A Py hon FDTD me hod Algo i hm o 1D Plana
Acous ic Wa e P opaga ion: Simula ing High-F equency
Ul asound in he B ain and Beyond
Nuno A.T.C. Fe nandes1[0000-0003-1848-726X]*, Ana A iei a1[0000-0003-4824-682X], Be ina
Hinckel2[0000-0003-3963-5241], Filipe Sil a1[0000-0003-3596-3328], Ana Leal1[0000-0002-2844-491X],
Ósca Ca alho1[0000-0002-9447-8739]
1 Cen e o mic oelec icomechanical sys ems (CMEMS), Uni e si y o Minho, 4800-058
Guima ães, Po ugal
2 Depa men o O hopaedic Su ge y, William Beaumon Hospi al, Royal Oak, MI, USA
[email p o ec ed]
Abs ac . Non-in asi e echniques, such as high- equency ul asound, ha e
eme ged as p omising he apeu ic ools o neu ological diso de s, including
Pa kinson’s disease and Alzheime ’s disease. By a ge ing speci ic b ain e-
gions, ul asound s imula ion modula es neu al ac i i y and induces bene icial
physiological esponses. Howe e , simula ing high- equency acous ic wa e
p opaga ion in biological issues p esen s compu a ional challenges due o he
high spa ial and empo al esolu ion equi ed o sa is y he low Cou an -
F ied ichs-Lewy (CFL) condi ion o nume ical s abili y and accu acy. This pa-
pe in oduces a no el Py hon algo i hm op imized o plana wa e p opaga ion,
enabling e icien one-dimensional simula ions o high- equency ul asound.
U ilizing he ini e di e ence ime domain (FDTD) me hod, he algo i hm in-
co po a es ma e ial-speci ic p ope ies, including densi y, sound speed, and e-
quency-dependen a enua ion, o model he e ogeneous issue s uc u es such as
skin, bone, ce eb ospinal luid, and b ain issue. The me hod accu a ely cap-
u es key acous ic phenomena, such as impedance misma ching and wa e e-
lec ion, acili a ing de ailed analysis o ene gy ansmission and abso p ion in
complex biological in e aces. The algo i hm’s pe o mance is compa ed wi h
COMSOL Mul iphysics, which is inhe en ly limi ed o wo and h ee-
dimensional acous ic wa e p opaga ion. By educing he p oblem o one dimen-
sion, he p oposed me hod simpli ies compu a ional complexi y while p ese -
ing key wa e in e ac ions, enabling ea ly-s age analysis wi h lowe compu a-
ional cos s. Beyond biomedical applica ions, his app oach is b oadly applica-
ble o any sys em go e ned by acous ic wa e equa ions. By signi ican ly educ-
ing compu a ional demands, i accele a es p elimina y s udies o wa e p opaga-
ion h ough mul ilaye ed media, con ibu ing o he de elopmen o e icien
ul asound-based he apeu ic models and ad ancing acous ic esea ch.
Keywo ds: Fini e-Di e ence Time-Domain, Ul asound P opaga ion, Biologi-
cal Tissue Acous ics, T ansc anial Ul asound, Acous ic A enua ion, Neu o-
modula ion, Simula ion.
2
1 In oduc ion
Ul asound is a powe ul modali y o bo h medical diagnos ics and non-des uc i e
es ing (NDT) due o i s abili y o pene a e laye ed media and p o ide eal- ime eed-
back on in e nal s uc u es [1], [2]. Ul asound is widely used in biomedical applica-
ions o imaging [3], he apy [4], and eal- ime moni o ing o bo h so and ha d is-
sues [5], o e ing a non-in asi e and e sa ile diagnos ic and he apeu ic ool. In in-
dus ial NDT, ul asonic ansduce s a e widely used in indus ial non-des uc i e
es ing o de ec in e nal laws in me allic componen s, ensu ing s uc u al in eg i y
wi hou damaging he ma e ial [6], [7]. Despi e he di e ences in applica ion, bo h
domains sha e common challenges ela ed o acous ic wa e p opaga ion h ough he -
e ogeneous, mul ilaye ed ma e ials.
