Fundamental kinematics laws of interstitial fluid flows on vascular walls

Yjun Yin , ,Hongyi Li ,Gng Peng ,Xioin Yu ,Yiy Kong

a Department of Engineering Mechanics, Tsinghua University, Beijing 10 0 084, China

b Beijing Hospital, National Center of Gerontology, Institute of Geriatric Medicine, Chinese Academy of Medical Sciences, Beijing 100730, China

Keywords:Vascular walls Interstitial fluid flow Essential flowing properties Fundamental kinematics laws

ABSTRACT In the previous studies,the phenomenon that the interstitial fluid (ISF) can flow along tunica adventitia of the arteries and veins in both human and animal bodies was reported.On the basis of these studies,this paper aims to:(i) summarize the basic properties of the ISF flows in the walls of arteries and veins,(ii) combine the basic properties with axiomaticism and abstract the axiom for ISF flows,and (iii) propose three fundamental laws of the ISF flow,(i.e.,the existence law,the homotropic law and the reverse law).The three laws provide solid theoretical basement for exploring the kinematic patterns of interstitial fluid flow in the cardiovascular system.

According to classical physiology,the interstitial fluid (ISF),which accounts for 20% of the body weight,is entrapped locally by proteoglycan filaments and tissue gel,and thus,cannot flow freely.However,Li et al.[1] discovered that the ISF within human and animal bodies is able to flow.Moreover,the flow phenomenon has objective reality and universal existence,which also indicates that the flow of ISF is a predictable and regular kinematic process.This paper devotes to generalizing the kinematic regularity of ISF flow.

Feng et al.[2,3] also confirmed that:(1) the interstitial fluid flow pathways were pluralistic,i.e.,the ISF could flow along blood vessels,nerves,and well-organized subcutaneous fat interlobular septum;(2) the topological spaces of perivascular ISF flow pathways were diverse,that is,there were two kinds of topologically connected spaces,the interfacial zones within the perivascular connective tissue and the adventitia.In further study,we found that there may also be a third type of topological space for interstitial fluid flow a space between tunica adventitia and its covering fascia.

Thus,in terms of topology,the interstitial space within the perivascular connective tissues of arterial and venous vessels are all the spaces for ISF flows.The two types of interfacial flow spaces are all located on the vessel wall,and the interstitial flow space is located outside the vessel wall,which were collectively named as the "flow spaces on the blood vessel wall",and the flows there were termed "interstitial fluid flows on the blood vessel wall".

After clarifying the location and topology of the flow pathways,the critical question,i.e."Where does the tissue fluid flow?" is answered.The subsequent issue is "How does the ISF flow?".In other words,the fundamental rules of ISF flow should be explored.

This research focuses on the kinematics of ISF flows on the blood vessel walls.Specifically,it includes the following contents:(1) Several basic properties of ISF flows on the blood vessel walls are put forward.(2) The "cardiac tendency" postulate of ISF flowing along the blood vessel walls was extracted from the basic properties with the help of axiomatic thought.(3) The kinematic laws of ISF flows on the blood vessel walls were constructed from the"cardiac tendency" hypothesis.

With utilization of magnetic resonance imaging technology to display the tracer’s trajectory,the ISF flow was revealed by tracking the movement of the tracer [1,4] .

In healthy subjects,the tracer (GD-DTPA) was injected into their subcutaneous tissues.The injection sites were selected at the extremities.Figure 1 shows the trace trajectory of the right wrist and ankle after injection.It was found that the tracer (GD-DTPA) firstly diffused locally at the injection site and formed bright spots.3–5 min after injection,the tracer showed a bright line and moved upward in the direction of the arrow,which points toward the heart.After about 30–40 min,the tracer line gradually disappeared.Overall,the trajectory was continuous;locally,the trajectory was smooth.

Fig.1. Traces of tracer movement on the limbs. a–c,tracer on human arm; f, g legs; h, i ankle and foot.White arrows,the direction of ISF flow.White lines,tracer traces.

Earlier results showed that the tracer traces in Fig.1 were mostly blood vessels which meant the tracer injected near the blood vessel would migrate along the blood vessel.Not surprisingly,the tracer migration along the blood vessels seems to be a trivial phenomenon and not unexpected,since blood vessels are essentially the channels through which blood flows.However,Li[5] confirmed through a decisive experiment that the tracer did not flow inside the vascular cavity,but outside of it!

