the term with
 ett 
2
0
t0
t O
1,

in (3), dominates during the interval of time; t
the interval where the pulse is different from zero, that is the time O 1 around t0 ( t0 the time at
which the pulse is activated). Therefore, for a transition to take place from state 1 to state 2, the first
mean passage time of the state ought to be less than 1, the time interval when the pulse is in action. The
U
2
latter can be summarized as (Van-Kampen 1981; Van-Kampen 1987),  e2 e42 ≤ 1 for 
1.
Figure 4. Potential energy diagram for the two states of the healing process. The inset shows one
specific pathway for the transition.
3. Conclusion
The recovery of bone or tendon tissues, after ESWT treatment, is represented by a bi-stable non-linear
system. The complexity is reduced to a state A before application of ESWT, and new “healing” state B
induced by the energy pulses. The phenomenological nonlinear model represents the bistability, and
brings to the forefront the role of two parameters: ε, γ. The parameter ε can be associated with the
tissue type, while the parameter γ is directly related to the intensity of the applied shock wave. In
addition, the activation time increases when the amplitude of the shock wave diminishes. The main
objective has been achieved here; to develop a phenomenological approach to ESWT based on the
physics of bi-stable systems. The next step of the work is to tune the semi-empirical parameters to
treatment and move towards the most efficient dose for patients.
Acknowledgments
Financial assistance from the Natural Science and Engineering Research Council of Canada (NSERC)
is gratefully acknowledged, PCE acknowledges the financial support from FONDECYT project
3090045, the ring program ACT-24 of the Bicentenial Program.
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Figure captions
Figure 1. ( a & b), rotator cuff enthesis bone; (c), rotator cuff tendon sample, biopsies taken at 8 weeks
after shockwave application.
Figure 2: Effect of the external sinusoidal force shown in the phase space (top) and applied pulse in
time (bottom).
Figure 3. Hipercellularity illustration for a tendinopathy case, before (left) an after ESWT treatment.
Figure 3a. Non-healed humerus fracture shockwave-treated and final results.
Figure 4. Potential energy diagram for the two states of the healing process. The inset shows one
specific pathway for the transition.