Neu ological diso de s such as Alzheime 's disease, Pa kinson's disease, and s oke
a e he leading cause o disabili y and he second leading cause o dea h globally,
accoun ing o o e 10 million dea hs and 349 million disabili y-adjus ed li e yea s in
2019 [8], [9], [10]. T adi ional diagnos ic and he apeu ic app oaches o neu ological
diso de s o en ely on in asi e me hods, un a eling g owing need o non-in asi e,
p ecisely a ge ed echnologies such as ansc anial magne ic s imula ion and ocused
ul asound, which o e sa e al e na i es o modula ing neu al ac i i y and deli e -
ing ea men s [11], [12]. Among non-in asi e s imula ion echniques, Ul asound-
based echniques a e eme ging as p omising non-in asi e me hods o b ain s imula-
ion [13], acili a ing d ug deli e y ac oss he blood-b ain ba ie [14], and enabling
eal- ime neu ological moni o ing [15]. Accu a ely modeling he p opaga ion o ul a-
sound h ough c anial s uc u es, including he skin, skull, and unde lying b ain issue
is essen ial o op imizing he sa e y, p ecision, and e icacy o eme ging neu ological
acous ic echnologies. This esea ch is d i en by he b oade objec i e o ad ancing
non-in asi e b ain heal h diagnos ics and he apeu ic in e en ions h ough high-
ideli y simula ion o acous ic wa e beha io in ana omically and acous ically he e o-
geneous ce eb al en i onmen s.
Nume ical modeling using ini e-di e ence ime-domain (FDTD) me hods is wide-
ly ecognized as a powe ul app oach o simula ing acous ic wa e p opaga ion in
inhomogeneous media [16], [17], [18]. FDTD enables di ec compu a ion o p essu e
and eloci y ields wi h high spa ial and empo al esolu ion and is pa icula ly e ec-
i e in inco po a ing spa ially a ying ma e ial p ope ies, equency-dependen a en-
ua ion, and complex bounda y condi ions. S udies ha e demons a ed he e sa ili y
o FDTD in modeling nonlinea e ec s, issue-speci ic damping, and ealis ic sou ce
injec ion in bo h s a ic and dynamic acous ic en i onmen s, ein o cing i s ele ance
in biomedical and enginee ing acous ics applica ions [19], [20].
In his s udy, we p esen a one-dimensional FDTD model o ul asound p opaga-
ion h ough a simpli ied human head c oss-sec ion, including a piezoelec ic ans-
duce , a polyme ic ma ching laye , and sequen ial laye s o skin, bone, and b ain is-
sue. The model inco po a es equency-dependen a enua ion, impedance misma ch-
es, and spa ial il e ing o emula e ideal ansduce con igu a ions. The goal is o p o-
ide a ligh weigh ye ealis ic simula ion ool o in es iga ing acous ic beha io in
3
biological condi ions, which can suppo bo h medical de ice de elopmen and un-
damen al biophysics esea ch.
2 Theo e ical backg ound
In one dimension, he p opaga ion o longi udinal acous ic wa es in a medium is go -
e ned by wo coupled i s -o de equa ions: he momen um equa ion de i ed by New-
on’s i s law (Eq. (1)) and he con inui y equa ion de i ed by mass conse a ion (Eq.
(2)) [21].
∂c/∂ =-∂p/(ρ·∂x) (1)
∂p/∂ =-K·∂c/∂x (2)
Toge he , hese equa ions de ine a coupled sys em ha suppo s wa e p opaga ion.
Thei linea i y assumes small-ampli ude oscilla ions and neglec s nonlinea , he mal,
o iscous e ec s, making hem pa icula ly sui able o modeling diagnos ic o he -
apeu ic ul asound unde mode a e in ensi y condi ions.
When an acous ic wa e encoun e s a change in ma e ial p ope ies, pa o he
wa e is e lec ed, and pa is ansmi ed. The e lec ion and ansmission a e de e -
mined by he acous ic impedance, which can be simpli ied o plana wa es o Eq.
(3).