Proofed by contradiction,it could also be logically demonstrated that the tracer migration lied indeed outside the vascular lumen.The argument goes as follows.

Assuming that the trace line was inside the vascular lumen,i.e.,the tracer stained the blood.Well,due to the blood circulation period is tens of seconds,all the blood vessels will be brightened by the tracer within tens of seconds.This is however completely contradictory to the experimental phenomenon in Fig.1,which is specifically manifested in:(1) From the trajectory distribution,only some blood vessels (not all of them) are stained.(2) From the spatial perspective,only the upper blood vessels of the injection site were stained,while the lower blood vessels were not.(3) From the temporal perspective,the time interval from the injection to the appearance of the trajectory is several minutes,much longer than the blood circulation period.

In short,both spatially and temporally,the ISF flows outside the vascular lumen.

Figure 1 shows merely one of the typical experimental phenomena,from which,it has been possible to glimpse the essence behind phenomena.Nevertheless,it is never easy to find patterns directly from the ever-changing experimental phenomena.Hence,in the following sections,"empirical research paradigm and rational research paradigm" will be combined and axiomatization will be introduced to reveal the invariance and regularity of ISF flow along the blood vessel wall.Specifically,the basic propositions that can become axioms are abstracted from the experimental phenomena,and based on the axioms,the fundamental laws of ISF flows are deduced.

The above flowing images are presented in not only human beings,but also animals such as rabbit.Li et al.[1] injected fluorescent tracer next to the distal vessels of the lower extremity in live rabbits and carried out anatomical observation,they found that the tracers stained the outer membrane of the small saphenous vein(shown in Fig.2a).The frozen-section immunofluorescence imaging (Fig.2b) showed that the tracer is mainly distributed on the outer membrane of the small saphenous vein vessels (the circular fluorescent band) and the surrounding connective tissues,which indicated that the ISF flows along the wall of the small saphenous vein vessel,from the distal part of the lower extremity towards the buttock region.

Figure 3a depicts the anatomy of the femoral artery and saphenous vein,and it can be seen that the tracer stained both saphenous vein and femoral artery.The frozen-section immunofluorescence imaging (Fig.3b) shows that the fluorescence is mainly distributed on the outer membrane of the saphenous vein vessel(the circular fluorescent band),the outer membrane of the femoral artery vessel (the right-angled fluorescent band in the upper left)and the surrounding connective tissues.

Li et al.[1] showed the anatomy of the inferior vena cava in a living rabbit,and it could be noted that the outer membrane of the inferior vena cava emits fluorescence (Fig.A11 in Ref.[1]),which suggests that the tracer in the distal end,flows up through the wall of the lower limb vessels to the inferior vena cava and then along the wall of the inferior vena cava in the direction of the heart.

Fig. 2. a Stained trajectory on the outer membrane of the small saphenous vein vessel;b Cross-sectional of the frozen-section immunofluorescence imaging of the small saphenous vein.

Summarily,the flow images of interstitial fluid in the walls of the blood vessels of the extremities are exactly the same for both fluorescent tracers and GD-DTPA tracers,both for humans or animals.

From the previous experiments,we can generalize a number of kinematic properties of tissue fluid flows along the blood vessel wall.

Fig.3. a Fluorescence on the outer membrane of the femoral artery and saphenous vein vessel; b Cross-sectional of the frozen-section immunofluorescence imaging.

PropertyIis "cardiac tendency":On the blood vessel wall,the ISF always flows towards the direction of the heart.Alternatively,ISF flows along the walls of veins and arteries towards the heart,which is termed by "cardiac tendency of ISF flowing along blood vessel walls".

So far,Li et al.[6] has completed hundreds of trials on the blood vessels of the human upper and lower limbs,as well as the blood vessels of the fore limbs and hind limbs of rabbits and mice.He confirmed that the repetition rate of cardiac tendency is 100%,and there has never been a counterexample.