Z =ρc (3)
The misma ch o impedances can limi he amoun o p essu e ha is ansmi ed.
As such, he e lec ion and ansmission o a plana acous ic wa e can be desc ibed by
he e lec ion and ansmission coe icien p esen ed in Eq. (4) and Eq. (5) espec i e-
ly.
R =(Z2-Z1)/(Z2+Z1) (4)
T =2·Z2/(Z2+Z1) (5)
Acous ic a enua ion in so issues and bone is equency-dependen and is ypical-
ly cha ac e ized in uni s o dB/MHz/cm. To inco po a e his in o he simula ion, a -
enua ion coe icien s Np/m a e calcula ed as Eq (6).
α=(αdB/(20·log10e))· ( /106)·100 (6)
This a enua ion a ises om a combina ion o iscous, he mal, and s uc u al e-
laxa ion p ocesses wi hin biological issues, which con e a po ion o he acous ic
ene gy in o hea [22]. The equency dependence o his phenomenon is especially
ele an in biomedical applica ions, whe e highe equencies esul in g ea e ene gy
loss, di ec ly a ec ing he ampli ude, wa e o m shape, and pene a ion dep h o he
acous ic signal. Since hese dissipa i e e ec s a e no included in he heo e ical de i-
a ion o he linea acous ic wa e (Eq. (1) and Eq. (2)), a enua ion is in oduced as an
4
ad hoc co ec ion du ing he nume ical implemen a ion wi hin he FDTD upda e
s eps.
3 Me hodology
A i s -o de acous ic FDTD sol e was implemen ed in Py hon, using explici ime
in eg a ion and a s agge ed g id. Pa icle eloci y (Eq. (7)) and p essu e (Eq. (8)) a e
de ined a al e na ing spa ial posi ions o imp o e nume ical s abili y and educe dis-
pe sion.
cin+1/2= cin-1/2-(pi+1n-pin)·Δ /(ρi·Δx) (7)
pin+1/2= pin -(ci+1n+1/2-cin+1/2)·Δ ·Ki/Δx (8)
This o mula ion inhe en ly accoun s o e lec ion and ansmission a ma e ial in-
e aces, as he spa ial a ia ion o densi y and bulk modulus di ec ly go e ns wa e
beha io ac oss bounda ies.
Ma e ial pa ame e s we e spa ially assigned based on li e a u e alues [23], [24],
[25], and he ansduce s p esen in he labo a o y, as ollows in Table 1.
Table 1. Ma e ial pa ame e s used o simula ion.
Ma e ial
c (m/s)
ρ (kg/m3)
α (dB/MHz/cm)
Skin
1540
1100
1.0
Bone
4080
1900
20.0
B ain
1560
1040
0.8
T ansduce (PZT)
4200
7700
0
Ma ching Laye
2700
1500
0
The simula ion domain spans 18 cm, consis ing o a 6 cm b ain-side bu e , a 3 cm
biological in e es egion, and a 9 cm ansduce -side bu e o a oid e lec ions. The
ansduce and ma ching laye we e dimensioned acco ding o hei cen al equency.
The cen al 3 cm egion includes skin, bone, and b ain laye s, each wi h use -de ined
hicknesses and p ope ies, acco ding o he li e a u e [26]. Se e al egions o he
b ain we e s udied, and hei espec i e hicknesses a e p esen ed in Fig. 1.
The wa e was gene a ed using a single-cycle sine bu s modula ed by a Hann win-
dow o educe spec al leakage. The wa e o m was injec ed in o he pa icle eloci y
ield o minimize nume ical ins abili y. The ampli ude was scaled based on he ans-
duce impedance o ma ch a a ge p essu e alue. Fi e dis inc equencies we e im-
plemen ed: 0.5, 1, 2, 4 and 5 MHz.
Biological issue a enua ion was implemen ed as an exponen ial decay applied di-
ec ly o he p essu e ield du ing each upda e s ep. A enua ion coe icien s we e
con e ed om dB/MHz/cm o Nepe s/m and scaled by local speed o sound and ime
s ep. The a enua ion is implemen ed as an exponen ial decay ac o pe ime s ep (Eq,
(9)), which damps he p essu e ield (Eq. (10)).
5
Fig. 1. Ana omical map showing skin and skull hicknesses ac oss key c anial egions ele an
o ansc anial ul asound a ge ing.
γ =α·c·Δ (9)
pin= pin·γ (10)
To ensu e nume ical s abili y and accu acy, he spa ial esolu ion Δx was chosen
based on he sho es wa eleng h in he sys em Δx≤λmin/10. The ime s ep was se
acco ding o he Cou an –F ied ichs–Lewy condi ion Δ ≤CFL·Δx/cmax, wi h CFL=0.1.
Time-domain p essu e signals we e eco ded a se e al posi ions: a he ansduce -
skin ba ie , skin-bone ba ie , bone-b ain ba ie , and 5, 10 and 15 mm b ain dep h.
To enhance signal isibili y and supp ess nume ical noise, low-ampli ude alues (<10-
7 Pa) we e clipped, as hey will be ha d o de ec wi h ypical ins umen a ion.
4 Resul s
The esul s o he simula ions o he Py hon algo i hm and COMSOL simula ions a e
displayed in Fig. 2.
Ou indings clea ly demons a e he expec ed equency-dependen a enua ion o
acous ic p essu e as he wa e p opaga es h ough b ain issues. Highe equencies
(4–5 MHz) exhibi a much s eepe decline in acous ic p essu e ac oss all egions,
while lowe equencies (500 kHz–1 MHz) main ain ela i ely highe ampli udes
e en a g ea e dep hs. This aligns wi h heo e ical p edic ions and expe imen al da a,
as biological issue a enua ion inc eases app oxima ely linea ly wi h equency. The
compa ison be ween he Py hon FDTD simula ions and COMSOL ou pu s shows
6
s ong ag eemen ac oss all equencies and ana omical egions, alida ing he nume -
ical implemen a ion and con i ming ha bo h a enua ion and e lec ion/ ansmission
e ec s due o acous ic impedance misma ches a e accu a ely modelled.
Fig. 2. P essu e a enua ion ac oss b ain dep h o six c anial egions and mul iple equencies,
illus a ing equency-dependen losses and egional a iabili y in acous ic ansmission.
I is also possible o obse e he spa ial a ia ion in a enua ion ac oss di e en
c anial egions, e lec ing ana omical di e ences such as skull and skin hickness. Fo
ins ance, he mas oid and empo al egions, which ha e hinne skull sec ions, exhibi
less a enua ion, whe eas he on al and e ex egions show s eepe d ops in p es-
su e, consis en wi h dense bone and laye ed impedance. Since hese alues exclude
he p opaga ion pa h h ough he skin and skull, i 's impo an o ecognize ha a sig-
ni ican po ion o he acous ic signal is al eady a enua ed be o e i e en eaches
b ain issue. The skin and skull laye s, especially he co ical bone, p esen subs an ial
acous ic impedance misma ches and equency-dependen a enua ion, which esul in
e lec ion, sca e ing, and abso p ion o he incoming wa e. As a esul , he p essu es
plo ed a 0 mm b ain dep h do no ep esen he o iginal ansduce ou pu bu a he
a diminished signal ha has al eady unde gone conside able ene gy loss.
7
Ul asound neu omodula ion holds signi ican he apeu ic po en ial o a ange o
neu ological condi ions, pa icula ly when i s abili y o pene a e di e en c anial
egions is ma ched wi h he ana omical egion o he disease. Alzheime ’s disease, a
p og essi e neu odegene a i e diso de ma ked by memo y loss and cogni i e de-
cline, p ima ily a ec s he empo al and on al lobes [27], hese deepe egions bene-
i om lowe - equency ul asound (e.g., 500 kHz–1 MHz), which o e s be e pene-
a ion. F on o empo al demen ia, which al e s beha iou , pe sonali y, and language,
also a ge s hese on al and empo al s uc u es [28] and may simila ly bene i om
low- equency neu omodula ion. Pos e io co ical a ophy, a isual a ian o Alz-
heime ’s disease, a ec s he occipi al lobe, which is ela i ely supe icial [29] and
hus accessible o highe - equency ul asound ha o e s ine spa ial esolu ion.