The cardiac tendency holds not only in the walls of the blood vessels of the extremities,but also in the cerebrum and neck vessels.In recent years,the neurobiologists have studied the flow of ISF and cerebrospinal fluid (CSF) in the brain of animals [7,8].Carare et al.[7] demonstrated that ISF in cerebrum flowed extracranially along the walls of intracranial arterial vessels.Iliff et al.[8] confirmed that the ISF and CSF flowed out of the cerebrum in the perivascular pathway around intracranial venous vessels.As shown by the anatomy of the cerebrum and neck vascular tree,the cerebral vessels are topographically connected to the heart.Therefore,the ISF flowing out along the walls of the cerebral vessels must point to the direction of the heart.That is,the cardiac tendency also exists in the wall of the cerebrum and neck vessels.The schematic diagrams of the cardiac tendency of interstitial fluid flow in the walls of the extremities and cerebrum and neck vessels are marked in Fig.4.

Fig. 4. Cardiac tendency of ISF flow in the walls of the extremities,cerebrum and neck vessels.Green arrows:the flowing direction of ISF (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.).

A particularly intriguing question is whether there exists a flow of ISF in the walls of blood vessels within the major organs.If so,does the direction of flow still have the cardiac tendency? At present,we do not have an answer,and we are designing experiments that would work to clarify the image of the flow along the vessel wall in major organs.

In a word,"cardiac tendency" has a solid empirical foundation,and its objective reality and universal existence are guaranteed.

Property II is "continuity":The flow of ISF on the blood vessel wall is continuous,and its spatial distribution in the axial direction is uninterrupted.

Observing any cross section along the blood vessel,one can find the flow of ISF.In mechanics,the continuity here can be regarded as "spatial continuity".

Property III is "persistence":The flow of ISF on the blood vessel wall is long-term,and it is continuous in time domain.

At any given moment,observing any given cross-section of the blood vessel,one will see the flow of ISF on the wall.From a mechanical point of view,the persistence here can be regarded as"temporal continuity".

Combining the "temporal continuity" with the "spatial continuity",it can be asserted that the flow of ISF on the blood vascular wall has the "spatial and temporal continuity".In mechanics,continuity is a very fundamental logical basis,for only in continuous space-time could the calculus function.

Property IV is "high speed":The velocity of ISF flow on the blood vessel wall can reach up to 20 mm/s [6].This velocity,although far less than the flow rate of blood inside the arteriovenous lumen (tens of centimeters per second),is far higher than the diffusion speed of the ISF through the tissue gel (micrometers per second).

It should be noted that the ISF velocities at the kinds of flow spaces are not the same.The maximum velocity occurs mainly on the vascular fascia/adventitia interface.

Then we will seek an underlying logical ground for exploring the patterns of ISF flows,with the entry point being the cardiac tendency.

Among the above properties,cardiac tendency is a critical one,which tells that,the flow of ISF is a purposeful and orderly movement,rather than a chaotic and disorderly one.Therefore,starting from cardiac tendency nature,it is possible to reveal the order or law of ISF flow.

Among the properties,cardiac tendency is also the most important.Physiologically,there are two types of flow associated with the heart,one being blood flow,and the other lymph flow.The importance of these two types of flow is self-evident.It can be said that all the flow associated with the heart are of fundamental importance.

Among all the motion properties,cardiac tendency is still the most extraordinary one.In animals,continuous,persistent,and high-speed flow are not unusual.But the cardiac tendency flow is beyond our expectations.

The history of mechanics and physics teaches us that the more basic the nature,the closer it is to the nature;the more unexpected the nature,the more it can help us to blaze new trails.We expect that cardiac tendency will help us disclose deeper regularities or orders.

To this end,we introduce the idea of axiomatization,giving cardiac tendency the status of an axiom.The objectivity and universality of centripetalism,which have been verified by hundreds of experiments,are unquestionable,as mentioned in the previous section.Subsequently,cardiac tendency,as the lowest level of theoretical foundation,is solid and firm.

On account of the cardiac tendency postulate,the basic flowing laws of the ISF flow on the vessel walls could be logically derived,which was named as the three laws of kinematics:

(1) The first law is the Existence Law:There are two basic flows at both arteries and veins,one is the blood flow in the lumen,and the other is the flow of ISF on the vessel wall.

(2) The second law is the Homotropic Law:For veins,the flow direction of ISF (green arrow) on the vessel wall is exactly the same as the flow direction of blood (white arrow) in the lumen (Fig.5)

(3) The third law is the Reverse Law:For arteries,the flow direction of ISF (green arrow) on the vessel wall is completely opposite to the flow direction of blood (white arrow) in the lumen (Fig.6).