Simila ly, mul iple scle osis, a ch onic au oimmune disease ha damages he p o ec-
i e co e ing o ne e ibe s (myelin), o en a ec s mul iple, di usely loca ed b ain
a eas including he e ex egion [30], demanding ca e ul a ge ing s a egies. Finally,
schizoph enia and majo dep essi e diso de , which in ol e cogni i e dys unc ion
and emo ional dys egula ion, a e s ongly associa ed wi h he p e on al co ex, a
ela i ely supe icial egion, making i an ideal a ge o ansc anial ul asound in-
e en ions [31]. Adap ing ul asound pa ame e s o ma ch bo h he disease cha ac e -
is ics and he ana omical cons ain s o each egion ein o ces he p omise o neu o-
modula ion as a p ecise, non-in asi e he apeu ic ool.
Fig. 3. Compa ison o simula ion un imes o Py hon and COMSOL ac oss equencies.
The compa ison o simula ion un imes be ween Py hon and COMSOL illus a ed
in Fig. 3 clea ly demons a es he compu a ional e iciency o he Py hon-based
FDTD implemen a ion, pa icula ly as equency inc eases. While bo h me hods pe -
o m ela i ely quickly a lowe equencies, he un ime o COMSOL g ows expo-
nen ially wi h equency, eaching o e 3.57 hou s a 5 MHz, due o he inc eased
spa ial esolu ion and mesh e inemen equi ed o highe - equency wa e p opaga-
ion in ini e elemen models. In con as , he Py hon sol e main ains a much smalle
compu a ional oo p in , scaling mo e g ace ully wi h equency hanks o i s explici
ime-s epping and s uc u ed g id app oach. This e iciency makes he Py hon imple-
8
men a ion highly sui able o i e a i e s udies, pa ame ic sweeps, o eal- ime appli-
ca ions, especially when apid eedback is essen ial. The ade-o , o cou se, is ha
COMSOL o e s g ea e modelling lexibili y and p ecision o complex geome ies,
bu o s uc u ed 1D o laye ed media p oblems like ansc anial wa e modelling,
Py hon p o ides an excellen balance o speed and ideli y.
5 Conclusion
This s udy p esen ed a compu a ional amewo k o modelling ul asound wa e
p opaga ion h ough laye ed c anial issues using a 1D FDTD me hod. By inco po a -
ing spa ially a ying acous ic p ope ies, equency-dependen a enua ion, and ana-
omical a iabili y in skin and skull hickness, he model e ec i ely cap u es he
complex in e ac ions ha in luence ansc anial acous ic ansmission. Valida ion
agains COMSOL simula ions demons a ed s ong ag eemen in p essu e p o iles
ac oss mul iple c anial egions and equencies, while also highligh ing he subs an ial
compu a ional e iciency o he Py hon implemen a ion. The esul s con i m he c i i-
cal ole o equency selec ion in balancing spa ial esolu ion and pene a ion dep h,
wi h lowe equencies enabling deepe b ain access and highe equencies o e ing
mo e localized ene gy deli e y. Regional di e ences in a enua ion u he emphasize
he impo ance o pe sonalized acous ic a ge ing, pa icula ly in he con ex o neu-
omodula ion he apies o condi ions such as Alzheime ’s disease, epilepsy, and
dep ession. O e all, his ligh weigh simula ion app oach o e s a aluable ool o
explo ing and op imizing non-in asi e ul asound applica ions in b ain heal h moni-
o ing and in e en ion.
6 Acknowledgemen s
This wo k was unded by he Na ional Founda ion o Science and Technology o
Po ugal (FCT) unde he p ojec “B ainS imMap – Mapping and modelling he
ansmission p o ile o op omechanical wa es in he b ain o op imize ansc anial
s imula ion agains b ain diso de s" wi h he e e ence PTDC/EME-EME/1681/2021
and he indi idual PhD g an wi h e e ence 2022.11063.BD.
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