Fig.5. Relation between the direction of ISF flow (green arrow) on the venous vessel walls and the direction of blood flow (white arrow).“Heart”means the location or direction towards heart (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.).

Fig.6. Relation between the direction of ISF flow (green arrow) on the arterial vessel walls and the direction of blood flow (white arrow).“Heart”means the location or direction towards heart (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.).

The three laws describe the interrelationship between the flow of ISF on the outer wall of the blood vessel and the blood movement inside the blood vessel.Among them,the first law states the objective reality and universality of ISF flow.The second law and the third law stipulate the direction of ISF flow and reveal the internal correlation between the direction of ISF flow and the direction of blood flow.For veins,the blood in the lumen always moves towards the heart.According to the cardiac tendency postulate,the ISF flow will certainly direct to the heart,which is exactly the meaning of the Homotropic Law.For arteries,the blood in the lumen always leaves away from the heart.According to the cardiac tendency postulate,the ISF flow will certainly direct toward the heart,which is exactly the meaning of the Reverse Law.

The three Laws are the direct corollary of cardiac tendency postulate.As long as the cardiac tendency postulate is accepted,the three laws hold.Since the cardiac tendency holds for the three topological flow spaces on the blood vessel wall,it is certain that the three laws are also valid for the three topological flow spaces on the blood vessel wall.

Now that the cardiac tendency is summarized from the arteries and veins of the extremities,there is no doubt that the three laws hold on the walls of the arteries and veins of the extremities (the third-level blood vessels).

In future studies,we will show that the three laws also hold true for secondary and primary vessels.Finally,we will confirm that the three laws are universally true throughout the blood vessel tree in the human body.

As mentioned above,the logical premise of the three laws is the cardiac tendency postulate.As long as the cardiac tendency postulate is not falsified,the credibility of the three laws is guaranteed.It can be claimed that the three laws are expected to be the reliable theoretical basis for the kinematics of ISF flows.The aforementioned flow conservation law is only one of the simplest and most specific examples.In subsequent studies,we will give more general conservation laws.

The three laws reveal the certainty of the flow of ISF on the walls of blood vessels throughout the body.Wherever there are blood vessels,there are flows of ISF.The direction of blood flow is deterministic,so the direction of ISF flow is also deterministic.The whole picture of blood flow throughout the body is clear.It can be expected that the picture of ISF flow in the whole body can also be depicted clearly.The three laws describe the orderly flow of ISF on the walls of blood vessels throughout the body.Blood flow is ordered,and ISF flow is also ordered.The final clarification of the order of ISF flow is worth to be expected.

The three laws lay the theoretical foundation for the subsequent exploration.Based on the three laws,we can make the following prediction:the flow of ISF in the vessel wall must constitute a closed circuit,in other words,the flow must form a circulation.Because,the flow in Figs.5 and 6 is continuous and persistent,as shown by Property II and Property III,and the circulation,without which,the continuity and persistence will immediately loss of usefulness.

The above basic properties are all formal statements,generalizations of common experimental phenomena,and none of them involves "why".In contrast,the three laws tell one only "how ISF flows",but not "why ISF flows".The reason is clear:the purpose of this paper is the kinematics,instead of the dynamics.Kinematics only needs to answer "how to move" and dynamics needs to answer "why to move".In the classical mechanics system,the following basic principles are generally followed:kinematics first,dynamics second.

Currently,the three laws can only be described qualitatively,but lack quantitative depiction.Although they are only qualitative laws,they have laid the foundation for us to understand the deeper laws of ISF movement.In the following research,we will gradually refine the three laws to achieve a quantitative description of kinematics.Of the flow pathways,the flow at the vascular fascia/adventitia

interface,and the flow at the fiber/gel interfacial zone within the

adventitia of the blood vessel,are of particular interest.Obviously,the vascular fascia is topologically connected to the cardiac plasma membrane,and the vascular adventitia is topologically connected to the epicardium.It can be inferred that the interfacial flow pathway is topologically connected with the pericardial cavity.The topological connectivity can direct the flow of ISF to the heart.In following articles,we will regard "flow towards the heart" as a breakthrough to develop the theory of ISF circulation in the cardiovascular system.

DeclarationofCompetingInterest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grants 12050001,82050004,and 11672150).